Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

9.7Tension Field Beams 315

stressesmaybefoundapproximatelyfrom

(

σmb

σcrb

) 2

+

(

τm

τcr

) 2

= 1 (9.46)

whereσcrbis the critical value of bending stress withS=0,M
=0, andτcris the critical value of
shear stress whenS
=0andM=0. Once the critical stress is reached, the web starts to buckle and
cannotcarryanyincreaseincompressivestresssothat,aswehaveseeninSection9.7.1,anyadditional
loadiscarriedbytensionfieldaction.Itisassumedthattheshearandbendingstressesremainattheir
criticalvaluesτmandσmbandthatthereareadditionalstressesσtwhichareinclinedatanangleθto
thehorizontalandwhichcarryanyincreasesintheappliedload.Atcollapse—thatis,atultimateload
conditions—theadditionalstressσtreachesitsmaximumvalueσt(max),andthepanelisinthecollapsed
stateshowninFig.9.15.
Consider now the small rectangular element on the edge AW of the panel before collapse. The
stressesactingontheelementareshowninFig.9.16(a).Thestressesonplanesparalleltoandperpen-
diculartothedirectionofthebucklemaybefoundbyconsideringtheequilibriumoftriangularelements
withinthisrectangularelement.Initially,weshallconsiderthetriangularelementCDEwhichissub-
jectedtothestresssystemshowninFig.9.16(b)andisinequilibriumundertheactionoftheforcescorre-
spondingtothesestresses.NotethattheedgeCEoftheelementisparalleltothedirectionofthebucklein
theweb.
ForequilibriumoftheelementinadirectionperpendiculartoCE(seeSection1.6),

σξCE+σmbEDcosθ−τmEDsinθ−τmDCcosθ= 0

DividingbyCEandrearranging,wehave

σξ=−σmbcos^2 θ+τmsin2θ (9.47)

Similarly,byconsideringtheequilibriumoftheelementinthedirectionEC,wehave

τηξ=−

σmb

2

sin2θ−τmcos2θ (9.48)

Fig.9.16

Determination of stresses on planes parallel and perpendicular to the plane of the buckle.