Page 81 - Aircraft Stuctures for Engineering Student

66 Torsion of solid sections

Am. r = Tr/Ip where Ip = 7ra4/2,
de/& = 2T/&a4, w = 0 everywhere
P.3.2 Deduce a suitable warping function for the circular section bar of P.3.1 and
hence derive the expressions for stress distribution and rate of twist.
TY Tx Tr de T
Ans. @ = 0, rzx = -- -
IP ' rzs = -' I,
I,' rzy = - dz GI,
P.3.3 Show that the warping function @ = kxy, in which k is an unknown constant,
may be used to solve the torsion problem for the elliptical section of Example 3.1.

P.3.4 Show that the stress function
27 'I
(2 + y2) - - (X3 - 3XY ) - -2
1
2a
is the correct solution for a bar having a cross-section in the form of the equilateral
triangle shown in Fig. P.3.4. Determine the shear stress distribution, the rate of
twist and the warping of the cross-section. Find the position and magnitude of the
maximum shear stress.

Fig. P.3.4

Am.
dz
rZx = -GE (y + %)
dZ

a dB
r,, (at centre of each side) = - - G-
2dz
de - 15&T
-
dz-Ga4

w = L de (y3 - 3*y)
2a dz