Equilibrium Equation

Because we are analyzing the static behavior of this wing, it is appropriate to simplify the fundamental constitutive relationship of torsional deformation, Eq. (2.42), to read

__ de

T = GJ— (4.42)

dy

where GJ is the effective torsional stiffness and T is the twisting moment about the elastic axis. Now, a static equation of moment equilibrium about the elastic axis

can be obtained by equating the rate of change of twisting moment to the negative of the applied torque distribution. This is a specialization of Eq. (2.43) in which time-dependent terms are ignored, yielding

dT dy	dy dy
= – M	(4.43)

Recognizing that uniformity implies GJ is constant over the length; substituting Eqs. (4.37) into Eq. (4.35) to obtain the applied torque; and, finally, substituting the applied torque and Eq. (4.42) for the internal torque into the equilibrium equation, Eq. (4.43), we obtain

—d20 2

GJ—r = – qc2cmac – eqcci + Nmgd dy2

Eq. (4.41) now can be substituted into the equilibrium equation to yield an in­homogeneous, second-order, ordinary differential equation with constant coeffi cients

d2 в qcae 1 , 2 л

~г~2 + -==- в = -= (qc^mac + qcaear – Nmgd) dy GJ GJ

A complete description of this equilibrium condition requires specification of the boundary conditions. Because the surface is built in at the root and free at the tip, these conditions can be written as

y = 0: в = 0 (zero deflection)

„ de. . . (4.46)

y = t. dy = 0 (zero twisting moment)

Obviously, these boundary conditions are valid only for the clamped-free condition. The boundary conditions for other end conditions for beams in torsion are given in Section 3.2.2.