Text-book of mechanics . nd q now represent the normal and the shear-ing stress, respectively, across any plane inclined at anangle a to the direction of the stress resultant (oblique) stress across this plane is thus 5 = Vp2 + q2 = Vp^sin2a + p2cos2a, for the stresses S, p, and q are all distributed over thesame area. Fig. 78 shows the directions of these stresses. Exercise 173. Obtain the above results from the equi-librium of any element of the material considered withoutreferring to equations previously obtained. Exercise 174. Find the x and y components of the resul-tant stress S d
Text-book of mechanics . nd q now represent the normal and the shear-ing stress, respectively, across any plane inclined at anangle a to the direction of the stress resultant (oblique) stress across this plane is thus 5 = Vp2 + q2 = Vp^sin2a + p2cos2a, for the stresses S, p, and q are all distributed over thesame area. Fig. 78 shows the directions of these stresses. Exercise 173. Obtain the above results from the equi-librium of any element of the material considered withoutreferring to equations previously obtained. Exercise 174. Find the x and y components of the resul-tant stress S directly from the values of p and q in terms ofpi, p2, and a. STRESS, STRAIN, AND ELASTIC FAILURE *53 The x and y components of S as obtained fromEx. 174 are pj = - pi sin a and py = - p2 cos a. If these values are considered as the x and y coordi-nates of the end P of the vector S representing theresultant stress across the plane AB, Fig. 78, so that x = — pi sin a and y = — p2 cos a, we - ®+($- sin2 a + cos2 a = Therefore, the locus of P for all inclinations of theplane AB is an ellipse whose center is at the point atwhich the stresses are considered and whose semi-axesare equal in magnitude to and coincide in direction withthe principal stresses at this point. From the above discussion a geometrical construc-tion for obtaining the resultant stress on any planethrough any point, provided the principal stresses atthis point are known, is readily obtained. Fig. 79 illustrates the construction. Exercise 175. Show that the construction illustrated inFig. 79 gives S, the resultant stress on the plane AB indirection and magnitude. 154 MECHANICS OF MATERIALS Exercise 176. If one of the principal stresses, say pi,parallel to the a-axis becomes negative, , changes to acompression, what change in Fig. 70 will this necessitate ?
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