Bearing Capacity of the Semi-Infinite Body with a Projecting Part in the Shape of a Truncated Wedge (I)

Theoretical Analysis by a Slip-Line Method

Abstract

By use of slip-line method, the authors have been studying the bearing capacity of Coulomb plastic bodies in various shapes. In the practical problems such as the bearing capacity of farm soils or the indentation resistance of rigid punches into metals, the width of loaded materials is not generally wide enough to be regarded as a semi-infinite body. In the previous papers, the bearing capacity of the infinite body in the shape of a truncated wedge and that of quarter to semi-infinite bodies subjected to a strip load were elucidated and verified by some experiments.
Further, the bearing capacity of an infinite body in the shape of a truncated wedge with finite height has been analyzed in this paper. The assumed slip-line fields are as follows:
i) When
ω≤{1+2cos(μ-φ)/cosμexp(ξtanφ)}(≡ω^p), (1)
the Prandtl's field shown in Fig. 1 holds. In Eq. (1), ω designates a ratio of the loading width w to the top width W of the wedge, φ is an angle of internal friction and μ stands for
μ=π/4+φ/2(2)
ii) When
ω>ω^p…(3)
and
h≤tanμcosξexp(ξtanφ) (≡h_c), …4)
the slip-line field shown in Fig. 2 holds. In Eq. (4), h denotes a ratio of the loading width w to the wedge height H and ξ is a spreading angle of the spiral slip-line region.
iii) When
ω>ω^p…(5)
and
h>h_c, …(6)
the slip-line field shown in Fig. 3 holds. This slip-line field cannot be expressed analitically. Therefore the bearing capacity must be calculated by a numerical method.
iv) As ω or h becomes larger, the slip-line field shown in Fig. 4 holds. The bearing capacity q is given by
q=q₀{1+H/wtan(H-w/2H+tanγ)}, …(7)
where q₀ is the bearing capacity of semi-infinite body and γ is the angle given by the following equation.
γ=tan^-1(W-w/2H+tanΦ), …(8)
where Φ stands for a spreading angle of the wedge.
The calculated results of the bearing capacity are shown in Fig. 7 (a)-(c) for γ=0°, 30°, 60° and internal friction φ=0°, 15°, 30°. From these figures, it can be said that
1. The larger γ, the larger is the bearing capacity.
2. The bearing capacity is constant for a small value of h. Thereafter, in case of γ=0° the bearing capacity increases to constant value q₀. In case of γ>0° it increases to the crossing point of two curves calculated by the field shown in Fig. 3 and Fig. 4, and then decreases asymptotically to constant value q₀.