A Review of Bearing Capacity of Shallow Foundation on Clay Layered

Soils Using Numerical Method Masyitah Md Nujid Postgraduate Student

Faculty of Civil Engineering, Universiti Teknologi MARA (Pulau Pinang), 13500 Malaysia

e-mail: masyitahnujid@gmail.com

Mohd Raihan Taha

Professor and Head, Department of Civil and Structural Engineering, Universiti Kebangsaaan Malaysia, 43650 Malaysia

e-mail: profraihan@gmail.com

ABSTRACT The bearing capacity of layered soils studies have been conducted using different approach whether theoretical, experimental and combination of both. Many empirical equations have been developed from correlation between in situ data and/or laboratory data in order to understand soil deformation when subjected to loading. However the complex soil behaviour could not be understood only with raw/input data obtained from site investigation or correlation equation. Thus with helpful in numerical modelling, stress-strain soil behaviour is well predicted, design and interpreted using appropriate soil model. Finite element method becomes powerful tool in geotechnical modelling where soil bodies are discretised into small finite elements. The FEM deals with complex geometry, mixed boundary conditions and material nonlinearity. Studies on bearing capacity and factors affect soil movements of strip footing on clay layered soils are summarized and presented. Different factors in design and modeling simulation approach may give different calculated of bearing capacity on clay layered soils using numerical method. KEYWORDS: Bearing capacity; clay layered soils; and finite element method.

INTRODUCTION Most bearing calculations of shallow footing has been analysed using conventional theory in

which bearing capacity factors are adopted. The elasticity theory is implemented in analysis where soil is assumed to be isotropic, homogeneous and rigid for simplification in geotechnical engineering practice. However in natural soils are deposits in layers which the stiffness and strength increase linearly with depth due to increasing overburden pressure (Davis & Booker, 1973, Gibson, 1967, Stark & Booker, 1997). There are two cases being considered as inhomogeneous layer soil (Bandini & Pham, 2011, Szypcio, 2006, Bowles 1982). First case when top layer weaker than bottom layer and second case is top layer stronger than bottom layer. It is generally assumed that general bearing capacity failure occurs in the upper layer which calculation is based on soil strength parameters of top layer of strong soil with thickness of the

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Vol. 19 [2014], Bund. D 812 layer is large compare to width of footing (Bandini & Pham, 2011, Zhu, 2004). However if the thickness of the top layer is smaller to the foundation width, this approach may not be suitable. The failure mechanism may involve two or more layers if a foundation resting on surface of layered soils where thickness of top layer is not large and resting on weak soil or bedrock (Zhu, 2004, Verma et al., 2013).

Many studies in bearing capacity have been conducted using different approach in analytical and numerical methods. In present day numerical analysis such as finite element difference and finite element method become a powerful technique in geotechnical problems to analyze complex behavior of stress and strain due to external loading. Nonlinear stress and strain relationship in complex geomaterial of soil is easily being predicted in finite element method using appropriate constitutive equation. There are five steps involved in finite element method which are: (1) discretization of the continuum, (2) derivation of the stiffness relations of a finite element, (3) assembly of the elements to form global stiffness matrix equations, (4) solution for the nodal displacements and (5) determination of the stresses in the elements. The advantage of this method is that it can elegantly handle complex geometry, mixed boundary conditions and material nonlinearity.

In present study, this paper is attempted to summarize selected studies on bearing capacity on clay layered soils using numerical method in analysis. In addition this paper will look into factors which govern the behavior of layered soil.

Numerical Analysis In numerical analysis the bearing capacity factors are obtained from load-displacement curve.

The analysis is done in function of strength ratio, thickness ration and embedment effect. There are few methods available in numerical analysis such as finite element difference, finite element method, boundary element and discrete element. The behavior of layered soils can be modelled using linear to nonlinear constitutive model which available in finite element package of geotechnical application software such as PLAXIS, ABAQUS, FLAC and COMSOL Multiphysics.

