Difference between revisions of "OPTUMG2/Examples"

From OPTUM CE
Jump to navigation Jump to search
 
(31 intermediate revisions by the same user not shown)
Line 25: Line 25:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/drcr23.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/drcr23.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 1: Material behavior as function of Drainage and Time Scope. '''</span>
+
|+<span style="font-size:90%">'''Figure 1.1: Material behavior as function of Drainage and Time Scope.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 51: Line 51:
  
 
The following shortcut keys are available in OPTUM G2:
 
The following shortcut keys are available in OPTUM G2:
 +
 +
<div id="shortcuts">
  
 
{| class="wikitable"
 
{| class="wikitable"
Line 109: Line 111:
 
| Delete selected object (point, line, etc).
 
| Delete selected object (point, line, etc).
 
|}
 
|}
 +
 +
 +
</div>
 +
<span id="shortcuts" label="shortcuts">[shortcuts]</span>
  
 
==  INTRODUCTORY EXAMPLE  ==
 
==  INTRODUCTORY EXAMPLE  ==
  
In the following, the steps to setting up and solving a problem of limit analysis are detailed. The problem, shown in Figure 2, concerns a shallow foundation on top of a slope of cohesive-frictional soil. The task is to determine the bearing capacity of the footing, i.e. the maximum load, <math display="inline">q_u</math> (kN/m<math display="inline">^2</math>), that it can be subjected to. A surcharge load of <math display="inline">q_s = 10</math> kN/m<math display="inline">^2</math> acts of the top of the slope. The soil is modeled as a Mohr-Coulomb material with cohesion <math display="inline">c = 10</math> kPa, friction angle <math display="inline">\phi=20^\circ</math> and unit weight <math display="inline">\gamma=20</math> kN/m<math display="inline">^3</math>. The foundation is assumed perfectly rigid with a unit weight of <math display="inline">\gamma=23</math> kN/m<math display="inline">^3</math>.
+
In the following, the steps to setting up and solving a problem of limit analysis are detailed. The problem, shown in Figure 2.1, concerns a shallow foundation on top of a slope of cohesive-frictional soil. The task is to determine the bearing capacity of the footing, i.e. the maximum load, <math display="inline">q_u</math> (kN/m<math display="inline">^2</math>), that it can be subjected to. A surcharge load of <math display="inline">q_s = 10</math> kN/m<math display="inline">^2</math> acts of the top of the slope. The soil is modeled as a Mohr-Coulomb material with cohesion <math display="inline">c = 10</math> kPa, friction angle <math display="inline">\phi=20^\circ</math> and unit weight <math display="inline">\gamma=20</math> kN/m<math display="inline">^3</math>. The foundation is assumed perfectly rigid with a unit weight of <math display="inline">\gamma=23</math> kN/m<math display="inline">^3</math>.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/sbs_setup.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/sbs_setup.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 2: Shallow foundation on top of slope. '''</span>
+
|+<span style="font-size:90%">'''Figure 2.1: Shallow foundation on top of slope.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 126: Line 132:
 
===  Geometry  ===
 
===  Geometry  ===
  
When OPTUM G2 is started up, the Geometry ribbon is the default active one. This contains various tools for defining and manipulating geometry. It is shown in Figure 3.
+
When OPTUM G2 is started up, the Geometry ribbon is the default active one. This contains various tools for defining and manipulating geometry. It is shown in Figure 2.2.
  
 
<center>
 
<center>
Line 132: Line 138:
 
|<div class="image600px">https://wiki.optumce.com/figures/sbs_geomtab.png</div>
 
|<div class="image600px">https://wiki.optumce.com/figures/sbs_geomtab.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 3: Geometry ribbon. '''</span>
+
|+<span style="font-size:90%">'''Figure 2.2: Geometry ribbon.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 140: Line 146:
 
# Left-click the Lines button in the Geometry ribbon.
 
# Left-click the Lines button in the Geometry ribbon.
 
# Place the cursor at (0,0) and left-click. This defines the first point.
 
# Place the cursor at (0,0) and left-click. This defines the first point.
# To define the next point, a choice between input mode must be made. OPTUM G2 offers two modes: relative and absolute coordinate input (see Figure 4).<br />
+
# To define the next point, a choice between input mode must be made. OPTUM G2 offers two modes: relative and absolute coordinate input (see Figure 2.3).<br />
For relative coordinate input (the default setting), the x and y coordinates relative to the last defined point are entered. Moving counter-clockwise in Figure 2, the following coordinates should be entered: (31,0), (0,4), (-7,0), (-9,6),...This can be done either via the keyboard by pressing Enter for each entry or via the mouse by left-click.<br />
+
For relative coordinate input (the default setting), the x and y coordinates relative to the last defined point are entered. Moving counter-clockwise in Figure 2.1, the following coordinates should be entered: (31,0), (0,4), (-7,0), (-9,6),...This can be done either via the keyboard by pressing Enter for each entry or via the mouse by left-click.<br />
To enter the absolute coordinates shown in Figure 2 directly, use the ABS/REL button in the left bottom corner of the program window to toggle the input mode to ABS. The ABS/REL toggling can only be done once the first point has been defined.<br />
+
To enter the absolute coordinates shown in Figure 2.1 directly, use the ABS/REL button in the left bottom corner of the program window to toggle the input mode to ABS. The ABS/REL toggling can only be done once the first point has been defined.<br />
 
Zoom All (via the button in ribbon or in the upper right corner of the drawing canvas) can be used at any time.
 
Zoom All (via the button in ribbon or in the upper right corner of the drawing canvas) can be used at any time.
 
# Next, the foundation is defined. This is most easily done using the Rectangle tool. Click the tool, move the mouse cursor to position (10,8) and left-click to define the lower left point. Move the cursor to position (12,11) and left-click to define the second point. The foundation is then created along with the intersections between the foundation and the soil. The resulting line through the foundation can be deleted (select the line and press Del on the keyboard or right-click the line and select Delete) or left as it is.
 
# Next, the foundation is defined. This is most easily done using the Rectangle tool. Click the tool, move the mouse cursor to position (10,8) and left-click to define the lower left point. Move the cursor to position (12,11) and left-click to define the second point. The foundation is then created along with the intersections between the foundation and the soil. The resulting line through the foundation can be deleted (select the line and press Del on the keyboard or right-click the line and select Delete) or left as it is.
Line 148: Line 154:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/sbs_absrel.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/sbs_absrel.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 4: Relative (left) and absolute (right) coordinate input modes. '''</span>
+
|+<span style="font-size:90%">'''Figure 2.3: Relative (left) and absolute (right) coordinate input modes.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 156: Line 162:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/sbs_Geomfinishes.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/sbs_Geomfinishes.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 5: After definition of geometry. The line through the foundation created as a result of the automatic intersection between the slope and the foundation has been deleted. '''</span>
+
|+<span style="font-size:90%">'''Figure 2.4: After definition of geometry. The line through the foundation created as a result of the automatic intersection between the slope and the foundation has been deleted.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 164: Line 170:
 
===  Materials  ===
 
===  Materials  ===
  
To assign materials first switch to the Materials ribbon (see Figure 6).
+
To assign materials first switch to the Materials ribbon (see Figure 2.5).
  
 
<center>
 
<center>
Line 170: Line 176:
 
|<div class="image600px">https://wiki.optumce.com/figures/sbs_matrribbon.png</div>
 
|<div class="image600px">https://wiki.optumce.com/figures/sbs_matrribbon.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 6: Materials ribbon (only Solids shown). '''</span>
+
|+<span style="font-size:90%">'''Figure 2.5: Materials ribbon (only Solids shown).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 177: Line 183:
 
To assign the soil material, first select the gray surface corresponding to the soil by left-click of the mouse. The color then changes to magenta indicating selection. Next, click the Firm Clay-MC button in the ribbon. Similarly, select the foundation and click the Rigid button in the ribbon. Finish by clicking anywhere on the drawing canvas outside the geometry defined.<br />
 
To assign the soil material, first select the gray surface corresponding to the soil by left-click of the mouse. The color then changes to magenta indicating selection. Next, click the Firm Clay-MC button in the ribbon. Similarly, select the foundation and click the Rigid button in the ribbon. Finish by clicking anywhere on the drawing canvas outside the geometry defined.<br />
 
The material parameters can be changed either by selecting the material from the ribbon by mouse click or by selecting a surface to which the material has been assigned. The parameters then appear in the property window on the righthand side of the program window and can be edited via the keyboard.<br />
 
The material parameters can be changed either by selecting the material from the ribbon by mouse click or by selecting a surface to which the material has been assigned. The parameters then appear in the property window on the righthand side of the program window and can be edited via the keyboard.<br />
For the present problem, the unit weight of the foundation, i.e. of the Rigid material, must be changed from its default value of 0 to 23 kN/m<math display="inline">^3</math>. To do this select the foundation by mouse click and enter 23 into the appropriate field of the property window (see Figure 7).
+
For the present problem, the unit weight of the foundation, i.e. of the Rigid material, must be changed from its default value of 0 to 23 kN/m<math display="inline">^3</math>. To do this select the foundation by mouse click and enter 23 into the appropriate field of the property window (see Figure 2.6).
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/sbs_matrass.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/sbs_matrass.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 7: After assignment of materials. '''</span>
+
|+<span style="font-size:90%">'''Figure 2.6: After assignment of materials.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 189: Line 195:
 
===  Loads  ===
 
===  Loads  ===
  
To assign loads first switch to the Features ribbon (see Figure 8).
+
To assign loads first switch to the Features ribbon (see Figure 2.7).
  
 
<center>
 
<center>
Line 195: Line 201:
 
|<div class="image600px">https://wiki.optumce.com/figures/sbs_featrib.png</div>
 
|<div class="image600px">https://wiki.optumce.com/figures/sbs_featrib.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 8: Features ribbon (partial). '''</span>
+
|+<span style="font-size:90%">'''Figure 2.7: Features ribbon (partial).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 204: Line 210:
 
# The load acting on the foundation whose ultimate value (corresponding to collapse) is to be determined.
 
# The load acting on the foundation whose ultimate value (corresponding to collapse) is to be determined.
  
In OPTUM G2, these two types of loads are referred to as Fixed and Multiplier loads respectively (see Figure 8). Fixed loads always remain constant while Multiplier loads are magnified from their reference value to bring about a state of collapse. The factor by which the Multiplier loads should be magnified to bring about a state of collapse is also referred to as the collapse multiplier. Denoting this quantity by <math display="inline">\alpha</math>, the ultimate limit load is given by  
+
In OPTUM G2, these two types of loads are referred to as Fixed and Multiplier loads respectively (see Figure 2.7). Fixed loads always remain constant while Multiplier loads are magnified from their reference value to bring about a state of collapse. The factor by which the Multiplier loads should be magnified to bring about a state of collapse is also referred to as the collapse multiplier. Denoting this quantity by <math display="inline">\alpha</math>, the ultimate limit load is given by  
  
<math display="block">q_u = \alpha q_\text{mult}</math>
+
<math display="block">(2.1)\qquad
 +
    q_u = \alpha q_\text{mult}</math>
  
 
where <math display="inline">q_\text{mult}</math> is the multiplier load.<br />
 
where <math display="inline">q_\text{mult}</math> is the multiplier load.<br />
 
To assign loads, first select the line defining the top of the foundation and then click Multiplier Distributed in the Features ribbon. This assigns a distributed load of magnitude <math display="inline">-1</math> kN/m<math display="inline">^2</math>.<br />
 
To assign loads, first select the line defining the top of the foundation and then click Multiplier Distributed in the Features ribbon. This assigns a distributed load of magnitude <math display="inline">-1</math> kN/m<math display="inline">^2</math>.<br />
Next, select the line defining the ground surface to the left of the foundation. While pressing Shift, select the ground surface line to the right of foundation. Both lines should now be selected. Next, click Fixed Distributed in the Features ribbon. This assigns the surcharge loads (see Figure 9). Finally, in the property window on the right, change the magnitude of the loads to <math display="inline">-10</math> kN/m<math display="inline">^2</math>. Finish by clicking the drawing canvas anywhere outside the geometry defined.
+
Next, select the line defining the ground surface to the left of the foundation. While pressing Shift, select the ground surface line to the right of foundation. Both lines should now be selected. Next, click Fixed Distributed in the Features ribbon. This assigns the surcharge loads (see Figure 2.8). Finally, in the property window on the right, change the magnitude of the loads to <math display="inline">-10</math> kN/m<math display="inline">^2</math>. Finish by clicking the drawing canvas anywhere outside the geometry defined.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/sbs_loadass.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/sbs_loadass.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 9: After assignment of loads. '''</span>
+
|+<span style="font-size:90%">'''Figure 2.8: After assignment of loads.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 222: Line 229:
 
===  Analysis  ===
 
===  Analysis  ===
  
We are now in ready to run the analysis. The type of analysis is selected from the Stage Manager window shown on the right in Figure 9. The default analysis type is Limit Analysis which is what is required for this example.<br />
+
We are now in ready to run the analysis. The type of analysis is selected from the Stage Manager window shown on the right in Figure 2.8. The default analysis type is Limit Analysis which is what is required for this example.<br />
For each analysis type, a number of settings are available. These are shown in the lower half of the Stage Manager window (Figure 9). For the purpose of the present analysis, two settings are of interest:
+
For each analysis type, a number of settings are available. These are shown in the lower half of the Stage Manager window (Figure 2.8). For the purpose of the present analysis, two settings are of interest:
  
 
# The type of element to be used (Element Type under Settings. Default = Lower).
 
# The type of element to be used (Element Type under Settings. Default = Lower).
Line 230: Line 237:
 
Regarding the element type, we will use the default Lower element. This results in a rigorous lower bound on the ultimate limit load.<br />
 
Regarding the element type, we will use the default Lower element. This results in a rigorous lower bound on the ultimate limit load.<br />
 
The number of elements is changed to 1,000.<br />
 
The number of elements is changed to 1,000.<br />
Next, in order to compute an upper bound on the ultimate limit load (in addition to a lower bound), we will first create a copy of the single stage presently available. This is done via the clone button (second button from the left in the Stage Controls shown in Figure 10).<br />
+
Next, in order to compute an upper bound on the ultimate limit load (in addition to a lower bound), we will first create a copy of the single stage presently available. This is done via the clone button (second button from the left in the Stage Controls shown in Figure 2.9).<br />
In the new stage, the Element Type is changed to Upper and the No of Elements is set to 1,000. The situation is thereby as shown in Figure 11.
+
In the new stage, the Element Type is changed to Upper and the No of Elements is set to 1,000. The situation is thereby as shown in Figure 2.10.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/StageControls.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/StageControls.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 10: Stage Manager controls. '''</span>
+
|+<span style="font-size:90%">'''Figure 2.9: Stage Manager controls.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 243: Line 250:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/sbs_anal.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/sbs_anal.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 11: After cloning of Stage 1 and change of Element Type and No of Elements. '''</span>
+
|+<span style="font-size:90%">'''Figure 2.10: After cloning of Stage 1 and change of Element Type and No of Elements.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The problem, comprising two stages, is processed by clicking the Run Analysis button in the Stage Manager control panel (see Figure 10). The results of the analysis are displayed in the analysis log shown in Figure 12.
+
The problem, comprising two stages, is processed by clicking the Run Analysis button in the Stage Manager control panel (see Figure 2.9). The results of the analysis are displayed in the analysis log shown in Figure 2.11.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/sbs_log.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/sbs_log.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 12: Analysis log. '''</span>
+
|+<span style="font-size:90%">'''Figure 2.11: Analysis log.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 266: Line 273:
 
This means the true collapse load is given by  
 
This means the true collapse load is given by  
  
<math display="block">q_u = 268.4\pm 56.2~\text{kN/m}^2</math>
+
<math display="block">(2.2)\qquad
 +
    q_u = 268.4\pm 56.2~\text{kN/m}^2</math>
  
 
or:  
 
or:  
  
<math display="block">q_u = 268.4 \text{kN/m}^2\pm 20.9\%</math>
+
<math display="block">(2.3)\qquad
 +
    q_u = 268.4 \text{kN/m}^2\pm 20.9\%</math>
  
 
That is, the estimate of collapse load calculated as the mean value between the upper and lower bounds, <math display="inline">q_u = 268.4</math> kN/m<math display="inline">^2</math>, is in error by at most 20.9%. It may be on the safe side or on the unsafe side, but the error will not be greater than 20.9% either way. This worst case error can be reduced either by increasing the number of elements or by using mesh adaptivity, or by a combination of the two. Many of the examples in this manual make use of these possibilities.
 
That is, the estimate of collapse load calculated as the mean value between the upper and lower bounds, <math display="inline">q_u = 268.4</math> kN/m<math display="inline">^2</math>, is in error by at most 20.9%. It may be on the safe side or on the unsafe side, but the error will not be greater than 20.9% either way. This worst case error can be reduced either by increasing the number of elements or by using mesh adaptivity, or by a combination of the two. Many of the examples in this manual make use of these possibilities.
Line 276: Line 285:
 
===  Results  ===
 
===  Results  ===
  
After the analysis log window is closed, the program switches automatically to the Results ribbon and the situation is as shown in Figure 13.
+
After the analysis log window is closed, the program switches automatically to the Results ribbon and the situation is as shown in Figure 2.12.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/sbs_mesh.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/sbs_mesh.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 13: After analysis. '''</span>
+
|+<span style="font-size:90%">'''Figure 2.12: After analysis.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The default results set shown is the mesh. Note that OPTUM G2 does not require a separate mesh generation stage – the mesh is created automatically as part of the analysis. Using the menus and controls in the Results ribbon, various plots can be created. An example is shown in Figure 14 which shows the distribution of deviatoric strain <math display="inline">|\varepsilon_1-\varepsilon_3|</math> available under Strains.
+
The default results set shown is the mesh. Note that OPTUM G2 does not require a separate mesh generation stage – the mesh is created automatically as part of the analysis. Using the menus and controls in the Results ribbon, various plots can be created. An example is shown in Figure 2.13 which shows the distribution of deviatoric strain <math display="inline">|\varepsilon_1-\varepsilon_3|</math> available under Strains.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/sbs_res.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/sbs_res.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''</math> available under Strains. '''</span>
+
|+<span style="font-size:90%">'''Distribution of deviatoric strain <math style="inline"> | \varepsilon_1 - \varepsilon_3| </math> available under Strains.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 298: Line 307:
 
==  SHALLOW FOUNDATION 1  ==
 
==  SHALLOW FOUNDATION 1  ==
  
This example deals with the an eccentrically loaded foundation as shown in Figure 15. The soil is saturated clay and the analysis is to be performed assuming undrained conditions. For this purpose a total stress analysis approach is adopted. The soil is modeled by means of the Tresca model with an undrained shear strength <math display="inline">s_u=30</math> kPa and an undrained Young’s modulus of <math display="inline">E_u = 40</math> MPa. The foundation is modeled as Rigid material with a unit weight of 24 kN/m<math display="inline">^3</math>. The material properties are shown in the property window on the right in Figure 15.
+
This example deals with the an eccentrically loaded foundation as shown in Figure 3.1. The soil is saturated clay and the analysis is to be performed assuming undrained conditions. For this purpose a total stress analysis approach is adopted. The soil is modeled by means of the Tresca model with an undrained shear strength <math display="inline">s_u=30</math> kPa and an undrained Young’s modulus of <math display="inline">E_u = 40</math> MPa. The foundation is modeled as Rigid material with a unit weight of 24 kN/m<math display="inline">^3</math>. The material properties are shown in the property window on the right in Figure 3.1.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex01_Fig01_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex01_Fig01_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 15: Shallow foundation in Tresca USS soil. '''</span>
+
|+<span style="font-size:90%">'''Figure 3.1: Shallow foundation in Tresca USS soil.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 320: Line 329:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex01_lass.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex01_lass.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 16: Stage settings for lower bound limit analysis. The Time Scope is irrelevant for the Tresca model. '''</span>
+
|+<span style="font-size:90%">'''Figure 3.2: Stage settings for lower bound limit analysis. The Time Scope is irrelevant for the Tresca model.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 328: Line 337:
 
Running the analyses results in lower and upper bound collapse multipliers of 851.1 and 1017.4 respectively. In other words, the maximum vertical load that can be sustained is:  
 
Running the analyses results in lower and upper bound collapse multipliers of 851.1 and 1017.4 respectively. In other words, the maximum vertical load that can be sustained is:  
  
<math display="block">851.1 \times 1 \text{kN/m}^2 \leq q_u \leq 1017.4 \times 1 \text{kN/m}^2</math>
+
<math display="block">(3.1)\qquad
 +
    851.1 \times 1 \text{kN/m}^2 \leq q_u \leq 1017.4 \times 1 \text{kN/m}^2</math>
  
 
or, in terms of total force (the load works over 0.8 m):  
 
or, in terms of total force (the load works over 0.8 m):  
  
<math display="block">680.9 \text{kN/m} \leq Q_u \leq 813.9 \text{kN/m}</math>
+
<math display="block">(3.2)\qquad
 +
    680.9 \text{kN/m} \leq Q_u \leq 813.9 \text{kN/m}</math>
  
 
The result may also be stated as  
 
The result may also be stated as  
  
<math display="block">q_u =  934.2 \text{kN/m}^2  \pm 8.9\%</math>
+
<math display="block">(3.3)\qquad
 +
    q_u =  934.2 \text{kN/m}^2  \pm 8.9\%</math>
  
 
In other words, the error in the mean value between the upper and lower bounds is <math display="inline">\pm 8.9\%</math>.
 
In other words, the error in the mean value between the upper and lower bounds is <math display="inline">\pm 8.9\%</math>.
Line 347: Line 359:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex01_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex01_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 17: Stage settings for lower bound limit analysis mesh adaptivity. '''</span>
+
|+<span style="font-size:90%">'''Figure 3.3: Stage settings for lower bound limit analysis mesh adaptivity.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
Mesh adaptivity is defined under the category Mesh in the Stage Manager (see Figure 17). In the following, we will use 3 adaptivity steps together with the default option of Shear Dissipation as adaptivity control. This means that a total of 3 calculations will be carried out, each with a mesh adapted according to the previous distribution of the shear dissipation and such that the number of elements in the final mesh is equal to the number of elements specified in Settings (1,000 as before).<br />
+
Mesh adaptivity is defined under the category Mesh in the Stage Manager (see Figure 3.3). In the following, we will use 3 adaptivity steps together with the default option of Shear Dissipation as adaptivity control. This means that a total of 3 calculations will be carried out, each with a mesh adapted according to the previous distribution of the shear dissipation and such that the number of elements in the final mesh is equal to the number of elements specified in Settings (1,000 as before).<br />
 
The results of the analyses are:  
 
The results of the analyses are:  
  
<math display="block">860.0 \text{kN/m}^2 \leq q_u \leq 930.0 \text{kN/m}^2</math>
+
<math display="block">(3.4)\qquad
 +
      860.0 \text{kN/m}^2 \leq q_u \leq 930.0 \text{kN/m}^2</math>
  
 
or:  
 
or:  
  
<math display="block">q_u =  895.0 \text{kN/m}^2  \pm 3.9\%</math>
+
<math display="block">(3.5)\qquad
 +
    q_u =  895.0 \text{kN/m}^2  \pm 3.9\%</math>
  
 
which is a substantial improvement on the previous solution. Further improvements – at the expense of computational cost – can be achieved by increasing the number of elements.<br />
 
which is a substantial improvement on the previous solution. Further improvements – at the expense of computational cost – can be achieved by increasing the number of elements.<br />
The initial and adapted meshes for 1,000 elements are shown in Figure 18 along with the collapse solution.
+
The initial and adapted meshes for 1,000 elements are shown in Figure 3.4 along with the collapse solution.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex01_Fig03_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex01_Fig03_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 18: Initial and adapted meshes and collapse solution with intensity of dissipation (Upper element). '''</span>
+
|+<span style="font-size:90%">'''Figure 3.4: Initial and adapted meshes and collapse solution with intensity of dissipation (Upper element).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 382: Line 396:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex01_ep.png</div>
+
|<div class="image300px">https://wiki.optumce.com/figures/Ex01_ep.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 19: Stage settings Elastoplastic analysis with mesh adaptivity. The Time Scope is irrelevant for the Tresca model. '''</span>
+
|+<span style="font-size:90%">'''Figure 3.5: Stage settings Elastoplastic analysis with mesh adaptivity. The Time Scope is irrelevant for the Tresca model.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 390: Line 404:
 
As for Limit Analysis, mesh adaptivity can be used. Again, this feature is activated by setting Mesh Adaptivity = Yes. A number of fields then appears. Adaptivity Iterations has the same meaning as before and is set to 3. Adaptivity Frequency is relevant only if more than one load step is used and is left at the default value of 3. And as before, the Adaptivity Control is set to Shear Dissipation. In the case of Elastoplastic analysis, the control variable incorporates both shear dissipation and elastic energy.<br />
 
As for Limit Analysis, mesh adaptivity can be used. Again, this feature is activated by setting Mesh Adaptivity = Yes. A number of fields then appears. Adaptivity Iterations has the same meaning as before and is set to 3. Adaptivity Frequency is relevant only if more than one load step is used and is left at the default value of 3. And as before, the Adaptivity Control is set to Shear Dissipation. In the case of Elastoplastic analysis, the control variable incorporates both shear dissipation and elastic energy.<br />
 
Any elastoplastic analysis requires an initial state of stress. In the present example, no From stage is specified, and consequently, the initial stresses are calculated automatically (see Section I.II).<br />
 
Any elastoplastic analysis requires an initial state of stress. In the present example, no From stage is specified, and consequently, the initial stresses are calculated automatically (see Section I.II).<br />
The deformed configuration is shown in Figure 20 along with the distributions of shear dissipation and elastic energy. As expected, the plastic zones are less developed than at full collapse (compare to Figure 18).<br />
+
The deformed configuration is shown in Figure 3.6 along with the distributions of shear dissipation and elastic energy. As expected, the plastic zones are less developed than at full collapse (compare to Figure 3.4).<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex01_Fig04_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex01_Fig04_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 20: Deformations and distribution of shear dissipation (top) and elastic energy (bottom) from Elastoplastic analysis (displacements scaled by a factor of 30). '''</span>
+
|+<span style="font-size:90%">'''Figure 3.6: Deformations and distribution of shear dissipation (top) and elastic energy (bottom) from Elastoplastic analysis (displacements scaled by a factor of 30).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 403: Line 417:
 
The displacements at selected points can be accessed by mouse click. In this way, the displacements at the upper left edge of the foundation are found as:  
 
The displacements at selected points can be accessed by mouse click. In this way, the displacements at the upper left edge of the foundation are found as:  
  
<math display="block">\begin{array}{l}
+
<math display="block">(3.6)\qquad\begin{array}{l}
 
     u_x = -6.0 \text{mm}\\
 
     u_x = -6.0 \text{mm}\\
 
     u_y = -13.8 \text{mm}
 
     u_y = -13.8 \text{mm}
Line 417: Line 431:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex01_mep.png</div>
+
|<div class="image300px">https://wiki.optumce.com/figures/Ex01_mep.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 21: Stage settings for Multiplier Elastoplastic analysis with mesh adaptivity. The Time Scope is irrelevant for the Tresca model. '''</span>
+
|+<span style="font-size:90%">'''Figure 3.7: Stage settings for Multiplier Elastoplastic analysis with mesh adaptivity. The Time Scope is irrelevant for the Tresca model.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
 
In the following, we apply a multiplier load of 600 kN/m<math display="inline">^2</math> (such that a multiplier <math display="inline">\alpha = 1</math> corresponds to the state arrived at in the previous analysis). All other parameters are left at their default values except that the No of Elements is set to 1,000 and Mesh Adaptivity is used, again with default values. The Adaptivity Frequency (<math display="inline">=3</math>) here indicates that the mesh is adapted in load steps 1, 4, 7, etc. The specification of initial stresses follows that of the previous Elastoplastic analysis. No From stage is specified, implying that the initial stresses will be calculated automatically. For further details on Multiplier Elastoplastic analysis, please refer to the Analysis Manual.<br />
 
In the following, we apply a multiplier load of 600 kN/m<math display="inline">^2</math> (such that a multiplier <math display="inline">\alpha = 1</math> corresponds to the state arrived at in the previous analysis). All other parameters are left at their default values except that the No of Elements is set to 1,000 and Mesh Adaptivity is used, again with default values. The Adaptivity Frequency (<math display="inline">=3</math>) here indicates that the mesh is adapted in load steps 1, 4, 7, etc. The specification of initial stresses follows that of the previous Elastoplastic analysis. No From stage is specified, implying that the initial stresses will be calculated automatically. For further details on Multiplier Elastoplastic analysis, please refer to the Analysis Manual.<br />
The results of the analysis in terms of the displacement, stress, etc versus load multiplier can be plotted using the XY Plots tool located in the Results ribbon. In order to specify a point at which to collect such data during the analysis, the Result Point tool located in the Features ribbon can be used. In this case, a Result Point is defined (prior to running the analysis) at the top left corner of the foundation (see Figure 22).
+
The results of the analysis in terms of the displacement, stress, etc versus load multiplier can be plotted using the XY Plots tool located in the Results ribbon. In order to specify a point at which to collect such data during the analysis, the Result Point tool located in the Features ribbon can be used. In this case, a Result Point is defined (prior to running the analysis) at the top left corner of the foundation (see Figure 3.8).
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex01_Fig05_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex01_Fig05_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 22: Setup for Multiplier Elastoplastic analysis: Multiplier Distributed Load of <math display="inline">600</math> kN/m<math display="inline">^2</math> and Result Point located at the top left corner of the foundation (only a section of the full problem domain is shown). '''</span>
+
|+<span style="font-size:90%">'''Figure 3.8: Setup for Multiplier Elastoplastic analysis: Multiplier Distributed Load of <math display="inline">600</math> kN/m<math display="inline">^2</math> and Result Point located at the top left corner of the foundation (only a section of the full problem domain is shown).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 436: Line 450:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex01_mepld.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex01_mepld.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 23: Load-displacement curve created by the XY Plots tool. The actual data can be accessed via the Data tab that appears in the right bottom corner when the curve is selected. '''</span>
+
|+<span style="font-size:90%">'''Figure 3.9: Load-displacement curve created by the XY Plots tool. The actual data can be accessed via the Data tab that appears in the right bottom corner when the curve is selected.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
Using the XY Plots tool, the displacement <math display="inline">|u| = \sqrt{u_x^2+u_y^2}</math> is plotted as function of the load multiplier as shown in Figure 23. We note that the result previously found by means of Elastoplastic analysis (using a single load step), <math display="inline">|u|=\sqrt{0.006^2+0.0138^2}=0.01477</math> is in good agreement with the result of the Multiplier Elastoplastic analysis (which uses 8 load steps to reach a load multiplier of 1 versus only a single step in the previous analysis).<br />
+
Using the XY Plots tool, the displacement <math display="inline">|u| = \sqrt{u_x^2+u_y^2}</math> is plotted as function of the load multiplier as shown in Figure 3.9. We note that the result previously found by means of Elastoplastic analysis (using a single load step), <math display="inline">|u|=\sqrt{0.006^2+0.0138^2}=0.01477</math> is in good agreement with the result of the Multiplier Elastoplastic analysis (which uses 8 load steps to reach a load multiplier of 1 versus only a single step in the previous analysis).<br />
 
Similarly, the final load multiplier of around 1.5, corresponding to a total load of <math display="inline">1.5\times 600 = 900</math> kN/m<math display="inline">^2</math>, is in good agreement with the results of the Limit Analyses (<math display="inline">q_u=895\text{ kN/m}^2 \pm 3.9 \%</math>).
 
Similarly, the final load multiplier of around 1.5, corresponding to a total load of <math display="inline">1.5\times 600 = 900</math> kN/m<math display="inline">^2</math>, is in good agreement with the results of the Limit Analyses (<math display="inline">q_u=895\text{ kN/m}^2 \pm 3.9 \%</math>).
  
 
===  Variation of undrained shear strength with depth  ===
 
===  Variation of undrained shear strength with depth  ===
  
The use of a constant undrained shear strength is often a rather crude approximation to reality where one will usually observe an increase of shear strength with depth. In OPTUM G2, linear variations of all parameters can be specified via the righthand side icon that appears when any parameter field is selected (see Figure 24).<br />
+
The use of a constant undrained shear strength is often a rather crude approximation to reality where one will usually observe an increase of shear strength with depth. In OPTUM G2, linear variations of all parameters can be specified via the righthand side icon that appears when any parameter field is selected (see Figure 3.10).<br />
In the following, a shear strength varying from <math display="inline">s_u = 15</math> kPa at the top surface (at level of <math display="inline">y=16</math> m) and increasing by <math display="inline">5</math> kPa/m with depth is used. Such a variation is can be defined using the Material Parameter dialog shown in Figure 24.
+
In the following, a shear strength varying from <math display="inline">s_u = 15</math> kPa at the top surface (at level of <math display="inline">y=16</math> m) and increasing by <math display="inline">5</math> kPa/m with depth is used. Such a variation is can be defined using the Material Parameter dialog shown in Figure 3.10.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex01_SV.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex01_SV.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 24: Specification of linear distribution of <math display="inline">s_u</math>. '''</span>
+
|+<span style="font-size:90%">'''Figure 3.10: Specification of linear distribution of <math display="inline">s_u</math>.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 460: Line 474:
 
Running upper and lower bound limit analysis for this problem gives:  
 
Running upper and lower bound limit analysis for this problem gives:  
  
<math display="block">q_u =  833.5\pm 3.5\% \text{kN/m}^2</math>
+
<math display="block">(3.7)\qquad
 +
    q_u =  833.5\pm 3.5\% \text{kN/m}^2</math>
  
 
as compared to the value of <math display="inline">q_u =  895.0 \text{kN/m}^2</math> for a constant <math display="inline">s_u=30</math> kPa.<br />
 
as compared to the value of <math display="inline">q_u =  895.0 \text{kN/m}^2</math> for a constant <math display="inline">s_u=30</math> kPa.<br />
Finally, as a check that the correct distribution of <math display="inline">s_u</math> has been specified, the distribution of all material parameters can be visualized under Results (see Figure 25).
+
Finally, as a check that the correct distribution of <math display="inline">s_u</math> has been specified, the distribution of all material parameters can be visualized under Results (see Figure 3.11).
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex01_SV2.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex01_SV2.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 25: Variation of <math display="inline">s_u</math>. '''</span>
+
|+<span style="font-size:90%">'''Figure 3.11: Variation of <math display="inline">s_u</math>.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 482: Line 497:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex02_Setup.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex02_Setup.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 26: Shallow foundation in Firm Clay-MC. '''</span>
+
|+<span style="font-size:90%">'''Figure 4.1: Shallow foundation in Firm Clay-MC.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The material used in this example is the default material Firm Clay-MC (Mohr-Coulomb material with <math display="inline">E=25</math> MPa, <math display="inline">\nu=0.3</math>, <math display="inline">c = 10</math> kPa, <math display="inline">\phi=20^\circ</math>). The overall geometry, shown in Figure 26, is the same as that of the previous example. Again, the aim of the example is to conduct Limit Analyses to determine the bearing capacity and Elastoplastic analyses to determine settlements.<br />
+
The material used in this example is the default material Firm Clay-MC (Mohr-Coulomb material with <math display="inline">E=25</math> MPa, <math display="inline">\nu=0.3</math>, <math display="inline">c = 10</math> kPa, <math display="inline">\phi=20^\circ</math>). The overall geometry, shown in Figure 4.1, is the same as that of the previous example. Again, the aim of the example is to conduct Limit Analyses to determine the bearing capacity and Elastoplastic analyses to determine settlements.<br />
 
In contrast to the previous analysis, however, a static water table is defined 2.5 m below the ground surface. This can be done by using the Water Table tool in the Features ribbon.<br />
 
In contrast to the previous analysis, however, a static water table is defined 2.5 m below the ground surface. This can be done by using the Water Table tool in the Features ribbon.<br />
Also, in contrast to the previous example, it is important to distinguish between Long Term and Short Term conditions. That is, the Drainage Conditions of the Firm Clay material are Drained/Undrained, meaning that excess pore pressures are generated in the short term but not in the long term. The appropriate selection of long/short term conditions is made for each stage in the lower half of the Stage Manager under Time Scope (see Figure 27).<br />
+
Also, in contrast to the previous example, it is important to distinguish between Long Term and Short Term conditions. That is, the Drainage Conditions of the Firm Clay material are Drained/Undrained, meaning that excess pore pressures are generated in the short term but not in the long term. The appropriate selection of long/short term conditions is made for each stage in the lower half of the Stage Manager under Time Scope (see Figure 4.2).<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex02_TS.png</div>
+
|<div class="image300px">https://wiki.optumce.com/figures/Ex02_TS.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 27: Stage settings for a Short Term analysis with Mesh Adaptivity. '''</span>
+
|+<span style="font-size:90%">'''Figure 4.2: Stage settings for a Short Term analysis with Mesh Adaptivity.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
For all analyses involving hydraulic boundary conditions or Fluid materials, the hydraulic model needs to be considered. In OPTUM G2, all seepage calculations are carried out as general variably saturated analyses. As such, both the saturated hydraulic conductivities and the hydraulic model need to be considered. In the following, we use the default Linear hydraulic model (see the Materials manual) with default settings as indicated in Figure 26.<br />
+
For all analyses involving hydraulic boundary conditions or Fluid materials, the hydraulic model needs to be considered. In OPTUM G2, all seepage calculations are carried out as general variably saturated analyses. As such, both the saturated hydraulic conductivities and the hydraulic model need to be considered. In the following, we use the default Linear hydraulic model (see the Materials manual) with default settings as indicated in Figure 4.1.<br />
Finally, for any Short Term analysis, the initial stress state needs to be calculated. As described in the previous example, this may be done via a separate stage or, if no From stage is specified, it is done automatically using either the specialized K<math display="inline">_0</math> Analysis or an Elastoplastic Analysis as specified for the given material. In both cases, the pore pressures are calculated as part of the analysis. In the present example, the K<math display="inline">_0</math> Analysis is used with an earth pressure coefficient of <math display="inline">K_0=0.66</math> (see Figure 26).
+
Finally, for any Short Term analysis, the initial stress state needs to be calculated. As described in the previous example, this may be done via a separate stage or, if no From stage is specified, it is done automatically using either the specialized K<math display="inline">_0</math> Analysis or an Elastoplastic Analysis as specified for the given material. In both cases, the pore pressures are calculated as part of the analysis. In the present example, the K<math display="inline">_0</math> Analysis is used with an earth pressure coefficient of <math display="inline">K_0=0.66</math> (see Figure 4.1).
  
