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SOIL MECHANICS - CHAPTER 30

Năm 1892 Flamant có được giải pháp cho một tải đường thẳng đứng trên một đồng nhất đẳng hướng một nửa không gian tuyến tính đàn hồi, xem hình 30,1. Đây là hai chiều tương đương với vấn đề cơ bản của Boussinesq. Nó có thể được coi là chồng chất của một số lượng vô hạn của tải trọng điểm, thống nhất phân phối F dọc theo trục-y. Nguồn gốc được đưa ra trong | Chapter 30 FLAMANT In 1892 Flamant obtained the solution for a vertical line load on a homogeneous isotropic linear elastic half space see Figure 30.1. This is the two F ơxx iX Z x dimensional equivalent of Boussinesq s basic problem. It can be considered as the superposition of an infinite number of point loads uniformly distributed along the y-axis. A derivation is given in Appendix B. In this case the stresses in the x z-plane are z 2FZ3 2F cos3 B 30.1 n r4 nr ơxx sin2 B cos B 30.2 n r4 nr ơ 2FXZr sinBcos2 B. 30.3 n r4 nr Figure 30.1 Flamant s Problem. force per unit length so that F r has the dimension of a stress. In these equations r Vx2 z2. The quantity F has the dimension of a Expressions for the displacements are also known but these contain singular terms with a factor ln r. This factor is infinitely large in the origin and at infinity. Therefore these expressions are not so useful. On the basis of Flamant s solution several other solutions may be obtained using the principle of superposition. An example is the case of a uniform load of magnitude p on a strip of width 2a see Figure 30.2. In this case the stresses are ơzz B1 e2 sin B1 cos B1 sin e2 cos e2 30.4 n ơxx Bl e2 sin B1 cos B1 sin e2 cos e2 30.5 n ơxz cos2 e2 cos2 Bl . 30.6 n 168 Arnold Verruijt Soil Mechanics 30. FLAMANT 169 . Ụ1 . In the center of the plane for x 0 92 91. Then the stresses are x x 0 ơzz 2p 91 sin 91 cos 91 n 30.7 x 0 ơxx 91 sin 91 cos 91 n 30.8 x 0 ơxz 0. 30.9 z That the shear stress ơxz 0 for x 0 is a consequence of the symmetry of this case. The stresses ơxx and ơzz are shown in Figure 30.3 as functions Figure 30.2 Strip load. of the depth z. Both stresses tend towards zero for z x of course but the horizontal normal stress appears to tend towards zero much faster than the vertical normal stress. It also appears that at the surface the horizontal stress is equal to the vertical stress. At the surface this vertical stress is equal to the load p of course because that is a .