Advanced Structural Analysis Lecture 1 ‐ Module 1: Review of Basic Structural Analysis Devdas Menon Professor IIT Madras ([email protected]) National Programme on Technology Enhanced Learning (NPTEL) www.nptel.ac.in
AdvancedStructuralAnalysis
Lecture1Module1:ReviewofBasicStructuralAnalysis
DevdasMenonProfessorIITMadras
NationalProgrammeonTechnologyEnhancedLearning(NPTEL)
www.nptel.ac.in
AdvancedStructuralAnalysisModules
1. Reviewofbasicstructuralanalysis 1
2. Reviewofbasicstructuralanalysis 2
3. Basicmatrixconcepts
4. Matrixanalysisofstructureswithaxialelements
5. Matrixanalysisofbeamsandgrids
6. Matrixanalysisofplaneandspaceframes
7. Analysisofelasticinstabilityandsecondordereffects
Paperback (2009)Paperback (2008)Hardbound (2008) Hardbound (2009)
Module1:ReviewofbasicSA 1
StaticallyDeterminateStructures:Introduction;Analysisofsupportreactions,internalforcesintrusses,beams,cables,archesandframes;Analysisofslopesanddeflectionsinbeams.
WorkandEnergyMethods: Introduction;Principleofvirtualdisplacementsandvirtualforces;Reciprocaltheorems;Energyconcepts;Displacementbasedandforcebasedenergyprinciples;derivingstiffnessandflexibilitycoefficients.
Module1:ReviewofbasicStructuralAnalysis 1
Introduction
StaticallyDeterminateStructures
WorkandEnergyMethods
ForceMethods StaticallyIndeterminateStructures
DisplacementMethods KinematicallyIndeterminateStructures
PartI:Fundamentals
PartI:Fundamentals
PartII:ForceResponseinStatically
DeterminateStructures
PartII:ForceResponseinStatically
DeterminateStructures
PartIII:DisplacementResponseinStatically
DeterminateStructures
PartIII:DisplacementResponseinStatically
DeterminateStructures
PartIV:Statically
IndeterminateStructures:
ForceMethods
PartIV:Statically
IndeterminateStructures:
ForceMethods
PartV:KinematicallyIndeterminateStructures:
DisplacementMethods
PartV:KinematicallyIndeterminateStructures:
DisplacementMethods
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LOADS
(input)STRUCTURE
(system)
RESPONSE?
(output)
STRUCTURALDESIGN
StructuralAnalysis&Design
o Stabilityo Strengtho Stiffnesso EconomyoAesthetics
STRUCTURALANALYSIS
StructuralAnalysisistheapplicationofsolidmechanicstopredicttheresponse (intermsofforcesanddisplacements)ofagivenstructure(existingorproposed)subjecttospecifiedloads.
IntroductiontoStructuralAnalysis
STRUCTURELOAD RESPONSE
StructuralSystem StructuralIdealisation StructuralElements
StructuralSystem StructuralIdealisation StructuralElements
Joints&InternalStability Supports&ExternalStability StaticIndeterminacy KinematicIndeterminacy
Joints&InternalStability Supports&ExternalStability StaticIndeterminacy KinematicIndeterminacy
StructuralModelling
slab
Spaceframe
STRUCTURE
WALLS
SLABS
SHELLS
FOLDEDPLATES
SKELETAL SPATIAL
AxialSystem
GridBeam
SpaceTrussPlaneTruss
MODELLINGOF
STRUCTURES
MODELLINGOF
STRUCTURES
PlaneFrame
PlaneFrameElement
LINEAR(1D) PLANAR(2D) SPACE(3D)
SpaceFrame
SpaceFrameElement
Joints&Supports
Joint
(Connectionbetweenelements)
(Connectionbetweenelements)
Kinematicfunction:Controlrelativedisplacements
Staticfunction:Providefortransmissionofforces
Semirigid
Pinned
Rigid
(Freerelativerotation)
(Norelativerotationortranslation)
Hingedsupport
Rollersupport
FixedsupportSupportReactionsSupportReactions
Internalhingebehaviour
SB
SBVCVA
MA
AB C
P1 P2
BC
P2
AB
P1
DetailatB A B
P1 C
P2
childparent
SpaceandPlaneFrames
Spaceframe(3Danalysis)
Planeframes(2Danalysis)
SpaceFrameElement
PlaneframeElement
PlaneFramesandBeams
Full(plane)frameundergravityloading
Partial(substitute)frame
Bendingmoments
Shearforces
Continuousbeam
Beamelement
Grids(grillages)andBeams
Grid
Gridelement
Bendingmoments
Shearforces
Twistingmoments
GridelementreducestoBeamelementTorsional stiffnessnegligible
(Introducetorsional releasesatmemberends)
StaticIndeterminacy(ns)
Cause(kinematicsrelated):Structureisoverconstrainedoroverrigid
Effect(staticsrelated):Structureishyperstaticorstaticallyindeterminate
(ns =m +r 2j =2)
(ns =m +r 2j =2)
(m =9,r =3,j =6)m +r =2j=12
UnstableOverrigid
Justrigid (ns =0)(ns =0)
ForcesandDisplacements
SimplePlaneTruss:
m =11,r =3,j =7m +r =2j=14
F1
F2F3
F4F5
F6
F12F13
F7F8
F9F10
F11F14
1
2 3
4
5 6
8 9 10 11
7
D5 D6
D9
D10
D1
D2
11
D3
D43
Forces: external;internalFj(j 1,2,,14);Ni (i =1,2,,11)
N11
Displacements:deflections;deformationsDj (j 1,2,,14);ei (i =1,2,,11)
D5
D6
D9
D10
L11
L11+e11
Mustsatisfyequilibriumconditions
Mustsatisfycompatibilityconditions
KinematicIndeterminacy
Global(Structure)Coordinates:j=1,2,,14Restraints:D12 =0,D13 =0,D14 =0UnknownDisplacements:D1 ,D2 ,,D11Degreeofkinematicindeterminacynk =11(Degreeofstaticindeterminacyns =0!)
