Module 11 · Lesson 11.2
Thin-walled and open sections
Shear flow in closed boxes, and why an open section is hopeless in torsion.
Why this matters
Almost no real structural member is a solid circle. Box girders, tubes, channels and I-sections all behave differently, and the difference between a closed and an open section is one of the largest in all of structural engineering — often a factor of hundreds.
By the end of this lesson you should be able to
- Use the shear flow relationship for a thin-walled closed section
- Explain why closed sections are so much stiffer than open ones
- Recognise warping and when it matters
In a thin-walled closed section — a box or a tube — the shear stress can be taken as constant through the wall thickness. The product of stress and thickness is the shear flow q, and it is constant all the way round the perimeter, in the same way that the flow in a closed pipe circuit is constant.
What it calculates: The shear flow and shear stress in a thin-walled closed section carrying torque.
- q
- Shear flow, constant around the perimeter (N/mm)
- T
- Applied torque (N·mm)
- Aₘ
- Area enclosed by the mid-line of the wall (mm²)
- t
- Wall thickness at the point considered (mm)
This assumes
- The wall is thin compared with the overall dimensions
- The section is closed — a complete loop
- Elastic behaviour and pure torsion
In plain terms: The enclosed area does the work, not the amount of material. Note that Aₘ is the area the wall goes round, not the area of the wall itself — a common and expensive confusion. Because q is constant, the thinnest part of the wall carries the highest stress.
Worked example
Torsion of a rectangular box section
Given
- Rectangular hollow section, mid-line dimensions 200 mm × 100 mm
- Uniform wall thickness 5 mm
- Applied torque 10 kN·m
Find
The shear flow and the shear stress in the wall.
Assumptions
- Thin wall
- Closed section
- Elastic behaviour
Predict first
A steel tube and an identical tube with a full-length slit cut in it are both twisted. How do their torsional stiffnesses compare?
Warping is the other complication. A circular shaft does not warp: plane sections stay plane, which is why the simple theory is exact. Every non-circular section warps — the cross-section distorts out of its own plane when twisted.
If the warping is free to happen, the member simply twists more than the simple theory suggests. If it is restrained — at a fixed end, for example — longitudinal stresses develop. That is warping torsion, and it matters most for open sections such as I-beams.
Practice
A square hollow section has mid-line dimensions 150 mm × 150 mm and a wall thickness of 6 mm. It carries a torque of 18 kN·m. What is the shear stress in the wall?
Practice
A closed rectangular box section has mean dimensions 200 mm × 100 mm and a uniform wall thickness of 6.0 mm. It carries a torque of 12 kNm. What is the shear stress in the wall, in N/mm²?
Summary
- Closed thin-walled sections: q = T/2Aₘ is constant round the perimeter, τ = q/t
- The enclosed area governs, not the amount of material
- Cutting a closed section open destroys most of its torsional stiffness
- Non-circular sections warp; restrained warping produces longitudinal stresses
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint