10.2 Torsion and Warping Effects

Torsion is moment about a member's longitudinal axis. It may be obvious, as in a shaft, or hidden in an eccentric beam reaction. For July 2026, use the April 2024 Civil: Structural specification, current PE Civil Reference Handbook, ACI 318-14, and the AISC Steel Construction Manual 15th edition. The April 2027 editions are future-only.

Find the Torque and Its Path

An eccentric transverse force P at perpendicular offset e creates torque

T = Pe

provided the offset is measured from the axis or point through which the resisting shear acts. Draw the load, eccentricity, member axis, and reactions. Torque must pass through connections and supporting members into a stabilizing system. A numerical T with no connection or support path is incomplete.

For a beam cross-section, loading through the shear center can produce bending without twist in the ideal elastic model. The shear center equals the centroid for doubly symmetric sections but may lie elsewhere, even outside an open unsymmetric section. A load through the centroid is therefore not automatically torsion-free.

Equilibrium Versus Compatibility Torsion

Equilibrium torsion is necessary for static equilibrium. A canopy edge beam supporting an eccentric cantilever reaction may have no alternate path; removing torsional resistance would leave the load unsupported. The full demand must be traced, analyzed, and detailed.

Compatibility torsion develops because connected members are forced to deform together in a statically indeterminate system. A spandrel beam monolithic with transverse slab framing may attract torque as the system enforces joint rotation compatibility. Cracking or yielding can reduce torsional stiffness and redistribute actions if the governing ACI or AISC model permits it and the remaining load path and minimum detailing are adequate. “Compatibility” does not mean zero torsion by declaration. Determine whether redistribution is allowed; never discard equilibrium torsion.

A practical decision sequence is:

1. Take moments to calculate external torque and reactions.
2. Ask whether the structure remains in equilibrium if the member's torsional stiffness is released.
3. Identify section type, shear center, open or closed walls, and warping restraints.
4. Choose the applicable Saint-Venant, warping, concrete torsion, or code design model.
5. Calculate twist and stresses, including combined bending, shear, and axial effects.
6. Design the member and every torque-transfer connection using compatible demand and resistance formats.

Circular-Shaft Mechanics

For an elastic circular shaft,

τ = Tr/J and θ = TL/(GJ)

where J is the polar moment, r is radial position, G is shear modulus, and θ is twist in radians. For a solid circle, J = πd^4/32, and maximum stress occurs at r = d/2.

A 4.0 in diameter solid steel shaft carries T = 24 kip-in over length L = 10 ft. Use G = 11,200 ksi.

J = π(4^4)/32 = 25.13 in^4

τ_max = (24 kip-in)(2 in)/(25.13 in^4) = 1.91 ksi

θ = (24 kip-in)(120 in)/[(11,200 kip/in^2)(25.13 in^4)]

θ = 0.01023 rad = 0.586°

The unit cancellation is a useful check. The calculated stress and twist are elastic responses, not code capacities. A strength check, service twist limit, fatigue requirement, or connection check still uses its applicable criteria.

Noncircular and Thin-Walled Members

Do not insert a rectangle, I-shape, channel, or angle into circular Tr/J formulas using the polar moment of area. Noncircular sections use a torsion constant, commonly also denoted J, whose value and stress distribution come from the appropriate theory or reference table. The same symbol does not make it the circular polar moment.

Closed thin-walled tubes develop circulating shear flow and are generally efficient in torsion. Open sections such as channels, angles, and wide-flange shapes have small Saint-Venant torsional stiffness. Their cross-sections may warp: longitudinal fibers displace different amounts. If warping is free, associated longitudinal restraint stress can be small; if a fixed end, diaphragm, or connection restrains warping, warping normal stresses and bimoment develop. End conditions can therefore change response even when applied torque is unchanged.

In steel, torsion may combine Saint-Venant shear, warping normal and shear stresses, bending, and local effects. The AISC 15th-edition provisions and design aids control the required checks. Lateral-torsional buckling of a beam involves coupled lateral displacement and twist under bending; it is related to torsional properties but is not the same calculation as a shaft under pure torque.

Reinforced Concrete Torsion

After cracking, reinforced concrete torsion is represented by a space-truss-type mechanism involving closed transverse reinforcement, longitudinal reinforcement, and diagonal compression. Required equilibrium torsion must be carried with ACI 318-14 strength and detailing provisions. Compatibility torsion may receive different treatment only under the code's conditions. Ordinary open stirrups or flexural bars assumed without anchorage do not automatically form the required torsional cage. Combined shear and torsion can govern transverse reinforcement.

Exam Checks

Preserve torque units—kip-ft versus kip-in is a factor of twelve. Mark whether twist is radians or degrees. Confirm J belongs to the actual section theory, locate the shear center, and identify warping restraint. Most importantly, state why the torque exists. If it is required by equilibrium, no assumed redistribution can make it disappear.

For a torque diagram, show concentrated and distributed torques, solve support torque by equilibrium, and plot internal T along the member. Internal torque changes where external torque enters; a change in section stiffness changes twist rate, not equilibrium torque by itself.