Bearing Capacity of Strip Footings by FEA

2004 ABAQUS Users’ Conference 777

Bearing Capacity of Strip Footings on Two-layer Clay Soil by Finite Element Method

Ming Zhu

Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor

Abstract: Parametric study was carried out to evaluate the ultimate bearing capacity of a rough

strip footing resting on two-layer clay soil. Computations were performed by the commercial finite

element analysis software ABAQUS. The computational results are compared with published

lower bound and upper bound solutions by limit analysis.

Keywords: Bearing Capacity, Strip Footings, Two-layer Soil, Finite Element Method.

1. Introduction

Geotechnical engineers often deal with layered foundation soil, which is non-homogeneous in

nature but can be simplified in representation as distinct homogeneous layers for engineering

purposes. The failure mechanism of layered soil depends on the thickness and soil properties of

each layer. In some cases where the top layer is relatively thick and consists of weak soil, the

failure mechanism may be limited in the top layer only and the strength of the remaining lower

layers has no influence. In many other cases, however, the failure mechanism may involve two or

more layers.

Bearing capacity of strip footings on two-layer soil has received much attention in the literature.

Terzaghi and Peck (1948) first applied the concept of load spreading to analyze a strip footing on

sand overlying clay. This was followed by many other researchers. The methods can be classified

into four major groups: limit equilibrium method (Button, 1953; Reddy and Srinivasan, 1967;

Meyerhof, 1974), limit analysis approach (Chen and Davidson, 1973; Florkiewicz, 1989;

Michalowski and Shi, 1995; Merifield, et al., 1999; Michalowski, 2002; Shiau, et al., 2003), semi-

empirical approach (Brown and Meyerhof, 1969; Meyerhof and Hanna, 1978; Hanna and

Meyerhof, 1980), and finite element method (Griffiths, 1982; Burd and Frydman, 1997).

This paper focuses on the application of finite element method. Parametric study was carried out

to evaluate the ultimate bearing capacity of a rough strip footing resting on two-layer clay soil.

Computations were performed by the commercial finite element analysis software ABAQUS

(Version 6.3). The computational results are presented and compared with published lower bound

and upper bound solutions by limit analysis.

778 2004 ABAQUS Users’ Conference

2. Problem definition

In the following, the problem definition and the finite element model are discussed. A schematic

diagram of the plain strain problem is shown in Figure 1. The footing is sitting on a half space of

soil. The width of the footing is B. The thickness of the top layer is H. Clay is homogenous in each

layer with undrained shear strength c1 and c2 respectively. Soil is assumed to be weightless.

Figure 1. Schematic diagram of the problem.

Finite element mesh is illustrated in Figure 2. Due to symmetry, only half of the problem is

modeled. The width of the footing B is 2 m. The length of the finite element model is 12.5B, and

the height is 7.5B. The size of the finite element model is large enough to keep the boundary

conditions at the bottom and the right side from restricting the soil movement due to the footing

load.

Figure 2. Finite element mesh.


Soil is discretized with eight-node plane strain quadrilateral element with reduced integration

technique (element type CPE8R in ABAQUS). Soil immediately under the position of the footing

is discretized into 6 elements. Mesh near the edge of the footing is refined because of significant

displacement change in this region. The rough rigid footing is represented by appropriate

boundary conditions: to model the rough footing-soil interface, the horizontal displacement at the

nodes immediately under the position of the footing is restrained; to simulate the footing as a rigid

body, uniform downward displacement is applied at the nodes immediately under the position of

the footing so that they move downward by the same amount. The base of the mesh is fixed in

both horizontal and vertical directions. The two vertical sides are restrained in horizontal direction

only, allowing vertical movement.

Soil is modeled as an isotropic elastic-perfectly plastic material satisfying the Tresca failure

criterion. The following elastic properties are assumed (note that these values do not influence the

ultimate bearing capacity): Young’s modulus E = 20000kN/m2, Poisson’s ratio ν = 0.3. The

undrained shear strength of the top layer c1 is 20kN/m2. The undrained shear strength of the

second clay layer c2 varies according to the ratio c1/c2.

