Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

5.4Application to the Solution of Statically Indeterminate Systems 127

Table 5.4Tension positive

① ② ③ ④ ⑤ ⑥

Member L(mm) F(N) ∂F/∂R FL∂F/∂R Force(N)

AB 4000 4R/ 34 /3 64000R/ 9 − 700

BC 3000 R 1 3000 R − 525

CD 4000 4R/ 34 /3 64000R/ 9 − 700

DA 3000 R 1 3000 R − 525

AC 5000 − 5 R/ 3 − 5 /3 125000R/ 9 875

DB 5000 − 5 R/ 3 − 5 /3 125000R/ 9 875

= 48000 R

Fig.5.11

Analysis of a propped cantilever by the method of complementary energy.

Thetotalcomplementaryenergyofthesystemis,withthenotationofEq.(5.12),

C=

∫

L

∫M

0

dθdM−P (^) C−RB (^) B
where (^) Cand (^) BarethedeflectionsatCandB,respectively.Usually,inproblemsofthistype, (^) Bis
eitherazeroforarigidsupportoraknownamount(sometimesintermsofRB)forasinkingsupport.
Hence,forastationaryvalueofC,
∂C
∂RB

=

∫

L

dθ

∂M

∂RB

− (^) B= 0
fromwhichequationRBmaybefound;RBbeingcontainedintheexpressionforthebendingmomentM.
Obviously,thesameprocedureisapplicabletoabeamhavingamultiredundantsupportsystem—for
example,acontinuousbeamsupportingaseriesofloadsP 1 ,P 2 ,...,Pn.Thetotalcomplementaryenergy
ofsuchabeamwouldbegivenby

C=

∫

L

∫M

0

dθdM−

∑m

j= 1

Rj (^) j−
∑n
r= 1
Pr (^) r