# Bearing Capacity Factors for Eccentrically Loaded Strip Footings Using Variational Analysis

Authors: Xinggao Li
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Bearing capacity factors for eccentrically loaded strip smooth footings on homogenous cohesive frictional material are deduced by the variational limit equilibrium method and by assuming general shear failure along continuous curved slip surface. From the calculated results, the effective width rule suggested by Meyerhof for bearing capacity factors due to cohesion of soil is justified, and the superposition principle of bearing capacity for eccentrically loaded strip smooth footings is derived together with the bearing capacity factors for cohesion and unit weight of soil. The two factors are represented by soil strength parameters and eccentricity of load. The bearing capacity factor related to unit weight for cohesionless soil is less than that for cohesive frictional soil. The reason for this discrepancy lies in the existence of the soil cohesion, for the shape of the critical rupture surface of footing soil depends on both soil strength parameters rather than on friction angle alone in the previous limit equilibrium solutions. The contact between footing and soil is decided by both the load and the mechanical properties of soil. Under conditions of higher eccentricity and less strength properties of soil, part of the footing will separate from the underlying soil.


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## Body

## 1. Introduction

The ultimate bearing capacity of a surface strip footing, subjected to a vertical load and resting on a ponderable cohesive frictional soil, has been studied by numerous investigators. Based on the solution techniques used, analytical solutions to the bearing capacity problem can be classified into three groups, namely, slip-line method [1–4], limit analysis [4–9], and limit equilibrium method [10]. In recent years, numerical methods, such as the finite element method [11, 12] and the finite difference method [13], have been widely used to compute the bearing capacity of strip footings. Most of these studies assume that the load applied to the footing is symmetric, and a few investigators deal with solutions to eccentrically loaded strip footings. A useful hypothesis was suggested by Meyerhof [10] to account for eccentricity of load, in which the footing width is reduced by twice the eccentricity to its “effective” size. This hypothesis was examined by Michalowski and You [14] using the kinematic approach of limit analysis for associative materials, and it is found that the effective width rule yields a bearing capacity equivalent to that calculated based on the assumption that the footing is smooth. In this paper, the variational limit equilibrium (VLE) method, which does not depend on the associative flow rule, is employed to study bearing capacities of eccentrically loaded strip smooth footings. Based on calculated results, the validity of the effective width rule is justified from another approach, and the bearing factors for eccentrically loaded strip smooth footings with respect to mechanical properties of footing soil and eccentricities of load are presented and analyzed.The VLE method was first put forward by Kopácsy [15]. Thereafter, many authors [16–27] used it to deal with problems of soil stability analysis. Compared with previous limit equilibrium method, the VLE solutions of bearing capacity problems make no prior assumptions with respect to the shape of the rupture surface of footing soil, as well as the distribution of normal stress along it. It is necessary to point out that Garber and Baker [17] and Dixit and Mandal [23] had studied the bearing capacity of the symmetrically loaded strip smooth footing using the VLE method. Whereas influences of the eccentricities of load on bearing capacities of strip smooth footings are investigated herein using the variational limit equilibrium method and analysis of contact between footing and soil with respect to the eccentricities of load and strength parameters of soil are analyzed.


## 2. Problem Definition

A strip footing is put on a homogeneous and isotropic soil mass with horizontal ground surface, as given in Figure1. The footing is l in width, and the underlying soil has an effective unit weight γ and shear strength parameters c and φ (cohesion and friction angle of soil). Acting on the footing soil is the vertical downward load (Q) with its point of application offsetting d from the right side of the footing. The load Q that the footing soil can bear safely is to be found by the variational limit equilibrium method.Figure 1
Surface footing onc-φ soil.To simplify the analysis, the following assumptions are made.(1)
The problem is considered to be a two-dimensional plane strain problem;(2)
The footing-soil interface is smooth.(3)
The failure of the footing soil system is characterized by the existence of a well-defined failure pattern, which consists of a continuous slip surface connecting one edge of the footing to the ground surface, and failure is accompanied by a substantial rotation of the foundation.(4)
Mohr-Coulomb’s failure criterion is assumed to be applicable along a potential slip surface of footing soil.


## 3. Mathematical Statement of the Problem

### 3.1. Limiting Equilibrium Equations of Footing Soil

As shown in Figure2, the soil underneath the footing is in the limiting equilibrium state, and E is the origin of the coordinate system XEY and y(x) is the equation of the slip surface OA. Taking the soil in OCAO area as an isolated body, the three equations of static equilibrium can be established. By ∑X=0, ∑Y=0, and ∑ME=0, one gets
(1)Q-∫s(τsinα+σcosα)ds+∫xOxA(γy)dx=0,∫s(σsinα-τcosα)ds=0,(2)Qd-∫s[(σsinα-τcosα)y+(τsinα+σcosα)x]ds+∫xOxA(xyγ)dx=0,
where σ and τ are the normal stress and tangential stress on the slip surface, respectively, xO and xA are the x coordinates of points O and A, s is the arc length along the curve y(x), and tanα=dy/dx. Equations (1) represent conditions of vertical and horizontal force equilibrium and (2) represents the condition of moment equilibrium of the sliding soil mass. The moments are taken about point O, the origin of the coordinate system XEY.Figure 2
Force diagram of surface footing.Mohr-Coulomb’s failure criterion is satisfied along the slip surfacey(x), which connects point O, one of the footing edges, to point A at the ground surface. Consider
(3)τ(x)=c+σ(x)tanφ.
Along the slip surface y(x), there exist
(4)dssinα=dy=y′dx,dscosα=dx.
Introducing (3) and (4) into (1) and (2), it follows that
(5)Q=∫xOxA[(c+σtanϕ)y′+σ-γy]dx,(6)∫xOxA[σ(y′-tanϕ)-c]dx=0,(7)d∫xOxA[(c+σtanϕ)y′+σ]dx-d∫xOxA(γy)dx-∫xOxA{[σ(y′-tanϕ)-c]y}dx-∫xOxA{[σ(tanϕy′+1)+cy′]x}dx+∫xOxA(xyγ)dx=0.Thus, the three equations, (5)–(7), of the limiting equilibrium of the sliding body OCAO are derived. In the present formulation, no constitutive law beyond Mohr-Coulomb’s yield condition is included. Consequently, no constrains are placed on the character of the critical rupture surface, y(x), and the critical normal stress along it, σ(x), except for the overall equilibrium of the falling section.


