Bearing Capacity Factors for Eccentrically Loaded Strip Footings Using Variational Analysis

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.

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 Figure 1. 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 on 𝑐-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 Figure 2, 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 ∑𝑀_𝐸=0, one gets 𝑄−_𝑠(𝜏sin𝛼+𝜎cos𝛼)𝑑𝑠+^𝑥𝐴_𝑥𝑂𝛾𝑦𝑑𝑥=0,_𝑠(𝜎sin𝛼−𝜏cos𝛼)𝑑𝑠=0,𝑄𝑑−_𝑠)()𝑥+(𝜎sin𝛼−𝜏cos𝛼𝑦+𝜏sin𝛼+𝜎cos𝛼𝑑𝑠_{𝑥𝐴𝑥𝑂}𝑥𝑦𝛾𝑑𝑥=0, (1)(2) where 𝜎 and 𝜏 are the normal stress and tangential stress on the slip surface, respectively, 𝑥_𝑂 and 𝑥_𝐴 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 surface 𝑦(𝑥), which connects point 𝑂, one of the footing edges, to point 𝐴 at the ground surface. Consider 𝜏(𝑥)=𝑐+𝜎(𝑥)tan𝜑. (3) Along the slip surface 𝑦(𝑥), there exist 𝑑𝑠sin𝛼=𝑑𝑦=𝑦^′𝑑𝑥,𝑑𝑠cos𝛼=𝑑𝑥. (4) Introducing (3) and (4) into (1) and (2), it follows that 𝑄=^𝑥𝐴_𝑥𝑂𝑐+𝜎tan𝜙𝑦^′+𝜎−𝛾𝑦𝑑𝑥,_{𝑥𝐴𝑥𝑂}𝜎𝑦^′𝑑−tan𝜙−𝑐𝑑𝑥=0,_{𝑥𝐴𝑥𝑂}𝑐+𝜎tan𝜙𝑦^′+𝜎𝑑𝑥−𝑑^𝑥𝐴_𝑥𝑂−𝛾𝑦𝑑𝑥_{𝑥𝐴𝑥𝑂}𝜎𝑦^′𝑦−−tan𝜙−𝑐𝑑𝑥_{𝑥𝐴𝑥𝑂}𝜎tan𝜙𝑦^′+1+𝑐𝑦^′𝑥+𝑑𝑥_{𝑥𝐴𝑥𝑂}𝑥𝑦𝛾𝑑𝑥=0. (5)(6)(7)

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 (𝑥_𝑂,0), 𝑥_𝑂 is undetermined; another endpoint is point 𝐴 at the ground surface, whose coordinate is (𝑥_𝐴,0), 𝑥_𝐴 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 𝐽^∗=^𝑥𝐴_𝑥𝑂𝐹𝑑𝑥,𝐹=𝐹₀+𝜆₁𝐹₁+𝜆₂𝐹₂,𝐹₀=𝑐+𝜎tan𝜙𝑦^′𝐹+𝜎−𝛾𝑦,₁=𝜎𝑦^′,𝐹−tan𝜙−𝑐₂=𝑥𝑦𝛾−𝜎𝑦^′𝑦−𝜎−tan𝜙−𝑐tan𝜙𝑦^′+1+𝑐𝑦^′𝑥, (8)(9) 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 functional 𝐹𝜕𝐹−𝑑𝜕𝜎𝑑𝑥𝜕𝐹𝜕𝜎^′=0, (10)where 𝜎^′=𝑑𝜎/𝑑𝑥𝜕𝐹−𝑑𝜕𝑦𝑑𝑥𝜕𝐹𝜕𝑦^′=0, (11)(2) the integration constraint equations: (6) and (7),(3) transversality conditions of variable endpoints(i) at Point 𝑂𝐹−𝑦^′𝜕𝐹𝜕𝑦^′−𝜎^′𝜕𝐹𝜕𝜎^′_{𝑥=𝑥𝑂}𝛿𝑥_𝐴+𝜕𝐹𝜕𝜎^′_{𝑥=𝑥𝑂}𝛿𝜎_𝑂=0, (12)(ii) at point 𝐴𝐹−𝑦^′𝜕𝐹𝜕𝑦^′−𝜎^′𝜕𝐹𝜕𝜎^′_{𝑥=𝑥𝐴}𝛿𝑥_𝐴+𝜕𝐹𝜕𝜎^′_{𝑥=𝑥𝐴}𝛿𝜎_𝐴=0, (13)where 𝛿 is the variational operator and 𝜎_𝑂=𝜎(𝑥=𝑥_𝑂), 𝜎_𝐴=𝜎(𝑥=𝑥_𝐴).

