BEARING CAPACITY OF STRIP FOOTING
BEARING CAPACITY OF STRIP FOOTING This document describes an example that has been used to verify the bearing capacity of strip footing in PLAXIS. F 1m
½B
cref
ν = 0.495 G = 100 · c
φ=0◦
No tension cut-off
c
Figure 1 Problem geometry
Used version: •
PLAXIS 2D - Version 2011
•
PLAXIS 3D - Version 2012
Input: Calculations are carried out for a rough and a smooth footing. The geometry of the 2D models is shown in Figure 2. Because of symmetry, only half of the geometry is modelled using 15-node elements. Note that for the smooth case, the x-direction of the prescribed displacement is set to free whereas in the rough case the x-direction of the prescribed displacement is set to fixed.
1m
2m
3m
Figure 2 Model geometry (PLAXIS 2D)
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The geometry of the 3D model is shown in Figure 3. The strip footing is defined as a surface prescribed displacement. A soil cluster and a surface are defined underneath and at the right of the strip footing respectively to enable local control of the mesh. A line prescribed displacement fixed in the x and y directions is defined between (0; 0; 2) and (0; 1; 2). 1m
0.5 m 2.5 m
1m
0.4 m
1.6 m
Figure 3 Model geometry (PLAXIS 3D)
Materials: The material properties are shown in Figure 1. The Mohr-Coulomb model is used to model the behavior of the soil in order to be consistent with the conventional foundation design (Potts & Zdravkovi´c (2001)). The cohesion at the soil surface, cref , is taken 1 kN/m2 . In the Advanced settings, the cohesion gradient, cinc , is set equal to 2 kN/m2 /m, using a reference level, yref = 0 m (= top of layer). The stiffness at the top is given by Eref = 299 kN/m2 and the increase of stiffness with depth is defined by Einc = 598 kN/m2 /m. Meshing: In the 2D model the Medium option is selected for the Global coarseness. The point at the left and at the right of the prescribed displacement are refined with a Local element size factor of 0.5 and 0.05 respectively. The resulting finite element mesh is shown in Figure 4. In the 3D model the Coarse option is selected for the Global coarseness. The soil cluster below the footing and the surface prescribed displacement representing the footing are refined with a Local element size factor of 0.25. The surface at the right of the footing is refined with a Local element size factor of 0.1. The resulting finite element mesh is shown in Figure 5. Calculations: In the Initial phase zero initial stresses are generated by using the K0 procedure with Σ -Mweight equal to zero. The prescribed displacement is activated in a separate phase. A vertical displacement of -0.1 is applied. In the case of the smooth footing the horizontal prescribed displacement is set to Free. In the case of the rigid footing the horizontal prescribed displacement is fixed. The calculation type is Plastic analysis and a Tolerated error of 0.001 is defined. The Reset displacements to zero option is selected and the Additional steps parameter is set to 500.
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BEARING CAPACITY OF STRIP FOOTING
Figure 4 The finite element mesh(PLAXIS 2D)
Figure 5 The finite element mesh (PLAXIS 3D)
Output: In PLAXIS 2D model the calculated maximum average vertical stress under the smooth footing is 7.831 kN/m2 , giving a bearing capacity of 15.662 kN/m. For the rough footing this is 9.174 kN/m2 , giving a bearing capacity of 18.358 kN/m. The computed load-displacement curves are shown in Figure 7. In PLAXIS 3D model the calculated maximum average vertical stress under the smooth footing is 8.095 kN/m2 , giving a bearing capacity of 16.19 kN/m. For the rough footing this is 9.820 kN/m2 , giving a bearing capacity of 19.640 kN/m. The load-displacement curves are shown in Figure 8. Verification: The analytical solution derived by Davis & Booker (1973) for the mean ultimate vertical stress beneath the footing, pmax , is:
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Figure 6 Deformed mesh (PLAXIS 2D - Smooth case) 10
Fz [ kN/m ]
8
6
4
2
Smooth case Rough case
0
0.00
0.02
0.04
uz [m]
0.06
0.08
0.10
Figure 7 Comparison of results for smooth and rough footing (PLAXIS 2D)
pmax =
F B × cinc = β (2 + π)cref + 4 B
where B is the footing width and β is a factor that depends on the footing roughness and the rate of increase of clay strength with depth. The appropriate values of β in this case are 1.27 for the smooth footing and 1.48 for the rough footing. The analytical solution therefore gives average vertical stresses at collapse of 7.8 kN/m2 for the smooth footing and 9.1 kN/m2 for the rough footing. The errors in the PLAXIS 2D solution are 0.40% and 0.81% respectively. The errors in the PLAXIS 3D solution are 3.64% and 7.91% respectively.
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BEARING CAPACITY OF STRIP FOOTING
10
Fz [ kN/m ]
8
6
4
2
Smooth case Rough case
0 0.00
0.02
0.06
0.04
0.08
0.10
uz [m] Figure 8 Comparison of results for smooth and rough footing (PLAXIS 3D)
REFERENCES [1] Davis, E.H., Booker, J.R. (1973). The effect of increasing strength with depth on the bearing capacity of clays. Geotechnique, 23(4), 551–563. [2] Potts, D.M., Zdravkovi´c, L. (2001). Finite element analysis in geotechnical engineering application. Thomas Telford, London.
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