### 3.2. Variational Analysis

Observing (5)–(7), one realizes that the load Q is a functional, a function of two functions, y(x) and σ(x), and is transcribed as the constrained variational extreme-value problem of undetermined boundaries. One variable endpoint of the slip surface is point O shown in Figure 2, whose coordinate is (xO,0), xO is undetermined; another endpoint is point A at the ground surface, whose coordinate is (xA,0), xA is undetermined. According to the variational method of the functional with constraints, the following auxiliary functional J* is constructed to convert the above restrained extremal problem into that without constraints. Consider
(8)J*=∫xOxAFdx,(9)F=F0+λ1F1+λ2F2,F0=(c+σtanϕ)y′+σ-γy,F1=[σ(y′-tanϕ)-c],F2=xyγ-[σ(y′-tanϕ)-c]y-[σ(tanϕy′+1)+cy′]x,
where λ1 and λ2 are the Lagrange undetermined multipliers. The equation of the slip surface y(x) and the normal stress distribution σ(x) on the slip surface must satisfy the following conditions:(1)
Euler’s differential equations for the functionalF(10)∂F∂σ-ddx(∂F∂σ′)=0,where σ′=dσ/dx(11)∂F∂y-ddx(∂F∂y′)=0,(2)
the integration constraint equations: (6) and (7),(3)
transversality conditions of variable endpoints
(i)
at PointO(12)(F-y′∂F∂y′-σ′∂F∂σ′)|x=xOδxA+∂F∂σ′|x=xOδσO=0,(ii)
at pointA(13)(F-y′∂F∂y′-σ′∂F∂σ′)|x=xAδxA+∂F∂σ′|x=xAδσA=0,where δ is the variational operator and σO=σ(x=xO), σA=σ(x=xA).Solving Euler’s equations (10) and (11), the equations of the slip surface and the normal stress distribution along the slip surface can be derived. Introducing the following coordinate transformation:
(14)x-1λ2=rcosθ=x0,y-λ1λ2=rsinθ=y0,
where (r,θ) is polar coordinate system in the Cartesian coordinate system X0O0Y0 with respect to point O0 (see Figure 2). Introducing (9) into (10) and using the coordinate transformation, (14), the following equation results:
(15)r=r0exp[(θ0-θ)tanφ],
where (r0,θ0) are undetermined coefficients. Similarly, the combination of (11), (9), (13), and (14) yields
(16)σ=Const·e2θtanφ-ctanφ+γ·rO·exp[(θO-θ)tanφ]9tan2φ+1(sinθ-3tanφcosθ),
in which Const is the integration constant and is expressed as
(17)Const=(ctanθA1-tanφtanθA+ctanφ-γ·rO·exp[(θO-θA)tanφ]9tan2φ+1×(sinθA-3tanφcosθA)ctanθA1-tanφtanθA)×(exp(2θAtanφ))-1.
Equation (15) is the equation of the slip surface, and (16) and (17) are the expressions of the normal stress on the slip surface.


### 3.3. Expressions of the LoadQ

The following nondimensional variables are introduced to make analysis convenient:(18)rO¯=rOl,c¯=c(γl),ψ=tanφ.
Equations (14)–(18) are substituted into (5) and (6) resulting in
(19)Q¯=Qγl2=f(rO¯,θO,θA,c¯,ψ)=-(rO¯)2·{sin2θO(1-3ψ2)4(9ψ2+1)cosθOsinθAexp[(θA-θO)ψ]-(cos2θO+2)ψ9ψ2+1+sin2θO(1-3ψ2)4(9ψ2+1)-sin2θO4}+rO¯c¯·cosθOψ-(rO¯)2·exp[3(θO-θA)ψ]-{sinθOψsinθA-3ψ2cosθA9ψ2+1+cosθO3ψcosθA-sinθA9ψ2+1ψsinθA-3ψ2cosθA9ψ2+1}-rO¯c¯·exp[2(θO-θA)ψ]×{tanθAcosθO-ψsinθO1-ψtanθA+cosθOψ-sinθO},(20)rO¯4exp[2(θO-θA)ψ](1+cos2θA)+rO¯9ψ2+1×[ψsin2θO+(1-3ψ2)cos2θO4-3(1+ψ2)4]-c¯exp[2(θO-θA)ψ](cosθO+sinθOψ)+c¯exp[(θO-θA)ψ]×(sinθAψ+tanθA1-ψtanθA+cosθA)+rO¯exp[3(θO-θA)ψ](9ψ2+1)×[(sinθA-3ψcosθA)(ψcosθO+sinθO)]-c¯exp[2(θO-θA)ψ](ψcosθO+sinθO)×tanθA1-ψtanθA+c¯ψsinθO=0,
where f(rO¯,θO,θA,c¯,ψ) is an auxiliary function. Equations (14)–(19) are substituted into (7) resulting in
(21)d¯=dl=i(rO¯,θO,θA,c¯,ψ)-j(rO¯,θO,θA,c¯,ψ)f(rO¯,θO,θA,c¯,ψ),
in which i(rO¯,θO,θA,c¯,ψ) and j(rO¯,θO,θA,c¯,ψ) are complex auxiliary functions listed in the appendix.