Analysis is performed under displacement control. Increments of vertical displacement are applied

at the nodes immediately under the position of the footing. The footing load is computed as the

summation of the vertical reaction forces at these nodes divided by the footing area, which is

expressed as

22

2

wB

RFq

+

=∑

(1)

where, q is the footing load, RF2 is the vertical component of the reaction force at each node

immediately under the position of the footing, B is the footing width, w is the width of the element

immediately adjacent to the edge of the footing, as shown in Figure 2. Equation (1) implies that

the edge of the strip footing is extended to the middle of the element immediately adjacent to the

edge of the footing. This is to account for the true footing area as a consequence of the finite

element model. A similar approach of radius extension for circular footings can be found in the

paper by Erickson and Drescher (2001). The loading process continues until the load-displacement

curve reaches a clear plateau which indicates soil failure. The load corresponding to the plateau is

the ultimate bearing capacity qu of the footing. The bearing capacity factor Nc* is then calculated

by

1

*

c

qN u

c = (2)

where, c1 is the undrained shear strength of the top clay.

The selection of the element size w affects the finite element solutions. Smaller elements around

the edge of footing will yield more accurate solutions because of the above-mentioned significant

displacement change (Day and Potts, 2000). However, this requires larger computational time.

Furthermore, divergence of the solutions was observed as the element size w becomes very small.


Homogenous clay soil, where c1 equals c2, was studied in order to investigate the influence of w.

The exact solution of Nc* for a strip footing over homogenous clay soil is (π+2). Study shows Nc

*

increases as w decreases: w=B/16 yields a factor of 5.086, which is 1.08% lower than the exact

solution, w=B/20 yields a factor of 5.127, which is 0.28% lower, and w=B/24 yields a factor of

5.150, which is 0.16% higher. Bearing capacity factors for w/B less than 1/24 are not available

because of divergence of the solutions. Element width w=B/22.2 has been adopted in this study,

which yields a factor of 5.146.

3. Results and discussion

Parametric study was carried out to investigate the bearing capacity factor Nc* as a function of H/B

and c1/c2. Seven ratios of H/B were considered: 0.125, 0.25, 0.5, 0.75, 1.0, 1.25, and 2.0. Nine

ratios of c1/c2 were considered: 5, 4, 3, 2, 1.5, 1, 0.8, 0.5, and 0.2. In total, 63 computations were

performed. The bearing capacity factors Nc* are presented in Table 1, and also shown graphically

in Figures 3.

0

1

2

3

4

5

6

7

8

9

10

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

H/B

Nc

*

C1/C2=0.2

C1/C2=0.5

C1/C2=0.8

C1/C2=1

C1/C2=1.5

C1/C2=2

C1/C2=3

C1/C2=4

C1/C2=5

Figure 3. Bearing capacity factor*

cN .

For cases where the top layer is weaker than the bottom layer (c1/c2 < 1), Nc* decreases as H/B

increases. For cases where the top layer is stronger than the bottom layer (c1/c2 > 1), Nc* increases

as H/B increases. Nc* approaches 5.146 for all cases, which indicates that the failure mechanism is

limited in the top layer and the whole soil can be treated as a homogenous soil using the properties

of the top soil only. According to Michalowski (2002), there exists a so-called critical depth where

the strength of the bottom layer does not affect the bearing capacity. In Figure 3, the critical depth

is the depth H where the curve of Nc* reaches 5.146. For strong-over-soft clay profile (c1/c2 > 1),

the larger the ratio c1/c2 is, the larger the critical depth. Whereas, for soft-over-strong clay profile

(c1/c2 < 1), the critical depth seems to be a constant around 0.75B. This observation is consistent

with the finding by Michalowski (2002).


(a) H/B=0.125 (b) H/B=0.75

(c) H/B=1.25 (d) H/B=2

Figure 4. Zones of plastic strain at failure for strong-over-soft clay (c1/c2 = 2).

