Calculate torsional stiffness (k = GJ/L) for shafts, springs, and torsion bars. Includes deflection, natural frequency, and length comparison table.
Torsional stiffness measures how much torque is needed to twist a shaft by one radian: k = GJ/L, where G is the shear modulus, J is the torsional constant, and L is the shaft length. Higher stiffness means less angular deflection under load — critical for drive shafts, coupling hubs, and precision positioning systems.
The relationship is directly analogous to linear spring stiffness (k = EA/L for axial, k = F/x for springs). Doubling the diameter increases torsional stiffness by 16× (since J scales as D⁴), while doubling the length halves it. This strong diameter dependence means that small increases in shaft size dramatically reduce twist.
Torsional stiffness also determines natural frequency: a shaft with a flywheel at one end oscillates at f = √(k/I)/(2π). If this frequency matches an excitation source (engine RPM, gear mesh), resonance causes dangerous vibrations. This calculator provides stiffness, deflection, natural frequency, and series/parallel combinations for system design.
Drive train engineers size shafts to meet stiffness requirements. Vehicle suspension uses torsion bars as springs. Robotics needs precise rotational stiffness for positioning. This calculator gives immediate answers without manual J calculation. Keep these notes focused on your operational context. Tie the context to the calculator’s intended domain. Use this clarification to avoid ambiguous interpretation.
k = GJ/L (N·m/rad). θ = T/k (rad). Natural frequency: f = √(k/I)/(2π) Hz. Series: 1/k_total = Σ(1/kᵢ). Parallel: k_total = Σkᵢ.
Result: k = 3,003 N·m/rad, deflection = 1.91°
A 25mm solid steel shaft, 800mm long: J = π(25)⁴/32 = 38,350 mm⁴. k = 79300 × 38350 / 800 / 10⁶ = 3,003 N·m/rad. Under 100 N·m: θ = 100/3003 = 0.0333 rad = 1.91°.
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The torque required to twist a shaft by one radian: k = T/θ = GJ/L. Units are N·m/rad or lb·in/rad. Higher k means stiffer — less twist for the same torque.
Torsional rigidity is GJ (N·m²) — a property of the cross-section and material, independent of length. Torsional stiffness is GJ/L — includes the shaft length.
J scales as D⁴ (for circular shafts), so doubling diameter increases stiffness 16×. This is the most powerful lever for increasing torsional stiffness.
Flexible couplings add their own torsional stiffness in series. The total system stiffness is always less than the stiffest element: 1/k_total = 1/k_shaft + 1/k_coupling.
Shear modulus G decreases slightly with temperature (~5% per 100°C for steel). This reduces torsional stiffness proportionally.
A straight bar fixed at one end and loaded in torsion at the other — acts as a rotational spring with rate k = GJ/L. Used in vehicle suspensions, especially heavy-duty applications.