## 3.1. Limiting Equilibrium Equations of Footing Soil

As shown in Figure2, the soil underneath the footing is in the limiting equilibrium state, and E is the origin of the coordinate system XEY and y(x) is the equation of the slip surface OA. Taking the soil in OCAO area as an isolated body, the three equations of static equilibrium can be established. By ∑X=0, ∑Y=0, and ∑ME=0, one gets
(1)Q-∫s(τsinα+σcosα)ds+∫xOxA(γy)dx=0,∫s(σsinα-τcosα)ds=0,(2)Qd-∫s[(σsinα-τcosα)y+(τsinα+σcosα)x]ds+∫xOxA(xyγ)dx=0,
where σ and τ are the normal stress and tangential stress on the slip surface, respectively, xO and xA are the x coordinates of points O and A, s is the arc length along the curve y(x), and tanα=dy/dx. Equations (1) represent conditions of vertical and horizontal force equilibrium and (2) represents the condition of moment equilibrium of the sliding soil mass. The moments are taken about point O, the origin of the coordinate system XEY.Figure 2
Force diagram of surface footing.Mohr-Coulomb’s failure criterion is satisfied along the slip surfacey(x), which connects point O, one of the footing edges, to point A at the ground surface. Consider
(3)τ(x)=c+σ(x)tanφ.
Along the slip surface y(x), there exist
(4)dssinα=dy=y′dx,dscosα=dx.
Introducing (3) and (4) into (1) and (2), it follows that
(5)Q=∫xOxA[(c+σtanϕ)y′+σ-γy]dx,(6)∫xOxA[σ(y′-tanϕ)-c]dx=0,(7)d∫xOxA[(c+σtanϕ)y′+σ]dx-d∫xOxA(γy)dx-∫xOxA{[σ(y′-tanϕ)-c]y}dx-∫xOxA{[σ(tanϕy′+1)+cy′]x}dx+∫xOxA(xyγ)dx=0.Thus, the three equations, (5)–(7), of the limiting equilibrium of the sliding body OCAO are derived. In the present formulation, no constitutive law beyond Mohr-Coulomb’s yield condition is included. Consequently, no constrains are placed on the character of the critical rupture surface, y(x), and the critical normal stress along it, σ(x), except for the overall equilibrium of the falling section.


## 3.2. Variational Analysis

Observing (5)–(7), one realizes that the load Q is a functional, a function of two functions, y(x) and σ(x), and is transcribed as the constrained variational extreme-value problem of undetermined boundaries. One variable endpoint of the slip surface is point O shown in Figure 2, whose coordinate is (xO,0), xO is undetermined; another endpoint is point A at the ground surface, whose coordinate is (xA,0), xA is undetermined. According to the variational method of the functional with constraints, the following auxiliary functional J* is constructed to convert the above restrained extremal problem into that without constraints. Consider
(8)J*=∫xOxAFdx,(9)F=F0+λ1F1+λ2F2,F0=(c+σtanϕ)y′+σ-γy,F1=[σ(y′-tanϕ)-c],F2=xyγ-[σ(y′-tanϕ)-c]y-[σ(tanϕy′+1)+cy′]x,
where λ1 and λ2 are the Lagrange undetermined multipliers. The equation of the slip surface y(x) and the normal stress distribution σ(x) on the slip surface must satisfy the following conditions:(1)
Euler’s differential equations for the functionalF(10)∂F∂σ-ddx(∂F∂σ′)=0,where σ′=dσ/dx(11)∂F∂y-ddx(∂F∂y′)=0,(2)
the integration constraint equations: (6) and (7),(3)
transversality conditions of variable endpoints
(i)
at PointO(12)(F-y′∂F∂y′-σ′∂F∂σ′)|x=xOδxA+∂F∂σ′|x=xOδσO=0,(ii)
at pointA(13)(F-y′∂F∂y′-σ′∂F∂σ′)|x=xAδxA+∂F∂σ′|x=xAδσA=0,where δ is the variational operator and σO=σ(x=xO), σA=σ(x=xA).Solving Euler’s equations (10) and (11), the equations of the slip surface and the normal stress distribution along the slip surface can be derived. Introducing the following coordinate transformation:
(14)x-1λ2=rcosθ=x0,y-λ1λ2=rsinθ=y0,
where (r,θ) is polar coordinate system in the Cartesian coordinate system X0O0Y0 with respect to point O0 (see Figure 2). Introducing (9) into (10) and using the coordinate transformation, (14), the following equation results:
(15)r=r0exp[(θ0-θ)tanφ],
where (r0,θ0) are undetermined coefficients. Similarly, the combination of (11), (9), (13), and (14) yields
(16)σ=Const·e2θtanφ-ctanφ+γ·rO·exp[(θO-θ)tanφ]9tan2φ+1(sinθ-3tanφcosθ),
in which Const is the integration constant and is expressed as
(17)Const=(ctanθA1-tanφtanθA+ctanφ-γ·rO·exp[(θO-θA)tanφ]9tan2φ+1×(sinθA-3tanφcosθA)ctanθA1-tanφtanθA)×(exp(2θAtanφ))-1.
Equation (15) is the equation of the slip surface, and (16) and (17) are the expressions of the normal stress on the slip surface.


