J. Cent. South Univ. (2014) 21: 3983−3992 DOI: 10.1007/s11771-014-2386-5

Numerical and experimental analyses for bearing capacity of rigid strip footing subjected to eccentric load

LU Liang(卢谅)1, 2, WANG Zong-jian(王宗建)3, K. ARAI4

1. School of Civil Engineering, Chongqing University, Chongqing 400045, China;

2. Key Laboratory of New Technology for Construction of Cities in Mountain Area of Ministry of Education (Chongqing University), Chongqing 400045, China;

3. Institute of Geotechnical Engineering, Chongqing Jiaotong University, Chongqing 400074, China; 4. NPO Research Institute of Ground Disaster Prevention of Fukui Region, Fukui 910-0004, Japan

© Central South University Press and Springer-Verlag Berlin Heidelberg 2014

Abstract: A footing may get an eccentric load caused by earthquake or wind, thus the bearing capacity of footing subjected to eccentric load become a fundamental geotechnical problem. The conventional limit equilibrium method used for this problem usually evaluates the material properties only by its final strength. But the classical finite element method (FEM) does not necessarily provide a clear collapse mechanism associated with the yield condition of elements. To overcome these defects, a numerical procedure is proposed to create an explicit collapse mode combining a modified smeared shear band approach with a modified initial stress method. To understand the practical performance of sand foundation and verify the performance of the proposed procedure applied to the practical problems, the computing results were compared with the laboratory model tests results and some conventional solutions. Furthermore, because the proposed numerical procedure employs a simple elasto-plastic model which requires a small number of soil parameters, it may be applied directly to practical design works. Key words: bearing capacity; eccentric load; slip surface; shear band; finite element method; model test

1 Introduction

The bearing capacity under eccentric loading is a fundamental geotechnical problem in engineering practices. For instance, when performing the pseudo-static analysis of seismic bearing capacity, a footing receives a moment load caused by horizontal forces acting on a structure, which is replaced by an eccentric load. In practical design works, two simple methods have been used for eccentrically loaded footings. One method assumes the linear distribution of footing pressure, and verifies the maximum pressure within the bearing capacity calculated for centrally loaded footing of the same width. The other method was developed by MEYERHOF which regards an eccentrically loaded footing as a centrally loaded footing [1]. Since the method is conservative for cohesive soils and overestimate the bearing capacity for frictional soils [2], many researchers have made considerable efforts to this bearing capacity problem.

A lot of model tests on footings were carried out to

assess the actual behavior of strip footing subjected to combined vertical, moment and horizontal loads [3−6]. Most researches successfully discussed the changes of slip surface with increasing eccentricity and verified the reduction of ultimate bearing capacity due to load eccentricity. PRAKASH et al [7] used the concept of one sided failure mechanism to prove the bearing capacity of eccentrically loaded strip footings considerably affected by the value of e/B. The common method based on the limit equilibrium method that defines the load conditions through failure locus usually does not obtain a reasonable explanation of application of the MEYERHOF solution. MICHALOWSKI et al [8] examined the MEYERHOF’s suggestion which is a reasonable account of eccentricity in bearing capacity calculations using the kinematic approach of limit analysis. HOULSBY et al [9] introduced a new scaling procedure which plays the same role for upper bounds as the effective-width method. Many researches also used limit analysis to provide suitable bearing capacity coefficients by failure mechanisms [10−12]. But subjected to Mohr-Coulomb material, the limit analysis

Foundation item: Projects(cstc2012jjA0510, cstc2013jcyjA30014) supported by Chongqing Natural Science Foundation in China; Project(CDJZR12200011)

supported by the Fundamental Research Funds for the Central Universities in China; Project(KJTD201305) supported by the Innovation Team Building Programs of Chongqing Universities in China; Project supported by the Scientific Research Foundation for the Returned Oversea Chinese Scholars

Received date: 2013−05−29; Accepted date: 2013−11−03 Corresponding author: LU Liang, Lecturer, PhD; Tel: +86−18602319075; E-mail: luliangsky@163.com

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based on upper or lower bound theorems in plasticity has not completely overcome the difficulty that the limit theorems cannot be proven without the normality rule in plasticity.

