1.1 static and kinematic indeterminacy

How to calculate degree of static Indeterminacy and kinematic indeterminacy.
Reference used : Structural analysis by R C Hibbler

1. TYPES OF THE STRUCTURE
MASS STRUCTURES
Mass structures are those which resist Load
by virtue of their weight.
Framed structures
A framed structures resist applied Load by
virtue of its geometry
Basic elements of framed structures
Rods, beams, columns, plate or slabs

2. Plane frame
A structure having all its member in one plane is called plane frame

3. Space frame
A structure having a member in three dimensions is called space frame

4. Structures
Structureisanassemblageof
numberofcomponentslikeslab,
beams,columnsandfoundations
whichremaininequilibrium
Anystructureisdesignedforthe
stressresultantsofbendingmoment,
shearforce,deflection,torsional
stresses,andaxialstresses.

5. Ifthesemoments,shearsandstresses
areevaluatedatvariouscriticalsections,
thenbasedonthese,theproportioning
canbedone.
Evaluationofthesestresses,moments
andforcesandplottingthemforthat
structuralcomponentisknownas
Structuralanalysis.
Determinationofdimensionsforthesecomponentsofthese
stressesandproportioningisknownasdesign.

6. CLASSIFICATION OF STRUCTURES
Statically Determinate structure
•The structure for which the reactions at the supports & internal
forces in the members can be found out by the conditions of
static equilibrium, is called a statically determinate structure.
•Example of determinate structures are:
simply supported beams, cantilever beams, single and double
overhanging beams, three hinged arches , etc.
•There are three basic conditions of static equilibrium:
ΣH = 0
ΣV = 0
ΣM = 0

7. Staticallyindeterminatestructure
•Thestructureforwhichthereactionsatthesupports&theinternalforcesinthe
memberscannotbefoundoutbytheconditionsofstaticequilibrium,iscalledstatically
indeterminatestructure.
•Examplesofindeterminatestructuresare:fixedbeams,continuousbeams,fixedarches,
twohingedarches,portals,multistoriedframes,etc.
•Ifequationsofstaticequilibriumarenotsufficienttodeterminealltheunknown
reactions(vertical,horizontal&momentreactions)actingonthestructure,itiscalled
externallyindeterminatestructureorexternallyredundantstructure.
•Ifequationsofstaticequilibriumarenotsufficienttodeterminealltheinternalforces
andmomentsinthememberofthestructure,eventhoughalltheexternalforcesacting
onthestructureareknown,iscalledinternallyindeterminatestructureorinternally
redundantstructure.

8. Foracoplanarstructurethereareat
mostthreeequilibriumequationsfor
eachpart,sothatifthereisatotalofn
partsandrforceandmomentreaction
components,wehave
Inparticular,ifastructureisstatically
indeterminate,theadditionalequations
neededtosolvefortheunknown
reactionsareobtainedbyrelatingthe
appliedloadsandreactionstothe
displacementorslopeatdifferentpoints
onthestructure
Theseequations,whicharereferredto
ascompatibilityequations,mustbe
equalinnumbertothedegreeof
indeterminacyofthestructure.
Compatibilityequationsinvolvethe
geometricandphysicalpropertiesof
thestructure

11. ClassifyeachofthebeamsshowninFigs.athroughdasstaticallydeterminateor
staticallyindeterminate.
Ifstaticallyindeterminate,reportthenumberofdegreesofindeterminacy.Thebeams
aresubjectedtoexternalloadingsthatareassumedtobeknownandcanactanywhere
onthebeams.

13. Classify each of the pin-connected structures shown in Figs. a through d as statically determinate or statically
indeterminate.
If statically indeterminate, report the number of degrees of indeterminacy. The structures are subjected to arbitrary
external loadings that are assumed to be known and can act anywhere on the structures.

15. Classify each of the frames shown in Figs. 2–22a through 2–22c as statically determinate or statically
indeterminate. If statically indeterminate, report the number of degrees of indeterminacy. The frames are
subjected to external loadings that are assumed to be known and can act anywhere on the frames.

