Page 77 - Aircraft Stuctures for Engineering Student

62 Torsion of solid sections

Fig. 3.8 Equilibrium of element of membrane.
or

(3.24)

Equation (3.24) must be satisfied at all points within the boundary of the membrane.
Furthermore, at all points on the boundary
w=o (3.25)
and we see that by comparing Eqs (3.24) and (3.25) with Eqs (3.11) and (3.7) w is
analogous to q5 when q is constant. Thus if the membrane has the same external
shape as the cross-section of the bar then
W(X,Y) = +(X,Y)
and

The analogy now being established, we may make several useful deductions relating
the deflected form of the membrane to the state of stress in the bar.
Contour lines or lines of constant w correspond to lines of constant q5 or lines of
shear stress in the bar. The resultant shear stress at any point is tangential to the
membrane contour line and equal in value to the negative of the membrane slope,
awlan, at that point, the direction n being normal to the contour line (see
Eq. (3.16)). The volume between the membrane and the xy plane is

Vol= jjwdxdy

and we see that by comparison with Eq. (3.8)
T = 2V0l