Boussinesk, France, based on the elastic theory, derived an elastic mechanical solution of stress and displacement caused by any point in an elastic half-space when a vertical concentrated force acts on the surface of the elastic half-space (above). The solutions of six stress components and three displacement components at any point M(x, y, z) in half space (equivalent to foundation) are as follows: (above formula)
Because soil is granular, it can't bear tensile stress generally. In soil mechanics, the law of stress symbol is the same as that of elasticity, but the law of stress symbol is opposite to that of elasticity. That is to say, the compressive stress is positive and the tensile stress is negative. The shear stress in the counterclockwise direction is positive.
In the plane strain problem, on any cross section perpendicular to the axis, the three strains related to Z coordinate are all equal to 0, namely εz, γzx and γ zy = 0; Three strains independent of Z coordinate are not equal to 0, namely ε x, ε y, γ xy≠0. There are only three strain components parallel to the xy plane at any point.
Many machine parts can be approximately calculated as plane strain problems, such as spline shafts, rollers in needle bearings and so on.