Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

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

Thedirectstressonanelementδsinthewallofthesectionisthen,fromEq.(15.45),

σ=EαT(x,y)tδs

Equations(15.46)through(15.48)thenbecome

NT=

∫

A

EαT(x,y)tds (15.52)

MxT=

∫

A

EαT(x,y)tyds (15.53)

MyT=

∫

A

EαT(x,y)txds (15.54)

Example 15.16
If, in the beam section of Example 15.15, the temperature change in the upper flange is 2T 0 but in
the web varies linearly from 2T 0 at its junction with the upper flange to zero at its junction with the
lowerflangedeterminethevaluesofthestressresultants;thetemperaturechangeinthelowerflange
remainszero.

Thetemperaturechangeatanypointinthewebisgivenby

Tw= 2 T 0 (a+y)/ 2 a=

T 0

a

(a+y)

Then,fromEqs.(15.49)and(15.52),

NT=Eα 2 T 0 at+

∫a

−a

Eα

T 0

a

(a+y)tds

thatis, NT=EαT 0

{

2 at+

1

a

[

ay+

y^2

2

]a

−a

}

whichgives

NT= 4 EαT 0 at

Notethat,inthiscase,theanswerisidenticaltothatinExample15.15,whichistobeexpected,sincethe
averagetemperaturechangeinthewebis(2T 0 + 0 )/ 2 =T 0 ,whichisequaltotheconstanttemperature
changeinthewebinExample15.15.