3.2 force method

Calculation of displacement in indeterminate structures by Flexibility method.
Reference used : Structural analysis by R C Hibbler

1. Analysis of Statically Indeterminate Structures
by the Force Method
Advantages and Disadvantages.
Thestaticallyindeterminate,fixed-supported
beaminFig.awillbesubjectedtoamaximum
momentofMmax=PL/8,
whereasthesamebeam,whensimply
supported,Fig.b,willbesubjectedtotwice
themoment,thatis,Mmax=PL/4.
Asaresult,thefixed-supportedbeamhasone
fourththedeflectionandonehalfthestressat
itscenteroftheonethatissimplysupported.

2. Anotherimportantreasonforselectinga
staticallyindeterminatestructureisbecauseit
hasatendencytoredistributeitsloadtoits
redundantsupportsincaseswherefaulty
designoroverloadingoccurs.
Methods of Analysis.
Whenanalyzinganyindeterminatestructure,
itisnecessarytosatisfyequilibrium,
compatibility,and force-displacement
requirementsforthestructure
Forastaticallyindeterminatestructure,they
aretheforceorflexibilitymethod,andthe
displacementorstiffnessmethod.

3. Force Method.
ithassometimesbeenreferredtoasthecompatibilitymethodorthemethodof
consistentdisplacements
Thismethodconsistsofwritingequationsthatsatisfythecompatibilityandforce-
displacementrequirementsforthestructureinordertodeterminetheredundant
forces.
Oncetheseforceshavebeendetermined,theremainingreactiveforcesonthe
structurearedeterminedbysatisfyingtheequilibriumrequirements.

4. Displacement Method.
Thedisplacementmethodofanalysisisbasedonfirstwritingforce-displacement
relationsforthemembersandthensatisfyingtheequilibriumrequirementsforthe
structure.Inthiscasetheunknownsintheequationsaredisplacements.
Oncethedisplacementsareobtained,theforcesaredeterminedfromthe
compatibilityandforcedisplacementequations.

5. Force Method of Analysis:
General Procedure
Thebeamisindeterminatetothefirstdegree.
Consequently,oneadditionalequationisnecessaryfor
solution.
Toobtainthisequation,wewillusetheprincipleof
superpositionandconsiderthecompatibilityof
displacementatoneofthesupports.
Thisisdonebychoosingoneofthesupportreactionsas
“redundant”andtemporarilyremovingitseffectonthe
beamsothatthebeamthenbecomesstaticallydeterminate
andstable.Thisbeamisreferredtoastheprimary
structure.

6. Herewewillremovetherestrainingactionofthe
rockeratB.Asaresult,theloadPwillcauseBtobe
displaceddownwardbyanamount∆Basshownin
Fig.b.
Bysuperposition,however,theunknownreactionat
B,i.e.,By,causesthebeamatBtobedisplaced∆BB
upward,Fig.c.
Herethefirstletterinthisdoublesubscriptnotation
referstothepoint(B)wherethedeflectionis
specified,andthesecondletterreferstothepoint
(B)wheretheunknownreactionacts.
ThenfromFigs.Athroughcwecanwritethe
necessarycompatibilityequationattherockeras

7. Whenwritteninthisformat,itcanbeseenthatthelinear
flexibilitycoefficientfBBisameasureofthedeflection
perunitforce,andsoitsunitsarem/N.
Thecompatibilityequationabovecanthereforebe
writtenintermsoftheunknownByas
ThusunknownBycabecalculated.

8. The compatibility equation for rotation at A therefore
requires