Geometry of Layered Soils

Geometry of layered soils in numerical modelling can be modelled whether in two dimensional (2D) or three dimensional (3D), plane strain problem with z-axis is zero, asymmetrical problem, and footing is founded on surface of elastic half space soil or footing is embedded in depth from ground surface. Figure 1 shows geometry of layered soils in modelling where width of footing B depends on type of shallow foundation. The width of strip footing is in range of 2 m to 5 m (Zhu, 2004, Han & Jiang, 2011). In order to consider effect of thickness ratio between top and bottom layers in layered soils, thickness of top layer, h play a vital role to indicate whether soil is homogeneous or inhomogeneous. If h < B the soil is inhomogeneous and h > B the soil is homogeneous. The identification of single layer or multiple layers soil indicate the calculation of bearing capacity and of footing (Bandini & Pham, 2011, Zhu, 2004, Han & Jiang, 2011, Szypcio, 2006, Bowles 1982 ).

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Figure 1: Geometry of Layered Soils Model

Zhu (2004) in his parametric study on undrained layered soils modelled the geometry of soil problem in finite element with length and height was 12.5B and 7.5B respectively. Han & Jiang (2011) modelled the soil geometry with 20 m (W) and 6 m (H). There was no strict rule to follow in order to draw the soil problem in finite element package. However the size of large finite element model was assigned to keep the boundary conditions at bottom and right side from restricting the soil movement due to footing load (Zhu, 2004).

Soil Properties and Soil Model

It was found that weightless soil in cohesive soil did not affect the predicted bearing capacity response in small deformation analysis but oppose with large deformation analysis (Wang & Carter, 2002). For strip footing lay on surface of layered soils, undrained shear strength of top and bottom layers are cu1 and cu2 respectively. Basically undrained shear strength cu1of top layer will be assigned with certain value and undrained shear strength of bottom layer cu2 varies according to the ratio cu1/cu2. This is due to consider effect of strength ratio in calculation bearing capacity of footing (Zhu, 2004, Han & Jiang, 2011). Gourvenec (2003) in studied undrained failure of circular footing on surface under combined load had considered the strength of soil increase linearly with depth in non-homogeneity of soil. A nonhomogenity factor was chosen as two to express the degree of soil strength. It was found that the ultimate vertical load increase in nonhomegeneity and ultimate horizontal load was not affected by foundation geometry or nonhomogeneity. This study was adopted theory as mentioned by Davis & Booker, 1973, Gibson, 1967, Stark & Booker, 1997) for layered soils. Most researchers (Bandini & Pham, 2011, Zhu, 2004, Han & Jiang, 2011,Gourvenec, 2003) found elasticity parameters which were Young’s Modulus and Poisson ratio did not affect so much in results as long these values remain constant throughout in analysis. For undrained clay layers typical ranges of values for modulus elasticity and Poisson’s ratio ware 10-20MPa and 0.3-0.5 (Zhu, 2004, Han & Jiang, 2011,Gourvenec, 2003). A value of 0.495 Poisson’s ratio for undrained clay was adopted in numerical analysis than

Layer 1 cu1, φ1

Layer 2 cu2, φ2

B

h H=2B

Vol. 19 [2014], Bund. D 814 conventional value (0.50) because to avoid numerical difficulties (Burd & Frydman, 1997). Table 1 shows typical range of values for static stress-strain modulus Es and Poisson’s ratio µ for selected soils (Bowles, 1982).

Table 1: Typical ranges of values for the static stress-strain modulus Es and Poisson’s

ratio for selected soils (Bowles, 1982)

Soil Es (MPa) µ Clay Very soft Soft Medium Hard Sandy

2-15 5-25 15-50 50-100 25-250

0.4-0.5 (saturated) 0.1-0.3 (unsaturated) - - - - 0.2-0.3

Glacial till Loose Dense Very dense

10-153 144-720 478-1440

- - -

Loess 14-57 0.1-0.3 Sand Silty Loose Dense Coarse (e=0.4-0.7) Fine-grained (e=0.4-0.7)

