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CE2351 STRUCTURAL ANALYSIS – II L T P C
3104
OBJECTIVE
This course is in continuation of Structural Analysis – Classical Methods. Here in advanced
method of analysis like Matrix method and Plastic Analysis are covered. Advanced topics such
as FE method and Space Structures are covered.
UNIT I FLEXIBILITY METHOD 12
Equilibrium and compatibility – Determinate vs Indeterminate structures – Indeterminacy Primary structure – Compatibility conditions – Analysis of indeterminate pin-jointed plane
frames, continuous beams, rigid jointed plane frames (with redundancy restricted to two).
UNIT II STIFFNESS MATRIX METHOD 12
Element and global stiffness matrices – Analysis of continuous beams – Co-ordinate
transformations – Rotation matrix – Transformations of stiffness matrices, load vectors and
displacements vectors – Analysis of pin-jointed plane frames and rigid frames( with redundancy
vertical to two)
UNIT III FINITE ELEMENT METHOD 12
Introduction – Discretisation of a structure – Displacement functions – Truss element – Beam
element – Plane stress and plane strain - Triangular elements
UNIT IV PLASTIC ANALYSIS OF STRUCTURES 12
Statically indeterminate axial problems – Beams in pure bending – Plastic moment of resistance
– Plastic modulus – Shape factor – Load factor – Plastic hinge and mechanism – Plastic
analysis of indeterminate beams and frames – Upper and lower bound theorems
UNIT V SPACE AND CABLE STRUCTURES 12
Analysis of Space trusses using method of tension coefficients – Beams curved in plan
Suspension cables – suspension bridges with two and three hinged stiffening girders
TOTAL: 60 PERIODS
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TEXT BOOKS
1. Vaidyanathan, R. and Perumal, P., “Comprehensive structural Analysis – Vol. I & II”, Laxmi
Publications, New Delhi, 2003
2. L.S. Negi & R.S. Jangid, “Structural Analysis”, Tata McGraw-Hill Publications, New Delhi,
2003.
3. BhaviKatti, S.S, “Structural Analysis – Vol. 1 Vol. 2”, Vikas Publishing House Pvt. Ltd., New
Delhi, 2008
REFERENCES
1. Ghali.A, Nebille,A.M. and Brown,T.G. “Structural Analysis” A unified classical and Matrix
approach” –5th edition. Spon Press, London and New York, 2003.
2. Coates R.C, Coutie M.G. and Kong F.K., “Structural Analysis”, ELBS and Nelson, 1990
3. Structural Analysis – A Matrix Approach – G.S. Pandit & S.P. Gupta, Tata McGraw Hill
2004.
4. Matrix Analysis of Framed Structures – Jr. William Weaver & James M. Gere, CBS
Publishers and Distributors, Delhi.
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CE2351
Structural Analysis II
CHAPTER 1
FLEXIBILITY METHOD
Equilibrium and compatibility – Determinate vs Indeterminate structures –
Indeterminacy -Primary structure – Compatibility conditions – Analysis of indeterminate
pin-jointed planeframes, continuous beams, rigid jointed plane frames (with redundancy
restricted to two).
1.1 INTRODUCTION
These are the two basic methods by which an indeterminate skeletal structure is
analyzed. In these methods flexibility and stiffness properties of members are employed.
These methods have been developed in conventional and matrix forms. Here conventional
methods are discussed.
Thegivenindeterminatestructureisfirstmadestaticallydeterminatebyintroducing
suitable numberof releases. The number of releases required is equal to
Introductionofreleasesresultsin
staticalindeterminacy∝s.
displacementdiscontinuitiesatthesereleases under the externally applied loads. Pairs
ofunknown
biactions(forces
andmoments)areappliedatthesereleasesinordertorestorethecontinuityorcompatibility
of
structure.
The computation of these unknown biactions involves solution of linear
simultaneousequations.Thenumberoftheseequationsisequaltostaticalindeterminacy∝s.
Aftertheunknownbiactionsarecomputedall
theinternalforcescanbecomputedintheentirestructureusingequationsofequilibriumandfreeb
odiesofmembers.Therequired displacements can also be computed using methods of
displacement computation.
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Inflexibilitymethodsinceunknownsareforces atthereleasesthemethodisalsocalled
force method.Since computation of displacement is also required at releases for
imposing conditions of compatibility the method is also called compatibility method. In
computationofdisplacementsuseismadeof flexibilityproperties,hence,themethodis also
called flexibility method.
