Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

5.3Application to Deflection Problems 115

Fig.5.3

Determination of the deflection of a point on a framework by the method of complementary energy.

whereλiistheextensionoftheithmember,Fiistheforceintheithmember,and (^) risthecorresponding
displacementoftherthloadPr.Fromtheprincipleofthestationaryvalueofthetotalcomplementary
energy,
∂C
∂P 2

=

∑k

i= 1

λi

∂Fi

∂P 2

− 2 = 0 (5.10)

fromwhich

2 =

∑k

i= 1

λi

∂Fi

∂P 2

(5.11)

Equation(5.10)isseentobeidenticaltotheprincipleofvirtualforcesinwhichvirtualforcesδFand
δPact through real displacementsλand. Clearly, the partial derivatives with respect toP 2 of the
constantloadsP 1 ,P 2 ,...,Pnvanish,leavingtherequireddeflection 2 astheunknown.Atthisstage,
before 2 canbeevaluated,theload–displacementcharacteristicsofthemembersmustbeknown.For
linearelasticity,

λi=

FiLi

AiEi

whereLi,Ai,andEiare the length, the cross-sectional area, and the modulus of elasticity of the
ith member, respectively. On the other hand, if the load–displacement relationship is of a nonlinear
form,say,

Fi=b(λi)c

inwhichbandcareknown,thenEq.(5.11)becomes

2 =

∑k

i= 1

(

Fi

b

) 1 /c

∂Fi

∂P 2

Thecomputationof 2 isbestaccomplishedintabularform,butbeforetheprocedureisillustratedby
anexample,someaspectsofthesolutionmeritdiscussion.