Torsion of Solid Noncircular Shafts

Torsion of solid noncircular shafts differs fundamentally from torsion of circular shafts. Cross-sections generally do not remain plane: they warp, and shear stress is distributed nonuniformly over the area.

Why the circular-shaft formula does not apply

For a circular section, the geometry is compatible with circular shear paths and \(\tau=T\rho/J\) can be used. For a rectangular, elliptical, or other solid noncircular section, the polar second moment of area alone defines neither torsional stiffness nor stress distribution.

Saint-Venant torsion

For a prismatic member, the twist rate is

\[\frac{d\varphi}{dx}=\frac{T}{GJ_t},\]

where \(T\) is torque, \(G\) is shear modulus, and \(J_t\) is the torsion constant. It depends on section shape and equals the polar second moment only for circular shafts. The shear-stress field follows from the Saint-Venant torsion problem, for example through Prandtl's stress function.

Solid rectangular section

Let \(a\) be the longer side and \(b\) the shorter side, so that \(a\ge b\). Engineering calculations commonly use

\[\tau_{\max}=\frac{T}{\alpha a b^2},\qquad J_t=\beta a b^3,\qquad \varphi=\frac{TL}{GJ_t}=\frac{TL}{\beta G a b^3}.\]

Maximum shear stress occurs at the midpoint of the long sides. It is zero at the corners and at the center. The coefficients \(\alpha\) and \(\beta\) depend on the aspect ratio:

\(a/b\)	\(\alpha\)	\(\beta\)
1.0	0.208	0.141
1.5	0.231	0.196
2.0	0.246	0.229
2.5	0.258	0.249
3.0	0.267	0.263
4.0	0.282	0.281
6.0	0.299	0.299
8.0	0.307	0.307
10.0	0.313	0.313
\(\infty\)	0.333	0.333

Linear interpolation may be used for intermediate \(a/b\). For a very narrow rectangle, \(a/b\gg1\), \(\alpha\approx\beta\approx1/3\), hence \(J_t\approx ab^3/3\) and \(\tau_{\max}\approx3T/(ab^2)\).

Approximation for the torsion constant

When tabulated data are unavailable, the following approximation is convenient for \(a\ge b\):

\[J_t\approx\frac{ab^3}{3}\left[1-0.63\frac{b}{a}+0.052\left(\frac{b}{a}\right)^5\right].\]

It closely reproduces the tabulated \(\beta\) values; for a square it gives \(J_t\approx0.141a^4\). The torsion constant must not be replaced by the rectangular section's polar second moment \(I_p=I_x+I_y\).

Strength and stiffness checks

The two criteria are checked separately:

\[\tau_{\max}\le\tau_{allow},\qquad \varphi=\frac{TL}{GJ_t}\le\varphi_{allow}.\]

If twist per unit length is limited, check \(\theta=T/(GJ_t)\le\theta_{allow}\). These relations apply away from torque application zones, abrupt section changes, and restraints. Fillets, keyways, and other discontinuities require a suitable stress-concentration factor.

Other solid noncircular sections

Elliptical, triangular, and other solid shapes use their own analytical solutions, tabulated coefficients, or numerical analysis. The general route remains the same: determine \(J_t\), calculate twist, and check maximum shear stress.

Scope

This section covers only solid noncircular sections. Open and closed thin-walled members require different models and belong to a separate advanced section.

Calculation route

1. Define the shape and dimensions of the solid section.
2. Obtain \(J_t\) and the maximum-stress coefficient from a formula or table.
3. Calculate maximum shear stress.
4. Calculate total twist or twist per unit length.
5. Check strength, stiffness, and stress concentration where relevant.