## 3.3. Expressions of the LoadQ

The following nondimensional variables are introduced to make analysis convenient:(18)rO¯=rOl,c¯=c(γl),ψ=tanφ.
Equations (14)–(18) are substituted into (5) and (6) resulting in
(19)Q¯=Qγl2=f(rO¯,θO,θA,c¯,ψ)=-(rO¯)2·{sin2θO(1-3ψ2)4(9ψ2+1)cosθOsinθAexp[(θA-θO)ψ]-(cos2θO+2)ψ9ψ2+1+sin2θO(1-3ψ2)4(9ψ2+1)-sin2θO4}+rO¯c¯·cosθOψ-(rO¯)2·exp[3(θO-θA)ψ]-{sinθOψsinθA-3ψ2cosθA9ψ2+1+cosθO3ψcosθA-sinθA9ψ2+1ψsinθA-3ψ2cosθA9ψ2+1}-rO¯c¯·exp[2(θO-θA)ψ]×{tanθAcosθO-ψsinθO1-ψtanθA+cosθOψ-sinθO},(20)rO¯4exp[2(θO-θA)ψ](1+cos2θA)+rO¯9ψ2+1×[ψsin2θO+(1-3ψ2)cos2θO4-3(1+ψ2)4]-c¯exp[2(θO-θA)ψ](cosθO+sinθOψ)+c¯exp[(θO-θA)ψ]×(sinθAψ+tanθA1-ψtanθA+cosθA)+rO¯exp[3(θO-θA)ψ](9ψ2+1)×[(sinθA-3ψcosθA)(ψcosθO+sinθO)]-c¯exp[2(θO-θA)ψ](ψcosθO+sinθO)×tanθA1-ψtanθA+c¯ψsinθO=0,
where f(rO¯,θO,θA,c¯,ψ) is an auxiliary function. Equations (14)–(19) are substituted into (7) resulting in
(21)d¯=dl=i(rO¯,θO,θA,c¯,ψ)-j(rO¯,θO,θA,c¯,ψ)f(rO¯,θO,θA,c¯,ψ),
in which i(rO¯,θO,θA,c¯,ψ) and j(rO¯,θO,θA,c¯,ψ) are complex auxiliary functions listed in the appendix.


## 4. Solutions

Referring to Figure2, one finds
(22)rOsinθO=rAsinθA.
Introducing the polar coordinates of point O and point A into the equation of the slip surface, (15), one obtains
(23)rO=rAexp[(θA-θO)tanφ].
The substitution of (23) into (22) yields
(24)sinθOexp(-θOtanφ)=sinθAexp(-θAtanφ).Now, we have four equations ((19)–(21) and (24)) and four unknowns (Q¯,rO¯,θO,θA), so the set of equations is consistent. To solve the set of equations would be very tedious, since (19)–(21) are too complex. So m-files in Matlab 7.0 are edited to realize the calculation. Firstly, the routine fsolve of MATLAB 7.0 is employed to solve (20), (21), and (24) for (rO¯,θO,θA) and then their solutions are substituted into (19) to calculate Q¯. The MATLAB routine fsolve can be used to solve sets of nonlinear algebraic equations using a quasi-Newton method [28].


## 5. Results and Discussions

The analysis presented shows that the shape of the critical surface is a log spiral, as obtained by Garber and Baker [17] as well as Dixit and Mandal [23]. Different from the work by these authors, the bearing capacity factors with respect to eccentricities of load are calculated, and the discrepancy between the bearing capacity factor related to unit weight for cohesionless soil and that for cohesive frictional soil is analyzed as well as the contact between footing and soil with respect to strength parameters of soil and eccentricities of load.The geometrical relation between the eccentricityg and the offset d of the load Q is explained in Figure 3, and one gets
(25)g=l2-d.Figure 3
Eccentricityg of the load Q.


### 5.1. Bearing Capacity Factors for Soil Cohesion

To determine the effects of material mechanical properties and eccentricities of load on bearing capacities of footing soil, extensive calculations are performed. Values of0.05l, 0.1l, 0.15l, 0.2l, 0.25l, 0.3l, 0.35l, 0.4l, 0.45l, and 0.5l for d; 5°, 10°, 15°, 20°, 25°, 30°, 35°, and 40° for φ; and 0.5, 1, 1.5, 2, 2.5, 3, 3.5, and 4 for c/(γl) are used in calculations. The calculated results for different combinations of d, φ, and c/(γl) are given in Figures 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13, and the regressed equations are listed at the right side of the figures, in which R2 represents the coefficient of determination, y represents the dimensionless bearing capacity Q/(γl2), and x represents the dimensionless cohesion of soil c/(γl). Obviously there exists the linear relation between the dimensionless bearing capacity Q/(γl2) and cohesion of soil c/(γl), which can be expressed as
(26)Q¯=Qγl2=cγlkc+12kγ=c¯kc+12kγ,
where kc is the slope of the fitted lines and is defined as the bearing capacity factor due to soil cohesion c; kγ is the twice of intercept of the fitted lines and defined as the bearing capacity factor related to soil unit weight γ. This equation is of the same form as the usual Terzaghi bearing capacity relation.Figure 4
Results ofQ/(γl2) when d=0.05l.Figure 5
Results ofQ/(γl2) when d=0.1l.Figure 6
Results ofQ/(γl2) when d=0.15l.Figure 7
Results ofQ/(γl2) when d=0.2l.Figure 8
Results ofQ/(γl2) when d=0.25l.Figure 9
Results ofQ/(γl2) when d=0.3l.Figure 10
Results ofQ/(γl2) when d=0.35l.Figure 11
Results ofQ/(γl2) when d=0.4l.Figure 12
Results ofQ/(γl2) when d=0.45l.Figure 13
Results ofQ/(γl2) when d=0.5l.For the special case ofd=l/2 as shown in Figure 13, the results of kc for φ=5°, 10°, 15°, 20°, 25°, and 30° are 7.094, 9.3208, 12.569, 17.482, 25.236, and 38.115, respectively, which agree well with the findings in Figure 5 of Garber and Baker [17]; the values of 0.5kγ for φ=25°, 30°, 35°, and 40° are 9.3807, 20.897, 49.37, and 127.36, respectively, and these values coincide with the results in Figure 6 of Garber and Baker [17]. Thus, the correctness of the previously derived equations and corresponding results is testified.From Figures4–13, the bearing capacity factors for cohesion of soil c varying with different values of d/l are obtained and plotted in Figure 14. According to the fitted equations and coefficients of determination listed at the right side of Figure 14 for different friction angles of the soil, the relationship of kc to d/l can be written as
(27)kc=kIdl=kId¯,
where kI is the coefficient related to friction angle of the soil and varied with φ. Meantime, we find
(28)kI=2kc(d=0.5l),
where kc(d=0.5l) is the bearing capacity factor due to cohesion of soil when d=0.5l. Substitution of (28) into (27) results in
(29)kc=2d¯kc(d=0.5l).
The combination of (29) and (25) yields
(30)kc=(1-2gl)kc(d=0.5l).
This finding of bearing capacity factors due to soil cohesion for eccentrically loaded strip smooth footings is the same as the suggestion by Meyerhof [10] and the equation attained by Michalowski and You [14]. Thus, the validity of the effective width rule is again verified from another different approach.Figure 14
Variations ofkc with d/l and φ.To understand the influence of the friction angleφ of soil on kI, variations of kI with friction angles of soil are presented in Figure 15. From the fitted equation and the calculated coefficient of determination in this figure, the following equation is obtained:
(31)kI=8.475e0.076φ,
where φ is in degrees. From (31) and (27), one gets
(32)kc=8.475e0.076φd¯=8.475e0.076φ(0.5-gl).
Thus, the bearing capacity factor kc due to soil cohesion is expressed as the function of the friction angle of soil and the eccentricity of load.Figure 15
Variations ofkI with φ.


