Spon Press PRESTRESSED CONCRETE DESIGN EUROCODES. University of Glasgow. Department of Civil Engineering. Prabhakara Bhatt LONDON AND NEW YORK

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1 PRESTRESSED CONCRETE DESIGN TO EUROCODES Prabhakara Bhatt Department of Civil Engineering University of Glasgow Spon Press an imprint of Taytor & Francfe LONDON AND NEW YORK

2 CONTENTS Preface xix Basic concepts Introduction Prestressed concrete Economics ofprestressed concrete 4 2. Technology of prestressing Methods of prestressing Pre-tensioning Debonding/blanketing of strands Deflecting/draping/harping of strands Loss of prestress at transfer Transmission length Example of calculation of transmission length Post-tensioning Post-tensioning anchors Loss of prestress at transfer External prestressing Unbonded systems Material properties Properties of concrete Compressive strength of concrete Tensile strength of concrete Defonnational properties Elastic moduli Creep coefficient Shrinkage Stress-strain relationship Parabolic-rectangular relationship Bi-linear relationship Confined concrete Permissible stresses in concrete Prestressing steel Relaxation Maximum Stress at Jacking Long-term loss of prestress References to Eurocode 2 clauses 40

3 viii Prestressed Concrete Design 4. Serviceability limit state design of pre-tensioned beams Design of prestressed concrete structure Beam design based on engineers' theory of bending Sign convention Example of beam designed based on engineer's theory of bending Development of SLS design equations Example of SLS design equations Magnel diagram Choice of prestress and eccentricity Stress check Debonding Choice of prestress and eccentricity at different sections Initial sizing of section Example of preliminary sizing Composite beam section Magnel equations for composite beam Shrinkage stress calculation Example of shrinkage stress calculation Magnel diagrams for a composite beam Choice of prestress and eccentricity at different sections Cracking Thermal stress calculation Heating Cooling Calculation of stresses due to thermal gradients Example of thermal stress calculation Thermal stress calculation: Heating Thermal stress calculation: Cooling Detailing References to Eurocode 1 and Eurocode 2 clauses Bonded post-tensioned structures Post-tensioned beams Cable profile in a post-tensioned beam Example of permitted cable zone Magnel equations Determination of maximum eccentricity Determination of cable zone Detailing of post-tensioned tendons Concept of equivalent loads General equation for equivalent loads General equation for distributed loads for a parabolic profile 91

4 Contents ix Drape of the cable Load balancing Reference to Eurocode 2 clauses Statically indeterminate post-tensioned structures Introduction Primary and secondary moments Prestressing of a propped cantilever Analysis to determine the moment distribution due to prestress Equivalent loads for a cable profile of a single parabola General equation for equivalent loads for a cable profileconsisting of three parabolic segments General equation for equivalent loads for a cable profile consisting of four parabolic segments Alternative profile consisting of three parabolas instead of four parabolas Alternative profile consisting of two parabolas instead of three parabolas Loss of prestress and equivalent loads Fixed end moments Fixed end moments for three-parabola cable profile Fixed end moments for four-parabola cable profile Analysis of a continuous beam for moment distribution due to prestress Distribution of shear force Cable profile consisting of linear variation between supports Determination of prestress and cable profile: Example ofa continuous bridge beam Analysis of the bridge Determination of prestress and eccentricity Refined analysis due to equivalent loads Fixed end moments for three parabola cable profile Fixed end moments for four parabola cable profile Moments at supports for the cable profile Choice of prestress at service Stress check at transfer and service Concordant cable profile Choice of tendon size and location of tendons Equivalent loads and shift in the centroidal axis Shift in the centroidal axis in box girders Equivalent loads and variable second moment of area Thermal stress analysis and continuous structures 139

5 strain strain strain strain PrestressedConcrete Design Thermal stress calculation: heating Thermal stress calculation: cooling Reduction of moment over support in continuous beams References to Eurocode 2 clauses Ultimate bending strength calculations Introduction Stress distribution at different stages of loading Stress - relationship for concrete Rectangular stress block in bending strength calculations Stress - relationship for steel Strain and stress in steel Prestress and pre-strain in steel Strain due to bending in steel Total strain and stress in steel The strain compatibility method Properties ofa trapezium Ultimate moment calculation of a bridge beam Ultimate moment calculation of a composite bridge beam Use of additional unstressed steel Stress-strain relationship for unstressed reinforcing 7.13 Example of ultimate moment calculation with stressed and steel 163 unstressed steels Calculation of Mu using tabular values Calculation of Mu for statically indeterminate beams Reference to Eurocode 2 clauses Analysis of cracked sections Introduction Cracked section analysis Cracked section analysis ofa double T-beam Stress relationship for concrete Stress relationship for steel Cracked section analysis Partially prestressed beam Composite beam Magnel diagram for composite beam Ultimate shear and torsional strength calculations Introduction Shear capacity of a section without shear reinforcement and uncracked in flexure 195

