Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

19.3 Effect of Idealization on the Analysis 541

19.3 EffectofIdealizationontheAnalysisofOpenandClosedSectionBeams...................

Theadditionofdirectstress-carryingboomstoopenandclosedsectionbeamswillclearlymodifythe
analyses presented in Chapters 15 through 17. Before considering individual cases, we shall discuss
theimplicationsofstructuralidealization.Generally,inanyidealization,differentloadingconditions
requiredifferentidealizationsofthesamestructure.InExample19.1,theloadingisappliedinavertical
plane.If,however,theloadinghadbeenappliedinahorizontalplane,theassumedstressdistribution
inthepanelsofthesectionwouldhavebeendifferent,resultingindifferentvaluesofboomarea.
Suppose that an open or closed section beam is subjected to given bending or shear loads and
that the required idealization has been completed. The analysis of such sections usually involves the
determinationoftheneutralaxispositionandthecalculationofsectionalproperties.Thepositionof
theneutralaxisisderivedfromtheconditionthattheresultantloadonthebeamcrosssectioniszero,
thatis,
∫

A

σzdA=0 (seeEq.(15.3))

The areaAin this expression is clearly the direct stress-carrying area. It follows that the centroid of
the cross section is the centroid of the direct stress-carrying area of the section, depending on the
degreeandmethodofidealization.Thesectionalproperties,Ixx,andsoon,mustalsorefertothedirect
stress-carryingarea.

19.3.1 Bending of Open and Closed Section Beams

The analysis presented in Sections 15.1 and 15.2 applies, and the direct stress distribution is given
byanyofEqs.(15.9),(15.18),or(15.19),dependingonthebeamsectionbeinginvestigated.Inthese
equations,thecoordinates(x,y)ofpointsinthecrosssectionarereferredtoaxeshavingtheirorigin
atthecentroidofthedirectstress-carryingarea.Furthermore,thesectionpropertiesIxx,Iyy,andIxyare
calculatedforthedirectstress-carryingareaonly.
Inthecasewherethebeamcrosssectionhasbeencompletelyidealizedintodirectstress-carrying
boomsandshear-stress-only-carryingpanels,thedirectstressdistributionconsistsofaseriesofdirect
stressesconcentratedatthecentroidsofthebooms.

Example 19.2
The fuselage section shown in Fig. 19.5 is subjected to a bending moment of 100kNm applied in
the vertical plane of symmetry. If the section has been completely idealized into a combination of
directstress-carryingboomsandshear-stress-only-carryingpanels,determinethedirectstressineach
boom.

The section has Cy as an axis of symmetry and resists a bending momentMx=100kNm.
Equation(15.18)thereforereducesto

σz=

Mx

Ixx

y (i)