Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

15.6 Temperature Effects 467

Fig.15.36

Beam section subjected to a temperature rise.

Wehaveseenthatthestrainproducedbyatemperaturechange Tisgivenby

ε=αT (seeEq.(1.55))

ItfollowsfromEq.(1.40)thatthedirectstressonanelementofcross-sectionalareaδAis

σ=EαTδA (15.45)

Consider now the beam section shown in Fig. 15.36 and suppose that a temperature variation Tis
appliedtothecompletecrosssection;thatis, Tisafunctionofbothxandy.
Thetotalnormalforceduetothetemperaturechangeonthebeamcrosssectionisthengivenby

NT=

∫∫

A

EαTdA (15.46)

Further,themomentsaboutthexandyaxesare

MxT=

∫∫

A

EαTydA (15.47)

and

MyT=

∫∫

A

EαTxdA, (15.48)

respectively.