 
===  Limit Analysis  ===
 
===  Limit Analysis  ===
  
Upper and lower bound limit analyses are conducted as described in the previous example, except that the Time Scope is Short Term. A total of 1,000 elements are used along with 3 adaptivity iterations (these settings are shown in Figure 27).<br />
+
Upper and lower bound limit analyses are conducted as described in the previous example, except that the Time Scope is Short Term. A total of 1,000 elements are used along with 3 adaptivity iterations (these settings are shown in Figure 4.2).<br />
 
This results in the following bounds on the limit load:  
 
This results in the following bounds on the limit load:  
  
<math display="block">667.2\text{ kN/m$^2$} \leq q_u\leq 727.7\text{ kN/m$^2$}</math>
+
<math display="block">(4.1)\qquad
 +
    667.2\text{ kN/m$^2$} \leq q_u\leq 727.7\text{ kN/m$^2$}</math>
  
 
or:  
 
or:  
  
<math display="block">q_u = 697.4\text{ kN/m$^2$} \pm 4.3\%</math>
+
<math display="block">(4.2)\qquad
 +
    q_u = 697.4\text{ kN/m$^2$} \pm 4.3\%</math>
  
 
Increasing the number of elements to 2,000 gives:  
 
Increasing the number of elements to 2,000 gives:  
  
<math display="block">q_u = 695.8\text{ kN/m$^2$} \pm 2.5\%</math>
+
<math display="block">(4.3)\qquad
 +
    q_u = 695.8\text{ kN/m$^2$} \pm 2.5\%</math>
  
 
and to 4,000:  
 
and to 4,000:  
  
<math display="block">q_u = 696.8\text{ kN/m$^2$} \pm 1.6\%</math>
+
<math display="block">(4.4)\qquad
 +
    q_u = 696.8\text{ kN/m$^2$} \pm 1.6\%</math>
  
 
at which point the solution is deemed sufficiently accurate. Note that while the error reduces three-fold by increasing the number of elements from 1,000 to 4,000, the calculated mean value increases by only slightly over 1%. This trend, that the mean value is rather more accurate than suggested by the error between the upper and lower bounds, is quite typical. It is a consequence of the fact that the errors made in the upper and lower bound calculations tend to be of a similar magnitude and thus cancel each other out in the mean value.<br />
 
at which point the solution is deemed sufficiently accurate. Note that while the error reduces three-fold by increasing the number of elements from 1,000 to 4,000, the calculated mean value increases by only slightly over 1%. This trend, that the mean value is rather more accurate than suggested by the error between the upper and lower bounds, is quite typical. It is a consequence of the fact that the errors made in the upper and lower bound calculations tend to be of a similar magnitude and thus cancel each other out in the mean value.<br />
The distribution of undrained shear strength is shown in Figure 28. For the Mohr-Coulomb model used in this example it is given by (see Materials Manual):  
+
The distribution of undrained shear strength is shown in Figure 4.3. For the Mohr-Coulomb model used in this example it is given by (see Materials Manual):  
  
<math display="block">s_u = c\cos\phi+\frac{1}{2}(1+K_0)\sin\phi\sigma_{v,0}'</math>
+
<math display="block">(4.5)\qquad
 +
    s_u = c\cos\phi+\frac{1}{2}(1+K_0)\sin\phi\sigma_{v,0}'</math>
  
 
where <math display="inline">\sigma_{v,0}'</math> is the initial vertical effective stress (positive in compression).<br />
 
where <math display="inline">\sigma_{v,0}'</math> is the initial vertical effective stress (positive in compression).<br />
Line 533: Line 553:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex2_su.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex2_su.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 28: Distribution of undrained shear strength <math display="inline">s_u</math>. '''</span>
+
|+<span style="font-size:90%">'''Figure 4.3: Distribution of undrained shear strength <math display="inline">s_u</math>.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The collapse mechanism and distribution of shear dissipation is shown in Figure 29. It follows that of the previous example quite closely.
+
The collapse mechanism and distribution of shear dissipation is shown in Figure 4.4. It follows that of the previous example quite closely.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex02_Collapse.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex02_Collapse.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 29: Collapse solution with intensity of shear dissipation. '''</span>
+
|+<span style="font-size:90%">'''Figure 4.4: Collapse solution with intensity of shear dissipation.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 553: Line 573:
 
===  Elastoplastic analysis  ===
 
===  Elastoplastic analysis  ===
  
Next, we aim to determine the deformations for a fixed load of <math display="inline">600</math> kN/m<math display="inline">^2</math> as shown in Figure 30. The load is assumed to be applied sufficiently rapidly for undrained conditions to be considered. It then remains on the foundation until all excess pore pressures have dissipated. The complete analysis must therefore consider both Short and Long Term Time Scopes.<br />
+
Next, we aim to determine the deformations for a fixed load of <math display="inline">600</math> kN/m<math display="inline">^2</math> as shown in Figure 4.5. The load is assumed to be applied sufficiently rapidly for undrained conditions to be considered. It then remains on the foundation until all excess pore pressures have dissipated. The complete analysis must therefore consider both Short and Long Term Time Scopes.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex02_EP.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex02_EP.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 30: Geometry, loads and material (top) and appearance of Stage Manager for Short Term (EP ST) and Long Term (EP LT) Elastoplastic analysis stages. '''</span>
+
|+<span style="font-size:90%">'''Figure 4.5: Geometry, loads and material (top) and appearance of Stage Manager for Short Term (EP ST) and Long Term (EP LT) Elastoplastic analysis stages.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
This is done via two separate stages. First an Elastoplastic analysis stage with Time Scope = Short Term is defined. This stage is then cloned and the Time Scope changed to Long Term. In addition, the From stage is chosen as the preceding Short Term stage. This means that the result of the first stage is used as input when processing the second stage. The Stage Manager for the two stages is shown in Figure 30. We note that it is possible to reset the displacements at the beginning of the Long Term stage that follows on from the Short Term stage. Since we are interested in the final total displacements, this option is not utilized and the default option of not resetting the displacements is used.<br />
+
This is done via two separate stages. First an Elastoplastic analysis stage with Time Scope = Short Term is defined. This stage is then cloned and the Time Scope changed to Long Term. In addition, the From stage is chosen as the preceding Short Term stage. This means that the result of the first stage is used as input when processing the second stage. The Stage Manager for the two stages is shown in Figure 4.5. We note that it is possible to reset the displacements at the beginning of the Long Term stage that follows on from the Short Term stage. Since we are interested in the final total displacements, this option is not utilized and the default option of not resetting the displacements is used.<br />
Generally speaking, for Elastoplastic analysis of Drained/Undrained materials under Short Term conditions, a nonassociated flow rule with a dilation angle of zero should be used in order to obtain reasonable results (see the Theory Manual). This setting is invoked via the material property window as indicated in Figure 30.<br />
+
Generally speaking, for Elastoplastic analysis of Drained/Undrained materials under Short Term conditions, a nonassociated flow rule with a dilation angle of zero should be used in order to obtain reasonable results (see the Theory Manual). This setting is invoked via the material property window as indicated in Figure 4.5.<br />
Running the two stages, both with 1,000 6-node Gauss elements and 3 adaptivity iterations, results in terms of the deformed configurations are shown in Figure 31. It is noted that the majority of the deformation occurs in the short term, i.e. in this case the effects of consolidation are relatively minor compared to the instantaneous deformations.<br />
+
Running the two stages, both with 1,000 6-node Gauss elements and 3 adaptivity iterations, results in terms of the deformed configurations are shown in Figure 4.6. It is noted that the majority of the deformation occurs in the short term, i.e. in this case the effects of consolidation are relatively minor compared to the instantaneous deformations.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image300px">https://wiki.optumce.com/figures/Ex02_Def.png</div>
+
|<div class="image600px">https://wiki.optumce.com/figures/Ex02_Def.png</div>
 
|-
 
|-
|+<span style="font-size:92%">''' = (u_x^2+u_y^2)^\frac{1 }{2}</math> (displacements scaled by a factor of 30).'''</span>
+
|+<span style="font-size:90%">'''Figure 4.6: Deformations and distribution of <math display="inline"> \|u\| = (u_x^2+u_y^2)^\frac{1}{2}</math> (displacements scaled by a factor of 30).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 580: Line 600:
 
==  SETTLEMENT OF STRIP FOOTING ON ELASTIC SOIL  ==
 
==  SETTLEMENT OF STRIP FOOTING ON ELASTIC SOIL  ==
  
The following example demonstrates the capabilities of OPTUM G2 in computing upper and lower bounds not only on the ultimate limit load, but also the elastic energy and, in turn, on the deformations. The example under consideration is shown in Figure 32. It involves a centrally loaded weightless rigid strip footing on an elastic soil. The footing and the soil are modeled using the default materials Rigid and Linear Elastic respectively.
+
The following example demonstrates the capabilities of OPTUM G2 in computing upper and lower bounds not only on the ultimate limit load, but also the elastic energy and, in turn, on the deformations. The example under consideration is shown in Figure 5.1. It involves a centrally loaded weightless rigid strip footing on an elastic soil. The footing and the soil are modeled using the default materials Rigid and Linear Elastic respectively.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex31_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex31_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 32: Shallow foundation on an elastic soil. '''</span>
+
|+<span style="font-size:90%">'''Figure 5.1: Shallow foundation on an elastic soil.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 592: Line 612:
 
Using dimensional analysis arguments, it may be shown that the vertical displacement of the footing can be expressed as:  
 
Using dimensional analysis arguments, it may be shown that the vertical displacement of the footing can be expressed as:  
  
<math display="block">(1)\qquad
+
<math display="block">(5.1)\qquad
 
     u_v = \beta\frac{q B}{E}</math>
 
     u_v = \beta\frac{q B}{E}</math>
  
 
where <math display="inline">q</math> is the footing pressure, <math display="inline">B</math> is the footing width, <math display="inline">E</math> is Young’s modulus and <math display="inline">\beta</math> is a parameter that, for fixed geometric dimensions, depends only on Poisson’s ratio.<br />
 
where <math display="inline">q</math> is the footing pressure, <math display="inline">B</math> is the footing width, <math display="inline">E</math> is Young’s modulus and <math display="inline">\beta</math> is a parameter that, for fixed geometric dimensions, depends only on Poisson’s ratio.<br />
 
As is discussed in detail in the Theory Manual, the elements Lower and 6-node FE provide bounds on the elastic energy. The elastic energy is equal to the external energy which in this case is given simply by the footing pressure times the vertical displacement which is constant along the footing. As such, the Lower element will provide an overestimate of the vertical displacement while the 6-node FE will provide an underestimate.<br />
 
As is discussed in detail in the Theory Manual, the elements Lower and 6-node FE provide bounds on the elastic energy. The elastic energy is equal to the external energy which in this case is given simply by the footing pressure times the vertical displacement which is constant along the footing. As such, the Lower element will provide an overestimate of the vertical displacement while the 6-node FE will provide an underestimate.<br />
In the following, the default Young’s modulus of <math display="inline">E=30</math> MPa is used and the load applied is <math display="inline">q=150</math> kN/m<math display="inline">^2</math>. With the vertical displacement calculated, Eq. 1 is used to determine <math display="inline">\beta</math>. For both types of elements, a total of 10,000 elements are used. The analysis can either be carried out using the Elastoplastic or the Elastic analysis types.<br />
+
In the following, the default Young’s modulus of <math display="inline">E=30</math> MPa is used and the load applied is <math display="inline">q=150</math> kN/m<math display="inline">^2</math>. With the vertical displacement calculated, Eq.5.1 is used to determine <math display="inline">\beta</math>. For both types of elements, a total of 10,000 elements are used. The analysis can either be carried out using the Elastoplastic or the Elastic analysis types.<br />
The results are shown in Table 2.
+
The results are shown in Table 5.1.
 +
 
 +
<div id="Ex31_Tab01">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 2: Upper and lower bounds on parameter <math display="inline">\beta</math> (Eqn. Eq. 1) for elastic settlement of shallow foundation in an elastic soil using 10,000 elements.
+
|+ Table 5.1: Upper and lower bounds on parameter <math display="inline">\beta</math> (Eqn. Eq.5.1) for elastic settlement of shallow foundation in an elastic soil using 10,000 elements.
 
|align="center"|  <math display="inline">\nu</math>  
 
|align="center"|  <math display="inline">\nu</math>  
 
|align="center"| Lower bound
 
|align="center"| Lower bound
Line 681: Line 703:
 
|}
 
|}
  
To give an indication of the error in typical analyses, the problem is re-analyzed using 100 elements. The results are shown in Figure 33 and indicate that the 6-node FE is somewhat more accurate than the Lower bound element. This observation is quite general and holds for most problems.
+
 
 +
</div>
 +
To give an indication of the error in typical analyses, the problem is re-analyzed using 100 elements. The results are shown in Figure 5.2 and indicate that the 6-node FE is somewhat more accurate than the Lower bound element. This observation is quite general and holds for most problems.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex31_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex31_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 33: Parameter <math display="inline">\beta</math> using 100 Lower and 6-node FE. '''</span>
+
|+<span style="font-size:90%">'''Figure 5.2: Parameter <math display="inline">\beta</math> using 100 Lower and 6-node FE.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 695: Line 719:
 
==  STRIP LOAD ON GIBSON SOIL  ==
 
==  STRIP LOAD ON GIBSON SOIL  ==
  
The following example concerns the problem of a strip load on a semi-infinite domain of a Gibson soil, i.e. an elastic material with a Young’s modulus that increases linearly with depth from a value of zero at the top surface. In the following, rather than using an actual semi-infinite domain, a sufficiently large one is used (see Figure 34). We note that the effect of the domain size decreases rather rapidly as the stiffness increases with depth (as opposed to the case where Young’s modulus is constant and the deformation at the top surface is a function of the domain depth).
+
The following example concerns the problem of a strip load on a semi-infinite domain of a Gibson soil, i.e. an elastic material with a Young’s modulus that increases linearly with depth from a value of zero at the top surface. In the following, rather than using an actual semi-infinite domain, a sufficiently large one is used (see Figure 6.1). We note that the effect of the domain size decreases rather rapidly as the stiffness increases with depth (as opposed to the case where Young’s modulus is constant and the deformation at the top surface is a function of the domain depth).
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex32_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex32_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 34: Strip load on Gibson soil. '''</span>
+
|+<span style="font-size:90%">'''Figure 6.1: Strip load on Gibson soil.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
For <math display="inline">q = 10</math> kN/m<math display="inline">^2</math>, <math display="inline">\nu = 0.5</math> and an increase in Young’s modulus of 0.3 MPa/m, <bib id="Gibson:1967" /> gives the exact solution of a uniform displacement of <math display="inline">u = 0.05</math> m underneath the load. With 10,000 6-node FE, we obtain a maximum displacement of <math display="inline">0.0494</math> m which is in error by about 1%. The vertical deformation field is shown in Figure 35.
+
For <math display="inline">q = 10</math> kN/m<math display="inline">^2</math>, <math display="inline">\nu = 0.5</math> and an increase in Young’s modulus of 0.3 MPa/m, <bib id="Gibson:1967" /> gives the exact solution of a uniform displacement of <math display="inline">u = 0.05</math> m underneath the load. With 10,000 6-node FE, we obtain a maximum displacement of <math display="inline">0.0494</math> m which is in error by about 1%. The vertical deformation field is shown in Figure 6.2.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex32_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex32_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 35: Strip load on Gibson soil: vertical deformation field. '''</span>
+
|+<span style="font-size:90%">'''Figure 6.2: Strip load on Gibson soil: vertical deformation field.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 719: Line 743:
 
==  FOOTING ON SOIL WITH USER DEFINED MATERIAL DATA  ==
 
==  FOOTING ON SOIL WITH USER DEFINED MATERIAL DATA  ==
  
In some cases, it may be necessary to use a spatial variation of certain material parameters that cannot easily be described by simple analytical expressions. OPTUM G2 caters for this scenario by allowing for arbitrary distributions of all material parameters to be used as demonstrated in the following. The problem setup is shown in Figure 36. The limit load of the footing is to be determined with the undrained shear strength, <math display="inline">s_u</math>, being imported from a data set external to OPTUM G2.
+
In some cases, it may be necessary to use a spatial variation of certain material parameters that cannot easily be described by simple analytical expressions. OPTUM G2 caters for this scenario by allowing for arbitrary distributions of all material parameters to be used as demonstrated in the following. The problem setup is shown in Figure 7.1. The limit load of the footing is to be determined with the undrained shear strength, <math display="inline">s_u</math>, being imported from a data set external to OPTUM G2.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/user01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/user01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 36: Footing on soil with user defined material data (top) and specification of input data (bottom). '''</span>
+
|+<span style="font-size:90%">'''Figure 7.1: Footing on soil with user defined material data (top) and specification of input data (bottom).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 732: Line 756:
 
With the input defined, upper and lower bound Limit Analyses are conducted using 5,000 elements and 3 adaptivity iterations. The results are a bearing capacity of  
 
With the input defined, upper and lower bound Limit Analyses are conducted using 5,000 elements and 3 adaptivity iterations. The results are a bearing capacity of  
  
<math display="block">q_u = 241.1 \text{kN/m}^2\pm 0.6\%</math>
+
<math display="block">(7.1)\qquad
 +
    q_u = 241.1 \text{kN/m}^2\pm 0.6\%</math>
  
The parameter map can be visualized under Results via the Material Parameters drop-down. The distribution of <math display="inline">s_u</math> is shown in Figure 37. The collapse mechanism (lower bound) is shown in Figure 38.
+
The parameter map can be visualized under Results via the Material Parameters drop-down. The distribution of <math display="inline">s_u</math> is shown in Figure 7.2. The collapse mechanism (lower bound) is shown in Figure 7.3.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/user03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/user03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 37: Variation of <math display="inline">s_u</math> from user defined input data. '''</span>
+
|+<span style="font-size:90%">'''Figure 7.2: Variation of <math display="inline">s_u</math> from user defined input data.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 746: Line 771:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/user04.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/user04.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 38: Collapse mechanism and distribution of shear dissipation (Element Type = Lower). '''</span>
+
|+<span style="font-size:90%">'''Figure 7.3: Collapse mechanism and distribution of shear dissipation (Element Type = Lower).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 756: Line 781:
 
==  SLOPE STABILITY – LONG TERM  ==
 
==  SLOPE STABILITY – LONG TERM  ==
  
In this example we consider the stability of a layered slope as shown in Figure 39. The layers comprise the default materials Firm Clay and Stiff Clay. In the following, only long term analysis is considered. The ground water table is assumed to be located well beneath the toe of the slope.
+
In this example we consider the stability of a layered slope as shown in Figure 8.1. The layers comprise the default materials Firm Clay and Stiff Clay. In the following, only long term analysis is considered. The ground water table is assumed to be located well beneath the toe of the slope.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex02_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex02_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 39: Layered slope. '''</span>
+
|+<span style="font-size:90%">'''Figure 8.1: Layered slope.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 769: Line 794:
 
A common definition involves the actual unit weight in relation to the unit weight that will lead to incipient collapse:  
 
A common definition involves the actual unit weight in relation to the unit weight that will lead to incipient collapse:  
  
<math display="block">\text{FS}_\gamma = \frac{\gamma_\text{cr}}{\gamma}</math>
+
<math display="block">(8.1)\qquad
 +
    \text{FS}_\gamma = \frac{\gamma_\text{cr}}{\gamma}</math>
  
 
where <math display="inline">\gamma</math> is the actual unit weight and <math display="inline">\gamma_\text{cr}</math> is the unit weight that will lead to incipient collapse. Alternatively, this factor of safety may be defined in terms of gravitational accelerations:  
 
where <math display="inline">\gamma</math> is the actual unit weight and <math display="inline">\gamma_\text{cr}</math> is the unit weight that will lead to incipient collapse. Alternatively, this factor of safety may be defined in terms of gravitational accelerations:  
  
<math display="block">\text{FS}_g = \frac{g_\text{cr}}{g}</math>
+
<math display="block">(8.2)\qquad
 +
    \text{FS}_g = \frac{g_\text{cr}}{g}</math>
  
 
where <math display="inline">g</math> is the actual gravitation acceleration (=9.8 m/s<math display="inline">^2</math> by default) and <math display="inline">g_\text{cr}</math> is the gravitational acceleration that will lead to incipient collapse.<br />
 
where <math display="inline">g</math> is the actual gravitation acceleration (=9.8 m/s<math display="inline">^2</math> by default) and <math display="inline">g_\text{cr}</math> is the gravitational acceleration that will lead to incipient collapse.<br />
 
Another common definition involves the material strengths in relation to the strengths that will lead to incipient collapse. In particular, for Mohr-Coulomb materials the factor of safety may be defined as  
 
Another common definition involves the material strengths in relation to the strengths that will lead to incipient collapse. In particular, for Mohr-Coulomb materials the factor of safety may be defined as  
  
<math display="block">(2)\qquad
+
<math display="block">(8.3)\qquad
 
     \text{FS}_s = \frac{c}{~ c_\text{cr}} = \frac{\tan\phi}{~ \tan\phi_\text{cr}}</math>
 
     \text{FS}_s = \frac{c}{~ c_\text{cr}} = \frac{\tan\phi}{~ \tan\phi_\text{cr}}</math>
  
Line 787: Line 814:
 
===  Gravity Multiplier  ===
 
===  Gravity Multiplier  ===
  
The assessment of the gravity based factor of safety using Limit Analysis is realized by setting Multiplier = Gravity under Settings in the lower half of the Stage Manager window (see Figure 39). This type of limit analysis ignores all multiplier loads and magnifies the gravitational acceleration (and thereby the unit weight) until a state of failure is attained. The resulting collapse multiplier is the sought Factor of Safety <math display="inline">\text{FS}_g</math>.<br />
+
The assessment of the gravity based factor of safety using Limit Analysis is realized by setting Multiplier = Gravity under Settings in the lower half of the Stage Manager window (see Figure 8.1). This type of limit analysis ignores all multiplier loads and magnifies the gravitational acceleration (and thereby the unit weight) until a state of failure is attained. The resulting collapse multiplier is the sought Factor of Safety <math display="inline">\text{FS}_g</math>.<br />
Long Term analysis is specified by the Time Scope field in the lower half of the Stage Manager window. For this and subsequent calculations we will use 1,000 elements and Mesh Adaptivity with 3 Adaptivity Iterations and Shear Dissipation as the Adaptivity Control. Lower and upper bound calculations are defined in separate stages by setting Element Type to Lower and Upper respectively. These settings are all shown in Figure 39.<br />
+
Long Term analysis is specified by the Time Scope field in the lower half of the Stage Manager window. For this and subsequent calculations we will use 1,000 elements and Mesh Adaptivity with 3 Adaptivity Iterations and Shear Dissipation as the Adaptivity Control. Lower and upper bound calculations are defined in separate stages by setting Element Type to Lower and Upper respectively. These settings are all shown in Figure 8.1.<br />
 
The results of the analyses are:  
 
The results of the analyses are:  
  
<math display="block">1.87 \leq \text{FS}_g \leq 1.94</math>
+
<math display="block">(8.4)\qquad
 +
    1.87 \leq \text{FS}_g \leq 1.94</math>
  
 
or:  
 
or:  
  
<math display="block">\text{FS}_g = 1.90 \pm 0.04</math>
+
<math display="block">(8.5)\qquad
 +
    \text{FS}_g = 1.90 \pm 0.04</math>
  
from which we can conclude that the slope is stable in the long term. The collapse solution is shown in Figure 40.
+
from which we can conclude that the slope is stable in the long term. The collapse solution is shown in Figure 8.2.
  
 
===  Strength Reduction  ===
 
===  Strength Reduction  ===
  
The objective of Strength Reduction analysis is to determine a set of reduced parameters that lead to incipient collapse, i.e. that imply a gravity multiplier equal to 1. For the Mohr-Coulomb model, the strengths are reduced according to Eqn. Eq. 2. The calculations are carried out as a series of limit analyses. Hence, Strength Reduction analysis is usually significantly more expensive than a single Gravity Multiplier Limit Analysis.<br />
+
The objective of Strength Reduction analysis is to determine a set of reduced parameters that lead to incipient collapse, i.e. that imply a gravity multiplier equal to 1. For the Mohr-Coulomb model, the strengths are reduced according to Eqn. Eq.8.3. The calculations are carried out as a series of limit analyses. Hence, Strength Reduction analysis is usually significantly more expensive than a single Gravity Multiplier Limit Analysis.<br />
 
Strength Reduction analysis is carried out by choosing this analysis in the Stage Manager window. Using 1,000 elements and 3 adaptivity steps as in the previous analysis, we obtain:  
 
Strength Reduction analysis is carried out by choosing this analysis in the Stage Manager window. Using 1,000 elements and 3 adaptivity steps as in the previous analysis, we obtain:  
  
<math display="block">1.34 \leq \text{FS}_s \leq 1.37</math>
+
<math display="block">(8.6)\qquad
 +
    1.34 \leq \text{FS}_s \leq 1.37</math>
  
 
or:  
 
or:  
  
<math display="block">\text{FS}_s = 1.35 \pm 0.02</math>
+
<math display="block">(8.7)\qquad
 +
    \text{FS}_s = 1.35 \pm 0.02</math>
  
 
from which we again conclude that the slope is stable in the long term, though with a numerically smaller factor safety than in the previous analysis.<br />
 
from which we again conclude that the slope is stable in the long term, though with a numerically smaller factor safety than in the previous analysis.<br />
The lower bound collapse solutions for the two analyses are shown in Figure 40. We note that the displacements (or velocities) are discontinuous which is a particular feature of the Lower bound element. The two collapse mechanisms are quite similar, with the differences stemming from the fact that the Strength Reduction analysis implies failure under a friction angle of <math display="inline">\phi_\text{cr} = \arctan[\tan (20^\circ)/1.35]\simeq 15^\circ</math> while the gravity multiplier mechanism corresponds to failure with the original value of <math display="inline">\phi=20^\circ</math>.
+
The lower bound collapse solutions for the two analyses are shown in Figure 8.2. We note that the displacements (or velocities) are discontinuous which is a particular feature of the Lower bound element. The two collapse mechanisms are quite similar, with the differences stemming from the fact that the Strength Reduction analysis implies failure under a friction angle of <math display="inline">\phi_\text{cr} = \arctan[\tan (20^\circ)/1.35]\simeq 15^\circ</math> while the gravity multiplier mechanism corresponds to failure with the original value of <math display="inline">\phi=20^\circ</math>.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex02_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex02_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 40: Gravity Multiplier Limit Analysis and Strength Reduction collapse solutions for layered slope (lower bound) with intensity of total dissipation. '''</span>
+
|+<span style="font-size:90%">'''Figure 8.2: Gravity Multiplier Limit Analysis and Strength Reduction collapse solutions for layered slope (lower bound) with intensity of total dissipation.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 827: Line 858:
 
==  SLOPE STABILITY – SHORT TERM  ==
 
==  SLOPE STABILITY – SHORT TERM  ==
  
This example concerns the slope previously introduced in the previous section (shown again in Figure 41). While the long term stability was considered in the previous section, the aim of the present example is to determine the short term stability. Again, the ground water table is assumed to be located well beneath the toe of the slope. At the same time, it is assumed that the degree of saturation throughout the slope is sufficient for excess pore pressures to develop, i.e. for the material to respond in an undrained manner under short term conditions everywhere.
+
This example concerns the slope previously introduced in the previous section (shown again in Figure 9.1). While the long term stability was considered in the previous section, the aim of the present example is to determine the short term stability. Again, the ground water table is assumed to be located well beneath the toe of the slope. At the same time, it is assumed that the degree of saturation throughout the slope is sufficient for excess pore pressures to develop, i.e. for the material to respond in an undrained manner under short term conditions everywhere.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/ExSLOPEII_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/ExSLOPEII_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 41: Layered slope. '''</span>
+
|+<span style="font-size:90%">'''Figure 9.1: Layered slope.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 844: Line 875:
 
This approach is the most direct and convenient. The initial stresses are here determined automatically using the approach described in detail in the previous section. Using Strength Reduction analysis with 1,000 elements and 3 adaptivity iterations leads to the following estimate of the strength based factor of safety:  
 
This approach is the most direct and convenient. The initial stresses are here determined automatically using the approach described in detail in the previous section. Using Strength Reduction analysis with 1,000 elements and 3 adaptivity iterations leads to the following estimate of the strength based factor of safety:  
  
<math display="block">1.47 \leq \text{FS}_s \leq 1.49</math>
+
<math display="block">(9.1)\qquad
 +
    1.47 \leq \text{FS}_s \leq 1.49</math>
  
 
or:  
 
or:  
  
<math display="block">\text{FS}_s = 1.48 \pm 0.01</math>
+
<math display="block">(9.2)\qquad
 +
    \text{FS}_s = 1.48 \pm 0.01</math>
  
This can be compared to the long term factor of approximately 1.34 in the previous section. In conclusion, the slope is slightly more stable in the short term than in the long term. The collapse mechanism is shown in Figure 42.
+
This can be compared to the long term factor of approximately 1.34 in the previous section. In conclusion, the slope is slightly more stable in the short term than in the long term. The collapse mechanism is shown in Figure 9.2.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex19_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex19_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 42: Collapse mechanism (short term Strength Reduction analysis). '''</span>
+
|+<span style="font-size:90%">'''Figure 9.2: Collapse mechanism (short term Strength Reduction analysis).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 862: Line 895:
 
===  Initial stresses by excavation  ===
 
===  Initial stresses by excavation  ===
  
An alternative approach to the determination of the initial stresses is to begin with an initially rectangular domain and then excavate the necessary material to create the slope. OPTUM G2 is particularly suited for this type of task. The different stages involved and the task of linking these to each other is explained in Figure 44.<br />
+
An alternative approach to the determination of the initial stresses is to begin with an initially rectangular domain and then excavate the necessary material to create the slope. OPTUM G2 is particularly suited for this type of task. The different stages involved and the task of linking these to each other is explained in Figure 9.4.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex19_Fig03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex19_Fig03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 43: Initial vertical and horizontal stresses. '''</span>
+
|+<span style="font-size:90%">'''Figure 9.3: Initial vertical and horizontal stresses.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 875: Line 908:
 
The resulting factor of safety is given by  
 
The resulting factor of safety is given by  
  
<math display="block">1.39 \leq \text{FS}_s \leq 1.41</math>
+
<math display="block">(9.3)\qquad
 +
    1.39 \leq \text{FS}_s \leq 1.41</math>
  
 
or:  
 
or:  
  
<math display="block">\text{FS}_s = 1.40 \pm 0.01</math>
+
<math display="block">(9.4)\qquad
 +
    \text{FS}_s = 1.40 \pm 0.01</math>
  
which is slightly lower than was obtained using the automatic approach. The initial stress distributions resulting from each of the approaches are shown in Figure 43. The differences here are ultimately reflected in the corresponding factors of safety. It should be noted that there is no general rule that one approach will imply a higher strength than the other. Indeed, in most cases the result will, as in this case, be fairly similar.<br />
+
which is slightly lower than was obtained using the automatic approach. The initial stress distributions resulting from each of the approaches are shown in Figure 9.3. The differences here are ultimately reflected in the corresponding factors of safety. It should be noted that there is no general rule that one approach will imply a higher strength than the other. Indeed, in most cases the result will, as in this case, be fairly similar.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex19_Fig04.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex19_Fig04.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 44: Stages in lower and upper bound short term strength reduction using elastoplastic analysis to excavate the slope. '''</span>
+
|+<span style="font-size:90%">'''Figure 9.4: Stages in lower and upper bound short term strength reduction using elastoplastic analysis to excavate the slope.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 894: Line 929:
 
==  STABILITY OF SLOPE SUBJECTED TO UNCONFINED SEEPAGE  ==
 
==  STABILITY OF SLOPE SUBJECTED TO UNCONFINED SEEPAGE  ==
  
The following example considers the stability of the slope shown in Figure 45. The seepage pressure distribution and the location of the phreatic surface are not known a priori except that the groundwater table is located at the foot of the slope and the conditions at a distance of 15 m from the crest of the slope correspond to hydrostatic conditions. These boundary conditions are imposed using the Water Table tool available in the Features ribbon. In particular, by clicking any point on a vertical line, a constant head with value equal to the position is imposed.
+
The following example considers the stability of the slope shown in Figure 10.1. The seepage pressure distribution and the location of the phreatic surface are not known a priori except that the groundwater table is located at the foot of the slope and the conditions at a distance of 15 m from the crest of the slope correspond to hydrostatic conditions. These boundary conditions are imposed using the Water Table tool available in the Features ribbon. In particular, by clicking any point on a vertical line, a constant head with value equal to the position is imposed.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/SeepSlope_Setup.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/SeepSlope_Setup.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 45: Slope subjected to unconfined seepage. '''</span>
+
|+<span style="font-size:90%">'''Figure 10.1: Slope subjected to unconfined seepage.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 908: Line 943:
 
The results of the analyses are a strength based factor of safety bracketed by:  
 
The results of the analyses are a strength based factor of safety bracketed by:  
  
<math display="block">1.48 \leq \text{FS}_s\leq 1.51</math>
+
<math display="block">(10.1)\qquad
 +
    1.48 \leq \text{FS}_s\leq 1.51</math>
  
 
or:  
 
or:  
  
<math display="block">\text{FS}_s = 1.49\pm 0.02</math>
+
<math display="block">(10.2)\qquad
 +
    \text{FS}_s = 1.49\pm 0.02</math>
  
The saturation distribution is shown in Figure 46. The collapse solution is shown in Figure 47. It is interesting to note the mechanism which comprises two independent slip lines.
+
The saturation distribution is shown in Figure 10.2. The collapse solution is shown in Figure 10.3. It is interesting to note the mechanism which comprises two independent slip lines.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/SeepSlope_Sat.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/SeepSlope_Sat.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 46: Degree of saturation (red corresponds to <math display="inline">S=1</math> and blue to <math display="inline">S=0</math>). '''</span>
+
|+<span style="font-size:90%">'''Figure 10.2: Degree of saturation (red corresponds to <math display="inline">S=1</math> and blue to <math display="inline">S=0</math>).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 929: Line 966:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/SeepSlope_Collapse.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/SeepSlope_Collapse.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 47: Collapse solution (Lower). '''</span>
+
|+<span style="font-size:90%">'''Figure 10.3: Collapse solution (Lower).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 939: Line 976:
 
==  STABILITY OF SLOPE SUBJECTED TO RAPID DRAWDOWN  ==
 
==  STABILITY OF SLOPE SUBJECTED TO RAPID DRAWDOWN  ==
  
This example examines the effects of rapid drawdown on slope stability. The problem setup is sketched in Figure 48. From the initial state shown in the figure, the water level is lowered suddenly. Due to the loss of the stabilizing effect of the water, the factor of safety will decrease.
+
This example examines the effects of rapid drawdown on slope stability. The problem setup is sketched in Figure 11.1. From the initial state shown in the figure, the water level is lowered suddenly. Due to the loss of the stabilizing effect of the water, the factor of safety will decrease.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex37_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex37_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 48: Slope subjected to rapid drawdown. '''</span>
+
|+<span style="font-size:90%">'''Figure 11.1: Slope subjected to rapid drawdown.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 954: Line 991:
  
 
The stability of the slope in the short time is first evaluated. In doing so, it is assumed that the seepage pressures in the slope do no change at all from the initial state. In the water, on the other hand, a hydrostatic distribution is assumed at all times. Provided that the permeability of the material is sufficiently low, this is a reasonable assumption. As with any other analysis, excess pore pressures are generated automatically as a consequence of the Time Scope being Short Term and the Drainage Conditions of the materials being Drained/Undrained.<br />
 
The stability of the slope in the short time is first evaluated. In doing so, it is assumed that the seepage pressures in the slope do no change at all from the initial state. In the water, on the other hand, a hydrostatic distribution is assumed at all times. Provided that the permeability of the material is sufficiently low, this is a reasonable assumption. As with any other analysis, excess pore pressures are generated automatically as a consequence of the Time Scope being Short Term and the Drainage Conditions of the materials being Drained/Undrained.<br />
In evaluating the short term stability of the slope, the following strategy is used. First, stage is defined to compute the initial stresses using Initial Stress analysis (see Figure 49). Then a limit analysis stage, with a specific lowering of the dam water level, is defined. This stage uses the initial stage as From stage and has Time Scope = Short Term. Moreover, no-flow boundary conditions are imposed on the face of the slope to achieve a hydrostatic state of seepage pressure both in the slope and in the water (see Figure 49).<br />
+
In evaluating the short term stability of the slope, the following strategy is used. First, stage is defined to compute the initial stresses using Initial Stress analysis (see Figure 11.2). Then a limit analysis stage, with a specific lowering of the dam water level, is defined. This stage uses the initial stage as From stage and has Time Scope = Short Term. Moreover, no-flow boundary conditions are imposed on the face of the slope to achieve a hydrostatic state of seepage pressure both in the slope and in the water (see Figure 11.2).<br />
  
  
Line 961: Line 998:
 
|<div class="image600px">https://wiki.optumce.com/figures/Ex37_Fig02.png</div>
 
|<div class="image600px">https://wiki.optumce.com/figures/Ex37_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 49: Seepage pressures (kPa) before (left) and after rapid drawdown to <math display="inline">d = 2</math> m. '''</span>
+
|+<span style="font-size:90%">'''Figure 11.2: Seepage pressures (kPa) before (left) and after rapid drawdown to <math display="inline">d = 2</math> m.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
In the following, Limit Analysis with Multiplier = Gravity is used to evaluate the factor of safety. Using 2,000 Lower and Upper elements with 3 adaptivity iterations, the result, in terms of the gravity based factor of safety versus drawdown level, are shown in Figure 50. We see that the stability decreases as the drawdown level increases. The collapse solutions corresponding to <math display="inline">d=0</math> m and <math display="inline">d=5</math> m are shown in Figure 53.
+
In the following, Limit Analysis with Multiplier = Gravity is used to evaluate the factor of safety. Using 2,000 Lower and Upper elements with 3 adaptivity iterations, the result, in terms of the gravity based factor of safety versus drawdown level, are shown in Figure 11.3. We see that the stability decreases as the drawdown level increases. The collapse solutions corresponding to <math display="inline">d=0</math> m and <math display="inline">d=5</math> m are shown in Figure 11.4.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex37_Fig04_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex37_Fig04_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 50: Upper and lower bound Gravity based Factors of Safety, FS<math display="inline">_g</math>. '''</span>
+
|+<span style="font-size:90%">'''Figure 11.3: Upper and lower bound Gravity based Factors of Safety, FS<math display="inline">_g</math>.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 977: Line 1,014:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex37_Fig03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex37_Fig03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 51: Collapse mechanisms resulting from Gravity Multiplier Limit Analysis before (top) and after (bottom) drawdown by 5 m. '''</span>
+
|+<span style="font-size:90%">'''Figure 11.4: Collapse mechanisms resulting from Gravity Multiplier Limit Analysis before (top) and after (bottom) drawdown by 5 m.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 987: Line 1,024:
 
Finally, the stability of the slope in the long term is investigated with respect to the maximum drawdown level, <math display="inline">d = 6</math> m. The no-flow boundary conditions on the face of the slope at the bottom of the reservoir are here removed and the long term steady state pore pressures are calculated as part of the Limit Analysis. Using the same element and adaptivity settings as above, we obtain:  
 
Finally, the stability of the slope in the long term is investigated with respect to the maximum drawdown level, <math display="inline">d = 6</math> m. The no-flow boundary conditions on the face of the slope at the bottom of the reservoir are here removed and the long term steady state pore pressures are calculated as part of the Limit Analysis. Using the same element and adaptivity settings as above, we obtain:  
  
<math display="block">\text{FS}_g = 2.07\pm 0.06</math>
+
<math display="block">(11.1)\qquad
 +
    \text{FS}_g = 2.07\pm 0.06</math>
  
The pore pressure and collapse solutions are shown in Figure 52.
+
The pore pressure and collapse solutions are shown in Figure 11.5.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex37_Fig05.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex37_Fig05.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 52: Long term pore pressure (top) and collapse (bottom) solutions for <math display="inline">d=6</math> m. '''</span>
+
|+<span style="font-size:90%">'''Figure 11.5: Long term pore pressure (top) and collapse (bottom) solutions for <math display="inline">d=6</math> m.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,003: Line 1,041:
 
==  SLOPE WITH PRE-EXISTING FAULT  ==
 
==  SLOPE WITH PRE-EXISTING FAULT  ==
  
This example considers the stability of a slope with a pre-existing fault as shown in Figure 54. In OPTUM G2, this scenario may be modeled by the use of Shear Joints. These are essentially infinitely thin patches of material sandwiched between the surrounding material. As such, Shear Joints can be assigned any of the properties that apply to solid domains of a finite extent, including materials, drainage conditions, etc.
+
This example considers the stability of a slope with a pre-existing fault as shown in Figure 12.1. In OPTUM G2, this scenario may be modeled by the use of Shear Joints. These are essentially infinitely thin patches of material sandwiched between the surrounding material. As such, Shear Joints can be assigned any of the properties that apply to solid domains of a finite extent, including materials, drainage conditions, etc.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex20b_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex20b_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 54: Slope with pre-existing fault. '''</span>
+
|+<span style="font-size:90%">'''Figure 12.1: Slope with pre-existing fault.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,015: Line 1,053:
 
Running Long Term Gravity Multiplier Limit Analysis using 5,000 elements (Lower and Upper) and 3 adaptivity iterations results in the following gravity based factors of safety:  
 
Running Long Term Gravity Multiplier Limit Analysis using 5,000 elements (Lower and Upper) and 3 adaptivity iterations results in the following gravity based factors of safety:  
  
<math display="block">\begin{array}{ll}
+
<math display="block">(12.1)\qquad 
 +
 
 +
    \begin{array}{ll}
 
     \text{Without fault}:& ~\text{FS}_g = 2.38\pm 0.04\\
 
     \text{Without fault}:& ~\text{FS}_g = 2.38\pm 0.04\\
 
     \text{With fault}:& ~\text{FS}_g = 1.57\pm 0.04
 
     \text{With fault}:& ~\text{FS}_g = 1.57\pm 0.04
 
     \end{array}</math>
 
     \end{array}</math>
  
The collapse solutions are shown in Figure 55 and reflect the significant reduction in strength resulting from the presence of the fault.<br />
+
The collapse solutions are shown in Figure 12.2 and reflect the significant reduction in strength resulting from the presence of the fault.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex20b_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex20b_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 55: Collapse solutions and total dissipation field for slopes with and without fault. '''</span>
+
|+<span style="font-size:90%">'''Figure 12.2: Collapse solutions and total dissipation field for slopes with and without fault.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,035: Line 1,075:
 
==  STABILITY OF SEISMICALLY LOADED SLOPE  ==
 
==  STABILITY OF SEISMICALLY LOADED SLOPE  ==
  
The stability of geostructures under seismic loading is often assessed via so-called pseudo-static analysis. The earthquake induced forces are here represented by body accelerations that are applied statically. Usually, one considers a situation where the vertical acceleration is kept fixed at <math display="inline">g_v = 9.8</math> m/s<math display="inline">^2</math> while the horizontal acceleration is increased until failure. The resulting ratio between the vertical and horizontal accelerations, <math display="inline">k_c = g_h/g_v</math>, is referred to as the critical seismic coefficient. In the following, a slope with the geometry shown in Figure 56 is considered.
+
The stability of geostructures under seismic loading is often assessed via so-called pseudo-static analysis. The earthquake induced forces are here represented by body accelerations that are applied statically. Usually, one considers a situation where the vertical acceleration is kept fixed at <math display="inline">g_v = 9.8</math> m/s<math display="inline">^2</math> while the horizontal acceleration is increased until failure. The resulting ratio between the vertical and horizontal accelerations, <math display="inline">k_c = g_h/g_v</math>, is referred to as the critical seismic coefficient. In the following, a slope with the geometry shown in Figure 13.1 is considered.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex20_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex20_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 56: Seismically loaded slope. '''</span>
+
|+<span style="font-size:90%">'''Figure 13.1: Seismically loaded slope.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,051: Line 1,091:
 
We begin by analyzing a problem previously analyzed by <bib id="Loukidis:2003" />. The material is of the Mohr-Coulomb type with <math display="inline">c=20</math> kPa, <math display="inline">\phi=30^\circ</math> and has a unit weight of 20 kN/m<math display="inline">^3</math>. Limit Analysis with 1,000 elements and 3 adaptivity steps is used. No excess pore pressures are generated so Time Scope is set to Long Term (although seismic forces of course work in the short term). For these settings we find:  
 
We begin by analyzing a problem previously analyzed by <bib id="Loukidis:2003" />. The material is of the Mohr-Coulomb type with <math display="inline">c=20</math> kPa, <math display="inline">\phi=30^\circ</math> and has a unit weight of 20 kN/m<math display="inline">^3</math>. Limit Analysis with 1,000 elements and 3 adaptivity steps is used. No excess pore pressures are generated so Time Scope is set to Long Term (although seismic forces of course work in the short term). For these settings we find:  
  
<math display="block">0.427 \leq k_c \leq 0.439</math>
+
<math display="block">(13.1)\qquad
 +
    0.427 \leq k_c \leq 0.439</math>
  
 
These results are in very good agreement with those of <bib id="Loukidis:2003" /> who for a total of four different hand calculation methods (Spencer’s, Bishop’s simplified, Sarma’s, and log-spiral upper bound) found values of <math display="inline">k_c</math> ranging from <math display="inline">0.426</math> to <math display="inline">0.432</math>. In addition, a conventional finite element analysis performed by Loukidis et al. resulted in <math display="inline">k_c=0.433</math> while upper and lower bound limit analyses of the kind used in OPTUM G2 gave bounds similar to the ones found in the present analysis.<br />
 
These results are in very good agreement with those of <bib id="Loukidis:2003" /> who for a total of four different hand calculation methods (Spencer’s, Bishop’s simplified, Sarma’s, and log-spiral upper bound) found values of <math display="inline">k_c</math> ranging from <math display="inline">0.426</math> to <math display="inline">0.432</math>. In addition, a conventional finite element analysis performed by Loukidis et al. resulted in <math display="inline">k_c=0.433</math> while upper and lower bound limit analyses of the kind used in OPTUM G2 gave bounds similar to the ones found in the present analysis.<br />
Line 1,059: Line 1,100:
  
 
Next, we consider seismic excitation in the short term. As for other problems (footings, slopes, etc), a common approach is to use a total stress analysis where the original Mohr-Coulomb criterion is replaced by a Tresca criterion with the undrained shear strength as the single parameter. If limited data is available, it is tempting to use a constant undrained shear strength that, by some reasonable estimate, would represent the average undrained shear strength encountered in the slope. However, as will be demonstrated in the following, this approach is not appropriate and will lead to unrealistic results.<br />
 
Next, we consider seismic excitation in the short term. As for other problems (footings, slopes, etc), a common approach is to use a total stress analysis where the original Mohr-Coulomb criterion is replaced by a Tresca criterion with the undrained shear strength as the single parameter. If limited data is available, it is tempting to use a constant undrained shear strength that, by some reasonable estimate, would represent the average undrained shear strength encountered in the slope. However, as will be demonstrated in the following, this approach is not appropriate and will lead to unrealistic results.<br />
The overall geometry is the same as above, except that the geometric parameters <math display="inline">a</math> and <math display="inline">b</math> are varied. The material is Tresca with an undrained shear strength of <math display="inline">s_u=150</math> kPa. In the first example, the critical seismic coefficient is independent of <math display="inline">a</math> and <math display="inline">b</math> provided they are sufficiently large. However, assuming a purely cohesive material alters this in such a way that the critical seismic coefficient comes to depend directly on <math display="inline">a</math> and <math display="inline">b</math>. In the following this is illustrated by increasing <math display="inline">a</math> and <math display="inline">b</math> proportionally, i.e <math display="inline">b</math> is increased while the ratio <math display="inline">a/b = 2.5</math> is maintained. The effect is as shown in Figure 57: the collapse mechanism extends to the right boundary and bottom of the domain regardless of its dimensions and the critical seismic coefficient gradually decreases to zero.<br />
+
The overall geometry is the same as above, except that the geometric parameters <math display="inline">a</math> and <math display="inline">b</math> are varied. The material is Tresca with an undrained shear strength of <math display="inline">s_u=150</math> kPa. In the first example, the critical seismic coefficient is independent of <math display="inline">a</math> and <math display="inline">b</math> provided they are sufficiently large. However, assuming a purely cohesive material alters this in such a way that the critical seismic coefficient comes to depend directly on <math display="inline">a</math> and <math display="inline">b</math>. In the following this is illustrated by increasing <math display="inline">a</math> and <math display="inline">b</math> proportionally, i.e <math display="inline">b</math> is increased while the ratio <math display="inline">a/b = 2.5</math> is maintained. The effect is as shown in Figure 13.2: the collapse mechanism extends to the right boundary and bottom of the domain regardless of its dimensions and the critical seismic coefficient gradually decreases to zero.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex20_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex20_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 57: Effect of extending domain for a slope with constant undrained shear strength. '''</span>
+
|+<span style="font-size:90%">'''Figure 13.2: Effect of extending domain for a slope with constant undrained shear strength.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,075: Line 1,116:
 
Using these settings and assuming the soil is the default Stiff Clay-MC material, we obtain  
 
Using these settings and assuming the soil is the default Stiff Clay-MC material, we obtain  
  
<math display="block">0.183 \leq k_c \leq 0.188</math>
+
<math display="block">(13.2)\qquad
 +
    0.183 \leq k_c \leq 0.188</math>
  
The collapse solution is shown in Figure 58.<br />
+
The collapse solution is shown in Figure 13.3.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex20_Fig03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex20_Fig03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 58: Short term collapse mechanism for slope of Stiff Clay-MC material. '''</span>
+
|+<span style="font-size:90%">'''Figure 13.3: Short term collapse mechanism for slope of Stiff Clay-MC material.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,092: Line 1,134:
 
==  STABILITY OF RETAINING WALL  ==
 
==  STABILITY OF RETAINING WALL  ==
  
This example concerns the stability of a retaining wall as shown in Figure 59. The soil is modeled using the default material Medium Sand-MC (a purely frictional Mohr-Coulomb material with a friction angle of 35<math display="inline">^\circ</math> and a bulk unit weight of <math display="inline">16</math> kN/m<math display="inline">^3</math>). The retaining wall is modeled as a Rigid material with a unit weight of <math display="inline">22</math> kN/m<math display="inline">^3</math>.<br />
+
This example concerns the stability of a retaining wall as shown in Figure 14.1. The soil is modeled using the default material Medium Sand-MC (a purely frictional Mohr-Coulomb material with a friction angle of 35<math display="inline">^\circ</math> and a bulk unit weight of <math display="inline">16</math> kN/m<math display="inline">^3</math>). The retaining wall is modeled as a Rigid material with a unit weight of <math display="inline">22</math> kN/m<math display="inline">^3</math>.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex03_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex03_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 59: Retaining wall. '''</span>
+
|+<span style="font-size:90%">'''Figure 14.1: Retaining wall.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The interface between the soil and the wall is modeled by means of Shear Joints. These elements may be specified by using the Shear Joint tool under Features. Once a Shear Joint has been applied to an edge, its material may be changed either by drag-and-drop or by select-and-assign in the same way as for solid materials applied to surfaces. It should be borne in mind that materials are ''global'' in the sense that any modifications made in one stage affects all stages. In the following example where we wish to investigate the influence of the soil-wall friction coefficient, we thus need a number of independent materials. In Figure 59 these materials appear as ‘Interface 0 deg’,...,‘Interface 35 deg’. These materialss are of the same type as the soil, i.e. Solids of Material Type Mohr-Coulomb. Physically, a Shear Joint may be thought of as an infinitely thin layer of solid material. In OPTUM G2, they are modeled as such and all the parameters that are relevant to usual solids are thus relevant to Shear Joints (although some of them, unit weight for example, will have no influence on their physical behaviour).<br />
+
The interface between the soil and the wall is modeled by means of Shear Joints. These elements may be specified by using the Shear Joint tool under Features. Once a Shear Joint has been applied to an edge, its material may be changed either by drag-and-drop or by select-and-assign in the same way as for solid materials applied to surfaces. It should be borne in mind that materials are ''global'' in the sense that any modifications made in one stage affects all stages. In the following example where we wish to investigate the influence of the soil-wall friction coefficient, we thus need a number of independent materials. In Figure 14.1 these materials appear as ‘Interface 0 deg’,...,‘Interface 35 deg’. These materialss are of the same type as the soil, i.e. Solids of Material Type Mohr-Coulomb. Physically, a Shear Joint may be thought of as an infinitely thin layer of solid material. In OPTUM G2, they are modeled as such and all the parameters that are relevant to usual solids are thus relevant to Shear Joints (although some of them, unit weight for example, will have no influence on their physical behaviour).<br />
 
As in the previous example the stability can be gauged either via Limit Analysis with Multiplier set to Gravity or via Strength Reduction analysis. Regarding the former option it is often observed for purely frictional materials that the gravity collapse multiplier is either zero (unstable) or infinite (stable). As such, little quantitative information is obtained. We therefore opt to use Strength Reduction analysis for this example.<br />
 
As in the previous example the stability can be gauged either via Limit Analysis with Multiplier set to Gravity or via Strength Reduction analysis. Regarding the former option it is often observed for purely frictional materials that the gravity collapse multiplier is either zero (unstable) or infinite (stable). As such, little quantitative information is obtained. We therefore opt to use Strength Reduction analysis for this example.<br />
 
Strength Reduction analysis essentially consists of a sequence of limit analyses with the strength being reduced or increased according to the last determined state until a reduction factor implying a collapse multiplier close to unity is obtained.<br />
 
Strength Reduction analysis essentially consists of a sequence of limit analyses with the strength being reduced or increased according to the last determined state until a reduction factor implying a collapse multiplier close to unity is obtained.<br />
Since the Drainage condition of the Medium Sand is Always Drained, only Long Term analysis is relevant. In the following we compute upper and lower bounds on the strength reduction factor (the factor of safety, <math display="inline">\text{FS}_s</math>, introduced in the previous examples) using 2,000 elements without mesh adaptivity. Four different soil-wall interface angles are used: <math display="inline">\phi_\text{i} = 0</math>, <math display="inline">15^\circ</math>, <math display="inline">25^\circ</math> and <math display="inline">35^\circ</math>. The results are shown in Table 3.<br />
+
Since the Drainage condition of the Medium Sand is Always Drained, only Long Term analysis is relevant. In the following we compute upper and lower bounds on the strength reduction factor (the factor of safety, <math display="inline">\text{FS}_s</math>, introduced in the previous examples) using 2,000 elements without mesh adaptivity. Four different soil-wall interface angles are used: <math display="inline">\phi_\text{i} = 0</math>, <math display="inline">15^\circ</math>, <math display="inline">25^\circ</math> and <math display="inline">35^\circ</math>. The results are shown in Table 14.1.<br />
 +
 
  
 +
<div id="Ex03_Tab01">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 3: Strength reduction factors (<math display="inline">\text{FS }_s</math>) for retaining wall as function of soil-wall interface friction angle using 2,000 elements.
+
|+ Table 14.1: Strength reduction factors (<math display="inline">\text{FS}_s</math>) for retaining wall as function of soil-wall interface friction angle using 2,000 elements.
 