(a) H/B=0.125 (b) H/B=0.50

(c) H/B=0.75 (d) H/B=1

Figure 5. Zones of plastic strain at failure for soft-over-strong clay1 (c1/c2 = 0.8).

1 Bright color inside the failure zones indicates region with lager plastic strain


Figure 4 illustrates the zones of plastic strain at failure for strong-over-soft clay profile. First, the

failure mechanism goes deeper and wider as H/B increases with both layers being involved. After

the thickness of the top layer exceeds a critical depth, the failure mechanism shrinks into the top

layer. For situations where the top layer is weaker than the bottom layer, the failure mechanism

does not change very much, as shown in Figure 5. The failure mechanism tends to be limited in

the top layer as H/B increases.

(a) strong-over-soft clay (c1/c2 = 2)

(b) soft-over-strong clay (c1/c2 = 0.8)

Figure 6. Vectors of displacement at failure (H/B = 0.75).


Displacement fields at failure for both strong-over-soft clay and soft-over-strong clay are shown in

Figure 6. The vectors indicate both the direction and the magnitude of soil movement. Given the

same thickness of the top layer, strong-over-soft clay has a larger area of soil movement than soft-

over-strong clay. Inspection of Figure 6(b) shows that the displacement at the top surface of the

first element adjacent to the edge of the footing changes direction abruptly, from vertically

downward to more than 45○ to the horizontal axis. The displacement change is smoother in the

case where the top clay is stronger than the bottom clay and less heave at the soil surface is

observed, as shown in Figure 6(a).

The finite element solutions are compared with published lower bound and upper bound solutions

by limit analysis in Table 1 and also in Figure 7. As is expected, the finite element solutions are

lying between the lower bound and upper bound solutions and show a favorable agreement with

their averages.

4. Final remarks

Parametric study was carried out to evaluate the bearing capacity of a strip footing over two-layer

clay soil. Finite element solutions for different combinations of layer thickness and soil strength

are presented in both tabular and graphical forms. At the same strength ratio, the bearing capacity

factor decreases as thickness of the top layer increases for a soft-over-strong clay profile, whereas

an inverse trend for a strong-over-soft clay profile. There exists a critical depth where the shear

strength of the bottom layer does not affect the bearing capacity and failure mechanism is

restricted only in the top layer. Different failure mechanisms and displacement fields are observed

for strong-over-soft clay profile and soft-over-strong clay profile. A comparison of the finite

element solutions with published limit analysis solutions shows a good agreement.

(a) H/B=0.125

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5

C1/C2

Nc

*

FEM

Low er bound (Merif ield, 1999)

Upper bound (Merif ield, 1999)

Average of low er and upper bounds

(Merif ield, 1999)

(b) H/B=0.25

0

1

2

3

4

5

6

7

0 1 2 3 4 5

C1/C2

Nc

*

FEM

Low er bound (Merifield, 1999)



(Merif ield, 1999)


(c) H/B=0.50

0

1

2

3

4

5

6

0 1 2 3 4 5

C1/C2

Nc

*

FEM




(Merif ield, 1999)

(d) H/B=0.75

0

1

2

3

4

5

6

0 1 2 3 4 5

C1/C2

Nc

*

FEM




(Merif ield, 1999)

(e) H/B=1.0

0

1

2

3

4

5

6

0 1 2 3 4 5

C1/C2

Nc

*

FEM




(Merif ield, 1999)

(f) H/B=1.5

0

1

2

3

4

5

6

0 1 2 3 4 5

C1/C2

Nc

*

FEM


Upper bound (Merifield, 1999)


(Merif ield, 1999)

(g) H/B=2.0

0

1

2

3

4

5

6

0 1 2 3 4 5

C1/C2

Nc

*

FEM




(Merif ield, 1999)

Figure 7. Comparison of bearing capacity factors.


