Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

5.11 Temperature Effects 157

Fig.5.31

(a) Linear temperature gradient applied to beam element; (b) bending of beam element due to temperature
gradient.

Wemaynowapplytheprincipleofthestationaryvalueofthetotalcomplementaryenergyinconjunction

withtheunitloadmethodtodeterminethedeflection (^) Te,duetothetemperatureofanypointofthe
beamshowninFig.5.30.Wehaveseenthattheprecedingprincipleisequivalenttotheapplicationof
theprincipleofvirtualworkwherevirtualforcesactthroughrealdisplacements.Therefore,wemay
specifythatthedisplacementsarethoseproducedbythetemperaturegradient,whilethevirtualforce
system is the unit load. Thus, the deflection (^) Te,Bof the tip of the beam is found by writing down
theincrementintotalcomplementaryenergycausedbytheapplicationofavirtualunitloadatBand
equatingtheresultingexpressiontozero(seeEqs.(5.7)and(5.12)).Thus,
δC=

∫

L

M 1 dθ− (^1) Te,B= 0
or
(^) Te,B=

∫

L

M 1 dθ (5.31)

whereM 1 isthebendingmomentatanysectionduetotheunitload.SubstitutingfordθfromEq.(5.30),
wehave

(^) Te,B=

∫

L

M 1

αt

h

dz (5.32)

wheretcan vary arbitrarily along the span of the beam but only linearly with depth. For a beam
supporting some form of external loading, the total deflection is given by the superposition of the
temperaturedeflectionfromEq.(5.32)andthebendingdeflectionfromEq.(5.21);thus,

=

∫

L

M 1

(

M 0

EI

+

αt

h

)

dz (5.33)