Torsion and Shafts Explained: Torque, Shear Stress, and Twist

What Torsion Does to a Shaft

Torque is a twisting moment applied about a shaft's longitudinal axis, and under it, imaginary flat cross-sections that were perpendicular to the axis before loading rotate relative to one another, like a stack of coins twisted against each other. The center of a solid circular shaft, right on the axis, does not rotate relative to its neighbors at all, so it carries essentially no shear stress. Material farther from the axis is forced to slide past its neighbors more, so shear stress climbs steadily from zero at the center to a maximum at the outer surface.

This distribution is the torsional twin of the bending stress picture: in bending, stress is zero at the neutral axis and maximum at the outer fiber; in torsion, stress is zero at the center and maximum at the outer radius. In both cases the material doing the least work sits right on the axis, which is the key idea behind hollow shaft design, covered next.

• Shaft axis (center): zero shear stress, since there is no relative rotation there.
• Outer surface: maximum shear stress, since material there rotates the most relative to its neighbors.
• Torque climbs with transmitted power and drops with rotational speed, so slow, high-power shafts see the largest torques.

The Torsion Formula

The shear stress at any radius in a circular shaft under torque is given by τ = T·r / J, where T is the applied torque, r is the radial distance from the shaft's axis to the point of interest, and J is the polar moment of inertia, a geometric property describing how the cross-section's area is distributed around the axis of twist. As with bending, design usually cares most about the outer surface, so r is typically taken as the shaft's outer radius, c.

Every term plays a distinct role. T comes from the power and speed the shaft transmits, or directly from an applied load. J comes purely from the shaft's geometry: a solid circular shaft, a hollow tube, and a shaft with keyways all have different J values even at the same outer diameter. Getting J right is essential, because it appears in the denominator: a shaft with a larger J carries the same torque at proportionally lower shear stress.

τ = T·r / J — shear stress grows with torque and with distance from the shaft's axis, and shrinks as the polar moment of inertia of the cross-section grows.

Polar Moment of Inertia: Why Hollow Shafts Work So Well

For a solid circular shaft of radius c, the polar moment of inertia is J = π·c⁴ / 2, and for a hollow shaft with outer radius c(o) and inner radius c(i) it becomes J = π·(c(o)⁴ − c(i)⁴) / 2. Because J depends on radius to the fourth power, material near the center of a solid shaft contributes very little, exactly matching the observation that stress there is nearly zero. Removing that lightly stressed core material and relocating it to the outer diameter, turning a solid shaft into a hollow one of larger outer radius, can increase torsional strength and stiffness dramatically for the same or even less total weight.

Aircraft drive shafts and racing driveshafts are frequently made as hollow tubes for exactly this reason, trading a modest increase in outer diameter for a large reduction in weight at the same torsional capacity.

• Solid circular shaft: J = π·c⁴ / 2.
• Hollow circular shaft: J = π·(c(o)⁴ − c(i)⁴) / 2.
• Because J scales with radius to the fourth power, small increases in outer diameter produce large gains in torsional capacity.

Angle of Twist

Torque does not just create stress; it also rotates one end of a shaft relative to the other, an effect quantified by the angle of twist, φ = T·L / (J·G). Here L is the shaft's length, J is the same polar moment of inertia used in the stress formula, and G is the shear modulus of the material, a stiffness property analogous to the elastic modulus used in tension and bending but specific to shear loading.

Angle of twist matters wherever precise rotation or alignment is important, such as in machine tool spindles, control linkages, or long transmission shafts, even when the stress itself is comfortably below yield. A shaft can be safe on stress alone and still twist enough to throw off timing or alignment, which is why twist is checked as its own separate limit rather than assumed from a stress check.

φ = T·L / (J·G) — twist grows with torque and length, and shrinks as the section's polar moment of inertia and the material's shear modulus grow.

Sizing a Shaft: Stress and Twist Together

A complete shaft design checks two separate limits, not one. The strength limit compares the maximum shear stress, τ = T·c / J, against an allowable shear stress derived from the material's shear strength divided by a safety factor. The stiffness limit compares the angle of twist over the shaft's length against whatever rotational tolerance the application demands, which is an entirely independent requirement with its own allowable value, often expressed in degrees per unit length for long shafts.

In many real designs the stiffness limit governs before the strength limit does, especially for long, slender shafts, so sizing purely to stress can leave a shaft that twists more than the machine can tolerate even though it will never yield. Keyways, shoulders, and other geometric discontinuities also concentrate stress locally well above the smooth-shaft prediction, so those features are typically checked separately using stress concentration factors rather than folded into the basic torsion formula.