Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

5.4Application to the Solution of Statically Indeterminate Systems 123

Fig.5.8

Analysis of a statically indeterminate framework by the method of complementary energy.

tensileforceRduetotheexternalloading.Thetotalcomplementaryenergyoftheframeworkis,with
thenotationofEq.(5.9),

C=

∑k

i= 1

∫Fi

0

λidFi−P

Hence,

∂C

∂R

=

∑k

i= 1

λi

∂Fi

∂R

= 0 (5.15)

or,assuminglinearelasticity,

1

AE

∑k

i= 1

FiLi

∂Fi

∂R

= 0 (5.16)

ThesolutionisnowcompletedinTable5.2,where,asinTable5.1,positivesignsindicatetension.
Hence,fromEq.(5.16),

4.83RL+2.707PL= 0

or

R=−0.56P

Substitution forRin column③of Table 5.2 gives the force in each member. Having determined the
forcesinthemembers,thenthedeflectionofanypointontheframeworkmaybefoundbythemethod
describedinSection5.3.
Unlikethestaticallydeterminatetype,staticallyindeterminateframeworksmaybesubjectedtoself-
straining.Thus,internalforcesarepresentbeforeexternalloadsareapplied.Suchasituationmaybe
causedbyalocaltemperaturechangeorbyaninitiallackoffitofamember.Supposethatthemember

BDoftheframeworkofFig.5.8isshortbyaknownamount (^) Rwhentheframeworkisassembledbut