7-21 10-24 48-81

0.2-0.4 - - - 0.15 0.25

Sand and gravel Loose Dense

48-144 96-192

Shale 144-14400 Silt 2-20 0.3-0.35 Rock - 0.1-0.4 Ice - 0.36 Concrete - 0.15

In finite element package of geotechnical application, user can choose soil model from linear problem to nonlinear complex problem of soil behaviour. Most available soil models in all applications software are linear elastic (Von Mises, Tresca), Mohr-Coulomb and elastoplastic (Drucker-Prager, Lade-Duncan, Matsuoka-Nakai, Modified Cam Clay, PLAXIS Soft Soil, PLAXIS Hardening Soil). Brinkgreve (2005) listed five aspects in choosing constitutive soil model in order to model complexity of soil behaviour are; 1) influence of water, 2) stiffness of soil, 3) irreversible deformation, 4) soil strength and 5) time-dependent. Due to shortcoming problem in parameters selection, it was easy to select Mohr-Coulmb (Zhu, 2004, Potts, 2003) and Tresca models (Han & Jiang, 2011, Bandini & Pham, 2011) in analysis and modelled soil problem as an isotropic elastic-perfectly plastic material satisfying the Tresca failure criterion and the Mohr-Coulomb failure criterion. There are five parameters involve in the first order of Mohr-Coulomb model in analysis (Es, µ - stiffness parameters, c, φ - strength parameters and ψ-dilation angle). The differences between the Tresca and the Mohr-Coulomb model are failure criterion and strength parameters involved in analysis. Table 2 shows advantages and limitation of failure models (Chen & Baladi, 1985).

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Table 2: Advantages and limitations of failure soil models (Chen & Baladi, 1985)

Advantages Limitations One-parameter model

1. Von Mises - simple and smooth

-only for undrained saturated soils (total stress)

2. Tresca - simple

-only for undrained saturated soils (total stress) and corners

3. Lade-Duncan - simple, effect of intermediate principle

stress and smooth

-only for cohesionless soils

Two-parameters model 1. Mohr-Coulomb

- Simple and its validity is well established for many soils

-corners and neglect effect of intermediate principal stress

2. Drucker-Prager - Simple, smooth and can match with

Mohr-Coulomb with proper choice of constants

-circular deviatoric trace which contradicts experiments for cohesionless soils

3. Lade’s - Simple, smooth, curve meridian and

wider range of pressures than the other criteria

-only for cohesionless soils

Boundary Condition and Mesh Discretization

The finite element solutions were affected by selection boundary conditions and mesh refinement and size (Bandini & Pham, 2011, Han & Jiang, 2011). Soil symmetrical in geometry only half of it was modelled and applying prescribed displacements at nodes to the right of centerline (Griffith, 1982) and a fixed boundary condition was assigned at bottom of soil problem (Griffith, 1982, Zhu, 2004, Han & Jiang, 2011). The two vertical boundaries were restrained from horizontal movement and only vertical directions were free (Zhu, 2004 and Han & Jiang, 2011). Bandini & Pham (2011) assigned fixed condition at bottom and right side of geometry model and roller condition at asymmetry and footing side for embedded footing problem. For soil - rough footing interaction, the horizontal displacement at the grid points under the base of the footing was restrained (Griffith, 1982, Zhu, 2004, Han & Jiang, 2011). In contrast to soil – smooth footing interaction, the horizontal displacements at the nodes were allow moving freely and vertically were displaced. For non-uniform mesh discretization, coarse element mesh would be assigned at bottom, left and right sides of soil geometry model. Under footing base, the soil element would be discretized into medium fine and near at edge of footing, it was very fine mesh because of significant displacement change in this region (Zhu, 2004, Bandini & Pham, 2011). The element sizes affect the finite element solution where divergence could occur at very small size but require larger computational time (Griffiths, 1982). Similarly studied by Zhu (2004) which adopted eight and six node plane strain quadrilateral element had done trial and error to choose right element size to yield a factor of Nc* at conventional value for strip footing over homogeneous clay soil. It showed that the Nc* increase as element size decrease. Thus element width equal to B/22 which yields a factor of 5.146.

Vol. 19 [2014], Bund. D 816 Analysis of Bearing Capacity

Griffith (1982) described techniques in analyzing bearing pressure which could be done in finite element package. The techniques adopted in finite element method were load control or displacement control. For perfectly flexible foundation, load control allows the applied stresses to be adjusted finely by the application of nodal forces and failure under this technique requires number of iterations per load step and time consuming whereas rationale in choosing displacement control due to numerical control and physical reality. The displacement control was applicable for perfectly rigid footing where equilibrium and yield satisfied with little iteration even failure had reached. This technique was adopted in ABAQUS and FLAC studies conducted by (Zhu, 2004, Bandini & Pham, 2011) and Han & Jiang (2011).

