37. A circular shaft of radius r in the figure has a moment of inertia I and polar moment of inertia J.

The shaft is under torsion Tz in the positive z-axis and bending moment Mv in the positive x-axis.

The material is mild steel with yield strength of 2.8 MPa. Use only the given coordinate system for your calculations.

(a) If Tz and Mv are gradually increased, which point (or points) will fail first among four points (A, B, C, and £>)? Identify all.

(b) Construct stress matrix [ct^ at point A in xyz-coordinates in terms of given parameters (i.e., Tz, Mx, /, J, and r).

(c) Calculate three principal stresses at point B in terms of given parameters.

(d) When the principal stresses at point C are ctj = 1, cr2 = 0, and <73 = —2 MPa, calculate safety factors (1) from maximum shear stress theory and (2) from distortion energy theory.