Table 1. Bearing capacity factor *

cN

H/B C1/C2 Finite Element

Method Lower bound

(Merifield, 1999) Upper bound

(Merifield, 1999) Average (Merifield, 1999)

Upper bound (Michalowski, 2002)

5 1.387 1.30 1.55 1.43 1.520 4 1.656 1.56 1.82 1.69 1.766 3 2.083 1.97 2.27 2.12 2.166 2 2.896 2.73 3.09 2.91 2.933

1.5 3.674 3.45 3.93 3.69 3.681 1 5.146 4.94 5.32 5.13 5.141

0.8 6.173 5.87 6.36 6.11 6.217 0.5 8.560 7.78 8.55 8.16 9.391

0.125

0.2 9.355 7.78 8.55 8.17 9.842

5 1.655 1.60 1.85 1.73 1.906 4 1.944 1.87 2.12 1.99 2.149 3 2.376 2.27 2.56 2.42 2.533 2 3.159 3.01 3.34 3.17 3.246

1.5 3.875 3.65 4.08 3.87 3.912 1 5.146 4.94 5.32 5.13 5.141

0.8 5.932 5.51 6.25 5.88 5.991 0.5 6.519 5.99 6.52 6.26 6.561

0.25

0.2 6.520 5.99 6.52 6.26 6.561

5 2.212 2.16 2.44 2.30 2.579 4 2.500 2.44 2.74 2.59 2.817 3 2.933 2.84 3.16 3.00 3.177 2 3.656 3.52 3.89 3.70 3.800

1.5 4.256 4.07 4.48 4.28 4.329 1 5.146 4.94 5.32 5.13 5.141

0.8 5.411 4.98 5.49 5.24 5.313 0.5 5.427 4.98 5.49 5.24 5.313

0.5

0.2 5.427 4.98 5.49 5.24 5.313

5 2.743 2.64 2.98 2.81 3.190 4 3.039 2.96 3.28 3.12 3.420 3 3.462 3.36 3.72 3.54 3.750 2 4.122 4.00 4.37 4.18 4.299

1.5 4.614 4.44 4.94 4.69 4.708 1 5.146 4.94 5.32 5.13 5.141

0.8 5.163 4.98 5.49 5.24 5.141 0.5 5.163 4.98 5.49 5.24 5.141

0.75

0.2 5.163 4.98 5.49 5.24 5.141

5 3.259 3.10 3.54 3.32 3.768 4 3.559 3.46 3.83 3.65 3.988 3 3.966 3.89 4.24 4.07 4.299 2 4.557 4.44 4.82 4.63 4.746

1.5 4.942 4.77 5.18 4.97 5.046 1 5.146 4.94 5.32 5.13 5.141

0.8 5.146 4.94 5.30 5.12 5.141 0.5 5.146 4.94 5.30 5.12 5.141

1.00

0.2 5.146 4.94 5.30 5.12 5.141


Table 1. Bearing capacity factor *

cN (continued)

H/B C1/C2 Finite Element

Method Lower bound

(Merifield, 1999) Upper bound

(Merifield, 1999) Average (Merifield, 1999)

Upper bound (Michalowski, 2002)

5 3.765 4.295 4 4.060 4.497 3 4.431 4.767 2 4.926 5.141

1.5 5.145 5.141 1 5.146 5.141

0.8 5.146 5.141 0.5 5.146 5.141

1.25

0.2 5.146 5.141

5 4.233 3.89 4.56 4.23 4.863 4 4.494 4.24 4.84 4.54 4.965 3 4.816 4.69 5.15 4.92 5.187 2 5.145 4.87 5.31 5.09 5.141

1.5 5.145 4.87 5.31 5.09 5.141 1 5.146 4.94 5.32 5.13 5.141

0.8 5.146 4.94 5.30 5.12 5.141 0.5 5.146 4.94 5.30 5.12 5.141

1.50

0.2 5.146 4.94 5.30 5.12 5.141

5 4.956 4.61 5.32 4.96 4 5.142 4.81 5.32 5.06 3 5.145 4.81 5.27 5.04 2 5.145 4.81 5.27 5.04

1.5 5.145 4.81 5.26 5.04 1 5.146 4.94 5.32 5.13

0.8 5.146 4.94 5.30 5.12 0.5 5.146 4.94 5.30 5.12

2.00

0.2 5.146 4.94 5.30 5.12

5. References

1. Brown, J. D. and G. G. Meyerhof, “Experiment Study of Bearing Capacity in Layered Clays,”

Proc. of 7th Int. Conf. on Soil Mechanics and Foundation Engineering, Mexico, vol. 2, pp. 45-

51, 1969.

