Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

468 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams

We have noted that beam sections in aircraft structures are generally thin walled so that
Eqs. (15.46) through (15.48) may be more easily integrated for such sections by dividing them into
thinrectangularcomponentsaswedidwhencalculatingsectionproperties.WethenusetheRiemann
integration technique in which we calculate the contribution of each component to the normal force
andmomentsandsumthemtodetermineeachresultant.Equations(15.46),(15.47),and(15.48)then
become

NT=EαTAi (15.49)

MxT=EαTy ̄iAi (15.50)

MyT=EαTx ̄iAi (15.51)

in whichAiis the cross-sectional area of a component andxi andyi are the coordinates of its
centroid.

Example 15.15
The beam section shown in Fig. 15.37 is subjected to a temperature rise of 2T 0 in its upper flange,
a temperature rise ofT 0 in its web, and zero temperature change in its lower flange. Determine the
normal force on the beam section and the moments about the centroidalxandyaxes. The beam
section has a Young’s modulusE and the coefficient of linear expansion of the material of the
beamisα.

FromEq.(15.49),

NT=Eα( 2 T 0 at+T 02 at)= 4 EαatT 0

FromEq.(15.50),

MxT=Eα[2T 0 at(a)+T 02 at( 0 )]= 2 Eαa^2 tT 0

andfromEq.(15.51),

MyT=Eα[2T 0 at(−a/ 2 )+T 02 at( 0 )]=−Eαa^2 tT 0

NotethatMyTisnegative,whichmeansthattheupperflangewouldtendtorotateoutofthepaper
aboutthewebwhichagreeswithatemperatureriseforthispartofthesection.Thestressescorresponding
totheabovestressresultantsarecalculatedinthenormalwayandareaddedtothoseproducedbyany
appliedloads.
Insomecases,thetemperaturechangeisnotconvenientlyconstantinthecomponentsofabeam
section and must then be expressed as a function ofxandy. Consider the thin-walled beam section
showninFig.15.38andsupposethatatemperaturechange T(x,y)isapplied.