The discrete rigid body rotation of reinforced concrete ...€¦ · In Chapter 5the rigid body rotation model is described in detail and an iterative solution , technique presented

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The discrete rigid body rotation of reinforced concrete beams using

partial interaction and shear friction theory

Matthew Haskett

B.E. Civil and Structural Engineering (Hons)

A thesis submitted for the degree of doctor of philosophy

Department of Civil, Environment and Mining Engineering

The University of Adelaide Australia July 2010

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STATEMENT OF ORIGINALITY This work contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution. To the best of my knowledge and belief it contains no material previously published or written by another person, except where due reference has been made in the text. The author acknowledges that copyright of published works contained within this thesis resides with the copyright holder(s) of those works. I give consent to this copy of my thesis, when deposited in the University Library, being available for loan and photocopying. ___________________________________ July 2010 Matthew Haskett

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ACKNOWLEDGEMENTS The content of this thesis owes much to the research passion, knowledge and guidance of Professor Deric Oehlers. I thank him deeply for always being available to provide thoughtful insight, recommend alternative solutions, and always being available to discuss any research issue, even when overseas on holiday. I would also like to thank all the other academics who have helped me with this thesis in their fields of expertise: Associate Professor Rudolf Seracino, Dr. Mohamed Ali, Professor Mike Griffith and Dr. Chengqing Wu.

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INTRODUCTION In reinforced concrete members, two types of deformations exist: deformations due to curvature distribution and deformation due to discrete localised conditions such as cracks. The latter is the subject of this thesis, which presents a new approach for evaluating the discrete moment-rotation relationship of reinforced concrete members. This thesis is a collection of submitted, accepted or published papers from internationally recognised Journals, where the titles of Chapters 1 through 13 reflect the titles of the Journal papers. Each chapter takes the following format: the key theory and results from each journal paper are presented in a short synopsis, after which the journal paper is presented in full. This provides the reader, if desired, with the ability to understand the research in full by only reading the synopses of each chapter. In Chapter 1, the peripheral areas of shear friction theory, partial interaction theory and rigid body displacement are combined to quantify the moment-rotation response of any reinforced concrete member. It is discussed how partial interaction theory is used to model the behaviour of the reinforcement, shear friction theory the behaviour of the concrete, and that both these behaviours are combined through a rigid body displacement profile. This rigid body rotation approach is a structural mechanics model, and hence can be used to quantify the moment-rotation response of any reinforced concrete member (the subject of this thesis), and amongst other things the shear capacity of a concrete member and the influence of confinement on member behaviour (not covered in or the subject of this thesis). Importantly, closed form solution can be developed for all failure mechanisms, and flexural failure closed form solutions are presented in this thesis. In Chapters 2-4, the partial interaction behaviour of steel reinforcing bars and externally bonded and near surface mounted FRP plates is described in detail. Specifically, Chapters 2 and 3 quantify the bond characteristics of steel reinforcing bars and embedded near surface mounted (NSM) fibre reinforced polymer (FRP) plates. Previously published pull test data is analysed to determine the local bond stress-slip relationship of deformed steel reinforcing bars, and experimental testing of embedded NSM FRP plates performed and analysed to determine the influence of embedment on the bond behaviour of embedded NSM FRP plates. In Chapter 4, partial interaction mathematical expressions are developed to model the elasto-plastic load-slip (P-∆) behaviour of steel reinforcing bars. These mathematical expressions are later used to model the behaviour of externally bonded (EB) and NSM steel plates. These mathematical load-slip expressions are a critical component of the moment-rotation analysis technique. In Chapter 5, the rigid body rotation model is described in detail and an iterative solution technique presented. The predicted moment-rotation behaviour of reinforced concrete beams is compared to that obtained experimentally with excellent accuracy. In Chapter 6, a closed form solution technique is presented for determining the moment-rotation response of an unplated or plated reinforced concrete member. This is a more user friendly analysis technique and can be used directly to determine the moment and rotation

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at failure without developing the entire moment-rotation response from the start of loading to failure. In Chapter 7, the influence of bond on the rotation capacity of both unplated, externally bonded and near surface mounted plated reinforced concrete members is assessed, where it is shown that bond is a critical component of ductility. In Chapters 8-11, an application of the moment-rotation model is presented. Moment redistribution expressions are derived for propped cantilevers and continuous members from elementary structural mechanics, and it is shown that the moment redistribution capacity of a member is proportional to the moment and rotation capacity of the section. Various examples of moment redistribution are presented, and finally a method to design for moment redistribution is presented where it is shown that the non-hinging region needs to remain elastic for the hinge to redistribute its maximum moment. In Chapters 12 and 13, the shear friction behaviour of concrete is examined in detail and a method for extracting the shear friction parameters of initially uncracked hydrostatically confined concrete is presented. A generic expression for the shear friction parameters of initially uncracked concrete is developed, where the shear stress is a function of the normal stress across and displacement of the sliding plane, and the compressive strength of concrete. A previously developed expression for the shear friction parameters of initially cracked concrete is modified so that the shear stress is expressed as a function of the normal stress across and displacement of the sliding plane and the compressive strength of concrete. Expressing the shear friction parameters in this way allows bounds to be developed using Mattock’s shear stress limits. These bounds for both initially cracked and uncracked concrete are expressed mathematically for the generic shear friction parameters. The development of these shear friction parameters and bounds will allow the rigid body rotation model to consider sliding failure of concrete, and quantify the effect of confinement through stirrups or FRP wrapping on the moment-rotation response.

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TABLE OF CONTENTS STATEMENT OF ORIGINALITY .............................................................................. ii ACKNOWLEDGEMENTS ......................................................................................... iii INTRODUCTION ........................................................................................................ iv TABLE OF CONTENTS ............................................................................................. vi LIST OF FIGURES .................................................................................................... viii NOTATION .................................................................................................................. x CHAPTER 1:................................................................................................................. 1 SYNOPSIS- A GENERIC UNIFIED REINFORCED CONCRETE MODEL ............ 1 JOURNAL PAPER - A GENERIC UNIFIED REINFORCED CONCRETE MODEL4 CHAPTER 2:............................................................................................................... 39 SYNOPSIS - LOCAL AND GLOBAL BOND CHARACTERISTICS OF STEEL

REINFORCING BARS ........................................................................... 39 JOURNAL PAPER - LOCAL AND GLOBAL BOND CHARACTERISTICS OF STEEL

REINFORCING BARS ........................................................................... 46 CHAPTER 3:............................................................................................................... 62 SYNOPSIS - EMBEDDING NSM FRP PLATES FOR IMPROVED IC DEBONDING

RESISTANCE ......................................................................................... 62 JOURNAL PAPER - EMBEDDING NSM FRP PLATES FOR IMPROVED IC

DEBONDING RESISTANCE ................................................................ 66 CHAPTER 4:............................................................................................................... 87 SYNOPSIS - YIELD PENETRATION HINGE ROTATION IN REINFORCED

CONCRETE BEAMS ............................................................................. 87 JOURNAL PAPER - YIELD PENETRATION HINGE ROTATION IN REINFORCED

CONCRETE BEAMS ............................................................................. 93 CHAPTER 5:............................................................................................................. 110 SYNOPSIS - RIGID BODY MOMENT-ROTATION MECHANISM FOR

REINFORCED CONCRETE BEAM HINGES .................................... 110 JOUNAL PAPER - RIGID BODY MOMENT-ROTATION MECHANISM FOR

REINFORCED CONCRETE BEAM HINGES .................................... 116 CHAPTER 6:............................................................................................................. 142 SYNOPSIS - THE DISCRETE ROTATION IN REINFORCED CONCRETE BEAMS

............................................................................................................... 142 JOURNAL PAPER – THE DISCRETE ROTATION IN REINFORCED CONCRETE

BEAMS ................................................................................................. 147 CHAPTER 7:............................................................................................................. 166 SYNOPSIS - INFLUENCE OF BOND ON THE HINGE ROTATION OF FRP PLATED

BEAMS ................................................................................................. 166 JOURNAL PAPER – INFLUENCE OF BOND ON THE HINGE ROTATION OF FRP

PLATED BEAMS ................................................................................. 171 CHAPTER 8:............................................................................................................. 187 SYNOPSIS - ANALYSIS OF MOMENT REDISTRIBUTION IN FRP PLATED RC

BEAMS ................................................................................................. 187 JOURNAL PAPER – ANALYSIS OF MOMENT REDISTRIBUTION IN FRP PLATED

RC BEAMS ........................................................................................... 192

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CHAPTER 9:............................................................................................................. 216 SYNOPSIS - MOMENT REDISTRIBUTION IN REINFORCED CONCRETE

MEMBERS ........................................................................................... 216 JOURNAL PAPER – MOMENT REDISTRIBUTION IN REINFORCED CONCRETE

MEMBERS ........................................................................................... 221 CHAPTER 10:........................................................................................................... 243 SYNOPSIS - DESIGN FOR MOMENT REDISTRIBUTION IN RC BEAMS

RETROFITTED WITH STEEL PLATES ............................................ 243 JOURNAL PAPER – DESIGN FOR MOMENT REDISTRIBUTION IN RC BEAMS

RETROFITTED WITH STEEL PLATES ............................................ 251 CHAPTER 11:........................................................................................................... 273 SYNOPSIS - DESIGN FOR MOMENT REDISTRIBUTION IN FRP PLATED RC

MEMBERS ........................................................................................... 273 JOURNAL PAPER – DESIGN FOR MOMENT REDISTRIBUTION IN FRP PLATED

RC BEAMS ........................................................................................... 278 CHAPTER 12:........................................................................................................... 302 SYNOPSIS - EVALUATING THE SHEAR-FRICTION RESISTANCE ACROSS

SLIDING PLANES IN CONCRETE .................................................... 302 JOURNAL PAPER – EVALUATING THE SHEAR-FRICTION RESISTANCE

ACROSS SLIDING PLANES IN CONCRETE ................................... 309 CHAPTER 13:........................................................................................................... 331 SYNOPSIS - THE SHEAR-FRICTION AGGREGATE INTERLOCK RESISTANCE

ACROSS SLIDING PLANES IN CONCRETE ................................... 331 JOURNAL PAPER – THE SHEAR-FRICTION AGGREGATE INTERLOCK

RESISTANCE ACROSS SLIDING PLANES IN CONCRETE .......... 337 CHAPTER 14:........................................................................................................... 369 CONCLUSIONS AND RECOMMENDATIONS ................................................... 369

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LIST OF FIGURES Figure A (Figure 10 in Chapter 1) - Rigid body deformations due to flexure ..................... 2Figure B (Figure 3 in Chapter 2) - Bi-linear and linear descending local τ−δ relationships

........................................................................................................................... 39Figure C (Figure 5 in Chapter 2) – Numerical simulation technique ................................ 41Figure D (Figure 9 in Chapter 2) - Accuracy of various τ−δ relationships ........................ 42Figure E (Figure 8 in Chapter 2) - Theoretical τ−δ relationship for deformed steel

reinforcing bars ................................................................................................. 43Figure F (Figure 6 in Chapter 3) – Experimental P-∆ response for Test Series 2 ............. 62Figure G (Figure 10 in Chapter 3) – Influence of cover on 2Gf ......................................... 63Figure H – Failure planes (Lper) and aspect ratios of various reinforcement types ............ 64Figure I (Figure 5 in Chapter 4) – Comparison of σ−∆ response ...................................... 87Figure J – Idealised stress-strain relationship of steel ........................................................ 87Figure K (Figure 13 in Chapter 4) – Comparison of rotations due to yield penetration .... 90Figure L (Figure 7 in Chapter 5) – Moment-rotation analysis ......................................... 110Figure M (Figure 3 in Chapter 5) – Stress-strain relationship of concrete under

compression ..................................................................................................... 111Figure N (Figure 10 in Chapter 5) – Specimen C1 moment-rotation .............................. 112Figure O (Figure 18 in Chapter 5) – Variation in sslide with stirrup confinement ............ 113Figure P (Figure 9 in Chapter 7) – Moment-rotation response for unplated and plated

150mm deep RC slab ...................................................................................... 167Figure Q (Figure 10 in Chapter 7) – M-θ for 600mm deep section ................................. 168Figure R (Figure 6 in Chapter 8) - Hinge at supports: continuous beam with UDL ........ 188Figure S (Figure 12 in Chapter 8) – Comparison of moment redistribution capacities ... 190Figure T (Figure 13 in Chapter 9) KMR response with varying bar diameter ................... 217Figure U (Figure 14 in Chapter 9) - KMR response with interface bond strength for 400

mm beam ......................................................................................................... 218Figure V (Figure 17 in Chapter 9) - KMR response with neutral axis depth factor for 400

mm beam ......................................................................................................... 219Figure W (Figure 1 in Chapter 10)– Hinge at midspan: propped cantilever with point load

......................................................................................................................... 243Figure X (Figure 8 in Chapter 10) – Slip and bond stress distribution for various loads 247Figure Y (Figure 9 in Chapter 10) – Influence of plate dimension on MR of EB plates . 249Figure Z (Figure 4 in Chapter 11) – Various moment-rotation responses (not to scale) . 273Figure AA (Figure 11 in Chapter 11) - Relative rotation and moment contributions for

both regions for a given static moment ........................................................... 275Figure BB (Figure 6 in Chapter 12) – Equilibrium of a wedge and cylinder deformations

......................................................................................................................... 303Figure CC (Figure 7 in Chapter 12) – Resolutions of deformations ................................ 304Figure DD (Figure 5 in Chapter 12) – Idealised concrete axial stress-strain relationship

......................................................................................................................... 305Figure EE (Figure 8 in Chapter 12) – Shear friction parameters for initially uncracked

concrete: fco 50MPa up to ssoft=70%σresidual ..................................................... 306Figure FF (Figure 2 in Chapter 12) – Normalised shear transfer as a function of normal

stress for initially cracked planes .................................................................... 307

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Figure GG (Figure 5 in Chapter 13) – Influence of normal confining stress on shear stress ......................................................................................................................... 331

Figure HH (Figure 6 in Chapter 13) – Influence of confinement on -mτ ......................... 332Figure II (Figure 7 in Chapter 13) – Influence of confinement on cτ .............................. 333

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NOTATION A cross sectional area ael development length required to yield the steel reinforcement afract additional embedment length required to increase the load in the

reinforcement from the yield to fracture Arebar cross sectional area of a reinforcing bar B bond force c cohesive component of Mohr-Coulomb failure plane c cover cτ shear friction parameter d depth of FRP or steel plate dasc depth of the non-softening region of concrete db diameter of reinforcing bar dprism diameter of a concrete prism drebar distance from the crack apex to the location of the reinforcing bars ds/dx slip strain dsoft depth of the softening region of concrete E elastic modulus Ec Elastic modulus of concrete EI flexural rigidity Es Young’s modulus of steel Esh strain hardening modulus of steel fc compressive strength of concrete fco unconfined compressive strength of concrete fy yield stress Gf interfacial fracture energy of bond stress-slip relationship hcr crack height in rigid body rotation model hcr crack separation across a sliding plane hplate crack width at the location of the plate hrebar crack width at the location of the reinforcing bar hsoft vertical displacement due to the formation of a softening wedge kb bond stress factor KMR moment redistribution factor ku neutral axis parameter L span length Ldeb length over which debonding can occur without a drop in axial load Lhinge plastic hinge length Lper perimeter of failure plane of axial reinforcing Lprism length of a concrete prism Lsoft length of the softening region of concrete m frictional component of Mohr-Coulomb failure plane M moment Mcap moment capacity Mh moment at the support in a continuous member Mhog moment at support in a continuous member