Kinematic Approach

For bearing capacity on layered soils problem upper bound solution with kinematic approach is chosen with simple assumption applicable. This approach yields collapse failure load are very close to the true collapse loads on elasto-plastic bodies (Burd & Frydman, 1997). The theorem adopted power dissipated by kinematic admissible field velocity equated by power dissipated externally to resemble true load for rigorous upper bound. This approach satisfies compatibility condition, the flow rule and boundary condition (Shiau et al, 2003). Kinematic analyses may be demonstrated to provide upper bound solutions only for the case of associated materials, whereas the dilation angle of real granular soils is generally substantially less than the angle of friction. An analysis that adopts a dilation angle equal to the friction angle will also provide an upper bound to the correct solution for a non-associated material, but this upper bound will probably become increasingly conservative as the difference between the two angles increases. A modified friction angle will be used in kinematic solution for a non-associated material (Burd & Frydman, 1997). Kinematic solution is good compared to conventional analysis because the construction of collapse mechanisms is based on laboratory.

Factors Affecting Bearing Capacity of Shallow Foundation on

Clay Layered Soils A few studies have been conducted to estimate bearing capacity failure of a rough strip

footing resting on two clay layers with a parametric study on effect of ratio of the top layer thickness to the width of the footing, the cohesion ratio of the top to bottom soil layer, influence of inclined bedrock and footing embedment (Zhu 2004, Merifield & Nguyen, 2006, Bandini & Pham 2011, Han & Jiang 2011).

Effect of Thickness Ratio and Strength Ratio

Zhu (2004), Bandini & Pham (2011), Merifield & Nguyen (2006) in parametric studies used ABAQUS finite element software to evaluate the bearing capacity factor Nc* in function of strength ratio and thickness ratio of top layer for strong overlay weak clay layers and weak overlay strong clay layers. For weak over strong clay layers (cu1/cu2 < 1) and thickness ratio (h/B >1) the Nc* decrease and for strong over weak clay layers (cu1/cu2>1) and the thickness ratio (h/B>1) the Nc* increase. The results indicated the Nc* affected by strength ratio and at critical depth, the failure mechanism restricted only in top layer and shear strength at bottom layer did

Vol. 19 [2014], Bund. D 817 not contribute to bearing capacity failure. The failure mechanism and displacement field profile depended on layered soil.

Effect of Inclined Bedrock

Figure 2 shows numerical model in FLAC 2D software where B, Hd and α were footing width, minimum depth from the edge of the footing base to the bedrock and bedrock inclination angle (Han & Jiang,2011). It was found that insignificant influence of bedrock inclination when the depth to width ratio equal to one and modified bearing capacity Nc* was closely to theoretical value (Table 3). The modified bearing capacity factor Nc* decreased as the inclination angle of the underlying inclined bedrock increased and at same inclination angle of the inclined bedrock, the bearing capacity factor decreased as the depth to width ratio increased. The results showed well compared between limit analysis (Merifield, 1999) and finite element software (Zhu, 2004) for soft clay over firm soil.

Figure 2: Numerical model (Han & Jiang, 2011)

Figure 2: Numerical model (Han & Jiang, 2011)

Table 3: Modified bearing capacity factors Nc* (Han & Jiang, 2011)

α (°) Hd/B

0 10 20 30 45

0.125 8.981 7.074 6.026 6.026 5.804 0.25 6.508 5.958 5.595 5.595 5.560 0.5 5.404 5.315 5.298 5.329 5.306 1.0 5.211 5.210 5.211 5.206 5.208

Effect of Footing Embedment

Figure 3 shows numerical model in ABAQUS software conducted by (Bandini & Pham, 2011). They did a series of parametric study on effect of embedment (D/B) varying from 0 to 1 on the ultimate bearing capacity of rigid strip footing on surface and embedded (D/B) in two layered clay soils with relatively different shear strengths (cu1/cu2).

Soil Bedrock

B

α

Hd

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Figure 3: Numerical model for D/B = 0.5 and H/B = 0.5 (Bandini & Pham, 2011)

Figure 4 shows the finite element results for surface and embedded footings in were within

the narrow ranges of the rigorous lower and upper bounds solutions comparing from previous study and the results consistent with (Zhu, 2004, Merifield, 2011) for embedment ratio (D/B = 0) where cu1/cu2< 1 and H/B< 1, the Nc* is affected by the relative thickness of the upper clay layer and for cu1/cu2<1 and H/B>1 general shear failure occurs in the weaker layer. For cu1/cu2>1 there was a limit value of cu1/cu2 after which partial or full punching through the upper clay and yielding of the weaker clay occur. The finite element simulation results of the study conducted above were compared from upper, average and lower bound limit analysis showed a good agreement (Zhu, 2004, Merifield, 1999) and semi empirical approach found to be very conservative.

