1. Structure Analysis II

2. S TRUCTURAL A NALYSIS II CE 1352 By R.REVATHI, M. Tech., STRUCTURAL Lecturer Department of Civil Engineering PITS VI Semester 2011-2012

3. O BJECTIVES This Course aims at teaching the students the concept of analyzing indeterminate structure using classical and up to date methods. It provides students with an understanding of the methods of analyzing indeterminate structure: –The force method of analysis –The Displacement method Slope deflection Moment distribution Stiffness Method (An Introduction to The Finite Element Method)

4. UNIT I FLEXIBILITY METHOD FOR INDETERMINATE FRAMES 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 MATRIX STIFFNESS METHOD 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. UNIT III FINITE ELEMENT METHOD Introduction – Discretisation of a structure – Displacement functions – Truss element – Beam element – Plane stress and plane strain Triangular elements.

5. UNIT IV PLASTIC ANALYSIS OF STRUCTURES 9 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 9 Analysis of Space trusses using method of tension coefficients – Beams curved in plan Suspension cables – Cables with two and three hinged stiffening girders.

6. L: 45 T: 15 Total: 60 TEXT BOOKS 1. Coates R.C., Coutie M.G. and Kong F.K., “Structural Analysis”, ELBS and Nelson, 1990. 2. Negi, L.S. and Jangid, R.S., “Structural Analysis”, Tata McGraw-Hill Publications, 2003. REFERENCES 1. Ghali, A., Nebille, A.M. and Brown, T.G., “Structural Analysis” A Unified Classical and Matrix approach”, 5th Edition, Spon Press, 2003. 2. Vazirani Vaidyanathan, R. and Perumail, P., “Comprehensive Structural Analysis – Vol. I and II”, Laxmi Publications, 2003

7. I NTRODUCTION DETERMINATE What is statically DETERMINATE structure? When all the forces (reactions) in a structure can be determined from the equilibrium equations its called statically determinate structure Structure having unknown forces equal to the available equilibrium equations No. of unknown = 3 No. of equilibrium equations = 3 3 = 3 thus statically determinate No. of unknown = 6 No. of equilibrium equations = 6 6 = 6 thus statically determinate

8. I NTRODUCTION INETERMINATED What is statically INETERMINATED structure Structure having more unknown forces than available equilibrium equations Additional equations needed to solve the unknown reactions No. of unknown = 4 No. of equilibrium equations = 3 4  3 thus statically Indeterminate No. of unknown = 10 No. of equilibrium equations = 9 10  9 thus statically Indeterminate

9. I NDETERMINATE S TRUCTURE Why we study indeterminate structure Most of the structures designed today are statically indeterminate Reinforced concrete buildings are considered in most cases as a statically indeterminate structures since the columns & beams are poured as continuous member through the joints & over the supports More stable compare to determinate structure or in another word safer. In many cases more economical than determinate. The comparison in the next page will enlighten more

10. C ONTRAST Indeterminate StructureDeterminate Structure 10 Deflection PP 4 1 Stress PP 2 1 Considerable compared to indeterminate structure Generally smaller than determinate structure Less moment, smaller cross section & less material needed High moment caused thicker member & more material needed

11. C ONTRAST Indeterminate StructureDeterminate Structure Stability in case of over load PP Plastic Hinge  Support will not develop the horizontal force & moments that necessary to prevent total collapse  No load redistribution  When the plastic hinge formed certain collapse for the system  Will develop horizontal force & moment reactions that will hold the beam  Has the tendency to redistribute its load to its redundant supports  When the plastic hinge formed the system would be a determinate structure

12. C ONTRAST Indeterminate StructureDeterminate Structure Temperature PP Differential Displacement PP No effect & no stress would be developed in the beam No effect & no stress would be developed Serious effect and stress would be developed in the beam Serious effect and stress would be developed

13. Distinctive Features of Suspension Bridge Major element is a flexible cable, shaped and supported in such a way that it transfers the loads to the towers and anchorageMajor element is a flexible cable, shaped and supported in such a way that it transfers the loads to the towers and anchorageflexible cableflexible cable This cable is commonly constructed from High Strength wires, either spun in situ or formed from component, spirally formed wire ropes. In either case allowable stresses are high of the order of 600 MPAThis cable is commonly constructed from High Strength wires, either spun in situ or formed from component, spirally formed wire ropes. In either case allowable stresses are high of the order of 600 MPA The deck is hung from the cable by Hangers constructed of high strength ropes in tensionThe deck is hung from the cable by Hangers constructed of high strength ropes in tension As in the long spans the Self-weight of the structures becomes significant, so the use of high strength steel in tension, primarily in cables and secondarily in hangers leads to an economical structure.As in the long spans the Self-weight of the structures becomes significant, so the use of high strength steel in tension, primarily in cables and secondarily in hangers leads to an economical structure. The economy of the cable must be balanced against the cost of the associated anchorage and towers. The anchorage cost may be high where foundation material is poorThe economy of the cable must be balanced against the cost of the associated anchorage and towers. The anchorage cost may be high where foundation material is poor