!align="center"| <math display="inline">\phi_\text{i}</math> (<math display="inline">^\circ</math>)
 
!align="center"| <math display="inline">\phi_\text{i}</math> (<math display="inline">^\circ</math>)
 
!align="center"| Lower
 
!align="center"| Lower
Line 1,147: Line 1,191:
 
|}
 
|}
  
 +
 +
</div>
 
We see that the soil-wall interface friction angle has a rather marked effect on the stability of the wall. Also, despite the relatively low number of elements, the gaps between the upper and lower bound solutions are quite moderate.
 
We see that the soil-wall interface friction angle has a rather marked effect on the stability of the wall. Also, despite the relatively low number of elements, the gaps between the upper and lower bound solutions are quite moderate.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image300px">https://wiki.optumce.com/figures/Ex03_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex03_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 60: Collapse solution for retaining wall with <math display="inline">\phi_\text{i }=15^\circ</math> (upper bound) with intensity of plastic multiplier.'''</span>
+
|+<span style="font-size:90%">'''Figure 14.2: Collapse solution for retaining wall with <math display="inline">\phi_\text{i}=15^\circ</math> (upper bound) with intensity of plastic multiplier.'''</span>
 
|}
 
|}
 
</center>
 
</center>
 
==  STABILITY OF STEM WALL  ==
 
==  STABILITY OF STEM WALL  ==
  
The next example is, on the surface, relatively uncomplicated and can be handled in much the same way as the previous example. It concerns the stability of a stem wall as shown in Figure 61. The soil consists of the default Medium Sand-MC material (a purely frictional Mohr-Coulomb material with <math display="inline">\phi=35^\circ</math>) and the wall is modeled as Rigid material with a unit weight of <math display="inline">20</math> kN/m<math display="inline">^3</math>.<br />
+
The next example is, on the surface, relatively uncomplicated and can be handled in much the same way as the previous example. It concerns the stability of a stem wall as shown in Figure 15.1. The soil consists of the default Medium Sand-MC material (a purely frictional Mohr-Coulomb material with <math display="inline">\phi=35^\circ</math>) and the wall is modeled as Rigid material with a unit weight of <math display="inline">20</math> kN/m<math display="inline">^3</math>.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex04_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex04_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 61: Stem wall. '''</span>
+
|+<span style="font-size:90%">'''Figure 15.1: Stem wall.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,171: Line 1,217:
 
As in the previous example, Strength Reduction analysis is used to determine the strength based factor of safety, <math display="inline">\text{FS}_s</math>. We begin by calculating upper and lower bounds for a fixed number of elements without using mesh adaptivity.<br />
 
As in the previous example, Strength Reduction analysis is used to determine the strength based factor of safety, <math display="inline">\text{FS}_s</math>. We begin by calculating upper and lower bounds for a fixed number of elements without using mesh adaptivity.<br />
  
 +
 +
<div id="Ex04_Tab01">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 4: Strength reduction factors for stem wall with and without subdivision.
+
|+ Table 15.1: Strength reduction factors for stem wall with and without subdivision.
 
| No Elem
 
| No Elem
 
|align="center"| Lower
 
|align="center"| Lower
Line 1,229: Line 1,277:
 
|}
 
|}
  
The results, shown in left half of Table 4, reveal a very significant gap between the upper and lower bound solutions up to 16,000 elements where the gap suddenly reduces to an acceptable magnitude. More precisely, the lower bounds up to the largest number of elements considered are very poor and in all cases predict that the structure is far from being stable. The upper bounds, on the other hand, display a less erratic convergence behaviour.<br />
+
 
 +
</div>
 +
The results, shown in left half of Table 15.1, reveal a very significant gap between the upper and lower bound solutions up to 16,000 elements where the gap suddenly reduces to an acceptable magnitude. More precisely, the lower bounds up to the largest number of elements considered are very poor and in all cases predict that the structure is far from being stable. The upper bounds, on the other hand, display a less erratic convergence behaviour.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex04_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex04_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 62: Manual subdivision of wall (left) and mesh resulting from specifying a minimum element size of 0.25 m for the wall (right). '''</span>
+
|+<span style="font-size:90%">'''Figure 15.2: Manual subdivision of wall (left) and mesh resulting from specifying a minimum element size of 0.25 m for the wall (right).'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
 
On closer examination of the meshes produced in the various runs, it is observed that only a single layer of elements across the width of the wall is produced for the 1,000 to 8,000 element runs. When the number of elements reaches 16,000, two layers of elements are present in most of the wall. This phenomenon, that rigid domains may affect lower bound solutions adversely if they are not resolved properly, is well-known. Indeed, since the lower bound method requires that the stress fields satisfy the strong form of the equilibrium equations, all parts of the domain – rigid and well as deformable – must necessarily be represented with enough elements to accommodate the exact stress distribution to within a reasonable degree of accuracy. The upper bound method, on the other hand, balances the internal and external work rates and since no work is dissipated in rigid parts, their resolution is of less importance as clearly seen from the results.<br />
 
On closer examination of the meshes produced in the various runs, it is observed that only a single layer of elements across the width of the wall is produced for the 1,000 to 8,000 element runs. When the number of elements reaches 16,000, two layers of elements are present in most of the wall. This phenomenon, that rigid domains may affect lower bound solutions adversely if they are not resolved properly, is well-known. Indeed, since the lower bound method requires that the stress fields satisfy the strong form of the equilibrium equations, all parts of the domain – rigid and well as deformable – must necessarily be represented with enough elements to accommodate the exact stress distribution to within a reasonable degree of accuracy. The upper bound method, on the other hand, balances the internal and external work rates and since no work is dissipated in rigid parts, their resolution is of less importance as clearly seen from the results.<br />
In this case – and in general – the only remedy to improving the lower bound solutions is to use more elements to discretize the wall. With OPTUM G2, this is most easily done by subdividing the wall as shown in Figure 63. This subdivision guarantees at least two elements across the width of the wall and improves the results dramatically as summarized in Table 4. It should also be noted that mesh adaptivity will be of little utility unless the wall is subdivided so that a reasonable initial solution, on the basis of which the subsequent mesh is adapted, is available.<br />
+
In this case – and in general – the only remedy to improving the lower bound solutions is to use more elements to discretize the wall. With OPTUM G2, this is most easily done by subdividing the wall as shown in Figure 15.2. This subdivision guarantees at least two elements across the width of the wall and improves the results dramatically as summarized in Table 15.1. It should also be noted that mesh adaptivity will be of little utility unless the wall is subdivided so that a reasonable initial solution, on the basis of which the subsequent mesh is adapted, is available.<br />
As an alternative to manual subdivision, the Mesh Size tool available under Features can be used to specify a minimum element length. As a general rule, the minimum element length should be one third to one half of the wall thickness. For the present example, a minimum element size of 0.25 m (half the wall thickness) leads to the desired layer of two elements across the wall thickness (see Figure 63) and thereby to satisfactory results.
+
As an alternative to manual subdivision, the Mesh Size tool available under Features can be used to specify a minimum element length. As a general rule, the minimum element length should be one third to one half of the wall thickness. For the present example, a minimum element size of 0.25 m (half the wall thickness) leads to the desired layer of two elements across the wall thickness (see Figure 15.2) and thereby to satisfactory results.
  
 
==  STABILITY OF CANTILEVER SHEET PILE WALL  ==
 
==  STABILITY OF CANTILEVER SHEET PILE WALL  ==
  
This example deals with a cantilever sheet pile wall supporting a wide excavation as shown in Figure 64. The soil is modeled using the default material Medium Sand (a purely frictional Mohr-Coulomb material with a friction angle of 35<math display="inline">^\circ</math>). The sheet pile wall is modeled using Plate elements. These elements couple with the solid elements used in the soil domain. In the following example, we use the default P800 plate.<br />
+
This example deals with a cantilever sheet pile wall supporting a wide excavation as shown in Figure 16.1. The soil is modeled using the default material Medium Sand (a purely frictional Mohr-Coulomb material with a friction angle of 35<math display="inline">^\circ</math>). The sheet pile wall is modeled using Plate elements. These elements couple with the solid elements used in the soil domain. In the following example, we use the default P800 plate.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image300px">https://wiki.optumce.com/figures/Ex05_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex05_Fig01.png</div>
 
|-
 
|-
 
|}
 
|}
Line 1,261: Line 1,311:
 
# By using the Plate tool available in the Features ribbon. In that case, the material assigned is the first material from the left in the Beams category under the Materials ribbon (‘P800’ by default).
 
# By using the Plate tool available in the Features ribbon. In that case, the material assigned is the first material from the left in the Beams category under the Materials ribbon (‘P800’ by default).
  
When Plates are assigned to segments that are part of domains to which solid materials have already been assigned, Plate interfaces are automatically generated. These appear as <math display="inline">\oplus</math> or <math display="inline">\ominus</math> and can be modified by selecting a given Plate through the property window (see Figure 64). Besides assigning arbitrary Solid materials to the interfaces, it is possible to specify a Reduction Factor such that the interface strengths are reduced as compared to those of the parent material. For the Mohr-Coulomb model, the reduced interface strengths, <math display="inline">c_\text{i}</math> and <math display="inline">\phi_\text{i}</math>, are given by  
+
When Plates are assigned to segments that are part of domains to which solid materials have already been assigned, Plate interfaces are automatically generated. These appear as <math display="inline">\oplus</math> or <math display="inline">\ominus</math> and can be modified by selecting a given Plate through the property window (see Figure 16.1). Besides assigning arbitrary Solid materials to the interfaces, it is possible to specify a Reduction Factor such that the interface strengths are reduced as compared to those of the parent material. For the Mohr-Coulomb model, the reduced interface strengths, <math display="inline">c_\text{i}</math> and <math display="inline">\phi_\text{i}</math>, are given by  
  
<math display="block">c_\text{i} = r c,  ~\phi_\text{i} = r\phi</math>
+
<math display="block">(16.1)\qquad
 +
    c_\text{i} = r c,  ~\phi_\text{i} = r\phi</math>
  
 
where <math display="inline">c</math> and <math display="inline">\phi</math> are the strengths of the parent Solid material and <math display="inline">r</math> is the Reduction Factor. Alternatively, the interface strength reduction can be applied to <math display="inline">c</math> and <math display="inline">\tan\phi</math> by changing the setting under Project/Physical Parameters.<br />
 
where <math display="inline">c</math> and <math display="inline">\phi</math> are the strengths of the parent Solid material and <math display="inline">r</math> is the Reduction Factor. Alternatively, the interface strength reduction can be applied to <math display="inline">c</math> and <math display="inline">\tan\phi</math> by changing the setting under Project/Physical Parameters.<br />
Line 1,273: Line 1,324:
 
# Or the strengths of the structural elements (Plates in this example) are reduced until the structure is at a state of incipient collapse. All other strengths are kept constant.
 
# Or the strengths of the structural elements (Plates in this example) are reduced until the structure is at a state of incipient collapse. All other strengths are kept constant.
  
Either of the two options can be specified via the Settings category in the lower half of the Stage Manager windows (see Figure 65).<br />
+
Either of the two options can be specified via the Settings category in the lower half of the Stage Manager windows (see Figure 16.2).<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex05_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex05_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 65: Stage settings for Strength Reduction with respect to solids (left) and structural elements (right). '''</span>
+
|+<span style="font-size:90%">'''Figure 16.2: Stage settings for Strength Reduction with respect to solids (left) and structural elements (right).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,286: Line 1,337:
 
===  Strength Reduction – Solids  ===
 
===  Strength Reduction – Solids  ===
  
We first use the Reduce Strength in Solids approach. The results are shown in Table 5.
+
We first use the Reduce Strength in Solids approach. The results are shown in Table 16.1.
 +
 
 +
<div id="Ex05_Tab01">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 5: Strength reduction factors (Solids) for cantilever sheet pile wall as function of soil-wall interface Reduction Factor, <math display="inline">r = \tan\phi_\text{i }/\tan\phi</math>, using 2,000 elements. <math display="inline">^{*)}</math>Sheet pile wall supported vertically.
+
|+ Table 16.1: Strength reduction factors (Solids) for cantilever sheet pile wall as function of soil-wall interface Reduction Factor, <math display="inline">r = \tan\phi_\text{i}/\tan\phi</math>, using 2,000 elements. <math display="inline">^{*)}</math>Sheet pile wall supported vertically.
 
!align="center"| <math display="inline">r</math>
 
!align="center"| <math display="inline">r</math>
 
!align="center"| Lower
 
!align="center"| Lower
Line 1,326: Line 1,379:
 
|}
 
|}
  
 +
 +
</div>
 
We see that the soil-wall interface strength has some influence on the overall Factor of Safety. Moreover, the error in the estimated Factor or Safety (mean value between the upper and lower bounds) is in all cases below <math display="inline">5\%</math>.<br />
 
We see that the soil-wall interface strength has some influence on the overall Factor of Safety. Moreover, the error in the estimated Factor or Safety (mean value between the upper and lower bounds) is in all cases below <math display="inline">5\%</math>.<br />
 
The case of <math display="inline">r = 0</math> (perfectly smooth wall) is somewhat pathological. In this case there is no strength available to sustain the weight of the wall vertically. As such, the Factor of Safety is zero. However, assuming that some vertical resistance is available even for a perfectly smooth wall – and accounting for this via a vertical support at the bottom of the wall – gives a finite Factor of Safety indicating stability.<br />
 
The case of <math display="inline">r = 0</math> (perfectly smooth wall) is somewhat pathological. In this case there is no strength available to sustain the weight of the wall vertically. As such, the Factor of Safety is zero. However, assuming that some vertical resistance is available even for a perfectly smooth wall – and accounting for this via a vertical support at the bottom of the wall – gives a finite Factor of Safety indicating stability.<br />
Some collapse solutions are shown in Figure 66 where the effects of interface friction are clearly seen.
+
Some collapse solutions are shown in Figure 16.3 where the effects of interface friction are clearly seen.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex05_Fig04_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex05_Fig04_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 66: Collapse solutions for sheet pile wall (upper bound) with intensity of plastic multiplier. '''</span>
+
|+<span style="font-size:90%">'''Figure 16.3: Collapse solutions for sheet pile wall (upper bound) with intensity of plastic multiplier.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,340: Line 1,395:
 
===  Strength Reduction – Structs  ===
 
===  Strength Reduction – Structs  ===
  
Next, we use the Reduce Strength in Structs approach. The results are shown in Table 6. We see that the strength reduction factors (or factors of safety) follow the same trend as in for the Reduce Strength in Solids approach though the factors are rather larger. These results indicate that the wall strength is not fully utilized and that the wall dimensions could be reduced substantially without affecting the overall stability of the system.
+
Next, we use the Reduce Strength in Structs approach. The results are shown in Table 16.3. We see that the strength reduction factors (or factors of safety) follow the same trend as in for the Reduce Strength in Solids approach though the factors are rather larger. These results indicate that the wall strength is not fully utilized and that the wall dimensions could be reduced substantially without affecting the overall stability of the system.
 +
 
 +
<div id="Ex05_Tab02">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 6: Strength reduction factors (Structs) for cantilever sheet pile wall as function of soil-wall interface Reduction Factor, <math display="inline">r = \tan\phi_\text{i }/\tan\phi</math>, using 2,000 elements. <math display="inline">^{*)}</math>Sheet pile wall supported vertically.
+
|+ Table 16.3: Strength reduction factors (Structs) for cantilever sheet pile wall as function of soil-wall interface Reduction Factor, <math display="inline">r = \tan\phi_\text{i}/\tan\phi</math>, using 2,000 elements. <math display="inline">^{*)}</math>Sheet pile wall supported vertically.
 
!align="center"| <math display="inline">r</math>
 
!align="center"| <math display="inline">r</math>
 
!align="center"| Lower
 
!align="center"| Lower
Line 1,380: Line 1,437:
 
|}
 
|}
  
 +
 +
</div>
 
In summary, the system may fail in two different ways: either due to failure of the soil or due to failure of the wall. The Reduce Strength in Solids option picks up the former type of failure while Reduced Strength in Structs picks up the latter. In general, it is recommended that both types of failure are considered. In particular, it may be quite unsafe to rely only on the Reduce Strength in Structs option.
 
In summary, the system may fail in two different ways: either due to failure of the soil or due to failure of the wall. The Reduce Strength in Solids option picks up the former type of failure while Reduced Strength in Structs picks up the latter. In general, it is recommended that both types of failure are considered. In particular, it may be quite unsafe to rely only on the Reduce Strength in Structs option.
  
 
==  STABILITY OF STRUTTED SHEET PILE WALL  ==
 
==  STABILITY OF STRUTTED SHEET PILE WALL  ==
  
This example deals with a sheet pile wall as shown in Figure 67. The soil is modeled using the default material Loose Sand-MC (a purely frictional Mohr-Coulomb material with a friction angle of 30<math display="inline">^\circ</math>). The wall is supported by a strut modeled as a Fixed End Anchor. These elements are equivalent one-dimensional bar elements and are assigned using the Fixed End Anchor tool from the Features ribbon. The materials that can be assigned to Fixed End Anchors are those of the Connector type. In the present case the default Connector materials C1000 (yield force <math display="inline">n_0 = 1,000</math> kN/m) is used.
+
This example deals with a sheet pile wall as shown in Figure 17.1. The soil is modeled using the default material Loose Sand-MC (a purely frictional Mohr-Coulomb material with a friction angle of 30<math display="inline">^\circ</math>). The wall is supported by a strut modeled as a Fixed End Anchor. These elements are equivalent one-dimensional bar elements and are assigned using the Fixed End Anchor tool from the Features ribbon. The materials that can be assigned to Fixed End Anchors are those of the Connector type. In the present case the default Connector materials C1000 (yield force <math display="inline">n_0 = 1,000</math> kN/m) is used.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex06_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex06_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 67: Strutted sheet pile wall. The upper strut (Fixed End Anchor) is selected and its properties are shown in the window on the right. '''</span>
+
|+<span style="font-size:90%">'''Figure 17.1: Strutted sheet pile wall. The upper strut (Fixed End Anchor) is selected and its properties are shown in the window on the right.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The sheet pile wall is modeled using the AZ25 beam from the Sheet Piles materials library (see Figure 68). This profile has a yield moment of 775.71 kNm/m.
+
The sheet pile wall is modeled using the AZ25 beam from the Sheet Piles materials library (see Figure 17.2). This profile has a yield moment of 775.71 kNm/m.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex06_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex06_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 68: Sheet Piles material library with AZ25 selected. '''</span>
+
|+<span style="font-size:90%">'''Figure 17.2: Sheet Piles material library with AZ25 selected.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,407: Line 1,466:
 
  The analyses are carried out using Strength Reduction analysis, first with the Reduce Strength in Solids approach and then with the Reduce Strength in Structs approach. In the latter case, both the strength of the wall and of the struts are reduced (see the previous example for details). Upper and lower bounds are computed using 2,000 elements with 3 adaptivity iterations. The results of the analyses are:  
 
  The analyses are carried out using Strength Reduction analysis, first with the Reduce Strength in Solids approach and then with the Reduce Strength in Structs approach. In the latter case, both the strength of the wall and of the struts are reduced (see the previous example for details). Upper and lower bounds are computed using 2,000 elements with 3 adaptivity iterations. The results of the analyses are:  
  
<math display="block">\begin{array}{lcl}
+
<math display="block">(17.1)\qquad 
 +
 
 +
    \begin{array}{lcl}
 
     \text{Reduce Strength in Solids:}&~& \text{FS} = 1.39\pm 0.01\\
 
     \text{Reduce Strength in Solids:}&~& \text{FS} = 1.39\pm 0.01\\
 
     \text{Reduce Strength in Structs:}&~& \text{FS} = 3.15\pm 0.07
 
     \text{Reduce Strength in Structs:}&~& \text{FS} = 3.15\pm 0.07
 
     \end{array}</math>
 
     \end{array}</math>
  
As in the previous example, the Reduce Strength in Structs approach gives a rather more flattering assessment of the safety of the structure. This differences in factors of safety between the two approaches also manifest themselves in the associated collapse solutions as shown in Figure 69. We here see that while the Solids approach implies that a single yield hinge is formed, the Structs approach gives rise to a rather different collapse mechanism involving two yield hinges. The moment distributions confirm these observations: in the Solids case the maximum moment is approximately <math display="inline">775</math> kNm/m corresponding to full utilization of the wall strength. Conversely, in the Structs approach the moment at collapse is about 241 kNm/m such that with a strength reduction factor of about 3.15, the wall is at the point of yielding (<math display="inline">775/3.15 \simeq 241</math>).
+
As in the previous example, the Reduce Strength in Structs approach gives a rather more flattering assessment of the safety of the structure. This differences in factors of safety between the two approaches also manifest themselves in the associated collapse solutions as shown in Figure 17.3. We here see that while the Solids approach implies that a single yield hinge is formed, the Structs approach gives rise to a rather different collapse mechanism involving two yield hinges. The moment distributions confirm these observations: in the Solids case the maximum moment is approximately <math display="inline">775</math> kNm/m corresponding to full utilization of the wall strength. Conversely, in the Structs approach the moment at collapse is about 241 kNm/m such that with a strength reduction factor of about 3.15, the wall is at the point of yielding (<math display="inline">775/3.15 \simeq 241</math>).
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex06_Fig03_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex06_Fig03_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 69: Collapse solutions and moment distributions (Upper). '''</span>
+
|+<span style="font-size:90%">'''Figure 17.3: Collapse solutions and moment distributions (Upper).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,426: Line 1,487:
 
==  STABILITY OF ANCHORED SHEET PILE WALL  ==
 
==  STABILITY OF ANCHORED SHEET PILE WALL  ==
  
This example deals with a sheet pile wall as shown in Figure 70. The soil is modeled using the default material Medium Sand-MC (a purely frictional Mohr-Coulomb material with a friction angle of 35<math display="inline">^\circ</math>). The wall is modeled using Plate elements with the default P800 profile. The anchoring system is modeled by a combination of Connectors (which do not interact with the soil) and Geogrids (which do interact with the soil and are used to account for grouting). It is important, especially when using elements of type Lower, that the Geogrid and Connector elements are perfectly aligned. Any misalignment will lead to inferior results (though still rigorously bounded if the elements Lower and Upper/B are used).
+
This example deals with a sheet pile wall as shown in Figure 18.1. The soil is modeled using the default material Medium Sand-MC (a purely frictional Mohr-Coulomb material with a friction angle of 35<math display="inline">^\circ</math>). The wall is modeled using Plate elements with the default P800 profile. The anchoring system is modeled by a combination of Connectors (which do not interact with the soil) and Geogrids (which do interact with the soil and are used to account for grouting). It is important, especially when using elements of type Lower, that the Geogrid and Connector elements are perfectly aligned. Any misalignment will lead to inferior results (though still rigorously bounded if the elements Lower and Upper/B are used).
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex01_Grout.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex01_Grout.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 70: Anchored sheet pile wall. '''</span>
+
|+<span style="font-size:90%">'''Figure 18.1: Anchored sheet pile wall.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,438: Line 1,499:
 
As in the previous examples, the stability of the structure is gauged via Strength Reduction analysis using the Reduce Strength in Solids approach. Upper and lower bounds are calculated using 2,000 elements without mesh adaptivity to obtain an estimate of the factor of safety given by  
 
As in the previous examples, the stability of the structure is gauged via Strength Reduction analysis using the Reduce Strength in Solids approach. Upper and lower bounds are calculated using 2,000 elements without mesh adaptivity to obtain an estimate of the factor of safety given by  
  
<math display="block">\text{FS}= 1.32\pm 0.05</math>
+
<math display="block">(18.1)\qquad
 +
    \text{FS}= 1.32\pm 0.05</math>
  
The upper bound collapse mechanism is shown in Figure 71. As seen, the system fails by a combination of wall yielding and the Geogrids being pulled out.<br />
+
The upper bound collapse mechanism is shown in Figure 18.2. As seen, the system fails by a combination of wall yielding and the Geogrids being pulled out.<br />
 
In the above analysis, the Square yield criterion is used. This means that the moments and normal forces in the wall are limited independently according to:  
 
In the above analysis, the Square yield criterion is used. This means that the moments and normal forces in the wall are limited independently according to:  
  
<math display="block">\left|\frac{m}{m_p}\right|\leq 1, ~\left|\frac{n}{n_p}\right|\leq 1</math>
+
<math display="block">(18.2)\qquad
 +
    \left|\frac{m}{m_p}\right|\leq 1, ~\left|\frac{n}{n_p}\right|\leq 1</math>
  
 
where <math display="inline">m_p</math> and <math display="inline">n_p</math> are the yield moments and forces respectively.<br />
 
where <math display="inline">m_p</math> and <math display="inline">n_p</math> are the yield moments and forces respectively.<br />
Line 1,450: Line 1,513:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex02_Grout.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex02_Grout.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 71: Collapse solution (Upper). '''</span>
+
|+<span style="font-size:90%">'''Figure 18.2: Collapse solution (Upper).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,458: Line 1,521:
 
A more conservative estimate of the strength of the wall may be realized by use of the Diamond yield criterion:  
 
A more conservative estimate of the strength of the wall may be realized by use of the Diamond yield criterion:  
  
<math display="block">\left|\frac{m}{m_p}\right| + \left|\frac{n}{n_p}\right|\leq 1</math>
+
<math display="block">(18.3)\qquad
 +
    \left|\frac{m}{m_p}\right| + \left|\frac{n}{n_p}\right|\leq 1</math>
  
The two yield criteria are shown in Figure 72.
+
The two yield criteria are shown in Figure 18.3.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex07_Fig03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex07_Fig03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 72: Square and Diamond yield criteria. '''</span>
+
|+<span style="font-size:90%">'''Figure 18.3: Square and Diamond yield criteria.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,472: Line 1,536:
 
However, in this case, the difference in results between the two yield criteria is very minor:  
 
However, in this case, the difference in results between the two yield criteria is very minor:  
  
<math display="block">\begin{array}{lcl}
+
<math display="block">(18.4)\qquad
 +
    \begin{array}{lcl}
 
     \text{Square:}&&1.274 \leq \text{FS} \leq 1.377\\
 
     \text{Square:}&&1.274 \leq \text{FS} \leq 1.377\\
 
     \text{Diamond:}&&1.271 \leq \text{FS} \leq 1.377
 
     \text{Diamond:}&&1.271 \leq \text{FS} \leq 1.377
Line 1,479: Line 1,544:
 
==  SHEET PILE WALL SUBJECTED TO SEEPAGE PRESSURES  ==
 
==  SHEET PILE WALL SUBJECTED TO SEEPAGE PRESSURES  ==
  
This example considers the effects of seepage pressures on a sheet pile wall as shown in Figure 73. The wall, which is impermeable, supports a 10 m excavation with a variable water depth, <math display="inline">d_w</math>. Due to the difference in head between the left and right hand sides of the wall (for <math display="inline">d_w<10</math> m), a flow will occur around the tip of the wall and into the excavation (see Figure 74).
+
This example considers the effects of seepage pressures on a sheet pile wall as shown in Figure 19.1. The wall, which is impermeable, supports a 10 m excavation with a variable water depth, <math display="inline">d_w</math>. Due to the difference in head between the left and right hand sides of the wall (for <math display="inline">d_w<10</math> m), a flow will occur around the tip of the wall and into the excavation (see Figure 19.2).
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex38_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex38_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 73: Sheet pile wall subjected to seepage pressures. '''</span>
+
|+<span style="font-size:90%">'''Figure 19.1: Sheet pile wall subjected to seepage pressures.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,491: Line 1,556:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex38_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex38_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 74: Flow vectors and effective vertical unit weight, <math display="inline">\gamma_y'</math> (kN/m<math display="inline">^3</math>), for <math display="inline">d_w=2</math> m. '''</span>
+
|+<span style="font-size:90%">'''Figure 19.2: Flow vectors and effective vertical unit weight, <math display="inline">\gamma_y'</math> (kN/m<math display="inline">^3</math>), for <math display="inline">d_w=2</math> m.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,499: Line 1,564:
 
A key quantity of interest are the effective unit weights: (available under Pore Pressures in the Results ribbon):  
 
A key quantity of interest are the effective unit weights: (available under Pore Pressures in the Results ribbon):  
  
<math display="block">\gamma_x' = -\frac{\partial p_s}{\partial x},  \gamma_y' = \gamma_\text{sat}-\frac{\partial p_s}{\partial y}</math>
+
<math display="block">(19.1)\qquad
 +
    \gamma_x' = -\frac{\partial p_s}{\partial x},  \gamma_y' = \gamma_\text{sat}-\frac{\partial p_s}{\partial y}</math>
 +
 
 +
For the present problem, the vertical component is of particular interest. With a saturated unit weight of <math display="inline">19</math> kN/m<math display="inline">^3</math>, the effective unit weight corresponding to a hydrostatic pore pressure distribution is <math display="inline">\gamma'_y = 9.2</math> kN/m<math display="inline">^3</math>. From Figure 19.2, we see an increase of effective unit weight from about 11 kN/m<math display="inline">^3</math> on the upper half of the wall to a maximum of about 15 kN/m<math display="inline">^3</math> close to the tip. Conversely, the effective unit weight decreases on the right hand side of the wall to a minimum value of about 3 kN/m<math display="inline">^3</math>. At the bottom of the excavation, the effective unit weight is about 6 kN/m<math display="inline">^3</math> meaning that there is no immediate danger of piping as would be implied by <math display="inline">\gamma_y' \leq 0</math>.<br />
 +
The stability evaluation proceeds by running Strength Reduction analyses for different water depths, <math display="inline">d_w</math>. In this example, we have used fairly coarse meshes comprising 1,000 elements (Lower or Upper). The results are shown in Table 19.1. As expected, we see a gradual decrease in the factor of safety as the water level in the excavation decreases. For <math display="inline">d_w = 0</math>, the system is very close to the limit of stability. Indeed, the lower bound analysis suggests instability while the upper bound analysis suggests the opposite. Whether or not the system is stable or unstable can be settled by increasing the number of elements and possibly using mesh adaptivity.
  
For the present problem, the vertical component is of particular interest. With a saturated unit weight of <math display="inline">19</math> kN/m<math display="inline">^3</math>, the effective unit weight corresponding to a hydrostatic pore pressure distribution is <math display="inline">\gamma'_y = 9.2</math> kN/m<math display="inline">^3</math>. From Figure 74, we see an increase of effective unit weight from about 11 kN/m<math display="inline">^3</math> on the upper half of the wall to a maximum of about 15 kN/m<math display="inline">^3</math> close to the tip. Conversely, the effective unit weight decreases on the right hand side of the wall to a minimum value of about 3 kN/m<math display="inline">^3</math>. At the bottom of the excavation, the effective unit weight is about 6 kN/m<math display="inline">^3</math> meaning that there is no immediate danger of piping as would be implied by <math display="inline">\gamma_y' \leq 0</math>.<br />
+
<div id="Ex38_Tab01">
The stability evaluation proceeds by running Strength Reduction analyses for different water depths, <math display="inline">d_w</math>. In this example, we have used fairly coarse meshes comprising 1,000 elements (Lower or Upper). The results are shown in Table 7. As expected, we see a gradual decrease in the factor of safety as the water level in the excavation decreases. For <math display="inline">d_w = 0</math>, the system is very close to the limit of stability. Indeed, the lower bound analysis suggests instability while the upper bound analysis suggests the opposite. Whether or not the system is stable or unstable can be settled by increasing the number of elements and possibly using mesh adaptivity.
 
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 7: Strength reduction factors, <math display="inline">\text{FS }</math>, for different water levels, <math display="inline">d_w</math>.
+
|+ Table 19.1: Strength reduction factors, <math display="inline">\text{FS}</math>, for different water levels, <math display="inline">d_w</math>.
 