To overcome the defects of conventional limit equilibrium method which evaluates the material properties only by its final strength and limit analysis in which the normality rule may not hold for soil, the finite element methods developed. MASSIH et al [13] computed the failure loads of a rough rigid strip footing subjected to eccentric loading using a finite difference approach approximate to those given by the limit analysis for ponderable soil and greater than those by the limit analysis for imponderable soil. Many researches indicate that the finite element method does not necessarily provide a reasonable collapse mechanism [14−15]. In some finite element methods, HU et al [16] monitored a closely confined failure mode by the flow of soil represented by the incremental displacement vectors. Clearly, it is not denoted by the stress conditions usually used in the conventional design methods. Because it is not easy to duplicate the rotation of principal stresses from the below part of footing to the peripheral region in bearing capacity problems. Adaptive finite element method appears to require a lot of numerical efforts and to contain a certain numerical difficulty in some cases [16−17].

The aim of this work is to propose a numerical procedure to estimate the bearing capacity of strip footing under eccentric loading by an actually formed failure mode. Based on Mohr-Coulomb yield criterion with a simple non-associated plastic flow rule, the numerical procedure attempts to provide an appropriate bearing capacity which is supported by an explicit collapse mechanism based on a smeared shear band approach and a modified initial stress method. The collapse mode is represented by the stress yield condition of soil elements as analogous with the conventional design method. In addition, the proposed numerical procedure requires a small number of soil parameters to reduce the numerical efforts and to be applied directly to practical design works. Laboratory model test results are included to examine the numerical characteristics and applicability of the proposed procedure. 2 Numerical procedure

Combining a modified smear shear band approach and a modified initial stress method [18−19], a numerical procedure is proposed to determine the bearing capacity of eccentric loaded rigid footing directly by an explicit shape of collapse mode. The conditions to get such a collapse mode are as follows: 1) assume an active wedge below footing; 2) treat the yielding mass as a stratified

material resulting from the smeared shear band approach, and 3) perform rigorously the nonlinear FE analysis based on the modified initial stress method. The proposed procedure employs a simple constitutive model which requires a small number of material parameters so that it may be applied to practical design works. 2.1 Analysis model

To relate the proposed procedure to conventional stability analysis, Mohr-Coulomb and Coulomb yield criteria are employed respectively to plane strain soil mass and friction interface between structure and soil.

Subjected to Coulomb interface, we employ a simple non-associated flow rule or plastic potential QC defined as

QC=τstg (1) where g is a hypothetical parameter which is not cited actually, because QC is used only by its differential form. Referring to the literature for the thin layer element, an elasto-plastic stress−strain relationship is given as [20] epep δ][δ εDσ

T

CT

CCep

}{][

}{}{][

σD

σσDDD

FFQ

1

C

}{][

σ

DQ

where [D] is elastic matrix under plane strain conditions. Based on the Coulomb yield criterion stC F

tant c , the following equation can be determined:

epst

ept

eps

st

t

s

δ

δ

δ

0)21)(1(

tan)1(

)21)(1(

tan

0)21)(1(

)1(

)21)(1(

0)21)(1()21)(1(

)1(

δ

δ

δ

EE

EE

EE

(2) where {δσ} and {δεep} are stress and elasto-plastic strain increments, respectively; [Dep] denotes elasto-plastic referred to coordinate s−t in Fig. 1(a), in which the plus sign (+) is for the positive τst and minus sign (−) for negative τst specified in Section 2.2.2; E is elastic modulus; μ is Poisson ratio; is friction angle.

2.2 Modified shear band approach

When shearing a finite size of soil element, it is well known that we often observe a shear band or slip surface as shown in Fig. 1(b). Since our main concern is to get a practical design procedure, a most fundamentalexpression of the inclination of angle of shear band is

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Fig. 1 Definition of shear band: (a) Coordinates of interface element; (b) Formation of shear band; (c) Stratified material

π/4+/2. 2.2.1 Proposal of shear band approach

Modifying the smeared shear band approach proposed by PIETRUSZCZAK et al [21] and combining a simple plastic flow rule for Mohr-Coulomb soil mass, the proposed procedure enables to get a distinct collapse mode analogous to that assumed in the common classic theory by assuming the yield soil element as a stratified element shown in Fig. 1(c). According to the assumption, the proposed procedure considers its elasto- plastic matrix [Dep] to be equal to that as given by Eq. (2) based on an elasto-plastic stress−strain relationship, which is different from the elastic-perfectly plastic model assumed in the previous works [18−19]. 2.2.2 Direction of shear band

