5.5Unit Load Method 139fromwhich
(^) C=
∑k
i= 1
λi
∂Fi
∂Pf
asbefore (5.19)
IfinsteadofthearbitrarydummyloadPfwehadplacedaunitloadatC,thentheloadintheithlinearly
elasticmemberwouldbe
Fi=
∂Fi
∂Pf
1
Therefore,theterm∂Fi/∂PfinEq.(5.19)isequaltotheloadintheithmemberduetoaunitloadatC,
andEq.(5.19)maybewrittenas
(^) C=
∑k
i= 1
Fi,0Fi,1Li
AiEi
(5.20)
whereFi,0is the force in theith member due to the actual loading andFi, 1 is the force in theith
memberduetoaunitloadplacedatthepositionandinthedirectionoftherequireddeflection.Thus,
in Example 5.1, columns④and⑥in Table 5.1 would be eliminated, leaving column⑤asFB,1and
column⑦asFD,1.Obviouslycolumn③isF 0.
Similarexpressionsfordeflectionduetobendingandtorsionoflinearstructuresfollowfromthe
well-knownrelationshipsbetweenbendingandrotationandtorsionandrotation.Hence,foramember
oflengthLandflexuralandtorsionalrigiditiesEIandGJ,respectively,
(^) B.M=
∫
LM 0 M 1
EI
dz (^) T=
∫
LT 0 T 1
GJ
dz (5.21)whereM 0 isthebendingmomentatanysectionproducedbytheactualloadingandM 1 isthebending
moment at any section due to a unit load applied at the position and in the direction of the required
deflection.Thesameappliestotorsion.
Generally,sheardeflectionsofslenderbeamsareignoredbutmaybecalculatedwhenrequiredfor
particularcases.Ofgreaterinterestinaircraftstructuresisthecalculationofthedeflectionsproducedby
thelargeshearstressesexperiencedbythin-walledsections.ThisproblemisdiscussedinChapter19.
Example 5.7
AsteelrodofuniformcircularcrosssectionisbentasshowninFig.5.19,ABandBCbeinghorizontal
andCDbeingvertical.ThearmsAB,BC,andCDareofequallength.TherodisencastréatA,and
theotherendDisfree.AuniformlydistributedloadcoversthelengthBC.Findthecomponentsofthe
displacementofthefreeendDintermsofEIandGJ.
Since the cross-sectional areaAand modulus of elasticityEare not given, we shall assume that
displacements due to axial distortion are to be ignored. We place, in turn, unit loads in the assumed
positivedirectionsoftheaxesxyz.