!align="right"| <math display="inline">d_w</math> (m)
 
!align="right"| <math display="inline">d_w</math> (m)
 
!align="center"| Lower
 
!align="center"| Lower
Line 1,549: Line 1,617:
 
|}
 
|}
  
It is interesting to note that the mode of failure changes fundamentally at a water level of around <math display="inline">5</math> m. For higher water levels, the moments in the wall are very limited and the mode of failure involves a rigid rotation of the wall about the connector point. As the water level decreases, the wall strength becomes fully utilized and a yield hinge forms approximately halfway down the wall. Some examples are shown in Figure 75.
+
 
 +
</div>
 +
It is interesting to note that the mode of failure changes fundamentally at a water level of around <math display="inline">5</math> m. For higher water levels, the moments in the wall are very limited and the mode of failure involves a rigid rotation of the wall about the connector point. As the water level decreases, the wall strength becomes fully utilized and a yield hinge forms approximately halfway down the wall. Some examples are shown in Figure 19.3.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex38_Fig03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex38_Fig03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 75: Bending moments (kNm/m) and modes of failure for high (left) and low (right) water levels. '''</span>
+
|+<span style="font-size:90%">'''Figure 19.3: Bending moments (kNm/m) and modes of failure for high (left) and low (right) water levels.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,563: Line 1,633:
 
==  ACCRETIONARY WEDGE  ==
 
==  ACCRETIONARY WEDGE  ==
  
This example is concerned with the maximum force that can be exerted on a wedge as shown in Figure 76. This is a classical problem in structural geology where it has been studied quite extensively – analytically, numerically and experimentally [see <bib id="Souloumiac:2010" /> and references therein].
+
This example is concerned with the maximum force that can be exerted on a wedge as shown in Figure 20.1. This is a classical problem in structural geology where it has been studied quite extensively – analytically, numerically and experimentally [see <bib id="Souloumiac:2010" /> and references therein].
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex08_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex08_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 76: Accretionary wedge. '''</span>
+
|+<span style="font-size:90%">'''Figure 20.1: Accretionary wedge.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,575: Line 1,645:
 
Upper and lower bounds on the maximum tectonic force are computed using the 2,000 elements and 3 adaptivity steps. The result in terms of the collapse multiplier is:  
 
Upper and lower bounds on the maximum tectonic force are computed using the 2,000 elements and 3 adaptivity steps. The result in terms of the collapse multiplier is:  
  
<math display="block">24.44\leq \alpha\leq 24.57</math>
+
<math display="block">(20.1)\qquad
 +
    24.44\leq \alpha\leq 24.57</math>
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex08_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex08_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 77: Lower and upper bound collapse solutions with intensity of shear dissipation. '''</span>
+
|+<span style="font-size:90%">'''Figure 20.2: Lower and upper bound collapse solutions with intensity of shear dissipation.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
 
Even though the exact value of the maximum force usually is of minor interest in structural geology applications, the very tight bounds obtained do indicate that the calculated solutions are close to the exact one, both in terms of the force and in terms of the collapse kinematics.<br />
 
Even though the exact value of the maximum force usually is of minor interest in structural geology applications, the very tight bounds obtained do indicate that the calculated solutions are close to the exact one, both in terms of the force and in terms of the collapse kinematics.<br />
The collapse solutions are shown in Figure 77. Both the upper and lower bound calculations reveal the characteristic V-shaped failure mechanism observed by many others in both calculations and experiments. We note that the lower bound solution tends to be somewhat more localized which is quite common.<br />
+
The collapse solutions are shown in Figure 20.2. Both the upper and lower bound calculations reveal the characteristic V-shaped failure mechanism observed by many others in both calculations and experiments. We note that the lower bound solution tends to be somewhat more localized which is quite common.<br />
 
Also shown in the figures is the intensity of plastic shear dissipation. This quantity is defined as:  
 
Also shown in the figures is the intensity of plastic shear dissipation. This quantity is defined as:  
  
<math display="block">D_s = \boldsymbol\sigma_s^\text{T}\dot{\boldsymbol\varepsilon}_s^p</math>
+
<math display="block">(20.2)\qquad
 +
    D_s = \boldsymbol\sigma_s^\text{T}\dot{\boldsymbol\varepsilon}_s^p</math>
  
 
where  
 
where  
  
<math display="block">\boldsymbol\sigma_s = \boldsymbol\sigma-\boldsymbol mp, ~\dot{\boldsymbol\varepsilon}_s^p = \boldsymbol\varepsilon-\frac{1}{3}\boldsymbol m\dot{\varepsilon}_v^p</math>
+
<math display="block">(20.3)\qquad
 +
    \boldsymbol\sigma_s = \boldsymbol\sigma-\boldsymbol mp, ~\dot{\boldsymbol\varepsilon}_s^p = \boldsymbol\varepsilon-\frac{1}{3}\boldsymbol m\dot{\varepsilon}_v^p</math>
  
 
with <math display="inline">p = \frac{1}{3}\boldsymbol m^\text{T}\boldsymbol\sigma</math> and <math display="inline">\dot{\varepsilon}_v^p = \boldsymbol m^\text{T}\dot{\boldsymbol\varepsilon}^p</math> being the mean stress and volumetric plastic strain rates respectively and <math display="inline">\boldsymbol m = (1,1,1,0,0,0)^\text{T}</math>. The shear dissipation is particularly relevant for purely frictional materials where the total dissipation, <math display="inline">D=\boldsymbol\sigma^\text{T}\dot{\boldsymbol\varepsilon}_v^p</math>, is zero and where the plastic multiplier field may be somewhat unreliable, especially in lower bound calculations. Hence, the shear dissipation is also the preferred control variable for mesh adaptivity.
 
with <math display="inline">p = \frac{1}{3}\boldsymbol m^\text{T}\boldsymbol\sigma</math> and <math display="inline">\dot{\varepsilon}_v^p = \boldsymbol m^\text{T}\dot{\boldsymbol\varepsilon}^p</math> being the mean stress and volumetric plastic strain rates respectively and <math display="inline">\boldsymbol m = (1,1,1,0,0,0)^\text{T}</math>. The shear dissipation is particularly relevant for purely frictional materials where the total dissipation, <math display="inline">D=\boldsymbol\sigma^\text{T}\dot{\boldsymbol\varepsilon}_v^p</math>, is zero and where the plastic multiplier field may be somewhat unreliable, especially in lower bound calculations. Hence, the shear dissipation is also the preferred control variable for mesh adaptivity.
Line 1,599: Line 1,672:
 
==  FISSURED MOHR-COULOMB SLOPE  ==
 
==  FISSURED MOHR-COULOMB SLOPE  ==
  
This example concerns the stability of a slope of fissured Mohr-Coulomb material as shown in Figure 78. We here use the default material Stiff Clay-MC (<math display="inline">c = 20</math> kPa, <math display="inline">\phi=22^\circ</math>) as the base material. A single fissure plane is then added via the property window on the right at an angle of <math display="inline">\alpha=45^\circ</math> (see Figure 78). The fissure plane has Mohr-Coulomb parameters <math display="inline">c = 0</math> and <math display="inline">\phi=22^\circ</math>.<br />
+
This example concerns the stability of a slope of fissured Mohr-Coulomb material as shown in Figure 21.1. We here use the default material Stiff Clay-MC (<math display="inline">c = 20</math> kPa, <math display="inline">\phi=22^\circ</math>) as the base material. A single fissure plane is then added via the property window on the right at an angle of <math display="inline">\alpha=45^\circ</math> (see Figure 21.1). The fissure plane has Mohr-Coulomb parameters <math display="inline">c = 0</math> and <math display="inline">\phi=22^\circ</math>.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex09_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex09_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 78: Slope of fissured Mohr-Coulomb material. '''</span>
+
|+<span style="font-size:90%">'''Figure 21.1: Slope of fissured Mohr-Coulomb material.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,612: Line 1,685:
 
To investigate the effect of the fissure plane, we conduct both Gravity Multiplier Limit Analysis and Strength Reduction for the intact Stiff Clay material and for the material with the fissures included. Only Long Term analysis is considered. Using 2,000 elements and 3 adaptivity iterations, the gravity based factor of safety is determined as:  
 
To investigate the effect of the fissure plane, we conduct both Gravity Multiplier Limit Analysis and Strength Reduction for the intact Stiff Clay material and for the material with the fissures included. Only Long Term analysis is considered. Using 2,000 elements and 3 adaptivity iterations, the gravity based factor of safety is determined as:  
  
<math display="block">\begin{array}{lclcl}
+
<math display="block">(21.1)\qquad
 +
    \begin{array}{lclcl}
 
     \text{Intact:}&&2.07 \leq \text{FS}_g \leq 2.13  &~\text{or:}~&\text{FS}_g = 2.10\pm 0.03\\
 
     \text{Intact:}&&2.07 \leq \text{FS}_g \leq 2.13  &~\text{or:}~&\text{FS}_g = 2.10\pm 0.03\\
 
     \text{Fissured:}&&1.52 \leq \text{FS}_g \leq 1.64 &~\text{or:}~&\text{FS}_g = 1.58\pm 0.06
 
     \text{Fissured:}&&1.52 \leq \text{FS}_g \leq 1.64 &~\text{or:}~&\text{FS}_g = 1.58\pm 0.06
 
     \end{array}</math>
 
     \end{array}</math>
  
The corresponding collapse solutions are shown in Figure 79 where the effects of the fissures are quite evident, especially on the upper half of the slope.<br />
+
The corresponding collapse solutions are shown in Figure 21.2 where the effects of the fissures are quite evident, especially on the upper half of the slope.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex09_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex09_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 79: Collapse solutions and intensity of shear dissipation. '''</span>
+
|+<span style="font-size:90%">'''Figure 21.2: Collapse solutions and intensity of shear dissipation.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,630: Line 1,704:
 
Next, strength based factors of safety are calculated using Strength Reduction analysis. In this analysis, the strength parameters of both the parent material and of the fissures are reduced to obtain a state of incipient collapse. The results are:  
 
Next, strength based factors of safety are calculated using Strength Reduction analysis. In this analysis, the strength parameters of both the parent material and of the fissures are reduced to obtain a state of incipient collapse. The results are:  
  
<math display="block">\begin{array}{lclcl}
+
<math display="block">(21.2)\qquad
 +
    \begin{array}{lclcl}
 
     \text{Intact:}&&1.46 \leq \text{FS}_s \leq 1.50  &~\text{or:}~&\text{FS}_s = 1.48\pm 0.02\\
 
     \text{Intact:}&&1.46 \leq \text{FS}_s \leq 1.50  &~\text{or:}~&\text{FS}_s = 1.48\pm 0.02\\
 
     \text{Fissured:}&&1.17 \leq \text{FS}_s \leq 1.21 &~\text{or:}~&\text{FS}_s = 1.19\pm 0.02
 
     \text{Fissured:}&&1.17 \leq \text{FS}_s \leq 1.21 &~\text{or:}~&\text{FS}_s = 1.19\pm 0.02
Line 1,636: Line 1,711:
  
 
Again, we see a drop in strength for the fissured material and, as is usually the case, the strength based factors of safety are somewhat lower than the gravity based factors of safety.<br />
 
Again, we see a drop in strength for the fissured material and, as is usually the case, the strength based factors of safety are somewhat lower than the gravity based factors of safety.<br />
Finally, we vary the angle of the failure plane and calculate strength based factors of safety. The results of this analysis are shown in Figure 80. We conclude that the slope is stable in the range <math display="inline">-15^\circ\leq \alpha \leq 55^\circ</math>.<br />
+
Finally, we vary the angle of the failure plane and calculate strength based factors of safety. The results of this analysis are shown in Figure 21.3. We conclude that the slope is stable in the range <math display="inline">-15^\circ\leq \alpha \leq 55^\circ</math>.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex09_Fig03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex09_Fig03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 80: Variation of strength based factor of safety with fissure plane angle. '''</span>
+
|+<span style="font-size:90%">'''Figure 21.3: Variation of strength based factor of safety with fissure plane angle.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,649: Line 1,724:
 
==  HOEK-BROWN SLOPE 1  ==
 
==  HOEK-BROWN SLOPE 1  ==
  
The Hoek-Brown failure criterion is commonly used to assess the strength and stability of fractured rock masses. In this example, we consider the stability of a slope of Hoek-Brown material as shown in Figure 81. To validate the results, two procedures for approximating the Hoek-Brown criterion by an equivalent Mohr-Coulomb criterion are discussed.
+
The Hoek-Brown failure criterion is commonly used to assess the strength and stability of fractured rock masses. In this example, we consider the stability of a slope of Hoek-Brown material as shown in Figure 22.1. To validate the results, two procedures for approximating the Hoek-Brown criterion by an equivalent Mohr-Coulomb criterion are discussed.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex10_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex10_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 81: Slope of Hoek-Brown material. '''</span>
+
|+<span style="font-size:90%">'''Figure 22.1: Slope of Hoek-Brown material.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,678: Line 1,753:
 
First, the stability of the slope is assessed by means of Limit Analysis with Multiplier = Gravity (Settings in the lower half of the Stage Manager window). We use 2,000 elements and 3 adaptivity iterations to calculate upper and lower bounds of the gravity based factor of safety. The results are:  
 
First, the stability of the slope is assessed by means of Limit Analysis with Multiplier = Gravity (Settings in the lower half of the Stage Manager window). We use 2,000 elements and 3 adaptivity iterations to calculate upper and lower bounds of the gravity based factor of safety. The results are:  
  
<math display="block">\begin{array}{lclcl}
+
<math display="block">(22.1)\qquad
 +
    \begin{array}{lclcl}
 
     &&3.05 \leq \text{FS}_g \leq 3.14  &~\text{or:}~&\text{FS}_g = 3.09\pm 0.05
 
     &&3.05 \leq \text{FS}_g \leq 3.14  &~\text{or:}~&\text{FS}_g = 3.09\pm 0.05
 
     \end{array}</math>
 
     \end{array}</math>
  
The collapse solution is shown in Figure 82. We see that the collapse mechanism is given in terms of a rather well defined curved slip line extending from the toe of the slope to the upper surface.<br />
+
The collapse solution is shown in Figure 22.2. We see that the collapse mechanism is given in terms of a rather well defined curved slip line extending from the toe of the slope to the upper surface.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex10_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex10_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 82: Collapse mechanism and intensity of shear dissipation (Lower). '''</span>
+
|+<span style="font-size:90%">'''Figure 22.2: Collapse mechanism and intensity of shear dissipation (Lower).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,695: Line 1,771:
 
Since it is the stresses in the slip line that are determining for the overall strength of the slope, it is reasonable to estimate the value of <math display="inline">\sigma_{3,\text{max}}</math> on the basis of the stresses in the slip line. In the present example, this is done in an approximate manner by mouse click under Results. The corresponding stresses are displayed in the window on the right. In this way, the maximum value of the minor principal stress in the slip line is determined as approximately 110 kPa. This value is used as <math display="inline">\sigma_{3,\text{max}}</math> which together with the Hoek-Brown material data gives the following Mohr-Coulomb parameters:  
 
Since it is the stresses in the slip line that are determining for the overall strength of the slope, it is reasonable to estimate the value of <math display="inline">\sigma_{3,\text{max}}</math> on the basis of the stresses in the slip line. In the present example, this is done in an approximate manner by mouse click under Results. The corresponding stresses are displayed in the window on the right. In this way, the maximum value of the minor principal stress in the slip line is determined as approximately 110 kPa. This value is used as <math display="inline">\sigma_{3,\text{max}}</math> which together with the Hoek-Brown material data gives the following Mohr-Coulomb parameters:  
  
<math display="block">c = 16.0 \text{kPa}, ~\phi = 23.7^\circ, ~k_t = 11.6 \text{kPa}</math>
+
<math display="block">(22.2)\qquad
 +
    c = 16.0 \text{kPa}, ~\phi = 23.7^\circ, ~k_t = 11.6 \text{kPa}</math>
  
 
The corresponding gravity based factor of safety using equivalent Mohr-Coulomb parameters is determined as:  
 
The corresponding gravity based factor of safety using equivalent Mohr-Coulomb parameters is determined as:  
  
<math display="block">\begin{array}{lclcl}
+
<math display="block">(22.3)\qquad
 +
    \begin{array}{lclcl}
 
     &&2.92 \leq \text{FS}_g \leq 3.07  &~\text{or:}~&\text{FS}_g = 3.00\pm 0.08
 
     &&2.92 \leq \text{FS}_g \leq 3.07  &~\text{or:}~&\text{FS}_g = 3.00\pm 0.08
 
     \end{array}</math>
 
     \end{array}</math>
Line 1,709: Line 1,787:
 
For the Hoek-Brown model, Strength Reduction is carried out by reducing the parameters <math display="inline">\sigma_{ci}</math> and <math display="inline">m_i</math> proportionally until incipient collapse, i.e. until a gravity multiplier of 1 is attained. We again use 2,000 elements and 3 adaptivity iterations to calculate the following estimate of the strength based factor of safety:  
 
For the Hoek-Brown model, Strength Reduction is carried out by reducing the parameters <math display="inline">\sigma_{ci}</math> and <math display="inline">m_i</math> proportionally until incipient collapse, i.e. until a gravity multiplier of 1 is attained. We again use 2,000 elements and 3 adaptivity iterations to calculate the following estimate of the strength based factor of safety:  
  
<math display="block">\begin{array}{lclcl}
+
<math display="block">(22.4)\qquad
 +
    \begin{array}{lclcl}
 
     &&1.77 \leq \text{FS}_s \leq 1.80  &~\text{or:}~&\text{FS}_s = 1.78\pm 0.02
 
     &&1.77 \leq \text{FS}_s \leq 1.80  &~\text{or:}~&\text{FS}_s = 1.78\pm 0.02
 
     \end{array}</math>
 
     \end{array}</math>
  
The collapse solution is shown in Figure 83.<br />
+
The collapse solution is shown in Figure 22.3.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex10_Fig03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex10_Fig03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 83: Collapse mechanism and intensity of shear dissipation (Lower). '''</span>
+
|+<span style="font-size:90%">'''Figure 22.3: Collapse mechanism and intensity of shear dissipation (Lower).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,726: Line 1,805:
 
This solution corresponds to a gravity multiplier of 1 which is attained by reducing <math display="inline">\sigma_{ci}</math> and <math display="inline">m_i</math> by a factor of approximately 1.78. A verification may thus be carried out by determining equivalent Mohr-Coulomb parameters for this state and checking that the resulting strength based factor of safety is in reasonable proximity of unity. Using the same procedure as above, we estimate <math display="inline">\sigma_{3,\text{max}}</math> as approximately 40 kPa. Using this value in conjuction with the reduced Hoek-Brown parameters of <math display="inline">\sigma_{ci} = 30,000/1.78 =  16,854</math> kPa, <math display="inline">m_i = 2.0/1.78 = 1.12</math> gives the following equivalent Mohr-Coulomb parameters:  
 
This solution corresponds to a gravity multiplier of 1 which is attained by reducing <math display="inline">\sigma_{ci}</math> and <math display="inline">m_i</math> by a factor of approximately 1.78. A verification may thus be carried out by determining equivalent Mohr-Coulomb parameters for this state and checking that the resulting strength based factor of safety is in reasonable proximity of unity. Using the same procedure as above, we estimate <math display="inline">\sigma_{3,\text{max}}</math> as approximately 40 kPa. Using this value in conjuction with the reduced Hoek-Brown parameters of <math display="inline">\sigma_{ci} = 30,000/1.78 =  16,854</math> kPa, <math display="inline">m_i = 2.0/1.78 = 1.12</math> gives the following equivalent Mohr-Coulomb parameters:  
  
<math display="block">c = 7.8 \text{kPa}, ~\phi = 21.2^\circ, ~k_t = 11.6 \text{kPa}</math>
+
<math display="block">(22.5)\qquad
 +
    c = 7.8 \text{kPa}, ~\phi = 21.2^\circ, ~k_t = 11.6 \text{kPa}</math>
  
 
The corresponding strength based factor of safety is determined as:  
 
The corresponding strength based factor of safety is determined as:  
  
<math display="block">\begin{array}{lclcl}
+
<math display="block">(22.6)\qquad
 +
    \begin{array}{lclcl}
 
     &&1.02 \leq \text{FS}_s \leq 1.04  &~\text{or:}~&\text{FS}_s = 1.03\pm 0.01
 
     &&1.02 \leq \text{FS}_s \leq 1.04  &~\text{or:}~&\text{FS}_s = 1.03\pm 0.01
 
     \end{array}</math>
 
     \end{array}</math>
Line 1,755: Line 1,836:
 
==  HOEK-BROWN SLOPE 2  ==
 
==  HOEK-BROWN SLOPE 2  ==
  
To further verify the Hoek-Brown model, we now consider a higher and steeper slope with a set of Hoek-Brown parameters that implies a significantly higher strength than in the previous example. The problem is shown in Figure 84.
+
To further verify the Hoek-Brown model, we now consider a higher and steeper slope with a set of Hoek-Brown parameters that implies a significantly higher strength than in the previous example. The problem is shown in Figure 23.1.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex11_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex11_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 84: Slope of Hoek-Brown material. '''</span>
+
|+<span style="font-size:90%">'''Figure 23.1: Slope of Hoek-Brown material.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,769: Line 1,850:
 
Using 2,000 elements and 3 adaptivity iterations we first calculate the following upper and lower bounds of the gravity based factor of safety:  
 
Using 2,000 elements and 3 adaptivity iterations we first calculate the following upper and lower bounds of the gravity based factor of safety:  
  
<math display="block">\begin{array}{lclcl}
+
<math display="block">(23.1)\qquad
 +
    \begin{array}{lclcl}
 
     &&2.75 \leq \text{FS}_g \leq 2.97  &~\text{or:}~&\text{FS}_g = 2.86\pm 0.11
 
     &&2.75 \leq \text{FS}_g \leq 2.97  &~\text{or:}~&\text{FS}_g = 2.86\pm 0.11
 
     \end{array}</math>
 
     \end{array}</math>
Line 1,775: Line 1,857:
 
Next, using the same procedure as in the previous example, we estimate a value of <math display="inline">\sigma_{3,\text{max}}=40</math> kPa and calculate the following equivalent Mohr-Coulomb parameters:  
 
Next, using the same procedure as in the previous example, we estimate a value of <math display="inline">\sigma_{3,\text{max}}=40</math> kPa and calculate the following equivalent Mohr-Coulomb parameters:  
  
<math display="block">c = 65.7 \text{kPa}, ~\phi = 56.6^\circ, ~k_t = 23.0 \text{kPa}</math>
+
<math display="block">(23.2)\qquad
 +
    c = 65.7 \text{kPa}, ~\phi = 56.6^\circ, ~k_t = 23.0 \text{kPa}</math>
  
 
The corresponding gravity based factor of safety determined as:  
 
The corresponding gravity based factor of safety determined as:  
  
<math display="block">\begin{array}{lclcl}
+
<math display="block">(23.3)\qquad
 +
    \begin{array}{lclcl}
 
     &&2.66 \leq \text{FS}_g \leq 2.93  &~\text{or:}~&\text{FS}_g = 2.80\pm 0.14
 
     &&2.66 \leq \text{FS}_g \leq 2.93  &~\text{or:}~&\text{FS}_g = 2.80\pm 0.14
 
     \end{array}</math>
 
     \end{array}</math>
Line 1,789: Line 1,873:
 
Using Strength Reduction analysis with 2,000 elements and 3 adaptivity iterations, the following estimate of the factor of safety for the native Hoek-Brown model is determined:  
 
Using Strength Reduction analysis with 2,000 elements and 3 adaptivity iterations, the following estimate of the factor of safety for the native Hoek-Brown model is determined:  
  
<math display="block">\begin{array}{lclcl}
+
<math display="block">(23.4)\qquad
 +
    \begin{array}{lclcl}
 
     &&2.27 \leq \text{FS}_s \leq 2.39  &~\text{or:}~&\text{FS}_s = 2.33\pm 0.06
 
     &&2.27 \leq \text{FS}_s \leq 2.39  &~\text{or:}~&\text{FS}_s = 2.33\pm 0.06
 
     \end{array}</math>
 
     \end{array}</math>
Line 1,795: Line 1,880:
 
Next, estimating <math display="inline">\sigma_{3,\text{max}}</math> at 30 kPa and using the reduced Hoek-Brown parameters <math display="inline">\sigma_{ci} = 40,000/2.32 = 17,241</math> kPa and <math display="inline">m_i = 10/2.32 = 4.31</math> gives the following equivalent Mohr-Coulomb parameters:  
 
Next, estimating <math display="inline">\sigma_{3,\text{max}}</math> at 30 kPa and using the reduced Hoek-Brown parameters <math display="inline">\sigma_{ci} = 40,000/2.32 = 17,241</math> kPa and <math display="inline">m_i = 10/2.32 = 4.31</math> gives the following equivalent Mohr-Coulomb parameters:  
  
<math display="block">c = 39.7 \text{kPa}, ~\phi = 44.7^\circ, ~k_t = 23.0 \text{kPa}</math>
+
<math display="block">(23.5)\qquad
 +
    c = 39.7 \text{kPa}, ~\phi = 44.7^\circ, ~k_t = 23.0 \text{kPa}</math>
  
 
Using these parameters, the strength based factor of safety is determined as:  
 
Using these parameters, the strength based factor of safety is determined as:  
  
<math display="block">\begin{array}{lclcl}
+
<math display="block">(23.6)\qquad
 +
    \begin{array}{lclcl}
 
     &&0.97 \leq \text{FS}_s \leq 0.99  &~\text{or:}~&\text{FS}_s = 0.98\pm 0.01
 
     &&0.97 \leq \text{FS}_s \leq 0.99  &~\text{or:}~&\text{FS}_s = 0.98\pm 0.01
 
     \end{array}</math>
 
     \end{array}</math>
  
 
which is sufficiently close to 1.0 that the verification may be considered successful.<br />
 
which is sufficiently close to 1.0 that the verification may be considered successful.<br />
The collapse solutions for the Hoek-Brown models for the two analyses – Gravity Multiplier Limit Analysis and Strength Reduction – are shown in Figure 85. We here see that the former solution corresponds to a higher equivalent friction angle than the latter.<br />
+
The collapse solutions for the Hoek-Brown models for the two analyses – Gravity Multiplier Limit Analysis and Strength Reduction – are shown in Figure 23.2. We here see that the former solution corresponds to a higher equivalent friction angle than the latter.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex11_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex11_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 85: Collapse solutions and intensity of shear dissipation (Lower). The dashed lines indicate the Strength Reduction slip line. '''</span>
+
|+<span style="font-size:90%">'''Figure 23.2: Collapse solutions and intensity of shear dissipation (Lower). The dashed lines indicate the Strength Reduction slip line.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,817: Line 1,904:
 
==  UNDERGROUND CAVITY IN HOEK-BROWN MATERIAL  ==
 
==  UNDERGROUND CAVITY IN HOEK-BROWN MATERIAL  ==
  
This example deals with a rectangular underground cavity in a Hoek-Brown material as shown in Figure 86. Note that the boundary conditions imply symmetry about the left vertical edge. The Hoek-Brown parameters are: <math display="inline">\text{GSI}=50</math>, <math display="inline">\sigma_{ci}=10,000</math> kPa, <math display="inline">m_i = 10</math>, <math display="inline">D=0</math>.
+
This example deals with a rectangular underground cavity in a Hoek-Brown material as shown in Figure 24.1. Note that the boundary conditions imply symmetry about the left vertical edge. The Hoek-Brown parameters are: <math display="inline">\text{GSI}=50</math>, <math display="inline">\sigma_{ci}=10,000</math> kPa, <math display="inline">m_i = 10</math>, <math display="inline">D=0</math>.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex13c_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex13c_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 86: Rectangular cavity in Hoek-Brown material (symmetry utilized to model half the problem). '''</span>
+
|+<span style="font-size:90%">'''Figure 24.1: Rectangular cavity in Hoek-Brown material (symmetry utilized to model half the problem).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,829: Line 1,916:
 
As in the previous examples, both Strength Reduction and Limit Analyses (with Multiplier = Gravity Multiplier) are conducted. In both cases, 2,000 Lower and Upper elements are used with 3 adaptivity iterations. The results are as follows:  
 
As in the previous examples, both Strength Reduction and Limit Analyses (with Multiplier = Gravity Multiplier) are conducted. In both cases, 2,000 Lower and Upper elements are used with 3 adaptivity iterations. The results are as follows:  
  
<math display="block">\begin{array}{lcl}
+
<math display="block">(24.1)\qquad 
 +
 
 +
    \begin{array}{lcl}
 
     \text{Limit Analysis (Gravity Multiplier):~}&& \text{FS}_g = 2.25\pm 0.14\\
 
     \text{Limit Analysis (Gravity Multiplier):~}&& \text{FS}_g = 2.25\pm 0.14\\
 
     \text{Strength Reduction:~}        &&\text{FS}_s = 4.20\pm 0.14
 
     \text{Strength Reduction:~}        &&\text{FS}_s = 4.20\pm 0.14
 
     \end{array}</math>
 
     \end{array}</math>
  
In contrast to the previous examples, the Strength Reduction analysis here implies a greater level of safety than the Gravity Multiplier Limit Analysis. This is due to the fact that the failure (see Figure 87) is primarily of the tensile type with the roof collapsing into the cavity. The tensile strength implied by the Hoek-Brown criterion is given by <math display="inline">\sigma_t = s\sigma_{ci}/m_b</math> where <math display="inline">m_b</math> and <math display="inline">s</math> are related to GSI and <math display="inline">D</math> and <math display="inline">m_b</math> is proportional to <math display="inline">m_i</math> (see the Materials Manual). In the Strength Reduction analysis used in OPTUM G2, <math display="inline">\sigma_{ci}</math> and <math display="inline">m_i</math> are reduced proportionally, meaning that the tensile strength remains unaltered. This situation is analogous to that of the Mohr-Coulomb criterion where the tensile strength is <math display="inline">\sigma_t = c/\tan\phi</math> and thus is unaffected by an equal and simultaneous reduction of <math display="inline">c</math> and <math display="inline">\tan\phi</math>.<br />
+
In contrast to the previous examples, the Strength Reduction analysis here implies a greater level of safety than the Gravity Multiplier Limit Analysis. This is due to the fact that the failure (see Figure 24.2) is primarily of the tensile type with the roof collapsing into the cavity. The tensile strength implied by the Hoek-Brown criterion is given by <math display="inline">\sigma_t = s\sigma_{ci}/m_b</math> where <math display="inline">m_b</math> and <math display="inline">s</math> are related to GSI and <math display="inline">D</math> and <math display="inline">m_b</math> is proportional to <math display="inline">m_i</math> (see the Materials Manual). In the Strength Reduction analysis used in OPTUM G2, <math display="inline">\sigma_{ci}</math> and <math display="inline">m_i</math> are reduced proportionally, meaning that the tensile strength remains unaltered. This situation is analogous to that of the Mohr-Coulomb criterion where the tensile strength is <math display="inline">\sigma_t = c/\tan\phi</math> and thus is unaffected by an equal and simultaneous reduction of <math display="inline">c</math> and <math display="inline">\tan\phi</math>.<br />
 
In conclusion: while the strength based factor of safety, <math display="inline">\text{FS}_s</math>, in most cases is smaller than the gravity based factor of safety, <math display="inline">\text{FS}_g</math>, exceptions do exist and can be expected to be most pronounced for problems dominated by tensile failure.
 
In conclusion: while the strength based factor of safety, <math display="inline">\text{FS}_s</math>, in most cases is smaller than the gravity based factor of safety, <math display="inline">\text{FS}_g</math>, exceptions do exist and can be expected to be most pronounced for problems dominated by tensile failure.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex13c_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex13c_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 87: Failure mechanism for underground cavity in Hoek-Brown material showing roof collapse into the cavity using Limit Analysis (left) and Strength Reduction analysis (right). '''</span>
+
|+<span style="font-size:90%">'''Figure 24.2: Failure mechanism for underground cavity in Hoek-Brown material showing roof collapse into the cavity using Limit Analysis (left) and Strength Reduction analysis (right).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,849: Line 1,938:
 
==  STRIP FOOTING ON MOHR-COULOMB SAND  ==
 
==  STRIP FOOTING ON MOHR-COULOMB SAND  ==
  
This example concerns the classic problem of the bearing capacity of a strip footing on a deep layer of sand as shown in Figure 88. Despite its apparent simplicity, this problem, also referred to as the <math display="inline">N_\gamma</math> problem, is widely recognized as being problematic to deal with. Firstly, the combination of a free surface and a purely frictional material causes problems for most conventional Newton-Raphson based finite element schemes. Consequently, it is often necessary to introduce some not insignificant cohesion. Secondly, the point at the edge of the footing is a singular point and failure to address this fact may lead to quite erroneous results. Thirdly, the problem is extremely sensitive to the friction angle. For example, the bearing capacity is more than doubled between <math display="inline">\phi = 30^\circ</math> and <math display="inline">\phi = 35^\circ</math> and tripled between <math display="inline">\phi = 35^\circ</math> and <math display="inline">\phi = 41^\circ</math>. Finally, it should be noted that the problem does not have a closed form solution and that some of the formulas cited in the literature come with not insignificant errors. In the following, we will use the solutions provided by <bib id="Martin:2005N" /> resulting from direct numerical integration of the ODE derived by von <bib id="Karman:1926" />. These solutions are for all practical purposes exact.
+
This example concerns the classic problem of the bearing capacity of a strip footing on a deep layer of sand as shown in Figure 25.1. Despite its apparent simplicity, this problem, also referred to as the <math display="inline">N_\gamma</math> problem, is widely recognized as being problematic to deal with. Firstly, the combination of a free surface and a purely frictional material causes problems for most conventional Newton-Raphson based finite element schemes. Consequently, it is often necessary to introduce some not insignificant cohesion. Secondly, the point at the edge of the footing is a singular point and failure to address this fact may lead to quite erroneous results. Thirdly, the problem is extremely sensitive to the friction angle. For example, the bearing capacity is more than doubled between <math display="inline">\phi = 30^\circ</math> and <math display="inline">\phi = 35^\circ</math> and tripled between <math display="inline">\phi = 35^\circ</math> and <math display="inline">\phi = 41^\circ</math>. Finally, it should be noted that the problem does not have a closed form solution and that some of the formulas cited in the literature come with not insignificant errors. In the following, we will use the solutions provided by <bib id="Martin:2005N" /> resulting from direct numerical integration of the ODE derived by von <bib id="Karman:1926" />. These solutions are for all practical purposes exact.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex12_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex12_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 88: Strip footing on Mohr-Coulomb sand (symmetry utilized to model half the problem). '''</span>
+
|+<span style="font-size:90%">'''Figure 25.1: Strip footing on Mohr-Coulomb sand (symmetry utilized to model half the problem).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 1,863: Line 1,952:
 
* A cohesion identically equal to zero can be used. Note that the Shear Dissipation, rather than the Total Dissipation, is the quantity of interest, both for mesh adaptivity and in terms of visualizing the results.
 
* A cohesion identically equal to zero can be used. Note that the Shear Dissipation, rather than the Total Dissipation, is the quantity of interest, both for mesh adaptivity and in terms of visualizing the results.
 
* Upper and lower bounds may be calculated as for all other problems. This gives a direct measure of the error in the numerical solution. Furthermore, it is often observed that the mean between the upper and lower bounds furnish a good estimate of the exact solution – even if the gap between the bounds is significant.
 
* Upper and lower bounds may be calculated as for all other problems. This gives a direct measure of the error in the numerical solution. Furthermore, it is often observed that the mean between the upper and lower bounds furnish a good estimate of the exact solution – even if the gap between the bounds is significant.
* The singularity at the footing edge may be handled using the Mesh Fan tool available under Features. This feature constructs a fan of elements around the singularity which often leads to improved solutions, especially for lower bound elements. An example is shown in Figure 89.
+
* The singularity at the footing edge may be handled using the Mesh Fan tool available under Features. This feature constructs a fan of elements around the singularity which often leads to improved solutions, especially for lower bound elements. An example is shown in Figure 25.2.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex12_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex12_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 89: Mesh fan (<math display="inline">10^\circ</math>) at footing edge. '''</span>
+
|+<span style="font-size:90%">'''Figure 25.2: Mesh fan (<math display="inline">10^\circ</math>) at footing edge.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The bearing capacity of a centrally loaded strip footing on sand as shown in Figure 88, can be expressed as  
+
The bearing capacity of a centrally loaded strip footing on sand as shown in Figure 25.1, can be expressed as  
  
<math display="block">q_u = \frac{1}{2} B \gamma N_\gamma</math>
+
<math display="block">(25.1)\qquad
 +
    q_u = \frac{1}{2} B \gamma N_\gamma</math>
  
 
where <math display="inline">N_\gamma</math> is the bearing capacity factor which depends on the friction angle. For the Loose Sand-MC, Medium Sand-MC, and Dense Sand-MC default materials these are:  
 
where <math display="inline">N_\gamma</math> is the bearing capacity factor which depends on the friction angle. For the Loose Sand-MC, Medium Sand-MC, and Dense Sand-MC default materials these are:  
  
<math display="block">\begin{array}{llcl}
+
<math display="block">(25.2)\qquad 
 +
 
 +
    \begin{array}{llcl}
 
     \text{Loose Sand} &\text{($\phi=30^\circ$)}  & : & N_\gamma = 14.7543\\
 
     \text{Loose Sand} &\text{($\phi=30^\circ$)}  & : & N_\gamma = 14.7543\\
 
     \text{Medium Sand}&\text{($\phi=35^\circ$)} & : & N_\gamma = 34.4761\\
 
     \text{Medium Sand}&\text{($\phi=35^\circ$)} & : & N_\gamma = 34.4761\\
Line 1,892: Line 1,984:
 
* Adaptivity, mesh fan.
 
* Adaptivity, mesh fan.
  
The mesh adaptivity calculations are carried out using 4 adaptivity iterations starting with 1,000 elements (see Section 1). The mesh fan is <math display="inline">10^\circ</math> in all cases.<br />
+
The mesh adaptivity calculations are carried out using 4 adaptivity iterations starting with 1,000 elements (see Section 25.1). The mesh fan is <math display="inline">10^\circ</math> in all cases.<br />
The results are shown in Tables 8-4. We here see that while the introduction of a mesh fan does improve the solutions, its benefit is relatively greater for coarser meshes. Mesh adaptivity, on the other hand, is very effective (and also more expensive), with or without mesh fan. Furthermore, the mean value between the upper and lower bound solutions offer a very good estimate of the exact solution. Indeed, the error is rarely greater than <math display="inline">5\%</math>, even when the respective bounds are rather poor. Moreover, they are generally on the safe side, indicating that the lower bound solutions usually are slightly less accurate than the upper bound solutions. The finest mesh and the collapse solution for the dense sand in case c) are shown in Figure 90. The concentration of deformation at the singularity is here quite apparent.<br />
+
The results are shown in Tables 25.1-4. We here see that while the introduction of a mesh fan does improve the solutions, its benefit is relatively greater for coarser meshes. Mesh adaptivity, on the other hand, is very effective (and also more expensive), with or without mesh fan. Furthermore, the mean value between the upper and lower bound solutions offer a very good estimate of the exact solution. Indeed, the error is rarely greater than <math display="inline">5\%</math>, even when the respective bounds are rather poor. Moreover, they are generally on the safe side, indicating that the lower bound solutions usually are slightly less accurate than the upper bound solutions. The finest mesh and the collapse solution for the dense sand in case c) are shown in Figure 25.3. The concentration of deformation at the singularity is here quite apparent.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex12_Fig03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex12_Fig03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 90: Mesh and collapse solution with intensity of shear dissipation for Medium Sand (lower bound). '''</span>
+
|+<span style="font-size:90%">'''Figure 25.3: Mesh and collapse solution with intensity of shear dissipation for Medium Sand (lower bound).'''</span>
 
|}
 
|}
 
</center>
 
</center>
 +
 +
<div id="Ex12_Tab01d">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 11: Limit loads, <math display="inline">q_u/q_\text{exact }</math>, for footing on Mohr-Coulomb sand. d) Adaptivity, mesh fan.
+
|+ Table 25.7: Limit loads, <math display="inline">q_u/q_\text{exact}</math>, for footing on Mohr-Coulomb sand. d) Adaptivity, mesh fan.
 
| No Elem
 
| No Elem
 
|align="center"| Lower
 
|align="center"| Lower
Line 1,991: Line 2,085:
 
|}
 
|}
  
 <br />
 
 <br />
 
  
 +
</div>
 +
 +
 +
<div id="Ex12_Tab01d">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 11: Limit loads, <math display="inline">q_u/q_\text{exact }</math>, for footing on Mohr-Coulomb sand. d) Adaptivity, mesh fan.
+
|+ Table 25.7: Limit loads, <math display="inline">q_u/q_\text{exact}</math>, for footing on Mohr-Coulomb sand. d) Adaptivity, mesh fan.
 
| No Elem
 
| No Elem
 
|align="center"| Lower
 
|align="center"| Lower
Line 2,082: Line 2,178:
 
|}
 
|}
  
 <br />
 
 <br />
 
  
 +
</div>
 +
 +
 +
<div id="Ex12_Tab01d">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 11: Limit loads, <math display="inline">q_u/q_\text{exact }</math>, for footing on Mohr-Coulomb sand. d) Adaptivity, mesh fan.
+
|+ Table 25.7: Limit loads, <math display="inline">q_u/q_\text{exact}</math>, for footing on Mohr-Coulomb sand. d) Adaptivity, mesh fan.
 
| No Elem
 
| No Elem
 
|align="center"| Lower
 
|align="center"| Lower
Line 2,173: Line 2,271:
 
|}
 
|}
  
 <br />
 
 <br />
 
  
 +
</div>
 +
 +
 +
<div id="Ex12_Tab01d">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 11: Limit loads, <math display="inline">q_u/q_\text{exact }</math>, for footing on Mohr-Coulomb sand. d) Adaptivity, mesh fan.
+
|+ Table 25.7: Limit loads, <math display="inline">q_u/q_\text{exact}</math>, for footing on Mohr-Coulomb sand. d) Adaptivity, mesh fan.
 
|align="center"| No Elem
 
|align="center"| No Elem
 
|align="center"| Lower
 
|align="center"| Lower
Line 2,264: Line 2,364:
 
|}
 
|}
  
===  On mesh adaptivity  ===
 
  
In this example, we have deviated from the default mesh adaptivity settings by using 4 adaptivity steps instead of the default 3. This and other settings related to mesh adaptivity may be modified via the Mesh categories under Stage Manager and Project (see Figure 91).
+
</div>
 +
===  On mesh adaptivity  ===
 +
 
 +
In this example, we have deviated from the default mesh adaptivity settings by using 4 adaptivity steps instead of the default 3. This and other settings related to mesh adaptivity may be modified via the Mesh categories under Stage Manager and Project (see Figure 25.4).
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex12_Fig05.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex12_Fig05.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 91: Settings related to mesh adaptivity. The settings under Stage Manager (left) are stage specific whereas the settings under Project (right) global and apply to all stages. '''</span>
+
|+<span style="font-size:90%">'''Figure 25.4: Settings related to mesh adaptivity. The settings under Stage Manager (left) are stage specific whereas the settings under Project (right) global and apply to all stages.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,287: Line 2,389:
 
==  STRIP FOOTING ON GSK SAND  ==
 
==  STRIP FOOTING ON GSK SAND  ==
  
This example follows on from the previous one, now using the GSK model with the parameters shown in Figure 92
+
This example follows on from the previous one, now using the GSK model with the parameters shown in Figure 26.1
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex13_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex13_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 92: Strip footing on GSK sand (symmetry utilized to model half the problem). '''</span>
+
|+<span style="font-size:90%">'''Figure 26.1: Strip footing on GSK sand (symmetry utilized to model half the problem).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,299: Line 2,401:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex13_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex13_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 93: GSK failure envelope. '''</span>
+
|+<span style="font-size:90%">'''Figure 26.2: GSK failure envelope.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The GSK model involves a curved Mohr-Coulomb envelope (see Figure 93) and is particularly relevant at low stress levels where the apparent friction angle may be significantly higher than at higher stress levels. The nonlinearity of the GSK failure envelope means that the effective friction angle is stress dependent. This in turn means that the bearing capacity of the footing no longer scales linearly with footing width as predicted by the Mohr-Coulomb model. In the following, we investigate the effective footing width dependence for a GSK sand with <math display="inline">\phi_1=40^\circ</math> and <math display="inline">\phi_2=33^\circ</math> corresponding to a medium sand (see the Material Manual). Upper and lower bounds are calculated using 8,000 elements and 4 adaptivity iterations.<br />
+
The GSK model involves a curved Mohr-Coulomb envelope (see Figure 26.2) and is particularly relevant at low stress levels where the apparent friction angle may be significantly higher than at higher stress levels. The nonlinearity of the GSK failure envelope means that the effective friction angle is stress dependent. This in turn means that the bearing capacity of the footing no longer scales linearly with footing width as predicted by the Mohr-Coulomb model. In the following, we investigate the effective footing width dependence for a GSK sand with <math display="inline">\phi_1=40^\circ</math> and <math display="inline">\phi_2=33^\circ</math> corresponding to a medium sand (see the Material Manual). Upper and lower bounds are calculated using 8,000 elements and 4 adaptivity iterations.<br />
 
For each footing width, we express the bearing capacity as  
 
For each footing width, we express the bearing capacity as  
  
<math display="block">q_u(B) = \frac{1}{2} B \gamma N_\gamma(B)</math>
+
<math display="block">(26.1)\qquad
 +
    q_u(B) = \frac{1}{2} B \gamma N_\gamma(B)</math>
  
The results are shown in Figure 94. We see that the effective bearing capacity factor ranges between the Mohr-Coulomb factors corresponding to <math display="inline">\phi_1=40^\circ</math> and <math display="inline">\phi_2=33^\circ</math>. The former limit is approached for <math display="inline">B\rightarrow 0</math> while the latter limit is approached as the footing width tends to infinity.<br />
+
The results are shown in Figure 26.3. We see that the effective bearing capacity factor ranges between the Mohr-Coulomb factors corresponding to <math display="inline">\phi_1=40^\circ</math> and <math display="inline">\phi_2=33^\circ</math>. The former limit is approached for <math display="inline">B\rightarrow 0</math> while the latter limit is approached as the footing width tends to infinity.<br />
The collapse mechanisms are in all cases similar to the one shown in Figure 90.
+
The collapse mechanisms are in all cases similar to the one shown in Figure 25.3.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex13_Fig03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex13_Fig03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 94: Effective bearing capacity factor for medium GSK sand as function of footing width. '''</span>
+
|+<span style="font-size:90%">'''Figure 26.3: Effective bearing capacity factor for medium GSK sand as function of footing width.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,325: Line 2,428:
 
==  STRIP FOOTING ON HOEK-BROWN MATERIAL  ==
 
==  STRIP FOOTING ON HOEK-BROWN MATERIAL  ==
  
In this example, a strip footing on a Hoek-Brown material is considered. The setup, shown in Figure 95, is similar to the previous two examples.
+
In this example, a strip footing on a Hoek-Brown material is considered. The setup, shown in Figure 27.1, is similar to the previous two examples.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex13b_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex13b_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 95: Strip footing on Hoek-Brown material. '''</span>
+
|+<span style="font-size:90%">'''Figure 27.1: Strip footing on Hoek-Brown material.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,337: Line 2,440:
 
Assuming a weightless material, the bearing capacity can be expressed as  
 
Assuming a weightless material, the bearing capacity can be expressed as  
  
<math display="block">q_u = N_{\sigma}\sigma_{ci}</math>
+
<math display="block">(27.1)\qquad
 +
    q_u = N_{\sigma}\sigma_{ci}</math>
  
where <math display="inline">N_{\sigma}</math> is the bearing capacity factor which depends on the material data (<math display="inline">\text{GSI}</math>, <math display="inline">m_i</math> and <math display="inline">D</math>) and <math display="inline">\sigma_{ci}</math> is the compressive strength of the intact rock. We recall that the Hoek-Brown criterion implies an increase of strength with increasing <math display="inline">\text{GSI}</math> and <math display="inline">m_i</math> (see Figure 96).<br />
+
where <math display="inline">N_{\sigma}</math> is the bearing capacity factor which depends on the material data (<math display="inline">\text{GSI}</math>, <math display="inline">m_i</math> and <math display="inline">D</math>) and <math display="inline">\sigma_{ci}</math> is the compressive strength of the intact rock. We recall that the Hoek-Brown criterion implies an increase of strength with increasing <math display="inline">\text{GSI}</math> and <math display="inline">m_i</math> (see Figure 27.3).<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex13b_Fig04.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex13b_Fig04.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 97: Bearing capacity factors for strip footing on weightless Hoek-Brown material. '''</span>
+
|+<span style="font-size:90%">'''Figure 27.4: Bearing capacity factors for strip footing on weightless Hoek-Brown material.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
Upper and lower bounds on the footing pressure (and thereby on the bearing capacity factor) are calculated using Limit Analysis for a range of <math display="inline">m_i</math> and GSI. In these calculations, 5,000 elements with 3 adaptivity iterations and 1,000 initial elements are used. In addition, a Mesh Fan with a fan angle of <math display="inline">30^\circ</math> is used. The results are shown in Figure 97 together with the solutions of <bib id="Serrano:2000" /> and <bib id="Merifield:2006" /> which have been verified as being within close proximity to the exact solutions. The accuracy of the present results follows that of Section 2 (Mohr-Coulomb footing on sand): the upper bounds tend to be somewhat more accurate than the lower bounds and the accuracy decreases with increasing material strength.<br />
+
Upper and lower bounds on the footing pressure (and thereby on the bearing capacity factor) are calculated using Limit Analysis for a range of <math display="inline">m_i</math> and GSI. In these calculations, 5,000 elements with 3 adaptivity iterations and 1,000 initial elements are used. In addition, a Mesh Fan with a fan angle of <math display="inline">30^\circ</math> is used. The results are shown in Figure 27.4 together with the solutions of <bib id="Serrano:2000" /> and <bib id="Merifield:2006" /> which have been verified as being within close proximity to the exact solutions. The accuracy of the present results follows that of Section 25 (Mohr-Coulomb footing on sand): the upper bounds tend to be somewhat more accurate than the lower bounds and the accuracy decreases with increasing material strength.<br />
 
Finally, we consider an example of a ponderable material with a unit weight of <math display="inline">\gamma = 20</math> kN/m<math display="inline">^3</math>. The Hoek-Brown parameters are taken from the book of Hoek (jointed quartz mica schist encountered at the Nathpa Jhakri Hydroelectric project in Himachel Pradesh, India): <math display="inline">\sigma_{ci} = 30</math> MPa, <math display="inline">m_i = 15</math>, <math display="inline">{\boldsymbol{G}}SI=65</math>, <math display="inline">D=0</math>. This rock mass is described as being of average quality. With the same mesh settings as used above, we obtain a footing pressure of:  
 
Finally, we consider an example of a ponderable material with a unit weight of <math display="inline">\gamma = 20</math> kN/m<math display="inline">^3</math>. The Hoek-Brown parameters are taken from the book of Hoek (jointed quartz mica schist encountered at the Nathpa Jhakri Hydroelectric project in Himachel Pradesh, India): <math display="inline">\sigma_{ci} = 30</math> MPa, <math display="inline">m_i = 15</math>, <math display="inline">{\boldsymbol{G}}SI=65</math>, <math display="inline">D=0</math>. This rock mass is described as being of average quality. With the same mesh settings as used above, we obtain a footing pressure of:  
  
<math display="block">81.0 \text{MPa} \pm 3.2\%</math>
+
<math display="block">(27.2)\qquad
 +
    81.0 \text{MPa} \pm 3.2\%</math>
  
 
The collapse solution is shown below.
 
