Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

596 CHAPTER 22 Wings

Fig.22.10

Moment equilibrium ofRth cell.

Bycomparingwiththepuretorsioncase,wededucethat

dθ

dz

=

1

2 ARG

⎛

⎝−qs,0,R− 1 δR−1,R+qs,0,RδR−qs,0,R+ 1 δR+1,R+

∮

R

qb

ds

t

⎞

⎠ (22.10)

inwhichqbhaspreviouslybeendetermined.ThereareNequationsofthetype(22.10)sothatafurther
equation is required to solve for theN+1 unknowns. This is obtained by considering the moment
equilibriumoftheRthcellinFig.22.10.
ThemomentMq,RproducedbythetotalshearflowaboutanyconvenientmomentcenterOisgivenby

Mq,R=

∮

qRp 0 ds (seeSection17.1)

Substituting forqRin terms of the “open section” shear flowqband the redundant shear flowqs,0,R,
wehave

Mq,R=

∮

R

qbp 0 ds+qs,0,R

∮

R

p 0 ds

or

Mq,R=

∮

R

qbp 0 ds+ 2 ARqs,0,R

The sum of the moments from the individual cells is equivalent to the moment of the externally
appliedloadsaboutthesamepoint.Thus,forthewingsectionofFig.22.8,

Sxη 0 −Syξ 0 =

∑N

R= 1

Mq,R=

∑N

R= 1

∮

R

qbp 0 ds+

∑N

R= 1

2 ARqs,0,R (22.11)