Bearing Capacity Factors for Eccentrically Loaded Strip Footings Using Variational Analysis

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 byMeyerhof 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.


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][2][3][4], limit analysis [4][5][6][7][8][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][17][18][19][20][21][22][23][24][25][26][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.

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  in width, and the underlying soil has an effective unit weight  and shear strength parameters  and  (cohesion and friction angle of soil).Acting on the footing soil is the vertical downward load () with its point of application offsetting  from the right side of the footing.The load  that the footing soil can bear safely is to be found by the variational limit equilibrium method.
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.

Limiting Equilibrium Equations of Footing Soil.
As shown in Figure 2, the soil underneath the footing is in the limiting equilibrium state, and  is the origin of the coordinate system XEY and () is the equation of the slip surface .Taking the soil in  area as an isolated body, the three equations of static equilibrium can be established.By ∑  = 0, ∑  = 0, and ∑   = 0, one gets where  and  are the normal stress and tangential stress on the slip surface, respectively,   and   are the  coordinates of points  and ,  is the arc length along the curve (), and tan  = /.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 , the origin of the coordinate system XEY.Mohr-Coulomb's failure criterion is satisfied along the slip surface (), which connects point , one of the footing edges, to point  at the ground surface.Consider  () =  +  () tan . ( Along the slip surface (), there exist Introducing ( 3) and ( 4) into ( 1) and ( 2), it follows that ∫ Thus, the three equations, ( 5)-( 7), of the limiting equilibrium of the sliding body  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, (), and the critical normal stress along it, (), except for the overall equilibrium of the falling section.

Variational Analysis.
Observing (5)- (7), one realizes that the load  is a functional, a function of two functions, () and (), and is transcribed as the constrained variational extreme-value problem of undetermined boundaries.One variable endpoint of the slip surface is point  shown in Figure 2, whose coordinate is (  , 0),   is undetermined; another endpoint is point  at the ground surface, whose coordinate is (  , 0),   is undetermined.According to the variational method of the functional with constraints, the following auxiliary functional  * is constructed to convert the above restrained extremal problem into that without constraints.Consider where  1 and  2 are the Lagrange undetermined multipliers.The equation of the slip surface () and the normal stress distribution () on the slip surface must satisfy the following conditions: (1) Euler's differential equations for the functional where (2) the integration constraint equations: ( 6) and ( 7), (3) transversality conditions of variable endpoints (ii) at point where  is the variational operator and   = ( =   ),   = ( =   ).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: where (, ) is polar coordinate system in the Cartesian coordinate system  0  0  0 with respect to point  0 (see Figure 2).Introducing ( 9) into (10) and using the coordinate transformation, ( 14), the following equation results: where ( 0 ,  0 ) are undetermined coefficients.Similarly, the combination of ( 11), ( 9), (13), and ( 14) yields in which Const is the integration constant and is expressed as Equation ( 15) is the equation of the slip surface, and ( 16) and ( 17) are the expressions of the normal stress on the slip surface.

Solutions
Referring to Figure 2, one finds Introducing the polar coordinates of point  and point  into the equation of the slip surface, (15), one obtains The substitution of ( 23) into (22) Now, we have four equations (( 19)-( 21) and ( 24)) and four unknowns (,   ,   ,   ), 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 (  ,   ,   ) and then their solutions are substituted into (19) to calculate .The MATLAB routine fsolve can be used to solve sets of nonlinear algebraic equations using a quasi-Newton method [28].

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  and the offset  of the load  is explained in Figure 3, and one gets where   is the slope of the fitted lines and is defined as the bearing capacity factor due to soil cohesion ;   is the twice of intercept of the fitted lines and defined as the bearing capacity factor related to soil unit weight .This equation is of the same form as the usual Terzaghi bearing capacity relation.
From Figures 4-13, the bearing capacity factors for cohesion of soil  varying with different values of / 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   to / can be written as where   is the coefficient related to friction angle of the soil and varied with .Meantime, we find where   ( = 0.5) is the bearing capacity factor due to cohesion of soil when  = 0.5.Substitution of ( 28) into ( 27) results in The combination of ( 29) and ( 25) yields 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.
To understand the influence of the friction angle  of soil on   , variations of   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: where  is in degrees.From (31) and ( 27), one gets Thus, the bearing capacity factor   due to soil cohesion is expressed as the function of the friction angle of soil and the eccentricity of load.

Bearing Capacity Factors Related to Unit Weight of Soil.
There are two methods that can be employed to calculate the bearing capacity factors   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  = 0, and let simplified equations ( 19)-( 21) and ( 24) be utilized to find results of   .In the second method, the intercepts of fitted lines in   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.

ZCS Method.
Results of bearing capacity factors   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   and /, which can be written as where   is the coefficient related to the friction angle of soil and is varied with .To grasp the impact of the friction angle of soil on   , variations of   with friction angles of soil are given in Figure 17.From the fitted equation and the calculated coefficient of determination, the following equation results: where  is in degrees.Substitution of (34) into (33) results in From ( 35) and ( 25), one gets Thus, the bearing capacity factor   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.

IFL Method.
Results of bearing capacity factors   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  2 in Figure 18, the following equation can be set up:  where   is the coefficient related to friction angle of soil and is varied with .To get the influence of the friction angle of soil on   , variations of   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: From (37) and (38), one obtains Thus, the bearing capacity factor   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.Comparisons of results of   from the ZCS method and the IFL method are presented in Figure 20, which is helpful to understand the effects of soil cohesion  on   .Obviously results of   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 (, ) rather than on friction angle  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 cohesivefrictional 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.

Contact between Footing and Soil.
With the solutions of ( 19)-( 21) and ( 24) for (  ,   ,   ), one can decide the  coordinates of point  and point  in the coordinate system  0  0  0 .To present the relative position between the footing and the underneath rotating soil, the  coordinate of point  in the coordinate system  0  0  0 should be given.From the moment equilibrium equation of the rotation body  about point  0 , the following equation results: where   ,   ,   , and   are solutions of   ,   ,   , and   , and  is given in Figure 2. From (23), one obtains From ( 40) and (41), the solution to  is gotten.With this solution and from Figure 2, the coordinates of point  and point  in the coordinate system  0  0  0 are found.
As typical examples to show relative positions and contact between the footing and the rotating soil, the slip surfaces when  = 15 ∘ , 25 ∘ , and 35 ∘ , / = 0.1, 0.5, and /() = 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 , one end point of the slip surface.When point  is at the left side of point , the left side of the footing, the footing will rotate together with the underlying soil.Otherwise, when point  is at the right side of point , part  of the footing will separate from the footing soil.As also observed from Figures 21-26, with higher eccentricity  (lower /) 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 , is always at the right side of the footing, which is also presented in Figures 21-26.

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   due to cohesion of soil and   related to unit weight of soil can be expressed as

Figure 3 :
Figure 3: Eccentricity  of the load .

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.
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.