The collapse solution is shown below.
Line 2,359: Line 2,464:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image300px">https://wiki.optumce.com/figures/Ex13b_Fig05.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex13b_Fig05.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 98: Collapse solution for ponderable Hoek-Brown material (<math display="inline">\sigma_{ci } = 30</math> MPa, <math display="inline">m_i = 15</math>, <math display="inline">{\boldsymbol{G}}SI=65</math>, <math display="inline">D=0</math>).'''</span>
+
|+<span style="font-size:90%">'''Figure 27.5: Collapse solution for ponderable Hoek-Brown material (<math display="inline">\sigma_{ci} = 30</math> MPa, <math display="inline">m_i = 15</math>, <math display="inline">{\boldsymbol{G}}SI=65</math>, <math display="inline">D=0</math>).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,368: Line 2,473:
 
==  COMBINED LOADING OF SHALLOW FOUNDATION  ==
 
==  COMBINED LOADING OF SHALLOW FOUNDATION  ==
  
This example considers the combined loading of a shallow foundation as shown in Figure 99. The foundation is embedded in a Tresca material, taken to represent undrained conditions using a total stress analysis. The undrained shear strength is <math display="inline">s_u = 35</math> kPa.
+
This example considers the combined loading of a shallow foundation as shown in Figure 28.1. The foundation is embedded in a Tresca material, taken to represent undrained conditions using a total stress analysis. The undrained shear strength is <math display="inline">s_u = 35</math> kPa.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex14_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex14_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 99: Combined loading of shallow foundation. '''</span>
+
|+<span style="font-size:90%">'''Figure 28.1: Combined loading of shallow foundation.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,381: Line 2,486:
 
In OPTUM G2, moments and concentrated forces may be applied to Plates and Geogrids only. Hence, a weightless Rigid Plate is defined on the top surface of the foundation, which itself is of Rigid material with a unit weight of 20 kN/m<math display="inline">^3</math>. The horizontal force and moment are specified as a Multiplier Concentrated Load with components:  
 
In OPTUM G2, moments and concentrated forces may be applied to Plates and Geogrids only. Hence, a weightless Rigid Plate is defined on the top surface of the foundation, which itself is of Rigid material with a unit weight of 20 kN/m<math display="inline">^3</math>. The horizontal force and moment are specified as a Multiplier Concentrated Load with components:  
  
<math display="block">{1.1}
+
<math display="block">(28.1)\qquad  {1.1}
 
     \begin{array}{lcl}
 
     \begin{array}{lcl}
 
     \text{Force X} &=& \cos\theta\\
 
     \text{Force X} &=& \cos\theta\\
Line 2,388: Line 2,493:
 
     \end{array}</math>
 
     \end{array}</math>
  
where <math display="inline">\theta</math> is varied from <math display="inline">0</math> to <math display="inline">180^\circ</math> is a number of steps, each generating a separate analysis. When specifying the concentrated force components the in-built calculator (see Figure 100) is useful. This is opened via the calculator icon in the right side of any numerical field.<br />
+
where <math display="inline">\theta</math> is varied from <math display="inline">0</math> to <math display="inline">180^\circ</math> is a number of steps, each generating a separate analysis. When specifying the concentrated force components the in-built calculator (see Figure 28.2) is useful. This is opened via the calculator icon in the right side of any numerical field.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex14_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex14_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 100: Calculator. '''</span>
+
|+<span style="font-size:90%">'''Figure 28.2: Calculator.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,401: Line 2,506:
 
As in previous examples, the calculations are organized in a series of independent stages, each corresponding to a particular value of <math display="inline">\theta</math>. The resulting ultimate horizontal force and moment are given by  
 
As in previous examples, the calculations are organized in a series of independent stages, each corresponding to a particular value of <math display="inline">\theta</math>. The resulting ultimate horizontal force and moment are given by  
  
<math display="block">{1.1}
+
<math display="block">(28.2)\qquad  {1.1}
 
     \begin{array}{lcl}
 
     \begin{array}{lcl}
 
     H = \alpha_u\cos\theta\\
 
     H = \alpha_u\cos\theta\\
Line 2,408: Line 2,513:
  
 
where <math display="inline">\alpha_u</math> is the collapse multiplier generated by the analysis.<br />
 
where <math display="inline">\alpha_u</math> is the collapse multiplier generated by the analysis.<br />
Using 2,000 elements with 3 adaptivity iterations and using both upper and lower bound elements, the <math display="inline">H</math>-<math display="inline">M</math> surface shown in Figure 101 may be generated (the actual plotting is done externally on the basis of the computed results).<br />
+
Using 2,000 elements with 3 adaptivity iterations and using both upper and lower bound elements, the <math display="inline">H</math>-<math display="inline">M</math> surface shown in Figure 28.3 may be generated (the actual plotting is done externally on the basis of the computed results).<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex14_Fig03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex14_Fig03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 101: Upper and lower bound <math display="inline">H</math>-<math display="inline">M</math> envelopes for <math display="inline">V = 200</math> kN/m. '''</span>
+
|+<span style="font-size:90%">'''Figure 28.3: Upper and lower bound <math display="inline">H</math>-<math display="inline">M</math> envelopes for <math display="inline">V = 200</math> kN/m.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,425: Line 2,530:
 
The type of analysis conducted above allows for tensile total stresses. If the material is not able to sustain tension under drained conditions, this implies that tensile total stresses can be realized only as a consequence of excess pore pressure corresponding to suction. To see this, recall the principle of effective stress:  
 
The type of analysis conducted above allows for tensile total stresses. If the material is not able to sustain tension under drained conditions, this implies that tensile total stresses can be realized only as a consequence of excess pore pressure corresponding to suction. To see this, recall the principle of effective stress:  
  
<math display="block">\sigma' = \sigma - p_e</math>
+
<math display="block">(28.3)\qquad
 +
    \sigma' = \sigma - p_e</math>
  
 
where both the stresses and the excess pore pressure are positive corresponding to tension. If no tensile stresses are allowed under drained conditions, the failure criterion must necessarily be such that <math display="inline">\sigma'\leq 0</math>. As such, <math display="inline">\sigma>0</math> implies <math display="inline">p_e>0</math> corresponding to suction. Whether or not this suction can reasonably be excepted to be established or the material is able to sustain some tension under drained conditions depends on the particular circumstances, but it is clearly on the safe side to take neither into account. This scenario is accounted for by including a tension cut-off with <math display="inline">\phi_t = 90^\circ</math> and <math display="inline">k_t = 0</math>.<br />
 
where both the stresses and the excess pore pressure are positive corresponding to tension. If no tensile stresses are allowed under drained conditions, the failure criterion must necessarily be such that <math display="inline">\sigma'\leq 0</math>. As such, <math display="inline">\sigma>0</math> implies <math display="inline">p_e>0</math> corresponding to suction. Whether or not this suction can reasonably be excepted to be established or the material is able to sustain some tension under drained conditions depends on the particular circumstances, but it is clearly on the safe side to take neither into account. This scenario is accounted for by including a tension cut-off with <math display="inline">\phi_t = 90^\circ</math> and <math display="inline">k_t = 0</math>.<br />
In the following we rerun the previous analyses with a tension cut-off included. All settings are the same except that the Adaptivity Control is changed to Total Dissipation which is often more appropriate when a tension cut-off is applied. The modified settings are shown in Figure 102.
+
In the following we rerun the previous analyses with a tension cut-off included. All settings are the same except that the Adaptivity Control is changed to Total Dissipation which is often more appropriate when a tension cut-off is applied. The modified settings are shown in Figure 28.4.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex14_Fig04.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex14_Fig04.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 102: Modified settings for analysis with tension cut-off. '''</span>
+
|+<span style="font-size:90%">'''Figure 28.4: Modified settings for analysis with tension cut-off.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The modified <math display="inline">H</math>-<math display="inline">M</math> envelopes are shown in Figure 103. We see that the new envelopes imply somewhat of a decrease in strength with the exact magnitude depending to the particular <math display="inline">H/M</math> ratio.<br />
+
The modified <math display="inline">H</math>-<math display="inline">M</math> envelopes are shown in Figure 28.5. We see that the new envelopes imply somewhat of a decrease in strength with the exact magnitude depending to the particular <math display="inline">H/M</math> ratio.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex14_Fig05.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex14_Fig05.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 103: Upper and lower bound <math display="inline">H</math>-<math display="inline">M</math> envelopes for <math display="inline">V = 200</math> kN/m with tension cut-off. '''</span>
+
|+<span style="font-size:90%">'''Figure 28.5: Upper and lower bound <math display="inline">H</math>-<math display="inline">M</math> envelopes for <math display="inline">V = 200</math> kN/m with tension cut-off.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,453: Line 2,559:
 
==  COMBINED LOADING OF PIPELINE  ==
 
==  COMBINED LOADING OF PIPELINE  ==
  
This example deals with the combined loading of a pipeline idealized as a plane strain cylinder (see Figure 104). The pipeline is partially embedded in a Tresca soil with an undrained shear strength of <math display="inline">s_u = 30</math> kPa. The pipeline is situated on the seabed and in the total stress analysis that follows we assume an effective soil unit weight of <math display="inline">\gamma' = 10</math> kN/m<math display="inline">^3</math> whereas the effective unit weight of the pipeline is <math display="inline">2</math> kN/m<math display="inline">^3</math>. The loading consists of a combination of horizontal load <math display="inline">(H)</math> and vertical load (<math display="inline">V</math>). As in the previous example, the application of concentrated loads is facilitated by including weightless Plate elements as shown in Figure 104.<br />
+
This example deals with the combined loading of a pipeline idealized as a plane strain cylinder (see Figure 29.1). The pipeline is partially embedded in a Tresca soil with an undrained shear strength of <math display="inline">s_u = 30</math> kPa. The pipeline is situated on the seabed and in the total stress analysis that follows we assume an effective soil unit weight of <math display="inline">\gamma' = 10</math> kN/m<math display="inline">^3</math> whereas the effective unit weight of the pipeline is <math display="inline">2</math> kN/m<math display="inline">^3</math>. The loading consists of a combination of horizontal load <math display="inline">(H)</math> and vertical load (<math display="inline">V</math>). As in the previous example, the application of concentrated loads is facilitated by including weightless Plate elements as shown in Figure 29.1.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex15_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex15_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 104: Pipeline subjected to combined vertical and horizontal loading. '''</span>
+
|+<span style="font-size:90%">'''Figure 29.1: Pipeline subjected to combined vertical and horizontal loading.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,466: Line 2,572:
 
As in the previous example, we aim to determine the full <math display="inline">V</math>-<math display="inline">H</math> failure envelope. This is done by means of Limit Analysis using multiplier loads  
 
As in the previous example, we aim to determine the full <math display="inline">V</math>-<math display="inline">H</math> failure envelope. This is done by means of Limit Analysis using multiplier loads  
  
<math display="block">{1.1}
+
<math display="block">(29.1)\qquad  {1.1}
 
     \begin{array}{lcl}
 
     \begin{array}{lcl}
 
     \text{Force X} &=& \sin\theta\\
 
     \text{Force X} &=& \sin\theta\\
Line 2,475: Line 2,581:
 
for a range of <math display="inline">\theta</math> in interval <math display="inline">0</math> to <math display="inline">180^\circ</math>. The resulting vertical and horizontal forces are then given by:  
 
for a range of <math display="inline">\theta</math> in interval <math display="inline">0</math> to <math display="inline">180^\circ</math>. The resulting vertical and horizontal forces are then given by:  
  
<math display="block">{1.1}
+
<math display="block">(29.2)\qquad  {1.1}
 
     \begin{array}{lcl}
 
     \begin{array}{lcl}
 
     V = \alpha_u\cos\theta\\
 
     V = \alpha_u\cos\theta\\
Line 2,494: Line 2,600:
 
# Maximum shear stress at the interface: either zero or equal to that of the surrounding soil.
 
# Maximum shear stress at the interface: either zero or equal to that of the surrounding soil.
  
Regarding the first point, the rigid plates connecting to the rigid cylinder may be constrained against rotation (as indicated in Figure 104) by applying a Plate BC to the center of the cross.<br />
+
Regarding the first point, the rigid plates connecting to the rigid cylinder may be constrained against rotation (as indicated in Figure 29.1) by applying a Plate BC to the center of the cross.<br />
Regarding the two latter points, there are number of possibilities for modeling the soil-pipeline interface, some of which are summarized in Figure 105, namely:
+
Regarding the two latter points, there are number of possibilities for modeling the soil-pipeline interface, some of which are summarized in Figure 29.2, namely:
  
 
* Rough/full tension: this model assumes that the interface properties are identical to those of the surrounding soil. In that case, no provisions need to be taken. However, one may include a shear joint of the same material as the soil. For some elements, this may improve the results slightly.
 
* Rough/full tension: this model assumes that the interface properties are identical to those of the surrounding soil. In that case, no provisions need to be taken. However, one may include a shear joint of the same material as the soil. For some elements, this may improve the results slightly.
Line 2,507: Line 2,613:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex15_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex15_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 105: Soil-structure interface models with associated plastic flow vectors. '''</span>
+
|+<span style="font-size:90%">'''Figure 29.2: Soil-structure interface models with associated plastic flow vectors.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The <math display="inline">V</math>-<math display="inline">H</math> diagrams for each of these interface models are shown in Figure 106 (mean values between upper and lower bounds using 2,000 elements and 3 adaptivity iterations). While it is obvious that the rough/full tension and smooth/no tension cases constitute upper and lower bounds, respectively, on the strength, the intermediate options are more complex to categorize. Thus, in some regions, the smooth/full tension interface is more favorable than the rough/no tension interface and vice versa.
+
The <math display="inline">V</math>-<math display="inline">H</math> diagrams for each of these interface models are shown in Figure 29.3 (mean values between upper and lower bounds using 2,000 elements and 3 adaptivity iterations). While it is obvious that the rough/full tension and smooth/no tension cases constitute upper and lower bounds, respectively, on the strength, the intermediate options are more complex to categorize. Thus, in some regions, the smooth/full tension interface is more favorable than the rough/no tension interface and vice versa.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex15_Fig03b.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex15_Fig03b.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 106: <math display="inline">V</math>-<math display="inline">H</math> diagrams for four different soil-pipeline interface models. '''</span>
+
|+<span style="font-size:90%">'''Figure 29.3: <math display="inline">V</math>-<math display="inline">H</math> diagrams for four different soil-pipeline interface models.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
Finally, it should be noted that while the <math display="inline">V</math>-<math display="inline">H</math> diagrams give the critical pair of <math display="inline">(V,H)</math> for any direction of loading, they also contain information about the failure modes for a given pair of <math display="inline">(V,H)</math> leading to failure. This is illustrated in Figure 107. Considering a critical <math display="inline">(V,H)</math>, the associated mode of deformation follows from the normality rule in the sense that the vector of incremental displacement <math display="inline">(u_V, u_H)</math> at failure is normal to the <math display="inline">(V,H)</math> curve.
+
Finally, it should be noted that while the <math display="inline">V</math>-<math display="inline">H</math> diagrams give the critical pair of <math display="inline">(V,H)</math> for any direction of loading, they also contain information about the failure modes for a given pair of <math display="inline">(V,H)</math> leading to failure. This is illustrated in Figure 29.4. Considering a critical <math display="inline">(V,H)</math>, the associated mode of deformation follows from the normality rule in the sense that the vector of incremental displacement <math display="inline">(u_V, u_H)</math> at failure is normal to the <math display="inline">(V,H)</math> curve.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex15_Fig04.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex15_Fig04.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 107: <math display="inline">V</math>-<math display="inline">H</math> diagram for the rough/no tension case and failure modes for selected ultimate <math display="inline">(V,H)</math> . '''</span>
+
|+<span style="font-size:90%">'''Figure 29.4: <math display="inline">V</math>-<math display="inline">H</math> diagram for the rough/no tension case and failure modes for selected ultimate <math display="inline">(V,H)</math> .'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,535: Line 2,641:
 
==  BEARING CAPACITY OF SKIRTED FOUNDATION  ==
 
==  BEARING CAPACITY OF SKIRTED FOUNDATION  ==
  
This example concerns the bearing capacity of a skirted foundation as shown in Figure 108. The foundation is subjected to a central point load inclined at 15<math display="inline">^\circ</math> with the vertical. The foundation is modeled as a Rigid material with a unit weight of <math display="inline">24</math> kN/m<math display="inline">^3</math> while the skirts are modeled as plates with a yield moment of 800 kNm/m (corresponding to the default P800 Plate material). As in the previous examples, the concentrated vertical force and moment are transferred to the foundation via a rigid weightless plate. A plate of this type is also used at the bottom of the foundation to connect the skirts. In the following, we consider two types of connections between the skirts and the bottom foundation plate: a standard rigid connection and a hinged connection as shown in Figure 108. The soil is modeled as a Tresca material with an undrained shear strength of <math display="inline">s_u = 30</math> kPa. Finally, a soil-skirt interface strength reduction factor of <math display="inline">r = 0.5</math> is assumed.
+
This example concerns the bearing capacity of a skirted foundation as shown in Figure 30.1. The foundation is subjected to a central point load inclined at 15<math display="inline">^\circ</math> with the vertical. The foundation is modeled as a Rigid material with a unit weight of <math display="inline">24</math> kN/m<math display="inline">^3</math> while the skirts are modeled as plates with a yield moment of 800 kNm/m (corresponding to the default P800 Plate material). As in the previous examples, the concentrated vertical force and moment are transferred to the foundation via a rigid weightless plate. A plate of this type is also used at the bottom of the foundation to connect the skirts. In the following, we consider two types of connections between the skirts and the bottom foundation plate: a standard rigid connection and a hinged connection as shown in Figure 30.1. The soil is modeled as a Tresca material with an undrained shear strength of <math display="inline">s_u = 30</math> kPa. Finally, a soil-skirt interface strength reduction factor of <math display="inline">r = 0.5</math> is assumed.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex16_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex16_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 108: Skirted foundation (the right skirt is selected and its properties shown in the window on the right). '''</span>
+
|+<span style="font-size:90%">'''Figure 30.1: Skirted foundation (the right skirt is selected and its properties shown in the window on the right).'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
Regarding the connection between the skirts and the foundation, some care must be exercised. Firstly, it should be recognized the plates connect to solids only along segments and not at points. In order to establish a connection between the skirts and the foundation, a weightless and rigid plate is placed at the bottom of the foundation. This plate connects to the skirts at its end points and thus establishes the necessary connection between the skirts and the foundation (see Figure 109).<br />
+
Regarding the connection between the skirts and the foundation, some care must be exercised. Firstly, it should be recognized the plates connect to solids only along segments and not at points. In order to establish a connection between the skirts and the foundation, a weightless and rigid plate is placed at the bottom of the foundation. This plate connects to the skirts at its end points and thus establishes the necessary connection between the skirts and the foundation (see Figure 30.2).<br />
Secondly, regarding the mechanics of the connection. It may be impractical to construct a perfectly rigid connection that transmits the full moment between the skirt and the bottom foundation plate. In OPTUM G2, the flexibility of the connection may be taken into account by adding a hinge to the end of the skirts. Hinges are defined through the property window that appears when a plate element is selected (see Figure 108). The properties of the hinge are given in terms of special Hinge materials that can be modified via the Materials ribbon. In the present example, we will use the default Hinge material which comes with a yield moment of 0, meaning that no moment is transmitted between the lower foundation plate and the skirts.<br />
+
Secondly, regarding the mechanics of the connection. It may be impractical to construct a perfectly rigid connection that transmits the full moment between the skirt and the bottom foundation plate. In OPTUM G2, the flexibility of the connection may be taken into account by adding a hinge to the end of the skirts. Hinges are defined through the property window that appears when a plate element is selected (see Figure 30.1). The properties of the hinge are given in terms of special Hinge materials that can be modified via the Materials ribbon. In the present example, we will use the default Hinge material which comes with a yield moment of 0, meaning that no moment is transmitted between the lower foundation plate and the skirts.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex16_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex16_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 109: Plate-plate connection. The skirt on the left is connected rigidly to the bottom foundation plate while the right skirt is hinged. '''</span>
+
|+<span style="font-size:90%">'''Figure 30.2: Plate-plate connection. The skirt on the left is connected rigidly to the bottom foundation plate while the right skirt is hinged.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
In the following, Limit Analysis is used to assess the bearing capacity of the foundation for different skirt depths. A total of 5,000 Lower and Upper elements are used along with 3 adaptivity iterations. The results of the analyses are shown in Figure 110. While the rigid foundation-skirt connection leads to a higher strength than the hinged connection, the latter still increases the bearing capacity substantially as compared to the case of a standard foundation without skirts. Selected collapse solutions are shown in Figure 111.
+
In the following, Limit Analysis is used to assess the bearing capacity of the foundation for different skirt depths. A total of 5,000 Lower and Upper elements are used along with 3 adaptivity iterations. The results of the analyses are shown in Figure 30.3. While the rigid foundation-skirt connection leads to a higher strength than the hinged connection, the latter still increases the bearing capacity substantially as compared to the case of a standard foundation without skirts. Selected collapse solutions are shown in Figure 30.4.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex16_Fig03_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex16_Fig03_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 110: Bearing capacity of skirted foundation as function of skirt depth. '''</span>
+
|+<span style="font-size:90%">'''Figure 30.3: Bearing capacity of skirted foundation as function of skirt depth.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,569: Line 2,675:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex16_Fig04_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex16_Fig04_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 111: Failure modes for skirted foundations. '''</span>
+
|+<span style="font-size:90%">'''Figure 30.4: Failure modes for skirted foundations.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,579: Line 2,685:
 
==  LOAD-DISPLACEMENT ANALYSIS – INTRODUCTORY EXAMPLE  ==
 
==  LOAD-DISPLACEMENT ANALYSIS – INTRODUCTORY EXAMPLE  ==
  
The following example introduces load-displacement analysis using the Multiplier Elastoplastic analysis type. This analysis type is based on an algorithm that is unique to OPTUM G2. In the following it is verified on the basis of previous results of <bib id="Sloan:2000" /> using a traditional algorithm. The verification example is shown in Figure 112. Two cases are considered: associated flow with <math display="inline">\phi=30^\circ</math> and nonassociated flow where the dilation angle is reduced to <math display="inline">\psi=20^\circ</math>. In both cases, the soil of type Mohr-Coulomb with <math display="inline">c=1</math> kPa, <math display="inline">E=1.04</math> MPa and <math display="inline">\nu=0.3</math>. Both the soil and the footing are weightless.
+
The following example introduces load-displacement analysis using the Multiplier Elastoplastic analysis type. This analysis type is based on an algorithm that is unique to OPTUM G2. In the following it is verified on the basis of previous results of <bib id="Sloan:2000" /> using a traditional algorithm. The verification example is shown in Figure 31.1. Two cases are considered: associated flow with <math display="inline">\phi=30^\circ</math> and nonassociated flow where the dilation angle is reduced to <math display="inline">\psi=20^\circ</math>. In both cases, the soil of type Mohr-Coulomb with <math display="inline">c=1</math> kPa, <math display="inline">E=1.04</math> MPa and <math display="inline">\nu=0.3</math>. Both the soil and the footing are weightless.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex23b_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex23b_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 112: Centrally loaded strip footing on a weightless Mohr-Coulomb soil. '''</span>
+
|+<span style="font-size:90%">'''Figure 31.1: Centrally loaded strip footing on a weightless Mohr-Coulomb soil.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,592: Line 2,698:
  
 
Multiplier Elastoplastic analysis requires that one or more Multiplier Loads (Concentrated, Distributed, or Body) are defined. The magnitude of these loads will then be increased until the bearing capacity is exhausted or else, by appropriate settings, until a predefined level of loading or displacement has been reached. The relevant settings for this purpose are available under the Advanced Settings in the Stage Manager. Fixed Loads, on the other hand, will remain at their specified value throughout the analysis.<br />
 
Multiplier Elastoplastic analysis requires that one or more Multiplier Loads (Concentrated, Distributed, or Body) are defined. The magnitude of these loads will then be increased until the bearing capacity is exhausted or else, by appropriate settings, until a predefined level of loading or displacement has been reached. The relevant settings for this purpose are available under the Advanced Settings in the Stage Manager. Fixed Loads, on the other hand, will remain at their specified value throughout the analysis.<br />
The load stepping is controlled via three parameters accessible under Settings in the Stage Manager: <math display="inline">N_E</math>, <math display="inline">N_P</math>, and <math display="inline">\beta</math>. Considering a typical load-displacement curve as shown in Figure 113, the response will initially be approximately linear elastic and then gradually become more and more plastic. <math display="inline">N_E</math> and <math display="inline">N_P</math> specifies the approximate number of steps in these two regions. The parameter <math display="inline">\beta</math> is used for automatic adjustment of the load step from step to step. For <math display="inline">\beta=1</math>, the step size (measured in terms of work) will not vary while higher values of <math display="inline">\beta</math> implies a more aggressive strategy with the step size being increased for parts of the curve where there is little change. For most problems, the default parameters, <math display="inline">N_E=10</math>, <math display="inline">N_P=10</math>, and <math display="inline">\beta=5</math>, are a good starting point and often little improvement is observed as a result of increasing the number of steps, i.e. increasing <math display="inline">N_E</math> and <math display="inline">N_P</math>.<br />
+
The load stepping is controlled via three parameters accessible under Settings in the Stage Manager: <math display="inline">N_E</math>, <math display="inline">N_P</math>, and <math display="inline">\beta</math>. Considering a typical load-displacement curve as shown in Figure 31.2, the response will initially be approximately linear elastic and then gradually become more and more plastic. <math display="inline">N_E</math> and <math display="inline">N_P</math> specifies the approximate number of steps in these two regions. The parameter <math display="inline">\beta</math> is used for automatic adjustment of the load step from step to step. For <math display="inline">\beta=1</math>, the step size (measured in terms of work) will not vary while higher values of <math display="inline">\beta</math> implies a more aggressive strategy with the step size being increased for parts of the curve where there is little change. For most problems, the default parameters, <math display="inline">N_E=10</math>, <math display="inline">N_P=10</math>, and <math display="inline">\beta=5</math>, are a good starting point and often little improvement is observed as a result of increasing the number of steps, i.e. increasing <math display="inline">N_E</math> and <math display="inline">N_P</math>.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex21_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex21_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 113: Typical load-displacement curve (left) and Settings (Stage Manager). '''</span>
+
|+<span style="font-size:90%">'''Figure 31.2: Typical load-displacement curve (left) and Settings (Stage Manager).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,618: Line 2,724:
 
===  Results  ===
 
===  Results  ===
  
Using 500 15-node Gauss elements, <math display="inline">N_E=15</math> and <math display="inline">N_P=5</math>, the load-displacement response shown in Figure 114 is computed. As seen, the agreement between the computed solution and that of <bib id="Sloan:2000" /> is excellent.<br />
+
Using 500 15-node Gauss elements, <math display="inline">N_E=15</math> and <math display="inline">N_P=5</math>, the load-displacement response shown in Figure 31.3 is computed. As seen, the agreement between the computed solution and that of <bib id="Sloan:2000" /> is excellent.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex23b_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex23b_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 114: Computed load-displacement response of strip footing and comparison to <bib id="Sloan:2000 " />. '''</span>
+
|+<span style="font-size:90%">'''Figure 31.3: Computed load-displacement response of strip footing and comparison to <bib id="Sloan:2000" />.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The deformations at the end of the analyses and the patterns of shear dissipation are shown in 115. We note that the size of the domain used by <bib id="Sloan:2000" /> in the associated case is somewhat too small to be considered semi-infinite although this is irrelevant to the conclusions of the present analysis.
+
The deformations at the end of the analyses and the patterns of shear dissipation are shown in 31.4. We note that the size of the domain used by <bib id="Sloan:2000" /> in the associated case is somewhat too small to be considered semi-infinite although this is irrelevant to the conclusions of the present analysis.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/SloanVerDef.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/SloanVerDef.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 115: Deformations and distributions of shear dissipation. '''</span>
+
|+<span style="font-size:90%">'''Figure 31.4: Deformations and distributions of shear dissipation.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,643: Line 2,749:
 
==  LOAD-DISPLACEMENT ANALYSIS OF FOUNDATION IN MC CLAY  ==
 
==  LOAD-DISPLACEMENT ANALYSIS OF FOUNDATION IN MC CLAY  ==
  
This example concerns the load-displacement analysis of a shallow foundation as shown in Figure 116. The soil is the default Firm Clay material and the foundation is modeled as a Rigid material with a unit weight of 22 kN/m<math display="inline">^3</math>. The analyses are carried out as a Multiplier Elastoplastic analysis where all Multipliers are incremented until the bearing capacity is exhausted or the process terminated due to a specified load level or displacement having been reached (see the previous section for details). The soil-foundation interface is modeled using a Shear Joint. The material is of the same type as the Firm Clay material used to model the soil but the strengths have been reduced to <math display="inline">c_\text{interface} = 0.5\times c_\text{soil} = 5</math> kPa and <math display="inline">\phi_\text{interface} = \arctan[0.5\tan(\phi_\text{soil})] = 10.3^\circ</math>.
+
This example concerns the load-displacement analysis of a shallow foundation as shown in Figure 32.1. The soil is the default Firm Clay material and the foundation is modeled as a Rigid material with a unit weight of 22 kN/m<math display="inline">^3</math>. The analyses are carried out as a Multiplier Elastoplastic analysis where all Multipliers are incremented until the bearing capacity is exhausted or the process terminated due to a specified load level or displacement having been reached (see the previous section for details). The soil-foundation interface is modeled using a Shear Joint. The material is of the same type as the Firm Clay material used to model the soil but the strengths have been reduced to <math display="inline">c_\text{interface} = 0.5\times c_\text{soil} = 5</math> kPa and <math display="inline">\phi_\text{interface} = \arctan[0.5\tan(\phi_\text{soil})] = 10.3^\circ</math>.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex21_Fig01_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex21_Fig01_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 116: Shallow foundation in Firm Clay. '''</span>
+
|+<span style="font-size:90%">'''Figure 32.1: Shallow foundation in Firm Clay.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
===  Elastic parameters ===
+
'''Elastic parameters'''
  
 
For this analysis, and in contrast to the most of the problems discussed so far, both elastic and plastic parameters must be considered. The former can be defined in two ways: either via Parameter Set A which requires specification of Young’s modulus <math display="inline">E</math> and Poisson’s ratio <math display="inline">\nu</math>, or via Parameter Set B which requires specification of the bulk modulus <math display="inline">K</math> and the shear modulus <math display="inline">G</math>. The relation between the two sets of parameters is given in the Materials Manual. Parameter Set A is the default. It should be noted that the two sets are not linked, i.e. a change to one set will not affect the parameters in the other set.
 
For this analysis, and in contrast to the most of the problems discussed so far, both elastic and plastic parameters must be considered. The former can be defined in two ways: either via Parameter Set A which requires specification of Young’s modulus <math display="inline">E</math> and Poisson’s ratio <math display="inline">\nu</math>, or via Parameter Set B which requires specification of the bulk modulus <math display="inline">K</math> and the shear modulus <math display="inline">G</math>. The relation between the two sets of parameters is given in the Materials Manual. Parameter Set A is the default. It should be noted that the two sets are not linked, i.e. a change to one set will not affect the parameters in the other set.
  
===  Flow rule ===
+
'''Flow rule'''
  
 
In the previous examples involving Limit Analysis and Strength Reduction analysis, the flow rule has been assumed associated. Indeed, the framework of limit analysis hinges crucially on this type of flow rule. However, as is well known, the associated flow rule tends to overestimate the dilation actually observed experimentally for typical geomaterials. In contrast to Limit Analysis, Multiplier Elastoplastic analysis opens the possibility of using any flow rule. For the Mohr-Coulomb criterion, the flow rule is specified via the dilation angle, <math display="inline">\psi</math>, as described in the Materials Manual. In summary, <math display="inline">\psi=\phi</math> implies associated flow (and thus, for most materials, excessive dilation) while <math display="inline">\psi=0</math> implies zero volumetric plastic strains. Under undrained conditions, the associated flow rule is particularly problematic as it leads to an infinite limit load, as will any dilation angle greater than zero.
 
In the previous examples involving Limit Analysis and Strength Reduction analysis, the flow rule has been assumed associated. Indeed, the framework of limit analysis hinges crucially on this type of flow rule. However, as is well known, the associated flow rule tends to overestimate the dilation actually observed experimentally for typical geomaterials. In contrast to Limit Analysis, Multiplier Elastoplastic analysis opens the possibility of using any flow rule. For the Mohr-Coulomb criterion, the flow rule is specified via the dilation angle, <math display="inline">\psi</math>, as described in the Materials Manual. In summary, <math display="inline">\psi=\phi</math> implies associated flow (and thus, for most materials, excessive dilation) while <math display="inline">\psi=0</math> implies zero volumetric plastic strains. Under undrained conditions, the associated flow rule is particularly problematic as it leads to an infinite limit load, as will any dilation angle greater than zero.
  
===  Multiplier Elastoplastic analysis settings ===
+
'''Multiplier Elastoplastic analysis settings'''
  
 
The Multiplier Elastoplastic settings are the default ones. Since no From stage is specified, the initial stresses will be computed automatically using an Initial Stress analysis. This analysis aims to find a stress state that satisfies yield and equilibrium while satisfying the initial stress conditions <math display="inline">\sigma_x'=\sigma_z' = K_0\sigma_y'</math> to the greatest possible extent (see the Analysis Manual for details). In this example, the earth pressure coefficient is <math display="inline">K_0 = 0.66</math>.
 
The Multiplier Elastoplastic settings are the default ones. Since no From stage is specified, the initial stresses will be computed automatically using an Initial Stress analysis. This analysis aims to find a stress state that satisfies yield and equilibrium while satisfying the initial stress conditions <math display="inline">\sigma_x'=\sigma_z' = K_0\sigma_y'</math> to the greatest possible extent (see the Analysis Manual for details). In this example, the earth pressure coefficient is <math display="inline">K_0 = 0.66</math>.
Line 2,669: Line 2,775:
 
We begin by a Short Term analysis using default settings for all parameters related to load stepping. The default element for Multiplier Elastoplastic analysis is the 6-node Gauss element. 2,000 of such elements are used.<br />
 
We begin by a Short Term analysis using default settings for all parameters related to load stepping. The default element for Multiplier Elastoplastic analysis is the 6-node Gauss element. 2,000 of such elements are used.<br />
 
Upon solving, distributions of stresses, strains, displacements, etc corresponding to the final state can be plotted in the same way as for the previous analysis types.<br />
 
Upon solving, distributions of stresses, strains, displacements, etc corresponding to the final state can be plotted in the same way as for the previous analysis types.<br />
In addition, various data at the location of the Result Point are stored and can subsequently be plotted via the XY Plots tool under Report in the Results ribbon (see Figure 117).<br />
+
In addition, various data at the location of the Result Point are stored and can subsequently be plotted via the XY Plots tool under Report in the Results ribbon (see Figure 32.2).<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex21_Fig03_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex21_Fig03_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 117: Load-displacement curves for Short Term analysis using the XY Plots tool. '''</span>
+
|+<span style="font-size:90%">'''Figure 32.2: Load-displacement curves for Short Term analysis using the XY Plots tool.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
 
Upon entering the XY Plots window, the stage for which results should be plotted is first selected. Then one of a number of results sets are selected and finally variables from these result sets are plotted using Add button. For the X axis, we choose the variable set Point 1 (Solid), corresponding to the previously defined Result point, and from this set choose the variable <math display="inline">{| class="wikitable"\text{u}|}</math> (the length of the displacement vector, <math display="inline">{| class="wikitable"\text{u}|}=\sqrt{\text{u}_\text{x}^2+\text{u}_\text{y}^2})</math> which in this case is the absolute value of the vertical displacement. For the Y axis, the set General is chosen and from this set Load Multiplier is chosen. The resulting curve thus shows the foundation displacement (positive downwards) versus the load multiplier which, with a reference Multiplier Distributed load of 1, is equal to the load <math display="inline">q</math> (kN/m<math display="inline">^2</math>).<br />
 
Upon entering the XY Plots window, the stage for which results should be plotted is first selected. Then one of a number of results sets are selected and finally variables from these result sets are plotted using Add button. For the X axis, we choose the variable set Point 1 (Solid), corresponding to the previously defined Result point, and from this set choose the variable <math display="inline">{| class="wikitable"\text{u}|}</math> (the length of the displacement vector, <math display="inline">{| class="wikitable"\text{u}|}=\sqrt{\text{u}_\text{x}^2+\text{u}_\text{y}^2})</math> which in this case is the absolute value of the vertical displacement. For the Y axis, the set General is chosen and from this set Load Multiplier is chosen. The resulting curve thus shows the foundation displacement (positive downwards) versus the load multiplier which, with a reference Multiplier Distributed load of 1, is equal to the load <math display="inline">q</math> (kN/m<math display="inline">^2</math>).<br />
Finally, the deformations and shear dissipation distribution for the nonassociated material at the final state are shown in Figure 118. We see that the collapse field is consistent with a dilation angle of <math display="inline">\psi=0</math>.
+
Finally, the deformations and shear dissipation distribution for the nonassociated material at the final state are shown in Figure 32.3. We see that the collapse field is consistent with a dilation angle of <math display="inline">\psi=0</math>.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex21_Fig05_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex21_Fig05_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 118: Deformed geometry (scaled by a factor of 10) and shear dissipation distribution for the final state of the analysis. '''</span>
+
|+<span style="font-size:90%">'''Figure 32.3: Deformed geometry (scaled by a factor of 10) and shear dissipation distribution for the final state of the analysis.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The final collapse load arrived at in the above analysis could also have been computed using Limit Analysis. Moreover, upper and lower bounds bracketing the true solution could have been computed using the Upper and Lower elements respectively. Alternatively, these elements may be used in a Multiplier Elastoplastic analysis to not only bracket the ultimate strength but also the stiffness of the system. An example is shown in Figure 119 where 2,000 elements have been used in all the analyses.
+
The final collapse load arrived at in the above analysis could also have been computed using Limit Analysis. Moreover, upper and lower bounds bracketing the true solution could have been computed using the Upper and Lower elements respectively. Alternatively, these elements may be used in a Multiplier Elastoplastic analysis to not only bracket the ultimate strength but also the stiffness of the system. An example is shown in Figure 32.4 where 2,000 elements have been used in all the analyses.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex21_UBLB.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex21_UBLB.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 119: Short Term Multiplier Elastoplastic analysis using 6-node Gauss, Lower and Upper elements. '''</span>
+
|+<span style="font-size:90%">'''Figure 32.4: Short Term Multiplier Elastoplastic analysis using 6-node Gauss, Lower and Upper elements.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,705: Line 2,811:
 
Next, the two cases, <math display="inline">\psi=\phi</math> and <math display="inline">\psi=0</math>, are considered in the long term. The necessary switch is easily achieved by changing the Time Scope to Long Term (Settings in the Stage Manager).<br />
 
Next, the two cases, <math display="inline">\psi=\phi</math> and <math display="inline">\psi=0</math>, are considered in the long term. The necessary switch is easily achieved by changing the Time Scope to Long Term (Settings in the Stage Manager).<br />
 
Regarding the calculations, it should be noted that while calculations with associated material requires only a single solution in each load step, nonassociated analyses require a number of iterations in each load step, each of which is approximately equal to a single associated calculation. As such, analyses with nonassociated materials are usually somewhat more expensive than the equivalent associated calculation. In the present example, the nonassociated analysis requires on average 3 solutions per load step in comparison to the single solution per load step of the associated analysis.<br />
 
Regarding the calculations, it should be noted that while calculations with associated material requires only a single solution in each load step, nonassociated analyses require a number of iterations in each load step, each of which is approximately equal to a single associated calculation. As such, analyses with nonassociated materials are usually somewhat more expensive than the equivalent associated calculation. In the present example, the nonassociated analysis requires on average 3 solutions per load step in comparison to the single solution per load step of the associated analysis.<br />
The resulting load-displacement curves are shown in Figure 120. In this case, the difference in bearing capacity is rather moderate while the difference in displacement at certain load levels is more pronounced. In both regards, the nonassociated material is the more conservative, i.e. it is both less stiff and has less strength than the equivalent associated material. While it can be shown that the strength of an associated material is always greater than or equal to that of the nonassociated material, the same cannot be said about the deformations: for some problems (such as the present one), the associated flow rule will imply smaller displacements while for other problems the trend is the opposite. It is also noted that the load-displacement curve in the nonassociated case displays an apparent softening. This is a consequence of the flow rule alone and not of numerical artifacts or material softening (see the Theory Manual).<br />
+
The resulting load-displacement curves are shown in Figure 32.5. In this case, the difference in bearing capacity is rather moderate while the difference in displacement at certain load levels is more pronounced. In both regards, the nonassociated material is the more conservative, i.e. it is both less stiff and has less strength than the equivalent associated material. While it can be shown that the strength of an associated material is always greater than or equal to that of the nonassociated material, the same cannot be said about the deformations: for some problems (such as the present one), the associated flow rule will imply smaller displacements while for other problems the trend is the opposite. It is also noted that the load-displacement curve in the nonassociated case displays an apparent softening. This is a consequence of the flow rule alone and not of numerical artifacts or material softening (see the Theory Manual).<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex21_Fig04_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex21_Fig04_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 120: Load-displacement curves for Long Term analysis. '''</span>
+
|+<span style="font-size:90%">'''Figure 32.5: Load-displacement curves for Long Term analysis.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,718: Line 2,824:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex21_Fig06_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex21_Fig06_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 121: Deformed geometry (to scale) and plastic multiplier field for final state of analyses. '''</span>
+
|+<span style="font-size:90%">'''Figure 32.6: Deformed geometry (to scale) and plastic multiplier field for final state of analyses.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The deformations and plastic multiplier fields for the two cases are shown in Figure 121. These reveal somewhat different modes of collapse and degrees of dilation (for example at the vertical soil-foundation interface for the associated material). It is also noted that the plastic strains (as gauged by the plastic multiplier field) are somewhat more localized than for the associated material.
+
The deformations and plastic multiplier fields for the two cases are shown in Figure 32.6. These reveal somewhat different modes of collapse and degrees of dilation (for example at the vertical soil-foundation interface for the associated material). It is also noted that the plastic strains (as gauged by the plastic multiplier field) are somewhat more localized than for the associated material.
  