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Ms moment at mid-span in a continuous member Mst static moment Myield moment at yield of steel mτ shear friction parameter P point load Pasc force in the non-softening region of concrete Pcc total compressive force in the concrete PIC load at which intermediate crack debonding commences Pplate axial force in a plate Pr force in the reinforcement Preinf-bar axial force in a reinforcing bar Psoft force in the softening region of concrete Rpl dimensionless parameter that indicates if yield occurs before

debonding in EB and NSM plates s slip sp stirrup spacing sslide displacement of the softening wedge at which uncontrollable

sliding occurs ssoft displacement of the softening region of concrete vc Poisson’s ratio of concrete w applied uniformly distributed load wb width of a reinforced concrete beam %c volumetric percentage weight loss due to corrosion α angle the softening wedge of concrete forms ∆ global displacement ∆deb(end) displacement at the end of debonding allowing for localised

debonding ∆fract displacement (slip) at fracture of axial reinforcement ∆L displacement (slip) of axial reinforcement at the loaded end from a

pull test ∆plate slip of a plate (steel or FRP) across a crack front ∆rebar slip of a reinforcing bar across a crack front ∆sliding slip at sliding failure ∆w proportional increase in applied load due to moment redistribution ∆yield slip at which steel reinforcement yields ∆yp-ult slip at fracture of the reinforcing bar χfract curvature at bar fracture δ local displacement (slip) δ1 slip at which the peak bond stress is first achieved in local bond

stress-slip relationship δaxial axial displacement in a concrete prism under hydrostatic

confinement (δlat)sm lateral displacement of wedge in a prism δmax peak slip from the bond stress-slip relationship ε strain

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εaxial axial strain in a concrete prism εc strain in concrete εcone strain in the “cone region” of a concrete prism εfract strain at bar fracture εIC strain at commencement of debonding εlat lateral strain in a concrete prism εpk concrete strain at the achievement of the peak compressive strength εr strain in the reinforcement εwedge strain in the softening wedge of concrete ϕf aspect ratio of failure plane of axial reinforcement λel elastic partial interaction parameter λsh plastic partial interaction parameter θ rotation θcap rotation capacity θemp empirical rotation from plastic hinge length θfract-limit rotation at fracture of reinforcing bar θh rotation at support θs rotation at midspan θtotal total rotation in a continuous member due to moment redistribution θyield rotation at yield of steel κ initial stiffness of the bond stress-slip relationship σ stress σlat lateral confinement σN stress normal to a sliding plane σsoft softening stress σresidual residual stress in concrete τ shear stress τmax peak bond stress from the bond stress-slip relationship τN shear stress along a sliding plane

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CHAPTER 1:

SYNOPSIS- A GENERIC UNIFIED REINFORCED CONCRETE MODEL

The ability of a reinforced concrete beam to rotate is extremely important at both the serviceability and ultimate limit states as it affects deflection, moment redistribution and the absorption of energy. The rotation in a reinforced concrete beam can be considered to have two components: the continuous rotation which occurs in the homogenous portions of the beam and which can be determined by straightforward integration of the curvature; and the discrete rotation due to the rigid body displacement at the location of cracks. The region where this discrete rotation occurs is often referred to as the “plastic hinge”, and has historically been very difficult to quantify. The behaviour of this “hinge” region is the subject of this chapter, and the thesis in general. Most of the current approaches for determining the moment-rotation relationship, evaluating the shear capacity, and the effect of lateral confinement on reinforced concrete members are empirically based. The current approach for determining the moment-rotation response of reinforced concrete members is a strain based analysis and determines a curvature at failure from an assumed linear strain profile. It assumes full interaction between the axial reinforcement and the adjacent concrete, and hence cannot allow for the formation of flexural cracks. The rotation at failure is simply the curvature at failure integrated over an empirically derived hinge length. This curvature approach, which is often referred to as a “curvature hinge length” approach in this thesis, cannot allow for the formation of flexural cracks since it is a full interaction analysis, and requires the use of empirically derived hinge lengths to convert a curvature to a rotation. These hinge length expressions vary widely, and do not provide good accuracy from outside the bounds that they were developed. Thus, a new approach that is independent of hinge length and can allow for the partial interaction behaviour of the reinforcement is required to address the deficiencies in the current empirical approach and to determine the moment rotation response of reinforced concrete members. The rigid body rotation analysis combines partial interaction and shear friction theory with a rigid body rotation. The bond characteristics (τ-δ relationship) of the reinforcement are required, and the global load-slip (P-∆) relationship, which is a function of the material parameters and the bond characteristics, must also be known. The softening behaviour of the concrete is also required since softening wedges form in the concrete to accommodate the rigid body rotation induced by the cracking. This softening behaviour is modelled using shear friction theory. The idealised rigid body deformation due to flexural cracking is shown in Figure A for a reinforced concrete beam with both externally bonded plates and reinforcing bars. Slip of the reinforcement relative to the concrete, Δrebar and Δplate, allows the crack to open and the crack width to increase from zero to hrebar and hplate. The corresponding load in the bar and plate, Preinf-bar and Pplate, for the crack width is evaluated using partial interaction

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theory and is a function of the material properties and the bond characteristics (τ−δ relationship) of the reinforcement. The rotation due to slip of the reinforcement must also be accommodated in the concrete. This rigid body rotation model models this deformation, or contraction of the concrete, through the formation of softening wedges, shown shaded in Figure A, where the softening deformation is proportional to the rigid body rotation θ. Ultimately, these two behaviours, the slip of reinforcement and the corresponding contraction of concrete in the softening wedge, together allow a discrete rotation θ.

∆rebar θrigid body rotation

concrete softening zone crack

face

rigid body rotation

wedge

ssoft

hrebar

reinforcing bar

externally bonded plate

hplate

∆plate

Preinf-rebar

Preinf-plate

hcr

Figure A (Figure 10 in Chapter 1) - Rigid body deformations due to flexure

Failure of the reinforced concrete member occurs when one of two limits is achieved: the slip of either of the reinforcements (∆reinf-bar or ∆plate) exceeds the slip capacity of the bar or plate (as a lower limit debonding failure at δmax or fracture at ∆fract and which are both explained later), or the displacement of the softening wedge (ssoft in Figure A) exceeds the sliding capacity of the concrete. These limits to rotation are explored in detail in Chapters 4, 5 and 6. This rigid body rotation model has the ability to explain many aspects of reinforced concrete behaviour: the reason the neutral axis depth factor will never truly quantify the moment redistribution because the approach is part of a family of curves; how compression wedges that form in flexural members as well as in compression members are not an illusion but a shear friction mechanism that can be quantified allowing the residual strength of confined concrete to be quantified; how stirrups increase the flexural ductility; and how FRP wrap does not increase the flexural strength but increases the ductility

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Statement of Authorship A generic unified reinforced concrete model Deric John Oehlers1, Mohamed Ali M.S. 2, Michael C. Griffith3, Matthew Haskett4 and Wade Lucas5 1Professor Deric J. Oehlers School of Civil, Environmental and Mining Engineering University of Adelaide Wrote the final manuscript, and supervised development of the model SIGNED____________________ 2Dr. Mohamed Ali M.S. Senior Research Associate, School of Civil, Environmental and Mining Engineering University of Adelaide Assisted in manuscript preparation and review 3Associate Professor Michael C. Griffith School of Civil, Environmental and Mining Engineering University of Adelaide Manuscript review 4Mr. Matthew Haskett PhD student School of Civil, Environmental and Mining Engineering University of Adelaide Developed the flexural rigid body rotation model SIGNED____________________ 5Mr. Wade Lucas PhD student School of Civil, Environmental and Mining Engineering University of Adelaide Modified the flexural rigid body rotation model for shear Accepted Proceedings ICE, Structures and Buildings 2/06/10

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A NOTE:

This publication is included on pages 4-38 in the print copy of the thesis held in the University of Adelaide Library.

Oehlers, D.J., Mohamed Ali, M.S., Griffith, M.C., Haskett, M. & Lucas, W. (2010) A generic unified reinforced concrete model. Proceedings of the ICE - Structures and Buildings, accepted for print, June 2010,

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CHAPTER 2:

SYNOPSIS - LOCAL AND GLOBAL BOND CHARACTERISTICS OF STEEL REINFORCING BARS

In Chapter 1, the rigid body rotation model developed over the course of this research was presented, where it was explained that two distinct areas of material behaviour are required for this model: the concrete behaviour is modelled by shear friction theory and the reinforcement behaviour by partial interaction theory. In this chapter, the local bond characteristics (τ−δ relationship) of ribbed reinforcing bars are developed from the analysis of previously published experimental load-slip (P-∆) pull test responses. In determining the bond stress-slip relationship of a perfectly elastic material like FRP, pull tests are generally conducted on specimens with long embedment lengths. This long embedment length allows the maximum debonding load of the perfectly elastic material to be achieved, where it has been shown mathematically that the debonding load is a function of the material parameters: the contact area of the reinforcement with the concrete (Lper), the elastic modulus of the reinforcement (E), the cross sectional area of the reinforcement (A) and the interfacial fracture energy of the local bond characteristics (Gf). The interfacial fracture energy is shown shaded in Figure B for a linear descending τ−δ relationship.

Figure B (Figure 3 in Chapter 2) - Bi-linear and linear descending local τ−δ relationships

Mathematically, the debonding load for a perfectly elastic material like FRP is:

EALδτEALG2P permaxmaxperfIC == Equation 1 Knowing the material properties E and A and Lper, , the interfacial fracture energy Gf can be determined from the experimental IC debonding failure load. Hence, when full embedment is provided the term τmaxδmax, can be determined from Equation 1 and the experimental failure load PIC. However, the relative values of τmax and δmax cannot be

Linear descending model

τ

δ δ1

Interfacial fracture energy, Gf

Bi-linear model

κ1

κ2

τmax

δmax

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determined this way. The slip δmax in Figure B corresponds to the experimental loaded end slip (∆L) at the first attainment of the IC debonding load. Thus, the value of τmax can be determined. Hence, for a perfectly elastic material like FRP the magnitudes of τmax and δmax from the local bond stress-slip relationship shown in Figure B can be determined directly through pull tests with full embedment. The shape of the bond stress-slip relationship, that is the relative stiffness κ in Figure B, can be determined from the stiffness of the experimental load-slip P-Δ response. Conversely, accurate bond stress-slip relationships for steel reinforcing bars have not been developed for various reasons: encased steel reinforcing bars are difficult to strain gauge and hence to directly measure the bond stress, the presence of yielding before debonding significantly alters the behaviour of the reinforcing bar, and the significant bond capacity of deformed bars has not necessitated the development of accurate expressions. To prevent yielding, researchers often conduct pull tests on specimens with short embedment length. Therefore, the method used to determine the bond characteristics of a perfectly elastic material like FRP can not be repeated to determine the bond stress-slip relationship of steel reinforcement. A different approach is required. A partial interaction numerical simulation technique is required to determine the theoretical global load-slip (P-∆) response for varying material characteristics: shape of reinforcing, size of reinforcing, embedment length, and most importantly the bond stress-slip relationships. The theoretical load-slip response from this numerical simulation is then subsequently compared to the experimentally recorded response and the bond stress-slip relationship continually modified until the theoretical and experimental load-slip responses are identical. When this occurs, we know the assumed bond stress-slip relationship is appropriate. The numerical simulation works in the following manner, and is shown in Figure C:

1. A strain is fixed at the loaded end Position 0, ε(0), as shown in Figure C. Hence the plate force P(0) from the material properties.

2. Corresponding to this fixed strain ε(0) and corresponding load P(0) at Position 0, a slip at the loaded end Position 0 is assumed or guessed, i.e. s(0) = ∆(0) and the following iterative routine is used to find ∆(0) for P(0).

3. As the segment lengths are made very small, the slip is assumed constant over the segment. Hence the bond stress τ(0) which can be derived from the local bond characteristics such as in Figures 2 and 3 is also constant.

4. The bond force acting over the first segment length is B(0) = τ(0)Lperdx. 5. Hence the load in the reinforcing bar at the end of the first segment is P(1)=P(0)-

B(0).

6. The corresponding strain in the plate is ( ) ( )( )pAE

1P1ε = where EAp is the axial

rigidity of the reinforcement and the corresponding strain in the concrete at the

end of the first segment is ( ) ( )( )c

c AE1P1ε −= .

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7. Hence, the slip strain is )0(ε)0(εdx

)0(dsc−= .

8. By integration, the change in slip over the first segment is ( ) dxdx

)0(ds0sΔ ∫= .

9. Therefore, the slip at the beginning of the second segment is s(1) = s(0) – Δs(0). 10. The numerical procedure is repeated over the subsequent segments until the

known boundary conditions are attained, for example the slip strain and the slip must both be equal to zero at the same position for the initial guess of the loaded end slip ∆L.

Figure C (Figure 5 in Chapter 2) – Numerical simulation technique

An example of the use of the numerical simulation technique in Figure C is presented in Figure D. The experimental and numerical simulation P-∆ responses are clearly labelled in the legend. The experimental and theoretical P-∆ responses of deformed steel reinforcing bars are compared in Figure D, where short embedment length has been provided to prevent the occurrence of yielding.

∫=

=∆1

0

)()0(x

dxdxdss ∫

=

=∆2

1

)()1(x

dxdxdss

First Segment Second Segment

Guess − ∆(0)

Fix − ε(0)

Posn 0 Posn 1 Posn 2

dx

Pconc(0) = -P(0)

δ(1)= δ(0)−∆s(0)

τ(0) = fnδ(0)

∆(0) = δ(0)

B(0)=τ(0)dxLper

εc(0)= −P(0)/(AcEc)

P(1)=P(0)-B(0)

εc(1)= −P(1)/(AcEc)

P(2)=P(1)-B(1)

τ(1) = fnδ(1)

B(1)=τ(1)dxLper

dx

{ })0()0()0( cdxds εε −= { })1()1()1( cdx

ds εε −=

δ(2)= δ(1)−∆(δ(1))

P(0)=ε(0)EA P continue

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Test Series 3 Eligehausen, Popov et al. (1983)

0

50000

100000

150000

200000

250000

0 2 4 6 8 10 12 14 16Free end slip (mm)

Load

(N)

Specimen 3.1 - db=19mmSpecimen 3.1 - NumericalSpecimen 3.2 - db=25mmSpecimen 3.2 - NumericalSpecimen 3.3 - db=32mmSpecimen 3.3 - Numerical

Figure D (Figure 9 in Chapter 2) - Accuracy of various τ−δ relationships

For pull tests with short embedment lengths, after the peak load is achieved the load continues to reduce approximately linearly for increasing loaded end slip. Mathematically, it can be shown that the P-∆ response unloads towards the coordinates (δmax,0), where δmax is from the local bond stress-slip relationship in Figure B, independent of embedment length. Hence, from the experimental P-∆ response the value of δmax can be determined directly, even when a short embedment length is provided to prevent yield. The value of τmax can not be determined directly from the global P-∆ response in Figure D. For very short embedment lengths, the average experimental shear stress is a suitable approximation of the value of τmax. However, as the embedment length increases the average shear stress reduces and begins to underestimate the value of τmax. To determine the value of τmax, the numerical simulation technique described earlier is used with varying values of τmax until the peak experimental load and the peak load from the numerical simulation as in Figure D are the same. When this occurs, the values of τmax and δmax are both known. In Figure D the peak experimental loads are not exactly equal to the peak loads from the numerical simulation because the “average” τ−δ relationship for steel reinforcing bars shown in Figure E has been used in the numerical simulations. This does confirm the accuracy of the t-d relationship shown in Figure E. The initial bond stress stiffness, shown as κ1 or κ2 in Figure B can also be determined using this numerical simulation approach. The bond stress stiffness is varied in the numerical simulations until the initial stiffness of the P-∆ response from the numerical simulation matches the stiffness of the experimental P-∆ response as in Figure D. When

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the loading stiffness from the numerical simulation and experimental tests are similar then the value of δ1 in Figure B is known. In the case of deformed steel reinforcing bars, the initial stiffness of the local bond stress-slip relationship is best modelled using a power expression. From this approach, the local bond stress-slip relationship for deformed steel reinforcing bars is shown in Figure E.