Generally, a set of two shear bands or slip surfaces A−A′ and B−B′ as shown in Fig. 2(a), are possible for a finite soil element according to the principal stress state. Considering the formation of active wedge, take the right half of it for instance, the shear band within the active wedge below footing assumed as B−B', the angle shown in Fig. 2(b) of shear band to the horizon is determined as α−θ on the contrary, the shear band outside of the active wedge is A−A', which corresponding angle β is –α−θ, where is angle of the major principal stress from

Fig. 2 Direction regulation of shear band (a) and slip surface

(b) in an element

vertical axis. The same principle applies in the other half of active wedge. Note that compressive stress is defined as positive here and that shear stress τst is negative along α−θ line and positive along –α–θ line. 2.3 Modified initial stress method

The original initial stress method is based on an iterative procedure. From mathematical viewpoint, it is a special application of the modified Newton- Raphson method. This method treats the nonlinearity as piecewise linear, and does not create the collapse mode even though assuming the stratified material. These difficulties are avoided by introducing a modified initial stress method, which finds directly the initial stresses without iterative procedure. Figure 3(a) defines the actual stress of initial state {σI}, yield stress {σA}, actual stress of plastic equilibrium state {σB}, elastic stress {σE}, virtual initial stress {σ0}, total strain {ε}, elastic strain {εe}, and elasto-plastic strain {εep}. Yield stress {σA} is isolated by NAYAK et al [22]. To determine the direction of shear band, the major principal stress is determined by using yield stress {σA}, and uses it throughout the succeeding loading stages.

Firstly, initial stress vector in s−t coordinate is

{σst0}={σs0, σt0, τst0}T (3)

When the stress state of an element has attained to

yield state at the present loading stage, the basic equation in the initial stress method is given as

{r}i={σst0}i−[T]i

−1{{σE}i−({σA}i+{δσB}i)} ={σst0}i−[T]i

−1([D]i−[Dep]i)([B]i{[K]−1({δf}+ Σj[B]j

T [T]j {σst0}j Aj)}i−{εe}i) =0 (4)

where {σE}={σ}n−1+[D][B]{δu}, {r}i is residual, [B]i is matrix for calculating strain components from nodal displacements, [K] is global stiffness matrix, [T] is coordinate transform matrix, {δf} is load increment vector, Aj is area of the element, and suffixes i and j denote element numbers. {εe} in Eq. (4) and Fig. 3 is

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Fig. 3 Modified initial stress method: (a) Initial stress method;

(b) Initial stress method at a loading stage

calculated as [D]−1({σA}−{σ}n−1).

Referring to Eq. (2), components in [Dep] is the same as the components of elastic matrix [D] except the third row components. This means that both σs0 and σt0 in Eq. (3) vanish both in interface and plane strain yield elements. The application of Eq. (4) reduces the numerical effort and clarifies the mechanical meaning of initial stresses. For instance, Eq. (4) for finite element i with respect to unknown {σst0}j is given as

δij−[T]i

−1([D]i−[Dep]i)[B]i{[K]−1[B]jT[T]jAj}i3(0, 0, τst0)j=

[T]i−1([D]i−[Dep]i)([B]i{[K]−1{δf}i−{εe}i})3 (5)

where |·|3 denotes the third component of vector, etc. When solving Eq. (4), the constant numbers of unknown τst0 must be assumed. Thus, the following additional iteration is required for determining the yield finite elements and tensile elements. The numerical steps during a typical loading stage are summarized as follows. 1) Performing an elastic analysis by using actual load increment {δf }, calculate {σE} and {δε} in Fig. 3(b); 2) Find the yield finite elements in which {σE} violates the yield criterion; 3) For the yield elements, calculate yield stress {σA} both from {σE} and the preceding stress state; 4) Concerning {σA}, calculate the direction of the major principal stress θ, and find shear band inclination angle according to Section 2.2.2; 5) Determine {σst0} by solving Eq. (4); 6) Find the yield elements once more by performing an elastic analysis by use of both {δf} and {σst0} determined at step 5). When finding new yield elements, determine {σst0} subjected to the total yield elements including the new yield elements. Repeat this procedure until no new yield element is found; 7) Calculate necessary state variables, based on the final results obtained at step 6), {σB}, settlements, and so on. Note that a stress state is assumed to move along the yield surface after yielding. To avoid the tensile stress occurs in some finite elements, it is necessary to constraint σt. In each loading stage, giving δσt≥0 and repeat the above procedure until no new tensile element is found.

