Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

4.2 Principle of Virtual Work 93

force,S;abendingmoment,M;andatorque,T,producedbysomeexternalloadingsystemactingon
thestructureofwhichthememberispart.Thestressdistributionscorrespondingtotheseinternalforces
arerelatedtoanaxissystemwhoseorigincoincideswiththecentroidofareaofthecrosssection.We
shall,infact,beusingthesestressdistributionsinthederivationofexpressionsforinternalvirtualwork
inlinearlyelasticstructuressothatitislogicaltoassumethesameoriginofaxeshere;weshallalso
assumethattheyaxisisanaxisofsymmetry.Initially,weshallconsiderthenormalforce,N.
Thedirectstress,σ,atanypointinthecrosssectionofthememberisgivenbyσ=N/A.Therefore,
thenormalforceontheelementδAatthepoint(z,y)is

δN=σδA=

N

A

δA

Supposenowthatthestructureisgivenanarbitraryvirtualdisplacementwhichproducesavirtualaxial
strain,εv, in the element. The internal virtual work,δwi,N, done by the axial force on the elemental
lengthofthememberisgivenby

δwi,N=

∫

A

N

A

dAεvδx

which,since

∫

AdA=A,reducesto

δwi,N=Nεvδx (4.9)

Inotherwords,thevirtualworkdonebyNistheproductofNandthevirtualaxialdisplacementof
theelementofthemember.ForamemberoflengthL,thevirtualwork,wi,N,doneduringthearbitrary
virtualstrainisthen

wi,N=

∫

L

Nεvdx (4.10)

Forastructurecomprisinganumberofmembers,thetotalinternalvirtualwork,Wi,N,donebyaxial
forceisthesumofthevirtualworkofeachofthemembers.Therefore,

wi,N=

∑∫

L

Nεvdx (4.11)

NotethatinthederivationofEq.(4.11),wehavemadenoassumptionregardingthematerialproperties
ofthestructuresothattherelationshipholdsfornonelasticaswellaselasticmaterials.However,fora
linearlyelasticmaterial—inotherwords,onethatobeysHooke’slaw—wecanexpressthevirtualstrain
intermsofanequivalentvirtualnormalforce:

εv=

σv

E

=

Nv

EA

Therefore,ifwedesignatetheactualnormalforceinamemberbyNA,Eq.(4.11)maybeexpressedin
theform

wi,N=

∑∫

L

NANv

EA

dx (4.12)