---
title: "Define shear force and bending moment"
subject: "Strength of Materials"
chapter: 1
key: "semester_3/strength_of_materials:define-shear-force-bending-moment"
source: https://www.bctengineer.com/q/semester_3/strength_of_materials/define-shear-force-bending-moment
site: BCT Engineering
---

# Define shear force and bending moment

Strength of Materials · Chapter 1: Axial Forces, Shearing Forces and Bending Moments

- Asked in: 2073 Shrawan, 4 marks; 2072 Chaitra, 4 marks (asked 2 times)
- Video: [Shear Force and Bending Moment Made EASY!](https://www.youtube.com/watch?v=dnIhP4ndtoE) — Math Physics Engage
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/define-shear-force-bending-moment
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-1

## Question

Define shear force and bending moment at a section of beam.

## Short answer

Shear force at a section is the algebraic sum of vertical forces on one side, tending to slide the beam; positive if the left part is pushed upward. Bending moment is the algebraic sum of moments of forces on one side about the section; positive when sagging (compression top, tension bottom).

## Answer

#### Shear force (SF)
The shear force at a section of a beam is the algebraic sum of all the vertical forces (loads and reactions) acting on either side of the section. It is the force tending to slide one part of the beam relative to the other along the section, and it is the same, in magnitude, whether taken from the left or the right part. Unit: N or kN.

**Sign convention:** shear force is positive when the part to the left of the section is pushed upward relative to the right part (i.e. the resultant of the vertical forces on the left of the section acts upward); this produces a clockwise shearing effect on the element.

#### Bending moment (BM)
The bending moment at a section is the algebraic sum of the moments, about the centroid of that section, of all the forces (loads and reactions) acting on either side of the section. It is the moment that tends to bend the beam and is resisted by the internal stresses. Unit: N·m or kN·m.

**Sign convention:** bending moment is positive (sagging) when it bends the beam concave upward, causing compression in the top fibres and tension in the bottom fibres; it is negative (hogging) when the beam bends concave downward, with tension on top.

```text
 Positive (sagging)         Negative (hogging)
   ⌣  tension at bottom       ⌢  tension at top
```

**Example:** for a simply supported beam of span $L$ with a central load $P$, the shear force is $+P/2$ on the left half and $-P/2$ on the right half, and the bending moment at mid-span is $+PL/4$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The shear force at a section is:
   A. The sum of moments of all forces on one side about the section
   B. The algebraic sum of all vertical forces on either side of the section
   C. The sum of horizontal forces on one side
   D. The maximum stress in the beam
   Answer: B. Shear force is the algebraic sum of the vertical loads and reactions on one side of the section. (Key point: Definition of shear force)

2. A positive (sagging) bending moment causes:
   A. Compression in the bottom fibres and tension on top
   B. Tension in the top fibres and the beam concave downward
   C. No stress on the section
   D. Compression in the top fibres and tension in the bottom fibres
   Answer: D. Sagging bends the beam concave upward, putting the top in compression and the bottom in tension. (Key point: Sagging bending moment)

3. For a simply supported beam with a central load P, what are the shear force and mid-span moment?
   A. +P/2 on the left half, -P/2 on the right half; M = +PL/4
   B. +P on both halves; M = PL/2
   C. +P/2 on the left half, -P/2 on the right half; M = PL/8
   D. Zero shear everywhere; M = PL/4
   Answer: A. Each reaction is P/2, so V = +P/2 then -P/2, and the mid-span moment is (P/2)(L/2) = PL/4. (Key point: Example SF and BM)

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