1 Introduction An Introductory SIGMA/W Example This is a fairly simple introductory example. The primary purpose is to demonstrate to new SIGMA/W users how to get started, to introduce the usual type of data required to do a SIGMA/W analysis, and to illustrate what can be done with SIGMA/W. The problem is quite simple, so that it can be analyzed with the GeoStudio Student Edition. This problem is about computing the stress changes in the ground beneath a round tank filled with a fluid. Problem configuration and description When studying a problem that is symmetric about a vertical axis, the three-dimensional effects can be analyzed with a two-dimensional finite element mesh, as illustrated by the following sketch. A special axisymmetric formulation in SIGMA/W for this case considers the stresses in the X-Y plane as well as the circumferential stresses. This view is selected with the Set Units and Scale command. Figure 1shows the corresponding SIGMA/W problem setup. The diameter of the tank is 5 m and the applied pressure is kpa. The soil region is 3 m wide and 5 m high. The upper 5 m of soil has different properties than the underlying m. SIGMA/W Example File: SIGMA Tutorial.doc (pdf) (gsz) Page 1 of 7
Tank File Name: SIGMA Tutorial.gsz Round tank - kpa pressure Upper soil 1 1 1 Lower soil Symmetric axis - - 1 1 1 3 3 3 3 3 Figure 1 Problem configuration 3 Geometry regions The intention is to have a finer mesh right under the tank where the stress changes will be the highest. Also, different regions are required to accommodate the two soil types. It is for this reason that we use four mesh regions, as shown in Figure. Also, the weight of the tank must be applied as boundary condition and specified boundary pressures must be applied on region edges. Consequently, Edge -1 of Region 1 needs to subdivided into Edge -3 and 3-1. This is done by adding a Point at the location of the outer edge of the tank (Point 3). 3 5 1 3 1 1 1 1 9 7 - - 1 1 1 3 3 3 3 3 Figure Geometry regions and points SIGMA/W Example File: SIGMA Tutorial.doc (pdf) (gsz) Page of 7
Boundary conditions Along the left symmetric axis we want to allow vertical movement (settlement) but no lateral (horizontal) movement. Along the bottom and right vertical side, it is considered that these boundaries are far enough away from the applied surface pressure that they can be fixed in both the X and Y directions. SIGMA/W has default boundary conditions you can use for this. You must also create a boundary condition representing the tank pressure. In this case, we can use a stress-type boundary. It is specified as a Y-stress boundary with a magnitude (action) of - kpa. The negative value means it is pointing downward. Figure 3 shows the boundary conditions. Now it is evident why it was necessary to subdivide the region edge so that the tank pressure is applied only under the tank. 1 1 1 - - 1 1 1 3 3 3 3 3 Figure 3 Boundary conditions 5 Material properties The soil will be treated as being linear elastic with the following properties: Upper soil: E = 3 kpa and Poisson s ratio ν =.5 Lower soil: E = kpa and Poisson s ratio ν =.5 No pore-pressures are considered, so we can treat these properties as Total Stress Parameters. The soil colors can be selected and changed as you wish. Finite element mesh Now that the geometry, boundary conditions and material properties are specified and applied, we should look at the mesh and make modifications is necessary. This is done using the Draw Properties command. A global element size of m is about right for this analysis, except right under the footing. Earlier it was noted that a finer mesh is desirable under the footing. This can be achieved by selecting Region 1 and specifying that the element size in this region should be.5 (half) of the global dimension. In addition, a decision was made to use -noded higher order elements in the two regions on the left side of the problem. Further away to the right, the stress change will be less, and therefore -node quadrilateral elements are consider adequate in the right two regions. SIGMA/W Example File: SIGMA Tutorial.doc (pdf) (gsz) Page 3 of 7
The results mesh is shown in the next diagram (Figure ). Figure The finite element mesh Worth noting is that you can set up and define the entire problem before you need to think about the mesh. 7 Verifying With the Tools command you can Verify the problem. This means that SIGMA/W checks that sufficient data has been supplied to compute a solution. It does not mean it will be the correct solution but it will be a solution. For example, Verify can alert you to the fact that no boundary conditions have been specified or no material properties have been applied. It is up to you to judge whether the solution is or is not correct. For example, are the solution trends correct and are the material properties appropriate? Solve The next step is to click on the SOLVE button and compute the solution to the finite element equations. 9 Vertical stresses Earlier it was noted that the object here is to compute the stress changes in the ground due to the weight of the fluid in the tank. The best way to look at this is to create a contour plot of the Y-stresses, as in Figure 5. The contour labels are in units of kpa. The contours from a pressure bulb, such as are typically shown in most introductory Soil Mechanics textbooks, suggest that the analysis has been correctly set up. SIGMA/W Example File: SIGMA Tutorial.doc (pdf) (gsz) Page of 7
Note that the kpa contour right under the footing is somewhat irregular and squiggly. This is primarily due to numerical noise and nothing to worry about. Ideally, the kpa contour is right at the contact, but in a numerical approximation such as a finite element analysis it is not possible to achieve this ideal condition. Tank Round tank - 35 1 15 1 ( ) 1 5 Figure 5 Symmetric axis Vertical stress change contours Vertical settlement Figure shows the vertical settlements as a deformed mesh at a X exaggeration. The maximum settlement is about.7 m (7 mm) under the center of the tank. SIGMA/W Example File: SIGMA Tutorial.doc (pdf) (gsz) Page 5 of 7
Tank Round tank - kpa pressure File Name: SIGMA Tutorial.gsz Upper soil 1 1 1 Symmetric axis Lower soil - - 1 1 1 3 3 3 3 3 Figure Settlement as a deformed mesh at X exaggeration 11 Graphing In addition to looking at the results as contour plots or deformed mesh plots as above, there are multiple ways of looking at the results with the Graph command in SIGMA/W. Figure 7 shows just one typical example. This figure gives the lateral (x) displacement along a vertical profile at the outer edge of the tank. 5 Lateral displacemnt Y (m) 15 5 -. -.........1 X-Displacement (m) Figure 7 Lateral displacement profile SIGMA/W Example File: SIGMA Tutorial.doc (pdf) (gsz) Page of 7
Demonstration movie A movie that goes through all the steps in the example is also available as an aide to learning how to use SIGMA/W. Watching the movie is an excellent way to get started. 13 Concluding remarks This simple introductory example has all the components of a finite element analysis, which are primarily geometry, boundary conditions and material properties. The example also illustrates how this type of problem can be easily and conveniently analyzed with SIGMA/W. Moreover, it clearly demonstrates how the results can be visualized with colorful contour plots and graphs. If you take the time to duplicate this analysis by going through all the steps, you will have a very good concept of what is required to do a SIGMA/W analysis, and what can be achieved with this powerful software product. SIGMA/W Example File: SIGMA Tutorial.doc (pdf) (gsz) Page 7 of 7