Problem

Many of the problems for Sections 2.1-2.4 require the determination of principal stresses...

Many of the problems for Sections 2.1-2.4 require the determination of principal stresses and maximum shear stress, as well as the location of the planes on which they act. These quantities are required input in design and failure criteria for structural and mechanical systems.

Using transformation equations of plane stress (see Problem 2.14), determine σXX and σXY for the X axis located 1.00 rad clockwise from the x axis. The nonzero stress components are given in Problem 2.20.

Problem 2.14

Consider a state of stress in which the nonzero stress components are σxx, σyy, σzz, and σxy. Note that this is not a state of plane stress since σzz ≠ 0. Consider another set of coordinate axes (X, Y, Z), with the Z axis coinciding with the z axis and the X axis located counterclockwise through angle θ from the x axis. Show that the transformation equations for this state of stress are identical to Eq. 2.30 or 2.31 for plane stress.

Problem 2.20

In Problems 2.18 through 2.21, the Z axis for the transformed axes coincides with the z axis for the volume element on which the known stress components act.

The nonzero stress components are σxx = 80 MPa, σzz = −60 MPa, and σxy = 30 MPa. Determine the principal stresses and maximum shear stress. Determine the angle between the X axis and the x axis when the X axis is in the direction of the principal stress with largest absolute magnitude.

(2.30)