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

The correct options representing the bending stress distribution after plastic yielding are the elastoplastic stress distribution (shown in image_4.png) and the fully plastic stress distribution (shown in image_3.png).

Step-by-Step Explanation:

1. Understanding the Beam Bending and Material Behavior
As shown in the first diagram (image_0.png), the beam is subjected to a pure bending moment $M$ causing sagging. This means that:
- The fibers above the neutral axis are in compression (negative stress).
- The fibers below the neutral axis are in tension (positive stress).
The stress-strain ($\sigma -\epsilon$) curve is elastic-perfectly plastic, meaning:
- For strain magnitude below the yield strain, the stress is linear: $\sigma =E\epsilon$.
- Once the yield stress ${\sigma }_Y$ (or $-{\sigma }_Y$ in compression) is reached, the stress remains constant at ${\sigma }_Y$ (or $-{\sigma }_Y$) even as strain continues to increase.

2. Progression of Stress Distribution after Yielding
Under pure bending, the strain distribution across the cross-section AA remains linear because plane sections remain plane:
$\epsilon \left(y\right)=\kappa y$
where $\kappa$ is the curvature and $y$ is the distance from the neutral axis.

Case A: Partially Yielded (Elastoplastic) State (image_4.png)
As the bending moment increases beyond the elastic limit, the outermost fibers yield first since they experience the maximum strain.
- For the outer fibers where strain exceeds yield strain, stress is capped at the yield stress:
$\sigma =-{\sigma }_Y$ (at the top fibers)
$\sigma ={\sigma }_Y$ (at the bottom fibers)
- For the inner core closer to the neutral axis, the strain is still below the yield strain. Therefore, the material in this region behaves elastically, resulting in a linear stress distribution.
This combined state corresponds directly to the stress profile shown in image_4.png, where a linear elastic core is bordered by flat plastic regions of stress $-{\sigma }_Y$ at the top and ${\sigma }_Y$ at the bottom.

Case B: Fully Plastic State (image_3.png)
If the bending moment is further increased to the fully plastic limit moment, the elastic core shrinks to a negligible width, and almost the entire cross-section yields.
- The upper half of the cross-section is entirely under a constant compressive yield stress of $-{\sigma }_Y$.
- The lower half of the cross-section is entirely under a constant tensile yield stress of ${\sigma }_Y$.
This corresponds to the step-function stress distribution shown in image_3.png.