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?

Correct Answer:
- Maximum shear stress theory predicts that the material fails
- Maximum normal stress theory predicts that the material does not fail

Step 1: Identify the Given Data
From the problem statement and the provided image, we have the principal stresses at point P in the solid:
As labeled in the image, the principal stress values are:
σ₁ = 70 MPa
σ₂ = -70 MPa
σ₃ = 0 MPa
The yield stress of the material is:
S_yt = 100 MPa
To apply failure theories properly, we sort the principal stresses in descending order of their algebraic values:
σ_max = 70 MPa (maximum principal stress)
σ_mid = 0 MPa (intermediate principal stress)
σ_min = -70 MPa (minimum principal stress)

Step 2: Analysis using Maximum Shear Stress (Tresca) Theory
According to the maximum shear stress theory, yielding (failure) occurs when the maximum shear stress in the component exceeds the maximum shear stress at the yield point in a simple tension test.
The maximum shear stress developed at point P is given by:

$${\tau }_{\max }=\frac{{\sigma }_{\max }-{\sigma }_{\min }}{2}$$

Substituting the values from the image calculation:

$${\tau }_{\max }=\frac{70-(-70)}{2}=\frac{140}{2}=70\text{MPa}$$

The allowable shear stress at yield under pure tension is:

$${\tau }_{\mathrm{yield}}=\frac{S_{\mathrm{yt}}}{2}=\frac{100}{2}=50\text{MPa}$$

Comparing the two shear stress values:

$${\tau }_{\max }>{\tau }_{\mathrm{yield}}\Rightarrow 70\text{MPa}>50\text{MPa}$$

Since the maximum shear stress at point P exceeds the allowable shear stress at the yield limit, the Maximum shear stress theory predicts that the material fails.

Step 3: Analysis using Maximum Normal Stress (Rankine) Theory
According to the maximum normal stress theory, failure occurs when the absolute value of the maximum principal stress exceeds the yield strength under simple tension or compression.
Here, the absolute maximum normal stress is:

$${\sigma }_{\mathrm{absolute\_max}}=\max \left(\right|{\sigma }_1|,|{\sigma }_2|,|{\sigma }_3\left|\right)=\max (70,70,0)=70\text{MPa}$$

Comparing this with the yield strength (S_yt = 100 MPa):

$$70\text{MPa}<100\text{MPa}$$

Since the magnitude of all principal stresses is less than the yield strength of the material, the Maximum normal stress theory predicts that the material does not fail (remains safe).