Structural Analysis Fundamentals / Shear in beams
Shear in unsymmetrical sections and the shear centre
Why a channel twists when you load it through its centroid.
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Read the lesson register and outstanding review notesWhy this matters
Load a channel vertically through its centroid and it twists. Nothing is wrong with the channel, and nothing is wrong with your load — the shear stresses inside it have a resultant that does not pass through the centroid, and the difference is a torque. The point you must load through is the shear centre, and for a channel it is not even inside the section.
By the end of this lesson you should be able to
- Apply τ = VQ/It to a thin-walled open section
- Trace the shear flow round a channel and find its resultant
- Define the shear centre and locate it for a channel
- Explain the effect of shear on the assumptions of simple bending theory
What you should already know
- The shear formula τ = VQ/It (this module)
- Shear flow q = VQ/I (this module)
- Unsymmetrical bending and principal axes (Module 9)
For the channel considered here, the vertical load acts in a principal bending direction. The thin-walled elastic shear-flow expression is q = VQ/I. More general unsymmetrical bending requires both first moments and the section inertia terms.
Starting at a free flange tip, q is zero. It rises linearly towards the web because the flange first moment accumulates linearly. In the web, the flow varies quadratically with height. The lower flange mirrors the upper flange. The flange flows act horizontally and form a couple, which determines the offset of the shear centre.
Add up what those flows do.
The two flange flows are horizontal, equal in magnitude, and opposite in direction — one runs inwards along the top flange, the other outwards along the bottom. Equal and opposite forces on parallel lines a distance h apart form a couple, of magnitude Fflange × h.
The web flow gives a vertical resultant equal to the applied shear V.
So the internal shear stresses are statically equivalent to a vertical force V plus a torque. For the section to carry the load without twisting, the applied load must produce exactly that same torque about the same point — which means it cannot act through the centroid. It must act through a point offset from the web, on the opposite side from the flanges.
That point is the shear centre.
Predict first
A channel section is used as a beam and loaded vertically through its centroid. What happens?
Euler–Bernoulli beam theory neglects shear deformation. A real shear-stress distribution produces shear strain and cross-section warping. For a sufficiently slender beam away from concentrated loads and supports, this may have little effect on the bending response.
For deeper members or materials with low shear stiffness, include shear deformation and consider whether a two- or three-dimensional stress model is needed. There is no universal span-to-depth ratio that validates all these approximations; loading, restraints, section form and the required accuracy matter.
Worked example
Shear flow and the shear centre of a channel
Given
- Thin-walled channel: web 300 mm deep, flanges 100 mm wide, uniform thickness t = 8 mm
- Dimensions taken to the wall centrelines
- Vertical shear force V = 120 kN through the section
Find
The flange shear flow, the flange force, and the distance from the web to the shear centre.
Practice
A thin-walled channel has I = 54.0 × 10⁶ mm⁴. At the flange–web junction the first moment of the flange about the neutral axis is Q = 120 × 10³ mm³. Under a shear force of 120 kN, what is the shear flow there, in N/mm?
Practice
The flange is 100 mm wide and the shear flow varies linearly from zero at the tip to 266.7 N/mm at the web. What is the total horizontal force in one flange, in kN?
Practice
For that channel under 120 kN of shear, how far from the web centreline is the shear centre, in mm?
Summary
- Shear flow q = VQ/I is easier than stress for thin-walled sections; start from a free edge where q = 0
- In a channel the flange flows are horizontal and opposite, forming a couple
- The shear centre is where the resultant of the internal shear flows acts
- For a doubly symmetric section it coincides with the centroid; for a channel it lies outside the section
- For an angle it is at the intersection of the leg centrelines
- Loading away from the shear centre applies a torque to a section that is poor in torsion
- Shear deformation and warping become more significant as beam slenderness falls; assess the model against geometry, material, loading and restraints