Page 479 - Aircraft Stuctures for Engineering Student

460 Structural constraint

The direct stress distribution uA(= PA/A) follows, i.e.
sinh Xz
UA = - (11.31)

The distribution of load in the edge booms is obtained by substituting for PA from
Eq. (11.30) in Eq. (11.26), thus
SYB (. I A sinhk)
PB = - (1 1.32)
h(2B + A) 2BX coshXL
whence
A sinhXz
ffB = - (1 1.33)
Finally, from either pairs of Eqs (11.25) and (11.30) or (11.24) and (11.32)
SY A
cosh Xz
'=2h(2B+A) ('-a)
(11.34)
so that the shear stress distribution T(= q/t) is
SYA
cosh Xz
= 2ht(2B + A) (' - a) (1 1.35)
Elementary theory (Chapter 9) gives

and

SY A
= 2h(2B + A)
so that, as in the case of the torsion of a four boom rectangular section beam, the
solution comprises terms corresponding to elementary theory together with terms
representing the effects of shear lag and structural constraint.
Many wing structures are spliced only at the spars so that the intermediate stringers
are not subjected to bending stresses at the splice. The situation for a six boom
rectangular section beam is then as shown in Fig. 11.17. The analysis is carried out
in an identical manner to that in the previous case except that the boundary condi-
tions for the central stringer are PA = 0 when z = 0 and z = L. The solution is

PA = - (1 1.36)
h(2B + A)
SyB ( AL sinhXz)
PB = - z+- 7 (1 1.37)
h(2B + A) 2B sinhXL
SyA (1-XL- cash Xz
= 2h(2B + A) (1 1.38)
where X2 = Gt(2B + A)/dEAB. Examination of Eq. (1 1.38) shows that q changes sign
when cosh Xz = (sinh xL)/XL, the solution of which gives a value of z less than L, i.e.