Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

5.10 The Reciprocal Theorem 151

fromwhich

vB=

2 WL^3

π^4 EI

=0.02053

WL^3

EI

(iv)

Theexactexpressionforthemidspandisplacement[Ref.3]is

vB=

WL^3

48 EI

=0.02083

WL^3

EI

(v)

Comparingtheexact(Eq.(v))andapproximateresults(Eq.(iv)),weseethatthedifferenceisless
than 2 percent. Further, the approximate displacement is less than the exact displacement, since, by
assuming a displaced shape, we have, in effect, forced the beam into taking that shape by imposing
restraint;thebeamisthereforestiffer.

5.9 PrincipleofSuperposition..........................................................................

Anextremelyusefulprincipleusedintheanalysisoflinearlyelasticstructuresisthatofsuperposition.
Theprinciplestatesthatifthedisplacementsatallpointsinanelasticbodyareproportionaltotheforces
producingthem—thatis,thebodyislinearlyelastic—theeffectonsuchabodyofanumberofforces
isthesumoftheeffectsoftheforcesappliedseparately.Weshallmakeimmediateuseoftheprinciple
inthederivationofthereciprocaltheoreminthefollowingsection.

5.10 TheReciprocalTheorem............................................................................

The reciprocal theorem is an exceptionally powerful method of analysis of linearly elastic structures
andisaccreditedinturntoMaxwell,Betti,andRayleigh.However,beforeweestablishthetheorem,we
firstconsiderausefulpropertyoflinearlyelasticsystemsresultingfromtheprincipleofsuperposition.
The principle enables us to express the deflection of any point in a structure in terms of a constant
coefficientandtheappliedloads.Forexample,aloadP 1 appliedatapoint1inalinearlyelasticbody
producesadeflection 1 atthepointgivenby

1 =a 11 P 1

in which theinfluenceorflexibilitycoefficienta 11 is defined as the deflection at the point 1 in the
directionofP 1 ,producedbyaunitloadatthepoint1appliedinthedirectionofP 1 .Clearly,ifthebody
supportsasystemofloadssuchasthoseshowninFig.5.26,eachoftheloadsP 1 ,P 2 ,...,Pncontributes
to the deflection at the point 1. Thus, thecorresponding deflection 1 at the point 1 (i.e., the total
deflectioninthedirectionofP 1 producedbyalltheloads)isthen

1 =a 11 P 1 +a 12 P 2 +···+a 1 nPn

wherea 12 isthedeflectionatthepoint1inthedirectionofP 1 ,producedbyaunitloadatthepoint2
inthedirectionoftheloadP 2 ,andsoon.Thecorrespondingdeflectionsatthepointsofapplicationof