Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

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

eitherendofST,weseethatthemomentresultantabouttheneutralaxisofthestressesonallsuchfibers
mustbeequivalenttotheappliednegativemomentM;thatis,

M=−

∫

A

E

y^2

R

dA

or

M=−

E

R

∫

A

y^2 dA (15.6)

Theterm

∫

Ay

(^2) dAisknownasthesecondmomentofareaofthecrosssectionofthebeamaboutthe
neutralaxisandisgiventhesymbolI.RewritingEq.(15.6),wehave

M=−

EI

R

(15.7)

or,combiningthisexpressionwithEq.(15.2)

M

I

=−

E

R

=

σz

y

(15.8)

FromEq.(15.8),weseethat

σz=

My

I

(15.9)

The direct stress,σz, at any point in the cross section of a beam is therefore directly proportional to
thedistanceofthepointfromtheneutralaxisandsovarieslinearlythroughthedepthofthebeamas
shown,forthesectionJK,inFig.15.5(b).Clearly,forapositivebendingmomentσzispositive—thatis,
tensile—whenyispositiveandcompressive(i.e.,negative)whenyisnegative.Thus,inFig.15.5(b),

σz,1=

My 1

I

(compression) σz,2=

My 2

I

(tension) (15.10)

Furthermore,weseefromEq.(15.7)thatthecurvature,1/R,ofthebeamisgivenby

1

R

=

M

EI

(15.11)

andisthereforedirectlyproportionaltotheappliedbendingmomentandinverselyproportionaltothe
productEIwhichisknownastheflexuralrigidityofthebeam.

Example 15.1
The cross section of a beam has the dimensions shown in Fig. 15.6(a). If the beam is subjected to a
negativebendingmomentof100kNmappliedinaverticalplane,determinethedistributionofdirect
stressthroughthedepthofthesection.