6 Contents xi Example of calculation of shear capacity of a section without shear reinforcement and uncracked in bending Checking for start of cracked section Shear capacity of a section without shear reinforcement and cracked in bending Example of calculation of shear capacity of a section without shear reinforcement and cracked in bending Design of shear reinforcement Derivation of equations (9.7) and (9.11) Procedure for shear link design Design of a beam not needing design shear reinforcement Design of a beam needing design shear reinforcement Shear capacity of a composite beam Effective web width in the presence of ducts Interface shear between web and flange in T-sections Example of reinforcement calculation for interface shear between web and flange Interface shear between precast beam and cast in-situ slab Design for torsion Spacing of torsion reinforcement Design for combined shear force and torsion Warping torsion References to Eurocode 2 clauses Calculation of crack widths Introduction Exposure classes Recommended values of maximum crack width Minimum steel areas Example of minimum steel area calculation Calculation of crack width, wk Crack spacing, Sr, max Example of crack width and spacing calculation Example of a partially prestressed beam Example of minimum steel area calculation Example of width and spacing of crack Control of cracking without direct calculation References to Eurocode 2 clauses Loss of prestress Introduction Immediate loss of prestress 241

7 Prestressed Concrete Design Elastic loss in pre-tensioned beams Example of elastic loss calculation Elastic loss in post-tensioned beams Loss of prestress due to friction and wobble Derivation of loss of prestress due to friction Example of calculation of loss of prestress due to friction and wobble Calculation of 6 for different profiles Loss due to draw-in of wedges Example of loss of prestress due to draw-in Loss of prestress due to creep, shrinkage and relaxation Example of final loss calculation References to Eurocode 2 clauses 259 Design of slabs Introduction Typical beam and slab depths Effective span of slabs for different support conditions One-way spanning slabs Design of a one-way spanning slab Analysis for applied loading Choice of prestress Calculation of losses Calculation of correct equivalent loads Calculation of moment distribution at service Calculation of stress distribution at service Calculation of stress distribution at transfer Edge-supported two-way spanning slab Design of a two-way spanning slab Flat slabs Methods of analysis of flat slabs Example of the design of flat slab Finite element analysis of flat slab Results of anal ys is for dead load Results of analysis for dead plus live load pattern Results of analysis for dead plus live load pattern Results of analysis for dead plus live load pattern Results of analysis for dead plus live load pattern Finite element analysis of a strip of flat slab Results of analysis for dead load Results of analysis for dead plus live load partem Results of analysis for dead plus live load pattern Results of analysis for dead plus live load pattern Results of analysis for dead plus live load pattern 4 321

8 Contents xiii Comparison between the results of analysis of full slab and a strip of slab Eurocode 2 recommendations for equivalent frame analysis Grillage analysis for irregular column layout Example of design of flat slab-frame Results of analysis of slab-f rame model Moment distribution due to prestress Cable profile Calculation of loss of prestress Calculation of loss due to friction and wobble per cable Calculation of loss due to wedge draw-in Calculation of prestress at service Determination of number of cables Fixed end moments due patch loads and concentrated force and couple Equivalent loads and fixed end moments Equivalent loads and fixed end moments at transfer Equivalent loads and fixed end moments at service Moment distribution due to equivalent loads at transfer Moment distribution due to equivalent Moment distribution due to equivalent loads loads at service 340 and external loads at transfer Moment distribution due to equivalent loads and external loads at service Stress distribution in the slab at transfer and service stages Moment in end columns Ultimate limit state moment calculations Moment envelopes Parasitic moments Parasitic moments: example Ultimate moment capacity Detailing of steel Eurocode 2 recommendations for detailing of steel References to Eurocode 2 clauses Design for punching shear Punching shear failure Punching shear stress calculation Critical shear perimeter Effect of holes near the column Example Columns with capitals 364