Figure E (Figure 8 in Chapter 2) - Theoretical τ−δ relationship for deformed steel reinforcing bars

Figure E indicates the absence of a frictional component; however the experimental results in Figure D and also CEB MC90 both indicate that a frictional component of bond does exists. This frictional component was ignored in our research for two reasons: to enable closed form solutions to be developed, and since experimentally the frictional component develops at large slips, for example approximately 10mm in Figure D, failure would reasonable be expected to occur in typical reinforced concrete sections well before the friction component is developed. To complete this research, the influence of corrosion on the bond properties of reinforcing bars is studied. It is shown that corrosion increases the peak bond stress τmax, but reduces the range of slips where bond stress can be transferred (δmax). The influence of corrosion on the interfacial fracture energy is given by Equation 2, where %c is the percentage weight loss due to corrosion.

( )c%19164G2 f −= Equation 2 Substituting Equation 2 for the term τmaxδmax in Equation 1, debonding will occur at the same time as yield according to Equation 3.

( )4

fdπEALc%19164P y

2b

perIC =−=

Slip - δ

cf5.2max =τ

Bond stress - τ

δ1 = 1.5mm

4.0

1max

= δ

δττ

δmax =15mm


44

Equation 3 As an example of the application of Equation 3, a 12mm diameter bar of yield stress 400MPa will debond prior to yield when the corrosion exceeds approximately 8% volumetric weight loss. Having determined the local bond stress-slip relationship of deformed steel reinforcing bars we now have the essential material parameter to allow us to model the load slip (P-∆) response of deformed steel bars. The mathematical P-∆ model, which uses the local bond stress-slip relationship in this chapter, is discussed in detail in Chapter 4. In the following chapter the local τ−δ relationship of embedded NSM FRP plates is determined using a procedure similar to that described in this chapter. For FRP plates yield does not occur, so full embedment can be provided experimentally. Hence, the interfacial fracture energy of the τ−δ relationship, τmaxδmax in Equation 1, can be determined directly from the load at failure.

45

Statement of Authorship Local and global bond characteristics of steel reinforcing bars by Matthew Haskett1, Deric John Oehlers2 and Mohamed Ali M.S. 3 1Mr. Matthew Haskett PhD student School of Civil and Environmental Engineering Adelaide University, Australia Wrote the manuscript, performed all analyses and developed model and theory SIGNED____________________

2Associate Professor Deric J. Oehlers Corresponding author:

School of Civil and Environmental Engineering Adelaide University Supervised the research and reviewed manuscript SIGNED____________________ 3 Dr. Mohamed Ali Research Associate School of Civil and Environmental Engineering Adelaide University, Australia Engineering Structures, Vol. 3(2), pp. 376-383

46

JOURNAL PAPER - LOCAL AND GLOBAL BOND CHARACTERISTICS OF STEEL REINFORCING BARS

Matthew Haskett, Deric John Oehlers and Mohamed Ali M.S. Abstract The development of an accurate local bond stress-slip (τ-δ) relationship for steel reinforcing bars has been hindered for two reasons: because steel reinforcing bars are internally encased, it is difficult to strain gauge the reinforcing bar to quantify the change in strain over a given length and, therefore, to directly measure the local τ-δ relationship; and because of the presence of yielding, which generally occurs before debonding and significantly influences the behaviour of the reinforcing bar. To overcome these difficulties associated with quantifying the local τ-δ relationship of steel reinforcing bars, theory developed to obtain the local τ-δ relationship from the global load slip (P-∆) response of FRP plated reinforced concrete structures has been modified in this research to account for short embedment lengths. Furthermore, in this paper, a link between the global load-slip (P-∆) response and the local τ-δ relationship of steel reinforcing bars, that is independent of embedment length, is presented for the first time. Additionally, the new theory is used to derive the local τ-δ relationship from test results and to illustrate the effect of corrosion on the local τ-δ relationship which quantifies the amount of corrosion to cause bar debonding in RC beams. Key Words: Interfacial fracture energy, FRP, corrosion, debonding, bond, reinforced concrete, pull test, reinforcing bars Introduction Pull tests are frequently used to determine the bond between steel reinforcing bars and the surrounding concrete. In the pull test, as in Figure 1, the edge of the concrete face acts as a crack front. When a load, P, is applied, this causes a slip, ∆L, at the crack face as shown and a slip at the free end, ∆F, also shown. This is referred to as the global (P-∆) response. The applied load, P, is resisted by interface shear stresses which depend on the interfacial bond characteristics between the reinforcing bar and the surrounding concrete, such as in Figure 2, where τ is the interface shear stress and ∆ is the interface slip. Hence τ−δ is the local or material bond characteristics in contrast to the global or structural (P-∆) response.

47

Figure 1 – Typical pull test

The local τ-δ response of steel reinforcing bars has been studied by many researchers, with most researchers (Azizinamini et al. 1993; Bolander et al. 1992; Darwin and Graham 1993; Malvar 1992) generally dividing the bond transfer into three distinct regions. These distinct regions are accommodated in the local bond stress-slip relationship proposed by Model Code 90 (CEB 1992), shown graphically in Figure 2 for “good bond conditions”.

Figure 2 – CEB Model Code 90 (CEB 1992) local bond stress-slip (τ−δ) relationship

As shown in Figure 2, CEB Model Code 90 (CEB 1992) suggests that there is a plateau over which the peak bond stress, τmax, is maintained. In the seminal research of Eligehausen et al.(1983), their results did not indicate that this plateau existed experimentally. Rather, their research indicated that the peak bond stress was achieved generally at free end slips, ΔF, of 1-2mm. Once the peak bond stress had been attained, the bond transferred across the bar-concrete interface reduced as the local slip increased. The initial stiffness in the local τ−δ relationship shown in Figure 2 is attributed to the adhesive component of bond. Once cracking commences, the adhesive component of bond has been overcome and mechanical interlock between the deformations and the

Slip - δ

cf5.2max =τ

Bond stress - τ

δ1 δ2

4.0

1max

= δ

δττ

δ3


Frictional component of bond

Adhesive component of bond

"δmax”

κ3

Crack face

Loaded end slip - ∆L Concrete

Local slip: δ Local stress: τ

Loaded end Free end Bond stress distribution

τmax

Critical bond length - Lcrit

Reinforcing bar

Circumference = Lper

Free end slip - ∆F

Load - P

Embedment length - L

48

surrounding concrete commences. This mechanical interlock continues until excessive local slip occurs such that the concrete between the bar deformations has sheared off. After this, only the frictional component of bond remains. This component commences at a local slip of δ3, as shown in Figure 2. Importantly, the area under the local τ−δ relationship in Figure 2 is the interfacial fracture energy Gf. This will be shown later to control the pull-out strength and is the reason why fully anchored reinforcing bars are much more likely to yield before debonding. In contrast, debonding generally occurs prior to fracture when modern materials like FRP are used. The high fracture strain of FRP, approximately 18,000 µε, and the reduced interfacial fracture energy of the local τ−δ relationship of EB and NSM FRP plates enables debonding to occur prior to fracture. As an example, the interfacial fracture energy of externally bonded (EB) FRP plates is approximately 1N/mm and 5N/mm for near surface mounted (NSM) FRP plates. In contrast, the interfacial fracture energy of steel reinforcing bars will be shown to be approximately 100 N/mm. This directly indicates the propensity of NSM and EB plating arrangements to debond before fracture, even in fully embedded conditions. This has enabled researchers to develop mathematical solutions that quantify the global load-slip (P-∆) response in terms of the local τ−δ relationship. In the following section the relationship between the global load-slip (P-∆) response and the local τ−δ relationship is discussed for EB and NSM plating arrangements as well as the development of a numerical model for interfacial slip which is used to analyse the behaviour of pull tests conducted with short embedment length. The numerical load-slip (P-∆) responses are then compared to experimental load-slip (P-∆) responses from pull tests conducted by Eligehausen, Popov et al (1983). Finally, this theory is applied to the analysis of corroded reinforcing bars, enabling the amount of corrosion required to cause IC debonding prior to yield in fully anchored conditions to be quantified. IC debonding resistance of FRP plates The absence of yielding and the significant fracture strains of FRP, in conjunction with the reduced bond performance between FRP and the surrounding concrete typically cause debonding failure in FRP plated structures, even in fully anchored conditions. This propensity to debond has allowed mathematical solutions to be developed that quantify the global load-slip (P-∆) response in terms of the local τ−δ relationship, where often, the local τ−δ relationship for FRP plates is considered to be a linear descending or a bilinear relationship as shown in Figure 3.

49

Figure 3 – Bi-linear and linear descending local τ−δ relationships

It is generally considered that the local τ−δ relationship is most accurately modelled by a bilinear relationship. According to this, Yuan et al. (2004) developed a mathematical relationship that quantified the global load-slip response (P-∆) in terms of a bilinear local τ−δ relationship. However, the presence of an elastic component in the bilinear local τ−δ relationship prevents the development of closed form mathematical solutions (Mohamed Ali et al. 2006a). Subsequently, Mohamed Ali, et al. (2006a) idealised the bilinear local τ−δ relationship as a linear descending relationship. This idealisation does not affect the ultimate pullout strength of the plating arrangement as the interfacial fracture energy is constant between the two different local τ−δ relationships, but it enables the development of a closed form solution. The peak load at which IC debonding commences is given by the following equation (Mohamed Ali et al. 2006b)

pperpperfIC EALEALGP maxmax2 δτ== Equation 1 where τmax, δmax and Gf are defined in Figure 3 and EAp is the axial rigidity of the reinforcement which could be an FRP plate or a steel reinforcing bar. The failure plane of the plating arrangement, Lper in Equation 1, is often considered to be the contact perimeter of the reinforcement with the surrounding concrete. As an example, if an FRP or steel reinforcing bar was used, the term Lper would be considered to be equal to the circumference of the bar, as shown in Figure 1. This peak IC debonding load can only be attained when sufficient bond length is provided. Also, increasing the bond length beyond this length does not increase the ultimate load. This bond length is often termed the “effective bond length” (Oehlers and Seracino 2004, Teng et al. 2002). When the effective (or critical) bond length is provided, the shear stress distribution varies from the maximum bond stress, τmax, to zero over the length of embedment. This distribution of shear stress and the concept of critical bond length are both shown in Figure 1.

Linear descending model

δ δ1

τ


Bi-linear model

κ1

κ2

τmax

δmax

50

The minimum bond length required such that the peak debonding load is attained, that is the effective bond length, for a linear descending local τ−δ relationship is given by the following equation (Mohamed Ali et al. 2006a; Mohamed Ali et al. 2006b).

p

percrit

EAL

L

max

max2δτ

π=

Equation 2 Essentially, the concept of effective bond length in plating is akin to the term “development length” in steel reinforced concrete design. Providing bond lengths less than the critical bond length or development length will not allow the section to achieve its maximum debonding resistance, although it may still yield if it is steel. Conversely, if length beyond the critical bond length or development length is provided, a factor of safety and additional ductility is incorporated into the design. The local parameters τmax and δmax in Figure 3 can be obtained directly from the global load-slip (P-∆) response when the specimens are fully anchored (that is the anchorage length is greater than Lcrit) such as the typical experimental global load-slip (P-∆) response for a NSM plating arrangement shown in Figure 4. The peak IC debonding load, shown as PIC, is attained after which the loaded end slip continues with no increase in load. This indicates that the effective or critical bond length has been provided. As PIC is known from the test results, τmaxδmax can be derived from Equation 1 as EAp and Lper depend on the geometric and material properties of the plate. When the IC debonding load is attained for the first time in Figure 4, the loaded end slip (∆L) equals δmax because the full bond stress distribution is first attained as in Figure 1. Subsequently τmax can be obtained from τmaxδmax and δmax. In summary, the local bond slip characteristics in Figure 3 can be obtained from global load-slip (P-∆) response, as in Figure 4, without the necessity for strain gauging which often affects the bond.

51

Global load-slip response

0

10

20

30

40

50

60

0 0.5 1 1.5 2∆L (mm)

Load

(kN

)

Figure 4 – Experimental global load-slip response of NSM pull test

This approach cannot be adopted to derive the local τ−δ relationship for pull tests with short embedment lengths, that is less than Lcrit. This is often the case with steel reinforcing bars because of the necessity to prevent yielding. Hence, the need for the following numerical model of pull tests. Partial interaction numerical model for pull tests The numerical model illustrated in Figure 5 is generic as it can accommodate any: local τ−δ relationship; failure plane (Lper); bar diameter (db); bar shape; length of embedment (L); cross sectional area of concrete (Ac); and stress-strain profile of the steel reinforcing bar. The following algorithm is used:

• A strain is fixed at the loaded end Position 0, ( )0ε ,as shown. Hence the plate force P(0) from the material properties.

• Corresponding to this fixed strain ( )0ε and corresponding load P(0) at Position 0, a slip at the loaded end Position 0 is assumed or guessed, i.e. s(0) = ( )0∆ and the following iterative routine is used to find ( )0∆ for P(0).

• As the segment lengths are made very small, the slip is assumed constant over the segment. Hence the bond stress ( )0τ which can be derived from the local bond characteristics such as in Figures 2 and 3 is also constant.

• The bond force acting over the first segment length is B(0) = ( )0τ Lperdx. • Hence the load in the reinforcing bar at the end of the first segment

is ( ) ( ) ( )001 BPP −= .

δmax

PIC

Commencement of debonding

52

• The corresponding strain in the plate is ( ) ( )( )pAEP 11 =ε where EAp is the axial

rigidity of the reinforcement and the corresponding strain in the concrete at the

end of the first segment is ( ) ( )( )c

c AEP 11 −=ε .

• Hence, the slip strain is )0()0()0(cdx

ds εε −= .

• By integration, the change in slip over the first segment is ( ) dxdx

dss )0(0 ∫=∆ .

• Therefore, the slip at the beginning of the second segment is s(1) = s(0) – Δs(0). • The numerical procedure is repeated over the subsequent segments until the

known boundary conditions are attained.

