Question

A beam is undergoing pure bending as shown in the figure. The stress (\(\sigma\))-strain (\(\epsilon\)) curve for the material is also given. The yield stress of the material is \(\sigma_Y\). Which of the option(s) given represent(s) the bending stress distribution at cross-section AA after plastic yielding? 

 

• A. A
• B. B
• C. C
• D. D

Hint:

For an elastic-perfectly plastic material in bending:
1. {Elastic:} Stress is linear, max stress \(< \sigma_Y\).
2. {Elasto-Plastic:} Outer fibers are at \(\sigma_Y\), inner core is linear. (Option C)
3. {Fully Plastic:} Entire section is at \(\sigma_Y\). (Option D)
The question asks for the state "after plastic yielding", which includes both the elasto-plastic and fully plastic conditions.

Answer 1

The Correct Option is C, D

Step 1: Understanding the Concept:
The question concerns the stress distribution in a beam made of an elastic-perfectly plastic material subjected to a bending moment that causes yielding.
- In the elastic region, stress is proportional to the distance from the neutral axis (\(\sigma = \dfrac{My}{I}\)).
- When the bending moment is large enough, the stress at the outermost fibers reaches the yield stress, \(\sigma_Y\). This is the onset of yielding.
- If the bending moment is increased further, the yielding progresses from the outer fibers inward toward the neutral axis. The region that has yielded is called the plastic zone, and the stress in this zone remains constant at \(\sigma_Y\). The inner region that has not yet yielded remains elastic and is called the elastic core.
- The theoretical limit is when the entire cross-section has yielded, which corresponds to the "fully plastic moment".
Step 2: Detailed Explanation:
Let's analyze the given options in the context of plastic bending:
- (A) and (B): These show a linear stress distribution, which is characteristic of purely elastic bending. The maximum stress is less than the yield stress \(\sigma_Y\). These represent the state before plastic yielding begins. Therefore, they are incorrect.
- (C): This diagram shows a stress distribution where the outer layers of the beam have yielded. In these outer layers, the stress is constant and equal to the yield stress (\(\sigma_Y\) in tension, \(-\sigma_Y\) in compression). There is still an inner "core" around the neutral axis where the stress is below \(\sigma_Y\) and varies linearly. This is a valid representation of the stress distribution for a moment that is greater than the yield moment but less than the fully plastic moment. This state is "after plastic yielding" has initiated. This is a correct representation.
- (D): This diagram shows a stress distribution where the entire cross-section has yielded. The stress is \(\sigma_Y\) throughout the top half (tension side) and \(-\sigma_Y\) throughout the bottom half (compression side). This represents the limiting case of plastic bending, corresponding to the fully plastic moment. This is also a valid state "after plastic yielding" has progressed through the entire section. This is a correct representation.
Step 3: Final Answer:
Both (C) and (D) are valid representations of the stress distribution after plastic yielding has occurred.
Step 4: Why This is Correct:
The phrase "after plastic yielding" can refer to any state from the moment yielding begins to the moment the section becomes fully plastic. Option (C) shows the elasto-plastic state, and option (D) shows the fully plastic state. Both are valid stages of plastic bending.