Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

156 CHAPTER 5 Energy Methods

5.11 TemperatureEffects.................................................................................

Auniformtemperatureappliedacrossabeamsectionproducesanexpansionofthebeam,asshownin
Fig. 5.29, provided there are no constraints. However, a linear temperature gradient across the beam
sectioncausestheupperfibersofthebeamtoexpandmorethanthelowerones,producingabending
strainasshowninFig.5.30withouttheassociatedbendingstresses,againprovidednoconstraintsare
present.
Consideranelementofthebeamofdepthhandlengthδzsubjectedtoalineartemperaturegradient
overitsdepth,asshowninFig.5.31(a).Theuppersurfaceoftheelement increasesinlengthtoδz( 1 +αt)
(seeSection1.15.1)whereαisthecoefficientoflinearexpansionofthematerialofthebeam.Thus,
fromFig.5.31(b),

R

δz

=

R+h

δz( 1 +αt)

giving

R=h/αt (5.29)

Also,

δθ=δz/R

sothatfromEq.(5.29),

δθ=

δzαt

h

(5.30)

Fig.5.29

Expansion of beam due to uniform temperature.

Fig.5.30

Bending of beam due to linear temperature gradient.