Page 79 - Aircraft Stuctures for Engineering Student

64 Torsion of solid sections

Although q5 does not disappear along the short-edges of the strip and therefore does
not give an exact solution, the actual volume of the membrane differs only slightly
from the assumed volume so that the corresponding torque and shear stresses are
reasonably accurate. Also, the maximum shear stress occurs along the long sides of
the strip where the contours are closely spaced, indicating, in any case, that conditions
in the end region of the strip are relatively unimportant.
The stress distribution is obtained by substituting Eq. (3.26) in Eqs (3.2), thus
de
rzy = 2Gx- , rzx = 0 (3.27)
dz
the shear stress varying linearly across the thickness and attaining a maximum
de
rzr,max fGt- (3.28)
=
dz
at the outside of the long edges as predicted. The torsion constant J follows from the
substitution of Eq. (3.26) into Eq. (3.13), giving
St3
J=- (3.29)
3
and
3T
-
-
rzy,max -
St3
These equations represent exact solutions when the assumed shape of the deflected
membrane is the actual shape. This condition arises only when the ratio s/t
approaches infinity; however, for ratios in excess of 10 the error is of the order of
only 6 per cent. Obviously the approximate nature of the solution increases as s/t
decreases. Therefore, in order to retain the usefulness of the analysis, a factor p is
included in the torsion constant, viz.

J=- Pt3
3
Values of p for Merent types of section are found experimentally and quoted in
various reference^^'^. We observe that as s/t approaches infinity p approaches unity.
The cross-section of the narrow rectangular strip of Fig. 3.9 does not remain plane
after loading but suffers warping displacements normal to its plane; this warping may
be determined using either of Eqs (3.10). From the first of these equations
aw de (3.30)
-_
ax - yz
since rzx = 0 (see Eqs (3.27)). Integrating Eq. (3.30) we obtain
d6
w = xy- + constant (3.31)
dz
Since the cross-section is doubly symmetrical w = 0 at x = y = 0 so that the constant
in Eq. (3.31) is zero. Therefore
d6
w=xy- (3.32)
dz
and the warping distribution at any cross-section is as shown in Fig. 3.10.