### 5.2. Bearing Capacity Factors Related to Unit Weight of Soil

There are two methods that can be employed to calculate the bearing capacity factorskγ related to unit weight of soil. The first method is termed as the zero cohesion of soil (ZCS) method, and the second is termed as the intercept of fitted line (IFL) method. In the first method, let c=0, and let simplified equations (19)–(21) and (24) be utilized to find results of kγ. In the second method, the intercepts of fitted lines in Figures 4–13 are defined as the bearing factors related to unit weight of soil. The first calculation method is fit for soils without cohesion, that is, cohesionless soils, and the second calculation method is suitable to cohesive-frictional soils.


#### 5.2.1. ZCS Method

Results of bearing capacity factorskγ related to unit weight of soil are found using ZCS method and are presented in Figure 16. Regression analysis shows that there exists the parabolic relation between kγ and d/l, which can be written as
(33)kγ=kII(dl)2=kII(d¯)2,
where kII is the coefficient related to the friction angle of soil and is varied with φ.Figure 16
Results ofkγ using ZCS method.To grasp the impact of the friction angle of soil onkII, variations of kII with friction angles of soil are given in Figure 17. From the fitted equation and the calculated coefficient of determination, the following equation results:
(34)kII=0.7799e0.1778φ,
where φ is in degrees. Substitution of (34) into (33) results in
(35)kγ=0.7799e0.1778φ(d¯)2.
From (35) and (25), one gets
(36)kγ=0.7799e0.1778φ(0.5-gl)2.
Thus, the bearing capacity factor kγ related to unit weight of soil for cohesionless soil is expressed as the function of the friction angle of soil and the eccentricity of load.Figure 17
Variations ofkII with φ.


#### 5.2.2. IFL Method

Results of bearing capacity factorskγ related to unit weight of soil using IFL method, which are twice the intercept of the fitted equations in Figures 4–13, are plotted in Figure 18. From the regressed equations and the calculated coefficients of determination R2 in Figure 18, the following equation can be set up:
(37)kγ=kIII(dl)1.961=kIIId¯1.961,
where kIII is the coefficient related to friction angle of soil and is varied with φ. To get the influence of the friction angle of soil on kIII, variations of kIII with friction angles of soil are presented in Figure 19. From the fitted equation and the calculated coefficient of determination in Figure 19, the following equation is gotten:
(38)kIII=1.132e0.168φ.
From (37) and (38), one obtains
(39)kγ=1.132e0.168φ(dl)1.961=1.132e0.168φ(0.5-gl)1.961.
Thus, the bearing capacity factor kγ related to unit weight of soil for cohesive-frictional soil is represented in terms of the friction angle of soil and the eccentricity of load.Figure 18
Variations ofkγ with d/l and φ.Figure 19
Variations ofkIII with φ.Comparisons of results ofkγ from the ZCS method and the IFL method are presented in Figure 20, which is helpful to understand the effects of soil cohesion c on kγ. Obviously results of kγ from the ZCS method are less than those from the IFL method for given friction angle of soil, and the differences between them decrease with the increase in friction angle of soil. The reason for such a difference is due to the existence of the cohesion for cohesive frictional soil, as the shape of the critical rupture surface of footing soil depends on both soil strength parameters (c,φ) rather than on friction angle φ alone in the previous limit equilibrium solutions, and the bearing capacity factor for unit weight of soil is closely related with the shape of the slip surface. Meantime, for real cohesive-frictional soils, the lower frictional soil usually has a higher cohesion, and the bearing factors related to unit weight of soil for lower frictional soils are more affected than those for soils with higher friction angles.Figure 20
Comparisons of results ofkγ from two methods.


