Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

19.4 Deflection of Open and Closed Section Beams 553

givingthesamesolutionasbefore.Notethatiftheunitlengthofbeamhadbeentakentobe1m,the
solutionwouldhavebeenq 12 =−6000N/m,q 23 =−12000N/m,andq 34 =−6000N/m.

19.3.5 Torsion of Open and Closed Section Beams

Nodirectstressesaredevelopedineitheropenorclosedsectionbeamssubjectedtoapuretorqueunless
axial constraints are present. The shear stress distribution is therefore unaffected by the presence of
booms,andtheanalysespresentedinChapter17apply.

19.4 DeflectionofOpenandClosedSectionBeams..................................................

Bending,shear,andtorsionaldeflectionsofthin-walledbeamsarereadilyobtainedbyapplicationof
theunitloadmethoddescribedinSection5.5.Thedisplacementinagivendirectionduetotorsionis
givendirectlybythelastofEqs.(5.21),thus,

(^) T=

∫

L

T 0 T 1

GJ

dz (19.14)

whereJ,thetorsionconstant,dependsonthetypeofbeamunderconsideration.Foranopensection
beam,JisgivenbyeitherofEqs.(17.11),whereasinthecaseofaclosedsectionbeam,J= 4 A^2 /(

∮

ds/t)
(Eq.(17.4))foraconstantshearmodulus.
Expressionsforthebendingandsheardisplacementsofunsymmetricalthin-walledbeamsmayalso
be determined by the unit load method. They are complex for the general case and are most easily
derivedfromfirstprinciplesbyconsideringthecomplementaryenergyoftheelasticbodyintermsof
stressesandstrainsratherthanloadsanddisplacements.InChapter5,weobservedthatthetheoremof
theprincipleofthestationaryvalueofthetotalcomplementaryenergyofanelasticsystemisequivalent
totheapplicationoftheprincipleofvirtualworkwherevirtualforcesactthroughrealdisplacements.We
maythereforespecifythatinourexpressionfortotalcomplementaryenergy,thedisplacementsarethe
actualdisplacementsproducedbytheappliedloads,whilethevirtualforcesystemistheunitload.
Considering deflections due to bending, we see, from Eq. (5.6), that the increment in total
complementaryenergyduetotheapplicationofavirtualunitloadis

−

∫

L

⎛

⎝

∫

A

σz,1εz,0dA

⎞

⎠dz+ (^1) M
whereσz,1isthedirectbendingstressatanypointinthebeamcrosssectioncorrespondingtotheunit
loadandεz,0isthestrainatthepointproducedbytheactualloadingsystem.Further, (^) Mistheactual
displacementduetobendingatthepointofapplicationandinthedirectionoftheunitload.Sincethe
systemisinequilibriumundertheactionoftheunitload,theaboveexpressionmustequalzero(see
Eq.(5.6)).Hence,
(^) M=

∫

L

⎛

⎝

∫

A

σz,1εz,0dA

⎞

⎠dz (19.15)