Page 76 - Aircraft Stuctures for Engineering Student

3.3 The membrane analogy 61

in Section 3.1 where, by reference to Fig. 3.3, we have

or

T = ."I dz 1 [ ($ + X)X - (2 - y)y] dxdy (3.22)
By comparison with Eq. (3.12) the torsion constant J is now, in terms of +

(3.23)

The warping function solution to the torsion problem reduces to the determination
of the warping function +which satisfies Eqs (3.20) and (3.21). The torsion constant
and the rate of twist follow from Eqs (3.23) and (3.22); the stresses and strains from
Eqs (3.19) and (3.18) and, finally, the warping distribution from Eq. (3.17).

"
___ ,..- ..... ,-,. _. . "
""
Prandtl suggested an extremely useful analogy relating the torsion of an arbitrarily
shaped bar to the deflected shape of a membrane. The latter is a thin sheet of material
which relies for its resistance to transverse loads on internal in-plane or membrane
forces.
Suppose that a membrane has the same external shape as the cross-section of a
torsion bar (Fig. 3.7(a)). It supports a transverse uniform pressure q and is restrained
along its edges by a uniform tensile force Nlunit length as shown in Fig. 3.7(a) and
(b). It is assumed that the transverse displacements of the membrane are small so
that N remains unchanged as the membrane deflects. Consider the equilibrium of an
element SxSy of the membrane. Referring to Fig. 3.8 and summing forces in the z direc-
tion we have

N

(a) (b)

Fig. 3.7 Membrane analogy: in-plane and transverse loading.