Bearing Capacity of an Infinite Body in the Shape of a Truncated Wedge (I)

Theoretical Analysis by a Slip-Line Method

Abstract

The bearing capacity of the semi-infinite body has been elucidated by L. Prandtl and R. Hill, applying a slip-line method. While, in most practical problems such as the bearing capacity of farm tractors or the indentation resistance of rigid punches to various machine elements, the width of material is not wide enough to be regarded as a semi-infinite body. In such circumstance, it is desirable that the bearing capacity of an infinite body with a limited width is elucidated.
In this paper, the bearing capacity of the infinite body in the shape of a truncated wedge was analyzed by use of the slip-line method. The assumed slip-line field is shown in Fig. 2, which is an extended one of the Prandtl's slip-line field for a semi-infinite body. As shown in this figure, this field is composed of straight slip-line regions in the shape of an isosceles triangle and of logarithmic spiral slip-line regions.
The geometrical configuration of this field is prescribed by four parameters and the relationship of these parameters is expressed by the following equation.
ω=[1-2{cos(π/4-φ/2)/cosμ(tanΦcosξ-sinξ)exp(ξtanφ)}]^-1 (1)
where ω is a ratio of the width of a material to that of a strip load, Φ is a half of spreading angle of a wedge, ξ is a spreading angle of the logarithmic spiral slip-line region and φ is an angle of internal friction. Denoting the value of ω at ξ=π/2 as ω^p, this value is given by
ω^p={1+2cos(π/4-φ/2)/cosμ·exp(π/2tanφ)}^-1 (2)
The bearing capacity is given as follows.
In case of φ=0, reffering to Fig. 3
q=2(1+ξ) (3)
In case of φ>0, reffering to Fig. 4
q=-cotφ{1+sinφ/1-sinφ·exp(-2ξtanφ)+1} (4)
where q is a non-dimensional bearing capacity (bearing capacity/cohesion). The value of ξ in these equations is calculated by substituting the values of ω, Φ and φ to the geometrical equation (1).
The calculated results of q are shown in Fig. 4 (a)-(d) for Φ=0°, 30°, 45°, 60° and φ=0°, 15°, 30°, 45°. As seen 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 the one 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 is this trend.
3. The larger φ, the larger is the rate of the increment of the bearing capacity to the one of Φ.