Page 74 - Aircraft Stuctures for Engineering Student

3.2 St. Venant warping function solution 59

from which (see Eq. (3.12))
7ra3b3
J= (VI
(a2 + b2)
The shear stress distribution is obtained in terms of the torque by substituting for
the product G dO/dz in Eq. (iii) from Eq. (iv) and then differentiating as indicated by
the relationships of Eqs (3.2). Thus
2TY 2 Tx
rZx = -~ TZY =
rub3 ’ 7ra3b
~
So far we have solved for the stress distribution, Eqs (vi), and the rate of twist,
Eq. (iv). It remains to determine the warping distribution w over the cross-section.
For this we return to Eqs (3.10) which become, on substituting from the above for
rzx, rzy and dO/dz
dw - 2Ty T (a2+b2) dw - 2Tx T (a2 +h2)
-
- --
-
dx 7rab3G+G m3b3 y7 % =-E 7ra3b3 X
or
(vii)

Integrating both of Eqs (vii)

T(b2 - a2) T(b2 - a2)
W= yx+fib), w= XY +f2 (XI
7ra3 b3 G nu3 b3 G
The warping displacement given by each of these equations must have the same value
at identical points (x,y). It follows thatfi(y) =f2(x) = 0. Hence

T(b2 - a2)
W= XY (viii)
nu3 b3 G
Lines of constant w therefore describe hyperbolas with the major and minor axes of
the elliptical cross-section as asymptotes. Further, for a positive (anticlockwise)
torque the warping is negative in the first and third quadrants (a > b) and positive
in the second and fourth.

v-- --I
3.2 St. Venant warping function-solution
In formulating his stress function solution Prandtl made assumptions concerned with
the stress distribution in the bar. The alternative approach presented by St. Venant
involves assumptions as to the mode of displacement of the bar; namely, that
cross-sections of a bar subjected to torsion maintain their original unloaded shape
although they may suffer warping displacements normal to their plane. The first
of these assumptions leads to the conclusion that cross-sections rotate as rigid
bodies about a centre of rotation or twist. This fact was also found to derive from
the stress function approach of Section 3.1 so that, referring to Fig. 3.4 and Eq.
(3.9), the components of displacement in the x and y directions of a point P in the