Text-book of mechanics . described in Ex. 151. Sketch these planes and stresses. Pure Shear. — Planes across which the only stress isa shearing stress are called planes of pure shear. To arrive at the conditions leading to pure shear, pin equation (A), p. 139, must be placed equal to zero,and the resulting equation solved for a. This value of 146 MECHANICS OF MATERIALS a when substituted in equation (B) gives the shearingstresses on the planes of pure shear. Exercise 160. Solve equation (A) for a. Instead of performing the operations indicated aboveand thus locating the plane of pure shear wit
Text-book of mechanics . described in Ex. 151. Sketch these planes and stresses. Pure Shear. — Planes across which the only stress isa shearing stress are called planes of pure shear. To arrive at the conditions leading to pure shear, pin equation (A), p. 139, must be placed equal to zero,and the resulting equation solved for a. This value of 146 MECHANICS OF MATERIALS a when substituted in equation (B) gives the shearingstresses on the planes of pure shear. Exercise 160. Solve equation (A) for a. Instead of performing the operations indicated aboveand thus locating the plane of pure shear with referenceto an arbitrarily selected axis of x, as shown in Figs. 70and 71, it will be found more convenient first to locatethe principal planes by means of equation (C) and thento locate the required planes with reference to theseprincipal planes. After a has been determined by means of tan 2 a = ———»P* ~ Pv a new element, at the point considered, having itsbounding planes parallel to the directions determined. by a, may be isolated. The normal stresses on thiselement, as indicated in Fig. 73, are found by means ofequations (D); the shearing stresses are zero. STRESS, STRAIN, AND ELASTIC FAILURE 147 The stresses p and q across any plane inclined at anangle a to the direction AB can be found by substitut-ing pi for px, p2 for pV) o for q, and a for a in equations(A) and (B), p. 139. Thus p = ^i^-2 - ^=^cos 2 a2 2 and q =^^^sin2a. 2 Exercise 161* Derive the above equations from anelement in equilibrium. To find the angles a which will locate the planes ofpure shear with reference to the axis AB (Fig. 73), putp = o, whence / Pi T PiCOS 2 a = 4 *- ) pi — pi which gives the required values. A substitution of these values of a into the generalequation for q yields •-^fV-fe^ Or q = V- />!/>.,, as the shear on the planes of pure shear. This result calls attention to the fact that eitherPi or p2 must be negative so that q may be real. Or Theorem V. Pure shear can on
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