Page 75 - Aircraft Stuctures for Engineering Student

60 Torsion of solid sections

cross-section are
u= -ey, u=ex

It is also reasonable to assume that the warping displacement w is proportional to the
rate of twist and is therefore constant along the length of the bar. Hence we may
define w by the equation

(3.17)

where $(x, y) is the warping function.
The assumed form of the displacements u, u and w must satisfy the equilibrium
and force boundary conditions of the bar. We note here that it is unnecessary to
investigate compatibility as we are concerned with displacement forms which are
single valued functions and therefore automatically satisfy the compatibility
requirement.
The components of strain corresponding to the assumed displacements are
obtained from Eqs (1.18) and (1.20) and are

E, = Ey = E, = yxy = 0
+y-=-+-=- du de(&) -- y)
dw
A~ ax az dz ax (3.18)

The corresponding components of stress are, from Eqs (1.42) and (1.46)
rzx=G-(--y) I
ax=u =,=rxy=0)
Y
de a$
dz ax b (3.19)
J
rzy = ce (3+x>
dz
ay
Ignoring body forces we see that these equations identically satisfy the first two of the
equilibrium equations (1.5) and also that the third is fuMled if the warping function
satisfies the equation

(3.20)

The direction cosine n is zero on the cylindrical surface of the bar and so the first
two of the boundary conditions (Eqs (1.7)) are identically satisfied by the stresses
of Eqs (3.19). The third equation simplifies to

(3.21)

It may be shown, but not as easily as in the stress function solution, that the shear
stresses defined in terms of the warping function in Eqs (3.19) produce zero resultant
shear force over each end of the bar'. The torque is found in a similar manner to that