Năm 1885 các nhà khoa học Pháp Boussinesq có được một giải pháp cho những căng thẳng và căng thẳng trong một không gian tuyến tính đồng nhất đẳng hướng một nửa đàn hồi, nạp bởi một lực lượng điểm theo chiều dọc trên bề mặt, xem hình 28,1. Một nguồn gốc của giải pháp này được đưa ra trong Phụ lục B, xem bất kỳ cuốn sách giáo khoa trên lý thuyết đàn hồi (ví dụ SP Timoshenko, Lý thuyết đàn hồi, khoản 123) | Chapter 28 BOUSSINESQ In 1885 the French scientist Boussinesq obtained a solution for the stresses and strains in a homogeneous isotropic linear elastic half space loaded by a vertical point force on the surface see Figure . A derivation of this solution is given in Appendix B see also any textbook on the theory of elasticity for instance . Timoshenko Theory of Elasticity paragraph 123 . The stresses are found to be P ơzz 3P z3 2nR5 ơrr Ễ pRf 1 2V rR z- ơzz ơỹỹ ơee . ơrr P 1 2v R z 2n R2 R z R Figure Point load on half space. The solution for the displacements is 3P rz2 2n R5 In these equations r is the cylindrical coordinate ơrz r Ự x2 y2 and R is the spherical coordinate R a x2 y2 z2. u . T 1 2v 1 Ĩ . 2nER R3 R ug 0 uz P 1 v 2 1 v z2 . z 2nER R2 . R z x 160 Arnold Verruijt Soil Mechanics 28. BOUSSINESQ 161 The vertical displacement of the surface is particularly interesting. This is z 0 Uz P 1 - V2 nER For R 0 this tends to infinity. At the point of application of the point load the displacement is infinitely large. This singular behavior is a consequence of the singularity in the surface load as in the origin the stress is infinitely large. That the displacement in that point is also infinitely large may not be so surprising. Another interesting quantity is the distribution of the stresses as a function of depth just below the point load . for r 0. This is found to be 3P r 0 ơzz _ 2 2nz2 r 0 P ơrr ơỹỹ 1 2v 2. 4nz2 These stresses decrease with depth of course. In engineering practice it is sometimes assumed as a first approximation that at a certain depth the stresses are spread over an area that can be found by drawing a line Figure Vertical normal stress ơzz. from the load under an angle of about 45 . That would mean that the vertical normal stress at a depth z would be P nz2 homogeneously over a circle of radius z. That appears to be incorrect the error is 50 if r 0 but the .
