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Arlon Online Engineering Library - TechRef! Stress & Strain Principal Stresses For the special case of a two-dimensional stress state, the equations for principal stress reduce to       The two nonzero values calculated from this equation are temporarily labeled a and b and the third value c is always zero in this case. Depending on their values, the three roots are then labeled according to the convention: algebraically largest = 1, algebraically smallest = 3, other = 2. Mohr's Circle - 2D Stress To construct a Mohr's circle, the following sign conventions are used. 1. Tensile normal stress components are plotted on the horizontal axis and are considered positive. Compressive normal stress components are negative. 2. For constructing Mohr's circle only, shearing stresses are plotted above the normal stress axis when the pair of shearing stresses, acting on opposite and parallel faces of an element, form a clockwise couple. Shearing stresses are plotted below the normal axis when the shear stresses form a counterclockwise couple. The circle drawn with the center on the normal stress (horizontal) axis with center, C, and radius, R, where The two nonzero principal stresses are then:      a = C + R      b = C - R The maximum inplane shear stress is in = R. However, the maximum shear stress considering three dimensions is always       Hooke's Law Three-dimensional case      x = (1/E)[x - (y + z)]     xy = xy/G      y = (1/E)[y - (z + x)]     yz = yz/G      z = (1/E)[z - (x + y)]     zx = zx/G Plane stress case (z = 0): Uniaxial case (y = z = 0):     x = Ex or = E, where x, y, z = normal stress x, y, z = normal strain xy, yz, zx = shear stress E = modulus of elasticity G = shear modulus = Poisson's ratio Elastic Strain Energy If the strain remains within the elastic limit, the work done during deflection (extension) of a member will be transformed into potential energy and can be recovered. If the final load is P and the corresponding elongation of a tension member is , then the total energy U stored is equal to the work W done during loading.      U = W = P/2 The strain energy per unit volume is      u = U/AL = 2/2E    (for tension)

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