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: 1𝑥−𝜆₂=𝑟cos𝜃=𝑥⁰,𝜆𝑦−¹𝜆₂=𝑟sin𝜃=𝑦⁰, (14) 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: 𝑟=𝑟₀exp𝜃₀−𝜃tan𝜑, (15) where (𝑟₀,𝜃₀) are undetermined coefficients. Similarly, the combination of (11), (9), (13), and (14) yields 𝜎=Const⋅𝑒^2𝜃tan𝜑−𝑐tan𝜑+𝛾⋅𝑟_𝑂⋅exp𝜃_𝑂−𝜃tan𝜑9tan²,𝜑+1sin𝜃−3tan𝜑cos𝜃 (16) in which Const is the integration constant and is expressed as Const=𝑐tan𝜃_𝐴1−tan𝜑tan𝜃_𝐴+𝑐tan𝜑−𝛾⋅𝑟_𝑂⋅exp𝜃_𝑂−𝜃_𝐴tan𝜑9tan²×𝜑+1sin𝜃_𝐴−3tan𝜑cos𝜃_𝐴×exp2𝜃_𝐴tan𝜑⁻¹. (17) 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 Load 𝑄

The following nondimensional variables are introduced to make analysis convenient: 𝑟_𝑂=𝑟_𝑂𝑙,𝑐𝑐=,𝛾𝑙𝜓=tan𝜑. (18) Equations (14)–(18) are substituted into (5) and (6) resulting in 𝑄𝑄=𝛾𝑙²=𝑓𝑟_𝑂,𝜃_𝑂,𝜃_𝐴,𝑐,𝜓=−𝑟_𝑂²⋅cos𝜃_𝑂sin𝜃_𝐴exp𝜃_A−𝜃_O𝜓−cos2𝜃_𝑂+2𝜓9𝜓²++1sin2𝜃_𝑂1−3𝜓²49𝜓²−+1sin2𝜃_𝑂4+𝑟_𝑂𝑐⋅cos𝜃_𝑂𝜓−𝑟_𝑂²⋅exp3𝜃_𝑂−𝜃_𝐴𝜓−sin𝜃_𝑂𝜓sin𝜃_𝐴−3𝜓²cos𝜃_𝐴9𝜓²+1+cos𝜃_𝑂3𝜓cos𝜃_𝐴−sin𝜃_𝐴9𝜓²−+1𝑟_𝑂𝑐⋅exp2𝜃_𝑂−𝜃_𝐴𝜓×tan𝜃_𝐴cos𝜃_𝑂−𝜓sin𝜃_𝑂1−𝜓tan𝜃_𝐴+cos𝜃_𝑂𝜓−sin𝜃_𝑂,𝑟_𝑂4exp2𝜃_𝑂−𝜃_𝐴𝜓1+cos2𝜃_𝐴+𝑟_𝑂9𝜓²×+1𝜓sin2𝜃_𝑂+1−3𝜓²cos2𝜃_𝑂4−31+𝜓²4−𝑐exp2𝜃_𝑂−𝜃_𝐴𝜓cos𝜃_𝑂+sin𝜃_𝑂𝜓+𝑐exp𝜃_𝑂−𝜃_𝐴𝜓×sin𝜃_𝐴𝜓+tan𝜃_𝐴1−𝜓tan𝜃_𝐴+cos𝜃_A+𝑟_𝑂exp3𝜃_𝑂−𝜃_𝐴𝜓9𝜓²×+1sin𝜃_𝐴−3𝜓cos𝜃_𝐴𝜓cos𝜃_𝑂+sin𝜃_𝑂−𝑐exp2𝜃_𝑂−𝜃_𝐴𝜓𝜓cos𝜃_𝑂+sin𝜃_𝑂×tan𝜃_𝐴1−𝜓tan𝜃_𝐴+𝑐𝜓sin𝜃_𝑂=0, (19)(20) where 𝑓(𝑟_𝑂,𝜃_𝑂,𝜃_𝐴,𝑐,𝜓) is an auxiliary function. Equations (14)–(19) are substituted into (7) resulting in 𝑑𝑑=𝑙=𝑖𝑟_𝑂,𝜃_𝑂,𝜃_𝐴,𝑐,𝜓−𝑗𝑟_𝑂,𝜃_𝑂,𝜃_𝐴,𝑐,𝜓𝑓𝑟_𝑂,𝜃_𝑂,𝜃_𝐴,𝑐,𝜓, (21) in which 𝑖(𝑟_𝑂,𝜃_𝑂,𝜃_𝐴,𝑐,𝜓) and 𝑗(𝑟_𝑂,𝜃_𝑂,𝜃_𝐴,𝑐,𝜓) are complex auxiliary functions listed in the appendix.