17. Toensuretheequilibriumofastructureorits
members,itisnotonlynecessarytosatisfythe
equationsofequilibrium,butthemembersmust
alsobeproperlyheldorconstrainedbytheir
supportsregardlessofhowthestructureisloaded.
Twosituationsmayoccurwheretheconditionsfor
properconstrainthavenotbeenmet.
Stability.
a) Partial Constraints.
Instabilitycanoccurifastructureoroneofits
membershasfewerreactiveforcesthanequationsof
equilibriumthatmustbesatisfied.Thestructure
thenbecomesonlypartiallyconstrained.
HeretheequationΣFx=0,willnotbesatisfiedfortheloadingconditions,
andthereforethememberwillbeunstable.

18. b) Improper Constraints.
Insomecasestheremaybeasmanyunknown
forcesasthereareequationsofequilibrium;
however,instabilityormovementofastructureor
itsmemberscandevelopbecauseofimproper
constrainingbythesupports.Thiscanoccurifall
thesupportreactionsareconcurrentatapoint
From the free-body diagram of the beam it is seen
that the summation of moments about point O will
not be equal to zero (Pd 0); thus rotation about
point O will take place.

19. Anotherwayinwhichimproperconstrainingleadsto
instabilityoccurswhenthereactiveforcesareall
parallel.AnexampleofthiscaseisshowninFig.
HerewhenaninclinedforcePisapplied,the
summationofforcesinthehorizontaldirectionwill
notequalzero.
Ingeneral,then,astructurewillbegeometrically
unstable—thatis,itwillmoveslightlyorcollapse
•iftherearefewerreactiveforcesthanequationsof
equilibrium;or
•ifthereareenoughreactions,instabilitywilloccurif
thelinesofactionofthereactiveforcesintersectata
commonpointorareparalleltooneanother.

20. Ifthestructureisunstable,itdoesnotmatterifitis
staticallydeterminateorindeterminate.Inallcases
suchtypesofstructuresmustbeavoidedinpractice.

21. ClassifyeachofthestructuresinFigs.athroughdasstableorunstable.Thestructuresaresubjected
toarbitraryexternalloadsthatareassumedtobeknown.

23. kinematic indeterminacy
Degrees of Freedom.
Whenastructureisloaded,specifiedpointsonit,callednodes,will
undergounknowndisplacements.
Thesedisplacementsarereferredtoasthedegreesoffreedomforthe
structure,andinthedisplacementmethodofanalysisitisimportantto
specifythesedegreesoffreedomsincetheybecometheunknowns
whenthemethodisapplied.
The number of these unknowns is referred to as the degree in which
the structure is kinematicallyindeterminate.

24. Todeterminethekinematicindeterminacywecanimagine
the
structuretoconsistofaseriesofmembersconnectedto
nodes,whichareusuallylocatedatjoints,supports,atthe
endsofamember,orwherethemembershaveasudden
changeincrosssection.
Inthreedimensions,eachnodeonaframeorbeam
canhaveatmostthreelineardisplacementsand
threerotationaldisplacements;and
intwodimensions,eachnodecanhaveatmosttwo
lineardisplacementsandonerotational
displacement.
ForexampleanyloadPappliedtothebeamwillcausenodeAonlytorotate(neglecting
axialdeformation),whilenodeBiscompletelyrestrictedfrommoving.Hencethebeam
hasonlyoneunknowndegreeoffreedom,θA,andisthereforekinematically
indeterminatetothefirstdegree.

25. ThebeaminFig.bhasnodesatA,B,andC,andso
hasfourdegreesoffreedom,designatedbythe
rotationaldisplacementsθA,θB,θC,andthevertical
displacement∆C.
Itiskinematicallyindeterminatetothefourth
degree.
ConsidernowtheframeinFig.c.
Again,ifweneglectaxialdeformationofthe
members,anarbitraryloadingPappliedtothe
framecancausenodesBandCtorotate,andthese
nodescanbedisplacedhorizontallybyanequal
amount.
Theframethereforehasthreedegreesoffreedom,
θB,θC,∆B,andthusitiskinematicallyindeterminate
tothethirddegree.