Figure 5– Graphical representation of the numerical analysis

As the process repeats itself over subsequent segment lengths, the axial stain ε and interface slip δ and slip-strain ds/dx are all determined. There are two boundary conditions that can be used to solve the initial guess Δ(0). For fully anchored plates, the boundary condition is 0== dx

dsδ and for short plates, that is plates with bond lengths

less than Lcrit, the boundary condition is ε=0 at the free end. Procedure for extraction of τ−δ using numerical simulation Focal point (δmax)

Guess − ∆(0)

Fix − ε(0)

Posn 0 Posn 1 Posn 2

dx

Pconc(0) = -P(0)

δ(1)= δ(0)−∆s(0)

τ(0) = fnδ(0)

∆(0) = δ(0)

B(0)=τ(0)dxLper

εc(0)= −P(0)/(AcEc)

P(1)=P(0)-B(0)

εc(1)= −P(1)/(AcEc)

P(2)=P(1)-B(1)

τ(1) = fnδ(1)

B(1)=τ(1)dxLper

dx

{ })0(ε)0(ε)0(dxds

c−= { })1(ε)1(ε)1(dxds

c−=

∫=

=1

0x

)dx(dxds)0(sΔ

δ(2)= δ(1)−∆(δ(1))

∫=

=2

1x

)dx(dxds)1(sΔ

P(0)=ε(0)EA P continue

First Segment Second Segment

53

Numerical simulations were performed using the process shown schematically in Figure 5. The results are shown in Figure 6 where the embedment lengths were varied from 50 mm to 500 mm. Once the critical bond length is provided, in this instance of 292mm, excess bond length does not increase the peak load, that is from point C onwards, as found experimentally as in Figure 4. However, when the bonded length is less than the critical bond length, increasing the bond length also increases the peak load. Importantly, all the unloading branches, whether they are from the short plates A-E and B-E or from long plates D-E and F-E, converge to the focal point at the global slip (ΔL) at E which is the local bond slip δmax in Figure 3. Very importantly, this allows δmax to be determined directly from non-fully anchored pull-tests with steel reinforcing bars. However, the value of the peak bond stress, τmax, still cannot be determined from a pull tests with short embedment. Instead the following procedure was developed to determine the magnitude of the peak bond stress from short embedment length pull tests.

Influence of embedment length on global P −∆ response

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Slip (mm)

Load

(N)

L=50mm

L=200mm

L=350mm L=500mmL=292mm

Figure 6– Influence of embedment length of global load-slip (P−∆) response

τmax from short embedment length The peak shear stress τmax can be obtained from pull-tests with short embedment lengths using an iterative procedure as δmax is now known. The peak load such as at A or B in Figure 6 can be used to derive the average bond stress over the embedment length τav. This has been found to be a good approximation for τmax for very short bond lengths but as would be expected reduces in accuracy with increasing bond lengths; for example from the variation in bond stress in Figure 1, it can be seen that τav would grossly underestimate the local bond stress τmax. To overcome this, τmax can be obtained from an

A

A

B

Focal point =δmax

C D F

E

54

iterative numerical analysis of the test specimens; for the local bond characteristics in Figure 3, τav is first used for τmax with the now known value of δmax and τmax varied until the numerical strength agrees with the experimental strength. Rising branch of τ−δ relationship Having determined the values of τmax and δmax, the numerical simulation of the global response can now be used to determine the rising branch in the local bond characteristics, such as in Figures 2 and 3. The rising branch of the global response such as in Figures 4 and 6 depends on the rising branch of the local bond characteristics. Numerical simulations were conducted with various shaped local τ-δ relationships. Three different local τ-δ relationship shapes were considered: linear descending in Figure 3: bilinear also in Figure 3; and a local τ−δ relationship with an initial stiffness of the CEB Model Code 90 relationship (CEB 1992) shown in Figure 2 which will be referred to as the “modified” CEB local τ-δ relationship. The frictional component of bond proposed by Model Code 90 (CEB 1992) shown in Figure 2 was ignored in this “modified” model, allowing closed form solutions to be developed. It is worth noting that the global slip value at the free end, ∆F in Figure 1, at which the maximum load P is attained experimentally is equal to δ1 in the bi-linear variation in Figure 3. This is because the rate of change in bond stress in the unloading branch is less than in the loading branch. Hence, for the peak load to be attained, the entire length of embedment is on the unloading branch of the local τ-δ relationship, with the free end subjected to a slip equal to δ1. The various τ−δ relationships are used in the numerical simulation of Test Series 1.1 by Eligehausen et al. (1983) and are compared in Figure 7. All three approaches, which use the local τmax and δmax values obtained from the experimental global load-slip (P-∆) response, model the experimental behaviour relatively accurately. Additionally, the initial stiffness of the numerical simulation using the “modified” CEB local τ-δ relationship matches the loading stiffness obtained experimentally. The linear descending local τ-δ relationship provides a peak load at minimal slip, whereas the peak load is attained experimentally at a noticeably greater slip. Hence, a linear descending local τ−δ relationship is too stiff to model the global load-slip (P-∆) response and is not considered further. The bilinear local τ-δ relationship can accurately predict the loaded end slip when the peak load is attained experimentally. However, the shape of the theoretical load-slip (P−∆) response does not match the experimental load-slip (P−∆) response as accurately as the “modified’ CEB local τ−δ relationship.

55

Test Series 1.1 Eligehausen, Popov et al. (1983)

0

20000

40000

60000

80000

100000

120000

140000

160000

0 2 4 6 8 10 12 14 16Free end slip (mm)

Load

(N)

Specimen 2 - ExperimentalBilinearModified CEBLinear

Figure 7 – Accuracy of various local τ−δ relationships

Essentially, the loading stiffness for the linear descending local τ−δ relationship, shown as κ1 in Figure 3, is too great and the loading stiffness for the bilinear local τ−δ relationship, shown as κ2 in Figure 3, is not great enough. Instead, a relationship with stiffness greater than κ2 and less than κ1 is required. This is provided by the “modified” Model Code 90 local τ−δ relationship with stiffness κ3, as shown in Figure 2. Hence, the “modified’ CEB local τ−δ relationship is adopted as the most suitable local τ−δ relationship to use in the analysis of test data. Derivation of τ−δ from experimental results Local τ−δ relationships were obtained from the experimental global load-slip (P−∆) responses of various pull tests (Eligehausen et al. 1983) using the processes described previously. The values of δ1, δmax and τmax in Figure 3 are given in columns 5 to 7 in Table 1. The average value of δ1 in col. 5 is 1.50 mm with a low coefficient of variation of 0.17. The average value of δmax in col. 6 is 14.1 mm. If the frictional component of the relationship proposed by Model Code 90 in Figure 2 was ignored and the descending branch extrapolated to the abscissa, shown in Figure 2 as “δmax”, then the value of “δmax” indicated by Model Code 90 (CEB 1992) is 15.7mm which is in good agreement with the 14.1 mm derived previously. Hence, a value of 15 mm is recommended for the theoretical local τ−δ relationship. The maximum bond stress attained experimentally in col. 7 is compared with the CEB Model Code value from Figure 2 in col. 8 i.e. cf5.2max =τ . The ratios are given in column 9 which has an average of 0.98 and a coefficient of variation of 0.092. Hence there is also very good agreement.

Table 1 – Accuracy of local “modified” CEB Model Code 90 τ−δ relationship

δ1

56

Series fc Le db δ1Exp δmaxExp τmaxExp MC 90τmax

τmaxExp/τmaxTheo

(1) (2) (3) (4) (5) (6) (7) (8) (9) 1.1 (1) 29.4 127 25.4 1.33 15 13.75 13.56 1.01 1.1 (2) 29.4 127 25.4 1.33 15 13.65 13.56 1.01 1.1 (3) 29.4 127 25.4 1.83 13 13.75 13.56 1.01 1.2 (1) 30.5 127 25.4 1.67 12.5 13.3 13.81 0.96 1.2 (2) 30.5 127 25.4 1.46 13 13.4 13.81 0.97 1.2 (3) 30.5 127 25.4 1.41 14 13.45 13.81 0.97 1.3 (1) 30.5 127 25.4 1.65 14 12.2 13.81 0.88 1.3 (2) 30.5 127 25.4 1.4 15 11.6 13.81 0.84 1.3 (3) 30.5 127 25.4 1.19 11.8 11.35 13.81 0.82 1.5 (1) 30.5 127 25.4 1.49 16.5 11.6 13.81 0.84 1.5 (2) 30.5 127 25.4 1.62 16 14.7 13.81 1.06 1.5 (3) 30.5 127 25.4 1.58 15 13 13.81 0.94 2.1 (1) 30 127 25.4 1.77 14 14.6 13.69 1.07 2.10 (1) 30 127 25.4 1.69 15 13.8 13.69 1.01 2.11 (1) 30 127 25.4 1.845 15 15.4 13.69 1.12 2.15 (1) 30 127 25.4 1.64 14 13.5 13.69 0.99 2.16 (1) 30 127 25.4 1.86 14 12.2 13.69 0.89 3.1 (1) 31.6 95 19 1.3 10.6 15.8 14.05 1.12 3.2 (1) 31.6 127 25.4 1.02 15 15.4 14.05 1.10 3.3 (1) 31.6 160 32 0.97 14.4 12.8 14.05 0.91 4.1 (1) 54.6 127 25.4 1.43 14 18.05 18.47 0.98 Average 1.50 14.13 Average 0.98 CV 0.17 0.097 CV 0.092 From the previous analysis of test data, the local τ−δ relationship in Figure 8 is recommended. This relationship has been used to model test results by Eligehausen, Popov et al. (1983) in Figure 9 where there can be seen to be good agreement.

57

Figure 8 – Theoretical τ−δ relationship for deformed steel reinforcing bars

Test Series 3 Eligehausen, Popov et al. (1983)

0

50000

100000

150000

200000

250000

0 2 4 6 8 10 12 14 16Free end slip (mm)

Load

(N)

Specimen 3.1 - db=19mmSpecimen 3.1 - NumericalSpecimen 3.2 - db=25mmSpecimen 3.2 - NumericalSpecimen 3.3 - db=32mmSpecimen 3.3 - Numerical

Figure 9 – Accuracy of proposed local τ−δ relationships

Example of application of new approach to corrosion The technique described previously will also be used to obtain the local τ−δ relationship from the global load-slip (P−∆) response to determine the influence of corrosion on the interfacial fracture energy. Specifically, the amount of corrosion, as a percentage weight loss, required to cause fully anchored steel reinforcing bars to debond prior to yielding is determined.

Slip - δ

cf5.2max =τ

Bond stress - τ

δ1 = 1.5mm

4.0

1max

= δ

δττ

δmax =15mm


58

Researchers have generally considered that the bond strength under low levels of corrosion is increased (Al-Sulaimaini et al. 1990) after which there is a reduction in bond strength as corrosion increases. This observation was supported by experimental tests by Cabrera and Ghoddoussi (1992), amongst others. Almusallam, Al-Gahtani et al. (1996) similarly reported increased bond capacity for corrosion up to 4% volumetric weight loss. These researchers all considered this increase in bond strength to be caused by the increased confinement provided to the reinforcing bar due to the by-products of corrosion. Therefore, researchers often considered that small amounts of corrosion were actually beneficial to the bond performance of steel reinforcing bars. Researchers, however, never considered the influence of corrosion on the interfacial fracture energy, Gf, of the local τ−δ relationship. Using the process described previously to obtain the local τ−δ relationship from the global load-slip (P−∆) response of corroded reinforcing bars, it was found that the small increases in bond strength under low levels of corrosion were actually offset by the reduction in the value of δmax in the local τ−δ relationship. The influence of corrosion on the values of τmax and δmax obtained from the global load-slip (P−∆) responses of steel reinforcing bars for various levels of corrosion is shown in Table 2.

Table 2 – Influence of corrosion on local τ−δ relationship

Corrosion τmax (Normalised) δmax 2Gf

% MPa mm N/mm

Al-Sulaimaini, Kaleemullah et

al. (1990) 10mm db

0 16 15 240 0.87 25 9 225 1.5 23 7 161 4.27 15 11 165 6.7 8 2.5 20 7.8 4 2 8

Almusallam, Al-Gahtani et al.

(1996)

0 16 10 160 3.6 18 5.5 99 4 18.5 5.5 101.75

4.78 16 4.5 72

5.1 13.8 3.75 51.75

7 4.5 - - 10 2.6 - -

Al-Sulaimaini, Kaleemullah et

al. (1990) 14mm db

1.62 19 3.7 70.3 2.75 15.5 3.5 54.25

5.45 5 0.75 3.75

59

For all levels of corrosion, there was an unequivocal reduction in the interfacial fracture energy of the local τ−δ relationship, as shown in Figure 10.

Comparison of Gf values for corrosion

0

50

100

150

200

250

300

0 2 4 6 8 10 12Corrosion (%)

2Gf =

τm

axδ

max

Al-Sulaimaini, Kaleemullah et al. (1990) 10mm db

Almusallam, Al-Gahtani et al. (1996)

Al-Sulaimaini, Kaleemullah et al. (1990) 14mm db

Figure 10 – Influence of corrosion on interfacial fracture energy

Figure 10 indicates that corrosion reduces the interfacial fracture energy for all levels of corrosion. This indicates that the reduction in the value of δmax offsets the increase in the bond strength under low levels of corrosion. The significant reduction in the interfacial fracture energy of corroded deformed steel reinforcing bars suggests that debonding (pullout) can occur prior to yield under advanced levels of corrosion even when full embedment was originally provided. It may be worth noting that there is a noticeable scatter in the magnitudes of the interfacial fracture energy between series but not within a series. This is because the material properties used across the three series was not constant. For example, the bar diameter and the compressive strength of the concrete varied. Using the results from Almusallam, Al-Gahtani et al. (1996), the influence of corrosion on the interfacial fracture energy is given by Equation 3, where %c is the percentage weight loss due to corrosion.

( )cG f %191642 −= Equation 3 Substituting Equation 3 for the term τmaxδmax in Equation 1, debonding will occur at the same time as yield according to Equation 4.

( )4

%191642

ybperIC

fdEALcP

π=−=

60

Equation 4 As an example of the application of Equation 4, a 12mm diameter bar of yield stress 400MPa will debond prior to yield when the corrosion exceeds approximately 8% volumetric weight loss. Conclusion and future research A technique has been developed for obtaining the local τ−δ relationship from the global load-slip (P−∆) response of steel reinforcing bars which is independent of embedment length. This technique recognises for the first time an analogy between the behaviour of FRP reinforcement and steel reinforcing bars. In particular, it demonstrates that theory developed for plated structures, such as IC debonding, is applicable to steel reinforcing bars. The new technique was used to derive a general local τ−δ relationship from test results as shown in Figure 8. Importantly, it suggests that the CEB Model Code 90 local τ−δ relationship is relatively accurate. The magnitude of the peak bond stress suggested by CEB Model Code 90 accurately models the peak loads attained experimentally. However, the presence of a region where the bond stress is constant for various local slips has been shown to not occur. Instead, once the peak bond stress is achieved, generally at a local slip of approximately 1.5mm, there is a reduction in bond stress. Finally, the frictional component of bond has not been considered in this research. Most researchers consider the frictional component of bond to be minimal. Ignoring the frictional component of bond allows implicit mathematical solutions to be developed, enabling the interfacial fracture energy of deformed reinforcing bars to be quantified. This theory was also applied to develop an expression for the influence of corrosion on the interfacial fracture energy of corroded steel reinforcing bars. It was shown that corrosion reduces the interfacial fracture energy significantly, and that debonding will occur prior to corrosion, even in fully embedded conditions, when the percentage corrosion exceeds approximately 8% for 12mm diameter bars with a yield strength of 400MPa.