3 Laboratory model tests 3.1 Test configuration

A series of laboratory model tests were executed in a soil container made of a steel frame, as shown in Fig. 4(a), having inside dimensions of 900 mm in width, 200 mm in thickness and 750 mm in depth. One of the sidewall of the soil container was constructed using composite glass plate with grids of 50 mm size for ease of observing the deformation of subsoil during testing. Considering the friction between subsoil and soil container, a thin rubber membrane is used which smeared with a thin layer of silicon grease on the surface of sidewalls of soil container. Since the walls of the test container were firmly held in position by steel melting and the wall friction was kept to minimum by using the thin rubber membrane, and since soil container was sufficiently rigid, the strains occurred only in the length and depth and the strain along width was ignored to remain plane strain conditions in the soil models.

Earth pressure sensors were installed in the subsoil as shown in Fig. 4(b) to measure the vertical earth pressure. One accelerometer was installed on the loading plate to determine the inclination of loading plate θ as sin−1(d/g), where d and g are the readings taken by the accelerometer respectively for inclined and horizontal states. During testing, the model footing was loaded on a certain point on the rigid loading plate with 150 mm in width using a Bellofram cylinder. The loading plate has a groove at the loading point so that the rod of cylinder should not slip out of the loading point. The eccentric distance e is specified as 20 mm or 50 mm. The settlement of the point loaded on the footing was measured by dial gauges placed on the footing. Many markers were settled on the sidewall of soil container as shown in Fig. 4(c) to observe the deformation of subsoil. In the figure, some color sands were also scattered on the soil layer in order to visualize the formation of active wedge under footing. A sand paper was glued onto the undersurface of loading plate, so that it may simulate rough condition and friction below the base of footing. 3.2 Test procedures

Using Toyoura dense sand in Japan, three series of model tests were performed by changing eccentric distance of load applied on footing. Case 1: Load was applied in the centerline of footing; Case 2: Eccentric load deviated 2 cm from the centerline, expressed as e= 2 cm; and Case 3: Eccentric load deviated 5 cm from the centerline, represented by e=5 cm. Table 1 lists the soil parameters, where c and f are Mohr-Coulomb strength parameters which were obtained by a direct shear test, E is elastic modulus back-calculated from the footing

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Fig. 4 Schematic view of test equipment (Unit: mm): (a) Soil container; (b) Installation of earth pressure sensors; (c) Distribution of

markers

Table 1 Soil properties

Parameter Value

c/kPa 0

f/(°) 41.4

E/kPa 3924

γ/(kN·m−3) 15.93

settlement observed in the model test, is Poisson ratio which was assumed to be an empirical value and is unit weight.

The soil model was produced using normal Toyoura sand by the spreading equipment, which is a vessel of triangular prism with a spindly opening. The spreading equipment was hung on a hoist in height of 1 m by which the sands drops into the soil container in the height of 0.625 m uniformly through a screen. The vessel was used to adjust the density of filling by altering the magnitude of opening. In the model tests, a vessel with 2 mm of opening settled on the height of 0.5 m above the surface of sand layer was selected for the dense sand. Note that during the process of production of foundation subsoil, some instruments such as the earth pressure sensors, markers and color sands, etc., were placed according to

their scheduled position. After completing the filling process, a loading plate

as a footing was put on the surface of sand model, and loads were applied on the loading plate or footing by a Bellofram cylinder. During loading, load was increased in a small increment until the failure occurs. Each increment of load was applied to the loading plate and kept constant for 3 min. Successively, the settlement of loading plate and the earth pressure corresponding to current load were measured. Repeat this procedure until large and sharp settlement occurs. 3.3 Test results

The test results of footing pressure loaded on the surface of loading plate versus settlement of loading plate, collapse mode of foundation and earth pressure distribution of soil and so on are compared with the analytic results later. Note that model tests executed in the plane strain condition, the unit of load is kN/m. The settlement means the value monitored by dial gauges at the point of loading. As shown in Fig. 5, the color sands evidently visualize the formation of active wedge in the dense sand and the markers monitor the deformation of soil.

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Fig. 5 Formation of active wedge and deformation of loading plate and subsoil: (a) Case 1; (b) Case 2; (c) Case 3

4 Numerical analysis and results discussions 4.1 Preparations of calculation

Numerical analysis is performed to the model test results to explain the practicality and applicability of the proposed procedure by comparing each group results. Employing the smeared shear band approach and the modified initial stress method, the procedure is particularly useful for representing a clear and full collapse mode and for identifying the distributions of stress and displacement. In the FE meshing, the footing or loading plate is modeled by beam elements and represented by its elastic modulus E=2.1108 kPa, cross area A=0.012 m2 and moment of inertia I=1.4410−7 m4. Without considering the intricate coupling interactions, some interface elements are simply set between footing and subsoil, in which shear modulus G is given as E/2(1+μ), where E and μ are elastic modulus and Poisson ratio of subsoil, respectively. The material parameters are given in Table 1. When considering anisotropic initial stresses of foundation soil, the proposed procedure provides very low bearing capacity compared with conventional solutions. Thus, we give isotropic initial stresses which are equal to the overburden pressure.

