Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

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

Further,ifeitherMyorMxiszero,then

σz=

Mx

Ixx

y or σz=

My

Iyy

x (15.21)

Equations (15.20) and (15.21) are those derived for the bending of beams having at least a singly
symmetricalcrosssection(seeSection15.1).ItmayalsobenotedthatinEq.(15.21)σz=0when,for
the first equation,y=0 and for the second equation whenx=0. Therefore, in symmetrical bending
theory,thexaxisbecomestheneutralaxiswhenMy=0andtheyaxisbecomestheneutralaxiswhen
Mx=0.Thus,weseethatthepositionoftheneutralaxisdependsontheformoftheappliedloadingas
wellasthegeometricalpropertiesofthecrosssection.
Thereexists,inanyunsymmetricalcrosssection,acentroidalsetofaxesforwhichtheproductsecond
momentofareaiszero(see[Ref.1]).Theseaxesarethenprincipalaxesandthedirectstressdistribution
referredtotheseaxestakesthesimplifiedformofEqs.(15.20)or(15.21).Itwouldthereforeappearthat
theamountofcomputationcanbereducediftheseaxesareused.Thisisnotthecase,however,unless
the principal axes are obvious from inspection, since the calculation of the position of the principal
axes, the principal sectional properties, and the coordinates of points at which the stresses are to be
determinedconsumesagreateramountoftimethandirectuseofEqs.(15.18)or(15.19)foranarbitrary
butconvenientsetofcentroidalaxes.

15.2.4 Position of the Neutral Axis

Theneutralaxisalwayspassesthroughthecentroidofareaofabeam’scrosssection,butitsinclination
α(see Fig. 15.12(b)) to thexaxis depends on the form of the applied loading and the geometrical
propertiesofthebeam’scrosssection.
Atallpointsontheneutralaxisthedirectstressiszero.Therefore,fromEq.(15.18),

0 =

(

MyIxx−MxIxy

IxxIyy−Ixy^2

)

xNA+

(

MxIyy−MyIxy

IxxIyy−Ixy^2

)

yNA,

wherexNAandyNAarethecoordinatesofanypointontheneutralaxis.Hence,

yNA

xNA

=−

MyIxx−MxIxy

MxIyy−MyIxy

or,referringtoFig.15.12(b)andnotingthatwhenαispositivexNAandyNAareofoppositesign

tanα=

MyIxx−MxIxy

MxIyy−MyIxy

(15.22)

Example 15.4
AbeamhavingthecrosssectionshowninFig.15.13issubjectedtoabendingmomentof1500Nm
in a vertical plane. Calculate the maximum direct stress due to bending stating the point at which
itacts.