Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

9.7Tension Field Beams 305

Fig.9.8

Diagonal tension field beam.

9.7.1 Complete Diagonal Tension

ThetheorypresentedhereisduetoWagner[Ref.13].ThebeamshowninFig.9.8(a)hasconcentrated
flangeareashavingadepthdbetweentheircentroidsandverticalstiffenerswhicharespaceduniformly
alongthelengthofthebeam.Itisassumedthattheflangesresisttheinternalbendingmomentatany
sectionofthebeam,whiletheweb,ofthicknesst,resiststheverticalshearforce.Theeffectofthisassump-
tionistoproduceauniformshearstressdistributionthroughthedepthoftheweb(seeSection19.3)at
anysection.Therefore,atasectionofthebeamwheretheshearforceisS,theshearstressτisgivenby

τ=

S

td

(9.14)

ConsidernowanelementABCDofthewebinapanelofthebeam,asshowninFig.9.8(a).Theelement
issubjectedtotensilestresses,σt,producedbythediagonaltensionontheplanesABandCD;theangle
ofthediagonaltensionisα.OnaverticalplaneFDintheelement,theshearstressisτandthedirect
stress isσz. Now, considering the equilibrium of the element FCD (Fig. 9.8(b)) and resolving forces
vertically,wehave(seeSection1.6)

σtCDtsinα=τFDt

whichgives

σt=

τ

sinαcosα

=

2 τ

sin2α

(9.15)

or,substitutingforτfromEq.(9.14)andnotingthatinthiscaseS=Watallsectionsofthebeam,

σt=

2 W

tdsin2α

(9.16)

Further,resolvingforceshorizontallyfortheelementFCD,

σzFDt=σtCDtcosα