Learning topic
Torsion of Solid and Hollow Circular Shafts
Learn torsion of solid and hollow circular shafts: torque, polar moment, shear stress, angle of twist, strength checks, and a worked shaft example.
Torsion is deformation of a member under moments acting about its longitudinal axis. For solid and hollow circular shafts, classical torsion theory assumes that cross-sections remain plane and rotate relative to one another.
Internal torque
The internal torque $T$ at a section is determined by the method of sections from moment equilibrium about the shaft axis. For a stepped shaft or a shaft carrying several applied torques, $T$ is determined separately for each segment and a torque diagram can be constructed.
Polar properties of the section
For a solid circular section of diameter $d$:
$$J_p=\frac{\pi d^4}{32},\qquad W_p=\frac{J_p}{d/2}=\frac{\pi d^3}{16}.$$
For a hollow circular section with outer diameter $D$ and inner diameter $d$:
$$J_p=\frac{\pi(D^4-d^4)}{32},\qquad W_p=\frac{J_p}{D/2}=\frac{\pi(D^4-d^4)}{16D}.$$
- $J_p$ — polar second moment of area;
- $W_p$ — polar section modulus;
- $D$, $d$ — section diameters.
The calculator below is for a solid circular section.
The polar second moment of area $J_p$ characterizes the geometric resistance of the section to torsion, while the polar section modulus $W_p$ is convenient for calculating the maximum shear stress.
Shear stresses
For a straight solid or hollow circular shaft under elastic torsion:
$$\tau(\rho)=\frac{T\rho}{J_p},\qquad \tau_{\max}=\frac{T}{W_p}.$$
If $T$, $G$, and $J_p$ are constant over a segment, the angle of twist is:
$$\varphi=\frac{TL}{GJ_p}.$$
- $T$ — torque;
- $\rho$ — radial distance from the shaft axis;
- $J_p$ — polar second moment of area;
- $W_p$ — polar section modulus;
- $G$ — shear modulus;
- $L$ — segment length;
- $\varphi$ — angle of twist, rad.
For a stepped shaft, the total twist is the algebraic sum $\varphi=\sum_i T_iL_i/(G_iJ_{p,i})$.
In a solid circular shaft, shear stress varies linearly with radius: $\tau=0$ at the axis and reaches its maximum at the outer surface. In a hollow shaft, material near the axis is removed, so for a given amount of material a hollow section can use material more efficiently in torsion.
Strength check
In a simple allowable-stress model, the strength condition is $|\tau_{\max}|\le[\tau]$. For a solid circular shaft, this relation can be used to select the required diameter from the known torque and allowable shear stress.
Stiffness check
Excessive twist may impair machine accuracy even when stresses are safe. Therefore, the total or specific angle of twist is compared with an allowable value. For a shaft consisting of several segments, the twists are added algebraically.
Example
For a solid shaft of diameter $d=40\ \text{mm}$ carrying $T=500\ \text{N·m}$, first convert the torque to N·mm: $T=500000\ \text{N·mm}$. The polar section modulus is:
$$W_p=\frac{\pi40^3}{16}\approx12566\ \text{mm}^3.$$
Thus $\tau_{\max}=500000/12566\approx39.8\ \text{MPa}$.
Limits of applicability
These relations apply primarily to solid and hollow circular members in the linear-elastic range. Noncircular sections have a different stress and deformation distribution in torsion.
About this topic
In circular and hollow shaft torsion, the plane sections assumption holds: cross-sections remain flat and rotate relative to one another. This section is devoted to calculating polar moments of inertia and section moduli, constructing torque and twist angle diagrams, and performing allowable shear stress strength and angular deformation stiffness calculations.