---
title: "Shear flow and shear centre in thin-wall beams"
subject: "Mechanics of Solids"
chapter: 10
key: "semester_5/mechanics_of_solids:shear-flow-shear-centre-thin-wall"
source: https://www.bctengineer.com/q/semester_5/mechanics_of_solids/shear-flow-shear-centre-thin-wall
site: BCT Engineering
---

# Shear flow and shear centre in thin-wall beams

Mechanics of Solids · Chapter 10: Shear Centers for Thin- Wall Beam Cross Sections

- Asked in: year not given, 6 marks
- Video: [Mechanics of Materials: Lesson 39 - Shear Flow Introduction, Example Problem](https://www.youtube.com/watch?v=eyYwKOFAgao) — Jeff Hanson
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/mechanics_of_solids/shear-flow-shear-centre-thin-wall
- Whole chapter: https://www.bctengineer.com/md/semester_5/mechanics_of_solids/chapter-10

## Question

Define shear flow and shear centre. Derive the expression for shear flow in a thin-walled open beam section, and explain the importance of locating the shear centre. State where the shear centre lies for an angle, a T-section and a channel section.

## Short answer

Shear flow q = τ t is shear force per unit length, derived as q = VQ/I. The shear centre is the point where a transverse load causes bending without twist; loads away from it induce torsion. For an angle, it lies at the leg junction; for a T-section, at the flange-web intersection; for a channel, outside the web away from flanges.

## Answer

#### Shear flow

For a thin-walled section, the shear stress is assumed constant across the thickness and parallel to the wall. **Shear flow** $q$ is the shear force per unit length of the wall:

$$
q = \tau\,t\quad (\text{N/mm})
$$

#### Derivation

Consider a beam element of length $dz$ under shear force $V$. Cut the section at a point and consider the part of the area $A^*$ beyond the cut, with first moment $Q = \int_{A^*} y\,dA$ about the neutral axis.

The bending moments on the two faces are $M$ and $M + dM$, so the normal forces on the cut part of the faces differ by

$$
dF = \frac{dM}{I}\,Q
$$

This is balanced by the shear force on the longitudinal cut, $q\,dz$:

$$
q\,dz = \frac{dM}{I}\,Q \quad\Rightarrow\quad \boxed{q = \frac{V\,Q}{I}}\quad\left(\text{since } V = \frac{dM}{dz}\right)
$$

and $\tau = \dfrac{VQ}{It}$. The shear flow is zero at a free edge and changes linearly in a flange of constant thickness (flange shear flow $q = V\,t\,s\,y_f/I$), and parabolically in a web.

#### Shear centre

The **shear centre** (flexural centre) $S$ is the point on the cross-section through which a transverse load must act so that the beam bends **without twisting**. Its position is found by equating the moment of the shear flow resultants about any point to the moment of the applied shear force $V$ about the same point.

#### Importance

If the load does not pass through $S$, the section twists as well as bends. Thin open sections have very low torsional stiffness, so the twist can be large and the stresses higher than for pure bending. Designers therefore place the load line through $S$ or add torsion allowance.

#### Location

| Section | Shear centre |
|---|---|
| Doubly symmetric (I, box, rectangle) | At the centroid |
| One axis of symmetry (channel, T) | On the axis of symmetry; channel: outside the web, on the side away from the flanges |
| T-section | At the intersection of the flange mid-line and the web mid-line (the shear flows in the two parts meet there) |
| Angle section (equal or unequal legs) | At the **junction** of the two legs (corner), as the shear flow resultants of both legs pass through that point |
| Channel (open towards right) | At distance $e$ to the left of the web centre-line, $e = \dfrac{3b^2t_f}{h\,t_w + 6bt_f}$ |

## Check yourself

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

1. Shear flow in a thin-walled open section is
   A. q = V Q / I
   B. q = V I / Q
   C. q = M y / I
   D. q = V / A
   Answer: A. Longitudinal equilibrium of the cut part gives q dz = (dM/I) Q, so q = VQ/I (tau = VQ/(It)). (Key point: Shear flow formula)

2. The shear centre is the point
   A. Where the centroid lies
   B. Through which a transverse load causes bending without twisting
   C. Where shear flow is maximum
   D. Where the neutral axis crosses the web
   Answer: B. A load through S produces no twist; its position follows from moment equilibrium of the shear flow resultants. (Key point: Definition of shear centre)

3. Locating the shear centre matters because
   A. Loads off S twist the section, and thin open sections have low torsional stiffness so twist and stresses can be large
   B. It reduces bending stress
   C. It makes shear flow constant
   D. It determines the neutral axis
   Answer: A. Designers place the load line through S or allow for torsion. (Key point: Importance)

4. The shear centre of an angle section lies
   A. At the centroid
   B. At the junction of the two legs
   C. Outside, away from the legs
   D. At the tip of the longer leg
   Answer: B. The shear flow resultants of both legs pass through the corner, so S is at the junction. (Key point: Location examples)

## Related in Mechanics of Solids

- [Torsion of a thin open channel section](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Aopen-channel-torsion-numerical) — Mechanics of Solids, chapter 9
- [Shear centre of a channel section](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Achannel-shear-centre-numerical) — Mechanics of Solids, chapter 10
- [Shear flow in a closed box beam](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Abox-beam-shear-flow-numerical) — Mechanics of Solids, chapter 10
- [Torsion of a rectangular hollow thin-wall box](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Athin-walled-box-torsion-numerical) — Mechanics of Solids, chapter 9

## Related in other subjects

- [Boundary shear stress and compound channel flow](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Ashear-stress-boundary-compound-channel-flow) — Hydraulics, chapter 6
- [Types of stresses in thin walled cylinders](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Atypes-of-stresses-thin-cylinder) — Strength of Materials, chapter 5
- [Pipe flow versus open channel flow](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Apipe-flow-vs-open-channel-flow) — Hydraulics, chapter 5, asked 2×
- [Conditions for uniform flow in open channel](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aconditions-of-uniform-flow) — Hydraulics, chapter 6, asked 2×