10. With respect to the coordinate system xyz, the state of stress at a point P in a solid is

-20 0 0 0 50 0 0 0 50 MPa

(a) m1, m2, and m3 are three mutually perpendicular vectors such that m1 makes 45° with both x- and y-axes and m3 is aligned with the z-axis. Compute the normal stresses on planes normal to m , m , and m .

(b) Compute two components of shear stress on the plane normal to m1 in the directions m2 and m .

(c) Is the vector n = {0, 1, l}r a principal direction of stress? Explain. What is the normal stress in the direction n?

(d) Draw an infinitesimal cube with faces normal to m1, m2, and m3 and display the stresses on the positive faces of this cube.

(e) Express the state of stress at the point P with respect to the x'y'z' coordinates system that is aligned with the vectors m1, m2, and m3.

(f) What are the principal stress and principal directions of stress at the point P with respect to the x'y'z' coordinates system? Explain.

(g) Compute the maximum shear stress at the point P. Which plane(s) does this maximum shear stress act on?