Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

516 CHAPTER 17 Torsion of Beams

Inadditiontowarpingacrossthethickness,thecrosssectionofthebeamwillwarpinasimilarmanner
tothatofaclosedsectionbeam.FromFig.16.3,

γzs=

∂w

∂s

+

∂vt

∂z

(17.15)

Referring the tangential displacementvtto the center of twist R of the cross section, we have from
Eq.(16.8)

∂vt

∂z

=pR

dθ

dz

(17.16)

Substitutingfor∂vt/∂zinEq.(17.15)gives

γzs=

∂w

∂s

+pR

dθ

dz

fromwhich

τzs=G

(

∂w

∂s

+pR

dθ

dz

)

(17.17)

Onthemidlineofthesectionwallτzs=0(seeEq.(17.9))sothatfromEq.(17.17)

∂w

∂s

=−pR

dθ

dz

Integratingthisexpressionwithrespecttosandtakingthelowerlimitofintegrationtocoincidewith
thepointofzerowarping,weobtain

ws=−

dθ

dz

∫s

0

pRds (17.18)

FromEqs.(17.14)and(17.18)itcanbeseenthattwotypesofwarpingexistinanopensectionbeam.
Equation(17.18)givesthewarpingofthemidlineofthebeam;thisisknownasprimarywarpingand
isassumedtobeconstantacrossthewallthickness.Equation (17.14)givesthewarpingofthebeam
acrossitswallthickness.Thisiscalledsecondary warping,isverymuchlessthanprimarywarping,
andisusuallyignoredinthethin-walledsectionscommontoaircraftstructures.
Equation(17.18)mayberewrittenintheform

ws=− 2 AR

dθ

dz

(17.19)

or,intermsoftheappliedtorque

ws=− 2 AR

T

GJ

(seeEq.(17.12)) (17.20)

inwhichAR=^12

∫s
0 pRdsistheareasweptoutbyagenerator,rotatingaboutthecenteroftwist,fromthe
pointofzerowarping,asshowninFig.17.11.Thesignofws,foragivendirectionoftorque,depends