Note for Structural Analysis-2 - SA-2 by Engineering Kings

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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 53 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. ww w.E asy En gin eer Downloaded ing .ne t

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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. ww w.E asy En gin eer ing 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. .ne t 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 SCE 1 Dept of Civil Downloaded

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Downloaded 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: ww w.E (1)Externally indeterminate but internally determinate (2)Externally determinate but internally indeterminate (3)Externallyand internallyindeterminate (4)Externally and internallydeterminate asy En 1.3.1.DETERMINATEVs INDETERMINATESTRUCTURES. gin Determinatestructurescanbesolvingusingconditionsofequilibriumalone(∑H=0;∑V=0 ;∑M=0). No otherconditions arerequired. eer Indeterminatestructurescannotbesolvedusingconditionsofequilibriumbecause(∑H≠0; ∑V≠0;∑M≠ 0).Additionalconditionsarerequiredforsolvingsuchstructures. Usuallymatrixmethods areadopted. ing .ne 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) SCE 2 t Dept of Civil

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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 ww If ∝s= 0 and if∝s<0 If Structure is statically indeterminate Structure is statically determinate Structure is a mechanism. w.E Itmaybenotedthatstructuremaybemechanismevenif ∝s >0ifthereleasesare presentinsuchawaysoastocausecollapseasmechanism.Thesituationofmechanism is unacceptable. asy En Statically Indeterminacy Itisdifferenceoftheunknownforces(internalforcesplusexternalreactions)andthe equations of equilibrium. Kinematic Indeterminacy Itisthenumberofpossiblerelativedisplacementsofthenodesinthedirectionsofstress resultants. gin eer ing 1.5 PRIMARY STRUCTURE Astructure formed bythe removingthe excess orredundant restraints froman indeterminatestructuremakingit staticallydeterminateis called primarystructure. This is required forsolvingindeterminatestructures byflexibilitymatrixmethod. Indeterminatestructure SCE PrimaryStructure 3 .ne t Dept of Civil