The principal stresses at a point P in a solid are 70 MPa, −70 MPa and 0. The yield stress of the material is 100 MPa. Which prediction(s) about material failure at P is/are CORRECT?

The correct answers are:
1. Maximum shear stress theory predicts that the material fails
2. Maximum normal stress theory predicts that the material does not fail

Step 1: Identify the Given Parameters
From the problem description and the visible labels in the attached image, we are given:
• Principal stresses: 70 MPa, −70 MPa, and 0 MPa. Let us arrange them in order:
${\sigma }_{\max }=70\text{ MPa}$
${\sigma }_{\mathrm{int}}=0\text{ MPa}$
${\sigma }_{\min }=-70\text{ MPa}$
• Yield stress of the material:
$S_{\mathrm{yt}}=100\text{ MPa}$

Step 2: Analysis using Maximum Normal Stress (Rankine's) Theory
According to the maximum normal stress theory, failure occurs when the absolute value of the maximum principal stress exceeds the yield strength:
$\left|{\sigma }_{\max }\right|\ge S_{\mathrm{yt}}$
Comparing the maximum tensile and compressive principal stresses to the yield strength:
$\left|70\text{ MPa}\right|<100\text{ MPa}$
$|-70\text{ MPa}|<100\text{ MPa}$
Since the magnitude of the principal stresses does not exceed the yield strength, the maximum normal stress theory predicts that the material does not fail.

Step 3: Analysis using Maximum Shear Stress (Tresca's) Theory
According to the maximum shear stress theory, failure occurs when the maximum shear stress (${\tau }_{\max }$) exceeds the yield strength in shear (${\tau }_y$).
The maximum shear stress developed at point P is given by:
${\tau }_{\max }=\frac{{\sigma }_{\max }-{\sigma }_{\min }}{2}$
Substituting the given principal stresses:
${\tau }_{\max }=\frac{70-(-70)}{2}=\frac{140}{2}=70\text{ MPa}$
The yield strength in shear (${\tau }_y$) is:
${\tau }_y=\frac{S_{\mathrm{yt}}}{2}=\frac{100}{2}=50\text{ MPa}$
Comparing the maximum shear stress to the yield shear strength:
${\tau }_{\max }\left(70\text{ MPa}\right)>{\tau }_y\left(50\text{ MPa}\right)$
Since the calculated maximum shear stress exceeds the allowable shear strength of the material, the maximum shear stress theory predicts that the material fails.