The problems for Sections 2.5-2.8 involve displacements, deformations, and strain states a...

The problems for Sections 2.5-2.8 involve displacements, deformations, and strain states at a point in a structural or machine member. These quantities, as with then stress counterparts, are important in design and failure criteria.

The nonzero strain components at a point in a machine member are ϵxx = 0.00180, ϵyy = −0.00108, and γxy = 2ϵxy = −0.00220. Using the transformation equations for plane strain (see Problem 2.54), determine the principal strain directions and principal strains.

Problem 2.54

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). Assume that ϵxx, ϵyy, and ϵxy for the (x, y) coordinate axes shown in Figure P2.54 are known. Let the (X, Y) coordinate axes be defined by a counterclockwise rotation through angle θ as indicated in Figure P2.54. Analogous to the transformation for plane stress, show that the transformation equations of plane strain are

ϵXX = ϵxx cos2 θ + ϵyy sin2 θ + 2ϵxy sin θ cos θ

ϵYY = ϵxx sin2 θ + ϵyy cos2 θ − 2 ϵxy sin θ cos θ

ϵXY = −ϵxx sin θ cos θ + ϵyy sin θ cos θ + ϵxy(cos2 θ − sin2 θ)

(See Eq. 2.30.)

FIGURE P2.54