Structural Analysis Fundamentals / module 11
Torsion
Twisting circular shafts from first principles, angle of twist, and why open sections are so poor in torsion.
What you will understand
- Derive the torsion relationship T/J = τ/r = Gθ/L from geometry and compatibility
- Calculate the polar second moment of area for solid and hollow shafts
- Find the angle of twist and understand torsional rigidity GJ
- Analyse thin-walled closed sections using shear flow
- Explain why circular-shaft theory must not be applied to open sections
Lessons
Deriving the torsion formula
From the geometry of twisting to T/J = τ/r = Gθ/L.
Thin-walled and open sections
Shear flow in closed boxes and the lower Saint-Venant stiffness of open sections.
Torque diagrams and plastic torsion
The fourth internal action, drawn like the others — and what a shaft does after it yields.
Non-circular sections, warping and the cost of a slit
Why circular-shaft theory fails everywhere else, and how much a single cut costs.
Module summary
- For a circular shaft, radii stay straight and shear strain grows linearly with radius
- T/J = τ/r = Gθ/L links torque, stress, geometry and twist
- J = πd⁴/32 solid, π(D⁴ − d⁴)/32 hollow — material near the surface does nearly all the work
- A hollow shaft is far more efficient than a solid one of the same weight
- GJ is torsional rigidity; the angle of twist is θ = TL/GJ
- Thin-walled closed sections carry torsion by shear flow q = T/2Aₘ and are vastly stiffer than open ones
Teaching basis and review status
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesCheck your understanding
Practice
A solid shaft of diameter 100 mm carries a torque of 8 kN·m. What is the maximum shear stress?
Check yourself
In T/J = τ/r = Gθ/L, what does J represent?
Check yourself
Why is a closed box section so much better in torsion than an open channel?