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.