Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

1.3 Equations of Equilibrium 7

Fig.1.

Sign conventions and notation for stresses at a point in a body.

1.3 EquationsofEquilibrium...........................................................................

Generally,exceptincasesofuniformstress,thedirectandshearstressesonoppositefacesofanelement
are not equal as indicated in Fig. 1.5 but differ by small amounts. Therefore if, say, the direct stress
actingonthezplaneisσz,thenthedirectstressactingonthez+δzplaneis,fromthefirsttwotermsof
aTaylor’sseriesexpansion,σz+(∂σz/∂z)δz.Wenowinvestigatetheequilibriumofanelementatsome
internalpointinanelasticbodywherethestresssystemisobtainedbythemethodjustdescribed.
InFig.1.6,theelementisinequilibriumunderforcescorrespondingtothestressesshownandthe
componentsofbodyforces(notshown).Surfaceforcesactingontheboundaryofthebody,although
contributingtotheproductionoftheinternalstresssystem,donotdirectlyfeatureintheequilibrium
equations.
Takingmomentsaboutanaxisthroughthecenteroftheelementparalleltothezaxis

τxyδyδz

δx

2

+

(

τxy+

∂τxy

∂x

δx

)

δyδz

δx

2

−τyxδxδz

δy

2

−

(

τyx+

∂τyx

∂y

δy

)

δxδz

δy

2

= 0