 
===  Limit Analysis  ===
 
===  Limit Analysis  ===
Line 2,732: Line 2,838:
 
====  Short Term analysis  ====
 
====  Short Term analysis  ====
  
Provided that identical material models, meshes, element types, etc are used in Multiplier Elastoplastic analysis and Limit Analysis, the final result in terms of the ultimate limit load will be the same in the two types of analysis. This point is illustrated by the results summarized in Table 12.<br />
+
Provided that identical material models, meshes, element types, etc are used in Multiplier Elastoplastic analysis and Limit Analysis, the final result in terms of the ultimate limit load will be the same in the two types of analysis. This point is illustrated by the results summarized in Table 32.1.<br />
In the above Multiplier Elastoplastic analysis, 2,000 elements of types 6-node Gauss, Lower, and Upper are used. Rerunning the problem using Limit Analysis a practically identical result is obtained, the slight difference being due to the fact that displacements in the Multiplier Elastoplastic analysis (about 0.2 cm, see Figure 117) still are such that a state of total collapse, in principle, has not be attained. However, by continuing the loading, the value obtained from Limit Analysis will eventually be replicated exactly.<br />
+
In the above Multiplier Elastoplastic analysis, 2,000 elements of types 6-node Gauss, Lower, and Upper are used. Rerunning the problem using Limit Analysis a practically identical result is obtained, the slight difference being due to the fact that displacements in the Multiplier Elastoplastic analysis (about 0.2 cm, see Figure 32.2) still are such that a state of total collapse, in principle, has not be attained. However, by continuing the loading, the value obtained from Limit Analysis will eventually be replicated exactly.<br />
 +
 
  
 +
<div id="Ex21_Tab01">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 12: Short Term limit loads using different elements and analysis types.
+
|+ Table 32.1: Short Term limit loads using different elements and analysis types.
 
! Mesh and Elements
 
! Mesh and Elements
 
! Analysis Type
 
! Analysis Type
Line 2,767: Line 2,875:
 
|}
 
|}
  
From the results of Table 12, we see that the original results computed using the 6-node Gauss element fall approximately in between the results of the Lower and Upper elements. As such, they can be regarded as being fairly accurate.
+
 
 +
</div>
 +
From the results of Table 32.1, we see that the original results computed using the 6-node Gauss element fall approximately in between the results of the Lower and Upper elements. As such, they can be regarded as being fairly accurate.
  
 
====  Long Term analysis  ====
 
====  Long Term analysis  ====
Line 2,775: Line 2,885:
  
  
<math display="block">\displaystyle c_D = 9.40 \text{kPa} ;      \displaystyle \phi_D = 18.88^\circ\\
+
<math display="block">(32.1)\qquad
 +
    \displaystyle c_D = 9.40 \text{kPa} ;      \displaystyle \phi_D = 18.88^\circ\\
 
</math>
 
</math>
  
Table 13 summarizes a variety of results obtained using these parameters as well as the original ones. These first of all confirm the fact that Limit Analysis and Multiplier Elastoplastic analysis provide identical bearing capacities in the case where the flow rule is associated. Secondly, in the case where the flow rule is nonassociated, Limit Analysis using Davis parameters furnish a reasonable, and in this case conservative, estimate of the bearing capacity.
+
Table 32.3 summarizes a variety of results obtained using these parameters as well as the original ones. These first of all confirm the fact that Limit Analysis and Multiplier Elastoplastic analysis provide identical bearing capacities in the case where the flow rule is associated. Secondly, in the case where the flow rule is nonassociated, Limit Analysis using Davis parameters furnish a reasonable, and in this case conservative, estimate of the bearing capacity.
 +
 
 +
<div id="Ex21_Tab02">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 13: Short Term limit loads using different elements and analysis types.
+
|+ Table 32.3: Short Term limit loads using different elements and analysis types.
 
! Mesh and Elements
 
! Mesh and Elements
 
! Flow Rule
 
! Flow Rule
Line 2,808: Line 2,921:
 
|}
 
|}
  
 +
 +
</div>
 
===  Summary  ===
 
===  Summary  ===
  
Line 2,821: Line 2,936:
 
==  LOAD-DISPLACEMENT ANALYSIS OF FOUNDATION IN MC SAND  ==
 
==  LOAD-DISPLACEMENT ANALYSIS OF FOUNDATION IN MC SAND  ==
  
This example (Figure 122) is similar to the previous one. Again, Multiplier Elastoplastic analysis is used to trace the load-displacement curve from the initial unloaded state to incipient collapse. The default strength parameters of the Medium Sand material are <math display="inline">c = 0</math>, <math display="inline">\phi = 35^\circ</math> and, in the case of nonassociated flow, <math display="inline">\psi=5^\circ</math>. In contrast to the previous example where the soil-foundation interface was modeled by means of a Shear Joint with reduced material parameters, we use Rigid Plate elements as shown in Figure 122. The interface strength reduction factor can then be applied directly without having to specify a separate material. In the following, we use a reduction factor of <math display="inline">r=0.5</math> throughout. The Drainage Conditions are Always Drained so the Time Scope (Long Term or Short Term) is immaterial.
+
This example (Figure 33.1) is similar to the previous one. Again, Multiplier Elastoplastic analysis is used to trace the load-displacement curve from the initial unloaded state to incipient collapse. The default strength parameters of the Medium Sand material are <math display="inline">c = 0</math>, <math display="inline">\phi = 35^\circ</math> and, in the case of nonassociated flow, <math display="inline">\psi=5^\circ</math>. In contrast to the previous example where the soil-foundation interface was modeled by means of a Shear Joint with reduced material parameters, we use Rigid Plate elements as shown in Figure 33.1. The interface strength reduction factor can then be applied directly without having to specify a separate material. In the following, we use a reduction factor of <math display="inline">r=0.5</math> throughout. The Drainage Conditions are Always Drained so the Time Scope (Long Term or Short Term) is immaterial.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex22_Fig01_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex22_Fig01_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 122: Shallow foundation in Medium Sand. '''</span>
+
|+<span style="font-size:90%">'''Figure 33.1: Shallow foundation in Medium Sand.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
 
We proceed by solving the problem for both the associated (<math display="inline">\psi=\phi</math>) and nonassociated (<math display="inline">\psi=5^\circ</math>) cases using 2,000 of 6-node Gauss elements. Mesh adaptivity with default settings is used. The initial stresses are computed automatically (Initial State = Default under Settings in the Stage Manager). In addition, to estimate the effects of nonassociativity without performing a full Multiplier Elastoplastic analysis, a Limit Analyses using Davis parameters is carried out, also with 2,000 6-node FE. In the case of <math display="inline">c=0</math>, <math display="inline">\phi=35^\circ</math>, <math display="inline">\psi=5^\circ</math> the Davis parameters are: <math display="inline">c_D = 0</math>, <math display="inline">\phi_D = 31.03^\circ</math>.<br />
 
We proceed by solving the problem for both the associated (<math display="inline">\psi=\phi</math>) and nonassociated (<math display="inline">\psi=5^\circ</math>) cases using 2,000 of 6-node Gauss elements. Mesh adaptivity with default settings is used. The initial stresses are computed automatically (Initial State = Default under Settings in the Stage Manager). In addition, to estimate the effects of nonassociativity without performing a full Multiplier Elastoplastic analysis, a Limit Analyses using Davis parameters is carried out, also with 2,000 6-node FE. In the case of <math display="inline">c=0</math>, <math display="inline">\phi=35^\circ</math>, <math display="inline">\psi=5^\circ</math> the Davis parameters are: <math display="inline">c_D = 0</math>, <math display="inline">\phi_D = 31.03^\circ</math>.<br />
The resulting load-displacement curves are shown in Figure 123 while the collapse solutions in the two cases are shown in Figure 124. The key features are:
+
The resulting load-displacement curves are shown in Figure 33.2 while the collapse solutions in the two cases are shown in Figure 33.3. The key features are:
  
 
* The load-displacement response in the two cases differs in several ways: the apparent stiffness in the nonassociated case is smaller, as is the strength. Moreover, the load-displacement curve for the nonassociated case displays a peak followed by an apparent softening to a residual level. Note, that no explicit material softening is included in the model; the apparent softening is a consequence of the flow rule alone.
 
* The load-displacement response in the two cases differs in several ways: the apparent stiffness in the nonassociated case is smaller, as is the strength. Moreover, the load-displacement curve for the nonassociated case displays a peak followed by an apparent softening to a residual level. Note, that no explicit material softening is included in the model; the apparent softening is a consequence of the flow rule alone.
Line 2,843: Line 2,958:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex22_Fig04_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex22_Fig04_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 123: Load-displacement curves for shallow foundation in Medium Sand. '''</span>
+
|+<span style="font-size:90%">'''Figure 33.2: Load-displacement curves for shallow foundation in Medium Sand.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,851: Line 2,966:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex22_Fig05_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex22_Fig05_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 124: Shear dissipation and collapse mechanism for associated and nonassociated materials. '''</span>
+
|+<span style="font-size:90%">'''Figure 33.3: Shear dissipation and collapse mechanism for associated and nonassociated materials.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,861: Line 2,976:
 
==  LOAD-DISPLACEMENT ANALYSIS OF FOUNDATION IN HMC SAND  ==
 
==  LOAD-DISPLACEMENT ANALYSIS OF FOUNDATION IN HMC SAND  ==
  
The following example considers the load-displacement analysis of a shallow foundation as shown in Figure 125. Three different analyses with three different materials are considered: Loose Sand-HMC, Medium Sand-HMC and Dense Sand-HMC. These are default materials available in OPTUM G2 and represent, approximately, a loose, medium and dense sand parameterized by the HMC model. The material parameters are as shown in Figure 125.
+
The following example considers the load-displacement analysis of a shallow foundation as shown in Figure 34.1. Three different analyses with three different materials are considered: Loose Sand-HMC, Medium Sand-HMC and Dense Sand-HMC. These are default materials available in OPTUM G2 and represent, approximately, a loose, medium and dense sand parameterized by the HMC model. The material parameters are as shown in Figure 34.1.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/emcfoot_new.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/emcfoot_new.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 125: Shallow foundation in HMC sand (top) and parameters for default materials Loose Sand-HMC, Medium Sand-HMC and Dense Sand-HMC (bottom). '''</span>
+
|+<span style="font-size:90%">'''Figure 34.1: Shallow foundation in HMC sand (top) and parameters for default materials Loose Sand-HMC, Medium Sand-HMC and Dense Sand-HMC (bottom).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,876: Line 2,991:
 
* Use of Taylor’s stress-dilatancy relation instead of the constant dilation in the Mohr-Coulomb model (the latter relation is an option in the HMC model). Whatever the flow rule chosen, the dilation angle at the ultimate limit state is given by <math display="inline">\psi</math>.
 
* Use of Taylor’s stress-dilatancy relation instead of the constant dilation in the Mohr-Coulomb model (the latter relation is an option in the HMC model). Whatever the flow rule chosen, the dilation angle at the ultimate limit state is given by <math display="inline">\psi</math>.
  
In the following, displacement controlled Elastoplastic analyses are conducted using 2,000 6-node Gauss elements with 20 steps. The final vertical displacement is 0.3 m and is specified as part of the Plate BC associated with the Rigid plate on top of the foundation (see Figure 125). The resulting load-displacement curves are shown in Figure 126. As for the standard Mohr-Coulomb model with a nonassociated flow rule, some apparent softening is observed, especially for the dense sand. The ultimate limit loads follow a similar pattern: they fall in between those obtained with Limit Analysis using the original friction angle and the Davis angle respectively and are closer to the former. Note, however, that since the problems considered are prone to localization, significantly lower bearing capacities could be obtained as a result of increasing the number of elements or using mesh adaptivity.
+
In the following, displacement controlled Elastoplastic analyses are conducted using 2,000 6-node Gauss elements with 20 steps. The final vertical displacement is 0.3 m and is specified as part of the Plate BC associated with the Rigid plate on top of the foundation (see Figure 34.1). The resulting load-displacement curves are shown in Figure 34.2. As for the standard Mohr-Coulomb model with a nonassociated flow rule, some apparent softening is observed, especially for the dense sand. The ultimate limit loads follow a similar pattern: they fall in between those obtained with Limit Analysis using the original friction angle and the Davis angle respectively and are closer to the former. Note, however, that since the problems considered are prone to localization, significantly lower bearing capacities could be obtained as a result of increasing the number of elements or using mesh adaptivity.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/emcld2_NEW.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/emcld2_NEW.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 126: Load-displacement curves for shallow foundation on HMC sand. '''</span>
+
|+<span style="font-size:90%">'''Figure 34.2: Load-displacement curves for shallow foundation on HMC sand.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
<div id="emctab">
+
<div id="tabemc">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 14: Approximate limit loads computed by means of Multiplier Elastoplastic analysis (MEP), Limit Analysis using the original friction angles (LA-Original), and Limit Analysis using the Davis angles (LA-Davis). The Davis angles are <math display="inline">\phi_D=26.6^\circ</math>, <math display="inline">31.0^\circ</math> and <math display="inline">35.5^\circ</math> for the loose, medium and dense sands respectively.
+
|+ Table 34.1: Approximate limit loads computed by means of Multiplier Elastoplastic analysis (MEP), Limit Analysis using the original friction angles (LA-Original), and Limit Analysis using the Davis angles (LA-Davis). The Davis angles are <math display="inline">\phi_D=26.6^\circ</math>, <math display="inline">31.0^\circ</math> and <math display="inline">35.5^\circ</math> for the loose, medium and dense sands respectively.
 
!
 
!
 
!align="center"| EP
 
!align="center"| EP
Line 2,913: Line 3,028:
  
 
</div>
 
</div>
  Finally, the deformations and plastic multiplier field for the Medium case are shown in Figure 127. The overall picture is here again similar to that of the standard Mohr-Coulomb model.<br />
+
<span id="tabemc" label="tabemc"></span>
 +
 
 +
Finally, the deformations and plastic multiplier field for the Medium case are shown in Figure 34.3. The overall picture is here again similar to that of the standard Mohr-Coulomb model.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/emcdef_new.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/emcdef_new.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 127: Deformations and plastic multiplier field for Medium Sand-HMC. '''</span>
+
|+<span style="font-size:90%">'''Figure 34.3: Deformations and plastic multiplier field for Medium Sand-HMC.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,928: Line 3,045:
 
==  PULL-OUT OF ANCHOR IN SAND  ==
 
==  PULL-OUT OF ANCHOR IN SAND  ==
  
This example concerns the pull-out of an anchor as shown in Figure 128. The soil is the default Loose Sand material and the anchor is modeled as a weightless Rigid Plate with rough interfaces. The boundary conditions used imply symmetry about the left vertical boundary. We again examine the effects of the flow rule using Multiplier Elastoplastic analysis for two cases: associated (<math display="inline">\psi=\phi</math>) and nonassociated with <math display="inline">\psi=0</math>. In both cases the cohesion is zero.<br />
+
This example concerns the pull-out of an anchor as shown in Figure 35.1. The soil is the default Loose Sand material and the anchor is modeled as a weightless Rigid Plate with rough interfaces. The boundary conditions used imply symmetry about the left vertical boundary. We again examine the effects of the flow rule using Multiplier Elastoplastic analysis for two cases: associated (<math display="inline">\psi=\phi</math>) and nonassociated with <math display="inline">\psi=0</math>. In both cases the cohesion is zero.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex23_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex23_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 128: Pull-out of anchor in Loose Sand. '''</span>
+
|+<span style="font-size:90%">'''Figure 35.1: Pull-out of anchor in Loose Sand.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The Settings for both the associated and the nonassociated analysis are as shown in Figure 128. In contrast to the previous example, we use slightly more ‘plastic’ steps (<math display="inline">N_P=15</math>) than ‘elastic’ steps (<math display="inline">N_E=10</math>). Also, mesh adaptivity with default settings (Adaptive Iterations = 3 and Adaptivity Frequency = 3 is used).<br />
+
The Settings for both the associated and the nonassociated analysis are as shown in Figure 35.1. In contrast to the previous example, we use slightly more ‘plastic’ steps (<math display="inline">N_P=15</math>) than ‘elastic’ steps (<math display="inline">N_E=10</math>). Also, mesh adaptivity with default settings (Adaptive Iterations = 3 and Adaptivity Frequency = 3 is used).<br />
The resulting load-displacement curves are shown in Figure 129. As in the previous example, nonassociativity implies a reduction in both strength and stiffness. However, in contrast to the previous examples, the Davis friction angle, <math display="inline">\phi_D = 26.5651^\circ</math> in this case, leads to an overestimate of the strength in the nonassociated case. This has previously been noted and discussed by <bib id="Krabbenhoft:2012IJNME" /> and the current results thus confirm that while the Davis parameters usually lead to a good estimate of the strength of nonassociated materials, it is not necessarily conservative.<br />
+
The resulting load-displacement curves are shown in Figure 35.2. As in the previous example, nonassociativity implies a reduction in both strength and stiffness. However, in contrast to the previous examples, the Davis friction angle, <math display="inline">\phi_D = 26.5651^\circ</math> in this case, leads to an overestimate of the strength in the nonassociated case. This has previously been noted and discussed by <bib id="Krabbenhoft:2012IJNME" /> and the current results thus confirm that while the Davis parameters usually lead to a good estimate of the strength of nonassociated materials, it is not necessarily conservative.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex23_Fig02_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex23_Fig02_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 129: Load-displacement curves for anchor pull-out in Loose Sand. '''</span>
+
|+<span style="font-size:90%">'''Figure 35.2: Load-displacement curves for anchor pull-out in Loose Sand.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
Finally, the deformations and plastic multiplier fields at the end of the analyses are shown in Figure 130. As in the previous examples, the effects of dilation are apparent.
+
Finally, the deformations and plastic multiplier fields at the end of the analyses are shown in Figure 35.3. As in the previous examples, the effects of dilation are apparent.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex23_Fig03_II.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex23_Fig03_II.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 130: Deformations (scaled by a factor of 5) and plastic multiplier fields. '''</span>
+
|+<span style="font-size:90%">'''Figure 35.3: Deformations (scaled by a factor of 5) and plastic multiplier fields.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,965: Line 3,082:
 
==  FAULT RUPTURE PROPAGATION IN SAND  ==
 
==  FAULT RUPTURE PROPAGATION IN SAND  ==
  
This example demonstrates the application of non-standard boundary conditions as well as the influence of the flow rule on the failure kinematics of frictional materials such as sands. The problem under consideration is inspired by the study of <bib id="Anastasopoulos:2007" />. The setup is as sketched in Figure 131. In the normal faulting configuration, the right and bottom walls are gradually moved downwards at a given inclination <math display="inline">\alpha_N</math> until a failure occurs. In the reverse faulting configuration, the walls are instead moved upwards at an inclination <math display="inline">\alpha_R</math> with the horizontal.
+
This example demonstrates the application of non-standard boundary conditions as well as the influence of the flow rule on the failure kinematics of frictional materials such as sands. The problem under consideration is inspired by the study of <bib id="Anastasopoulos:2007" />. The setup is as sketched in Figure 36.1. In the normal faulting configuration, the right and bottom walls are gradually moved downwards at a given inclination <math display="inline">\alpha_N</math> until a failure occurs. In the reverse faulting configuration, the walls are instead moved upwards at an inclination <math display="inline">\alpha_R</math> with the horizontal.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex23c_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex23c_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 131: Normal and reverse faulting setups. The gray walls are immovable while displacements are prescribed on the red walls. All walls are assumed rigid. '''</span>
+
|+<span style="font-size:90%">'''Figure 36.1: Normal and reverse faulting setups. The gray walls are immovable while displacements are prescribed on the red walls. All walls are assumed rigid.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,977: Line 3,094:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex27_Fig02a.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex27_Fig02a.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 132: Fault propagation problem (reverse faulting). The horizontal divisions are included for visualization purposes only. '''</span>
+
|+<span style="font-size:90%">'''Figure 36.2: Fault propagation problem (reverse faulting). The horizontal divisions are included for visualization purposes only.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
In OPTUM G2, displacement boundary conditions can be imposed with respect to structural elements only. For the present problem, the moveable walls are modeled as Rigid Plates as shown in Figure 132 while regular Supports are used for the immovable walls. Next, from the lower right corner, another Rigid Plate is extended downwards at a direction corresponding to the direction of faulting (<math display="inline">\alpha_N</math> or <math display="inline">\alpha_R</math>). At the end of this plate, displacement boundary conditions can be imposed in a local coordinate system to allow the whole Rigid Plate system to move only along the direction given by the extension plate.<br />
+
In OPTUM G2, displacement boundary conditions can be imposed with respect to structural elements only. For the present problem, the moveable walls are modeled as Rigid Plates as shown in Figure 36.2 while regular Supports are used for the immovable walls. Next, from the lower right corner, another Rigid Plate is extended downwards at a direction corresponding to the direction of faulting (<math display="inline">\alpha_N</math> or <math display="inline">\alpha_R</math>). At the end of this plate, displacement boundary conditions can be imposed in a local coordinate system to allow the whole Rigid Plate system to move only along the direction given by the extension plate.<br />
 
In the following, the normal and reverse fault configurations for <math display="inline">\alpha_N = \alpha_R = 60^\circ</math> are considered. The material is the default material Loose Sand with a nonassociated flow rule (<math display="inline">\phi=30^\circ</math>, <math display="inline">\psi=0^\circ</math>). The problems are analyzed by Multiplier Elastoplastic analysis using a total of 10,000 6-node Gauss elements.<br />
 
In the following, the normal and reverse fault configurations for <math display="inline">\alpha_N = \alpha_R = 60^\circ</math> are considered. The material is the default material Loose Sand with a nonassociated flow rule (<math display="inline">\phi=30^\circ</math>, <math display="inline">\psi=0^\circ</math>). The problems are analyzed by Multiplier Elastoplastic analysis using a total of 10,000 6-node Gauss elements.<br />
The deformations and plastic multiplier fields at the final state of the analyses are shown in Figure 133. The solutions are in good agreement with those observed experimentally by <bib id="Anastasopoulos:2007" />.<br />
+
The deformations and plastic multiplier fields at the final state of the analyses are shown in Figure 36.3. The solutions are in good agreement with those observed experimentally by <bib id="Anastasopoulos:2007" />.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex27x_Fig05.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex27x_Fig05.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 133: Normal (top) and reverse (bottom) fault configurations close to failure. '''</span>
+
|+<span style="font-size:90%">'''Figure 36.3: Normal (top) and reverse (bottom) fault configurations close to failure.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 2,998: Line 3,115:
 
==  CONFINED SEEPAGE AROUND IMPERMEABLE SHEET PILE  ==
 
==  CONFINED SEEPAGE AROUND IMPERMEABLE SHEET PILE  ==
  
In this example, we consider the confined seepage around a sheet pile wall as shown in Figure 134. It is assumed that the sheet pile wall is positioned under an 8 m wide impermeable dam. Rather than modeling the dam, the base of the dam is considered impermeable. Similarly, rather than explicitly modeling the water on the upstream side of the dam, equivalent fixed head boundary conditions are imposed. On the downstream side of the dam, the water table is maintained at ground level. The sheet pile wall is modeled as a Rigid Plate. In OPTUM G2, such elements may be either permeable or impermeable as indicated in Figure 134. In this example, the sheet pile is considered impermeable. The hydraulic model is taken as the Linear model with default settings and <math display="inline">k_x = k_y = 1</math> m/day.
+
In this example, we consider the confined seepage around a sheet pile wall as shown in Figure 37.1. It is assumed that the sheet pile wall is positioned under an 8 m wide impermeable dam. Rather than modeling the dam, the base of the dam is considered impermeable. Similarly, rather than explicitly modeling the water on the upstream side of the dam, equivalent fixed head boundary conditions are imposed. On the downstream side of the dam, the water table is maintained at ground level. The sheet pile wall is modeled as a Rigid Plate. In OPTUM G2, such elements may be either permeable or impermeable as indicated in Figure 37.1. In this example, the sheet pile is considered impermeable. The hydraulic model is taken as the Linear model with default settings and <math display="inline">k_x = k_y = 1</math> m/day.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex35_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex35_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 134: Confined seepage around sheet pile. '''</span>
+
|+<span style="font-size:90%">'''Figure 37.1: Confined seepage around sheet pile.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
This problem has been solved analytically by <bib id="PK:1962" /> for a range of geometries. The total flux, <math display="inline">Q</math>, from one side of the dam to the other can be determined from the charts in Figure 135. Comparisons between analytical and computed solutions for selected wall depths, <math display="inline">s</math>, are shown in Table 15. We see that the numerical and analytical solutions are in very good agreement already for the coarsest meshes comprising 1,000 elements. The pressure head distributions (for 16,000 elements) are shown in Figure 136.  
+
This problem has been solved analytically by <bib id="PK:1962" /> for a range of geometries. The total flux, <math display="inline">Q</math>, from one side of the dam to the other can be determined from the charts in Figure 37.2. Comparisons between analytical and computed solutions for selected wall depths, <math display="inline">s</math>, are shown in Table 37.1. We see that the numerical and analytical solutions are in very good agreement already for the coarsest meshes comprising 1,000 elements. The pressure head distributions (for 16,000 elements) are shown in Figure 37.3.  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex35_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex35_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 135: <bib id="PK:1962 " /> solution for confined seepage around sheet pile. '''</span>
+
|+<span style="font-size:90%">'''Figure 37.2: <bib id="PK:1962" /> solution for confined seepage around sheet pile.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
 
 
 
 
 +
 +
<div id="Ex35_Tab01">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 15: Comparison between analytical <bib id="PK:1962 " /> and computed fluxes <math display="inline">Q</math> (m<math display="inline">^3</math>/day/m) using 1,000 to 16,000 elements.
+
|+ Table 37.1: Comparison between analytical <bib id="PK:1962" /> and computed fluxes <math display="inline">Q</math> (m<math display="inline">^3</math>/day/m) using 1,000 to 16,000 elements.
 
|align="center"|
 
|align="center"|
 
|align="center"|
 
|align="center"|
Line 3,071: Line 3,190:
 
|}
 
|}
  
 +
 +
</div>
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex35_Fig03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex35_Fig03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 136: Pressure head distributions (m). '''</span>
+
|+<span style="font-size:90%">'''Figure 37.3: Pressure head distributions (m).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,081: Line 3,202:
 
===  Alternative modeling  ===
 
===  Alternative modeling  ===
  
Instead of modeling the problem by imposing relevant fixed head and no-flow boundary conditions to account for the reservoir and the dam, both may be modeled using the Water material from the Fluids category and a solid with Drainage = Non-Porous from the Solids category. This alternative problem setup is shown in Figure 137. The reservoir may here be defined as usual, by defining the geometry and assigning the relevant material, or it may be defined using the Water Table tool available in the Features ribbon. Whichever approach is used it is important to note that the top of the water domain must be defined as a zero pressure line (indicated by a blue triangle). The geometry of the dam is defined in the usual way and the Rigid material assigned. The default drainage condition for this material is Impermeable and placing the dam on top of the soil domain as shown will thus have the same effect as imposing a no-flow boundary condition as was done originally.<br />
+
Instead of modeling the problem by imposing relevant fixed head and no-flow boundary conditions to account for the reservoir and the dam, both may be modeled using the Water material from the Fluids category and a solid with Drainage = Non-Porous from the Solids category. This alternative problem setup is shown in Figure 37.4. The reservoir may here be defined as usual, by defining the geometry and assigning the relevant material, or it may be defined using the Water Table tool available in the Features ribbon. Whichever approach is used it is important to note that the top of the water domain must be defined as a zero pressure line (indicated by a blue triangle). The geometry of the dam is defined in the usual way and the Rigid material assigned. The default drainage condition for this material is Impermeable and placing the dam on top of the soil domain as shown will thus have the same effect as imposing a no-flow boundary condition as was done originally.<br />
 
Using this alternative modeling strategy, all domains including the reservoir and the dam are discretized by finite elements as shown below.
 
Using this alternative modeling strategy, all domains including the reservoir and the dam are discretized by finite elements as shown below.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex35_Fig04.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex35_Fig04.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 137: Confined seepage around sheet pile: alternative modeling and resulting mesh. '''</span>
+
|+<span style="font-size:90%">'''Figure 37.4: Confined seepage around sheet pile: alternative modeling and resulting mesh.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,095: Line 3,216:
  
 
As with all other analysis types, it is for Seepage analysis possible to adapt the mesh in a series of adaptivity iterations. In this case, the relevant Adaptivity Control variable is Flow. This ensures that the mesh is adapted on the basis of a combination of the ‘flow energy’, <math display="inline">\frac{1}{2}\boldsymbol q^\text{T}\boldsymbol K\boldsymbol q</math>, and a measure ensuring a reasonable concentration of elements around free surfaces (not relevant in the present problem).<br />
 
As with all other analysis types, it is for Seepage analysis possible to adapt the mesh in a series of adaptivity iterations. In this case, the relevant Adaptivity Control variable is Flow. This ensures that the mesh is adapted on the basis of a combination of the ‘flow energy’, <math display="inline">\frac{1}{2}\boldsymbol q^\text{T}\boldsymbol K\boldsymbol q</math>, and a measure ensuring a reasonable concentration of elements around free surfaces (not relevant in the present problem).<br />
The results of the analyses using 1,000 elements and 3 adaptivity iterations are shown in Figure 138. As seen, the critical regions are the edges of the dam and the bottom of the sheet pile. Also, note that the results in terms of the total flow are of an accuracy similar to or better than those obtained with 16,000 elements without adaptivity.
+
The results of the analyses using 1,000 elements and 3 adaptivity iterations are shown in Figure 37.5. As seen, the critical regions are the edges of the dam and the bottom of the sheet pile. Also, note that the results in terms of the total flow are of an accuracy similar to or better than those obtained with 16,000 elements without adaptivity.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex35_Fig06.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex35_Fig06.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 138: Adapted meshes (1,000 elements) and resulting fluxes. '''</span>
+
|+<span style="font-size:90%">'''Figure 37.5: Adapted meshes (1,000 elements) and resulting fluxes.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,107: Line 3,228:
 
==  UNCONFINED SEEPAGE THROUGH RECTANGULAR DAMS  ==
 
==  UNCONFINED SEEPAGE THROUGH RECTANGULAR DAMS  ==
  
In this example the classical problem of unconfined seepage through a rectangular dam is considered. The problem setup is shown in Figure 139. A dam of height <math display="inline">H_A=10</math> m and length <math display="inline">L=5</math> m is considered. On the left side, the water level is at <math display="inline">H_A = 10</math> m while on the right side it is at <math display="inline">H_B = 4</math> m. To account for these conditions, pressure boundary conditions are imposed with the pressures varying linearly from the top to <math display="inline">H_A\gamma_w = -980</math> kPa and <math display="inline">H_B\gamma_w = -39.2</math> kPa, respectively, on the left and right sides (note the sign of the boundary pressures). Finally, at the bottom, a no-flow boundary condition is imposed. The remaining boundaries for which no conditions have been imposed will be treated as seepage faces, i.e. boundaries at which a discharge at zero pressure can take place.
+
In this example the classical problem of unconfined seepage through a rectangular dam is considered. The problem setup is shown in Figure 38.1. A dam of height <math display="inline">H_A=10</math> m and length <math display="inline">L=5</math> m is considered. On the left side, the water level is at <math display="inline">H_A = 10</math> m while on the right side it is at <math display="inline">H_B = 4</math> m. To account for these conditions, pressure boundary conditions are imposed with the pressures varying linearly from the top to <math display="inline">H_A\gamma_w = -980</math> kPa and <math display="inline">H_B\gamma_w = -39.2</math> kPa, respectively, on the left and right sides (note the sign of the boundary pressures). Finally, at the bottom, a no-flow boundary condition is imposed. The remaining boundaries for which no conditions have been imposed will be treated as seepage faces, i.e. boundaries at which a discharge at zero pressure can take place.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex34_Fig04.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex34_Fig04.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 139: Unconfined seepage through rectangular dam (pressure boundary conditions). '''</span>
+
|+<span style="font-size:90%">'''Figure 38.1: Unconfined seepage through rectangular dam (pressure boundary conditions).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,119: Line 3,240:
 
Alternatively, instead of fixed pressure boundary conditions, fixed head boundary conditions may be used. Recall that the pressure head is defined as:  
 
Alternatively, instead of fixed pressure boundary conditions, fixed head boundary conditions may be used. Recall that the pressure head is defined as:  
  
<math display="block">h = y - \frac{p}{\gamma_w}</math>
+
<math display="block">(38.1)\qquad
 +
    h = y - \frac{p}{\gamma_w}</math>
  
 
where <math display="inline">y</math> is the vertical coordinate and <math display="inline">p</math> is the pressure (negative in the fully saturated range consistent with the sign convention used for stresses). Thus, with the pressures at left boundary varying as  
 
where <math display="inline">y</math> is the vertical coordinate and <math display="inline">p</math> is the pressure (negative in the fully saturated range consistent with the sign convention used for stresses). Thus, with the pressures at left boundary varying as  
  
<math display="block">p_A = -\gamma_w (H_A+y_0-y),</math>
+
<math display="block">(38.2)\qquad
 +
    p_A = -\gamma_w (H_A+y_0-y),</math>
  
 
where <math display="inline">y_0</math> is the vertical coordinate at the base of the dam, the equivalent boundary head is:  
 
where <math display="inline">y_0</math> is the vertical coordinate at the base of the dam, the equivalent boundary head is:  
  
<math display="block">h_A = H_A+y_0</math>
+
<math display="block">(38.3)\qquad
 +
    h_A = H_A+y_0</math>
  
The equivalent problem with these conditions imposed is shown in Figure 140. The vertical coordinate at the base of the dam is here <math display="inline">y_0 = 0</math> m.
+
The equivalent problem with these conditions imposed is shown in Figure 38.2. The vertical coordinate at the base of the dam is here <math display="inline">y_0 = 0</math> m.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex34_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex34_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 140: Unconfined seepage through rectangular dam (head boundary conditions). '''</span>
+
|+<span style="font-size:90%">'''Figure 38.2: Unconfined seepage through rectangular dam (head boundary conditions).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,141: Line 3,265:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex34_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex34_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 141: Flow through rectangular dam: location of phreatic surface for different <math display="inline">h^*</math> compared with the solution of <bib id="PK:1962 " />. '''</span>
+
|+<span style="font-size:90%">'''Figure 38.3: Flow through rectangular dam: location of phreatic surface for different <math display="inline">h^*</math> compared with the solution of <bib id="PK:1962" />.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The calculations are carried out with the Seepage analysis type using 10,000. The hydraulic model is the Linear model. The influence of the parameter <math display="inline">h^*</math> indicating the range of the unsaturated zone (see the Materials Manual) is shown in Figure 141. We see that the results obtained for <math display="inline">h^* = 0.1</math> m and <math display="inline">h^* = 1</math> m are quite similar and in good agreement with the analytical solution of <bib id="PK:1962" /> which assumes <math display="inline">h^*=0</math>. On the other hand, a value of <math display="inline">h^* = 10</math> m appears to be too high in this case although the basic features of the solution are still captured.<br />
+
The calculations are carried out with the Seepage analysis type using 10,000. The hydraulic model is the Linear model. The influence of the parameter <math display="inline">h^*</math> indicating the range of the unsaturated zone (see the Materials Manual) is shown in Figure 38.3. We see that the results obtained for <math display="inline">h^* = 0.1</math> m and <math display="inline">h^* = 1</math> m are quite similar and in good agreement with the analytical solution of <bib id="PK:1962" /> which assumes <math display="inline">h^*=0</math>. On the other hand, a value of <math display="inline">h^* = 10</math> m appears to be too high in this case although the basic features of the solution are still captured.<br />
 
Finally, the same analysis is repeated for a square dam with <math display="inline">L=10</math> m, <math display="inline">H_A=10</math> m and <math display="inline">H_B = 0</math> m. Again, the solutions for the smaller values of <math display="inline">h^*</math> are in good agreement with the analytical solution of <bib id="PK:1962" />.<br />
 
Finally, the same analysis is repeated for a square dam with <math display="inline">L=10</math> m, <math display="inline">H_A=10</math> m and <math display="inline">H_B = 0</math> m. Again, the solutions for the smaller values of <math display="inline">h^*</math> are in good agreement with the analytical solution of <bib id="PK:1962" />.<br />
  
Line 3,153: Line 3,277:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex34_Fig03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex34_Fig03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 142: Flow through square dam: location of phreatic surface for different <math display="inline">h^*</math> compared with the solution of <bib id="PK:1962 " />. '''</span>
+
|+<span style="font-size:90%">'''Figure 38.4: Flow through square dam: location of phreatic surface for different <math display="inline">h^*</math> compared with the solution of <bib id="PK:1962" />.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,163: Line 3,287:
 
==  FREE SURFACE FLOW THROUGH EARTH DAM  ==
 
==  FREE SURFACE FLOW THROUGH EARTH DAM  ==
  
This example considers the flow through a trapezoidal dam as shown in Figure 143. The hydraulic conductivity of the core of the dam (Zone 2) is different from that in the other parts of the dam (Zone 1). A drain is situated at the bottom right part of the dam. This example has been considered by <bib id="Kazemzadeh:2012" /> for a range of conductivities in the two zones as summarized in Table 16. We note that cases 5-9 involve an anisotropic hydraulic conductivity with <math display="inline">K_x>K_y</math> is all cases.
+
This example considers the flow through a trapezoidal dam as shown in Figure 39.1. The hydraulic conductivity of the core of the dam (Zone 2) is different from that in the other parts of the dam (Zone 1). A drain is situated at the bottom right part of the dam. This example has been considered by <bib id="Kazemzadeh:2012" /> for a range of conductivities in the two zones as summarized in Table 39.1. We note that cases 5-9 involve an anisotropic hydraulic conductivity with <math display="inline">K_x>K_y</math> is all cases.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex36_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex36_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 143: Inhomogeneous and anisotropic dam. '''</span>
+
|+<span style="font-size:90%">'''Figure 39.1: Inhomogeneous and anisotropic dam.'''</span>
 
|}
 
|}
 
</center>
 
</center>
 +
 +
<div id="Ex36_Tab01">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 16: Conductivities in Zones 1 and 2 and resulting fluxes at the water table compared to those obtained by <bib id="Kazemzadeh:2012 " />.
+
|+ Table 39.1: Conductivities in Zones 1 and 2 and resulting fluxes at the water table compared to those obtained by <bib id="Kazemzadeh:2012" />.
 
|align="center"| Case
 
|align="center"| Case
 
|align="center"|
 
|align="center"|
Line 3,264: Line 3,390:
 
|}
 
|}
  
The hydraulic model used by <bib id="Kazemzadeh:2012" /> corresponds to the Linear model with <math display="inline">h^*=0</math>, i.e. step saturation versus pressure and pressure versus conductivity functions. In the following, we approximate this situation by using <math display="inline">h^*=0.1</math> m. The drain at the bottom right of the dam is modeled by imposing a Fixed Pressure boundary condition with <math display="inline">p_s=0</math> as shown in Figure 143. Other boundary conditions are Tangential Supports on the left vertical boundary, which act as no-flow BCs, and actual no-flow BCs at the bottom boundary (any Supports could have been used here). On the remaining inclined boundary on the right side of the dam, no boundary conditions are imposed. This implies that these boundaries act as seepage faces allowing for an outward discharge at zero pressure.<br />
+
 
As with other analysis types, Seepage Analysis gives the possibility to adapt the mesh in a number of successive iterations. The relevant Adaptivity Control setting is Flow (see Figure 143). With this setting, the mesh is adapted on the basis of a combination of the ‘flow energy’, <math display="inline">\frac{1}{2}\boldsymbol q^\text{T}\boldsymbol K\boldsymbol q</math>, and a measure ensuring a reasonable concentration of elements around free surfaces.<br />
+
</div>
The results of the nine analyses in terms of the total rate of flow through the dam are shown in Table 16. These are seen to compare well with the results obtained by <bib id="Kazemzadeh:2012" />.<br />
+
The hydraulic model used by <bib id="Kazemzadeh:2012" /> corresponds to the Linear model with <math display="inline">h^*=0</math>, i.e. step saturation versus pressure and pressure versus conductivity functions. In the following, we approximate this situation by using <math display="inline">h^*=0.1</math> m. The drain at the bottom right of the dam is modeled by imposing a Fixed Pressure boundary condition with <math display="inline">p_s=0</math> as shown in Figure 39.1. Other boundary conditions are Tangential Supports on the left vertical boundary, which act as no-flow BCs, and actual no-flow BCs at the bottom boundary (any Supports could have been used here). On the remaining inclined boundary on the right side of the dam, no boundary conditions are imposed. This implies that these boundaries act as seepage faces allowing for an outward discharge at zero pressure.<br />
The distributions of saturation are shown in Figure 143 for selected cases. The effects of inhomogeneity (Case 1 vs. Case 4) and anisotropy (Case 4 vs. Case 7) are apparent.
+
As with other analysis types, Seepage Analysis gives the possibility to adapt the mesh in a number of successive iterations. The relevant Adaptivity Control setting is Flow (see Figure 39.1). With this setting, the mesh is adapted on the basis of a combination of the ‘flow energy’, <math display="inline">\frac{1}{2}\boldsymbol q^\text{T}\boldsymbol K\boldsymbol q</math>, and a measure ensuring a reasonable concentration of elements around free surfaces.<br />
 +
The results of the nine analyses in terms of the total rate of flow through the dam are shown in Table 39.1. These are seen to compare well with the results obtained by <bib id="Kazemzadeh:2012" />.<br />
 +
The distributions of saturation are shown in Figure 39.1 for selected cases. The effects of inhomogeneity (Case 1 vs. Case 4) and anisotropy (Case 4 vs. Case 7) are apparent.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex36_Fig03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex36_Fig03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 144: Saturation distributions for selected cases (red corresponds to <math display="inline">S=1</math> and blue to <math display="inline">S=0</math>). '''</span>
+
|+<span style="font-size:90%">'''Figure 39.2: Saturation distributions for selected cases (red corresponds to <math display="inline">S=1</math> and blue to <math display="inline">S=0</math>).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,279: Line 3,407:
 
==  EXCAVATION MODEL TEST  ==
 
==  EXCAVATION MODEL TEST  ==
  
The following problem concerns the simulation of a model excavation test carried out by <bib id="Tefera:2006" /> at NTNU in Trondheim, Norway. The model is shown schematically in Figure 145. It comprises a <math display="inline">3\times 4</math> m box filled with sand. At 1 m from the vertical boundary a 2.5 m wall has been installed. The soil in front of the wall is then excavated in a number of stages. In the course of the excavation, at a level of 0.91 m, a strut is inserted and prestressed to 5 kN/m.
+
The following problem concerns the simulation of a model excavation test carried out by <bib id="Tefera:2006" /> at NTNU in Trondheim, Norway. The model is shown schematically in Figure 40.1. It comprises a <math display="inline">3\times 4</math> m box filled with sand. At 1 m from the vertical boundary a 2.5 m wall has been installed. The soil in front of the wall is then excavated in a number of stages. In the course of the excavation, at a level of 0.91 m, a strut is inserted and prestressed to 5 kN/m.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex27_Fig01_c.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex27_Fig01_c.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 145: Excavation model test setup with indication of the material excavated in each stage. '''</span>
+
|+<span style="font-size:90%">'''Figure 40.1: Excavation model test setup with indication of the material excavated in each stage.'''</span>
 
|}
 
|}
 
</center>
 
</center>
 +
 +
<div id="Ex27_Tab01">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 17: Stages in excavation model test. <math display="inline">^{*) }</math>Initial Strength Reduction analysis reveals an upper bound strength reduction factor of 0.95.
+
|+ Table 40.1: Stages in excavation model test. <math display="inline">^{*)}</math>Initial Strength Reduction analysis reveals an upper bound strength reduction factor of 0.95.
 
