Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

1.9 Strain 21

ordirect strainsare associated with direct stressesσ and relate to changes in length, whileshear
strainsdefinechangesinangleproducedbyshearstresses.Thesestrainsaredesignated,withappro-
priate suffixes, by the symbols εandγ, respectively, and have the same sign as the associated
stresses.
ConsiderthreemutuallyperpendicularlineelementsOA,OB,andOCatapointOinadeformable
body.Theiroriginalorunstrainedlengthsareδx,δy,andδz,respectively.If,now,thebodyissubjected
toforcesthatproduceacomplexsystemofdirectandshearstressesatO,suchasthatinFig.1.6,then
thelineelementsdeformtothepositionsO′A′,O′B′,andO′C′asshowninFig.1.15.
ThecoordinatesofOintheunstrainedbodyare(x,y,z)sothatthoseofA,B,andCare(x+δx,y,z),
(x,y+δy,z),and(x,y,z+δz).ThecomponentsofthedisplacementofOtoO′paralleltothex,y,and
zaxesareu,v,andw.Thesesymbolsareusedtodesignatethesedisplacementsthroughoutthebook
andaredefinedaspositiveinthepositivedirectionsoftheaxes.Weagainusethefirsttwotermsofa
Taylor’sseriesexpansiontodeterminethecomponentsofthedisplacementsofA,B,andC.Thus,the
displacementofAinadirectionparalleltothexaxisisu+(∂u/∂x)δx.Theremainingcomponentsare
foundinanidenticalmannerandareshowninFig.1.15.
Wenowdefinedirectstraininmorequantitativeterms.IfalineelementoflengthLatapointina
bodysuffersachangeinlength L,thenthelongitudinalstrainatthatpointinthebodyinthedirection

Fig.1.15

Displacement of line elements OA, OB, and OC.