1. CTC / MTC 222 Strength of Materials Chapter 5 Torsional Shear Stress and Torsional Deformation

2. Chapter Objectives Define torque and compute the magnitude of torque exerted on a member subjected to torsional loading Compute the maximum shear stress in a member subjected to torsional loading Define polar moment of inertia and polar section modulus and compute their values for round shafts Compute the angle of twist of a member loaded in torsion Define the relationship between power, torque and rotational speed

3. Torsion Torsion – rotation or twist of a member about its longitudinal axis Torque – the load which causes this rotation (also called rotational moment or twisting moment) Torque = T = F x d F – applied force, d – distance from line of action of force to axis of rotation Units – Force x distance (same as moment) U. S. units – k-in, k-ft, lb-in, lb-ft SI units – N-m Application of a torque will cause torsional shear stress in the member The torque will also cause torsional deformation The angle of twist is dependent on the applied torque, the length of the member, its cross-sectional properties and the properties of the material

4. Torsional Shear Stress, τ When a member is subjected to an applied torque, an internal resisting torque is developed The internal resisting torque is the result of torsional shear stresses developed in the member Stress (and strain) vary linearly from the zero at the center of the member to a maximum value at the outside surface For a circular member, τ max = Tc / J T = applied torque, c = radius of cross section, J = polar moment of inertia Polar moment of Inertia, J = ∫ r 2 dA (Derivation in text) For a solid circular cross-section, J = π D 4 / 32 For a hollow circular cross-section, J = π (D o 4 - D i 4 ) / 32 Units – in 4, or mm 4 Stress at any radial position, r, in the bar can be calculated from: τ = τ max r / c = Tr / J

5. Torsional Shear Stress, τ Torsional shear stress equation, τ max = Tc / J Expression can be simplified by defining the polar section modulus, Z p = J / c, where c = r = D/2 For a solid circular cross-section, Z p = π D 3 / 16 For a hollow circular cross-section, Z p = π (D o 4 - D i 4 ) / (16D o ) Then, τ max = T / Z p Consistent units must be used, e.g., if Z p is in 3, and τ max is desired in psi, T must be in # - in If design shear stress, τ d is known, required polar section modulus can be calculated from: Z p = T / τ d (Error in textbook in Equation 5-17)

6. Torsional Deformation An applied torque will also cause torsional deformation The magnitude of torsional deformation, or angle of twist, is dependent on the applied torque, the length of the member, its cross-sectional properties and the properties of the material The angle of twist, in radians can be calculated from the equation: Θ = TL / JG, where T = applied torque, J = polar moment of inertia, and G – shear modulus of elasticity See Derivation in text, Section 5 -10 Recommended values of torsional stiffness, in terms of angle of twist per unit length are given in Table 5-2

7. Torque, Power and Rotational Speed Work (or energy) = force x distance U. S. units – k-in, k-ft, lb-in, lb-ft SI units – N-m (1 N-m = 1 Joule (J)) In a rotating shaft, an applied torque torque, T, rotates the shaft through an angular distance, Θ. In that case: Work = Torque x (angular distance) = T Θ Units of work (or energy) – Force x distance (same as moment) U. S. units – k-in, k-ft, lb-in, lb-ft SI units – N-m Power – the rate of transferring energy, or work done per unit time Power = work / time = torque x rotational speed P = T x n

8. Torque, Power and Rotational Speed Power in SI Units The standard unit for work is the joule, 1 joule = 1 N – M Power = work / time = joule / second = N-m / s = watt = W 1 kW = 1000W Standard unit for rotational speed (n) is radians per second. However, it is sometimes given in revolutions per minute (rpm). Use 1 minute = 60 s, and 1 revolution = 2 π radians to calculate conversion factor 1 rad / s = 9.549 rpm In the equation P = T x n Torque T is in N-M Rotational speed n is in rad /s Power P is then in N-M

9. Torque, Power and Rotational Speed Power in US Customary Units The standard unit for work is the lb-in, or lb-ft Power = work / time = lb-in / second, or lb-ft / second Power more commonly expressed in horsepower 1 hp = 550 lb-ft / s = 6600 lb–in / s Standard unit for rotational speed (n) is radians per second. However, it is sometimes given in revolutions per minute (rpm). Use 1 minute = 60 s, and 1 revolution = 2 π radians to calculate conversion factor 1 rad / s = 9.549 rpm In the equation P = T x n, If torque T is in lb-ft, and rotational speed n is in rad /s Power P is in lb-ft / s Convert to horsepower using 1 hp = 550 lb-ft / s

10. Torque, Power and Rotational Speed Power in US Customary Units Standard units for energy – force x distance lb-in, lb-ft, k-in, k-ft Power = energy / time = lb-in/s, or lb-ft/s Standard units for power is horsepower 1 hp = 550 lb-ft/s = 6600 lb-in/s The equation P = T x n can be used to calculate power, but the necessary conversions must be made If torque T is in lb-in, and rotational speed n is in rev/min (rpm) Power P in hp = T x n / 63000 If torque T is in lb-ft, and rotational speed n is in rev/min (rpm) Power P in hp = T x n / 5250