Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

562 CHAPTER 20 Wing Spars and Box Beams

Fig.20.1

Effect of taper on beam analysis.

The bending moment resultantsPz,1andPz,2are parallel to thezaxis of the beam. For a beam in
whichtheflangesareassumedtoresistallthedirectstresses,Pz,1=Mx/handPz,2=−Mx/h.Inthecase
wherethewebisassumedtobefullyeffectiveinresistingdirectstress,Pz,1andPz,2aredeterminedby
multiplyingthedirectstressesσz,1andσz,2foundusingEqs.(15.18)or(15.19)bytheflangeareasB 1
andB 2 .Pz,1andPz,2arethecomponentsinthezdirectionoftheaxialloadsP 1 andP 2 intheflanges.
ThesehavecomponentsPy,1andPy,2paralleltotheyaxisgivenby

Py,1=Pz,1

δy 1

δz

Py,2=−Pz,2

δy 2

δz

(20.1)

inwhich,forthedirectionoftapershown,δy 2 isnegative.Theaxialloadinflange①isgivenby

P 1 =(P^2 z,1+P^2 y,1)^1 /^2

SubstitutingforPy,1fromEq.(20.1),wehave

P 1 =Pz,1

(δz^2 +δy^21 )^1 /^2

δz

=

Pz,1

cosα 1

(20.2)

Similarly,

P 2 =

Pz,2

cosα 2

(20.3)

TheinternalshearforceSycomprisestheresultantSy,wofthewebshearflowstogetherwiththevertical
componentsofP 1 andP 2 .Thus,

Sy=Sy,w+Py,1−Py,2