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.
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 [
The VLE method was first put forward by Kopácsy [
A strip footing is put on a homogeneous and isotropic soil mass with horizontal ground surface, as given in Figure
Surface footing on
To simplify the analysis, the following assumptions are made. The problem is considered to be a two-dimensional plane strain problem; The footing-soil interface is smooth. 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. Mohr-Coulomb’s failure criterion is assumed to be applicable along a potential slip surface of footing soil.
As shown in Figure
Force diagram of surface footing.
Mohr-Coulomb’s failure criterion is satisfied along the slip surface
Thus, the three equations, (
Observing ( Euler’s differential equations for the functional the integration constraint equations: ( transversality conditions of variable endpoints at Point at point
where
where
Solving Euler’s equations (
The following nondimensional variables are introduced to make analysis convenient:
Referring to Figure
Now, we have four equations ((
The analysis presented shows that the shape of the critical surface is a log spiral, as obtained by Garber and Baker [
The geometrical relation between the eccentricity
Eccentricity
To determine the effects of material mechanical properties and eccentricities of load on bearing capacities of footing soil, extensive calculations are performed. Values of
Results of
Results of
Results of
Results of
Results of
Results of
Results of
Results of
Results of
Results of
For the special case of
From Figures
Variations of
To understand the influence of the friction angle
Variations of
There are two methods that can be employed to calculate the bearing capacity factors
Results of bearing capacity factors
Results of
To grasp the impact of the friction angle of soil on
Variations of
Results of bearing capacity factors
Variations of
Variations of
Comparisons of results of
Comparisons of results of
With the solutions of (
As typical examples to show relative positions and contact between the footing and the rotating soil, the slip surfaces when
Relative position between footing and slip surface when
Relative position between footing and slip surface when
Relative position between footing and slip surface when
Relative position between footing and slip surface when
Relative position between footing and slip surface when
Relative position between footing and slip surface when
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. 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. For cohesive-frictional soil, the bearing capacity factors of Calculations of For cohesionless soil, the bearing capacity factors The discrepancy between the bearing capacity factors 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.
Consider