CTC / MTC 222Strength of Materials

Chapter 5Torsional Shear Stress and

Torsional Deformation

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

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

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, τmax = Tc / J• T = applied torque, c = radius of cross section, J = polar moment of

inertia• Polar moment of Inertia, J = ∫ r2 dA (Derivation in text)

• For a solid circular cross-section, J = π D4 / 32• For a hollow circular cross-section, J = π (Do

4 - Di4 ) / 32

• Units – in4 , or mm4

• Stress at any radial position, r, in the bar can be calculated from:

• τ = τ max r / c = Tr / J

Torsional Shear Stress, τ

• Torsional shear stress equation, τmax = Tc / J• Expression can be simplified by defining the polar section

modulus, Zp= J / c, where c = r = D/2• For a solid circular cross-section, Zp = π D3 / 16• For a hollow circular cross-section, Zp = π (Do

4 - Di4 ) /

(16Do)

• Then, τmax = T / Zp • Consistent units must be used, e.g., if Zp is in3 , and τmax is

desired in psi, T must be in # - in

• If design shear stress, τd is known, required polar section modulus can be calculated from:

• Zp = T / τd (Error in textbook in Equation 5-17)

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

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

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

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