2. Burd, H. J. and S. Frydman, “Bearing Capacity of Plane-strain Footings on Layered Soils,”

Can. Geotech. J. Vol. 34, pp. 241-253, 1997.

3. Button, S. J., “the Bearing Capacity of Footings on a Two-layer Cohesive Subsoil,” Proc. of

the 3rd Int. Conf. on Soil Mechanics and Foundation Engineering, Zurich, vol. 1, pp. 332-335,

1953.

4. Chen, W. F. and H. L. Davidson, “Bearing Capacity Determination by Limit Analysis,” J.

Soil Mech. Found. Div., ASCE, vol. 99, no. 6, pp. 433-449, 1973.

5. Day, R.A. and D. M. Potts, “Discussion on ‘Observations on the computation of the Bearing

Capacity Factor Nγ by Finite Elements’ by Woodward & Griffiths, Geotechnique, Vol. 50,

No. 3, pp. 301-303.


6. Erickson, H. L. and A. Drescher, “Bearing Capacity of Circular Footings,” J. Geotech. Engrg.,

ASCE, vol. 128, no. 1, pp. 38-43, 2001.

7. Florkiewicz, A., “Upper Bound to Bearing Capacity of Layered Soils,” Can. Geotech. J., vol.

26, no. 4, pp. 730-736, 1989.

8. Griffiths, D. V., “Computation of Bearing Capacity on Layered Soil,” Proc. of 4th Int. Conf.

Num. Meth. Geomech., Z. Eisenstein, Ed., Balkema, Rotterdam, the Netherlands, pp. 163-

170, 1982.

9. Hanna, A. M. and G. G. Meyerhof, “Design Charts for Ultimate Bearing Capacity of

Foundations on Sand Overlying Soft Clay,” Can. Geotech. J., vol. 17, pp. 300-303, 1980.

10. Merifield, R. S., S. W. Sloan, and H. S. Yu, “Rigorous Plasticity Solutions for the Bearing

Capacity of Two-layered Clays,” Geotechnique, London, England, vol. 49, no. 4, pp. 471-

490, 1999.

11. Meyerhof, G. G., “Ultimate Bearing Capacity of Footings on Sand Layer Overlaying Clay,”

Can. Geotech. J., vol. 11, no. 2, pp. 223-229, 1974.

12. Meyerhof, G. G. and A. M. Hanna, “Ultimate Bearing Capacity of Foundations on Layered

Soils under Inclined Load,” Can. Geotech. J., vol. 15, pp. 565-572, 1978.

13. Michalowski, R. L., “Collapse Loads over Two-layer Clay Foundation Soils,” Soils and

Foundations, Vol. 42, No. 1, pp. 1-7, 2002.

14. Michalowski, R. L. and L. Shi, “Bearing Capacity of Footings over Two-layer Foundation

Soils,” J. Geotech. Engrg., ASCE, vol. 121, no. 5, pp. 421-428, 1995.

15. Reddy, A. S. and R. J. Srinivasan, “Bearing Capacity of Footings on Layered Clays,” J. Soil

Mech. Found. Div., ASCE, vol. 93, no. 2, pp. 83-99, 1967.

16. Shiau, J. S., A. V. Lyamin, and S. W. Sloan, “Bearing Capacity of a Sand Layer on Clay by

Finite Element Limit Analysis,” Can. Geotech. J., Vol. 40, pp. 900-915, 2003.

17. Terzaghi, K. and R. B. Peck, “Soil Mechanics in Engineering Practice,” New York:Wiley,

1948.

Bearing Capacity of Strip Footings by FEA

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