A principal stress method of stress analysis

Abstract

The equations of equilibrium in a stress field as given by Lamé are compared with those given by Cauchy. The ( P − Q and P − R) terms of the Lamé equations identified by the new words detend and double detend are invariant for any point in a stress field. The principal stresses are defined as real stress components. The ∂Xx ∂x terms of the Cauchy equations are considered as imaginary force components. Real and imaginary stress and force components are differentiated by the differences between ray vectors, as for example a force which has only one real value and direction, and diffuse vectors, as for example a gradient, which has a real value in any direction. Shear is differentiated from detend value by its variation with the direction in which it is taken. Shear is found to be an imaginary component of force parallel to a plane. The photoelastic equidetend (isochromatic) and isoclinic maps of a plane stress field define, for any point, a gradient triangle with the three maximum gradients, Δ( P − Q) ΔP and Δ − Q, for its sides. Five quantities, the detend ( P − Q), the ratio of the detend to its gradient ( r 4), the ratio ( r 3) of the isoclinic normal ds 3 to the stress axis direction change along its length dα, the direction of the equidetend normal (θ) and the orientation of an isoclinic line (ω) with reference to a principal stress direction, determine each gradient triangle. Five equations which define elements of the gradient triangle in terms of the five measurable quantities, are developed. Definition of the triangle can be complete with two of the five measurable quantities or two of the five equations omitted. Graphic integration of principal stress changes along any line in a stress field can be made by plotting the line straight and measuring the area swept out by the normal projections, along the line, of the sides of the gradient triangle, after it has been rotated through 90 degrees. Properties of the gradient triangle are used to solve several symmetrical stress problems and to integrate along two lines through a stress field.