Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

5.10 The Reciprocal Theorem 155

Fig.5.28

Model analysis of a fixed beam.

or

θA(c),2=

W

M

δ 1 (ii)

whereθA(c),2is the rotation at A due toWat C. Finally, the rotation at A due toMAat A is, from
Fig.5.28(a)and(c),

θA(c),3=

MA

M

θA (iii)

ThetotalrotationatAproducedbyMAatA,WatC,andMBatBis,fromEqs.(i),(ii),and(iii),

θA(c),1+θA(c),2+θA(c),3=

MB

M

θB+

W

M

δ 1 +

MA

M

θA=0(iv)

sincetheendAisrestrainedfromrotation.Similarly,therotationatBisgivenby

MB

M

θC+

W

M

δ 2 +

MA

M

θB=0(v)

SolvingEqs.(iv)and(v)forMAgives

MA=W

(

δ 2 θB−δ 1 θC

θAθC−θB^2

)

ThefactthatthearbitrarymomentMdoesnotappearintheexpressionfortherestrainingmoment
at A (similarly it does not appear inMB), produced by the loadW, indicates an extremely useful
applicationofthereciprocaltheorem,namelythemodelanalysisofstaticallyindeterminatestructures.
Forexample,thefixedbeamofFig.5.28(c)couldpossiblybeafull-scalebridgegirder.Itisthenonly
necessary to construct a model, say of Perspex, having the same flexural rigidityEIas the full-scale
beam and measure rotations and displacements produced by an arbitrary momentMto obtain fixing
momentsinthefull-scalebeamsupportingafull-scaleload.