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Structural Analysis Fundamentals / Torsion

Thin-walled and open sections

Shear flow in closed boxes and the lower Saint-Venant stiffness of open sections.

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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

For a single-cell thin-walled closed section under Saint-Venant torsion, take shear stress as approximately constant through each wall thickness. The shear flow q = τt is constant around that cell. Multicell sections need additional equilibrium and compatibility equations.

Bredt's shear flow for a closed section
q=τt=T2Amτ=T2Amtq = \tau t = \frac{T}{2A_m} \qquad \tau = \frac{T}{2A_m t}

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 a single closed cell — a complete loop
  • Elastic behaviour and pure torsion

In plain terms: Aₘ is the area enclosed by the wall mid-line, not the area of material in the wall. Both enclosed area and wall thickness affect torsional response. Since q is constant in this single-cell model, the thinnest wall has the greatest shear 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 longitudinal displacement of points in a cross-section during torsion. Circular shafts have no warping in the homogeneous Saint-Venant model; non-circular sections generally warp. Where that displacement varies or is restrained, additional longitudinal stresses can develop. Use the appropriate torsion constant and assess warping restraint, particularly for open sections.

    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

    • Single-cell closed thin-walled sections: q = T/2Aₘ is constant around the perimeter, τ = q/t
    • Enclosed mid-line area and wall thickness both influence closed-section torsion
    • Cutting a closed section open destroys most of its torsional stiffness
    • Non-circular sections warp; restrained warping produces longitudinal stresses