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GATE 2023 ME – Question 39

Mechanics of Materials · Shear force and bending moment diagrams · 2 marks · Multiple choice

A beam of length $L$ is loaded in the $xy$-plane by a uniformly distributed load, and by a concentrated tip load parallel to the $z$-axis, as shown in the figure. The resulting bending moment distributions about the $y$ and the $z$ axes are denoted by $M_y$ and $M_z$, respectively.

Which one of the options given depicts qualitatively CORRECT variations of $M_y$ and $M_z$ along the length of the beam?

a cantilever fixed at the left with a uniformly distributed load q acting downwards (in the $-y$ direction) and a tip load P pointing in the $-z$ direction. The four options show pairs of moment diagrams.
  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: (B) Option B in the figure

Explanation

The beam is a cantilever. The distributed load acts in the $y$-direction, so it bends the beam about the $z$-axis, and the moment is parabolic, $M_z \propto (L - x)^2$, largest at the fixed end and zero at the tip. The tip load acts in the $z$-direction and bends the beam about the $y$-axis, with a moment that is linear, $M_y \propto (L - x)$, also largest at the wall and zero at the tip. So $M_y$ is linear and $M_z$ is parabolic, which rules out A and D. The two loads act in opposite senses about their axes (the tip load P points towards $-z$ and q towards $-y$), so the moments have opposite signs, which is the case in B and not in C.