7.5 Torsion of Circular Cross-Section
235
or
τ =
M t
I P
x 2 + y 2
Therefore,
τ =
M t .r
I P
or
τ =
M t .r
J
(since J = I P )
where r is the radial distance of the point (x, y). Hence, all the results of the elementary
analysis are justified.
7.6 Torsion of Elliptical Cross-Section
Let the warping function is given by
ψ = Ax y
(7.16)
where A is a constant. This also satisfies the Laplace equation. The boundary
condition gives
(Ay − y)
dy
dS
− (Ax + x)
dx
dS
= 0
or
y( A − 1)
dy
dS
− x(A + 1)
dx
dS
= 0
i.e.
(A + 1)2x
dx
dS
− (A − 1)2y
dy
dS
= 0
or
235
or
τ =
M t
I P
x 2 + y 2
Therefore,
τ =
M t .r
I P
or
τ =
M t .r
J
(since J = I P )
where r is the radial distance of the point (x, y). Hence, all the results of the elementary
analysis are justified.
7.6 Torsion of Elliptical Cross-Section
Let the warping function is given by
ψ = Ax y
(7.16)
where A is a constant. This also satisfies the Laplace equation. The boundary
condition gives
(Ay − y)
dy
dS
− (Ax + x)
dx
dS
= 0
or
y( A − 1)
dy
dS
− x(A + 1)
dx
dS
= 0
i.e.
(A + 1)2x
dx
dS
− (A − 1)2y
dy
dS
= 0
or