14. Distinctive Features of Suspension Bridge The main cable is stiffened either by a pair of stiffening trusses or by a system of girders at deck level.The main cable is stiffened either by a pair of stiffening trusses or by a system of girders at deck level. This stiffening system serves to (a) control aerodynamic movements and (b) limit local angle changes in the deck. It may be unnecessary in cases where the dead load is great.This stiffening system serves to (a) control aerodynamic movements and (b) limit local angle changes in the deck. It may be unnecessary in cases where the dead load is great.stiffening system stiffening system The complete structure can be erected without intermediate staging from the groundThe complete structure can be erected without intermediate staging from the ground The main structure is elegant and neatly expresses its function.The main structure is elegant and neatly expresses its function. It is the only alternative for spans over 600m, and it is generally regarded as competitive for spans down to 300m. However, shorter spans have also been built, including some very attractive pedestrian bridgesIt is the only alternative for spans over 600m, and it is generally regarded as competitive for spans down to 300m. However, shorter spans have also been built, including some very attractive pedestrian bridges The height of the main towers can be a disadvantage in some areas; for example, within the approach road for an AIRPORTThe height of the main towers can be a disadvantage in some areas; for example, within the approach road for an AIRPORT

15. Components of a Suspension Bridge Anchor Block: Just looking at the figure we can compare it as a dead man having no function of its own other than its weight. Suspension girder: It is a girder built into a suspension bridge to distribute the loads uniformly among the suspenders and thus to reduce the local deflections under concentrated loads. Suspenders: a vertical hanger in a suspension bridge by which the road is carried on the cablesSuspenders: Tower: Towers transfers compression forces to the foundation through piers. Saddles: A steel block over the towers of a suspension bridge which acts as a bearing surface for the cable passing over it.Saddles: Cables: Members that take tensile forces and transmit it through saddles to towers and rest of the forces to anchorage block.Cables:

16. Suspension Bridge Suspension Bridge

17. Suspension Bridge Suspension Bridge

18. Suspension Bridge

19. The deck of a suspension bridge is usually suspended by vertical hangers, though, some bridges, following the example of the Severn bridge, use inclined ones to increase stability. But the structure is essentially flexible, and great effort must be made to withstand the effects of traffic and wind. If, for example, there is a daily flow of traffic across a bridge to a large city on one side, the live load can be asymmetrical, with more traffic on one side in the morning, and more traffic on the other side in the evening. This produces a periodic torsion, and the bridge needs to be strong enough to resist the possible effects of fatigue.

20. We can list the main parts of Suspension bridge Two towers Suspended structure Two main cables Many hanger cables Two terminal piers Four anchorages

21. 5. ANALYSIS OF INDETERMINATE STRUCTURES BY FORCE METHOD

22. 5.1 ANALYSIS OF INDETERMINATE STRUCTURES BY FORCE METHOD - AN OVERVIEW 5.2 INTRODUCTION 5.3 METHOD OF CONSISTENT DEFORMATION 5.4 INDETERMINATE BEAMS 5.5 INDETRMINATE BEAMS WITH MULTIPLE DEGREES OF INDETERMINACY 5.6 TRUSS STRUCTURES 5.7 TEMPERATURE CHANGES AND FABRICATION ERRORS

23. 5.2 INTRODUCTION 5.2 Introduction While analyzing indeterminate structures, it is necessary to satisfy (force) equilibrium, (displacement) compatibility and force- displacement relationships (a) Force equilibrium is satisfied when the reactive forces hold the structure in stable equilibrium, as the structure is subjected to external loads (b) Displacement compatibility is satisfied when the various segments of the structure fit together without intentional breaks, or overlaps (c) Force-displacement requirements depend on the manner the material of the structure responds to the applied loads, which can be linear/nonlinear/viscous and elastic/inelastic; for our study the behavior is assumed to be linear and elastic

24. 5.2 INTRODUCTION (C ONT ’ D ) Two methods are available to analyze indeterminate structures, depending on whether we satisfy force equilibrium or displacement compatibility conditions - They are: Force method and Displacement Method Force Method satisfies displacement compatibility and force- displacement relationships; it treats the forces as unknowns - Two methods which we will be studying are Method of Consistent Deformation and (Iterative Method of) Moment Distribution Displacement Method satisfies force equilibrium and force- displacement relationships; it treats the displacements as unknowns - Two available methods are Slope Deflection Method and Stiffness (Matrix) method