!align="center"| Stage
 
!align="center"| Stage
 
!align="center"| Excavation depth (m)
 
!align="center"| Excavation depth (m)
Line 3,358: Line 3,488:
 
|}
 
|}
  
The sequence of stages is shown in Table 17. With the exception of the first stage, which computes the initial stresses, these are all of type Elastoplastic and each stage uses the preceding stage as From stage.
+
 
 +
</div>
 +
The sequence of stages is shown in Table 40.1. With the exception of the first stage, which computes the initial stresses, these are all of type Elastoplastic and each stage uses the preceding stage as From stage.
  
 
===  Soil model  ===
 
===  Soil model  ===
  
Excavation analysis is an example of a problem type where the behaviour of the soil in loading and unloading may be important. As the excavation proceeds, the soil behind the wall will unload (active pressure) whereas the soil in front of the wall below the current excavation level will experience loading (passive pressure). The HMC model, which allows for the specification of separate loading and unloading moduli, is well suited to capture this behaviour. In the following, this model is used with the parameters reported by <bib id="Tefera:2006" />, see Table 18.
+
Excavation analysis is an example of a problem type where the behaviour of the soil in loading and unloading may be important. As the excavation proceeds, the soil behind the wall will unload (active pressure) whereas the soil in front of the wall below the current excavation level will experience loading (passive pressure). The HMC model, which allows for the specification of separate loading and unloading moduli, is well suited to capture this behaviour. In the following, this model is used with the parameters reported by <bib id="Tefera:2006" />, see Table 40.3.
 +
 
 +
<div id="Ex27_Tab02">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 18: Sand properties [after <bib id="Tefera:2006 " />].
+
|+ Table 40.3: Sand properties [after <bib id="Tefera:2006" />].
 
! '''Sand (HMC)'''
 
! '''Sand (HMC)'''
 
!
 
!
Line 3,411: Line 3,545:
 
|}
 
|}
  
 +
 +
</div>
 
While the HMC model is more advanced than the standard Mohr-Coulomb model, the latter model is in fact in many cases able to capture the key features of excavation analyses quite adequately. Indeed, the overall behaviour is governed mainly by the failing soil mass behind the wall and the stiffness of the wall rather than by the elastic properties of the soil. Therefore, in the absence of detailed information about the soil properties, use of the standard Mohr-Coulomb model – with reasonable estimates of the parameters – is generally quite reasonable. The following example explores the differences between the two models in more detail.
 
While the HMC model is more advanced than the standard Mohr-Coulomb model, the latter model is in fact in many cases able to capture the key features of excavation analyses quite adequately. Indeed, the overall behaviour is governed mainly by the failing soil mass behind the wall and the stiffness of the wall rather than by the elastic properties of the soil. Therefore, in the absence of detailed information about the soil properties, use of the standard Mohr-Coulomb model – with reasonable estimates of the parameters – is generally quite reasonable. The following example explores the differences between the two models in more detail.
  
 
===  Structural elements  ===
 
===  Structural elements  ===
  
The wall and strut are modeled using Plate and Connector elements respectively. The parameters are given in Table 19.
+
The wall and strut are modeled using Plate and Connector elements respectively. The parameters are given in Table 40.5.
 +
 
 +
<div id="Ex27_Tab03">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 19: Wall and strut properties [after <bib id="Tefera:2006 " />].
+
|+ Table 40.5: Wall and strut properties [after <bib id="Tefera:2006" />].
 
! '''Wall (Plate)'''
 
! '''Wall (Plate)'''
 
!
 
!
Line 3,452: Line 3,590:
 
|}
 
|}
  
 +
 +
</div>
 
====  Prestress  ====
 
====  Prestress  ====
  
As indicated in Table 17, the simulation proceeds by first excavating to a depth of 0.91 m in three stages. At this point, the Connector is inserted 0.5 m below the top level. By selecting the Connector its properties appear in the property window. Under Prestress, Apply Prestress is set to Yes and a Prestress Force of 5 kN/m is entered. This applies the specified prestress at the current stage in a single step. In the subsequent stages, the Connector is still present but with Apply Prestress set to No. An overview of the prestressing functionality is given in Figure 146.
+
As indicated in Table 40.1, the simulation proceeds by first excavating to a depth of 0.91 m in three stages. At this point, the Connector is inserted 0.5 m below the top level. By selecting the Connector its properties appear in the property window. Under Prestress, Apply Prestress is set to Yes and a Prestress Force of 5 kN/m is entered. This applies the specified prestress at the current stage in a single step. In the subsequent stages, the Connector is still present but with Apply Prestress set to No. An overview of the prestressing functionality is given in Figure 40.2.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex27_Fig02_res.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex27_Fig02_res.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 146: Prestressing of Connector. '''</span>
+
|+<span style="font-size:90%">'''Figure 40.2: Prestressing of Connector.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,466: Line 3,606:
 
===  Strength Reduction analysis  ===
 
===  Strength Reduction analysis  ===
  
Before the actual excavation simulation gets under way, it is useful to run Strength Reduction analyses for all stages to gauge the proximity to failure of each stage. The results of upper and lower bound analyses of this type are shown in Figure 147. Of particular note are the results for an excavation depth of 2.3 m. Both the lower, and more importantly the upper, bound analyses here reveal that the factor of safety is below unity. In other words, the system is not stable at this level and for the material parameters assumed. This is in contrast to the experiment. The present result is confirmed by the numerical analyses of <bib id="Tefera:2006" /> who reported ‘convergence problems’ for this level of excavation.
+
Before the actual excavation simulation gets under way, it is useful to run Strength Reduction analyses for all stages to gauge the proximity to failure of each stage. The results of upper and lower bound analyses of this type are shown in Figure 40.3. Of particular note are the results for an excavation depth of 2.3 m. Both the lower, and more importantly the upper, bound analyses here reveal that the factor of safety is below unity. In other words, the system is not stable at this level and for the material parameters assumed. This is in contrast to the experiment. The present result is confirmed by the numerical analyses of <bib id="Tefera:2006" /> who reported ‘convergence problems’ for this level of excavation.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex27_Fig02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex27_Fig02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 147: Upper and lower bound strength based factors of safety as function of excavation depth. '''</span>
+
|+<span style="font-size:90%">'''Figure 40.3: Upper and lower bound strength based factors of safety as function of excavation depth.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,478: Line 3,618:
 
===  Excavation analysis  ===
 
===  Excavation analysis  ===
  
The analysis proceeds by running the 9 stages indicated in Table 17. For all stages, 1,000 6-node Gauss elements are used with three adaptivity iterations. The results in terms of the horizontal displacement of the top of the wall are shown in Figure 148. Also shown are the experimental results as well as the results of numerical analysis of <bib id="Tefera:2006" /> using the so-called Hardening Soil (HS), a detailed description of which has been given by <bib id="Benz:2007" />.<br />
+
The analysis proceeds by running the 9 stages indicated in Table 40.1. For all stages, 1,000 6-node Gauss elements are used with three adaptivity iterations. The results in terms of the horizontal displacement of the top of the wall are shown in Figure 40.4. Also shown are the experimental results as well as the results of numerical analysis of <bib id="Tefera:2006" /> using the so-called Hardening Soil (HS), a detailed description of which has been given by <bib id="Benz:2007" />.<br />
The results shown in Figure 148 show a very good agreement between the HS and HMC models. As already discussed at length by <bib id="Tefera:2006" />, there are several points of discrepancy between the experiments and the results, the most noticeable of which are the underestimate of wall displacement at the beginning of the excavation and as a result of the application of prestress at a level of 0.91 m.<br />
+
The results shown in Figure 40.4 show a very good agreement between the HS and HMC models. As already discussed at length by <bib id="Tefera:2006" />, there are several points of discrepancy between the experiments and the results, the most noticeable of which are the underestimate of wall displacement at the beginning of the excavation and as a result of the application of prestress at a level of 0.91 m.<br />
 
<bib id="Tefera:2006" /> also monitored the force in the strut, from the initial application of the prestress to the end of the excavation. Again, there is some quantitative discrepancy between the experimental data and the predictions, especially towards the end of the excavation. However, the HMC model appears to make somewhat better predictions than the HS model used by <bib id="Tefera:2006" />. For example, at the first stage after the application of the prestress, the former predicts a decrease in force, consistent with the experimental data, while the latter predicts an increase.<br />
 
<bib id="Tefera:2006" /> also monitored the force in the strut, from the initial application of the prestress to the end of the excavation. Again, there is some quantitative discrepancy between the experimental data and the predictions, especially towards the end of the excavation. However, the HMC model appears to make somewhat better predictions than the HS model used by <bib id="Tefera:2006" />. For example, at the first stage after the application of the prestress, the former predicts a decrease in force, consistent with the experimental data, while the latter predicts an increase.<br />
Finally, the deformations and the intensity of plastic dissipation at various stages of the excavation are shown in Figure 149. We see that the distribution of plasticity varies quite significantly with excavation depth. The adaptive mesh procedure used ensures a high degree of accuracy while maintaining a moderate number of elements in each stage.
+
Finally, the deformations and the intensity of plastic dissipation at various stages of the excavation are shown in Figure 40.5. We see that the distribution of plasticity varies quite significantly with excavation depth. The adaptive mesh procedure used ensures a high degree of accuracy while maintaining a moderate number of elements in each stage.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex27_Fig04_d3.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex27_Fig04_d3.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 148: Horizontal displacement of top of wall as function of excavation depth (top) and strut force as function of excavation depth (bottom). '''</span>
+
|+<span style="font-size:90%">'''Figure 40.4: Horizontal displacement of top of wall as function of excavation depth (top) and strut force as function of excavation depth (bottom).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,493: Line 3,633:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex27_Fig05_c.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex27_Fig05_c.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 149: Deformations (scaled by a factor of 5) and plastic dissipation for the last four excavation stages. '''</span>
+
|+<span style="font-size:90%">'''Figure 40.5: Deformations (scaled by a factor of 5) and plastic dissipation for the last four excavation stages.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,503: Line 3,643:
 
==  EXCAVATION IN SAND – MOHR-COULOMB VS HMC  ==
 
==  EXCAVATION IN SAND – MOHR-COULOMB VS HMC  ==
  
The following problem is similar to the previous one. An excavation in sand, supported by a sheet pile wall, is to be conducted. The problem is sketched in Figure 150. The analysis begins by determining the geostatic stresses (with the sheet pile ‘wished in place’). 4 m of soil is then excavated in front of the wall in the first stage followed by another 4 m in the second stage. In the third stage, a strut is inserted 2 m below ground level and another 4 m excavated bringing the total excavation depth to 12 m. In the fourth stage, the strut is prestressed to a force of 500 kN/m. Finally, in the fifth stage, another 4 m of soil is excavated bringing the final excavation depth to 16 m.
+
The following problem is similar to the previous one. An excavation in sand, supported by a sheet pile wall, is to be conducted. The problem is sketched in Figure 41.1. The analysis begins by determining the geostatic stresses (with the sheet pile ‘wished in place’). 4 m of soil is then excavated in front of the wall in the first stage followed by another 4 m in the second stage. In the third stage, a strut is inserted 2 m below ground level and another 4 m excavated bringing the total excavation depth to 12 m. In the fourth stage, the strut is prestressed to a force of 500 kN/m. Finally, in the fifth stage, another 4 m of soil is excavated bringing the final excavation depth to 16 m.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/ExcMCvsEMC01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/ExcMCvsEMC01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 150: Excavation in sand: problem setup. '''</span>
+
|+<span style="font-size:90%">'''Figure 41.1: Excavation in sand: problem setup.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
 
As discussed in the previous example, excavation analyses are a class of problems where both loading (in front of the wall) and unloading (behind the wall) takes place and may be important to capture correctly. As such, the HMC model appears well suited in that it operates with separate stiffness moduli for initial elastoplastic loading and subsequent elastic unloading/reloading. However, successful application of the HMC model to any problem hinges crucially on relevant experimental data being available.
 
As discussed in the previous example, excavation analyses are a class of problems where both loading (in front of the wall) and unloading (behind the wall) takes place and may be important to capture correctly. As such, the HMC model appears well suited in that it operates with separate stiffness moduli for initial elastoplastic loading and subsequent elastic unloading/reloading. However, successful application of the HMC model to any problem hinges crucially on relevant experimental data being available.
 +
 +
<div id="ExcMCvsHMC01_tab">
  
 
{| class="wikitable"
 
{| class="wikitable"
|+ Table 20: HMC and Mohr-Coulomb parameters for the soil considered
+
|+ Table 41.1: HMC and Mohr-Coulomb parameters for the soil considered
 
! '''HMC'''
 
! '''HMC'''
 
!
 
!
Line 3,595: Line 3,737:
 
|}
 
|}
  
In the absence of such data, one could argue that the standard Mohr-Coulomb model – with a reasonable estimate of the parameters – might be more appropriate. The purpose of the following analyses is to gauge the sensitivity of the overall behaviour to the constitutive model and in particular the choice of stiffness moduli. For this purpose, the HMC and Mohr-Coulomb models are used with the parameters shown in Table 20. While the reference stiffness moduli used in the HMC model are <math display="inline">E_{50,\text{ref}}=25</math> MPa and <math display="inline">E_{ur,\text{ref}}=125</math> MPa, two runs are performed with the Mohr-Coulomb model using fixed Young’s moduli of <math display="inline">E=25</math> MPa and <math display="inline">E=125</math> MPa. It should be noted that these do not represent actual bounds on the HMC moduli as the latter are pressure dependent.<br />
+
 
 +
</div>
 +
In the absence of such data, one could argue that the standard Mohr-Coulomb model – with a reasonable estimate of the parameters – might be more appropriate. The purpose of the following analyses is to gauge the sensitivity of the overall behaviour to the constitutive model and in particular the choice of stiffness moduli. For this purpose, the HMC and Mohr-Coulomb models are used with the parameters shown in Table 41.1. While the reference stiffness moduli used in the HMC model are <math display="inline">E_{50,\text{ref}}=25</math> MPa and <math display="inline">E_{ur,\text{ref}}=125</math> MPa, two runs are performed with the Mohr-Coulomb model using fixed Young’s moduli of <math display="inline">E=25</math> MPa and <math display="inline">E=125</math> MPa. It should be noted that these do not represent actual bounds on the HMC moduli as the latter are pressure dependent.<br />
 
All calculations are performed on a mesh with 1,000 6-node Gauss elements. While this spatial discretization is somewhat coarse, it does not affect the main conclusions of the analysis.<br />
 
All calculations are performed on a mesh with 1,000 6-node Gauss elements. While this spatial discretization is somewhat coarse, it does not affect the main conclusions of the analysis.<br />
The results of the three calculations are shown in Figures 151–153. Generally, there is good agreement between all three calculations with the Mohr-Coulomb calculation using <math display="inline">E=125</math> MPa agreeing particularly well with the HMC calculation. All in all, it appears that the discrepancy between the different models is of a magnitude not larger than that resulting from uncertainty in the material parameters, geometry, exact method of excavation, etc.<br />
+
The results of the three calculations are shown in Figures 41.2–41.4. Generally, there is good agreement between all three calculations with the Mohr-Coulomb calculation using <math display="inline">E=125</math> MPa agreeing particularly well with the HMC calculation. All in all, it appears that the discrepancy between the different models is of a magnitude not larger than that resulting from uncertainty in the material parameters, geometry, exact method of excavation, etc.<br />
 
In conclusion: while the choice of stiffness moduli does have some effect on the overall behavior, it is, for this example, relatively moderate. This conclusion is consistent with the assertion that what mainly governs the behavior are the earth pressures behind and in front of the wall. And with these being a function of the friction angle, it is in fact the strength of the soil rather than its stiffness that governs the overall behavior of the problem. Note, however, that for problems where the extent of soil failure is limited and elasticity dominates, the elastic law and/or the exact value of the elastic parameters is of paramount importance. For the present example, this is observed in the first excavation stage where the Mohr-Coulomb model with <math display="inline">E=25</math> MPa predicts an excessive heave behind the wall.
 
In conclusion: while the choice of stiffness moduli does have some effect on the overall behavior, it is, for this example, relatively moderate. This conclusion is consistent with the assertion that what mainly governs the behavior are the earth pressures behind and in front of the wall. And with these being a function of the friction angle, it is in fact the strength of the soil rather than its stiffness that governs the overall behavior of the problem. Note, however, that for problems where the extent of soil failure is limited and elasticity dominates, the elastic law and/or the exact value of the elastic parameters is of paramount importance. For the present example, this is observed in the first excavation stage where the Mohr-Coulomb model with <math display="inline">E=25</math> MPa predicts an excessive heave behind the wall.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/exc_momdisp2.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/exc_momdisp2.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 151: Wall displacements (left column) and moments (right column). '''</span>
+
|+<span style="font-size:90%">'''Figure 41.2: Wall displacements (left column) and moments (right column).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,610: Line 3,754:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/exc_set.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/exc_set.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 152: Ground displacements behind wall (left column) and in front of wall (right column). '''</span>
+
|+<span style="font-size:90%">'''Figure 41.3: Ground displacements behind wall (left column) and in front of wall (right column).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,618: Line 3,762:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/exc_def.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Exc_def.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 153: Deformations and shear dissipation for HMC and Mohr-Coulomb models. '''</span>
+
|+<span style="font-size:90%">'''Figure 41.4: Deformations and shear dissipation for HMC and Mohr-Coulomb models.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,626: Line 3,770:
 
==  EXCAVATION WITH SEEPAGE  ==
 
==  EXCAVATION WITH SEEPAGE  ==
  
The following example demonstrates the process of excavation with a gradual lowering of the groundwater table within the excavation. The problem setup is shown in Figure 154. The material is the default Medium Sand-MC, though with a Young’s modulus set to <math display="inline">E=120</math> MPa.
+
The following example demonstrates the process of excavation with a gradual lowering of the groundwater table within the excavation. The problem setup is shown in Figure 42.1. The material is the default Medium Sand-MC, though with a Young’s modulus set to <math display="inline">E=120</math> MPa.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/excdryprob.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/excdryprob.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 154: Dry excavation in sand: problem setup. '''</span>
+
|+<span style="font-size:90%">'''Figure 42.1: Dry excavation in sand: problem setup.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,638: Line 3,782:
 
The excavation is to be performed in four stages. First to a level of 2 m, coinciding with the ground water table. Then a second stage of 2 m where the ground water table within the excavation is lowered to the bottom of the excavation, i.e. 4 m below ground level. In the third stage, a Fixed End Anchor is inserted before another 2 m is excavated, again with a 2 m lowering of the groundwater table within the excavation. Finally, in the fourth stage, the excavation is taken to the final depth of 8 m and the ground water table lowered accordingly.<br />
 
The excavation is to be performed in four stages. First to a level of 2 m, coinciding with the ground water table. Then a second stage of 2 m where the ground water table within the excavation is lowered to the bottom of the excavation, i.e. 4 m below ground level. In the third stage, a Fixed End Anchor is inserted before another 2 m is excavated, again with a 2 m lowering of the groundwater table within the excavation. Finally, in the fourth stage, the excavation is taken to the final depth of 8 m and the ground water table lowered accordingly.<br />
 
In OPTUM G2, seepage is automatically included as part of any analysis and is performed before any mechanical analysis. As such, the lowering of the groundwater table does not need to be considered in anymore detail than simply prescribing the groundwater table at the bottom of the excavation in each stage. This is most easily done using the Water Table Tool located under Flow BCs in Features, but may also be done using the Fixed Pressure or Fixed Head boundary conditions.<br />
 
In OPTUM G2, seepage is automatically included as part of any analysis and is performed before any mechanical analysis. As such, the lowering of the groundwater table does not need to be considered in anymore detail than simply prescribing the groundwater table at the bottom of the excavation in each stage. This is most easily done using the Water Table Tool located under Flow BCs in Features, but may also be done using the Fixed Pressure or Fixed Head boundary conditions.<br />
All stages are run using 2,000 6-node Gauss elements with 3 Adaptivity iterations. The results in terms of pressure heads and deformations are shown in Figure 155.
+
All stages are run using 2,000 6-node Gauss elements with 3 Adaptivity iterations. The results in terms of pressure heads and deformations are shown in Figure 42.2.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/excdrydefp.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/excdrydefp.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 155: Pressure heads (m) and deformations (scaled by a factor of 10) with shear dissipation. '''</span>
+
|+<span style="font-size:90%">'''Figure 42.2: Pressure heads (m) and deformations (scaled by a factor of 10) with shear dissipation.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,650: Line 3,794:
 
==  EXCAVATION IN MOHR-COULOMB CLAY  ==
 
==  EXCAVATION IN MOHR-COULOMB CLAY  ==
  
The following example concerns an excavation in clay as sketched in Figure 156. The three layers consist of the default materials Soft Clay-MC, Firm Clay-MC, and Stiff Clay-MC. The groundwater table is located 1 m below ground level.
+
The following example concerns an excavation in clay as sketched in Figure 43.1. The three layers consist of the default materials Soft Clay-MC, Firm Clay-MC, and Stiff Clay-MC. The groundwater table is located 1 m below ground level.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/excclay00_2.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/excclay00_2.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 156: Excavation in Mohr-Coulomb clay: problem setup. '''</span>
+
|+<span style="font-size:90%">'''Figure 43.1: Excavation in Mohr-Coulomb clay: problem setup.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
 
Besides the initial stage where the geostatic stress field is computed, the problem comprises three stages, each involving 2 m of excavation. The material is here assumed to behave in an undrained manner, i.e. the Time Scope is set to Short Term. Additionally, the seepage pressures must be considered. For a fine grained material, it is reasonable to assume that the initial hydrostatic seepage pressure does not change in the short term. Appropriate boundary conditions must therefore be imposed to ensure that the initial seepage pressure distribution remains unchanged. In the present example, where the wall is considered impermeable, this is done by imposing no-flow boundary conditions at the bottom of the excavation in each stage.<br />
 
Besides the initial stage where the geostatic stress field is computed, the problem comprises three stages, each involving 2 m of excavation. The material is here assumed to behave in an undrained manner, i.e. the Time Scope is set to Short Term. Additionally, the seepage pressures must be considered. For a fine grained material, it is reasonable to assume that the initial hydrostatic seepage pressure does not change in the short term. Appropriate boundary conditions must therefore be imposed to ensure that the initial seepage pressure distribution remains unchanged. In the present example, where the wall is considered impermeable, this is done by imposing no-flow boundary conditions at the bottom of the excavation in each stage.<br />
Finally, at a depth of 6 m, the excavation is left to consolidate with the groundwater table being maintained at the bottom of the excavation. In other words, the transition from short term conditions to long term conditions must be accounted for. This is done via a final stage where Time Scope is set to Long Term. The evolution of seepage pressures and the different boundary conditions imposed at the bottom of the excavation are shown in Figure 157.<br />
+
Finally, at a depth of 6 m, the excavation is left to consolidate with the groundwater table being maintained at the bottom of the excavation. In other words, the transition from short term conditions to long term conditions must be accounted for. This is done via a final stage where Time Scope is set to Long Term. The evolution of seepage pressures and the different boundary conditions imposed at the bottom of the excavation are shown in Figure 43.2.<br />
The deformations at various stages of the excavation are shown in Figure 158. We see that the deformations increase dramatically in the long term. Furthermore, the factor of safety against collapse decreases markedly (see Figure 159) and the excavation is in fact barely stable in the long term.
+
The deformations at various stages of the excavation are shown in Figure 43.3. We see that the deformations increase dramatically in the long term. Furthermore, the factor of safety against collapse decreases markedly (see Figure 43.4) and the excavation is in fact barely stable in the long term.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/excclay01_2.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/excclay01_2.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 157: Pore pressure distributions. '''</span>
+
|+<span style="font-size:90%">'''Figure 43.2: Pore pressure distributions.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,674: Line 3,818:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/excclaydef.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/excclaydef.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 158: Deformations (scaled by a factor of 5) and distribution of shear dissipation. '''</span>
+
|+<span style="font-size:90%">'''Figure 43.3: Deformations (scaled by a factor of 5) and distribution of shear dissipation.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,682: Line 3,826:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/excclayFS.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/excclayFS.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 159: Strength based factor of safety versus excavation depth. At a depth of 6 m the factor of safety decreases from about 1.3 in the short term to about 1.06 in the long term. '''</span>
+
|+<span style="font-size:90%">'''Figure 43.4: Strength based factor of safety versus excavation depth. At a depth of 6 m the factor of safety decreases from about 1.3 in the short term to about 1.06 in the long term.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,690: Line 3,834:
 
==  PRELOADED FOOTING ON CLAY  ==
 
==  PRELOADED FOOTING ON CLAY  ==
  
This problem considers the following scenario. A building was originally erected on strip footings such as the one shown in Figure 160. The soil was of the kind that could be described reasonably well by the default Firm Clay-MC material. The building was erected sufficiently rapidly for its loading of the foundations to be considered undrained. At the end construction, the building exerted a load of <math display="inline">q=45</math> kN/m<math display="inline">^2</math> on the foundations. This corresponds to approximately 70% of the short term bearing capacity. Many years then pass until one day it is decided to extend the building by another storey. The question is: can this be done?
+
This problem considers the following scenario. A building was originally erected on strip footings such as the one shown in Figure 44.1. The soil was of the kind that could be described reasonably well by the default Firm Clay-MC material. The building was erected sufficiently rapidly for its loading of the foundations to be considered undrained. At the end construction, the building exerted a load of <math display="inline">q=45</math> kN/m<math display="inline">^2</math> on the foundations. This corresponds to approximately 70% of the short term bearing capacity. Many years then pass until one day it is decided to extend the building by another storey. The question is: can this be done?
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/preload1_1.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/preload1_1.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 160: Problem setup (initial short term loading). '''</span>
+
|+<span style="font-size:90%">'''Figure 44.1: Problem setup (initial short term loading).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,702: Line 3,846:
 
To answer this question it should first be clarified that by ’many years having passed’, is meant a sufficient amount of time for all the excess pore pressures that were generated in response the initial loading to have dissipated. Assuming that to be the case, the effective stresses in the ground will most likely have increased and it is likely that the new short term bearing capacity in effect has been increased. To see this, recall that the undrained shear strength implied by the Mohr-Coulomb model is given by  
 
To answer this question it should first be clarified that by ’many years having passed’, is meant a sufficient amount of time for all the excess pore pressures that were generated in response the initial loading to have dissipated. Assuming that to be the case, the effective stresses in the ground will most likely have increased and it is likely that the new short term bearing capacity in effect has been increased. To see this, recall that the undrained shear strength implied by the Mohr-Coulomb model is given by  
  
<math display="block">s_u = c\cos\phi - \frac{1}{2}(\sigma_{x,0}'+\sigma_{y,0}')\sin\phi</math>
+
<math display="block">(44.1)\qquad
 +
    s_u = c\cos\phi - \frac{1}{2}(\sigma_{x,0}'+\sigma_{y,0}')\sin\phi</math>
  
where <math display="inline">-(\sigma_{x,0}'+\sigma_{y,0}')</math> is the initial effective pressure in the ground, i.e. the pressure in the ground prior to the (possible) extension of the building. With this quantity undoubtedly having increased, especially underneath the foundation, the undrained shear strength and thereby the overall bearing capacity must necessarily also have increased. The distributions of undrained shear strength prior to the original construction and prior to the planned construction are shown in Figure 161 (the undrained shear strength distribution is available under Initial Stresses in Results).<br />
+
where <math display="inline">-(\sigma_{x,0}'+\sigma_{y,0}')</math> is the initial effective pressure in the ground, i.e. the pressure in the ground prior to the (possible) extension of the building. With this quantity undoubtedly having increased, especially underneath the foundation, the undrained shear strength and thereby the overall bearing capacity must necessarily also have increased. The distributions of undrained shear strength prior to the original construction and prior to the planned construction are shown in Figure 44.2 (the undrained shear strength distribution is available under Initial Stresses in Results).<br />
 
The entire analysis is carried out using three stages. First a stage where the original 45 kN/m<math display="inline">^2</math> is applied in the short term. Then a stage where the original loading is maintained but where Time Scope = Long Term. This stage uses the first stage as From stage. Finally, a Limit Analysis stage that uses the previous stage as From stage and has Time Scope = Short Term. The result is an additional bearing capacity of 52 kN/m<math display="inline">^2</math>. In other words, the original bearing capacity (<math display="inline">q_u\approx 64</math> kN/m<math display="inline">^2</math>) has increased by some 50% (to <math display="inline">45+52=97</math> kN/m<math display="inline">^2</math>). This increase corresponds very well to that reported by <bib id="Lehane:2003" /> for a field study concerning a scenario similar to the one described in this example.
 
The entire analysis is carried out using three stages. First a stage where the original 45 kN/m<math display="inline">^2</math> is applied in the short term. Then a stage where the original loading is maintained but where Time Scope = Long Term. This stage uses the first stage as From stage. Finally, a Limit Analysis stage that uses the previous stage as From stage and has Time Scope = Short Term. The result is an additional bearing capacity of 52 kN/m<math display="inline">^2</math>. In other words, the original bearing capacity (<math display="inline">q_u\approx 64</math> kN/m<math display="inline">^2</math>) has increased by some 50% (to <math display="inline">45+52=97</math> kN/m<math display="inline">^2</math>). This increase corresponds very well to that reported by <bib id="Lehane:2003" /> for a field study concerning a scenario similar to the one described in this example.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/preload1_2.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/preload1_2.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 161: Undrained shear strength (kPa) before original construction (top) and prior to new construction (bottom). '''</span>
+
|+<span style="font-size:90%">'''Figure 44.2: Undrained shear strength (kPa) before original construction (top) and prior to new construction (bottom).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,719: Line 3,864:
 
==  SHALLOW FOUNDATION IN CLAY: EFFECTS OF PRELOADING  ==
 
==  SHALLOW FOUNDATION IN CLAY: EFFECTS OF PRELOADING  ==
  
The following example is similar to the previous one and is inspired by the work of <bib id="Gourvenec:2014" />. It considers the effect of preloading on the bearing capacity of a shallow foundation as shown in Figure 162. The material is Modified Cam Clay with <math display="inline">\phi=30^\circ</math>, <math display="inline">e_0=1.2</math>, <math display="inline">\kappa=0.044</math>, <math display="inline">\lambda=0.205</math>, <math display="inline">\nu=0.25</math> and <math display="inline">\gamma=18</math> kN/m<math display="inline">^2</math>. The foundation is of Rigid material with a unit weight of <math display="inline">\gamma=20</math> kN/m<math display="inline">^2</math>.
+
The following example is similar to the previous one and is inspired by the work of <bib id="Gourvenec:2014" />. It considers the effect of preloading on the bearing capacity of a shallow foundation as shown in Figure 45.1. The material is Modified Cam Clay with <math display="inline">\phi=30^\circ</math>, <math display="inline">e_0=1.2</math>, <math display="inline">\kappa=0.044</math>, <math display="inline">\lambda=0.205</math>, <math display="inline">\nu=0.25</math> and <math display="inline">\gamma=18</math> kN/m<math display="inline">^2</math>. The foundation is of Rigid material with a unit weight of <math display="inline">\gamma=20</math> kN/m<math display="inline">^2</math>.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/MCCPreload.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/MCCPreload.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 162: Shallow foundation in a Modified Cam Clay soil. '''</span>
+
|+<span style="font-size:90%">'''Figure 45.1: Shallow foundation in a Modified Cam Clay soil.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,732: Line 3,877:
 
In all analyses, the earth pressure coefficient is taken as:  
 
In all analyses, the earth pressure coefficient is taken as:  
  
<math display="block">K_0 = (1-\sin\phi)\text{OCR}^{\sin{\phi}}</math>
+
<math display="block">(45.1)\qquad
 +
    K_0 = (1-\sin\phi)\text{OCR}^{\sin{\phi}}</math>
  
 
Besides the initial Limit Analysis to determine the bearing capacity without preloading, each analysis comprises two stages:
 
Besides the initial Limit Analysis to determine the bearing capacity without preloading, each analysis comprises two stages:
Line 3,740: Line 3,886:
  
 
Both stages are run using 2,000 6-node Gauss elements with three Adaptivity Iterations used for the Limit Analysis stage.<br />
 
Both stages are run using 2,000 6-node Gauss elements with three Adaptivity Iterations used for the Limit Analysis stage.<br />
The results of the analyses are shown in Figure 163. As seen, the relative gain in bearing capacity decreases for increasing OCR. This result, which is in agreement with <bib id="Gourvenec:2014" />, is not surprising. While preloading will tend to increase the bearing capacity by virtue of the stresses in the ground increasing, the gain is greater when the OCR is also increased, i.e. when the yield surface hardens, and thereby expands, as a result of the preloading.
+
The results of the analyses are shown in Figure 45.2. As seen, the relative gain in bearing capacity decreases for increasing OCR. This result, which is in agreement with <bib id="Gourvenec:2014" />, is not surprising. While preloading will tend to increase the bearing capacity by virtue of the stresses in the ground increasing, the gain is greater when the OCR is also increased, i.e. when the yield surface hardens, and thereby expands, as a result of the preloading.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/mccpreres2.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/mccpreres2.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 163: Relative preload versus relative bearing capacity for different OCR. '''</span>
+
|+<span style="font-size:90%">'''Figure 45.2: Relative preload versus relative bearing capacity for different OCR.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,754: Line 3,900:
 
==  TUNNELING USING CONVERGENCE-CONFINEMENT METHOD  ==
 
==  TUNNELING USING CONVERGENCE-CONFINEMENT METHOD  ==
  
The following example considers the construction of a circular lined tunnel in weathered rock as shown in Figure 164. The rock is modeled as a Mohr-Coulomb material and the lining is modeled using Plate elements. Both sets of material parameters are shown in the figure below.
+
The following example considers the construction of a circular lined tunnel in weathered rock as shown in Figure 46.1. The rock is modeled as a Mohr-Coulomb material and the lining is modeled using Plate elements. Both sets of material parameters are shown in the figure below.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/Ex41_Fig01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Ex41_Fig01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 164: Circular tunnel. '''</span>
+
|+<span style="font-size:90%">'''Figure 46.1: Circular tunnel.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,766: Line 3,912:
 
Tunneling is an inherently three-dimensional problem and should ideally be modeled as such. However, a number of plane strain approaches have been developed and remain widely used in practice. These include the convergence-confinement, the progressive softening method, and volume loss control method. A thorough review of these methods is given by <bib id="Potts:2001" />.<br />
 
Tunneling is an inherently three-dimensional problem and should ideally be modeled as such. However, a number of plane strain approaches have been developed and remain widely used in practice. These include the convergence-confinement, the progressive softening method, and volume loss control method. A thorough review of these methods is given by <bib id="Potts:2001" />.<br />
 
The approximate plane strain methods all consider a cross section of the tunnel perpendicular to the direction of excavation. The critical issue is what cross section should be considered. One extreme would be to first excavate the tunnel and then, once a new stress state has been obtained, to insert the lining. Another extreme would be to have the lining wished in place from the outset and then excavate the soil inside the tunnel. In the former case, the ground settlements would be overestimated and the the latter the sectional forces in the lining would be overestimated. It is noted that both models effectively neglect 3D effects – the former by assuming an infinitely long unsupported tunnel and the latter by assuming an infinitely long tunnel supported at all cross sections. The approximate plane strain mentioned above methods all seek to account for 3D effects by establishing a reasonable compromise between these extremes.<br />
 
The approximate plane strain methods all consider a cross section of the tunnel perpendicular to the direction of excavation. The critical issue is what cross section should be considered. One extreme would be to first excavate the tunnel and then, once a new stress state has been obtained, to insert the lining. Another extreme would be to have the lining wished in place from the outset and then excavate the soil inside the tunnel. In the former case, the ground settlements would be overestimated and the the latter the sectional forces in the lining would be overestimated. It is noted that both models effectively neglect 3D effects – the former by assuming an infinitely long unsupported tunnel and the latter by assuming an infinitely long tunnel supported at all cross sections. The approximate plane strain mentioned above methods all seek to account for 3D effects by establishing a reasonable compromise between these extremes.<br />
In the following, the convergence-confinement method is used. The basic principles are as follows (see Figure 165). From an initial state, the tunnel is excavated while keeping the perimeter of the tunnel fully supported. This induces no changes to the stress state and hence no deformations. Next, the reactions on the tunnel perimeter are relaxed, i.e. reduced by a factor <math display="inline">\lambda</math> where <math display="inline">0\leq\lambda\leq 1</math>. Finally, the lining is inserted and all supports around the tunnel perimeter are removed. This induces further settlements.<br />
+
In the following, the convergence-confinement method is used. The basic principles are as follows (see Figure 46.2). From an initial state, the tunnel is excavated while keeping the perimeter of the tunnel fully supported. This induces no changes to the stress state and hence no deformations. Next, the reactions on the tunnel perimeter are relaxed, i.e. reduced by a factor <math display="inline">\lambda</math> where <math display="inline">0\leq\lambda\leq 1</math>. Finally, the lining is inserted and all supports around the tunnel perimeter are removed. This induces further settlements.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/TunnelRelax01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/TunnelRelax01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 165: Schematics of the convergence-confinement method [after <bib id="Vermeer:2003 " />]. '''</span>
+
|+<span style="font-size:90%">'''Figure 46.2: Schematics of the convergence-confinement method [after <bib id="Vermeer:2003" />].'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
In OPTUM G2, the convergence-confinement method may be set up using three stages following Figure 165:
+
In OPTUM G2, the convergence-confinement method may be set up using three stages following Figure 46.2:
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/tunnel03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/tunnel03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 166: Support relaxation in OPTUM G2. '''</span>
+
|+<span style="font-size:90%">'''Figure 46.3: Support relaxation in OPTUM G2.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
 
# An initial stage where the initial stresses are calculated using an Initial Stress analysis.
 
# An initial stage where the initial stresses are calculated using an Initial Stress analysis.
# A second stage where the tunnel is excavated. The tunnel perimeter is here fully supported and a Relaxation Factor, <math display="inline">\lambda</math>, is specified (see Figure 166).
+
# A second stage where the tunnel is excavated. The tunnel perimeter is here fully supported and a Relaxation Factor, <math display="inline">\lambda</math>, is specified (see Figure 46.3).
 
