Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

106 CHAPTER 4 Virtual Work and Energy Methods

These axial forces are constant along the length of each member so that for a truss comprisingn
members,Eq.(4.12)reducesto

Wi,N=

∑n

j= 1

FA,jFv,jLj

EjAj

(i)

inwhichFA,jandFv,jaretheactualandvirtualforcesinthejthmember,whichhasalengthLj,anarea
ofcross-sectionAj,andaYoung’smodulusEj.
SincetheforcesFv,jareduetoaunitload,weshallwriteEq.(i)intheform

Wi,N=

∑n

j= 1

FA,jF1,jLj

EjAj

(ii)

Also,inthisparticularexample,theareaofcrosssection,A,andYoung’smodulus,E,arethesamefor
allmemberssothatitissufficienttocalculate

∑n
j= 1 FA,jF1,jLjandthendividebyEAtoobtainWi,N.
Theforcesinthemembers,whetheractualorvirtual,maybecalculatedbythemethodofjoints.Note
thatthesupportreactionscorrespondingtothethreesetsofappliedloads(oneactualandtwovirtual)
mustbecalculatedbeforetheinternalforcesystemscanbedetermined.However,inFig.4.12(c),itis
clearfrominspectionthatF1,AB=F1,BC=F1,CD=+1,whiletheforcesinallothermembersarezero.
ThecalculationsarepresentedinTable4.1;notethatpositivesignsindicatetensionandnegativesigns
compression.
Thus,equatinginternalandexternalvirtualworkdone(Eq.(4.23)),wehave

1 δB,v=

1263.6× 106

200000 × 1800

hence

δB,v=3.51mm

Table 4.1

Member L(m) FA(kN) F1,B F1,D FAF1,BL(kNm) FAF1,DL(kNm)

AE 5.7 −84.9 −0.94 0 +451.4 0

AB 4.0 +60.0 +0.67 +1.0 +160.8 +240.0

EF 4.0 −60.0 −0.67 0 +160.8 0

EB 4.0 +20.0 +0.67 0 +53.6 0

BF 5.7 −28.3 +0.47 0 −75.2 0

BC 4.0 +80.0 +0.33 +1.0 +105.6 +320.0

CD 4.0 +80.0 +0.33 +1.0 +105.6 +320.0

CF 4.0 +100.0 0 0 0 0

DF 5.7 −113.1 −0.47 0 +301.0 0

∑

=+1263.6

∑

=+880.0