Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

1.15 Stress–Strain Relationships 29

Sofarwehavemadenoassumptionsregardingtheforce–displacementorstress–strainrelationship
inthebody.Thiswill,infact,provideuswiththerequiredsixequations,butbeforethesearederived,
itisworthwhiletoconsidersomegeneralaspectsoftheanalysis.
Thederivationoftheequilibrium,strain–displacement,andcompatibilityequationsdoesnotinvolve
anyassumptionastothestress–strainbehaviorofthematerialofthebody.Itfollowsthatthesebasic
equationsareapplicabletoanytypeofcontinuous,deformablebodynomatterhowcomplexitsbehavior
understress.Infact,weshallconsideronlythesimplecaseoflinearlyelasticisotropicmaterialsfor
whichstressisdirectlyproportionaltostrainandwhoseelasticpropertiesarethesameinalldirections.
Amaterialpossessingthesamepropertiesatallpointsissaidtobehomogeneous.
Particularcasesarisewheresomeofthestresscomponentsareknowntobezero,andthenumber
ofunknownsmaythenbenogreaterthantheremainingequilibriumequationsthathavenotidentically
vanished. The unknown stresses are then found from the conditions of equilibrium alone, and the
problemissaidtobestaticallydeterminate.Forexample,theuniformstressinthemembersupporting
atensileloadPinFig.1.3isfoundbyapplyingoneequationofequilibriumandaboundarycondition.
Thissystemisthereforestaticallydeterminate.
Staticallyindeterminatesystemsrequiretheuseofsome,ifnotall,oftheotherequationsinvolving
strain–displacement and stress–strain relationships. However, whether the system is statically deter-
minate or not, stress–strain relationships are necessary to determine deflections. The role of the six
auxiliarycompatibilityequationswillbediscussedwhenactualelasticityproblemsareformulatedin
Chapter2.
Wenowproceedtoinvestigatetherelationshipofstressandstraininathree-dimensional,linearly
elastic,isotropicbody.
Experimentsshowthattheapplicationofauniformdirectstress,sayσx,doesnotproduceanyshear
distortionofthematerialandthatthedirectstrainεxisgivenbytheequation

εx=

σx

E

(1.40)

whereEis a constant known as themodulus of elasticityorYoung’s modulus. Equation (1.40) is an
expressionofHooke’slaw.Further,εxisaccompaniedbylateralstrains

εy=−ν

σx

E

εz=−ν

σx

E

(1.41)

inwhichνisaconstanttermedPoisson’sratio.
For a body subjected to direct stressesσx,σy,andσz, the direct strains are from Eqs. (1.40) and
(1.41)andtheprincipleofsuperposition(seeChapter5,Section5.9)

εx=

1

E

[σx−ν(σy+σz)]

εy=

1

E

[σy−ν(σx+σz)]

εz=

1

E

[σz−ν(σx+σy)]

⎫

⎪⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎪

⎭

(1.42)