By Indrajit Chowdhury; Shambhu P Dasgupta
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Extra info for Dynamics of structure and foundation: a unified approach. I, Fundamentals
Let us now introduce the concept of a certain surface, called the stress surface define in the following manner. Set in the direction of n a vector whose length is r and say r= √ k , |σn | where k is a constant. 18) Coordinates of the end of this vector, x = r, y = mr and z = nr. 20) +ve indicates tension and −ve, a compression. When all three principal (we shall discuss them later) stresses have same sign, only one of the alternative sign is needed and the surface is an ellipsoid. When the principal stresses are not all of the same sign, both signs are needed and the surface consists of a hyperboloid of two surfaces, with a common asymptotic cone, which is the boundary surface, corresponding to the directions along which the stress is zero.
When all three principal (we shall discuss them later) stresses have same sign, only one of the alternative sign is needed and the surface is an ellipsoid. When the principal stresses are not all of the same sign, both signs are needed and the surface consists of a hyperboloid of two surfaces, with a common asymptotic cone, which is the boundary surface, corresponding to the directions along which the stress is zero.
10) τxz cos(p, x) + τyz cos(p, y) + (σz − σp ) cos(p, z) = 0 in which σp and p are unknowns. 11) it is obvious that all direction cosines cannot be zero, hence for a nontrivial solution, one should have σx − p τxy τxz τxy σy − p τyz τxz τyz = 0. e. invariant to coordinate transformation). 13), according to Descarte’s rule, should have three distinct roots. 10). These principal planes can be shown to be mutually orthogonal. 44 Dynamics of Structure and Foundation: 1. Fundamentals First strain invariant say J1 has an important physical meaning.