Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

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

Fig.15.12

Determination of neutral axis position and direct stress due to bending.

SubstitutingforεzinEq.(15.13),wehave

σz=

Eξ

ρ

(15.14)

The beam supports pure bending moments so that the resultant normal load on any section must be
zero.Hence,
∫

A

σzdA= 0

Therefore,replacingσzinthisequationfromEq.(15.14)andcancellingtheconstantE/ρgives
∫

A

ξdA= 0

thatis,thefirstmomentofareaofthecrosssectionofthebeamabouttheneutralaxisiszero.Itfollows
thattheneutralaxispassesthroughthecentroidofthecrosssectionasshowninFig.15.12(b),which
istheresultweobtainedforthecaseofsymmetricalbending.
SupposethattheinclinationoftheneutralaxistoCxisα(measuredclockwisefromCx),then

ξ=xsinα+ycosα (15.15)

andfromEq.(15.14),

σz=

E

ρ

(xsinα+ycosα) (15.16)