### 5.3. Contact between Footing and Soil

With the solutions of (19)–(21) and (24) for (rO¯,θO,θA), one can decide the x coordinates of point O and point A in the coordinate system X0O0Y0. To present the relative position between the footing and the underneath rotating soil, the x coordinate of point E in the coordinate system X0O0Y0 should be given. From the moment equilibrium equation of the rotation body OCAO about point O0, the following equation results:
(40)tl=rAe3¯3(1+9ψ2)1Q¯×{(sinθAe-3ψcosθAe)-exp[3(θAe-θOe)ψ]×(sinθOe-3ψcosθOe)}+rAe2¯2ψc¯1Q¯{exp[2(θAe-θOe)ψ]-1}-rAe¯61Q¯(rAe2¯cos2θAe-rOe2¯cos2θOe)sinθAe,
where rOe¯, rAe¯, θAe, and θOe are solutions of rO¯, rA¯, θA, and θO, and t is given in Figure 2. From (23), one obtains
(41)rOe¯=rAe¯exp[(θAe-θOe)ψ].
From (40) and (41), the solution to t is gotten. With this solution and from Figure 2, the coordinates of point E and point B in the coordinate system X0O0Y0 are found.As typical examples to show relative positions and contact between the footing and the rotating soil, the slip surfaces whenφ=15°, 25°, and 35°, d/l=0.1, 0.5, and c/(γl)=2 are plotted in Figures 21, 22, 23, 24, 25, and 26. The failure of footing soil is decided by both the load position and the mechanical properties of soil. Under higher eccentricity conditions, the footing will have a higher tilt and part of it may separate from the underlying soil, and this can be reflected from positions of point A, one end point of the slip surface. When point A is at the left side of point B, the left side of the footing, the footing will rotate together with the underlying soil. Otherwise, when point A is at the right side of point B, part AB of the footing will separate from the footing soil. As also observed from Figures 21–26, with higher eccentricity (lower d/l) and with less mechanical properties of soil the footing is more liable to separate from the underlying soil. Another finding of the calculated results is that the starting point of the slip surface, point O, is always at the right side of the footing, which is also presented in Figures 21–26.Figure 21
Relative position between footing and slip surface whenφ=15° and d/l=0.1.Figure 22
Relative position between footing and slip surface whenφ=15° and d/l=0.5.Figure 23
Relative position between footing and slip surface whenφ=25° and d/l=0.1.Figure 24
Relative position between footing and slip surface whenφ=25° and d/l=0.5.Figure 25
Relative position between footing and slip surface whenφ=35° and d/l=0.1.Figure 26
Relative position between footing and slip surface whenφ=35° and d/l=0.5.


## 5.1. Bearing Capacity Factors for Soil Cohesion

To determine the effects of material mechanical properties and eccentricities of load on bearing capacities of footing soil, extensive calculations are performed. Values of0.05l, 0.1l, 0.15l, 0.2l, 0.25l, 0.3l, 0.35l, 0.4l, 0.45l, and 0.5l for d; 5°, 10°, 15°, 20°, 25°, 30°, 35°, and 40° for φ; and 0.5, 1, 1.5, 2, 2.5, 3, 3.5, and 4 for c/(γl) are used in calculations. The calculated results for different combinations of d, φ, and c/(γl) are given in Figures 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13, and the regressed equations are listed at the right side of the figures, in which R2 represents the coefficient of determination, y represents the dimensionless bearing capacity Q/(γl2), and x represents the dimensionless cohesion of soil c/(γl). Obviously there exists the linear relation between the dimensionless bearing capacity Q/(γl2) and cohesion of soil c/(γl), which can be expressed as
(26)Q¯=Qγl2=cγlkc+12kγ=c¯kc+12kγ,
where kc is the slope of the fitted lines and is defined as the bearing capacity factor due to soil cohesion c; kγ is the twice of intercept of the fitted lines and defined as the bearing capacity factor related to soil unit weight γ. This equation is of the same form as the usual Terzaghi bearing capacity relation.Figure 4
Results ofQ/(γl2) when d=0.05l.Figure 5
Results ofQ/(γl2) when d=0.1l.Figure 6
Results ofQ/(γl2) when d=0.15l.Figure 7
Results ofQ/(γl2) when d=0.2l.Figure 8
Results ofQ/(γl2) when d=0.25l.Figure 9
Results ofQ/(γl2) when d=0.3l.Figure 10
Results ofQ/(γl2) when d=0.35l.Figure 11
Results ofQ/(γl2) when d=0.4l.Figure 12
Results ofQ/(γl2) when d=0.45l.Figure 13
Results ofQ/(γl2) when d=0.5l.For the special case ofd=l/2 as shown in Figure 13, the results of kc for φ=5°, 10°, 15°, 20°, 25°, and 30° are 7.094, 9.3208, 12.569, 17.482, 25.236, and 38.115, respectively, which agree well with the findings in Figure 5 of Garber and Baker [17]; the values of 0.5kγ for φ=25°, 30°, 35°, and 40° are 9.3807, 20.897, 49.37, and 127.36, respectively, and these values coincide with the results in Figure 6 of Garber and Baker [17]. Thus, the correctness of the previously derived equations and corresponding results is testified.From Figures4–13, the bearing capacity factors for cohesion of soil c varying with different values of d/l are obtained and plotted in Figure 14. According to the fitted equations and coefficients of determination listed at the right side of Figure 14 for different friction angles of the soil, the relationship of kc to d/l can be written as
(27)kc=kIdl=kId¯,
where kI is the coefficient related to friction angle of the soil and varied with φ. Meantime, we find
(28)kI=2kc(d=0.5l),
where kc(d=0.5l) is the bearing capacity factor due to cohesion of soil when d=0.5l. Substitution of (28) into (27) results in
(29)kc=2d¯kc(d=0.5l).
The combination of (29) and (25) yields
(30)kc=(1-2gl)kc(d=0.5l).
This finding of bearing capacity factors due to soil cohesion for eccentrically loaded strip smooth footings is the same as the suggestion by Meyerhof [10] and the equation attained by Michalowski and You [14]. Thus, the validity of the effective width rule is again verified from another different approach.Figure 14
Variations ofkc with d/l and φ.To understand the influence of the friction angleφ of soil on kI, variations of kI with friction angles of soil are presented in Figure 15. From the fitted equation and the calculated coefficient of determination in this figure, the following equation is obtained:
(31)kI=8.475e0.076φ,
where φ is in degrees. From (31) and (27), one gets
(32)kc=8.475e0.076φd¯=8.475e0.076φ(0.5-gl).
Thus, the bearing capacity factor kc due to soil cohesion is expressed as the function of the friction angle of soil and the eccentricity of load.Figure 15
Variations ofkI with φ.


