88 HANDBOOK OF MECHANICAL DESIGN 



If the front and rear spars are of the same material, Er = Ep, and cancel out. 

 In Fig. 16, E.A. is the center of resistance to bending, and in the figure 



a = j^ (18) 



^F ~r J^R 



The point E.A. is called the elastic center, and the locus of these points is called 

 the elastic axis. The torsional moment apphed to the wdng is the load times the dis- 

 tance of the center of gravity of the load to the elastic axis, i.e. Pic — a) in Fig. 16. 

 This will be the torsion that will be assumed resisted entirely by the box. 



For two spars acting in bending and interconnected only by pin-ended ribs, the 

 load P in Fig. 16 wdll be divided proportionally between the two spars, as foUows: 



P..=^^ . • (19) 



^2 = y (20) 



The root bending moments wUl be 



Mf, = P^-rL (21) 



Muo_ = PrX (22) 



This proportioning of the loads applies also when the spars offer but little resistance 

 to torsion and the ribs are rigidly connected. If the spars have high torsional rigidity 

 or if a box as in Fig. 13 is formed, the distribution approaches that given by Eqs. (16) 

 and (17) for Mr^ and Af,.i. 



If all torsion about E.A. is resisted by the box in torsional shear, there is complete 

 interaction between spars. If no torsion is resisted by the box, the interaction is zero. 

 The amount of interaction is obtained from 



C, = 1^* (23) 



where L = total length of uniform cross Bo = GJ when spars have relatively little 

 section of box resistance to torsion 



Bo = total of torsional stiffness Ao = IfIr/{If + h), if E is same for both 

 of two spars plus box spars 



Generally for a stressed skin box, ratio C, is such that the moment would divide as 

 in Eqs. (16) and (17), for all points along the span except the root. The difference 

 between the moment obtained by the two methods is 



M,.p = Mp, - Mfi (24) 



MeR = Mr, - Mr, (25) 



For any degree of interaction Cr between spars, the final bending moment in each 

 spar is 



M, = M,, - Cr{Mf2 - Mf,) (26) 



Mr = Mr, - Cr{Mr, - Mr,) (27) 



