GATE 2024 ME – Question 41

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

Which of the following beam(s) is/are statically indeterminate?

Four beams. (A) A beam with a pin support at the left end, a roller support part-way along and two downward loads $P_1$ and $P_2$, with an overhang at the right end. (B) A beam with a pin support at the left end, a roller support part-way along, an internal hinge near the right end and a couple $M$ applied at the right end. (C) A beam with a pin support at the left end, two roller supports to its right and loads $P_1$ and $P_2$. (D) A beam fixed at the left end with an internal hinge near the middle, a couple $M$ at the hinge, a load $P$ and a roller support at the right end.

Option A in the figure
Option B in the figure
Option C in the figure
Option D in the figure

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Show answer and explanation

Correct answer: (C) Option C in the figure

Explanation

Count the support reactions and compare them with the 3 equations of equilibrium, adding 1 equation for every internal hinge. In (A) a pin and a roller give 3 reactions, so it is determinate. In (B) the 3 reactions meet a hinge, which needs a fourth equation, so it has too few supports for the load, which makes it unstable, not indeterminate. In (C) a pin and two rollers give 4 reactions against 3 equations, so it is indeterminate to the first degree. In (D) the fixed support (3) and the roller (1) give 4 reactions, and the hinge gives a fourth equation, so it is determinate. Only (C) is statically indeterminate.