AdvancedStructuralAnalysis

Lecture1Module1:ReviewofBasicStructuralAnalysis

DevdasMenonProfessorIITMadras

(dmenon@iitm.ac.in)

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

B

a

s

i

c

S

t

r

u

c

t

u

r

a

l

A

n

a

l

y

s

i

s

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

E

l

o

a

d

a

x

i

a

l

t

e

n

s

i

o

n

s

t

r

e

s

s

(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