142 CHAPTER 5 Energy Methods
Weareassumingthatthetrussislinearlyelasticsothattherelativedisplacementofthecutendsof
thememberBD(ineffect,themovementofBandDawayfromortowardeachotheralongthediagonal
BD)maybefoundusing,say,theunitloadmethod.Thus,wedeterminetheforcesFa,j,inthemembers
producedbytheactualloads.Wethenapplyequalandoppositeunitloadstothecutendsofthemember
BD as shown in Fig. 5.20(c) and calculate the forces,F1,j, in the members. The displacement of B
relativetoD, (^) BD,isthengivenby
(^) BD=
∑n
j= 1
Fa,jF1,jLj
AE
(seeEq.(ii)inExample4.6)
Theforces,Fa,j,aretheforcesinthemembersofthereleasedtrussduetotheactualloadsandarenot,
therefore, the actual forces in the members of the complete truss. We shall therefore redesignate the
forcesinthemembersofthereleasedtrussasF0,j.Theexpressionfor (^) BDthenbecomes
(^) BD=
∑n
j= 1
F0,jF1,jLj
AE
(i)
Intheactualstructure,thisdisplacementispreventedbytheforce,XBD,intheredundantmemberBD.
If,therefore,wecalculatethedisplacement,aBD,inthedirectionofBDproducedbyaunitvalueof
XBD,thedisplacementduetoXBDwillbeXBDaBD.Clearly,fromcompatibility
(^) BD+XBDaBD=0(ii)
fromwhichXBDisfound,aBDisaflexibilitycoefficient.HavingdeterminedXBD,theactualforcesin
the members of the complete truss may be calculated by, say, the method of joints or the method of
sections.
InEq.(ii),aBDisthedisplacementofthereleasedtrussinthedirectionofBDproducedbyaunit
load.Thus,inusingtheunitloadmethodtocalculatethisdisplacement,theactualmemberforces(F1,j)
andthememberforcesproducedbytheunitload(Fl,j)arethesame.Therefore,fromEq.(i)
aBD=
∑n
j= 1
F1,^2 jLj
AE
(iii)
ThesolutioniscompletedinTable5.6.Fromthattable,
(^) BD=
2.71PL
AE
aBD=4.82L
AE
SubstitutingthesevaluesinEq.(i),wehave
2.71PL
AE+XBD
4.82L
AE
= 0
fromwhich
XBD=−0.56P (i.e.,compression)