Page 73 - Aircraft Stuctures for Engineering Student

58 Torsion of solid sections

Fig. 3.6 Torsion of a bar of elliptical cross-section.
The semi-major and semi-minor axes are u and b respectively, so that the equation of
its boundary is
x2 y'
-+-=I
u2 b2
If we choose a stress function of the form
4=c (:: -+--1 ;: )

then the boundary condition 4 = 0 is satisfied at every point on the boundary and the
constant C may be chosen to fuKl the remaining requirement of compatibility. Thus,
from Eqs (3.11) and (i)
2C(-$+$) = -2Gx de

or
de hb2
C = -G- (ii)
dz (a2 + b2)
giving
(iii)

Substituting this expression for 9 in Eq. (3.8) establishes the relationship between
the torque T and the rate of twist

T = -2G- de dz (a2 *b2 +b2) ('JJld.dy+~lly2~dy-Jldxdy)
a2
The first and second integrals in this equation are the second moments of area
Iyy = 7ru3b/4 and I,, = 7rub3/4, while the third integral is the area of the cross-section
A = nub. Replacing the integrals by these values gives
dB m3b3
T=G-
dz (a2 + b2)