Civil engineering mechanics & structural foundations of Shaft Torsion Testing.
Torsion represents a primary state of shear deformation occurring in structural drive shafts, crane hoist mechanisms, curved bridge box girders, and building spandrel beams supporting eccentrically loaded floor slabs. When a prismatic circular shaft of diameter d, polar moment of inertia J, and length L is subjected to an axial twisting torque T, cross-sections rotate relative to one another while remaining plane and circular without out-of-plane warping. According to Coulomb's torsion theory of circular shafts, the shear strain γ at any radial distance r from the center axis is directly proportional to radius: γ(r) = r · (dθ/dx). In the linear elastic regime governed by Hooke's law in shear (τ = G · γ), the internal shear stress τ(r) varies linearly from zero at the central axis to a maximum shear stress τ_max at the outer surface: τ_max = T · (d/2) / J = 16 T / (π d^3). Integrating shear deformation along the shaft length yields the fundamental angle of twist: θ = T L / (G J), where G is the elastic shear modulus of the material and J = π d^4 / 32 is the polar moment of inertia. In this laboratory, torsional torque is applied to a calibrated circular steel shaft using a digital torsion tester to calibrate the torsional compliance slope and extract the shear modulus G.
1. Coulomb's Theory of Torsion for Circular Shafts
Consider a straight, solid circular cylinder of radius $R = d/2$, length $L$, and polar second moment of area $J$ subjected to twisting couples $T$ at its distal ends. Due to rotational axisymmetry, cross-sections rotate as rigid disks about the longitudinal axis through angle $\theta(x)$ without radial distortion or axial out-of-plane warping.
Consider a cylindrical element at radial distance $r$ from the axis. An axial line element on this surface is sheared through an angle $\gamma(r)$, representing engineering shear strain:
\[ \gamma(r) = r \frac{d\theta}{dx} = \frac{r \theta}{L} \]In the linear elastic regime governed by Hooke's Law in shear ($\tau = G \gamma$), internal shear stress $\tau(r)$ is:
\[ \tau(r) = G \cdot \gamma(r) = G \cdot r \frac{\theta}{L} \]The internal resisting torque is obtained by integrating shear stress moments over cross-sectional area $A$:
\[ T = \int_A r \cdot \tau(r) \, dA = \frac{G \theta}{L} \int_A r^2 \, dA = \frac{G J \theta}{L} \]where $J = \int_A r^2 \, dA = \frac{\pi d^4}{32}$ is the polar moment of inertia.
2. Stress Distribution & Angle of Twist Formulation
Equating $\frac{G \theta}{L} = \frac{T}{J}$ yields the unified torsion formula for circular shafts:
\[ \frac{T}{J} = \frac{\tau(r)}{r} = \frac{G \theta}{L} \]Maximum shear stress $\tau_{max}$ occurs at the outermost surface fibers ($r = R = d/2$):
\[ \tau_{max} = \frac{T \cdot (d/2)}{J} = \frac{T \cdot (d/2)}{\pi d^4 / 32} = \frac{16 T}{\pi d^3} \]The total angle of twist $\theta$ (in radians) across gauge length $L$ is:
\[ \theta = \frac{T L}{G J} \]Converting $\theta$ from radians to degrees ($\theta^\circ = \theta \times \frac{180}{\pi}$):
\[ \theta^\circ = \left[ \frac{180 L}{\pi G J} \right] T = k_t \cdot T \]where $k_t = 0.0450\text{ deg/(N}\cdot\text{m)}$ represents the torsional compliance slope.
Viva voce questions & analytical theory
Why do circular cross-sections not experience out-of-plane warping under torsion, unlike rectangular sections?
Axisymmetry of circular shafts ensures that shear stresses are purely circumferential and shear strains purely tangential, meaning plane circular cross-sections rotate as rigid planes without axial warping; non-circular sections warp axially to satisfy zero shear stress boundary conditions on flat faces.
How does the elastic shear modulus G relate to Young's modulus E and Poisson's ratio ν?
For isotropic materials, continuum mechanics proves G = E / [2 (1 + ν)]; for structural steel with E ≈ 200 GPa and ν ≈ 0.28, G evaluates to approximately 78 to 80 GPa.
What is the characteristic fracture surface of a brittle vs ductile circular shaft under pure torsion?
Ductile metals fail along transverse shear planes (flat 90° cross-section) governed by maximum shear stress τ_max; brittle materials (e.g. cast iron or chalk) fail along a 45° helical spiral governed by maximum principal tensile stress σ_1 = τ_max.
Why are hollow structural circular tubes far more efficient in torsion than solid circular shafts?
In torsion, internal material near the neutral axis carries near-zero shear stress while contributing weight; moving material away from the axis to the outer perimeter maximizes polar moment of inertia J for a given cross-sectional area.
What is Saint-Venant's torsional constant J for a non-circular thin-walled closed tube?
According to Bredt's first and second bath formulas, shear stress is τ = T / (2 A_m t) and angle of twist is θ = (T L / 4 A_m^2 G) ∮ (ds / t), where A_m is the area enclosed by the mean perimeter profile.
Why must the torsion test machine permit free axial elongation of the specimen during twisting?
High torsional shear induces minor second-order longitudinal shortening (Poynting effect); restraining axial movement would induce parasitic axial tension that distorts pure torsion stress states.
Experimental precautions and laboratory safety
- Verify collet chucks are tightened symmetrically to prevent shaft slippage or eccentric bending during torque application.
- Ensure the rotary twist encoder is clamped rigidly to the shaft and does not slip against the polished steel surface.
- Do not exceed the elastic torsional yield limit (T_y = τ_y J / r) to prevent permanent torsional plastic set.
- Confirm the reaction head moves freely on low-friction axial guide rails to prevent parasitic axial tension.
Theoretical assumptions & boundary condition limits
- The civil engineering physical models apply within the specified calibrated operational window (20.0 to 140.0 N·m).
- Assumes homogeneous, isotropic continuum material behavior without unmodeled micro-fracture anomalies.
- Continuous sensor readings reflect ordinary least-squares regression across successive observations.
- Extreme inelastic yielding or cyclic degradation requires non-linear constitutive continuum damage mechanics.