m =11,r =3,j =7m +r =2j=14
1
23
45
6
1213
78
910
1114
SimplePlaneTruss:
Thedegreeofkinematicindeterminacymaybedefinedasthetotalnumberofdegreesoffreedom(independentdisplacementcoordinates)atthevariousjointsinaskeletalstructure.
KinematicIndeterminacy
BeamelementBeamelement
11
22 AA BB33
44
D1
D4D2
D3AA BB nk =2
CantileverCantilever
nk =0
FixedBeamFixedBeam
11
22
ContinuousBeamContinuousBeam nk =4
11 22
33
44
PlaneframeelementPlaneframeelement
BB
55
6644
22
33 AA11
55
nk =6nk =6
PortalframePortalframe
22
3311
6644
BoxframeBoxframenk =3nk =3
IgnoringaxialdeformationsIgnoringaxialdeformations
11 223311 33
55
441166
nk =6nk =6
Staticvs KinematicIndeterminacy
ns =(4 3) 3=36nk =(4 4) 3=48ns =(4 3) 3=36nk =(4 4) 3=48
Ignoringaxialdeformations:nk =16+4=20
Ignoringaxialdeformations:nk =16+4=20
11 171722 33 44
55 181866 77 88
99 19191010 1111 1212
1313 2020
1414 1515 1616
StructuralAnalysis
STRUCTURELOAD RESPONSE
SupportReactions InternalForceResultants Displacements
SupportReactions InternalForceResultants Displacements
DirectActions IndirectLoading DirectActions IndirectLoading ForceField
DisplacementField
EquilibriumofForces CompatibilityofDisplacements Forcedisplacementrelations
EquilibriumofForces CompatibilityofDisplacements Forcedisplacementrelations
UnderlyingPrinciples:
IndirectLoading
SupportDisplacements
ConstructionalErrors
EnvironmentalChanges
Caninduceaforceresponseifthestructureisoverrigid;otherwiseonlyadisplacementresponse!
SupportDisplacements
(a)cantileverbeam
A B
rotationalslipA
A
(c)proppedcantileverbeam
A B
rotationalslipA
(d)fixedbeam
A Bdifferentialsettlement
BA
(b)simplysupportedbeam
A B
differentialsettlement
BA
Overrigidstructures
(bendingmoments,shearforces,supportreactionsinduced)
Justrigidstructures
(rigidbodymotions;noforceresponse)
ConstructionalErrors
(a) Internallyjustrigid (b) internallyoverrigid
xx
xxA
B C
D
xx
xx
xx
xxA
B C
D
xx
xx
(AClongby5mm)
(noforceresponse)(diagonalsundercompression;
othermembersintension)
Lackoffitintrusses:
EnvironmentalChanges
L
=L(T)
Noforce
Justrigid
Temperatureeffect
Overrigid
L L
ks
Axialforce:N =kbar Axialforce:N =kbar
Axialforce:N =ks sAxialforce:N =ks s
00
Equilibriumofforcesensuresstaticadmissibility
Compatibilityofdisplacementsensureskinematicadmissibility
Forcedisplacementrelationsinvolvematerialstressstrainlaws
BasicRequirements
ForceResponse
Supportreactions
Internalforces
C
AB
P
30011
22
Supportreactions
PRC
RA300
RA 3 P
RC 2P 3
2300 1
Fij Actionofion j(appearsonfreebodyofj)
Fij =Fji(Newtonsthirdlaw)
RA
RC
A B
P
C
F1A FA1
F1BFB1
F2BFB2
FC2F2C
11
22
N1 3 P
N2 2P
Barforces
StaticallyAdmissibleFORCERESPONSE
LinearElasticBehaviour
SYSTEMStimulus Response?
P
L
L
+
e
TensionTest
highAlowL
low Ahigh L
P
e0
P/A
e/L0
A
manyrodspecimens(samematerial),variouslengthsanddiameters
E
1
Singlematerialproperty!
k P ki Niei
EA iLi
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(a)loaddeflection(structure)
(b) internalforcedeformation(trusselement)
(c)stressstrainrelation(material)
deflection elongation strain0 0 0
1E
11k ki
NiP
ei
Forcedisplacementrelations
DisplacementResponse
Deflections
Rotations
AB
P
30011
22
N1 3 P
N2 2P
e1
e2
Equilibrium:
e1 N
1k1
e2 N
2k2
Forcedisplacementrelation:A
C
B
B, hor
B, verB
Compatibility:
Kinematically AdmissibleDISPLACEMENTRESPONSE