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GATE 2023 ME – Question 24

Mechanics of Materials · Bending and shear stresses, shear centre · 1 mark · Multiple select

A beam is undergoing pure bending as shown in the figure. The stress ($\sigma$)-strain ($\varepsilon$) 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 beam with moments M at both ends, and the stress-strain curve of an elastic-perfectly plastic material that is linear up to $\sigma_Y$ and then flat. The options show the stress across the depth: (A) a linear distribution with a maximum of $\pm\sigma_Y/\sqrt{2}$, (B) a linear distribution with a maximum of $\pm\sigma_Y/2$, (C) a uniform stress of $-\sigma_Y$ over the upper half and $+\sigma_Y$ over the lower half, (D) a distribution that is linear near the middle and reaches $\pm\sigma_Y$ at the outer fibres, constant at $\pm\sigma_Y$ near the surfaces.
  1. Option A in the figure
  2. Option B in the figure
  3. Option C in the figure
  4. Option D in the figure

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Show answer and explanation

Correct answer: (C) Option C in the figure; (D) Option D in the figure

Explanation

In an elastic-perfectly plastic material the stress cannot exceed $\sigma_Y$. After yielding has started, the outer fibres carry $\pm\sigma_Y$, and the stress varies linearly in the inner elastic core (option D). When the moment is large enough for the whole section to yield, the stress is $+\sigma_Y$ on one side of the neutral axis and $-\sigma_Y$ on the other (option C). Options A and B are linear distributions with a peak below $\sigma_Y$, which describe a purely elastic state, with no yielding.