## 5.2. Bearing Capacity Factors Related to Unit Weight of Soil

There are two methods that can be employed to calculate the bearing capacity factorskγ related to unit weight of soil. The first method is termed as the zero cohesion of soil (ZCS) method, and the second is termed as the intercept of fitted line (IFL) method. In the first method, let c=0, and let simplified equations (19)–(21) and (24) be utilized to find results of kγ. In the second method, the intercepts of fitted lines in Figures 4–13 are defined as the bearing factors related to unit weight of soil. The first calculation method is fit for soils without cohesion, that is, cohesionless soils, and the second calculation method is suitable to cohesive-frictional soils.


### 5.2.1. ZCS Method

Results of bearing capacity factorskγ related to unit weight of soil are found using ZCS method and are presented in Figure 16. Regression analysis shows that there exists the parabolic relation between kγ and d/l, which can be written as
(33)kγ=kII(dl)2=kII(d¯)2,
where kII is the coefficient related to the friction angle of soil and is varied with φ.Figure 16
Results ofkγ using ZCS method.To grasp the impact of the friction angle of soil onkII, variations of kII with friction angles of soil are given in Figure 17. From the fitted equation and the calculated coefficient of determination, the following equation results:
(34)kII=0.7799e0.1778φ,
where φ is in degrees. Substitution of (34) into (33) results in
(35)kγ=0.7799e0.1778φ(d¯)2.
From (35) and (25), one gets
(36)kγ=0.7799e0.1778φ(0.5-gl)2.
Thus, the bearing capacity factor kγ related to unit weight of soil for cohesionless soil is expressed as the function of the friction angle of soil and the eccentricity of load.Figure 17
Variations ofkII with φ.


### 5.2.2. IFL Method

Results of bearing capacity factorskγ related to unit weight of soil using IFL method, which are twice the intercept of the fitted equations in Figures 4–13, are plotted in Figure 18. From the regressed equations and the calculated coefficients of determination R2 in Figure 18, the following equation can be set up:
(37)kγ=kIII(dl)1.961=kIIId¯1.961,
where kIII is the coefficient related to friction angle of soil and is varied with φ. To get the influence of the friction angle of soil on kIII, variations of kIII with friction angles of soil are presented in Figure 19. From the fitted equation and the calculated coefficient of determination in Figure 19, the following equation is gotten:
(38)kIII=1.132e0.168φ.
From (37) and (38), one obtains
(39)kγ=1.132e0.168φ(dl)1.961=1.132e0.168φ(0.5-gl)1.961.
Thus, the bearing capacity factor kγ related to unit weight of soil for cohesive-frictional soil is represented in terms of the friction angle of soil and the eccentricity of load.Figure 18
Variations ofkγ with d/l and φ.Figure 19
Variations ofkIII with φ.Comparisons of results ofkγ from the ZCS method and the IFL method are presented in Figure 20, which is helpful to understand the effects of soil cohesion c on kγ. Obviously results of kγ from the ZCS method are less than those from the IFL method for given friction angle of soil, and the differences between them decrease with the increase in friction angle of soil. The reason for such a difference is due to the existence of the cohesion for cohesive frictional soil, as the shape of the critical rupture surface of footing soil depends on both soil strength parameters (c,φ) rather than on friction angle φ alone in the previous limit equilibrium solutions, and the bearing capacity factor for unit weight of soil is closely related with the shape of the slip surface. Meantime, for real cohesive-frictional soils, the lower frictional soil usually has a higher cohesion, and the bearing factors related to unit weight of soil for lower frictional soils are more affected than those for soils with higher friction angles.Figure 20
Comparisons of results ofkγ from two methods.


## 5.2.1. ZCS Method

Results of bearing capacity factorskγ related to unit weight of soil are found using ZCS method and are presented in Figure 16. Regression analysis shows that there exists the parabolic relation between kγ and d/l, which can be written as
(33)kγ=kII(dl)2=kII(d¯)2,
where kII is the coefficient related to the friction angle of soil and is varied with φ.Figure 16
Results ofkγ using ZCS method.To grasp the impact of the friction angle of soil onkII, variations of kII with friction angles of soil are given in Figure 17. From the fitted equation and the calculated coefficient of determination, the following equation results:
(34)kII=0.7799e0.1778φ,
where φ is in degrees. Substitution of (34) into (33) results in
(35)kγ=0.7799e0.1778φ(d¯)2.
From (35) and (25), one gets
(36)kγ=0.7799e0.1778φ(0.5-gl)2.
Thus, the bearing capacity factor kγ related to unit weight of soil for cohesionless soil is expressed as the function of the friction angle of soil and the eccentricity of load.Figure 17
Variations ofkII with φ.