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Figure 4: Numerical results: a) for surface strip footings (D/B = 0) in two layer clay soils, and b) embedded footings in homogeneous clay (cu1 = cu2) compared with rigorous lower (LB) and upper

(UB) bounds (Bandini & Pham, 2011)

Failure Mechanism Failure mechanism is defined where line of velocity discontinuity and the soil mass above

that undergoes unrestricted flow at failure in comparison with the rest of the soil mass (Manoharan & Dasgupta, 1995). The modes of failures (Merifield & Nguyen, 2006) for clay layered soils can be categorized into three; i) general shear, ii) partial punching shear and iii) full punching shear. The modes of the failure mechanism depend on strength and thickness ratio. Merifield & Nguyen (2006) conducted a series of 2D and 3D numerical analysis on bearing capacity of two layered clays for different type of footing, strength and thickness layer. It was found for strip footing on layered clay with nonhomogeneous strength affect the failure mechanism pattern. Figure 5 shows a failure mechanism for strip footing on clay layers soil. For strong over soft clay layers with strength ratio (cu1 / cu2 > 2.5) full punching failure occurs. The vertical and horizontal restrict from entering bottom layer of soft soil and produce a deep zone of plastic shearing in the zone with small amount of bulging which could be found immediately adjacent to the footing and significant yielding below the upper layer. A small vertical movement of top layer to bottom layer is result for partial punching shear failure ( 1≤ cu1 / cu2 ≤ 2.5). For soft/weak overlay strong clay layer failure mechanism only happen in top layer as thickness ratio increase (Zhu, 2004 and Merifield & Nguyen, 2006).

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(a)

(b)

Figure 5: Vectors displacement at punching shear failure for cu1 / cu2 = 5: a) strong over soft clay (H/B = 0.5), and b) strong over soft clay (H/B = 1.0) (Merifield & Nguyen, 2006)

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Figure 6 shows vector displacement which consist of magnitude and direction of vector for soil movement by Zhu (2004). It was found for a given same thickness top layer with different strength indicated that strong over soft layer (c1 / c2 > 1) has smoother and large area of soil movement than soft over strong layer soil (c1 / c2 < 1). It was also found in soft over strong layer that the soil downward movement direction changed vertically to inclined 45° to the horizontal surface.

(a)

(b) Figure 6: Vectors displacement at failure for h/B = 0.75: a) strong over soft clay (c1 / c2 = 2), and b) soft over strong clay (c1 / c2 = 0.8) (Zhu, 2004)

Figure 7 and 8 show the displacement vectors of soil over horizontal and inclined bedrock subjected to vertical loading studied by Han & Jiang (2011). In horizontal bedrock layer, symmetrical heave was found near edge footing. At shallow depth the soil movement was bounded with hard bedrock at bottom layer and vector displacement area became deeper and large when bedrock was located at very deep. The wedge failure developed when the bedrock was located at a depth, Hd equal to width of footing B. For inclined bedrock it was found that unsymmetrical heave was formed on left and right side of displacement zones. The deeper of

Vol. 19 [2014], Bund. D 822 displacement zone was affected by angle of bedrock position and failure mechanism for inclined bedrock showed less effect at a larger depth.

(a)

(b)

Figure 7: Vectors displacement at failure with horizontal bedrock a) Hd/B = 0.125 and b) Hd/B = 1.0 (Han & Jiang, 2011)

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(a)

(b) Figure 8: Vectors displacement at failure with a inclined horizontal bedrock at inclination angle of α = 30° a) Hd/B = 0.125 and b) Hd/B = 1.0

SUMMARY AND CONCLUSION The paper summarized studies on influence factors which governed the bearing capacity of

clay layered soils using numerical method. Numerical analysis (finite difference and finite element method) is a computer simulation in prediction real soil behavior on field with certain assumption in analysis for examples are geometry, boundary condition, discretization element and size. Besides choosing right solution of constitutive relation of stress and strain in soil model, it also could predict the real soil behavior itself and interaction with structures. Advantages numerical analyses compared to conventional methods are accurately predicting real soil behavior

Vol. 19 [2014], Bund. D 824 with appropriate constitutive models and boundary conditions to various stages of construction. In addition to couple different physics phenomena like pore pressure, consolidation and swelling. It is superior analysis with short time require for analysis complex soil problems.

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