To get a global collapse mode, the proposed procedure requires to assume the shape of active wedge, because it is difficult to duplicate the rotation of principle stresses from the below part of footing to the peripheral region. We assume that angle of active wedge is π/4+f/2 regarding the vertical footing pressure as the major principle stress. Note that the active wedge in symmetry and shows only the case of footing under central loading, called as Case 1. However, when a footing is subjected to eccentric vertical load, the pattern of active wedge usually changes in its depth and its position of apex. At the present stage, it is open to question to isolate the shape of active wedge for the case of eccentric loading by considering mechanically reasonable basis. Thus, it is necessary to select the shape of active wedge which gives the minimum bearing capacity from various patterns of active wedge, to ensure that the bearing capacity given by the proposed procedure is conservative.

In the analysis, we discuss the patterns of active wedge in Fig. 6 by varying its depth and eccentricity to the central line. Figure 6(a) shows two patterns based on the change of eccentric distance from the apex to the center line of active wedge, where δ represents the eccentric distance of active wedge. Considering that the value of δ varies with the subdivision of FE mesh, δ1 is selected as 0.75 cm and δ2 is 3.5 cm for Case 2 of eccentric distance e is 2 cm; While δ1 is 2.5 cm and δ2 is 5 cm for Case 3 of e=5 cm in the analysis. Figure 6(b) gives three types of depth of active wedge, where H denotes the depth of active wedge of Case 1.

Fig. 6 Verification of shape of active wedge: (a) Eccentric

distance from apex to center active wedge; (b) Changing depth

of active wedge

4.2 Comparisons between numerical and

experimental results Figure 7 shows the yield regions of three cases

corresponding to their bearing capacities respectively in the dense Toyoura sand. Because the proposed procedure does not perfectly duplicate the strain localization behavior, the procedure cannot produce an infinite plastic shear flow of subsoil. The bearing capacity must be determined based on the distribution of yield elements. In the figure, the solid line within each finite element represents that the corresponding element has yielded and the inclination of solid line illustrates the direction of shear band formed in each yield element. Since the shear band means the slip surface in each finite element, the

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line connected by shear band is thought to correspond to a continuous slip surface as assumed in the limit equilibrium analysis. As shown in Fig. 7, when the current footing load reaches the denotative value, the yield elements begin to be connected to form a collapse mode. The corresponding value is determined as the bearing capacity in the analysis. This means that the bearing capacity is determined as the value when a full collapse mode is created for the first time. Table 2 shows the results of potential bearing capacity for all the cases changing the shape of active wedge as illustrated in Fig. 5. Analyzing all the results, the ratio of difference between the minimum solution for one case among the all potential bearing capacities and the nearest neighboring solution obtained from other cases is no more than 10%. This proves that the minimum solution calculated from the five patterns of active wedge in Fig. 5 is extremely close to the actual value, so it is

Fig. 7 Analytical yield regions represented by stress yield

elements: (a) Case 1; (b) Case 2; (c) Case 3

Table 2 Potential bearing capacity changing shapes of active

wedge (kN/m)

Eccentric distance Depth Case 2 Case 3

δ1 H — 13.52

δ1 1H/3 36.87 3.07

δ1 2H/3 41.78 4.30

δ2 H — 8.60

δ2 1H/3 33.80 3.07

δ2 2H/3 38.10 4.92

Ultimate bearing capacity — 33.80 3.07

representative to find the minimum solution from the five patterns of active wedge in Fig. 6 to determine the actual bearing capacity of rigid strip footing subjected to eccentric load. Based on the definition, in the following analysis, the potential bearing capacity of all shapes of active wedge for each case is calculated and then the minimum solution is selected as the actual bearing capacity. The represented yield regions in Fig. 7 are obtained corresponding to the actual bearing capacity determined by this method.

