Torsion
Torsion is twisting of a member by a torque T about its longitudinal axis. For a circular shaft, cross-sections remain plane and radii remain straight, giving linear shear strain γ = r·θ/L where θ is angle of twist over length L. With linear elastic isotropic material, shear stress τ = G·γ, so τ varies linearly from zero at center to maximum at outer radius: τ = T·r / J. Maximum at r=c: τ_max = T·c / J. J is polar second moment of area, for solid circle J = π·d⁴/32 = π·r⁴/2. For hollow circle J = π·(D⁴ - d⁴)/32. Angle of twist: θ = T·L / (G·J) in radians, where G is shear modulus. Torsional stiffness k_t = T/θ = G·J/L, torsional flexibility f = L/(G·J). Shear modulus relates to E and ν: G = E/[2(1+ν)]. For solid shaft, τ_max = 16·T/(π·d³) and θ = 32·T·L/(G·π·d⁴). For hollow, higher strength-to-weight because material near center carries low stress. Assumptions: prismatic circular shaft, small deformations, homogeneous linear elastic below proportional limit in shear, torque applied as pure couple, cross-section free to warp is prevented only for circular (non-circular sections warp and formula τ=Tr/J invalid; need α·β coefficients). Beyond yield, elastic formula overestimates, need plastic torsion T_p = 2π·τ_y·c³/3 for solid. Engineers use torsion for shaft sizing: τ_max ≤ τ_allow = τ_y/FS, θ/L ≤ allowable twist (e.g., 1°/m), and to avoid resonance with torsional natural frequency.
Knowledge Check
1.A solid circular shaft of radius 50 mm is subjected to a torque of 5 kN·m. What is the maximum shear stress? (J = πr⁴/2)
2.The angle of twist θ of a circular shaft under torque is given by:
3.In a circular shaft under torsion, the shear stress distribution across radius is:
4.For a solid circular shaft, the polar second moment of area (J) is:
5.A hollow circular shaft has outer diameter 100 mm and inner diameter 80 mm. The polar moment of inertia (J) is:
6.What is the relationship between shear modulus (G), Young's modulus (E), and Poisson's ratio (ν) for isotropic material?
7.A shaft length 2 m with G=80 GPa and J=1×10⁶ mm⁴ is subjected to torque 2 kN·m. What is angle of twist?
8.The maximum shear stress in a circular shaft under torsion occurs at:
9.In torsion, the shear strain γ at radius r is related to angle of twist θ by:
10.A solid shaft diameter 40 mm is subjected to torque producing max shear stress 60 MPa. What is applied torque?