25. 5.3 METHOD OF CONSISTENT DEFORMATION Solution Procedure: (i) Make the structure determinate, by releasing the extra forces constraining the structure in space (ii) Determine the displacements (or rotations) at the locations of released (constraining) forces (iii) Apply the released (constraining) forces back on the structure (To standardize the procedure, only a unit load of the constraining force is applied in the +ve direction) to produce the same deformation(s) on the structure as in (ii) (iv) Sum up the deformations and equate them to zero at the position(s) of the released (constraining) forces, and calculate the unknown restraining forces Types of Problems to be dealt: (a) Indeterminate beams ; (b) Indeterminate trusses ; and (c) Influence lines for indeterminate structures

26. 5.4 INDETERMINATE BEAMS 5.4.1 Propped Cantilever - Redundant vertical reaction released (i) Propped Cantilever : The structure is indeterminate to the first degree; hence has one unknown in the problem. (ii) In order to solve the problem, release the extra constraint and make the beam a determinate structure. This can be achieved in two different ways, viz., (a) By removing the vertical support at B, and making the beam a cantilever beam (which is a determinate beam); or (b) By releasing the moment constraint at A, and making the structure a simply supported beam (which is once again, a determinate beam).

27. 5.4 INDETERMINATE BEAMS (C ONT ’ D ) (a) Release the vertical support at B: The governing compatibility equation obtained at B is, f BB = displacement per unit load (applied in +ve direction) x y P P B C L/2 L CC BB = + BB RBRB  BB =R B *f BB Applied in +ve direction

28. 5.4 INDETERMINATE BEAM (Cont’d) 5.4.2 Propped cantilever - Redundant support moment released L P L/2 (b) Release the moment constraint at a: A B AA = AB P Primary structure + B A MAMA  A =M A  AA Redundant M A applied

29. To recapitulate on what we have done earlier, I.Structure with single degree of indeterminacy: (a) Remove the redundant to make the structure determinate (primary structure) (b) Apply unit force on the structure, in the direction of the redundant, and find the displacement (c) Apply compatibility at the location of the removed redundant A B RBRB AB  Bo f BB 5.4.3 OVERVIEW OF METHOD OF CONSISTENT DEFORMATION  B0 + f BB  R B = 0 P P

30. 5.5 INDETERMINATE BEAM WITH MULTIPLE DEGREES OF INDETERMINACY (a) Make the structure determinate (by releasing the supports at B, C and D) and determine the deflections at B, C and D in the direction of removed redundants, viz.,  BO,  CO and  DO A BCD E RBRB RCRC RDRD  B0  C0  D0 w/u.l

31. (b) Apply unit loads at B, C and D, in a sequential manner and determine deformations at B, C and D, respectively. A BCD E f BB f CB f DB 1 A BCD E f BC f CC f DC A BCD E f BD f CD f DD 1 1

32. (c ) Establish compatibility conditions at B, C and D  BO + f BB R B + f BC R C + f BD R D = 0  CO + f CB R B + f CC R C + f CD R D = 0  DO + f DB R B + f DC R C + f DD R D = 0

33. 5.4.2 When support settlements occur: Compatibility conditions at B, C and D give the following equations:  BO + f BB R B + f BC R C + f BD R D =  B  CO + f CB R B + f CC R C + f CD R D =  C  DO + f DB R B + f DC R C + f DD R D =  D A BCD E BB CC DD Support settlements w / u. l.

34. 5.5 TRUSS STRUCTURES (a)(a) Remove the redundant member (say AB) and make the structure a primary determinate structure The condition for stability and indeterminacy is: r+m>=<2j, Since, m = 6, r = 3, j = 4, (r + m =) 3 + 6 > (2j =) 2*4 or 9 > 8  i = 1 C 80 kN 60 kN AB D C 80 kN 60 kN AB D 12 Primary structure

35. 5.5 Truss Structures ( Cont’d) (b)Find deformation  ABO along AB:  ABO =  (F 0 u AB L)/AE F 0 = Force in member of the primary structure due to applied load u AB = Forces in members due to unit force applied along AB (c) Determine deformation along AB due to unit load applied along AB: (d) Apply compatibility condition along AB:  ABO +f AB,AB F AB =0 (d) Hence determine F AB

36. (e) Determine the individual member forces in a particular member CE by F CE = F CE0 + u CE F AB where F CE0 = force in CE due to applied loads on primary structure (=F 0 ), and u CE = force in CE due to unit force applied along AB (= u AB )

37. 5.6 TEMPERATURE CHANGES AND FABRICATION ERROR Temperature changes affect the internal forces in a structure Similarly fabrication errors also affect the internal forces in a structure (i) Subject the primary structure to temperature changes and fabrication errors. - Find the deformations in the redundant direction (ii) Reintroduce the removed members back and make the deformation compatible