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

Transcription

1 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.. 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 (98) 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, 9; Reddy and Srinivasan, 97; Meyerhof, 97), limit analysis approach (Chen and Davidson, 97; Florkiewicz, 989; Michalowski and Shi, 99; Merifield, et al., 999; Michalowski, ; Shiau, et al., ), semiempirical approach (Brown and Meyerhof, 99; Meyerhof and Hanna, 978; Hanna and Meyerhof, 98), and finite element method (Griffiths, 98; Burd and Frydman, 997). 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.). The computational results are presented and compared with published lower bound and upper bound solutions by limit analysis. ABAQUS Users Conference 777

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. 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 c and c respectively. Soil is assumed to be weightless. Figure. Schematic diagram of the problem. Finite element mesh is illustrated in Figure. Due to symmetry, only half of the problem is modeled. The width of the footing B is m. The length of the finite element model is.b, and the height is 7.B. 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. Finite element mesh. 778 ABAQUS Users Conference

3 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 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 = kn/m, Poisson s ratio ν =.. The undrained shear strength of the top layer c is kn/m. The undrained shear strength of the second clay layer c varies according to the ratio c /c. 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 RF q = () B w + where, q is the footing load, RF 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. Equation () 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 (). 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 q u of the footing. The bearing capacity factor N c is then calculated by where, c is the undrained shear strength of the top clay. N ABAQUS Users Conference 779 q c u c = () 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, ). However, this requires larger computational time. Furthermore, divergence of the solutions was observed as the element size w becomes very small.

4 Homogenous clay soil, where c equals c, was studied in order to investigate the influence of w. The exact solution of N c for a strip footing over homogenous clay soil is (π+). Study shows N c increases as w decreases: w=b/ yields a factor of.8, which is.8% lower than the exact solution, w=b/ yields a factor of.7, which is.8% lower, and w=b/ yields a factor of., which is.% higher. Bearing capacity factors for w/b less than / are not available because of divergence of the solutions. Element width w=b/. has been adopted in this study, which yields a factor of... Results and discussion Parametric study was carried out to investigate the bearing capacity factor N c as a function of H/B and c /c. Seven ratios of H/B were considered:.,.,.,.7,.,., and.. Nine ratios of c /c were considered:,,,,.,,.8,., and.. In total, computations were performed. The bearing capacity factors N c are presented in Table, and also shown graphically in Figures. 9 Nc 8 7 C/C=. C/C=. C/C=.8 C/C= C/C=. C/C= C/C= C/C= C/C= H/B Figure. Bearing capacity factor N. For cases where the top layer is weaker than the bottom layer (c /c < ), N c decreases as H/B increases. For cases where the top layer is stronger than the bottom layer (c /c > ), N c increases as H/B increases. N c approaches. 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 (), there exists a so-called critical depth where the strength of the bottom layer does not affect the bearing capacity. In Figure, the critical depth is the depth H where the curve of N c reaches.. For strong-over-soft clay profile (c /c > ), the larger the ratio c /c is, the larger the critical depth. Whereas, for soft-over-strong clay profile (c /c < ), the critical depth seems to be a constant around.7b. This observation is consistent with the finding by Michalowski (). c 78 ABAQUS Users Conference

5 (a) H/B=. (b) H/B=.7 (c) H/B=. (d) H/B= Figure. Zones of plastic strain at failure for strong-over-soft clay (c /c = ). (a) H/B=. (b) H/B=. (c) H/B=.7 (d) H/B= Figure. Zones of plastic strain at failure for soft-over-strong clay (c /c =.8). Bright color inside the failure zones indicates region with lager plastic strain ABAQUS Users Conference 78

6 Figure 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. The failure mechanism tends to be limited in the top layer as H/B increases. (a) strong-over-soft clay (c /c = ) (b) soft-over-strong clay (c /c =.8) Figure. Vectors of displacement at failure (H/B =.7). 78 ABAQUS Users Conference