61

References Almusallam, A. A., Al-Gahtani, A. S., and Aziz, A. R. (1996). "Effect of reinforcement corrosion on bond strength." Construction and Building Materials, 10(2), 123-129. Al-Sulaimaini, G. J., Kaleemullah, M., Basumbal, I. A., and Rasheeduzzafar. (1990). "Influence of corrosion and cracking on bond behaviour and strength of reinforced concrete members." ACI Structural Journal, 87(2), 220-231. Azizinamini, A., Stark, M., Roller, J. J., and Ghosh, S. K. (1993). "Bond performance of reinforcing bars embedded in high-strength concrete." ACI Structural Journal (American Concrete Institute), 90(5), 554-561. Bolander, J. J., Satake, M., and Hikosaka, H. (1992). "Bond degradation near developing cracks in reinforced concrete structures." 0023-6160. Cabrera, J. G., and Ghoddoussi, P. "The effect of reinforcement corrosion on the strength of steel/concrete bond." Bond in concrete: From research to practice, Riga, Latvia, 10.11-10.24. CEB. (1992). "CEB-FIP Model Code 90." London. Darwin, D., and Graham, E. K. (1993). "Effect of deformation height and spacing on bond strength of reinforcing bars." ACI Materials Journal (American Concrete Institute), 90(6), 646-657. Eligehausen, R., Popov, E. P., and Bertero, V. V. (1983). "Local bond stress-slip relationship of deformed bars under generalized excitations." UCB/EERC-83/23, Earthquake Engineering Research Center, University of California, Berkeley. Malvar, J. (1992). "Bond of reinforcement under controlled confinement." ACI Materials Journal, 89(6), 593-601. Mohamed Ali, M. S., Oehlers, D. J., and Seracino, R. (2006a). "Vertical shear interaction model between external FRP transverse plates and internal steel stirrups." Engineering Structures, 28(3), 381-389. Mohamed Ali, M. S., Oehlers, D. J., and Seracino, R. (2006b). "Interfacial stress transfer of near-surface mounted FFP-to-concrete joints." Engineering Structures, (yet to be published). Oehlers, D. J., and Seracino, R. (2004). Design of FRP and Steel Plated RC Structures, Elsevier, Oxford. Teng, J. G., Chen, J. F., Smith, S. T., and Lam, L. (2002). FRP Strengthened RC Structures, Wiley, London. Yuan, H., Teng, J. G., Seracino, R., Wu, Z. S., and Yao, J. (2004). "Full-range behavior of FRP-to-concrete bonded joints." Engineering Structures, 26(5), 553-565.

62

CHAPTER 3:

SYNOPSIS - EMBEDDING NSM FRP PLATES FOR IMPROVED IC DEBONDING RESISTANCE

The bond stress-slip relationship of deformed steel reinforcing bars was derived in Chapter 2. In this chapter, the influence of embedment on the bond behaviour of near surface mounted (NSM) plates is investigated. Embedding NSM plates is important because embedment can provide larger debonding strains, hence more ductility and strength, and it may also be the first step in improving the fire resistance of NSM plates. Twenty embedded NSM FRP specimens were produced and pull tests conducted for various levels of embedment. The experimental load-slip response was recorded. An example of the influence of embedment on the global P-∆ response is shown in Figure F, where the label TS2-6.0-C0 refers to “Test Series 2”, 6mm thick strips, and 0mm cover (i.e. conventional NSM arrangement). TS2-6.0-C30 refers to Test Series 2, 6mm thick plate and 30mm cover, etc.

0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5∆L (mm)

Load

(kN

)

TS2-6.0-C0

TS2-6.0-C10

TS2-6.0-C30B

TS2-6.0-C30 TS2-6.0-C40

TS2-6.0-C55

TS2-6.0-C20

Figure F (Figure 6 in Chapter 3) – Experimental P-∆ response for Test Series 2

From Figure F, it is clear as the cover provided to the NSM strip increases the load at failure generally continues to increase, and the peak debonding load is achieved at larger loaded end slips. According to the theory detailed in Chapter 2 for deformed steel reinforcing bars, the interfacial fracture energy of the τ−δ response (τmaxδmax) was determined directly (Equation 1 in Chapter 2) from the load at debonding for various levels of cover, since full embedment length was provided. The effect of cover on the interfacial fracture energy for various levels of embedment is shown in Figure G. Experimentally IC debonding did not always occur and this data is shown in Figure G as a grey circle. When debonding did not occur, the experimental fracture energy (2GfExp) provides a lower bound to the actual fracture energy. In Figure G the

63

abscissa is the dimensionless parameter (c+d)/d, where “c” is the cover and “d” the plate depth, and experimentally all plates were 10mm deep.

0

20

40

60

80

100

1 3 5 7(c+d)/d

2Gf E

xp

Test Series 1 - ICTest Series 2 - ICTest Series 3 - ICOther failure modes

Specimen TS2-6.0-30B

Figure G (Figure 10 in Chapter 3) – Influence of cover on 2Gf

From Figure G, it can be seen that as the cover increases the interfacial fracture energy and hence the debonding load continues to increase. The influence of cover on the IC debonding load of embedded NSM plates is:

( ) )EA(Ld

dcG2P per

2.1

0cfIC

+

= =

Equation 4 where E is the elastic modulus of the FRP, Ap is the cross sectional area of the plate, c is the cover (mm) provided to the strip, d is the plate depth (mm), and (2Gf)c=0 is the interfacial fracture energy of a NSM plate from previous research ( ) ( ) 6.0

c526.0

0cmaxmax0cf ffφ976.0δτG2 == == Equation 5 and fc is the compressive strength of concrete (MPa), Lper is the failure plane and is shown in Figure H and φf is the ratio df/bf in Figure H(b) for NSM plates.

LperLper

Ap of EB plate1 mm

1 mm

1 mm

1 mm

Ap of NSM plate

df

df

bf

bf

failure plane EB plate

failure plane NSM plate

concrete surface

concreteelement

bp tp

tp bp

Lper

1 mmdp

Abar of reinforcing bar

failure plane of reinforcing bar

(a) externally bonded plate (b) near surface mounted plate (c) reinforcing bar

64

Figure H – Failure planes (Lper) and aspect ratios of various reinforcement types The influence of cover on the individual parameters τmax and δmax for embedded NSM plates was also investigated. The influence of cover on δmax is:

( )55.0

0cmaxmax ddcδδ

+

= =

Equation 6 where (δmax)c=0 is

( )55.0

f

526.0f

0cmax φ078.0802.0φ976.0δ

+== (Units N and mm)

Equation 7 The influence of cover on the peak shear stress τmax is

( )

( )

( )( )

65.0

0cmax

0cf55.0

0cmax

2.1

0cf

max ddc

δG2

ddcδ

ddcG2

τ

+

=

+

+

==

=

=

=

Equation 8 The stiffness of the bilinear τ−δ response for embedded NSM plates was also determined by repeating the same process described in Chapter 2 for steel reinforcing bars. Hence, embedding plates significantly improves the bond performance of NSM FRP plates. Quantifying the local bond relationship mathematically will enable debonding loads and strains to be determined. Quantifying the influence of cover on the value δmax will also enable embedded NSM FRP plates to be analysed using the moment-rotation technique. In the following chapter the P-∆ relationship of deformed steel reinforcing bars is quantified mathematically in terms of the τ−δ relationship from Chapter 2.

65

Statement of Authorship Embedding NSM FRP plates for improved IC debonding resistance” By Deric John Oehlers1, Matthew Haskett2, Chengqing Wu3 and Rudolf Seracino4 Paper CC/2007/022906

1 Professor Deric J. Oehlers Corresponding author:

School of Civil, Environmental and Mining Engineering University of Adelaide Supervised research and reviewed manuscript SIGNED____________________ 2Mr. Matthew Haskett PhD student School of Civil, Environmental and Mining Engineering University of Adelaide Developed theory, wrote manuscript and performed all analyses SIGNED____________________ 3Dr. Chengqing Wu Lecturer School of Civil, Environmental and Mining Engineering University of Adelaide 4 Associate Professor Rudolf Seracino Department of Civil, Construction and Environmental Engineering North Carolina State University Raleigh, North Carolina United States of America 27695 Journal of Composites for Construction, Vol. 12(6), pp. 635-642

66

Oehlers, D.J., Haskett, M., Wu, C. & Seracino, R. (2008) Embedding NSM FRP plates for improved IC debonding resistance. Journal of composites for construction, v. 12(6), pp. 635-642

A NOTE:


A It is also available online to authorised users at:

A http://dx.doi.org/10.1061/(ASCE)1090-0268(2008)12:6(635)

A

http://dx.doi.org/10.1061/(ASCE)1090-0268(2008)12:6(635)

87

CHAPTER 4:

SYNOPSIS - YIELD PENETRATION HINGE ROTATION IN REINFORCED CONCRETE BEAMS

In Chapters 2 and 3, experimentally recorded P- relationships were used to quantify the local relationships of deformed reinforcing bars and embedded NSM plates. In this chapter, the characteristics of deformed steel reinforcing bars from Chapter 2 are used to model the global P- response of steel reinforcing bars mathematically, including the influence of yield. Two distinct loading stiffnesses are observed in the P- responses of steel reinforcing bars. The initial response is very stiff with a significant increase in load for a minimal increase in loaded end displacement. After yield, which in this example occurs at a stress of approximately 460MPa, the - response is far less stiff with a smaller increases in load for a given displacement. This behaviour is shown by the grey and black markers referred to as “Viwathanatepa (1979)” and shown as O-A in Figure I.

0

200

400

600

800

0 5 10 15Slip ( rebar) (mm)

Str

ess

(re

bar)

(MP

a)

1.25sqrt(fc)

2.5sqrt(fc)

Saatcioglu (1992)

Viwathanatepa (1979)

Figure I (Figure 5 in Chapter 4) – Comparison of response

The difference between the loading stiffness pre and post yield is attributed to the difference between the elastic Young’s modulus of steel (Es) and the strain hardening modulus of steel (Esh), both shown in Figure J. For convenience, the strain hardening modulus is assumed to be linear.

fy

ffract

y fract

Es

Esh

0

Figure J – Idealised stress-strain relationship of steel

cf25.1max

Bar yield

cf5.2max

A

B

C

D

O

88

The numerical simulation technique presented in Chapter 2 was used to compare the P-∆ response of steel reinforcing bars for various shaped τ−δ relationships: linear descending, bi-linear and curvilinear. It was shown numerically that a linear descending τ−δ relationship (Figure B) and a curvilinear τ−δ relationship (Figure E) provide very similar P-∆ responses. This is important because it means that we can approximate the “real” curvilinear τ−δ response of steel reinforcing bars in Figure E as a linear descending τ−δ response as in Figure B without a loss in accuracy. This simplification allows closed form mathematical solutions to be developed. Prior to yield the reinforcing bar behaves perfectly elastically. Hence, expressions previously developed to model the P-∆ response of FRP plates can be used to model the reinforcing bar behaviour up to yield. The load-slip relationship prior to yield is therefore:

−

=max

rebarmax

el

bmaxrebar δ

Δδarccossin

λdπτ

P

Equation 9 in which

sbmax

maxel Edδ

τ4λ =

Equation 10 for 0<∆rebar<∆yield where db is the bar diameter, τmax is from the local τ−δ relationship and where ∆yield is

[ ]{ }elelmaxyield aλcos1δΔ −= Equation 11 ael is the embedment length required to develop the yield stress fy. This is akin to the development length expressions for deformed steel reinforcing bars from codes worldwide.

el

max

byel

el λτ4

dfλarcsin

a

=

Equation 12 After bar yield strain hardening commences. The upper bound to the strain hardening capacity is the yield load plus the increase in strength due to strain hardening. Ignoring the bond stress required to yield the bar, which is an appropriate approximation since yield occurs at very small slips and there is significant interfacial fracture energy, the debonding load of a steel reinforcing bar is:

89

( ) rebarshpermaxmaxyrebarshIC AELδτfAEP += Equation 13 The load-slip response post yield is simply the load at which yield occurs, plus the increase in load due to the strain hardening behaviour:

+

−−=

4dπ

fδ

ΔΔ1arccossin

λdπτ

P2

by

max

yieldrebar

el

bmaxrebar

Equation 14 where

shbmax

maxsh Edδ

τ4λ =

Equation 15 Alternatively, fracture of the reinforcing bars may occur prior to debonding. The slip at which fracture occurs is simply the slip at yield plus the increase in slip due to strain hardening from fy to ffract in Figure J. The increase in slip from yield of the bar to fracture is

[ ]{ }fractshmaxfract aλcos1δΔ −= Equation 16 where afract is the additional embedment length required to increase the load from the yield load Py to the fracture load Pfract.

( )

sh

max

yfractbsh

fract λτ4

ffdλarcsin

a

−

=

Equation 17 Hence, the total slip at fracture of the reinforcing bar

[ ]{ } [ ]{ }fractshmaxelelmaxultyp aλcos1δaλcos1δΔ −+−=− Equation 18 From the rigid body rotation approach the rotation at fracture is simply the slip at fracture (∆yp-ult) divided by the crack height hcr.

cr

ultypitlimfract h

Δθ −

− =

Equation 19 Assuming a linear strain distribution, the curvature at bar fracture is simply

90

cr

fractfract h

εχ

Equation 20 where fract is the bar fracture strain in Figure J. According to the “curvature hinge length” approach, the rotation for a given empirical hinge length is simply the hinge length multiplied by the curvature

cr

fracthingeemp h

εLθ

Equation 21 The comparison of empirical rotations from Equation 21 to the rigid body rotations from Equation 19 is shown in Figure K for various empirically derived hinge lengths: Lhinge = 6db (Priestley and Park 1987) and Lhinge = 0.021dbfy (Panagiotakos and Fardis 2001).

0

1

2

3

4

6 10 14 18 22 26db (mm)

emp /

frac

t-li

mit

Priestley and Park 1.25sqrtfc

Priestley and Park 2.5sqrtfc

Panagiotakos and Fardis 1.25sqrtfc

Figure K (Figure 13 in Chapter 4) – Comparison of rotations due to yield penetration

Curve A-B and points C and D in Figure K are the comparisons with Priestley and Park’s Lhinge = 6db. The points in A-B were governed by bar fracture at yp-ult. Whereas, the points C and D were governed by bond failure in which case it was assumed that yp-ult = max where max = 15mm and max is 1.25√fc. It can be seen that there is reasonably good correlation between the partial-interaction and empirical curvatures particularly at high bar diameters. Doubling max gives the variation I-J which would mean that the partial-interaction model would be very conservative and which also illustrates the importance of choosing a bond characteristic that allows for premature debonding such as occurs through splitting and cone pull-out. According to these results, we consider the bond behaviour in a beam to be better approximated with the peak bond stress max 1.25√fc. This peak shear stress is used through this thesis.

