center of the loaded area at any depthz is given by
σz= q
1− 1
[(R/z)2+ 1]3/2
(3.11)
whereq (force/unit area) is the applied pressure, R the radius of the loaded circle, andz the depth below the center of the loaded circle at which the stress increase is calculated. The elastic solution for stress increase elsewhere within the semi-infinite soil mass (not under the center) may be found in Ahlvin and Ulery (1962).
X
Z
z Y
q (kN/m2)
R
∆σz
FIGURE 3.20 Stresses under the center of a uniformly loaded circular area.
Example 3.7 A pressure of 10 kPa is uniformly distributed on a circular area with R= 0.5 m. (a) Calculate the increase in vertical stress directly under the center of the applied load forz= 0 to 5 m. (b) Repeat your solution using the finite element method and assuming that the soil is linear elastic with E= 1 × 107 kPa and ν = 0.3.
SOLUTION: (a) For the increase in vertical stress directly under the center of the applied load forz= 0 to 5 m, we use (3.11):
σz= (10)
1− 1
[(0.5/z)2+ 1]3/2
Using this equation, we can calculate the increase in vertical stress as a function of z. The equation is plotted in Figure 3.21. Note that σz is large (10 kPa) near the surface but decreases very rapidly with depth.
(b) Finite element solution (filename: Chapter3 Example7.cae) For simplicity, the semi-infinite soil mass is assumed to be a cylinder 100 m in diameter and 50 m in height, as shown in Figure 3.22. The reason for using a cylindrical shape in this simulation is to take advantage of axisymmetry, in which we can utilize axisymmetric two-dimensional analysis instead of three-dimensional analysis. The 10-kPa pressure is applied at the top surface on a circular area with 0.5-m radius.
The purpose of the analysis is to calculate the increase in vertical stress within the soil mass due to the application of the 10-kPa pressure, and to compare with the analytical solution.
The two-dimensional axisymmetric finite element mesh used has 20 elements in the x-direction and 40 elements in the z-direction, as shown in Figure 3.22.
0 1 2 3 4 5 6 7 8 9 10 Vertical Stress (kPa)
0
1
2
3
4
5
Depth (m)
FEM, x= 0 m Boussinesq, x= 0 m
X
Z Y
q (kN/m2)
R
FIGURE 3.21 Comparison between FEM and analytical solution of the stresses under the center of a uniformly loaded circular area.
The finite element mesh is made finer in the zone around the pressurized circle where stress concentration is expected. The element chosen is a four-node bilinear axisymmetric quadrilateral element. The increase in vertical stress under the center of the pressurized circle is plotted as a function of depth as shown in Figure 3.21.
The figure shows excellent agreement between the stresses calculated using the analytical elastic solution and the finite element solution. Note that the finite element solution is not limited to finding the stresses under the center of the loaded circle.
It provides stresses, strains, and displacements at all nodal points within the loaded semi-infinite soil mass as well.
Vertical Stress Increase in a Layered Soil System The equations pre-sented above for point load (3.9), line load (3.10), and circularly loaded area (3.11) are based on the assumption that the underlying soil is homogeneous and infinitely thick. These equations are invalid for a soil system having several layers with vary-ing elastic moduli (i.e., nonhomogeneous), such as the one shown in Figure 3.23.
More complicated solutions based on the theory of elasticity are required for such cases. The following example is about a soil system with four different layers that resembles the structure of a highway pavement: an asphalt layer (top), a base layer, a subbase layer, and the existing soil (bottom). An analytical solution is not avail-able for such a system. Thus, the finite element method can be extremely helpful for such a system.
Axis of Symmetry
q= 10 kPa 100 m
1 m
50 m
Not to scale
FIGURE 3.22 Axisymmetric finite element mesh of the loaded circular area problem.
Example 3.8 Consider a system with four layers of varying stiffness and thick-ness as shown in Figure 3.23. A pressure of 10 kPa is uniformly distributed on a circular area with R = 0.5 m. Using the finite element method, calculate the increase in vertical stress directly under the center of the circular area forz= 0 to 5 m. Compare this stress increase with that obtained in Example 3.7 for a single homogeneous soil layer.
SOLUTION: Finite element solution (filename: Chapter3 Example8.cae) Similar to what we did in Example 3.7, we assume that the semi-infinite soil mass is a cylinder 100 m in diameter and 50 m in height as shown in Figure 3.22. The 10-kPa pressure is applied at the top surface on a circular area with an 0.5 m radius.
The purpose of the analysis is to calculate the increase in vertical stress within the stratified soil mass due to the application of a uniformly distributed load on a circular area, and to compare with the solution for a single homogeneous layer.
The two-dimensional axisymmetric finite element mesh used has 20 elements in thex-direction and 40 elements in the z-direction, as shown in Figure 3.22. The
mesh includes four layers with the elastic moduli shown in Figure 3.23. The finite element mesh is made finer in the zone around the pressurized circle, where stress concentration is expected. The element chosen is a four-node bilinear axisymmet-ric quadrilateral element. The increase in vertical stress under the center of the pressurized circle is plotted as a function of depth as shown in Figure 3.24. For comparison, the increase in vertical stress in a single homogeneous layer is included
X
Z q= 10 kN/m2
2R = 1 m
E1= 7 × 108 kPa, V1= 0.3 E2= 7 × 107 kPa, V2= 0.3 E3= 7 × 106 kPa, V3= 0.3
E4= 7 × 105 kPa, V4= 0.3 H1= 0.25 m
H4= 48.75 m H3= 0.5 m H2= 0.5 m
Asphalt Layer Base Layer Sub Base Layer
Existing Soil Layer
FIGURE 3.23 Stress increase in a layered soil system with a uniformly loaded circular area.
0 1 2 3 4 5 6 7 8 9 10
Vertical Stress (kPa) 0
1
2
3
4
5
Depth (m)
FEM
(Layered System) Boussinesq (One Layer)
FIGURE 3.24 Comparison between FEM and analytical solution of a layered system with a uniformly loaded circular area.
in the figure (taken from Example 3.7). When the two are compared, the beneficial effects of the stiff asphalt layer and the base layer are seen clearly. These two layers absorbed most of the damaging vertical stress increase. Only a small frac-tion of stress increase is passed on to the subbase and existing softer soil layers.
Thus, the asphalt layer acts as a shield that protects the underlying softer layers from excessive stress increases due to repeated traffic loads, which usually cause pavement rutting and cracking.