7 Displacement fields at failure for both strong-over-soft clay and soft-over-strong clay are shown in Figure. 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 softover-strong clay. Inspection of Figure (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 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 (a). The finite element solutions are compared with published lower bound and upper bound solutions by limit analysis in Table 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.. 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. Nc C/C (a) H/B=. Low er bound Upper bound Nc 7 Low er bound Upper bound C/C (b) H/B=. ABAQUS Users Conference 78

8 Nc Low er bound Nc Low er bound Upper bound C/C (c) H/B=. Upper bound C/C (d) H/B=.7 Nc Low er bound Nc Low er bound Upper bound C/C (e) H/B=. Upper bound C/C (f) H/B=. Nc Low er bound Upper bound C/C (g) H/B=. Figure 7. Comparison of bearing capacity factors. 78 ABAQUS Users Conference

9 H/B C /C Table. Bearing capacity factor Finite Element Method Lower bound Upper bound Average Upper bound (Michalowski, ) N c ABAQUS Users Conference 78

10 Table. Bearing capacity factor N c (continued) H/B C /C Finite Element Lower bound Upper bound Average Upper bound Method (Michalowski, ) References. Brown, J. D. and G. G. Meyerhof, Experiment Study of Bearing Capacity in Layered Clays, Proc. of 7 th Int. Conf. on Soil Mechanics and Foundation Engineering, Mexico, vol., pp. -, 99.. Burd, H. J. and S. Frydman, Bearing Capacity of Plane-strain Footings on Layered Soils, Can. Geotech. J. Vol., pp. -, Button, S. J., the Bearing Capacity of Footings on a Two-layer Cohesive Subsoil, Proc. of the rd Int. Conf. on Soil Mechanics and Foundation Engineering, Zurich, vol., pp. -, 9.. Chen, W. F. and H. L. Davidson, Bearing Capacity Determination by Limit Analysis, J. Soil Mech. Found. Div., ASCE, vol. 99, no., pp. -9, 97.. Day, R.A. and D. M. Potts, Discussion on Observations on the computation of the Bearing Capacity Factor N γ by Finite Elements by Woodward & Griffiths, Geotechnique, Vol., No., pp ABAQUS Users Conference

11 . Erickson, H. L. and A. Drescher, Bearing Capacity of Circular Footings, J. Geotech. Engrg., ASCE, vol. 8, no., pp. 8-,. 7. Florkiewicz, A., Upper Bound to Bearing Capacity of Layered Soils, Can. Geotech. J., vol., no., pp. 7-7, Griffiths, D. V., Computation of Bearing Capacity on Layered Soil, Proc. of th Int. Conf. Num. Meth. Geomech., Z. Eisenstein, Ed., Balkema, Rotterdam, the Netherlands, pp. - 7, Hanna, A. M. and G. G. Meyerhof, Design Charts for Ultimate Bearing Capacity of Foundations on Sand Overlying Soft Clay, Can. Geotech. J., vol. 7, pp. -, 98.. 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. 9, no., pp. 7-9, Meyerhof, G. G., Ultimate Bearing Capacity of Footings on Sand Layer Overlaying Clay, Can. Geotech. J., vol., no., pp. -9, 97.. Meyerhof, G. G. and A. M. Hanna, Ultimate Bearing Capacity of Foundations on Layered Soils under Inclined Load, Can. Geotech. J., vol., pp. -7, Michalowski, R. L., Collapse Loads over Two-layer Clay Foundation Soils, Soils and Foundations, Vol., No., pp. -7,.. Michalowski, R. L. and L. Shi, Bearing Capacity of Footings over Two-layer Foundation Soils, J. Geotech. Engrg., ASCE, vol., no., pp. -8, 99.. Reddy, A. S. and R. J. Srinivasan, Bearing Capacity of Footings on Layered Clays, J. Soil Mech. Found. Div., ASCE, vol. 9, no., pp. 8-99, 97.. 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., pp. 9-9,. 7. Terzaghi, K. and R. B. Peck, Soil Mechanics in Engineering Practice, New York:Wiley, 98. ABAQUS Users Conference 787