242
7 Torsion of Prismatic Bars
Fig. 7.8 Cross-section of a thin-walled tube and torque due to shear
Therefore,
M t = 2q A
(7.23)
where A is the area enclosed by the centre line of the tube. Equation (7.23) is generally
known as the “Bredt-Batho” formula.
Determination of the Twist of the Tube
In order to determine the twist of the tube, Castigliano’s theorem is used. Referring
to Fig. 7.8b, the shear force on the element is τ t dS = qdS. Due to shear strain γ ,
the force does work equal to U
i.e.
U =
1
2
(τ tdS)δ
=
1
2
(τ tdS)γ · dl
=
1
2
(τ tdS) · dl ·
τ
G
(since τ = Gγ )
=
τ
2 t
2 dSdl
2Gt
=
q
2 dSdl
2Gt
U =
M
2
t dl
8A 2 G
·
dS
t
Therefore, the total elastic strain energy is
7 Torsion of Prismatic Bars
Fig. 7.8 Cross-section of a thin-walled tube and torque due to shear
Therefore,
M t = 2q A
(7.23)
where A is the area enclosed by the centre line of the tube. Equation (7.23) is generally
known as the “Bredt-Batho” formula.
Determination of the Twist of the Tube
In order to determine the twist of the tube, Castigliano’s theorem is used. Referring
to Fig. 7.8b, the shear force on the element is τ t dS = qdS. Due to shear strain γ ,
the force does work equal to U
i.e.
U =
1
2
(τ tdS)δ
=
1
2
(τ tdS)γ · dl
=
1
2
(τ tdS) · dl ·
τ
G
(since τ = Gγ )
=
τ
2 t
2 dSdl
2Gt
=
q
2 dSdl
2Gt
U =
M
2
t dl
8A 2 G
·
dS
t
Therefore, the total elastic strain energy is