4. Solutions

Referring to Figure 2, one finds 𝑟_𝑂sin𝜃_𝑂=𝑟_𝐴sin𝜃_𝐴. (22) Introducing the polar coordinates of point 𝑂 and point 𝐴 into the equation of the slip surface, (15), one obtains 𝑟_𝑂=𝑟_𝐴exp𝜃_𝐴−𝜃_𝑂tan𝜑. (23) The substitution of (23) into (22) yields sin𝜃_𝑂exp−𝜃_𝑂tan𝜑=sin𝜃_𝐴exp−𝜃_𝐴tan𝜑. (24)

Now, we have four equations ((19)–(21) and (24)) and four unknowns (𝑄,𝑟_𝑂,𝜃_𝑂,𝜃_𝐴), 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 (𝑟_𝑂,𝜃_𝑂,𝜃_𝐴) 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 eccentricity 𝑔 and the offset 𝑑 of the load 𝑄 is explained in Figure 3, and one gets 𝑙𝑔=2−𝑑. (25)

Figure 3 

Eccentricity 𝑔 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 of 0.05𝑙, 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 𝑄𝑄=𝛾𝑙²=𝑐𝑘𝛾𝑙^𝑐+12𝑘_𝛾=𝑐𝑘_𝑐+12𝑘_𝛾, (26) where 𝑘_𝑐 is the slope of the fitted lines and is defined as the bearing capacity factor due to soil cohesion 𝑐; 𝑘_𝛾 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 of 𝑄/(𝛾𝑙2) when 𝑑=0.05𝑙.

Figure 5 

Results of 𝑄/(𝛾𝑙2) when 𝑑=0.1𝑙.

Figure 6 

Results of 𝑄/(𝛾𝑙2) when 𝑑=0.15𝑙.

Figure 7 

Results of 𝑄/(𝛾𝑙2) when 𝑑=0.2𝑙.

Figure 8 

Results of 𝑄/(𝛾𝑙2) when 𝑑=0.25𝑙.

Figure 9 

Results of 𝑄/(𝛾𝑙2) when 𝑑=0.3𝑙.

Figure 10 

Results of 𝑄/(𝛾𝑙2) when 𝑑=0.35𝑙.

Figure 11 

Results of 𝑄/(𝛾𝑙2) when 𝑑=0.4𝑙.

Figure 12 

Results of 𝑄/(𝛾𝑙2) when 𝑑=0.45𝑙.

Figure 13 

Results of 𝑄/(𝛾𝑙2) when 𝑑=0.5𝑙.

