Text-book of mechanics . s of Principal Stress. — In any stressed mate-rial surfaces across which pure normal stresses act maybe traced. These surfaces may be divided into two sets;across one set the stress will be tensile, across the otherit will be compressive. By reason of Theorem II, thesesets of surfaces will intersect at right angles. As an illustration, trace the surfaces of principalstress for a uniformly loaded simple beam, Fig. 72. The normal stress across vertical planes is given by Mythe equation p = -y- and the normal stress across hori-zontal planes is assumed to be zero. The she
Text-book of mechanics . s of Principal Stress. — In any stressed mate-rial surfaces across which pure normal stresses act maybe traced. These surfaces may be divided into two sets;across one set the stress will be tensile, across the otherit will be compressive. By reason of Theorem II, thesesets of surfaces will intersect at right angles. As an illustration, trace the surfaces of principalstress for a uniformly loaded simple beam, Fig. 72. The normal stress across vertical planes is given by Mythe equation p = -y- and the normal stress across hori-zontal planes is assumed to be zero. The shearingstress across horizontal and vertical planes is given by Assuming the axis of x horizontal, then in the notationof equations (A); (B), (C), and (D), px = p, py = o, STRESS, STRAIN, AND ELASTIC FAILURE 143 and q = q; so that equation (C), determining the direc-tions of the principal planes, becomes tan 2u = —*•P Along the plane AB, Fig. 72, q = o, thereforea = 0° or 900; evidently the planes inclined at 900 to. the axis of x are the planes of pure compression indicatedby heavy lines. At A and B both q = o and p = o, therefore a isindeterminate; p = o because on the section AD, M = o. Similarly, along the plane DC the lightly drawn linesindicate the planes of pure tension. Along AD, p = o (for M = o) and g is zero only atA and Z), therefore a = 45° or 135°. Here again theheavy lines indicate compression planes and the lightlines tension planes. Similarly, along BC, as shown in Fig. 72. Along the section GH we have q = o (for Qx = o) andp is zero only on NS. Hence the intersection of GHand NS is an indeterminate point, while at other pointsin the section a = 0° and 900. 144 MECHANICS OF MATERIALS Along NS, p = o, therefore a = 45° or 1350, asshown. Following the general directions obtained above, theheavy lines indicate surfaces of pure compression andthe light lines indicate surfaces of pure tension. As an application of these orthogonal cylindrical sur-faces, att
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