The nonzero strain components at a point in a loaded member are ϵ xx = XXXXXXXXXX, ϵ yy = XXXXXXXXXX, and ϒ xy = 2ϵ xy = XXXXXXXXXXUsing the results of Problem 2.35, determine the principal strain...

Problem 2.35

In many practical engineering problems, the state of strain is approximated by the condition that the normal and shear strains for some direction, say, the z direction, are zero, that is, ϵzz= ϵzx= ϵzy= 0 (plane strain). In Chapter 3, it is shown that analogously, ϵzx= ϵzy= 0, but ϵzz= 10 for members made of isotropic materials and loaded such that the state of stress may be approximated by the condition σzz= σzx= σzy= 0 (plane stress). Assume that ϵxx, ϵyy, and ϵxyfor the (x, y) coordinate axes shown in Fig. P2.35 are known. Let the (X, Y) coordinate axes be defined by a counterclockwise rotation through angle θ as indicated in Fig. P2.35. Analogous to the transformation for plane stress, show that the transformation equations of plane strain are ϵxx= ϵxxcos? θ + E sin2θ + 2ϵxysin θ cos θ and ϵyy= ϵxxsin θ cos θ + ϵyysin θ cos θ + ϵxy(cos2θ – sin2θ).