Bearing Capacity of Quarter to Semi Infinite Bodies (I)

Theoretical Analysis by a Slip-Line Method

Abstract

By use the slip-line method, the authors have been studying to estimate the bearing capacity of farm soils or the indentation resistance of rigid punches to various machine elements. In these problems the width of material was not usually wide enough to be regarded as a semi infinite body. In the previous paper, the bearing capacity of the infinite body in the shape of truncated wedge subjected to a single strip load was elucidated.
In this paper, bearing capacity of quarter to semi infinite bodies against a strip load was analyzed. The assumed slip-line field is shown in Fig. 2. As shown in this figure, this field is composed of a rigid region just under the load, two logarithmic spiral slip-line regions and two similar straight slip-line regions.
The geometrical configuration of this field is prescribed by four parameters and the relation ship of these parameters is expressed by the following equation.
ω={1-4/cosφ(cos2ξ+tanφsin2ξ)sin(μ-φ+ξ)cos(μ-φ)exp(ξtanφ)}^-1 (1)
where ω is a ratio of a half of load-width to a distance from the center of load to the side-edge of material, φ is a half of spreading angle of the infinite body, ξ is a spreading angle of the logarithmic spiral slip-line region and φ is the angle of internal friction. Especially denoting the value of ω at ξ=π/2 as ω^p, this value is given by
ω^p={1+4/cosφcas²(μ-φ)exp(π/2tanξ)}^-1 (2)
The bearing capacity is given as follows:
In case of φ=0,
q=1+2ξ-cos2ξ. (3)
In case of φ>0,
q=cotφ{1-sinφcos2ξ/1-sinφexp(2ξtanφ)-1} (4)
where q is a non-dimensional bearing capacity (bearing capacity/cohesion), The value of ξ in these equations can be calculated by substituting the values of ω, φ and φ to the geometrical equation (1).
The calculated results of q are shown in Fig. 5 (a)-(d) for φ=0°, 30°, 45°, 60°, and φ=0°, 15°, 30°, 45°. As shown in these figures, the calculated results show the following trends.
1. The larger φ, the larger is the rate of the increment of the bearing capacity to that of φ.
2. For ω≤ω^p the bearing capacity is constant, and ω>ω^p it decreases first abruptly and then decreases gradually with the increment of ω. The larger φ and the smaller φ, the stronger becomes this trend.
3. The larger φ, the larger becomes the rate of the increment of the bearing capacity to that of φ.