Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

5.3Application to Deflection Problems 119

mechanicsofthesolutionarethesameineithercase.WechoosethereforealinearM–θrelationshipas
thisisthecaseinthemajorityoftheproblemsweconsider.Hence,fromFig.5.5,

δθ=Kδz

or

dθ=

M

EI

dz

(

1

K

=

EI

M

fromsimplebeamtheory

)

wheretheproductmodulusofelasticity×secondmomentofareaofthebeamcrosssectionisknown
asthebendingorflexuralrigidityofthebeam.Also,

M=Pz

sothat
dM
dP

=z

Substitutionfordθ,M,anddM/dPinEq.(5.13)gives

(^) v=

∫L

0

Pz^2

EI

dz

or

(^) v=

PL^3

3 EI

Thefictitiousloadmethodoftheframeworkexamplemaybeusedinthesolutionofbeamdeflection
problemswherewerequiredeflectionsatpositionsonthebeamotherthanconcentratedloadpoints.

Suppose that we are to find the tip deflection (^) Tof the cantilever of the previous example in which
theconcentratedloadhasbeenreplacedbyauniformlydistributedloadofintensitywperunitlength
(seeFig.5.6).First,weapplyafictitiousloadPfatthepointwherethedeflectionisrequired.Thetotal
complementaryenergyofthesystemis

C=

∫

L

∫M

0

dθdM− (^) TPf−

∫L

0

wdz

Fig.5.6

Deflection of a uniformly loaded cantilever by the method of complementary energy.