For the special case of 𝑑=𝑙/2 as shown in Figure 13, the results of 𝑘_𝑐 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 0.5𝑘_𝛾 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 Figures 4–13, 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 𝑘_𝑐 to 𝑑/𝑙 can be written as 𝑘_𝑐=𝑘_𝐼𝑑𝑙=𝑘_𝐼𝑑, (27) where 𝑘_𝐼 is the coefficient related to friction angle of the soil and varied with 𝜑. Meantime, we find 𝑘_𝐼=2𝑘_𝑐(𝑑=0.5𝑙), (28) where 𝑘_𝑐(𝑑=0.5𝑙) is the bearing capacity factor due to cohesion of soil when 𝑑=0.5𝑙. Substitution of (28) into (27) results in 𝑘_𝑐=2𝑑𝑘_𝑐(𝑑=0.5𝑙). (29) The combination of (29) and (25) yields 𝑘_𝑐=1−2𝑔𝑙𝑘_𝑐(𝑑=0.5𝑙). (30) 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 of 𝑘𝑐 with 𝑑/𝑙 and 𝜑.

To understand the influence of the friction angle 𝜑 of soil on 𝑘_𝐼, variations of 𝑘_𝐼 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: 𝑘_𝐼=8.475𝑒^0.076𝜑, (31) where 𝜑 is in degrees. From (31) and (27), one gets 𝑘_𝑐=8.475𝑒^0.076𝜑𝑑=8.475𝑒^0.076𝜑𝑔0.5−𝑙. (32) Thus, the bearing capacity factor 𝑘_𝑐 due to soil cohesion is expressed as the function of the friction angle of soil and the eccentricity of load.

Figure 15 

Variations of 𝑘𝐼 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 factors 𝑘_𝛾 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 𝑘_𝛾. 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 factors 𝑘_𝛾 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 𝑘_𝛾 and 𝑑/𝑙, which can be written as 𝑘_𝛾=𝑘_𝐼𝐼𝑑𝑙²=𝑘_𝐼𝐼𝑑², (33) where 𝑘_𝐼𝐼 is the coefficient related to the friction angle of soil and is varied with 𝜑.

Figure 16 

Results of 𝑘𝛾 using ZCS method.

To grasp the impact of the friction angle of soil on 𝑘_𝐼𝐼, variations of 𝑘_𝐼𝐼 with friction angles of soil are given in Figure 17. From the fitted equation and the calculated coefficient of determination, the following equation results: 𝑘_𝐼𝐼=0.7799𝑒^0.1778𝜑, (34) where 𝜑 is in degrees. Substitution of (34) into (33) results in 𝑘_𝛾=0.7799𝑒^0.1778𝜑𝑑². (35) From (35) and (25), one gets 𝑘_𝛾=0.7799𝑒^0.1778𝜑𝑔0.5−𝑙². (36) Thus, the bearing capacity factor 𝑘_𝛾 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 of 𝑘𝐼𝐼 with 𝜑.

5.2.2. IFL Method

Results of bearing capacity factors 𝑘_𝛾 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: 𝑘_𝛾=𝑘_𝐼𝐼𝐼𝑑𝑙^1.961=𝑘_𝐼𝐼𝐼𝑑^1.961, (37) where 𝑘_𝐼𝐼𝐼 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 𝑘_𝐼𝐼𝐼, variations of 𝑘_𝐼𝐼𝐼 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: 𝑘_𝐼𝐼𝐼=1.132𝑒^0.168𝜑. (38) From (37) and (38), one obtains 𝑘_𝛾=1.132𝑒^0.168𝜑𝑑𝑙^1.961=1.132𝑒^0.168𝜑𝑔0.5−𝑙^1.961. (39) Thus, the bearing capacity factor 𝑘_𝛾 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 of 𝑘𝛾 with 𝑑/𝑙 and 𝜑.

Figure 19 

Variations of 𝑘𝐼𝐼𝐼 with 𝜑.

Comparisons of results of 𝑘_𝛾 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 𝑘_𝛾. Obviously results of 𝑘_𝛾 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 of 𝑘𝛾 from two methods.

