Question

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?

• A. Maximum normal stress theory predicts that the material fails
• B. Maximum shear stress theory predicts that the material fails
• C. Maximum normal stress theory predicts that the material does not fail
• D. Maximum shear stress theory predicts that the material does not fail

Hint:

For ductile materials, the Maximum Shear Stress (Tresca) and Distortion Energy (von Mises) theories are generally more accurate than the Maximum Normal Stress (Rankine) theory. The state of pure shear (\(\sigma_1 = -\sigma_3\)) is a classic case where the Rankine theory can be unconservative.

Answer 1

The Correct Option is B, C

Step 1: Understanding the Concept:
Theories of failure are used to predict the yielding of a ductile material under a complex state of stress, based on the material's yield strength (\(\sigma_Y\)) from a simple tensile test. We need to apply two such theories: the Maximum Normal Stress Theory (Rankine) and the Maximum Shear Stress Theory (Tresca).
Step 2: Key Formula or Approach:
Given principal stresses: \(\sigma_1 = 70\) MPa, \(\sigma_2 = 0\) MPa, \(\sigma_3 = -70\) MPa.
Given yield stress: \(\sigma_Y = 100\) MPa.
- Maximum Normal Stress Theory (Rankine): Failure occurs if the magnitude of the maximum or minimum principal stress equals or exceeds the yield strength. \[ \text{Condition for safety: } \max(|\sigma_1|, |\sigma_3|)<\sigma_Y \] - Maximum Shear Stress Theory (Tresca): Failure occurs if the maximum shear stress equals or exceeds the shear strength at yield, which is \(\sigma_Y / 2\). The maximum shear stress is \(\tau_{\max} = (\sigma_1 - \sigma_3) / 2\). \[ \text{Condition for safety: } \tau_{\max}<\frac{\sigma_Y}{2} \implies \frac{\sigma_1 - \sigma_3}{2}<\frac{\sigma_Y}{2} \implies \sigma_1 - \sigma_3<\sigma_Y \] Step 3: Detailed Calculation:
Analysis using Maximum Normal Stress Theory:
- Maximum principal stress magnitude: \(|\sigma_1| = |70| = 70\) MPa.
- Minimum principal stress magnitude: \(|\sigma_3| = |-70| = 70\) MPa.
- The controlling stress is 70 MPa.
- Compare with yield strength: \(70 \text{ MPa}<100 \text{ MPa}\).
- Since the maximum normal stress is less than the yield stress, this theory predicts that the material does not fail.
- Therefore, statement (C) is correct and (A) is incorrect.
Analysis using Maximum Shear Stress Theory:
- Calculate the maximum shear stress:
\[ \tau_{\max} = \frac{\sigma_1 - \sigma_3}{2} = \frac{70 - (-70)}{2} = \frac{140}{2} = 70 \text{ MPa} \] - Calculate the shear stress at yield: \[ \frac{\sigma_Y}{2} = \frac{100}{2} = 50 \text{ MPa} \] - Compare the stresses: \(70 \text{ MPa}>50 \text{ MPa}\).
- Since the maximum shear stress in the material exceeds the shear stress at yield, this theory predicts that the material fails.
- Therefore, statement (B) is correct and (D) is incorrect.
Step 4: Final Answer:
The correct predictions are that the Maximum shear stress theory predicts failure (B) and the Maximum normal stress theory predicts no failure (C).
Step 5: Why This is Correct:
The calculations correctly apply the failure criteria for both theories. The Rankine theory, being less conservative, predicts safety, while the Tresca theory, being more conservative for this stress state, predicts failure. Both predictions are correctly identified.