(1.25√fc)

(1.25√fc)

A

B C D

E

F G H

I

J

(2.5√fc)

91

The comparison with Panagiotakos and Fardis’ (2001) Lhinge = 0.021dbfy is shown in Figure K as line E-F and G and H where debonding also occurred. In this case, the partial-interaction rotations are reasonable at high bar diameters and overly conservative at small bar diameters. Bearing in mind the very large scatter of results reported by Panagiotakos and Fardis (2001) and the complexity of the problem plus the influence of the rotation in the non-hinge region, it is suggested that the partial-interaction model developed in this paper is in good agreement with the published empirical models. Equating the empirically derived rotation and the theoretical rotation from the rigid body rotation approach, the “equivalent” plastic hinge length is

( )fract

ultyp

ultyphinge εΔ

L −

−=

Equation 22 which is independent of the beam properties and only dependent on the reinforcement properties: the bond characteristics and stress-strain relationship of the reinforcement. This supports Priestley and Park’s and Panagiotakos and Fardis’ empirical hinge length expressions, which are independent of crack height or beam depth. It appears that the hinge length is inversely proportional to the bar fracture strain εfract. However, it should be remembered that the fracture slip ∆yp-ult depends on the fracture stress σfract and, hence, is also dependent on the bar fracture strain. In the following chapter a rigid body rotation, partial interaction and shear friction theory model is presented that quantifies the moment-rotation response of any reinforced concrete member. The concrete behaviour is modelled using shear friction theory, and the load in the steel reinforcing bars is quantified using the mathematical expressions presented in this chapter. Various failure criteria are considered, including concrete failure and reinforcing bar debonding or fracture, where the slip at bar fracture, ∆yp-ult, is quantified in this chapter.

92

Statement of Authorship Yield penetration hinge rotation in reinforced concrete beams By Matthew Haskett1, Deric John Oehlers2, Mohamed Ali M.S.3 and. Chengqing Wu4

1Mr. Matthew Haskett PhD student School of Civil and Environmental Engineering University of Adelaide Wrote manuscript, performed all analyses, developed mathematical expressions SIGNED____________________ 2Professor Deric J. Oehlers School of Civil and Environmental Engineering University of Adelaide Supervised research, manuscript review SIGNED____________________ 3Dr. Mohamed Ali M.S. Research Associate, School of Civil and Environmental Engineering University of Adelaide 4Dr. Chengqing Wu Lecturer, School of Civil and Environmental Engineering University of Adelaide ASCE Structural Journal, Vol. 135(2), pp. 130-138

93

A NOTE:



A http://dx.doi.org/10.1061/(ASCE)0733-9445(2009)135:2(130)

A

Haskett, M., Oehlers, D.J., Mohamed Ali, M.S. & Wu, C. (2009) Yield penetration hinge rotation in reinforced concrete beams. Journal of structural engineering, v. 135(2), pp. 130-138

http://dx.doi.org/10.1061/(ASCE)0733-9445(2009)135:2(130)

110

CHAPTER 5:

SYNOPSIS - RIGID BODY MOMENT-ROTATION MECHANISM FOR REINFORCED CONCRETE BEAM HINGES

In this chapter, current methods for determining the rotation of reinforced concrete beams are examined. The current method for determining the moment-rotation relationship of a reinforced concrete member requires the use of a empirically derived hinge lengths and assumes full interaction between the reinforcement and adjacent concrete in the determination of the moment-curvature relationship. A uni-linear strain profile is also assumed. This approach can clearly not accommodate the slip between the reinforcement and concrete that occurs in practice, and does not consider the influence of bond on the behaviour of the reinforcement. The rigid body rotation, RBR, model initially presented briefly in Chapter 1 considers the influence of bond of the reinforcement, allows for the development of cracks through its partial interaction nature, and does not require an empirically derived hinge length expression since the rotation is determined directly from the rigid body rotation profile. The rigid body rotation analysis technique idealisation is shown in Figure L.

∆rebar

dsoft

dsoft/2

dasc

α

θ

Pasc

Psoft

Prebar

ssoft

ssoft

hsoft

fc at εpk

concretestress

pivotal point

ascendingbranch9/24 dasc

A

Lsoft

C G

O

F

hcr

drebar

concretesofteningzone

concreteascendingzone

tensionzone

c

Figure L (Figure 7 in Chapter 5) – Moment-rotation analysis

The analysis technique works in the following way: For a given depth of softening (dsoft in Figure L) the force in the softening region of concrete is determined

( )( )

−

++=

αsinmαcosαsinαcosmαsinαcosσc

dwP latsoftbsoft

Equation 23

111

where wb is the width of the beam, c is the cohesive component of the Mohr-Coulomb failure plane which has a typical value of 0.17fc (Jensen 1975), and σlat is the lateral confinement to the wedge such as might be induced by stirrups. A conservative assumption would be to assume σlat= 0. The wedge force acts at dsoft/2 as uniform shear is assumed along the wedge interface. in Equation 23 and shown in Figure L is the angle the wedge forms and is a function of the frictional component of the Mohr-Coulomb failure plane (m), which for concrete has a typical value of 0.8, according to

1mmarctanα 2 Equation 24 For commonly accepted shear friction parameters, is between 27˚ and 37˚. The strain at the wedge interface is pk which corresponds to the peak compressive strength fc in Figure M. The displacement of the softening wedge ssoft is simply the integration of the slip-strain across the wedge which if the strain in the wedge is assumed to be small is simply the integration of the wedge strain pk over the length of the softening region Lsoft in Figure L.

pksoftsoft εLs Equation 25 The stress-strain relationship of concrete is shown in Figure M.

'stress'

'strain'

start (fc)

pk

soft

real materialproperties

effective materialproperties

@ ssoft

onset of softening

0

risingbranch

'rising branch' 'softening'

wedge Figure M (Figure 3 in Chapter 5) – Stress-strain relationship of concrete under compression

The compressive force in the concrete below the softening wedge is

ascbcasc dwf32P

Equation 26 which occurs a distance (9/24)dasc from the wedge interface. The steel reinforcing bar behaviour is modelled using the theory from Chapter 4. For a given displacement, reinf-bar in Figure L, the corresponding load in the reinforcing bar is determined. If reinf-bar is less than yield the corresponding load in the reinforcing

112

bar is from Chapter 4 Equation 9. If ∆reinf-bar is greater than ∆yield the corresponding load in the reinforcing bar is from Chapter 4 Equation 14. Failure occurs for ∆rebar greater than the minimum of δmax from the local τ−δ relationship or ∆yp-ult (Chapter 4 Equation 18). Hence for a given fixed depth of softening, dsoft, the force in the softening region (Equation 23) and the corresponding displacement of the softening wedge (Equation 25) can both be determined. The pivot point of this analysis is the softening slip ssoft, point C in Figure L. The angle θ in Figure L is assumed and constantly iterated until the algebraic sum of the forces Psoft (which is fixed), Pasc and Preinf-bar equals zero. Once equilibrium is achieved, the moment of the forces P can be taken to derive the moment M for the rotation θ. The depth of the softening region is subsequently gradually increased and the procedure repeated to get the complete moment-rotation response. This moment-rotation response has three limits: when the bar slip Δreinf-bar equals that to cause fracture Δyp-ult (Equation 19); or when the bar slip equals that to cause debonding which as a lower bound is given by δmax (15mm); or when the slip ssoft (Equation 25) reaches its sliding capacity sslide which is a material property that can be determined from tests. It is also worth noting that there is a fourth limit to the moment-rotation due to shear failure which this model can quantify but which is not the subject of this thesis. A typical moment-rotation response using this iterative procedure is shown in Figure N as the black solid line.

0

20

40

60

80

0.000 0.025 0.050 0.075 0.100

Rotation (radians)

Mom

ent (

kN-m

)

EXPERIMENTAL - SPECIMEN C1THEORETICAL - SPECIMEN C1FRACTUREEXPERIMENTAL FAILURE - FROM MOMENT AND ROTATION

Figure N (Figure 10 in Chapter 5) – Specimen C1 moment-rotation

The predicted moment-rotation response using this approach is compared to moment-rotation responses from previous research (Mattock 1964). An example is shown in Figure N as grey diamonds labelled “EXPERIMENTAL-SPECIMEN C1”. The various experimental moment-rotation responses were compared to those obtained from the

113

RBR model with good agreement. Various other examples are presented in the subsequent journal paper. Note that Mattock’s specimens contained compressive reinforcement. This compressive reinforcement was ignored in the theoretical analyses. As shown in Figure N, the theoretical approach provided very good agreement with the experimental results, even with this simplification. Mattock’s experimental results can be used to determine the sliding capacity of concrete, sslide. Experimentally failure always occurred in Mattock’s experiments through “concrete failure”. Hence Mattock’s rotations at failure can be used in a statistical analysis to determine the influence of confinement (in this example provided by stirrups rather than an FRP wrap) on the sliding capacity of concrete. The stirrup confinement σlat can be written as

pc

st)st(y

c

lat

sbfAσ2

fσ

=

Equation 27 where sp is the spacing of stirrups, σy(st) is the yield stress of the stirrups, and it is assumed that 2 stirrup legs confine the concrete. The relationship between wedge sliding at failure and confinement is shown in Figure O.

0.00

0.40

0.80

1.20

1.60

0.00 0.05 0.10 0.15 0.20 0.25σlat/fc

sslid

e(Ex

p)

Figure O (Figure 18 in Chapter 5) – Variation in sslide with stirrup confinement

From Figure O

42.0f

σ76.2s

c

latslide +=

Equation 28

114

which is in agreement with previous research on the influence of confinement on the sliding behaviour of concrete. Hence, knowing the limits to the reinforcement behaviour described in Chapter 4, where failure occurs either through debonding or fracture, and knowing the limit to sliding failure of the wedge from Equation 28, the failure limits to the rigid body rotation model are all known, and we can now model the rotation capacity at failure of any reinforced concrete member.

115

Statement of Authorship Rigid body moment-rotation mechanism for reinforced concrete beam hinges By Matthew Haskett1, Deric John Oehlers2, Mohamed Ali M.S.3 and. Chengqing Wu4

1Mr. Matthew Haskett PhD student School of Civil and Environmental Engineering University of Adelaide Wrote manuscript, developed all theory, performed all analyses SIGNED____________________ 2Professor Deric J. Oehlers School of Civil and Environmental Engineering University of Adelaide Supervised research, reviewed manuscript SIGNED____________________ 3Dr. Mohamed Ali M.S. Senior Research Associate, School of Civil and Environmental Engineering University of Adelaide 4Dr. Chengqing Wu Lecturer, School of Civil and Environmental Engineering University of Adelaide Engineering Structures, Vol. 31(5), pp. 1032-1041

116

JOUNAL PAPER - RIGID BODY MOMENT-ROTATION MECHANISM FOR REINFORCED CONCRETE BEAM HINGES

Matthew Haskett, Deric John Oehlers, Mohamed Ali M.S and. Chengqing Wu Abstract Structural engineers have long recognised the importance of the ductility of reinforced concrete members in design, that is the ability of the reinforced concrete member to rotate and consequently: redistribute moments; give prior warning of failure; absorb seismic, blast and impact loads; and control column drift. However, quantifying the rotational behaviour through structural mechanics has been found over a lengthy period of time to be a very complex problem so that empirical solutions have been developed which for a safe design are limited by the bounds of the test parameters from which they were derived. In this paper, a rigid body moment-rotation mechanism is postulated that is based on established shear-friction and partial-interaction research; it is shown to give reasonable correlation with test results as well as incorporating and quantifying the three major limits to rotation of concrete crushing and reinforcing bar fracture and debonding. Keywords: Reinforced c oncrete; b eam; hi nge; r otation, duc tility; s hear-friction; par tial-interaction. Introduction A reinforced concrete slab that has been subjected to a blast load (Wu et al 2008) is shown in Fig.1. Typically, the slab can be considered to consist of two distinct regions: the small hinge r egion where concrete crushing is visible, where wide flexural cracks occur, and where most of the permanent rotation is concentrated around the wide flexural cracks so that the trend of the moment distribution has little effect; and the non-hinge r egion which applies to most of the length so that it is affected by the trend of the moment distribution, where there are much narrower cracks, where, in particular, concrete crushing does not occur and where standard procedures of equilibrium and compatibility can be applied (Oehlers et al 2005 & 2007, Oehlers 2006 and Mohamed Ali et al 2008a). The importance of the ductility of reinforced concrete members, such as that shown in Fig.1, has been recognised for over fifty years (Baker 1956 and Mattock 1964) and there is still much ongoing research on the topic (Griffith et al 2005, Cai et al 2006, Gilbert and Smith 2006, Marefat et al 2006, and Vintzileou et al 2007). However, there has been a major difficulty in developing a mechanism that can quantify the rotation of the hinge because of the concrete softening shown in the stress profile σsoft in Fig.1 (Fantilli et al 1998 & 2002, , Debernardi and Taliano 2002). It can be shown (Oehlers 2006, and Oehlers et al 2007) that this reduction in the material concrete stress in the softening branch requires a hinge of zero length; this problem of using a concrete compressive stress that reduces with increasing strain was recognised in the sixties by Barnard (1964), Barnard and Johnson (1965), and also by Wood (1968) who referred to the problem as “Some controversial and curious developments in the plastic theory of structures”.

117

To find a solution to the zero hinge length problem, research has mainly concentrated on quantifying the hinge length empirically with examples shown in chronological order in Table. 1. The aim was to find a length of hinge over which the curvature from a full-interaction analysis could be integrated to give the correct rotation. It can be seen that the major hinge length variables in Table 1 are the depth of slab d, diameter of the reinforcing bar db and yield strength of the reinforcing bar which probably control the rotation of the hinge region in Fig.1, and the span z or L which denotes the contribution to the rotation in the non-hinge region. No doubt these empirical hinge lengths give good agreement within the bounds of the test results from which they were derived but Panagiotakos and Fardis (2001) showed that they have to be used with caution outside their experimental bounds. Recently, a more advanced moment-curvature approach has been developed by Fantilli et al (2007) as shear-friction has been used in simulating the moment-curvature post peak behaviour and, furthermore, as shear-friction has also been used to quantify the hinge length in terms of the dimensions of the softening wedge.

Table 1. Empirically derived hinge lengths

Researcher Reference Hinge length (lp) Hinge length variables Baker (1956) ( ) ddzk 4/1/ span, depth Sawyer (1964) zd 075.025.0 + span, depth Corley (1966) ( )dzdd /2.05.0 + span, depth Mattock (1967) zd 05.05.0 + span, depth Priestley and Park (1987) yb fdL 022.008.0 + span, bar diameter Panagiotakos and Fardis (2001)

yb fdL 021.018.0 + span, bar diameter

non-hinge region hinge region

σnon-soft

non-hinge region

non-softening stress profile

softening stress profile

θ θ

σsoft

primary flexural crack

Fig. 1 Deformation zones in reinforced concrete slab There is a limit to the accuracy in using a combination of moment-curvature with hinge length to determine the rotation. This is because moment-curvature is a measure of the sectional ductility of a beam cross section and not of the member ductility that is the rotation which is required. As an illustration, the moment curvature relationship is generally derived assuming full-interaction between all components of the beam so that there is a uni-linear strain profile. This is fine up to concrete cracking but once a crack forms the crack can only widen, as in Fig.1, if there is slip between the reinforcing bar and the concrete. This opening shown as θ concentrates the rotation around the flexural crack (Haskett et al 2008b, and Mohamed Ali et al 2008a) which now depends on the bond characteristics between the reinforcing bar and the concrete

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which moment-curvature relationships cannot cope with. Hence, there is a need for a model that can directly simulate the moment-rotation about this primary crack which is the subject of this paper. A rigid body rotation mechanism which uses the established shear-friction and partial-interaction techniques is first described and this is then compared with test results to show that it can simulate not only the moment-rotation of a hinge but also the limits imposed on this rotation due to concrete wedge failure, and reinforcing bar fracture or debonding. It needs to be emphasised that the aim of this paper is to illustrate a novel moment-rotation model and that it is recognised that further research is required in quantifying the material properties used in the model.

