Which of the following is correct relation between shear force (F) and bending moment (M)?

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HPCL Engineer Civil 23 Dec 2023 Official Paper
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  1. \( M=\frac{d F}{d x} \\\)
  2. \( F=\frac{d M}{d x} \\\)
  3. \( F=\frac{d^2 M}{d x^2} \\\)
  4. \( M=\frac{d^2 F}{d x^2} \)

Answer (Detailed Solution Below)

Option 2 : \( F=\frac{d M}{d x} \\\)
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Detailed Solution

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Explanation:

a) In a bending beam, a point of contra flexure is a location where the bending moment is zero (changes its sign). In a bending moment diagram, it is the point at which the bending moment curve intersects with the zero lines.

  • At the point of contra flexure, the bending moment is zero.

​b) The rate of change of bending moment at any section is equal to the shear force at that section.

  • \(\frac{{dM}}{{dx}} = F \Rightarrow dM = Fdx \Rightarrow M = \smallint Fdx\)

c) The rate of change of shear force at any section is equal to load intensity at that section.

  • \(\frac{{dF}}{{dx}} = \omega \Rightarrow dF = \omega dx \Rightarrow F = \smallint \omega dx\)

d) For bending moment to be maximum or minimum, shear force should change its sign.

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