# A final stage where the perimeter supports are replaced by a Plate to model the lining.
 
# A final stage where the perimeter supports are replaced by a Plate to model the lining.
  
The crucial issue in the convergence-confinement method is the selection of the relaxation factor <math display="inline">\lambda</math>. In practice it is either selected on the basis of experience and/or field calibration or by considering two values at either end of the spectrum, for example <math display="inline">\lambda = 0.3</math> and <math display="inline">\lambda = 0.7</math>, the idea being that the smaller value tends to overestimate the ground settlements while the larger value tends to overestimate the sectional forces in the lining. More generally, to aid in establishing the appropriate value of the relaxation factor, various quantities of interest can be examined as function of <math display="inline">\lambda</math>. Some examples are shown in Figures 167-5 where the ground settlement above the center of the tunnel and the maximum bending moment in the lining are plotted as function of <math display="inline">\lambda</math>.
+
The crucial issue in the convergence-confinement method is the selection of the relaxation factor <math display="inline">\lambda</math>. In practice it is either selected on the basis of experience and/or field calibration or by considering two values at either end of the spectrum, for example <math display="inline">\lambda = 0.3</math> and <math display="inline">\lambda = 0.7</math>, the idea being that the smaller value tends to overestimate the ground settlements while the larger value tends to overestimate the sectional forces in the lining. More generally, to aid in establishing the appropriate value of the relaxation factor, various quantities of interest can be examined as function of <math display="inline">\lambda</math>. Some examples are shown in Figures 46.4-5 where the ground settlement above the center of the tunnel and the maximum bending moment in the lining are plotted as function of <math display="inline">\lambda</math>.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/tunnel04_1.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/tunnel04_1.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 167: Ground settlement (<math display="inline">\Delta u_1</math>) as function of support relaxation factor. '''</span>
+
|+<span style="font-size:90%">'''Figure 46.4: Ground settlement (<math display="inline">\Delta u_1</math>) as function of support relaxation factor.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,803: Line 3,949:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/tunnel04_2.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/tunnel04_2.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 168: Bending moment as function of support relaxation factor. '''</span>
+
|+<span style="font-size:90%">'''Figure 46.5: Bending moment as function of support relaxation factor.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
  The final vertical displacements (after introduction of the lining) for a value of <math display="inline">\lambda = 0.3</math> are shown in Figure 169 while the normal forces and bending moments are shown in Figure 170.
+
  The final vertical displacements (after introduction of the lining) for a value of <math display="inline">\lambda = 0.3</math> are shown in Figure 46.6 while the normal forces and bending moments are shown in Figure 46.7.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/tunnel05.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/tunnel05.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 169: Final vertical displacements for <math display="inline">\lambda=0.3</math> (deformations scaled by a factor of 50). '''</span>
+
|+<span style="font-size:90%">'''Figure 46.6: Final vertical displacements for <math display="inline">\lambda=0.3</math> (deformations scaled by a factor of 50).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,821: Line 3,967:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/tunnel06.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/tunnel06.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 170: Normal forces and bending moments in lining for <math display="inline">\lambda=0.3</math>. '''</span>
+
|+<span style="font-size:90%">'''Figure 46.7: Normal forces and bending moments in lining for <math display="inline">\lambda=0.3</math>.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,831: Line 3,977:
 
==  EXCAVATION WITH UNDERDRAINAGE  ==
 
==  EXCAVATION WITH UNDERDRAINAGE  ==
  
The following example considers the excavation in a soil where the steady state seepage pressures deviate from the usual hydrostatic distribution. In particular, an under-drained pressure profile as indicated in Figure 171 is considered. This type of profile is commonly observed in the London area [see <bib id="Potts:2001" /> and references therein]
+
The following example considers the excavation in a soil where the steady state seepage pressures deviate from the usual hydrostatic distribution. In particular, an under-drained pressure profile as indicated in Figure 47.1 is considered. This type of profile is commonly observed in the London area [see <bib id="Potts:2001" /> and references therein]
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/ExcUnder01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/ExcUnder01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 171: Problem setup (top) and specification of user defined seepage pressure profile. The Flow BCs (Water Table and No-Flow conditions) are ignored when a user defined seepage pressure profile is specified. '''</span>
+
|+<span style="font-size:90%">'''Figure 47.1: Problem setup (top) and specification of user defined seepage pressure profile. The Flow BCs (Water Table and No-Flow conditions) are ignored when a user defined seepage pressure profile is specified.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The profile indicated in Figure 171 is defined by a set of points giving the pressure as function of depth, i.e. the <math display="inline">y</math> coordinate. The steps needed to import an arbitrary distribution of seepage pressure are illustrated in the figure above. In the lower half of stage manager under Advanced, Seepage Pressure = User is selected. The button on the right hand side of the field below opens the dialog shown on the right in the figure. Using the Import button, a file with the data can be loaded into the project. The input file must contain two columns with the <math display="inline">y</math> coordinate and the pressure value No particular ordering of the coordinates is needed, but the data are expected to cover the entire problem domain. Hence, a point at the very top of the problem domain is needed to specify that <math display="inline">p_s</math> in the upper sand layer. It is important that <math display="inline">p_s<0</math> be specified in areas that are deemed to be above the ground water table – a pore pressure of <math display="inline">p_s=0</math> will be interpreted a degree of saturation <math display="inline">S=1</math>. The data can be unloaded via the red cross next to the button that opens the Material Parameter window.<br />
+
The profile indicated in Figure 47.1 is defined by a set of points giving the pressure as function of depth, i.e. the <math display="inline">y</math> coordinate. The steps needed to import an arbitrary distribution of seepage pressure are illustrated in the figure above. In the lower half of stage manager under Advanced, Seepage Pressure = User is selected. The button on the right hand side of the field below opens the dialog shown on the right in the figure. Using the Import button, a file with the data can be loaded into the project. The input file must contain two columns with the <math display="inline">y</math> coordinate and the pressure value No particular ordering of the coordinates is needed, but the data are expected to cover the entire problem domain. Hence, a point at the very top of the problem domain is needed to specify that <math display="inline">p_s</math> in the upper sand layer. It is important that <math display="inline">p_s<0</math> be specified in areas that are deemed to be above the ground water table – a pore pressure of <math display="inline">p_s=0</math> will be interpreted a degree of saturation <math display="inline">S=1</math>. The data can be unloaded via the red cross next to the button that opens the Material Parameter window.<br />
The problem is analyzed as indicated in Figure 171 with an Initial Stress stage followed by four excavation stages of type Elastoplastic. Anchors are inserted in the third and fourth excavation stages. Finally, after the last excavation stage, a short term Strength Reduction analysis is carried out.<br />
+
The problem is analyzed as indicated in Figure 47.1 with an Initial Stress stage followed by four excavation stages of type Elastoplastic. Anchors are inserted in the third and fourth excavation stages. Finally, after the last excavation stage, a short term Strength Reduction analysis is carried out.<br />
In addition to the user defined profile, the problem is also run for a standard hydrostatic seepage pressure distribution, i.e. by selecting Seepage Pressures = Auto under Advanced in the stage manager. The results of the analyses in terms of the maximum horizontal displacement of the wall are shown in Figure 172. We see that the displacements are somewhat larger for the hydrostatic distribution.
+
In addition to the user defined profile, the problem is also run for a standard hydrostatic seepage pressure distribution, i.e. by selecting Seepage Pressures = Auto under Advanced in the stage manager. The results of the analyses in terms of the maximum horizontal displacement of the wall are shown in Figure 47.2. We see that the displacements are somewhat larger for the hydrostatic distribution.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/eudisp.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/eudisp.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 172: Maximum displacement of wall versus excavation depth. '''</span>
+
|+<span style="font-size:90%">'''Figure 47.2: Maximum displacement of wall versus excavation depth.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,855: Line 4,001:
 
The Strength Reduction analysis give the following results:  
 
The Strength Reduction analysis give the following results:  
  
<math display="block">\begin{array}{lcl}
+
<math display="block">(47.1)\qquad 
 +
 
 +
    \begin{array}{lcl}
 
     \text{Hydrostatic:}&~& \text{FS}_s = 1.03\pm 0.02\\
 
     \text{Hydrostatic:}&~& \text{FS}_s = 1.03\pm 0.02\\
 
     \text{Under-drained:}&~& \text{FS}_s = 1.16\pm 0.03
 
     \text{Under-drained:}&~& \text{FS}_s = 1.16\pm 0.03
Line 3,866: Line 4,014:
 
==  ONE-DIMENSIONAL CONSOLIDATION  ==
 
==  ONE-DIMENSIONAL CONSOLIDATION  ==
  
The following example introduces the various possibilities of the Consolidation analysis implemented in OPTUM G2. The problem is shown in Figure 173 and involves a 1 m by 1 m block of elastic material. The Young’s modulus is 30 MPa and the Poisson’s ratio has been set to <math display="inline">\nu=0</math> to facilitate a direct comparison with known analytical solutions. The block consolidates under a load of <math display="inline">q=100</math> kN/m<math display="inline">^2</math>.
+
The following example introduces the various possibilities of the Consolidation analysis implemented in OPTUM G2. The problem is shown in Figure 48.1 and involves a 1 m by 1 m block of elastic material. The Young’s modulus is 30 MPa and the Poisson’s ratio has been set to <math display="inline">\nu=0</math> to facilitate a direct comparison with known analytical solutions. The block consolidates under a load of <math display="inline">q=100</math> kN/m<math display="inline">^2</math>.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/consol_setup.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/Consol_setup.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 173: One-dimensional consolidation problem. '''</span>
+
|+<span style="font-size:90%">'''Figure 48.1: One-dimensional consolidation problem.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
 
The analysis proceeds by selecting Consolidation in the Analysis column of the stage manager. The stage settings then appear in the lower half of the window. As for all other analysis types, the Element Type and No of Elements must be selected. For this problem, 100 6-node Gauss elements are used, but any of the available elements are in principle applicable.<br />
 
The analysis proceeds by selecting Consolidation in the Analysis column of the stage manager. The stage settings then appear in the lower half of the window. As for all other analysis types, the Element Type and No of Elements must be selected. For this problem, 100 6-node Gauss elements are used, but any of the available elements are in principle applicable.<br />
The second category of the stage settings, Time Stepping, contains various options and parameters to control the time stepping. Three different time stepping schemes are available: Auto, Target/Time, Target/Degree (see Figure 174).
+
The second category of the stage settings, Time Stepping, contains various options and parameters to control the time stepping. Three different time stepping schemes are available: Auto, Target/Time, Target/Degree (see Figure 48.2).
  
 
<center>
 
<center>
Line 3,883: Line 4,031:
 
|<div class="image600px">https://wiki.optumce.com/figures/consol03.png</div>
 
|<div class="image600px">https://wiki.optumce.com/figures/consol03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 174: Time stepping settings: Auto (left), Target/Time (center), and Target/Degree (right). '''</span>
+
|+<span style="font-size:90%">'''Figure 48.2: Time stepping settings: Auto (left), Target/Time (center), and Target/Degree (right).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,901: Line 4,049:
 
Firstly, with the short and long term states available from Steps 2 and 3, a work based degree of consolidation can be defined as  
 
Firstly, with the short and long term states available from Steps 2 and 3, a work based degree of consolidation can be defined as  
  
<math display="block">U_W(t) = \frac{W(t)-W_{ST}}{W_{LT}-W_{ST}}\times 100\%</math>
+
<math display="block">(48.1)\qquad
 +
    U_W(t) = \frac{W(t)-W_{ST}}{W_{LT}-W_{ST}}\times 100\%</math>
  
 
where <math display="inline">W(t)</math> is the work at time <math display="inline">t</math> and <math display="inline">W_{ST}</math> and <math display="inline">W_{LT}</math> are the short and long term work respectively. This is perhaps the best general measure of the degree of consolidation. However, a complication is that <math display="inline">W_{LT}</math> is computed using a single step while the actual <math display="inline">W_{LT}</math> will depend on the exact time dependent process to reach the long term state. As such, the computed single step estimate may be somewhat inaccurate and values of <math display="inline">U_W</math> slightly above or below 100% at full consolidation may be encountered.<br />
 
where <math display="inline">W(t)</math> is the work at time <math display="inline">t</math> and <math display="inline">W_{ST}</math> and <math display="inline">W_{LT}</math> are the short and long term work respectively. This is perhaps the best general measure of the degree of consolidation. However, a complication is that <math display="inline">W_{LT}</math> is computed using a single step while the actual <math display="inline">W_{LT}</math> will depend on the exact time dependent process to reach the long term state. As such, the computed single step estimate may be somewhat inaccurate and values of <math display="inline">U_W</math> slightly above or below 100% at full consolidation may be encountered.<br />
 
For cases where the loading is applied to a rigid footing, the work based degree of consolidation is identical to that based on settlements:  
 
For cases where the loading is applied to a rigid footing, the work based degree of consolidation is identical to that based on settlements:  
  
<math display="block">U_u(t) = \frac{u(t)-u_{ST}}{u_{LT}-u_{ST}}\times 100\%</math>
+
<math display="block">(48.2)\qquad
 +
    U_u(t) = \frac{u(t)-u_{ST}}{u_{LT}-u_{ST}}\times 100\%</math>
  
 
where <math display="inline">u(t)</math> is the vertical settlement of the foundation and <math display="inline">u_{ST}</math> and <math display="inline">u_{LT}</math> are the short and long term settlements respectively. Again, the problem with this measure, as well as other displacement based measures, is that the final state is not readily estimated a priory.<br />
 
where <math display="inline">u(t)</math> is the vertical settlement of the foundation and <math display="inline">u_{ST}</math> and <math display="inline">u_{LT}</math> are the short and long term settlements respectively. Again, the problem with this measure, as well as other displacement based measures, is that the final state is not readily estimated a priory.<br />
 
Another quantity of interest is the average normalized excess pore pressure:  
 
Another quantity of interest is the average normalized excess pore pressure:  
  
<math display="block">\frac{P_e(t)}{P_{e,0}} = \frac{\displaystyle\int_V p_e(x,y,t) \text{d}V}{\displaystyle\int_V p_{e,0}(x,y) \text{d}V}</math>
+
<math display="block">(48.3)\qquad
 +
    \frac{P_e(t)}{P_{e,0}} = \frac{\displaystyle\int_V p_e(x,y,t) \text{d}V}{\displaystyle\int_V p_{e,0}(x,y) \text{d}V}</math>
  
 
where <math display="inline">p_{e,0}(x,y)</math> is the pore pressure distribution at <math display="inline">t=0</math> (computed in Step 2 of the above) and <math display="inline">p_{e}(x,y,t)</math> is the excess pore pressure distribution at time <math display="inline">t</math>. On the basis of this quantity, the degree of consolidation may defined as:  
 
where <math display="inline">p_{e,0}(x,y)</math> is the pore pressure distribution at <math display="inline">t=0</math> (computed in Step 2 of the above) and <math display="inline">p_{e}(x,y,t)</math> is the excess pore pressure distribution at time <math display="inline">t</math>. On the basis of this quantity, the degree of consolidation may defined as:  
  
<math display="block">U_P(t) = \left(1-\frac{P_e(t)}{P_{e,0}}\right)\times 100\%</math>
+
<math display="block">(48.4)\qquad
 +
    U_P(t) = \left(1-\frac{P_e(t)}{P_{e,0}}\right)\times 100\%</math>
  
 
This definition of the degree of consolidation tends to differ slightly from the work based degree of consolidation, <math display="inline">U_W</math>.<br />
 
This definition of the degree of consolidation tends to differ slightly from the work based degree of consolidation, <math display="inline">U_W</math>.<br />
Line 3,925: Line 4,077:
 
The problem of one-dimensional consolidation can be described in terms of a diffusion type equation involving the excess pore pressures as variables:  
 
The problem of one-dimensional consolidation can be described in terms of a diffusion type equation involving the excess pore pressures as variables:  
  
<math display="block">\frac{\partial p_e}{\partial t} = \left(C_v\frac{\partial^2p_e}{\partial y^2}\right)</math>
+
<math display="block">(48.5)\qquad
 +
    \frac{\partial p_e}{\partial t} = \left(C_v\frac{\partial^2p_e}{\partial y^2}\right)</math>
  
 
where the coefficient of consolidation, <math display="inline">C_v</math>, is given by  
 
where the coefficient of consolidation, <math display="inline">C_v</math>, is given by  
  
<math display="block">C_v = \frac{K E}{\gamma_w}</math>
+
<math display="block">(48.6)\qquad
 +
    C_v = \frac{K E}{\gamma_w}</math>
  
 
with <math display="inline">K</math> being the permeability and <math display="inline">\gamma_w</math> (<math display="inline">=9.8</math> kN/m<math display="inline">^3</math>) the unit weight of water.<br />
 
with <math display="inline">K</math> being the permeability and <math display="inline">\gamma_w</math> (<math display="inline">=9.8</math> kN/m<math display="inline">^3</math>) the unit weight of water.<br />
 
The solution to this equation, for constant <math display="inline">C_v</math>, is given by  
 
The solution to this equation, for constant <math display="inline">C_v</math>, is given by  
  
<math display="block">\frac{p_e(y,t)}{p_{e,0}}= \frac{p_e(y,t)}{q} = \frac{4}{\pi}\sum_{i=1}^\infty \frac{(-1)^{i-1}}{2i-1}\cos\left((2i-1)\frac{\pi}{2}\frac{y}{H}\right)\exp\left(-(2i-1)^2\frac{\pi^2}{4}\frac{C_v}{H^2}t\right)</math>
+
<math display="block">(48.7)\qquad
 +
    \frac{p_e(y,t)}{p_{e,0}}= \frac{p_e(y,t)}{q} = \frac{4}{\pi}\sum_{i=1}^\infty \frac{(-1)^{i-1}}{2i-1}\cos\left((2i-1)\frac{\pi}{2}\frac{y}{H}\right)\exp\left(-(2i-1)^2\frac{\pi^2}{4}\frac{C_v}{H^2}t\right)</math>
  
 
with <math display="inline">H</math> being the height of the block (<math display="inline">H=1</math> m in the present case).<br />
 
with <math display="inline">H</math> being the height of the block (<math display="inline">H=1</math> m in the present case).<br />
 
With the excess pore pressures determined, the effective vertical stress and thereby the vertical settlement can be determined as:  
 
With the excess pore pressures determined, the effective vertical stress and thereby the vertical settlement can be determined as:  
  
<math display="block">\frac{u_y(y,t)}{H} = \frac{q}{E}\left[1-\frac{8}{\pi^2}\sum_{i=1}^\infty \frac{(-1)^{i-1}}{(2i-1)^2}\sin\left((2i-1)\frac{\pi}{2}\frac{y}{H}\right)\exp\left(-(2i-1)^2\frac{\pi^2}{4}\frac{C_v}{H^2}t\right)\right]</math>
+
<math display="block">(48.8)\qquad
 +
    \frac{u_y(y,t)}{H} = \frac{q}{E}\left[1-\frac{8}{\pi^2}\sum_{i=1}^\infty \frac{(-1)^{i-1}}{(2i-1)^2}\sin\left((2i-1)\frac{\pi}{2}\frac{y}{H}\right)\exp\left(-(2i-1)^2\frac{\pi^2}{4}\frac{C_v}{H^2}t\right)\right]</math>
  
 
 
 
 
Line 3,945: Line 4,101:
 
===  Results  ===
 
===  Results  ===
  
The results of the analysis using the Auto scheme with 20 steps are shown in Figure 175. The agreement between numerical and analytical solution is very good and can be improved further by increasing the number of steps in the analysis.
+
The results of the analysis using the Auto scheme with 20 steps are shown in Figure 48.3. The agreement between numerical and analytical solution is very good and can be improved further by increasing the number of steps in the analysis.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/consol0102.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/consol0102.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 175: Evolution of excess pore pressure at the center of the block with time (left) and evolution of settlement of top surface with time (right). '''</span>
+
|+<span style="font-size:90%">'''Figure 48.3: Evolution of excess pore pressure at the center of the block with time (left) and evolution of settlement of top surface with time (right).'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,959: Line 4,115:
 
==  EFFECT OF CONSOLIDATION ON BEARING CAPACITY  ==
 
==  EFFECT OF CONSOLIDATION ON BEARING CAPACITY  ==
  
The following example is similar to that of Section 4. A shallow foundation in a Firm Clay-MC material (Figure 177) is loaded rapidly and then left to consolidated. After a period of time, the load is removed and the short term bearing capacity is determined. Following Example 4, we expect a short term bearing capacity that increases with consolidation time as a result of an increase in effective stresses, and hence an increase in the undrained shear strength, beneath the foundation.
+
The following example is similar to that of Section 44. A shallow foundation in a Firm Clay-MC material (Figure 49.1) is loaded rapidly and then left to consolidated. After a period of time, the load is removed and the short term bearing capacity is determined. Following Example 44, we expect a short term bearing capacity that increases with consolidation time as a result of an increase in effective stresses, and hence an increase in the undrained shear strength, beneath the foundation.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/preloadconsol01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/preloadconsol01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 177: Problem setup (consolidation stage). '''</span>
+
|+<span style="font-size:90%">'''Figure 49.1: Problem setup (consolidation stage).'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
 
The example commences by determining the short term bearing capacity from a standard Limit Analysis. The result is a limit load of approximately <math display="inline">Q_{0} = 280</math> kN/m. Next, a Consolidation analysis with a fixed load of <math display="inline">P = 200</math> kN/m, representing about 70% of the short term bearing capacity, is carried out. The Target time stepping scheme is used with Target = Degree and a specified target degree of consolidation, <math display="inline">U_1</math>, corresponding to a time <math display="inline">t_1</math>. Finally, a Limit Analysis stage, linked to the Consolidation stage, is used to determine the new short term bearing capacity, <math display="inline">Q(t_1)</math>.<br />
 
The example commences by determining the short term bearing capacity from a standard Limit Analysis. The result is a limit load of approximately <math display="inline">Q_{0} = 280</math> kN/m. Next, a Consolidation analysis with a fixed load of <math display="inline">P = 200</math> kN/m, representing about 70% of the short term bearing capacity, is carried out. The Target time stepping scheme is used with Target = Degree and a specified target degree of consolidation, <math display="inline">U_1</math>, corresponding to a time <math display="inline">t_1</math>. Finally, a Limit Analysis stage, linked to the Consolidation stage, is used to determine the new short term bearing capacity, <math display="inline">Q(t_1)</math>.<br />
The results of the analyses are shown in Figures 178-3. Figure 178 shows the evolution of bearing capacity with time while Figure 179 shows the relative gain as function of the degree of consolidation. It is observed that the gain increases approximately linearly with the work based degree of consolidation:  
+
The results of the analyses are shown in Figures 49.2-3. Figure 49.2 shows the evolution of bearing capacity with time while Figure 49.3 shows the relative gain as function of the degree of consolidation. It is observed that the gain increases approximately linearly with the work based degree of consolidation:  
  
<math display="block">\frac{Q(t)}{Q_0} \simeq 1+\left(\frac{Q_\infty}{Q_0}-1\right)\frac{U_W(t)}{100\%}</math>
+
<math display="block">(49.1)\qquad
 +
    \frac{Q(t)}{Q_0} \simeq 1+\left(\frac{Q_\infty}{Q_0}-1\right)\frac{U_W(t)}{100\%}</math>
  
where <math display="inline">Q_\infty</math> is the short term bearing capacity after full consolidation. Finally, we note that the gain for <math display="inline">U=100\%</math>, <math display="inline">Q(t)/Q_0 = Q_\infty/Q_0 =1.45</math>, corresponds approximately to that found in Example 4 for <math display="inline">\text{OCR}=1</math>. Similar gains for the applied level of preloading have been observed in full scale field experiments by <bib id="Lehane:2003" /> for a scenario similar to the one considered in this example.
+
where <math display="inline">Q_\infty</math> is the short term bearing capacity after full consolidation. Finally, we note that the gain for <math display="inline">U=100\%</math>, <math display="inline">Q(t)/Q_0 = Q_\infty/Q_0 =1.45</math>, corresponds approximately to that found in Example 44 for <math display="inline">\text{OCR}=1</math>. Similar gains for the applied level of preloading have been observed in full scale field experiments by <bib id="Lehane:2003" /> for a scenario similar to the one considered in this example.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/effconsol02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/effconsol02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 178: Relative bearing capacity gain, <math display="inline">Q/Q_0</math>, as function of time. '''</span>
+
|+<span style="font-size:90%">'''Figure 49.2: Relative bearing capacity gain, <math display="inline">Q/Q_0</math>, as function of time.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,986: Line 4,143:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/effconsol03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/effconsol03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 179: Relative bearing capacity gain, <math display="inline">Q/Q_0</math>, as function of degree of consolidation. '''</span>
+
|+<span style="font-size:90%">'''Figure 49.3: Relative bearing capacity gain, <math display="inline">Q/Q_0</math>, as function of degree of consolidation.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 3,996: Line 4,153:
 
==  EMBANKMENT CONSTRUCTION – PART 1  ==
 
==  EMBANKMENT CONSTRUCTION – PART 1  ==
  
The following two sections describe the analysis of an embankment (see Figure 180) to be constructed on a soft soil overlaying a layer of dense sand. The soft soil is modeled by means of the Modified Cam Clay model whereas the embankment fill and the dense sand layer are modeled as Mohr-Coulomb materials. For full details of the material parameters used, please refer to the accompanying input file that can be accessed via the welcome window in OPTUM G2 or via File/Examples.
+
The following two sections describe the analysis of an embankment (see Figure 50.1) to be constructed on a soft soil overlaying a layer of dense sand. The soft soil is modeled by means of the Modified Cam Clay model whereas the embankment fill and the dense sand layer are modeled as Mohr-Coulomb materials. For full details of the material parameters used, please refer to the accompanying input file that can be accessed via the welcome window in OPTUM G2 or via File/Examples.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/embank1_01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/embank1_01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 180: Embankment construction: problem setup. '''</span>
+
|+<span style="font-size:90%">'''Figure 50.1: Embankment construction: problem setup.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 4,027: Line 4,184:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/embank1_02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/embank1_02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 181: Vertical stage displacements for Stages 3 and 5. The deformations are scaled by a factor of 10. '''</span>
+
|+<span style="font-size:90%">'''Figure 50.2: Vertical stage displacements for Stages 3 and 5. The deformations are scaled by a factor of 10.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 4,035: Line 4,192:
 
The factors of safety are:  
 
The factors of safety are:  
  
<math display="block">\begin{array}{lcl}
+
<math display="block">(50.1)\qquad 
 +
 
 +
    \begin{array}{lcl}
 
     \text{I.} &\text{(Short Term):}& \text{FS}_s = 1.55\pm 0.02\\
 
     \text{I.} &\text{(Short Term):}& \text{FS}_s = 1.55\pm 0.02\\
 
     \text{II.} &\text{(Long Term):}& \text{FS}_s = 3.12\pm 0.03\\
 
     \text{II.} &\text{(Long Term):}& \text{FS}_s = 3.12\pm 0.03\\
Line 4,043: Line 4,202:
 
     \end{array}</math>
 
     \end{array}</math>
  
As expected, the long term stability decreases as the height of the embankment increases. For the short term stability, the increase in embankment height is compensated by the increase in undrained shear strength as a result of consolidation of the soft soil, to an extent that the short term stability at the completion of construction and dissipation of all excess pressure is greater than at any point during construction. The factors of safety and associated distributions of shear dissipation are shown in Figure 182.
+
As expected, the long term stability decreases as the height of the embankment increases. For the short term stability, the increase in embankment height is compensated by the increase in undrained shear strength as a result of consolidation of the soft soil, to an extent that the short term stability at the completion of construction and dissipation of all excess pressure is greater than at any point during construction. The factors of safety and associated distributions of shear dissipation are shown in Figure 50.3.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/embank1_03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/embank1_03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 182: Distributions of shear dissipation from Strength Reduction analysis. '''</span>
+
|+<span style="font-size:90%">'''Figure 50.3: Distributions of shear dissipation from Strength Reduction analysis.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 4,057: Line 4,216:
 
==  EMBANKMENT CONSTRUCTION – PART 2  ==
 
==  EMBANKMENT CONSTRUCTION – PART 2  ==
  
The second part of the example concerns the consolidation of the embankment after each construction stage, i.e. the dissipation of excess pore pressure with time. Two different situations are considered. The first one as sketched in the previous example and the second one with pre-installed drains underneath the embankment is shown in Figure 183. In OPTUM G2, drains may be modeled by means of the Zero Excess Pressure BC.
+
The second part of the example concerns the consolidation of the embankment after each construction stage, i.e. the dissipation of excess pore pressure with time. Two different situations are considered. The first one as sketched in the previous example and the second one with pre-installed drains underneath the embankment is shown in Figure 51.1. In OPTUM G2, drains may be modeled by means of the Zero Excess Pressure BC.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/embank2_01b.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/embank2_01b.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 183: Embankment with drains. '''</span>
+
|+<span style="font-size:90%">'''Figure 51.1: Embankment with drains.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 4,069: Line 4,228:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/embank2_02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/embank2_02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 184: Degree of consolidation versus time with and without drains. '''</span>
+
|+<span style="font-size:90%">'''Figure 51.2: Degree of consolidation versus time with and without drains.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
 
In each of the two situations (drains or no drains), the problem is modeled by using an Initial Stress stage. This is linked to a Consolidation stage accounting for the construction of the lower part of the embankment. Finally, a second Consolidation stage, accounting for the upper part of the embankment is defined and linked to the previous stage. In both Consolidation stages, the Target scheme with Degree = 90% is used. It should be noted that in the case where drains are used, these are included already in the Initial Stress stage. In other words, it is assumed that the drains have been placed well in advance of the actual construction and that a steady state seepage pressure distribution exists before construction.<br />
 
In each of the two situations (drains or no drains), the problem is modeled by using an Initial Stress stage. This is linked to a Consolidation stage accounting for the construction of the lower part of the embankment. Finally, a second Consolidation stage, accounting for the upper part of the embankment is defined and linked to the previous stage. In both Consolidation stages, the Target scheme with Degree = 90% is used. It should be noted that in the case where drains are used, these are included already in the Initial Stress stage. In other words, it is assumed that the drains have been placed well in advance of the actual construction and that a steady state seepage pressure distribution exists before construction.<br />
The degree of consolidation with time for each of the two situations is shown in Figure 184. As expected, the drains facilitate a significantly more rapid consolidation. Note also, the the rate of consolidation increases between the two construction stages. This is a consequence of the stress dependence of the Modified Cam Clay model which implies an increase in Young’s modulus with effective mean stress. As such, the coefficient of consolidation, <math display="inline">C_v = KE/\gamma_w</math>, increases accordingly.
+
The degree of consolidation with time for each of the two situations is shown in Figure 51.2. As expected, the drains facilitate a significantly more rapid consolidation. Note also, the the rate of consolidation increases between the two construction stages. This is a consequence of the stress dependence of the Modified Cam Clay model which implies an increase in Young’s modulus with effective mean stress. As such, the coefficient of consolidation, <math display="inline">C_v = KE/\gamma_w</math>, increases accordingly.
  
 
 
 
 
Line 4,082: Line 4,241:
 
==  CONSOLIDATION OF EXCAVATION  ==
 
==  CONSOLIDATION OF EXCAVATION  ==
  
This problem considers an excavation in front of a sheet pile wall as sketched in Figure 185. After the excavation – which is assumed to be performed sufficiently rapidly for no significant excess pore pressure dissipation to have taken place – a consolidation analysis to full consolidation is performed. The soil soil profile consists of 2 m of loose sand overlying a deep layer of clay.
+
This problem considers an excavation in front of a sheet pile wall as sketched in Figure 52.1. After the excavation – which is assumed to be performed sufficiently rapidly for no significant excess pore pressure dissipation to have taken place – a consolidation analysis to full consolidation is performed. The soil soil profile consists of 2 m of loose sand overlying a deep layer of clay.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/excconsol01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/excconsol01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 185: Problem setup. '''</span>
+
|+<span style="font-size:90%">'''Figure 52.1: Problem setup.'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The excavation is carried out in four stages as indicated in Figure 185. At the end of the excavation process, a consolidation stage is used to simulate the process of excess pore pressure dissipation to a final time of 200 days. From Figure 186 it is seen that the displacements resulting from excess pore pressure dissipation are of a similar magnitude to those induced in the short term as a result of the excavation, approximately 5 cm.
+
The excavation is carried out in four stages as indicated in Figure 52.1. At the end of the excavation process, a consolidation stage is used to simulate the process of excess pore pressure dissipation to a final time of 200 days. From Figure 52.2 it is seen that the displacements resulting from excess pore pressure dissipation are of a similar magnitude to those induced in the short term as a result of the excavation, approximately 5 cm.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/excconsol02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/excconsol02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 186: Maximum horizontal displacement. '''</span>
+
|+<span style="font-size:90%">'''Figure 52.2: Maximum horizontal displacement.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 4,105: Line 4,264:
  
 
Under undrained conditions in plane strain, the Mohr-Coulomb model is equivalent to the Tresca model with an undrained shear strength that depends on the initial effective stresses and the Mohr-Coulomb cohesion and friction angle (see the Theory and Materials Manuals of details). Under more general stress states, including those encountered in axisymmetry, this equivalence does not hold. For example, the Mohr-Coulomb model will display different undrained strengths in triaxial compression and extension. More generally, the undrained shear strength depends on the Lode angle. This characteristic cannot be reproduced by the Tresca model which operates with a single, Lode angle independent, undrained shear strength.<br />
 
Under undrained conditions in plane strain, the Mohr-Coulomb model is equivalent to the Tresca model with an undrained shear strength that depends on the initial effective stresses and the Mohr-Coulomb cohesion and friction angle (see the Theory and Materials Manuals of details). Under more general stress states, including those encountered in axisymmetry, this equivalence does not hold. For example, the Mohr-Coulomb model will display different undrained strengths in triaxial compression and extension. More generally, the undrained shear strength depends on the Lode angle. This characteristic cannot be reproduced by the Tresca model which operates with a single, Lode angle independent, undrained shear strength.<br />
In OPTUM G2 offers two models capable of accounting for Lode angle dependent undrained shear strength: the Generalized Tresca model and the AUS model. Both models use the undrained shear strengths in triaxial compression and extension as input parameters. Furthermore, for the Isotropic version of the AUS model, the yield surface is identical to that of the Generalized Tresca model (see Figure 187). In terms of plasticity, the key difference between the two model is the flow rule. While the Generalized Tresca model assumes associated flow, the flow rule of AUS model model is nonassociated with the plastic potential being that of von Mises (a circle in the deviatoric plane).<br />
+
In OPTUM G2 offers two models capable of accounting for Lode angle dependent undrained shear strength: the Generalized Tresca model and the AUS model. Both models use the undrained shear strengths in triaxial compression and extension as input parameters. Furthermore, for the Isotropic version of the AUS model, the yield surface is identical to that of the Generalized Tresca model (see Figure 53.1). In terms of plasticity, the key difference between the two model is the flow rule. While the Generalized Tresca model assumes associated flow, the flow rule of AUS model model is nonassociated with the plastic potential being that of von Mises (a circle in the deviatoric plane).<br />
 
In the following, the bearing capacity predictions of the Standard and Generalized Tresca models and the AUS model are examined for three different problems.
 
In the following, the bearing capacity predictions of the Standard and Generalized Tresca models and the AUS model are examined for three different problems.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image300px">https://wiki.optumce.com/figures/ussyc3.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/ussyc3.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 187: Generalized Tresca failure surface in the deviatoric plane (left) and in principal stress space for an intermediate value of <math display="inline">s_{ue }/s_{uc}</math> (right). The points indicated correspond to triaxial compression (TC), triaxial extension (TE) and plane strain (PS). No particular ordering of the principal stresses is assumed.'''</span>
+
|+<span style="font-size:90%">'''Figure 53.1: Generalized Tresca failure surface in the deviatoric plane (left) and in principal stress space for an intermediate value of <math display="inline">s_{ue}/s_{uc}</math> (right). The points indicated correspond to triaxial compression (TC), triaxial extension (TE) and plane strain (PS). No particular ordering of the principal stresses is assumed.'''</span>
 
|}
 
|}
 
</center>
 
</center>
 
===  Problem 1: Circular foundation  ===
 
===  Problem 1: Circular foundation  ===
  
The first problem concerns the bearing capacity of a circular foundation as shown in Figure 188.
+
The first problem concerns the bearing capacity of a circular foundation as shown in Figure 53.2.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/stgtaus_foot00.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/stgtaus_foot00.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 188: Problem 1: circular foundation. '''</span>
+
|+<span style="font-size:90%">'''Figure 53.2: Problem 1: circular foundation.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 4,129: Line 4,288:
 
For the Standard Tresca model, the bearing capacity is <bib id="Cox:1961" />:  
 
For the Standard Tresca model, the bearing capacity is <bib id="Cox:1961" />:  
  
<math display="block">q_{u,ST} = 6.05 s_u</math>
+
<math display="block">(53.1)\qquad
 +
    q_{u,ST} = 6.05 s_u</math>
  
 
Using 10,000 Lower/Upper elements with 3 adaptivity iterations, this solution is reproduced with negligible error.<br />
 
Using 10,000 Lower/Upper elements with 3 adaptivity iterations, this solution is reproduced with negligible error.<br />
 
For the Generalized Tresca model the result is exactly the same, regardless of the value of <math display="inline">s_{ue}</math> in relation to <math display="inline">s_{uc}</math>. Inspection of the Lode angle (available under Final Stresses in the Results tab) reveals a value equal to <math display="inline">-30^\circ</math> throughout the areas undergoing plasticity. In other words, the stress state corresponds to triaxial compression and the strength in extension has no influence on the results.<br />
 
For the Generalized Tresca model the result is exactly the same, regardless of the value of <math display="inline">s_{ue}</math> in relation to <math display="inline">s_{uc}</math>. Inspection of the Lode angle (available under Final Stresses in the Results tab) reveals a value equal to <math display="inline">-30^\circ</math> throughout the areas undergoing plasticity. In other words, the stress state corresponds to triaxial compression and the strength in extension has no influence on the results.<br />
For the AUS model on the other hand, the bearing capacity does depend on the <math display="inline">s_{uc}/s_{ue}</math> ratio as shown in Figure 189. As would be expected the bearing capacity increasing with increasing <math display="inline">s_{ue}/s_{uc}</math>. The fact that it does not equal the Tresca value for <math display="inline">s_{ue}/s_{uc}=1</math> is due to the effect of the flow rule (nonassociated von Mises for AUS and associated for Tresca).
+
For the AUS model on the other hand, the bearing capacity does depend on the <math display="inline">s_{uc}/s_{ue}</math> ratio as shown in Figure 53.3. As would be expected the bearing capacity increasing with increasing <math display="inline">s_{ue}/s_{uc}</math>. The fact that it does not equal the Tresca value for <math display="inline">s_{ue}/s_{uc}=1</math> is due to the effect of the flow rule (nonassociated von Mises for AUS and associated for Tresca).
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/stgtaus_foot01b.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/stgtaus_foot01b.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 189: Bearing capacity of circular foundation for the Generalized Tresca and AUS model relative to the Standard Tresca Model (mean values between upper and lower bounds). '''</span>
+
|+<span style="font-size:90%">'''Figure 53.3: Bearing capacity of circular foundation for the Generalized Tresca and AUS model relative to the Standard Tresca Model (mean values between upper and lower bounds).'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The failure mechanisms are in all cases similar to the familiar Prandtl mechanisms although some deviation is observed with the AUS model for low values of <math display="inline">s_{ue}/s_{uc}</math> (see Figure 190).
+
The failure mechanisms are in all cases similar to the familiar Prandtl mechanisms although some deviation is observed with the AUS model for low values of <math display="inline">s_{ue}/s_{uc}</math> (see Figure 53.4).
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/stgtaus_foot02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/stgtaus_foot02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 190: Failure mechanisms for the Tresca and AUS models. '''</span>
+
|+<span style="font-size:90%">'''Figure 53.4: Failure mechanisms for the Tresca and AUS models.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 4,157: Line 4,317:
 
===  Problem 2: Cylindrical excavation  ===
 
===  Problem 2: Cylindrical excavation  ===
  
The next problem concerns a cylindrical excavation as shown in Figure 191.<br />
+
The next problem concerns a cylindrical excavation as shown in Figure 53.5.<br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/stgtaus_cyl00.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/stgtaus_cyl00.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 191: Cylindrical excavation. '''</span>
+
|+<span style="font-size:90%">'''Figure 53.5: Cylindrical excavation.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 4,170: Line 4,330:
 
Using 10,000 Upper/Lower elements with 3 adaptivity iterations, the Standard Tresca model predicts a non-dimensional stability number equal to:  
 
Using 10,000 Upper/Lower elements with 3 adaptivity iterations, the Standard Tresca model predicts a non-dimensional stability number equal to:  
  
<math display="block">N_{ST} = \frac{s_u}{\gamma H} = 5.99\pm 0.02</math>
+
<math display="block">(53.2)\qquad
 +
    N_{ST} = \frac{s_u}{\gamma H} = 5.99\pm 0.02</math>
  
The predictions of the Generalized Tresca and AUS models are shown in Figure 192.
+
The predictions of the Generalized Tresca and AUS models are shown in Figure 53.6.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/stgtaus_cyl01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/stgtaus_cyl01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 192: Stability number of cylindrical excavation for the Generalized Tresca and AUS models relative to the Standard Tresca Model (mean values between upper and lower bounds). '''</span>
+
|+<span style="font-size:90%">'''Figure 53.6: Stability number of cylindrical excavation for the Generalized Tresca and AUS models relative to the Standard Tresca Model (mean values between upper and lower bounds).'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
In contrast to the previous example, there is now a marked dependence of the stability number on the <math display="inline">s_{ue}/s_{uc}</math> ratio. A plot (Figure 193) of the Lode angle at failure reveals a Lode angle of around zero along the main slip line (corresponding, approximately, to simple shear) and an angle of close to <math display="inline">30^\circ</math> (corresponding to triaxial extension) in large regions within the failing mass. This distribution of Lode angle is consistent with the observed decrease in strength with decreasing <math display="inline">s_{ue}/s_{uc}</math> for both the Generalized Tresca and the AUS model.
+
In contrast to the previous example, there is now a marked dependence of the stability number on the <math display="inline">s_{ue}/s_{uc}</math> ratio. A plot (Figure 53.7) of the Lode angle at failure reveals a Lode angle of around zero along the main slip line (corresponding, approximately, to simple shear) and an angle of close to <math display="inline">30^\circ</math> (corresponding to triaxial extension) in large regions within the failing mass. This distribution of Lode angle is consistent with the observed decrease in strength with decreasing <math display="inline">s_{ue}/s_{uc}</math> for both the Generalized Tresca and the AUS model.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image300px">https://wiki.optumce.com/figures/stgtaus_cyl02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/stgtaus_cyl02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 193: Failure mechanism and Lode angle distribution for AUS model with <math display="inline">s_{ue }/s_{uc}=0.5</math>.'''</span>
+
|+<span style="font-size:90%">'''Figure 53.7: Failure mechanism and Lode angle distribution for AUS model with <math display="inline">s_{ue}/s_{uc}=0.5</math>.'''</span>
 
|}
 
|}
 
</center>
 
</center>
 
===  Problem 3: Deep anchor  ===
 
===  Problem 3: Deep anchor  ===
  
The final problem concerns the pull-out capacity of a deep anchor as shown in Figure 194.
+
The final problem concerns the pull-out capacity of a deep anchor as shown in Figure 53.8.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/stgtaus_anchor02.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/stgtaus_anchor02.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 194: Deep anchor. '''</span>
+
|+<span style="font-size:90%">'''Figure 53.8: Deep anchor.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 4,205: Line 4,366:
 
Assuming rough interfaces between the soil and the anchor plate, the pull-out capacity is computed as (10,000 Upper/Lower elements with 3 adaptivity steps):  
 
Assuming rough interfaces between the soil and the anchor plate, the pull-out capacity is computed as (10,000 Upper/Lower elements with 3 adaptivity steps):  
  
<math display="block">\frac{q_{u,ST}}{s_u} = \frac{Q_{u,ST}}{\pi(D/2)^2s_u} = 13.07\pm 0.12</math>
+
<math display="block">(53.3)\qquad
 +
    \frac{q_{u,ST}}{s_u} = \frac{Q_{u,ST}}{\pi(D/2)^2s_u} = 13.07\pm 0.12</math>
  
 
 <br />
 
 <br />
The predictions of the Generalized Tresca and AUS models are shown in Figure 195.
+
The predictions of the Generalized Tresca and AUS models are shown in Figure 53.9.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/stgtaus_anchor01.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/stgtaus_anchor01.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 195: Pull-out capacity of deep anchor for the Generalized Tresca and AUS models relative to the Standard Tresca Model (mean values between upper and lower bounds). '''</span>
+
|+<span style="font-size:90%">'''Figure 53.9: Pull-out capacity of deep anchor for the Generalized Tresca and AUS models relative to the Standard Tresca Model (mean values between upper and lower bounds).'''</span>
 
|}
 
|}
 
</center>
 
</center>
  
The trend is here similar to the previous example with the pull-out capacity decreasing with decreasing <math display="inline">s_{ue}/s_{uc}</math>. A plot of the Lode angle (Figure 196) reveals that stress state below of the anchor plate anchor plate corresponds to approximately to triaxial extension (<math display="inline">\theta = +30^\circ</math>) while the stress above corresponds roughly to triaxial compression (<math display="inline">\theta = -30^\circ</math>). The result is a bearing capacity that depends on the <math display="inline">s_{ue}/s_{uc}</math> as shown in Figure 195.  <br />
+
The trend is here similar to the previous example with the pull-out capacity decreasing with decreasing <math display="inline">s_{ue}/s_{uc}</math>. A plot of the Lode angle (Figure 53.10) reveals that stress state below of the anchor plate anchor plate corresponds to approximately to triaxial extension (<math display="inline">\theta = +30^\circ</math>) while the stress above corresponds roughly to triaxial compression (<math display="inline">\theta = -30^\circ</math>). The result is a bearing capacity that depends on the <math display="inline">s_{ue}/s_{uc}</math> as shown in Figure 53.9.  <br />
  
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/stgtaus_anchor03.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/stgtaus_anchor03.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 196: Plastic strain (top) and Lode angle (bottom) distributions. '''</span>
+
|+<span style="font-size:90%">'''Figure 53.10: Plastic strain (top) and Lode angle (bottom) distributions.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 4,234: Line 4,396:
  
 
The following example concerns the AUS model. A set of test data are first fitted and the model is then applied to the simulation of a circular skirted foundation.<br />
 
The following example concerns the AUS model. A set of test data are first fitted and the model is then applied to the simulation of a circular skirted foundation.<br />
The test data used in the example is adapted from <bib id="Won:2013" />. It comprises triaxial compression and extension data. In OPTUM G2, these two tests are simulated using Multiplier Elastoplastic analysis under axisymmetric conditions as indicated in Figure 197. The Fixed loads here represent the initial axial and radials stresses while the axial Multiplier load is increased in the course of the analysis to reach the ultimate limit state.
+
The test data used in the example is adapted from <bib id="Won:2013" />. It comprises triaxial compression and extension data. In OPTUM G2, these two tests are simulated using Multiplier Elastoplastic analysis under axisymmetric conditions as indicated in Figure 54.1. The Fixed loads here represent the initial axial and radials stresses while the axial Multiplier load is increased in the course of the analysis to reach the ultimate limit state.
  
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px">https://wiki.optumce.com/figures/wontest0.png</div>
+
|<div class="image500px">https://wiki.optumce.com/figures/wontest0.png</div>
 
|-
 
|-
|+<span style="font-size:92%">'''Figure 197: Setups for triaxial compression and extension tests. '''</span>
+
|+<span style="font-size:90%">'''Figure 54.1: Setups for triaxial compression and extension tests.'''</span>
 
|}
 
|}
 
</center>
 
</center>
Line 4,246: Line 4,408:
 
<center>
 
<center>
 
{| style="caption-side:bottom; text-align:center"
 
{| style="caption-side:bottom; text-align:center"
|<div class="image600px&q