Figure 8 compares the experimental relationship between footing load and settlement with the numerical result of actual bearing capacity for three cases with different eccentricities, where Ti represents the bearing capacity define by the turning point of load−settlement curve obtained from the model tests, Qi is the bearing capacity calculated from numerical analysis, in which suffixes i corresponds respectively to Case i defined before. Some conventional limit equilibrium solutions are also shown, where QT denotes TERZAGHI solution only concerning central loading, QM2 and QM3 represent MEYERHOF solution corresponding to Cases 2 and 3. As shown in Fig. 8, the experimental bearing capacity is determined by the evident transition point of settlement. This is because when a global collapse mode is formed, the foundation comes into collapse state which induces a distinct settlement. The proposed procedure cannot

Fig. 8 Comparisons of numerical and experimental results with

conventional solutions

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provide a clear turning point of settlement as in the experiment. For Case 1 (central loading), no calculated bearing capacity is shown. Because, as shown in Fig. 7(a), the development of collapse mode in this case is extended to the boundary of soil container, and is restrained by limited space. This restriction of boundary may increase largely the calculated bearing capacity up to 240 kPa which is outside of the indicial load in Fig. 8. It is well known that the model tests should try to overcome the influence of boundary constraints by the optimization of model size, but the relative research [18] showed that an explicit active wedge cannot be formed for some sand if the width of loading plate is too small. Considering the effect of size, the loading plate of loading plate has been set at 150 mm wide in this work. So the restriction of boundary cannot be completely eliminated, the experiment also provides the bearing capacity higher than TERZAGHI solution. For Case 2 (e=2 cm), the proposed procedure obtains a similar bearing capacity with the experimental result. Compared with MEYERHOF solution, both the numerical analysis and experiment provide higher results due to the same reason, i.e., the development of slip surface is restrained by the boundary of soil container, as shown in Fig. 7(b). For Case 3 (e=5 cm), the proposed procedure and experiment provide the similar results, which are smaller than MEYERHOF solution. This is because, as shown in Fig. 7(c), the large eccentric distance induces a collapse mode formed in a little shallow region of subsoil. It is concluded from Fig. 8 that both the numerical analysis and the experiment show that the bearing capacity decreases with the increase in the eccentric distance.

Compared with Fig. 5(b) which shows the deformation of loading plate and subsoil for Case 2, Fig. 9(a) shows the displacement field in Case 2 which is monitored by the movement of markers (see Fig. 3). Figure 9(a) shows that the ground on the left side edge of loading plate has a larger settlement while the part on the right side deforms upward gradually from the level because of the effect of eccentric load. This result gives the similar tendency with the calculated displacement field shown in Fig. 9(b).

The shear strain is calculated by using the discretization procedure usually employed in FEM on the basis of monitored displacement of subsoil in Fig. 9(a). Considering the minimum quadrangle constituted by four markers as a finite element, strain components {εx, εy, εxy} are calculated from the nodal displacements as [B]×{U}, where [B] is matrix for calculating strain components from nodal displacements; {U} is monitored displacements in Fig. 9. Shear strain εs is defined as

εs=[{(εx−εm)2+(εy−εm)2+0.5γxy

2}2/3]1/2, εm=(εx+εy)/2 (6)

The collapse mode by shear strain calculated as

Fig. 9 Comparison of displacement fields for Case 2 between (a)

experimental results and (b) numerical analysis

above, as shown in Fig. 10(a), cannot be directly compared with the mode, as shown in Fig. 10(b), given by the proposed procedure. The proposed procedure represents the collapse mode by the distribution of yield elements and determines the bearing capacity as stated before. But, it is difficult to verify the calculated collapse mode from the results of laboratory model test, because the model test does not provide such a stress distribution. Though the collapse mode by shear strain calculated as above cannot be directly compared with the mode by the proposed procedure, but the accordance by the tendency of development of two different types of collapse mode is especially found. As an example, Fig. 10(a) shows the observed collapse mode by the shear strain distribution in Case 2, where the magnitude of shear strain increases in the density of shade. The shear strain obtained from experiment concentrates on the right of loading plate and most of them distribute in the shallow position, which is considerably accordance with the tendency of collapse mode represented by the distribution of stress yield elements as shown in Fig. 10(b).