## 5.2.2. IFL Method

Results of bearing capacity factorskγ related to unit weight of soil using IFL method, which are twice the intercept of the fitted equations in Figures 4–13, are plotted in Figure 18. From the regressed equations and the calculated coefficients of determination R2 in Figure 18, the following equation can be set up:
(37)kγ=kIII(dl)1.961=kIIId¯1.961,
where kIII is the coefficient related to friction angle of soil and is varied with φ. To get the influence of the friction angle of soil on kIII, variations of kIII with friction angles of soil are presented in Figure 19. From the fitted equation and the calculated coefficient of determination in Figure 19, the following equation is gotten:
(38)kIII=1.132e0.168φ.
From (37) and (38), one obtains
(39)kγ=1.132e0.168φ(dl)1.961=1.132e0.168φ(0.5-gl)1.961.
Thus, the bearing capacity factor kγ related to unit weight of soil for cohesive-frictional soil is represented in terms of the friction angle of soil and the eccentricity of load.Figure 18
Variations ofkγ with d/l and φ.Figure 19
Variations ofkIII with φ.Comparisons of results ofkγ from the ZCS method and the IFL method are presented in Figure 20, which is helpful to understand the effects of soil cohesion c on kγ. Obviously results of kγ from the ZCS method are less than those from the IFL method for given friction angle of soil, and the differences between them decrease with the increase in friction angle of soil. The reason for such a difference is due to the existence of the cohesion for cohesive frictional soil, as the shape of the critical rupture surface of footing soil depends on both soil strength parameters (c,φ) rather than on friction angle φ alone in the previous limit equilibrium solutions, and the bearing capacity factor for unit weight of soil is closely related with the shape of the slip surface. Meantime, for real cohesive-frictional soils, the lower frictional soil usually has a higher cohesion, and the bearing factors related to unit weight of soil for lower frictional soils are more affected than those for soils with higher friction angles.Figure 20
Comparisons of results ofkγ from two methods.


## 5.3. Contact between Footing and Soil

With the solutions of (19)–(21) and (24) for (rO¯,θO,θA), one can decide the x coordinates of point O and point A in the coordinate system X0O0Y0. To present the relative position between the footing and the underneath rotating soil, the x coordinate of point E in the coordinate system X0O0Y0 should be given. From the moment equilibrium equation of the rotation body OCAO about point O0, the following equation results:
(40)tl=rAe3¯3(1+9ψ2)1Q¯×{(sinθAe-3ψcosθAe)-exp[3(θAe-θOe)ψ]×(sinθOe-3ψcosθOe)}+rAe2¯2ψc¯1Q¯{exp[2(θAe-θOe)ψ]-1}-rAe¯61Q¯(rAe2¯cos2θAe-rOe2¯cos2θOe)sinθAe,
where rOe¯, rAe¯, θAe, and θOe are solutions of rO¯, rA¯, θA, and θO, and t is given in Figure 2. From (23), one obtains
(41)rOe¯=rAe¯exp[(θAe-θOe)ψ].
From (40) and (41), the solution to t is gotten. With this solution and from Figure 2, the coordinates of point E and point B in the coordinate system X0O0Y0 are found.As typical examples to show relative positions and contact between the footing and the rotating soil, the slip surfaces whenφ=15°, 25°, and 35°, d/l=0.1, 0.5, and c/(γl)=2 are plotted in Figures 21, 22, 23, 24, 25, and 26. The failure of footing soil is decided by both the load position and the mechanical properties of soil. Under higher eccentricity conditions, the footing will have a higher tilt and part of it may separate from the underlying soil, and this can be reflected from positions of point A, one end point of the slip surface. When point A is at the left side of point B, the left side of the footing, the footing will rotate together with the underlying soil. Otherwise, when point A is at the right side of point B, part AB of the footing will separate from the footing soil. As also observed from Figures 21–26, with higher eccentricity (lower d/l) and with less mechanical properties of soil the footing is more liable to separate from the underlying soil. Another finding of the calculated results is that the starting point of the slip surface, point O, is always at the right side of the footing, which is also presented in Figures 21–26.Figure 21
Relative position between footing and slip surface whenφ=15° and d/l=0.1.Figure 22
Relative position between footing and slip surface whenφ=15° and d/l=0.5.Figure 23
Relative position between footing and slip surface whenφ=25° and d/l=0.1.Figure 24
Relative position between footing and slip surface whenφ=25° and d/l=0.5.Figure 25
Relative position between footing and slip surface whenφ=35° and d/l=0.1.Figure 26
Relative position between footing and slip surface whenφ=35° and d/l=0.5.


## 6. Conclusions

The VLE method, valid for general failure mode of shear of footing soil, has been applied to the problem of bearing capacity of eccentrically loaded footings. Due to the nature of the VLE formulation, such a solution is independent of the details of a particular constitutive model and therefore realistically reflects the present state of uncertainty with respect to soil behavior. Based on calculated results of bearing capacity of footings, the conclusions drawn are as follows.(1)
The superposition principle of bearing capacity for eccentrically loaded strip smooth footings is derived, and the bearing capacity is represented by two factors for cohesion and unit weight of soil, respectively.(2)
For cohesive-frictional soil, the bearing capacity factors ofkc due to cohesion of soil and kγ related to unit weight of soil can be expressed as
(42)kc=8.475e0.076φ(0.5-gl),kγ=1.132e0.168φ(0.5-gl)1.961.Calculations ofkc show that the effective width rule suggested by Meyerhof [10] is valid for eccentrically loaded strip smooth footings.(3)
For cohesionless soil, the bearing capacity factorskγ due to unit weight of soil are less than those for cohesive-frictional soil and can be written as
(43)kγ=0.7799e0.1778φ(0.5-gl)2.(4)
The discrepancy between the bearing capacity factorskγ for cohesive-frictional soil and that for cohesionless soil lies in the effect of cohesion of soil, for the shape of the critical rupture surface of footing soil depends on both soil strength parameters (c,φ) rather than on friction angle φ alone in the previous limit equilibrium solutions.(5)
The contact between footing and soil is decided by both the load and the strength properties of soil. Under conditions of higher eccentricity and less strength properties of soil, part of the footing will separate from the underlying soil and one end point of the slip surface will go beyond the range of the footing width. Calculated results show that the starting point of the slip surface is always at the right side of the footing.



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*Source: 640273-2013-09-23.xml*