2. Rigid body moment-rotation mechanism 2.1 Hinges in RC beams The hinge in the negative or hogging region of a steel plated reinforced concrete beam is shown in Fig.2 (Park and Oehlers 2000). It can be seen from the discontinuity of the slope of the beam that most of the permanent rotation is concentrated in the hinge region where there are three wide cracks. Most of the rotation of the hinge is due to the opening up or rotation of the crack faces (shown as 2θ in Fig.1) of the three wide cracks in Fig.2 as the close spacing of the cracks ensures that the concrete tensile stresses between these three flexural cracks, and consequently the curvatures, are very small (Mohamed Ali et al. 2008a).

Fig. 2 Discontinuity of slope at reinforced concrete hinge A close up view of a positive or sagging hinge region in a steel plated reinforced concrete beam is shown in Fig.3 (Park and Oehlers 2000). It can be shown both theoretically (Haskett et al 2008b) and through numerical simulations (Oehlers et al 2005, and Mohamed Ali et al. 2008a) that these three wide cracks occur within the zone affected by yielding and strain hardening of the reinforcing bar. A close up view of a hinge in a sagging or positive region of a reinforced ultra-high-performance concrete beam subjected to blast loads is shown in Fig.4 (Wu et al 2008). In this case, the hinge rotation is concentrated at a single flexural crack.

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Fig. 3 Hinge flexural cracks

Fig. 4 Hinge rotation mechanism Numerical simulations of reinforcement debonding (Oehlers et al 2005; and Mohamed Ali M.S. 2008c) have shown that the hinge moment-rotation behaviour when there is a single hinge crack as in Fig.4 is equal to or a lower bound to that which occurs when there are multiple hinge cracks as in Fig.3. Hence the hinge mechanism associated with a single crack will be used as the basis for the following mathematical model. It may be worth noting that the problem of multiple cracks will be the next stage of the research; the effect of multiple cracks on the hinge rotation can already be simulated in numerical models (Oehlers et al 2005; and Mohamed Ali M.S. 2008c) and there are already some structural mechanics models available for multiple cracks (Chen et al 2007). 2.2 Idealised beam hinge The hinge region in Fig.4 is characterised by a major flexural crack in the tension zone and the formation of wedges of concrete in the compression zone (Daniell et al 2008) as idealised in Fig.5. For the flexural crack in Fig.5 to widen, the reinforcing bar must slip relative to the concrete Δreinf-bar and this depends on the interface bond-slip τ-δ characteristics which have been quantified elsewhere (Eligenhausen et al 1983, CEB 1992, and Haskett et al 2008a,). The relationship between the reinforcing bar force Preinf-bar and the slip Δreinf-bar depends not only on the bond-slip characteristics but also on the reinforcing bar material properties such as the yield and fracture strains and the strain-hardening modulus; the dependence of the P:Δ characteristics on the material properties has also been quantified elsewhere (Haskett et al 2008b) using

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partial-interaction theory developed for fibre reinforced polymer plated structures (Mohamed Ali et al 2008b). Furthermore, this partial-interaction analysis also gives the slip at bar fracture and local bar debonding which is the limit to crack widening. Hence for a given crack height hcr, the relationship between the bar force Preinf-bar and crack rotation θ can be derived. Shear-friction theory has been used to quantify the force Psoft in Fig. 5 that a wedge of depth dsoft and length Lsoft can resist (Oehlers et al 2008a). This research has been validated from comparison with concrete prism tests (Mohamed Ali et al 2008c). The limit to the force Psoft depends on the sliding capacity of the wedge that is when ssoft in Fig.5 reaches its slip capacity sslide (Mohamed Ali et al 2008c). Hence the compressive force Psoft and tensile force Preinf-bar and their limits on the hinge rotation in Figs. 4 and 5 have been quantified elsewhere (Haskett et al 2008b, Mohamed Ali et al 2008c and Oehlers et al 2008a) but what is required is the interaction between the compression and tension zones which is the subject of this paper.

∆reinf-bar

θ

θrigid body rotation

concrete softening wedge

crack face

rigid body rotation

rigid body rotation

hcr

dsoftPsoft

Preinf-bar

ssoft

primary flexural crack

Lsoft

Fig.5 Idealised Rigid Body Hinge Rotation The rigid body movements of the hinge in the photograph in Fig.4 are shown pictorially in Fig.6. The crack faces O-E and O-F can be considered as rigid bodies that rotate about the apex of the crack O by an angle 2θ. This rigid body rotation of the crack must be accommodated by the triangular shaded deformation O-A-B in the compression zone. Between the crack apex O and the compression wedge C-D, the concrete is in the ascending branch of its stress-strain relationship so that the shaded deformation O-C-D in this region is accommodated by the usual flexural strains in the concrete which, as shown, have a peak stress fc and peak strain εpk adjacent to the wedge interface (Mohamed Ali et al 2008c and Oehlers et al 2007, 2008a, 2008b). The concrete wedges are also assumed to be rigid bodies, hence the shaded deformations A-B-C-D are accommodated by rigid body movements of the wedges such that they effectively slide C-G and G-D; this slip ssoft is equal to Lsoftεpk (Oehlers et al 2008a, and Mohamed Ali 2008a) as the strain in the adjacent wedge can be assumed to be relatively small. The rigid body deformation in Fig.6 is shown in an idealised form in Fig. 7 which can be used to determine the moment-rotation relationship. It can be seen how the rigid body sliding displacement shown as ssoft is directly related to the slip of the reinforcement Δreinf-bar and it is this geometric relationship which allows the moment-rotation to be determined as well as the limits

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to this rotation due to uncontrollable sliding of the wedge at sslide and the slip at fracture of the reinforcement or the slip at debonding of the reinforcing bars.

primaryflexuralcrack

compressionwedge

rigid body rotation of crack faces

θ

α

θ

ssoft

Lsoft

ε = 0

ε = 0

d/2

d/2 ∆t

d

A B

C D

G

O

E F

wedgeplane

fc at εpk

concretestress

ssoft

hsoft

θ

d/2

Fig.6 Rigid body displacement

∆reinf-bar

dsoft

dsoft/2

dasc

α

θ

Pasc

Psoft

Preinf-bar

ssoft

ssoft

hsoft

fc at εpk

concretestress

pivotal point

ascendingbranch9/24 dasc

A

Lsoft

C G

O

F

hcr

drebar

concretesofteningzone

concreteascendingzone

tensionzone

c

Fig. 7 Moment-rotation analysis

The analysis shown in Fig.7 is a rigid body analysis and should not be confused with a strain profile. The right hand side of the beam in Fig.6 is subjected to the displacements given by line A-O-F and that on the left hand side to the displacements given by line B-O-E. The beam in Fig.7 can be separated into three zones: the softening zone A-C of depth dsoft where the force in the concrete is the force the wedge can resist Psoft and which depends on the shear-friction properties and which has a limit that does not depend on the force Psoft but on the sliding capacity of the

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wedge interface sslide; the compression zone C-O of depth dasc where the concrete is in its ascending portion of its stress-strain relationship, as shown, and where the peak stress is fc and the peak strain is εpk; and finally the tension zone O-F of crack height hcr where the tension forces in the reinforcing bars are governed by partial-interaction slip theory and where it is assumed that the tensile strength of the concrete is zero. The detailed analysis is given below. 3. Rigid body rotation analysis The rigid body rotation analysis depicted in Fig.7 consists of deriving the moment-rotation for increasing values of dsoft and is ideally suited for a spreadsheet analysis. For a specific value of dsoft, ssoft is determined and this is then used as a pivotal point to rotate the line A-F until longitudinal equilibrium of the forces P is achieved from which the moment can be determined. The equations for deriving these forces are given below. However, the derivation of these equations and the fundamental principles used in deriving them have been published elsewhere (Oehlers et al 2007 & 2008a, and Haskett et al 2008a & 2008b). 3.1 Concrete compression softening zone For a specific value of the depth of the wedge dsoft in Fig.7, the slip of the wedge is

pksoftsoft Ls ε= (1) where εpk is the strain at which the peak concrete stress fc is achieved in the concrete ascending branch as shown in Fig. 7 which for unconfined concrete has a typical value of 0.002, and where the length of the wedge Lsoft = dsoft/tanα and in which the angle of inclination of the wedge is given by

( )1arctan 2 ++−= mmα (2) in which m is the frictional component of the Mohr-Coulomb failure plane and which for concrete has a typical value of 0.8 (Walraven et al 1987, Duthinh 1999, and Mattock et al 1988) such that α in Eq.2 is typically 26o. When ssoft reaches the sliding capacity sslide which is a material property determined through tests, then the wedge fails by sliding. The force in the wedge is given by

( )( )

−

++=

αααααασ

sincossincossincos

mmcdwP lat

softbsoft (3)

where wb is the width of the beam, c is the cohesive component of the Mohr-Coulomb failure plane which has a typical value of 0.17fc (Jensen 1975), and σlat is the lateral confinement to the wedge such as might be induced by stirrups; a conservative assumption would be to assume σlat = 0. The wedge force acts at dsoft/2 as uniform shear is assumed along the wedge interface. It can be seen that the wedge does not fail when its strength Psoft in Eq. 3 is achieved but it fails when its slip ssoft in Eq.1 reaches the slip capacity sslide. This is analogous to the failure of mechanical stud shear connectors in composite steel and concrete beams

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(Oehlers and Bradford 1995; and Oehlers and Sved 1995) in which failure is not caused by the strength of the stud shear connector being exceeded but by the slip capacity of the shear connector being exceeded. It may also be worth noting that the force in the wedge Psoft in Eq.3, is a generic expression as it can be applied to any type of concrete in which the shear-friction material properties m and c can be derived by testing. 3.2 Concrete ascending compression zone Any appropriate variation of the ascending stress-strain relationship in Fig.7 can be used. It may be convenient to use a parabolic distribution such as Hognestad’s (1955) in which case the compressive force in the concrete below the wedge or softening region is given by

ascbcasc dwfP32

= (4)

which occurs at (9/24)dasc from the wedge interface. 3.3 Tension zone The slip between the reinforcing bar and the concrete in Fig.7 is half the crack width which and is shown as Δreinf-bar. Furthermore, because the bar slides through the concrete, the interface bond characteristics have the form shown in Fig.8, that is the interface bond shear stress τ depends on the interface slip δ. From partial-interaction theory (Haskett et al 2008b) which allows for the interface bond slip, the slip that is required to cause an individual reinforcing bar to yield is given by

( )( )elelyield aλδ cos1max −=∆ (5) where δmax is the interface bond slip capacity of the reinforcing bar in Fig.8 which typically has a value of 15mm (Haskett et al 2008a), where

bars

perel AE

L

max

max

δτ

λ = (6)

in which Lper and Abar are the perimeter and cross-sectional area of the bar respectively, τmax is the interface bond-slip shear capacity in Fig.8 which has a typical value of 1.25√f c for units of N and mm (Haskett et al 2008b), and Es the elastic modulus of the reinforcement steel in Fig.9, and where in Eq.5

el

per

elybar

el

LfA

aλ

τλ

= max

arcsin (7)

where fy is the yield strength of the reinforcing bar in Fig.9.

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τmax

δmax

τ

δ0

uni-linear

typical bi-linear

Fig. 8 Idealised bond slip

fy

ffract

εy εfract

Es

Esh

0

stress

strain

Fig. 9 Idealised steel properties The slip to cause an individual bar to fracture is given by

( )( )fractshyieldfract aλδ cos1max −+∆=∆ (8) where

barsh

persh AE

L

max

max

δτ

λ = (9)

in which Esh is the strain hardening modulus of the reinforcing bar in Fig.9, and where

( )

sh

per

shyfractbar

fract

LffA

aλ

τλ

−

=max

arcsin

(10)

where ffract is the ultimate strength of the reinforcement in Fig.9. The slip to cause a reinforcing bar to debond is simply equal to δmax. Hence, if the slip in the reinforcement Δreinf-bar in Fig. 7 is less than both Δyield in Eq.5 and δmax, then the reinforcing bar has neither yielded nor debonded so that the force in the reinforcing bar is given by

∆−= −

−−max

infmaxinf 1arccossin

δλτ barre

el

perelbarre

LP (11)

If the slip in the reinforcement Δreinf-bar in Fig. 7 is greater than Δyield in Eq.5 and less than that to cause fracture Δfract in Eq.8 and that to cause debonding δmax, then the force in the reinforcing bar that undergoes strain hardening is given by

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ybaryieldbarre

sh

pershbarre fA

LP +

∆−∆−= −

−−max

infmaxinf 1arccossin

δλτ

(12)

Obviously, if the slip reaches Δfract then the bar fractures and if it reaches δmax the bar debonds which are, therefore, two limitations to the rotation capacity, and the third limitation being the sliding capacity of the wedge sslide as previously discussed. Although the above equations appear complex, it may be worth noting that they simply give the relationship between the force in the bar Preinf-bar in Fig.7 and the slip at the crack interface Δreinf-bar, as well as the upper bounds to this slip which occurs when either the bar fractures at a slip Δreinf-bar equal to Δfract or debonds at a slip δmax. It may also be worth noting that the above equations are also generic as they can in theory be applied to any type of reinforcement such as round bars or externally bonded plates made with steel or fibre reinforced polymer (Haskett et al 2008a; Seracino et al 2007), and also to any type of concrete in which the material properties τmax and δmax can be determined from tests. 4. Comparison with test results The moment-rotation model in Section 3 underlines the complexity of the problem. It can be seen that to obtain the moment-rotation relationship, and in particular the limits to the rotation, requires an estimation of the shear-friction material properties of the concrete (m, c and sslide), and an estimation of the reinforcement material bond characteristics (τmax and δmax) both of which are rarely measured in practice, and also more straightforward material parameters (εpk, Es, Esh, ffract and fy) which are often not recorded. Quantifying many of these material properties, in particular the shear-friction and partial-interaction bond properties, are still the subject of extensive on-going research which makes the following comparison with test results difficult. However, an attempt is made by using general estimates of these material properties in order to illustrate that the rigid-body rotation component of the moment-rotation model is quite accurate considering that the softening shear-friction component has already been validated by comparison with 58 test results (Mohamed Ali et al 2008c) and the partial-interaction reinforcing bar component has also been validated by comparison with numerous beam and pull tests (Haskett et al 2008a & 2008b, and Mohamed Ali et al2008a). It is recognised that more research effort is required in quantifying the relatively novel material properties for the application of the moment-rotation model. The moment-rotation curve can be derived from the analysis depicted in Fig.7 where the rigid body displacement is represented by the line A-O-F. The angle θ is first fixed at a small value, then the rigid body displacement line A-O-B is moved up or down, maintaining θ at its fixed value, until the algebraic sum of the forces Psoft, Pasc and Preinf-bar equals zero. Once equilibrium is achieved, the moment of the forces P can be taken to derive the moment M at the chosen rotation θ. The angle θ can then be increased and the procedure repeated to get the moment-rotation response. This moment-rotation response has three limits: when the bar slip Δreinf-bar equals that to cause fracture Δfract in Eq.8; or when the bar slip equals that to cause debonding which as a lower bound is given by δmax in Fig.8; or when the slip ssoft in Fig.7, which can be derived from Eq.1, reaches its sliding capacity sslide which is a material property that can be determined from tests. It is also worth noting that there is a fourth limit to the

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moment-rotation due to shear failure which this model can quantify (Oehlers et al 2008c) but which is not the subject of this paper. 4.1 Mattock’s hinge tests In a comprehensive and invaluable study, Mattock (1964) investigated experimentally the rotational capacity of RC beams. The concrete strength fc, depth of beam d, percentage (%) reinforcement, yield stress of reinforcement σy, stirrup confinement ratio, and span of the beam L were varied as shown in Table 2. However, the bar diameter remained at 19 mm (No. 6 bars) and the width of the beam at 152 mm (6 inches). The moment-rotation model developed above has been compared with the thirty-one beam test results in which a concentrated load was applied at mid-span of simply supported beams. Mattock measured the rotation of the whole beam, that is the rotation in the hinge and non-hinge region in Fig.1. Mattock also measured the rotation over a length half the beam depth (d/2) on either side of the concentrated load as shown in Fig. 6; this rotation is the hinge rotation as shown in Figs.3 and 4 and which is modelled in the moment-rotation model in Fig.7. Hence, the moment-rotation model is compared directly with Mattock’s hinge rotations.