Figure 11 compares the vertical earth pressures calculated by the proposed procedure and the experimental results at two loading stages before and after collapse. As shown in Fig. 11(a), both numerical and experimental results show that before collapse, the earth pressure becomes higher on the left edge of loading plate than that on the right edge due to the eccentric load. Since the subsoil before collapse is assumed to be elastic

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Fig. 10 Comparison of collapse mode for Case 2 between shear

strain distribution from experiment (a) and yield elements

distribution from numerical analysis (b)

Fig. 11 Comparisons of vertical earth pressure distributions

between experimental and numerical results: (a) Loading stage

before failure (Case 2); (b) Loading stage after failure (Case 2);

(c) Loading stage after failure (Case 3)

state, the stresses mainly concentrate on the region just underneath the loading point. In Fig. 11(b), after collapse the distribution of earth pressure becomes approximately

symmetric. This suggests that the subsoil under the edge of loading plate comes into plastic state, and that the subsoil cannot sustain the concentrated stress. The calculated and monitored earth pressures in Case 3 are also compared in Fig. 11(c). Compared with Case 2, the monitored earth pressure on the left side is higher because of a larger effect of eccentricity. The earth pressures induced by the large eccentric distance, concentrate on the region below the loading point without the tendency of symmetry. The numerical results are lower than experimental those especially at the part of loading point. This result may be attributed two reasons: 1) the proposed procedure applies an ideal elasto-plastic constitutive model, it cannot reflect the gradual destruction of subsoil under rigid strip footing; 2) the proposed procedure does not sufficiently stimulate the phenomenon of stress concentration of particulate media for particulate soil media. 4.3 Discussion

The calculated and experimental results have the similar tendency that the bearing capacity decreases with increasing the load eccentricity. This is due to two reasons. One is the increase of eccentric distribution of contact pressure below footing by the eccentric load, as demonstrated by the earth pressure distribution in Fig. 11. The other is that the shape of active wedge induced by eccentric load makes the position of slip surface shallower than the one no eccentricity of load, as shown in Fig. 7. Figure 5 also shows that the active wedge becomes shallower due to eccentric load. As seen in Fig. 9, the calculated yield region tends to distribute more deeply below footing than the shear strain distribution monitored in the experiment. This is because the vertical pressure must reach lower subsoil due to the vertical equilibrium condition, and the pressure makes lower subsoil yield. The calculated displacement field also shows little deformation of lower subsoil. For the vertical earth pressures as shown in Fig. 11, there are distinct different tendency before and after failure. Since the subsoil is still in the elastic state before failure, while after failure, the subsoil near the top of active wedge induces the stress concentration due to the property of particulate media of the dense sands. The analysis cannot perfectly simulate this characteristic of subsoil, and it provides a lower value due to the assumption of elasto-plastic continuum. 5 Conclusions

1) A numerical procedure, based on Mohr- Coulomb and Coulomb yield criteria respectively for soil mass and friction interface between soil and footing, is presented for estimating the bearing capacity considering the

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stiffness of material and collapse pattern. To fill a gap existing between conventional stability analysis and classical FEM, the procedure provides a collapse mechanism analogous to a slip surface assumed in conventional stability analysis by assuming a shear band for yield element and by employing a modified initial stress method. Such a definition of collapse mode is different from most applications of FEM which tend to express the collapse mode by the distribution of shear strain or displacement. This characteristic indicates the possibility of applying the procedure to the stability analysis which takes stiffness and deformation of material into consideration, for instance, earth reinforcement problems.

2) In order to verify the precision and practicality of the proposed procedure, a series of laboratory model tests were conducted. It is found from the test results that the shapes and locations of active wedge at failure point are different for various eccentric conditions and the bearing capacity decreases with increasing the load eccentricity.

3) Most cases show that the proposed procedure is capable of simulating the subsoil behavior to a fairly good level of accuracy. Only few cases, for example, Case 1 shows that the proposed procedure provides a higher bearing capacity than conventional solution due to the restriction of boundary, approximate to the tendency obtained by model test. By comparing the calculated and experimental footing load-settlement curves, the procedure provides the bearing capacity which is in good agreement with the experiments in most cases. It further verifies that the procedure is able to give realistic predictions and supply a useful engineering tool for the design of foundation under eccentric loading. It is demonstrated that the proposed procedure can be applied quantitatively to experimental or actual bearing capacity problems, because it requires only some fundamental soil parameters such as elastic modulus and Mohr-Coulomb strength parameters. References [1] MEYERHOF G G. The bearing capacity of footings under eccentric

and inclined loads [C]// Proceedings 3rd International Conference

Soil Mechanics Foundations Engineering. Zurich: Organizing

Committee ICOSOMEF, 1953, 1: 440−445.

