Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

16.2 Shear of Open Section Beams 483

and

∂vt

∂z

=p

dθ

dz

−xRsinψ

dθ

dz

+yRcosψ

dθ

dz

(16.9)

AlsofromEq.(16.7)

∂vt

∂z

=p

dθ

dz

+

du

dz

cosψ+

dv

dz

sinψ (16.10)

ComparingthecoefficientsofEqs.(16.9)and(16.10),weseethat

xR=−

dv/dz

dθ/dz

yR=

du/dz

dθ/dz

(16.11)

16.2 ShearofOpenSectionBeams......................................................................

TheopensectionbeamofarbitrarysectionshowninFig.16.5supportsshearloadsSxandSysuchthat
thereisnotwistingofthebeamcrosssection.Forthisconditiontobevalid,theshearloadsmustboth
passthroughaparticularpointinthecrosssectionknownastheshearcenter.
Sincetherearenohoopstressesinthebeam,theshearflowsanddirectstressesactingonanelement
ofthebeamwallarerelatedbyEq.(16.2)—thatis,

∂q

∂s

+t

∂σz

∂z

= 0

Weassumethatthedirectstressesareobtainedwithsufficientaccuracyfrombasicbendingtheoryso
thatfromEq.(15.18)

∂σz

∂z

=

[(∂My/∂z)Ixx−(∂Mx/∂z)Ixy]

IxxIyy−Ixy^2

x+

[(∂Mx/∂z)Iyy−(∂My/∂z)Ixy]

IxxIyy−Ixy^2

y

Fig.16.5

Shear loading of open section beam.