5.3. Contact between Footing and Soil

With the solutions of (19)–(21) and (24) for (𝑟_𝑂,𝜃_𝑂,𝜃_𝐴), 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: 𝑡𝑙=𝑟³_𝐴𝑒31+9𝜓²1𝑄×sin𝜃_𝐴𝑒−3𝜓cos𝜃_𝐴𝑒−exp3𝜃_𝐴𝑒−𝜃_𝑂𝑒𝜓×sin𝜃_𝑂𝑒−3𝜓cos𝜃_𝑂𝑒+𝑟²_𝐴𝑒2𝜓𝑐1𝑄exp2𝜃_𝐴𝑒−𝜃_𝑂𝑒𝜓−−1𝑟_𝐴𝑒61𝑄𝑟²_𝐴𝑒cos²𝜃_𝐴𝑒−𝑟²_𝑂𝑒cos²𝜃_𝑂𝑒sin𝜃_𝐴𝑒, (40) where 𝑟_𝑂𝑒, 𝑟_𝐴𝑒, 𝜃_𝐴𝑒, and 𝜃_𝑂𝑒 are solutions of 𝑟_𝑂, 𝑟_𝐴, 𝜃_𝐴, and 𝜃_𝑂, and 𝑡 is given in Figure 2. From (23), one obtains 𝑟_𝑂𝑒=𝑟_𝐴𝑒exp𝜃_𝐴𝑒−𝜃_𝑂𝑒𝜓. (41) 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 when 𝜑=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 when 𝜑=15∘ and 𝑑/𝑙=0.1.

Figure 22 

Relative position between footing and slip surface when 𝜑=15∘ and 𝑑/𝑙=0.5.

Figure 23 

Relative position between footing and slip surface when 𝜑=25∘ and 𝑑/𝑙=0.1.

Figure 24 

Relative position between footing and slip surface when 𝜑=25∘ and 𝑑/𝑙=0.5.

Figure 25 

Relative position between footing and slip surface when 𝜑=35∘ and 𝑑/𝑙=0.1.

Figure 26 

Relative position between footing and slip surface when 𝜑=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 of 𝑘_𝑐 due to cohesion of soil and 𝑘_𝛾 related to unit weight of soil can be expressed as 𝑘_𝑐=8.475𝑒^0.076𝜑𝑔0.5−𝑙,𝑘_𝛾=1.132𝑒^0.168𝜑𝑔0.5−𝑙^1.961. (42) Calculations of 𝑘_𝑐 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 factors 𝑘_𝛾 due to unit weight of soil are less than those for cohesive-frictional soil and can be written as 𝑘_𝛾=0.7799𝑒^0.1778𝜑𝑔0.5−𝑙². (43)(4) The discrepancy between the bearing capacity factors 𝑘_𝛾 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.