9 xiv Prestressed Concrete Design 13.6 Calculation of punching shear stress ved under the action of a moment Punching shear stress under shear force and moment acting simultaneously Special cases of shear force and moment acting together Punching shear stress checks Example of punching shear capacity design Reference to Eurocode 2 clauses Loading on buildings Introduction Limit states Classification of actions Characteristic values of actions Design values of actions Combination of actions Combination of actions for ULS Values of y factors Examples of the use of y factors Combination of actions for SLS References to Eurocode 1 clauses Loading on bridges Introduction Notional Lanes Load models Load Model Load Model Load Model Load Model Dispersal of concentrated load Horizontal forces Breaking forces Centrifugal forces Loads on footways, cycle tracks and foot bridges Groups of traffic loads Combinations of actions for ULS Values of y factors Values of v / factors for road bridges Combinations of actions for SLS References to Eurocode 1 clauses 400

10 Contents 16. Analysis and design of bridge decks Introduction Balanced Cantilever Construction Methods of analysis Grillage analysis Aspects of behaviour ignored in grillage analysis Edge stiffening Torsional constant Torsional constant of solid sections Torsional constant of thin-walled closed hollow sections Example of analysis of a beam and slab deck Bending properties of precast beam Section properties of interior composite beam Section properties of end composite beam Torsion constant for composite beam Alternative expressions for approximate value of J for rectangular cross sections Section properties of transverse beams Material properties Calculation of live loads and bending moment distribution in beam elements: SLS Stresses due to shrinkage of slab Thermal stresses in the composite beam Thermal stresses: heating Thermal stresses: cooling Stress distribution at SLS due to external loads Magnel diagrams Stress checks Calculation of live loads and bending moment distribution in beam elements: ULS Self-weight moments Ultimate moment capacity: Mid-span section Ultimate shear force Analysis to determine maximum shear force along the span: Cases Analysis to determine maximum shear force along the span: Cases Summary of results Design of shear reinforcement Design ofa post-tensioned box girder bridge Calculation of moments at SLS Thermal stresses: Heating Thermal stresses: Cooling Determination of prestress and eccentricity Stress calculation at SLS 471

11 17. Lower bound approaches to design at ultimate limit state 493 xvi Prestressed Concrete Design Calculation ofmoments at ULS Calculation of moment capacity at ULS Calculation of shear force at ULS Calculation of twisting moment at ULS ODesign of shear and torsional reinforcement Longitudinal reinforcement to resist torsion Stress analysis of the deck Eurocode 2 rules for reinforcement at anchorages External and internal tendons: A comparison References to Eurocode 2 clauses Introduction Theory of Plasticity In-plane stresses Examples of reinforcement calculations Presence of prestressing cables Designs for a combination of in-plane and flexural forces Example of design for a combination of in-plane and flexural forces Criterion for cracking Out-of-plane shear Strut and tie method of design B and D Regions Saint Venant's principle An example of strut-tie modelling Design of struts Types of nodal zones Correct layout of struts and ties Correct layout of struts and ties: deep beam Correct layout of struts and ties: corbel Code recommendation for design of corbel Correct layout of struts and ties: half-joint Correct layout of struts and ties: end-block Reinforcement at frame corners Reference to Eurocode 2 clauses Design for earthquake resistance Introduction Ductility Types of structural systems 536

12 Contents 18.4 Behaviour factor, q Ductility classes A brief introduction to structural dynamics Single-degree-of-freedom system Multi-degree-of-freedom system Response to an acceleration of the base Vibration of an undamped free multi-degree-of-freedom system Calculation of eigenvalues Eigenvectors of [K-o2 M] Properties of eigenvectors Mode superposition: undamped forced response Mode superposition: damped forced response Mass participation factors and effective mass Mass participation factors: Example Response acceleration spectrum Design elastic response acceleration spectrum Elastic design spectrum: Eurocode Methods of analysis Lateral force method of analysis Lateral force method: example Modal response spectrum method of analysis Displacement spectrum Combining modal values: SRSS and CQC rules Rayleigh damping 'Resultant'storey level displacements 'Resultant'storey level forces Combination ofseismic action with other actions Basic principles of conceptual design Detailing for local ductility: beams Detailing for local ductility: columns Design shear force in beams and columns Design provisions for ductile walls Reference to Eurocode 8 clauses Miscellaneous topics Introduction Unbonded design Design of a post-tensioned box girder Calculation of live loadings at SLS Calculation oftotal loads at SLS Calculation of bendingm moments at SLS Bending stresses at SLS Thermal stresses: heating and cooling 575

13 xviii Prestressed Concrete Design 19.4 Determination of prestress Cracking moment Ultimate moment capacity Ultimate shear capacity Calculation of deflection References to Eurocode 2 clauses References 587 Index 591

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