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 \(T_{z}\) in the positive zaxis and bending moment \(M_{x}\) in the positive xaxis. The material is mild steel with a yield strength of \(2.8 \mathrm{MPa}\). Use only the given coordinate system for your calculations.

a. If \(T_{Z}\) and \(M_{\chi}\) are gradually increased, which point (or points) will fail first among four points \((A, B, C\), and \(D)\) ? Identify all.

b. Construct the stress matrix \([\boldsymbol{\sigma}]_{A}\) at point \(A\) in xyzcoordinates in terms of the given parameters (i.e., \(T_{Z}, M_{\chi}, I, J\), and \(r\) ).

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

d. When the principal stresses at point \(C\) are \(\sigma_{1}=1, \sigma_{2}=0\), and \(\sigma_{3}=-2 \mathrm{MPa}\), calculate safety factors (1) from maximum shear stress theory and (2) from distortion energy theory.

Transcribed Image Text:

2 y X . B Mx A