Chapter 7
Torsion of Prismatic Bars
7.1 Introduction
Most of the engineering components are required to withstand torsional or twisting
load. Usually, the components that transmit torque, such as propeller shaft and torque
tubes of power equipments, are tubular or circular in cross-section. The strength and
stiffness of these shafts and torque tubes of uniform circular cross-section can be
calculated using simple theory of torsion based on strength of materials. However,
slender members with other than circular cross-section are also often used. The
strength and stiffness of these members can be determined only by using more
sophisticated analyses.
From the study of elementary strength of materials, two important expressions
related to the torsion of circular bars were developed. They are
τ =
M t r
J
(7.1)
and
θ =
1
L
L
M t dz
G J
(7.2)
Here τ represents the shear stress, M t the applied torque, r the radius at which the
stress is required, G the shear modulus or modulus of rigidity, θ the angle of twist
per unit longitudinal length, L the length, and z the axial co-ordinate. Also, J is polar
moment of inertia which is defined by
A r
2 dA.
The following are the assumptions associated with the elementary approach in
deriving Eqs. (7.1) and (7.2).
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
T. G. Sitharam and L. Govindaraju, Theory of Elasticity,
https://doi.org/10.1007/978-981-33-4650-5_7
223
Torsion of Prismatic Bars
7.1 Introduction
Most of the engineering components are required to withstand torsional or twisting
load. Usually, the components that transmit torque, such as propeller shaft and torque
tubes of power equipments, are tubular or circular in cross-section. The strength and
stiffness of these shafts and torque tubes of uniform circular cross-section can be
calculated using simple theory of torsion based on strength of materials. However,
slender members with other than circular cross-section are also often used. The
strength and stiffness of these members can be determined only by using more
sophisticated analyses.
From the study of elementary strength of materials, two important expressions
related to the torsion of circular bars were developed. They are
τ =
M t r
J
(7.1)
and
θ =
1
L
L
M t dz
G J
(7.2)
Here τ represents the shear stress, M t the applied torque, r the radius at which the
stress is required, G the shear modulus or modulus of rigidity, θ the angle of twist
per unit longitudinal length, L the length, and z the axial co-ordinate. Also, J is polar
moment of inertia which is defined by
A r
2 dA.
The following are the assumptions associated with the elementary approach in
deriving Eqs. (7.1) and (7.2).
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
T. G. Sitharam and L. Govindaraju, Theory of Elasticity,
https://doi.org/10.1007/978-981-33-4650-5_7
223