In this case, the ultimate limit bearing capacity of circular footings will be investigated for a Tresca soil with vertical and centric loading with no overburden pressure. For this case, both three dimensional and two-dimensional versions of the program is going to be used.
5.2.1 Theoretical Solution
The solution for a circular foundation with centric loading on soil with the Tresca model is quite similar to the plane strain case. Ultimate limit bearing capacity can be defined as:
σv =Nc·Su+p (5.2)
Cox et al. [1961] calculated bearing capacity factor Nc, for a circular foundation by using the method of characteristics, giving it an exact solution of 6.05. Another solution suggested by Meyerhof [1963] suggests another solution, where the Nc is equal to 6.18, or roughly to3+π. Other solutions based on numerical analyses, such as Gourvenec and Randolph [2003], have suggested a bearing capacity factor equal to 6.05.
5.2.2 Computation Results of OptumG2 Runs
In this case, bearing capacity of a circular foundation on a Tresca soil was computed. The two-dimensional version of the Optum, with axisymmetry, will be used. 5 runs, with a different number of elements, were used to run lower and upper bound analyses.
5.2.2.1 Geometry and Meshing
Figure 5.7 shows the geometry and meshing for this case for a special case with 10.000 elements using regular meshing and meshing with 3 iterations of adaptive meshing. Soil body has 3 m width and 3 m height and there is a rigid foundation resting on it with 1 m radius.
5.2.2.2 Results for regular meshing
Figure 5.8 shows normalized bearing capacity factor, Nc, obtained for a different number of triangular elements. By increasing the number of elements, the average of two bounds is getting
(a) Geometry (b) Regular meshing (c) adaptive meshing
Figure 5.7: Geometry and meshing of case 2
close to the exact solution. The figure also shows how the absolute relative error is decreasing with increasing the number of triangular elements.
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Average of boubds Exact Solution
(a) Normalized ultimate load
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Figure 5.8: Results for regular meshing (without adaptive meshing) for different number of triangular elements
5.2.2.3 Results for meshing with adaptive meshing
Figure 5.9 shows the calculated bearing capacity factor for a different number of triangular elements with 3 iterative adaptive meshing. It can be seen than the solution is getting closer to 6.03, rather than 6.05. Furthermore, it can be seen that absolute relative error is significantly lower in comparison to regular meshing and the error reduces to a small amount by increasing the number of used elements.
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(a) Normalized ultimate load
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Figure 5.9: Results for case 2 with adaptive meshing for different number of triangular elements 5.2.2.4 Comparison of results
Figure 5.10 shows how the average of lower and upper bounds changes by increasing the num-ber of elements, for both regular meshing and adaptive meshing. Most importantly, the figure shows how the error is decreasing by increasing the number of elements. It also shows how the adaptive meshing is giving lower error than regular meshing for every number of elements.
5.2.3 Computation Results of OptumG3 Runs
Bearing capacity of a circular foundation on a Tresca soil was analyzed using the three-dimensional version of Optum. Three types of meshing, namely lower, upper, and mixed, was used to run the analyses. The mixed element type is a mixture of lower and upper element types, so the average of bounds is the average of lower and upper bounds, while the mixed solution is going to be reported separately. As the two-dimensional problem, a total of 5 sets of analyses, with a different number of elements, was computed. As the problem is symmetrical, only1/4of the problem was modeled.
5.2.3.1 Geometry and Meshing
Figure 5.11 shows the geometry and meshing used for this case. A soil body with dimensions of4×4×4(B×H×L) was used where a foundation with 1 m diameter and 1 m height is resting on the soil mass.
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Normal Meshing 3 iterative adaptive meshing Exact Solution
(a) Normalized ultimate load
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Number of triangular elements
Normal Meshing 3 iterative adaptive meshing
(b) Absolute relative error
Figure 5.10: Comparing the average of lower bound and upper bound between regular meshing and adaptive meshing
The figure also shows how 10.000 elements are distributed for regular meshing and adaptive meshing.
5.2.3.2 Results for regular meshing
Figure 5.12 shows the results of the analyses for regular meshing with OptumG3 with a different number of elements. It can be seen that results are getting closer as the number of the elements increasing, but the precision of the simulation does not increase to the level of the 2D version.
5.2.3.3 Results for meshing with adaptive meshing
Figure 5.13 shows the results of the analyses for the case where three iterations of adaptive meshing were used. The number of meshes was kept constant while through adaptive meshing their size and place would vary in the area with high shear dissipation.
It can be seen that by increasing the number of elements, results are getting closer to 6.08, rather than 6.05. The error of simulation is shown as well. Increasing the number of elements generally decreases the error, but it is not able to lower it to below 1%, as was seen in the 2D version.
(a) Geometry (b) Regular meshing (c) Adaptive meshing Figure 5.11: Geometry and meshing of case 1 (10.000 triangular elements)
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Figure 5.12: Results for regular meshing for different number of triangular elements
5.2.4 Comparison of results
Figure 5.14 shows a comparison of the results obtained from 2D and 3D analyses with adaptive meshing. Average of lower and upper bound for both 2D and 3D versions, alongside results of mixed solutions from 3D analyses are reported.
It can be seen that the results of the 3D analyses are generally reporting higher bearing capacity than the exact solution while the 2D version is doing the opposite. Furthermore, for 3D results, the mixed solution is overestimating the bearing capacity in comparison to the average of lower and upper bounds.
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Figure 5.13: Results of analyses with adaptive meshing for a different number of elements
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Figure 5.14: Comparing the results between 2D and 3D analyses (adaptive meshing)