Appendix

Consider𝑑𝑑=𝑙=∫^𝑥𝐵_𝑥𝐴𝜎𝑦^′−tan𝜙−𝑐𝑦−𝜎tan𝜙𝑦^′+1+𝑐𝑦^′𝑥𝑑𝑥𝛾𝑙³−∫^𝑥𝐵_𝑥𝐴𝑥𝑦𝛾𝑑𝑥𝛾𝑙³×𝑄𝛾𝑙²⁻¹, (A.1)where𝑖𝑟_𝑂,𝜃_𝑂,𝜃_𝑂,=∫𝑐,𝜓_{𝑥𝐴𝑥𝑂}𝜎𝑦^′−tan𝜙−𝑐𝑦−𝜎tan𝜙𝑦^′+1+𝑐𝑦^′𝑥𝑑𝑥𝛾𝑙³,𝑖𝑟_𝑂,𝜃_𝑂,𝜃_𝑂,𝑐,𝜓=𝑟_𝑂³𝑖₁𝜃_𝑂,𝜃_𝑂,𝜓+𝑐𝑟_𝑂²𝑖₂𝜃_𝑂,𝜃_𝑂,𝜓, (A.2)where 𝑖₁(𝜃_𝑂,𝜃_𝑂,𝜓) and 𝑖₂(𝜃_𝑂,𝜃_𝑂,𝜓) are auxiliary functions and defined as 𝑟_𝑂³𝑖₁𝜃_𝑂,𝜃_𝑂=,𝜓𝑟_𝑂³exp3𝜃_𝑂−𝜃_𝐴𝜓9𝜓²×+1sin𝜃_𝐴cos²𝜃_𝑂−𝜓sin𝜃_𝐴cos𝜃_𝑂sin𝜃_𝑂+𝑟_𝑂³sin𝜃_𝐴419𝜓²×𝜃+1exp_𝑂−𝜃_𝐴𝜓×3𝜓²cos2𝜃_𝑂−4𝜓sin2𝜃_𝑂−cos2𝜃_𝑂+𝑟_𝑂³1+𝜓²29𝜓²3𝜃+1exp_𝑂−𝜃_𝐴𝜓sin𝜃_𝐴−𝑟_𝑂³3𝜓1+𝜓²29𝜓²𝜃+1exp_𝑂−𝜃_𝐴𝜓sin𝜃_𝐴+𝑟_𝑂³sin𝜃_𝐴49𝜓²3𝜃+1exp_𝑂−𝜃_𝐴𝜓×4𝜓sin2𝜃_𝐴+cos2𝜃_𝐴−3𝜓²cos2𝜃_𝐴+𝑟_𝑂³169𝜓²3𝜃+1exp_𝑂−𝜃_𝐴𝜓×3𝜓²sin3𝜃_𝐴+4𝜓cos3𝜃_𝐴−sin3𝜃_𝐴+𝑟_𝑂³3𝜓1+𝜓²29𝜓²3𝜃+1exp_𝑂−𝜃_𝐴𝜓sin𝜃_𝐴+12𝑟_𝑂³1+𝜓²9𝜓²+1cos𝜃_𝑂−sin𝜃_𝑂+𝑟_𝑂³3𝜓1+𝜓²29𝜓²3𝜃+1exp_𝑂−𝜃_𝐴𝜓×cos𝜃_𝐴cos2𝜃_𝑂−cos2𝜃_𝐴+𝑟_𝑂³cos𝜃_𝑂9𝜓²2𝜃+1exp_𝑂−𝜃_𝐴𝜓×12𝜓cos2𝜃_𝐴−94𝜓²sin2𝜃_𝐴+𝑟_𝑂³1+𝜓²29𝜓²3𝜃+1exp_𝑂−𝜃_𝐴𝜓×sin𝜃_𝐴cos2𝜃_𝐴−cos2𝜃_𝑂+3𝜓𝑟_𝑂³exp3𝜃_𝑂−𝜃_𝐴𝜓9𝜓²×𝜓+12cos𝜃_𝐴sin2𝜃_𝑂−sin2𝜃_𝐴+3𝜓𝑟_𝑂³exp3𝜃_𝑂−𝜃_𝐴𝜓9𝜓²×+1−sin³𝜃_𝐴−cos²𝜃_𝑂cos𝜃_𝐴+𝑟_𝑂³cos𝜃_𝑂9𝜓²2𝜃+1exp_𝑂−𝜃_𝐴𝜓×𝜓sin²𝜃_𝐴−14sin2𝜃_𝐴+14cos𝜃_𝑂