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1.2 EQUILIBRIUM and COMPATABILITY CONDITIONS
Thethreeconditionsofequilibriumarethesumofhorizontalforces,verticalforcesandmom
ents at anyjoint should beequal to zero.
i.e.∑H=0;∑V=0;∑M=0
Forces should be in equilibrium
i.e.∑FX=0;∑FY=0;∑FZ=0
i.e.∑MX=0;∑MY=0;∑MZ=0
Displacement of a structure should be compatable
The compatibility conditions for the supports can be given as
1.Roller Support δV=0
2.Hinged Support δV=0, δH=0
3.Fixed Support δV=0, δH=0, δө=0
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CE2351
Structural Analysis II
1.3.DETERMINATE AND INDETERMINATE STRUCTURAL SYSTEMS
Ifskeletalstructureissubjectedtograduallyincreasingloads,withoutdistortingthe
initialgeometryofstructure,thatis,causingsmalldisplacements,thestructureissaidto be stable.
Dynamic loads and buckling or instability of structural system are not
consideredhere.Ifforthestablestructureitispossibletofindtheinternalforcesinall the members
constituting the structure and supporting reactions at all the supports providedfrom
staticallyequationsofequilibrium only,thestructureissaidtobe determinate.
Ifitispossibletodetermineallthesupport
reactionsfromequationsof
equilibrium
alonethestructureissaidtobeexternallydeterminateelseexternally indeterminate.If structureis
externallydeterminatebutitisnotpossible
todetermineall
internalforcesthenstructureissaidtobe
internallyindeterminate.
Thereforeastructural
systemmaybe:
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(1)Externally indeterminate but internally determinate
(2)Externally determinate but internally indeterminate
(3)Externallyand internallyindeterminate
(4)Externally and internallydeterminate
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1.3.1.DETERMINATEVs INDETERMINATESTRUCTURES.
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Determinatestructurescanbesolvingusingconditionsofequilibriumalone(∑H=0;∑V=0
;∑M=0). No otherconditions arerequired.
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Indeterminatestructurescannotbesolvedusingconditionsofequilibriumbecause(∑H≠0;
∑V≠0;∑M≠ 0).Additionalconditionsarerequiredforsolvingsuchstructures.
Usuallymatrixmethods areadopted.
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1.4 INDETERMINACYOF STRUCTURAL SYSTEM
The indeterminacy of a structure is measured as statically (∝s) or kinematical
(∝k)Indeterminacy.
∝s= P (M – N + 1) – r = PR– r ∝k= P (N – 1) + r – s+∝k= PM –c
P = 6 for space frames subjected to general loading
P = 3 for plane frames subjected to inplane or normal to plane loading.
N = Numberof nodes in structural system.
M=Numberofmembersofcompletelystiffstructurewhichincludesfoundationas
singlyconnectedsystem ofmembers.
Incompletelystiffstructurethereisnorelease
present.Insinglyconnectedsystem
ofrigidfoundationmembersthereisonlyoneroute
betweenanytwopointsinwhichtracksarenotretraced. Thesystemisconsidered comprising of
closed rings or loops.
R = Numberof loops or rings in completely stiff structure.
r = Number of releases in the system.
c = Number of constraints in the system.
R = (M – N + 1)
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CE2351
Structural Analysis II
For plane and space trusses∝sreduces
to:∝s=M- (NDOF)N+ P
M= Number ofmembers in completely stifftruss.
P = 6 and 3 for space and plane trussrespectively
N= Number of nodes in truss.
NDOF = Degrees of freedomat node which is 2 for plane truss and 3 for space truss.
For space truss∝s=M- 3N+ 6
For plane truss∝s= M- 2 N+ 3
Test for static indeterminacy of structural system
∝s> 0
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If
∝s= 0
and if∝s<0
If
Structure is statically indeterminate
Structure is statically determinate
Structure is a mechanism.
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Itmaybenotedthatstructuremaybemechanismevenif
∝s
>0ifthereleasesare
presentinsuchawaysoastocausecollapseasmechanism.Thesituationofmechanism
is
unacceptable.
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Statically Indeterminacy
Itisdifferenceoftheunknownforces(internalforcesplusexternalreactions)andthe
equations of equilibrium.
Kinematic Indeterminacy
Itisthenumberofpossiblerelativedisplacementsofthenodesinthedirectionsofstress
resultants.
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1.5 PRIMARY STRUCTURE
Astructure formed bythe removingthe excess orredundant restraints froman
indeterminatestructuremakingit staticallydeterminateis called primarystructure. This is
required forsolvingindeterminatestructures byflexibilitymatrixmethod.
Indeterminatestructure
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