Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

126 CHAPTER 5 Energy Methods

Fig.5.10

Framework of Example 5.4.

Example 5.4
Aplane,pin-jointedframeworkconsistsofsixbarsformingarectangleABCD4000mmby3000mm
with two diagonals, as shown in Fig. 5.10. The cross-sectional area of each bar is 200mm^2 ,and
theframeisunstressedwhenthetemperatureofeachmemberisthesame.Becauseofthelocalcon-
ditions, the temperature of one of the 3000mm members is raised by 30◦C. Calculate the resulting
forces in all the members if the coefficient of linear expansionαofthebarsis7× 10 −^6 /◦C.E=
200000N/mm^2.

Suppose that BC is the heated member; then the increase in length of BC= 3000 × 30 ×
7 × 10 −^6 =0.63mm.Therefore,fromEq.(5.17),

−0.63=

1

200 × 200000

∑k

i= 1

FiLi

∂Fi

∂R

(i)

Substitutionfromthesummationofcolumn⑤inTable5.4intoEq.(i)gives

R=

−0.63× 200 × 200000

48000

=−525N

Column⑥ofTable5.4isnowcompletedfortheforceineachmember.
Sofar,ouranalysishasbeenlimitedtosinglyredundantframeworks,althoughthesameprocedure
maybeadoptedtosolveamulti-redundantframeworkof,say,mredundancies.Therefore,insteadofa
singleequationofthetype(5.15),wewouldhavemsimultaneousequations

∂C

∂Rj

=

∑k

i= 1

λi

∂Fi

∂Rj

= 0 (j=1,2,...,m)

fromwhichthemunknownsR 1 ,R 2 ,...,Rmwouldbeobtained.TheforcesFinthemembersfollow,
beingexpressedinitiallyintermsoftheappliedloadsandR 1 ,R 2 ,...,Rm.
Othertypesofstaticallyindeterminatestructurearesolvedbytheapplicationoftotalcomplementary
energywithequalfacility.TheproppedcantileverofFig.5.11isanexampleofasinglyredundantbeam
structureforwhichtotalcomplementaryenergyreadilyyieldsasolution.