9𝜓²+1𝑟_𝑂³×3𝜓²sin2𝜃_𝑂−sin2𝜃_𝑂+4𝜓cos2𝜃_𝑂+1619𝜓²+1𝑟_𝑂³×sin3𝜃_𝑂−3𝜓²sin3𝜃_𝑂−4𝜓cos3𝜃_𝑂+𝑟_𝑂³19𝜓²3𝜃+1exp_𝑂−𝜃_𝐴𝜓×𝜓sin²𝜃_𝐴cos𝜃_𝐴+sin³𝜃_𝐴+𝑟_𝑂³3𝜓sin𝜃_𝐴cos𝜃_𝐴9𝜓²4𝜃+1×exp_𝑂−𝜃_𝐴𝜓𝜓cos𝜃_𝑂+sin𝜃_𝑂−𝑟_𝑂³sin²𝜃_𝐴9𝜓²4𝜃+1exp_𝑂−𝜃_𝐴𝜓×𝜓cos𝜃_𝑂+sin𝜃_𝑂+𝑟_𝑂³cos𝜃_𝑂9𝜓²2𝜃+1exp_𝑂−𝜃_𝐴𝜓3𝜓−1−𝜓²2,𝑐𝑟_𝑂²𝑖₂𝜃_𝑂,𝜃_𝑂=,𝜓𝑐𝑟_𝑂²sin𝜃_𝐴tan𝜃_𝐴1−𝜓tan𝜃_𝐴×exp3𝜃_𝑂−3𝜃_𝐴𝜓𝜓cos𝜃_𝑂+sin𝜃_𝑂+𝑐𝑟_𝑂²tan𝜃_𝐴21−𝜓tan𝜃_𝐴exp2𝜃_𝑂−𝜃_𝐴𝜓×𝜓sin2𝜃_𝑂−𝜓sin2𝜃_𝐴+−2𝑐𝑟_𝑂²1𝜓cos²𝜃_𝑂−12cos2𝜃_𝑂+𝑐𝑟_𝑂²cos2𝜃_𝑂𝜓exp2𝜃_𝑂−𝜃_𝐴𝜓+𝑐𝑟_𝑂²𝜓²tan𝜃_𝐴21−𝜓tan𝜃_𝐴×exp2𝜃_𝑂−𝜃_𝐴𝜓cos2𝜃_𝑂−cos2𝜃_𝐴−𝑐𝑟_𝑂²cos𝜃_𝑂tan𝜃_𝐴1−𝜓tan𝜃_𝐴×exp𝜃_𝑂−𝜃_𝐴𝜓𝜓sin𝜃_𝐴−cos𝜃_𝐴+𝑐𝑟_𝑂²𝜓2exp2𝜃_𝑂−𝜃_𝐴𝜓×cos2𝜃_𝑂−cos2𝜃_𝐴−𝑐𝑟_𝑂²sin𝜃_𝐴𝜓exp𝜃_𝑂−𝜃_𝐴𝜓×𝜓cos𝜃_𝑂+sin𝜃_𝑂+𝑐𝑟_𝑂²12exp2𝜃_𝑂−𝜃_𝐴𝜓×sin2𝜃_𝑂−sin2𝜃_𝐴+𝑐𝑟_𝑂²sin𝜃_𝐴𝜓exp3𝜃_𝑂−3𝜃_𝐴𝜓×𝜓cos𝜃_𝑂+sin𝜃_𝑂,𝑗𝑟_𝑂,𝜃_𝑂,𝜃_𝑂,=∫𝑐,𝜓_{𝑥𝐴𝑥𝑂}𝑥𝑦𝛾𝑑𝑥𝛾𝑙³=𝑟_𝑂³𝑗₁𝜃_𝑂,𝜃_𝑂=,𝜓𝑟_𝑂³exp𝜃_𝑂−𝜃_𝐴𝜓×sin𝜃_𝐴12cos²𝜃_𝐴−cos²𝜃_𝑂+𝑟_𝑂³exp2𝜃_𝑂−𝜃_𝐴𝜓cos𝜃_𝑂sin2𝜃_𝐴4−1+4𝜓𝑟_𝑂³exp3𝜃_𝑂−𝜃_𝐴𝜓×sin3𝜃_𝐴−12sin𝜃_𝐴−3𝜓²sin𝜃_𝐴−4𝜓cos𝜃_𝐴49𝜓²−+1sin𝜃_𝐴cos²𝜃_𝐴2+𝑟_𝑂³cos𝜃_𝑂−4𝜓sin3𝜃_𝑂+12sin2𝜃_𝑂cos𝜃_𝑂4+sin𝜃_𝑂−3𝜓²sin𝜃_𝑂−4𝜓cos𝜃_𝑂49𝜓².+1 (A.3)