[2] PEKER A, SALENCON J. Seismic bearing capacity of shallow strip

foundations on clay soils [C]// Proceedings of International

Workshop on Seismology and Earthquake Engineering, Mexico,

1991: 297−304.

[3] BUTTERFIELD R, GOTTARDI G. A complete three-dimentional

failure envelope for shallow footing on sand [J]. Géotechnique, 1994,

44(1): 181−184.

[4] MARTIN C M. Physical and numerical modeling of offshore

foundations under combined loads [D]. UK: University of Oxford,

1994.

[5] MAUGERI M, MUSUMECI G, NOVITA D, TAYLOR C A. Shaking

table test of failure of a shallow foundation subjected to an eccentric

load [J]. Soil Dynamics and Earthquake Engineering, 2000, 22:

435−444.

[6] PATRA C R, DAS, B M, BHOI M and SHIN E C. Eccentrically

loaded strip foundation on geogrid-reinforced sand [J]. Geotextiles

and Geomemberanes, 2006, 24: 254−259.

[7] PRAKASH S, SARAN S. Bearing capacity of eccentrically loaded

footings [J]. Journal of the Soil Mechanics and Foundations Division:

ASCE, 1971, 97(1): 95−117.

[8] MICHALOWSKI R L, YOU L. Effective width rule in calculations

of bearing capacity of shallow footings [J]. Computers and

Geotechnics, 1998, 23: 237−253.

[9] HOULSBY G T, PUZRIN A M. The bearing capacity of a strip

footing on clay under combined loading [C]// Proceedings of the

Royal Society of London: Series A. London, 1999, 455(1983):

893−916.

[10] DORMIEUX L, PECKER A. Seismic bearing capacity of foundation

on cohesionless soil [J]. Journal of Geotechnical Engineering, 1995,

121(3): 300−303.

[11] YANG Xiao-li, YIN Jian-hua, LI Liang. Influence of a nonlinear

failure criterion on the bearing capacity of a strip footing resting on

rock mass using a lower bound approach [J]. Canandian

Geotechnical Journal, 2003, 40(3): 702−707.

[12] YANG Xiao-li, SUI Zhi-rong. Seismic failure mechanisms for loaded

slopes with associated and nonassociated flow rules [J]. J Cent South

Univ Technol, 2008, 15: 276−279.

[13] MASSIN D S Y A, SOUBRA A H. Numerical simulations for the

bearing capacity of strip footings [J] Conference Proceedings,

Advances in Shallow Foundations, ASCE, 2007, 234(5): 1−10.

[14] de BORST R, VERMEER P A. Possibilities and limitations of finite

elements for limit analysis [J]. Géotechnique, 1984, 34(2): 199−210.

[15] FRYDMAN S, BURD H J. Numerical studies of bearing capacity

factor N [J]. Journal of Geotechnical and Geoenvironmental

Engineering: ASCE, 1997, 123(1): 20−29.

[16] HU Y, RANDOLPH M F. Deep penetration of shallow foundations

on non-homogeneous soil [J]. Soils and Foundations, 1998, 38(1):

241−246.

[17] ZIENKIEWICZ O C, HUANG M. Localization problems in

plasticity using finite elements with adaptive remeshing [J].

International Journal for Numerical and Analytical Methods in

Geomechanics, 1995, 19(2): 127−148.

[18] LU Liang, ARAI K, WANG Zong-jian. Laboratory model test and

numerical analysis of bearing capacity of rigid strip footing [J].

Journal of Applied Mechanics: JSCE, 2007, 10: 351−362.

[19] LU Liang ARAI K, WANG Zong-jian, NISHIYAMA R. Laboratory

model test and numerical analysis of bearing capacity of rigid strip

footing on slope [J]. Journal of Applied Mechanics: JSCE, 2008, 11:

399−410.

[20] ZIENKIEWICZ O C, VALLIAPPAN S, KING I P. Elastoplastic

solutions of engineering problems “initial stress”, finite element

approach [J]. International Journal for Numerical Methods in

Engineering, 1969, 1: 75−100.

[21] PIETRUSZCZAK S, MROZ Z. Finite element analysis of

deformation of strain-softening materials [J]. International Journal

for Numerical Methods in Engineering, 1981, 17: 327−334.

[22] NAYAK G C, ZIENKIEWICZ O C. Elasto-plastic stress analysis: A

generalization for various constitutive relations including strain

softening [J]. International Journal for Numerical Methods in

Engineering, 1972, 5: 113−135.

(Edited by HE Yun-bin)