Table 2: Mattock’s test results (1964) 1 2 3 4 5 6 7 8 9% MPa mm - mm MPa MPa MPa MPa%st fc d σlat/fc L σy Es σult Esh

A1 1.47% 38.2 279.4 0.07 1397 315 195811 576 5931A2 1.47% 42.3 279.4 0.03 2794 318 195811 576 5910A3 1.47% 41.0 279.4 0.03 5588 336 195811 576 5761A4 2.95% 42.9 279.4 0.12 1397 315 195811 576 5931A5 2.95% 39.6 279.4 0.06 2794 314 195811 576 5936A6 2.95% 41.1 279.4 0.03 5588 328 195811 576 5829B1 1.47% 43.0 533.4 0.06 2794 329 194432 576 5834B2 1.47% 41.8 533.4 0.03 5588 322 194432 576 5884B3 2.95% 42.9 552.5 0.12 2794 321 194432 576 5889B4 2.95% 42.8 552.5 0.06 5588 323 194432 576 5879C1 1.47% 27.4 279.4 0.09 1397 329 195122 576 5130C2 1.47% 26.0 279.4 0.06 2794 329 195122 576 5130C3 1.47% 25.6 279.4 0.05 5588 330 195122 576 5126C4 2.95% 25.9 279.4 0.19 1397 325 195122 576 5153C5 2.95% 23.4 279.4 0.11 2794 328 195122 576 5135C6 2.95% 27.4 279.4 0.05 5588 319 195122 576 5192D1 1.47% 26.7 533.4 0.10 2794 319 194432 576 5252D2 1.47% 25.6 533.4 0.05 5588 316 194432 576 5269D3 2.95% 26.0 552.5 0.19 2794 320 194432 576 5248D4 2.95% 26.9 552.5 0.09 5588 322 194432 576 5234E1 1.47% 27.9 279.4 0.13 1397 404 192364 707 7301E2 1.47% 28.3 279.4 0.12 2794 414 192364 707 7237E3 1.47% 29.8 279.4 0.12 5588 412 192364 707 7246F1 1.47% 41.2 279.4 0.08 1397 404 192364 707 7013F2 1.47% 41.4 279.4 0.09 2794 415 192364 707 6946F3 1.47% 42.9 279.4 0.09 5588 415 192364 707 6946G1 1.10% 27.4 533.4 0.13 2794 414 197190 707 7003G2 1.10% 28.3 533.4 0.13 5588 414 197190 707 7007G3 1.47% 28.8 533.4 0.12 2794 415 197190 707 6998G4 1.47% 27.2 533.4 0.14 5588 415 197190 707 6998G5 0.74% 27.4 533.4 0.14 5588 417 197190 707 6985

Beam

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Mattock gave moment-curvature results in his paper (1964). However, as the effective compression face strain is simply ssoft/(d/2) as shown on the left hand side of Fig.6 and the effective tension face strain is Δt/(d/2), then dividing the sum of these strains by the beam depth d gives the curvature χ. This expression can be rearranged as

dsd tsoft ∆+

=2

χ (13)

which is the rotation. Hence Mattock’s curvatures can by multiplied by d/2 to obtain the rotation of the hinge directly for comparison with the moment-rotation model. In the following comparisons, the moment-rotation model uses the reported test beam properties listed in Table 2. However, the concrete modulus has been assumed to be Ec=3320√fc = 6900 MPa (ACI Committee 363 1992), the maximum bond stress τmax is given by 1.25√f c which is half of the value recommended by Model Code 90 (CEB FIB 1992) as this reduced bond stress was found to give good correlation with beam tests (Haskett et al 2008b, and Eligehausen et al 1983) and the bond slip capacity as δmax = 15mm (Haskett et al 2008a) 4.2 Moment-rotations As discussed previously, three modes of failure are possible: sliding failure of the wedge which occurs when the softening wedge slip exceeds sslide,; fracture of the reinforcing bar when the slip of the bar exceeds ∆fract; or debonding of the bar when the slip exceeds δmax. Debonding of reinforcing bars is highly unlikely due to the significant slip capacity of deformed steel reinforcing bars and debonding was not observed either experimentally nor in the theoretical simulations. Mattock (1964) furnished six moment-curvature relationships which have been converted into moment-rotation relationships in Figs. 10 to 15 and where they are also compared with the theoretical moment-rotation relationship. The symbol in the graphs of a black cross with a grey box around it, which is labelled “experimental failure – from moment and rotation”, refers to the experimental value of the moment and rotation of the hinge of length d/2 tabulated by Mattock (1964). Mattock reported that experimental failure always occurred due to “concrete crushing after the tension reinforcement had yielded by varying amounts”. This type of ‘failure’ corresponds to the formation of the softening wedges as in Figs. 1 and 4 and not necessarily wedge sliding failure at a slip sslide as Mattock’s results do not show a falling branch, so it is difficult to compare Mattock’s rotation capacities directly with those obtained from the moment-rotation model. Furthermore, the analysis can, on occasion, be sensitive to the sliding capacity assumed. For example in Specimen C4 in Fig.13, it can be seen that if a sliding capacity of sslide=0.87mm is assumed (shown as a white dot) then a rotation at sliding failure of 0.06 radians is achieved which is in close agreement with the test data. Alternatively, if the sliding capacity sslide, is increased to 1.09mm (shown as a black dot) which allows a large amount of strain hardening and consequentially rotation to occur then the rotation at sliding failure is increased 14 fold to 0.85 radians.

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0

20

40

60

80

0.000 0.025 0.050 0.075 0.100

Rotation (radians)

Mom

ent (

kN-m

)


Fig. 10 Specimen C1 moment-rotation

0

20

40

60

80

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Rotation (radians)

Mom

ent (

kN-m

)



129

0

20

40

60

80

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Rotation (radians)

Mom

ent (

kN-m

)

EXPERIMENTAL - SPECIMEN C3THEORETICAL - SPECIMEN C3

FRACTUREEXPERIMENTAL FAILURE - FROM MOMENT AND ROTATION


0

20

40

60

80

100

120

140

0.00 0.20 0.40 0.60 0.80 1.00

Rotation (radians)

Mom

ent (

kN-m

)

EXPERIMENTAL - SPECIMEN C4THEORETICAL - SPECIMEN C4FRACTURESLIDING - 1.09mmEXPERIMENTAL FAILURE - FROM MOMENT AND ROTATIONSLIDING - 0.87mm


130

0

20

40

60

80

100

0.000 0.020 0.040 0.060 0.080 0.100

Rotation (radians)

Mom

ent (

kN-m

)


SLIDINGEXPERIMENTAL FAILURE - FROM MOMENT AND ROTATION


0

20

40

60

80

100

0.000 0.010 0.020 0.030 0.040

Rotation (radians)

Mom

ent (

kN-m

)


SLIDINGEXPERIMENTAL FAILURE - FROM MOMENT AND ROTATION


In the absence of the necessary material data, it may be best to compare the overall moment-rotation relationship in Figs. 10 to 15. Specimens C1, C2 and C3 in Figs. 10-12 were under-reinforced and there is reasonably good agreement particularly at high rotations. The discrepancy at the start where yielding first occurs is partly because the moment-rotation model uses a bi-linear steel stress-strain relationship as shown in Fig.9 so it tends to over-estimate the stiffness during yielding. The discrepancy is also because the moment-rotation model assumes a uni-linear bond-slip relationship as

131

shown in Fig.8 in order to develop a closed form solution which also tends to over-estimate the initial stiffness but has little effect at high rotations. Specimens C4, C5 and C6 in Figs. 13-15 were heavily reinforced and there is also good agreement when comparing the shapes except possibly for C4 in Fig.13 which may be an outlier as it is much stronger than expected. In general, it is suggested that there is reasonably good correlation in the moment-rotation relationships considering the huge scatter of results associated with quantifying the hinge rotations (Panagotiokas and Fardis 2001). Specimens C1, C2, and C3 in Figs. 10 to 12 were all significantly under-reinforced and fracture of the bars was predicted prior to wedge sliding failure at sslide. In under reinforced sections, the force that can be developed in the reinforcing bars is small, and this limits the depth of the softening wedge that can form, thereby reducing the probability of sliding failure. As the section became more heavily reinforced (sections C4, C5, and C6 in Figs. 13 to 15), larger tensile forces could be developed in the bars, allowing a greater depth of concrete wedge to form and hence increasing the probability of sliding failure at sslide. 4.3 Sliding rotation capacity Previous research (Martinez et al 1984, and Mattock and Hawkins 1972) has suggested that the unconfined sliding capacity of the concrete wedge (sslide) is approximately 0.4mm. This value for sslide may not be appropriate in the analysis of Mattock’s results (1964) because of confinement due to stirrups σlat which may increase sslide (Mohamed Ali et al 2008c) As an initial estimate and as a lower bound, the unconfined sliding capacity of the wedge was considered to be sslide=0.4mm in the rotation model. A comparison in Fig.16 of the test rotations in column 1 in Table 3 and the theoretical rotations in column 2 show that this value of sslide gives a lower bound. The direct comparison in column 3 shows that the experimental rotation is 3.2 times the theoretical rotation with a standard deviation of 2.0. Clearly the confinement provided by the stirrups significantly increases the sliding capacity beyond the unconfined wedge sliding capacity of 0.4mm.

Fig. 16 Theoretical rotations based on lower bound sslide

132

Table 3 Analysis of Mattock’s results (1964) 1 2 3 4 5 6 7 8 9 10 11 12

radians sslide=0.4mm kN-m kN-m kN-m kN-m sslide=2.5σlat/fc+0.6 Theo failure mode

θu(Exp) θu(Theo) My(Exp) Mult(Exp) My(Theo) Mult(Theo) θu(Theo)A1 3.60E-02 2.87E-02 1.25 42.2 73.4 41.1 70.2 1.03 1.05 2.460E-02 1.46 FRACTUREA2 2.05E-02 3.09E-02 0.66 44.3 63.6 39.8 71.4 1.11 0.89 2.164E-02 0.95 FRACTUREA3 1.68E-02 2.82E-02 0.60 44.3 59.2 42.8 70.7 1.04 0.84 1.888E-02 0.89 FRACTUREA4 3.05E-02 7.92E-03 3.85 76.4 124.1 77.3 119.6 0.99 1.04 3.659E-02 0.83 FRACTUREA5 2.07E-02 7.48E-03 2.77 76.3 100.5 78.0 113.8 0.98 0.88 3.522E-02 0.59 SLIDNGA6 1.18E-02 7.39E-03 1.60 78.3 86.3 79.8 113.6 0.98 0.76 2.443E-02 0.48 SLIDNGB1 2.77E-02 5.81E-03 4.77 177.7 263.2 172.6 287.3 1.03 0.92 9.854E-03 2.81 FRACTUREB2 1.52E-02 5.77E-03 2.63 170.3 212.3 169.9 286.3 1.00 0.74 1.074E-02 1.42 FRACTUREB3 2.18E-02 2.55E-03 8.53 298.9 427.4 322.9 447.7 0.93 0.95 8.149E-03 2.68 SLIDNGB4 1.20E-02 2.54E-03 4.72 296.8 330.3 322.5 434.8 0.92 0.76 5.909E-03 2.03 SLIDNGC1 7.18E-02 2.23E-02 3.22 41.8 69.4 41.9 64.6 1.00 1.08 3.679E-02 1.95 FRACTUREC2 2.58E-02 2.13E-02 1.21 42.1 58.8 42.0 63.9 1.00 0.92 4.204E-02 0.61 FRACTUREC3 1.87E-02 2.10E-02 0.89 40.3 51.3 41.6 63.4 0.97 0.81 4.139E-02 0.45 FRACTUREC4 2.97E-02 5.37E-03 5.53 78.3 115.2 71.8 88.5 1.09 1.30 3.596E-01 0.08 SLIDNGC5 1.30E-02 4.98E-03 2.61 80.5 89.2 69.3 78.8 1.16 1.13 6.481E-02 0.20 SLIDNGC6 1.25E-02 5.70E-03 2.19 72.8 75.5 72.2 87.0 1.01 0.87 2.760E-02 0.45 SLIDNGD1 2.02E-02 4.18E-03 4.84 161.5 221.3 160.7 255.1 1.01 0.87 1.940E-02 1.04 SLIDNGD2 1.44E-02 4.07E-03 3.54 163.6 198.7 162.0 238.5 1.01 0.83 1.270E-02 1.13 SLIDNGD3 1.47E-02 1.79E-03 8.22 302.3 367.3 286.7 330.8 1.05 1.11 8.831E-03 1.66 SLIDNGD4 7.90E-03 1.82E-03 4.33 305.0 324.2 292.1 332.5 1.04 0.98 5.157E-03 1.53 SLIDNGE1 2.77E-02 1.42E-02 1.96 50.9 78.6 49.7 74.8 1.02 1.05 4.879E-02 0.57 FRACTUREE2 2.79E-02 1.39E-02 2.01 52.7 73.7 51.2 75.3 1.03 0.98 4.498E-02 0.62 FRACTUREE3 1.55E-02 1.45E-02 1.07 51.5 62.6 51.8 76.4 0.99 0.82 4.028E-02 0.38 FRACTUREF1 3.66E-02 1.99E-02 1.84 53.3 87.7 53.0 83.9 1.01 1.05 2.983E-02 1.23 FRACTUREF2 2.58E-02 1.93E-02 1.34 53.8 77.8 53.1 83.9 1.01 0.93 2.758E-02 0.94 FRACTUREF3 2.28E-02 1.99E-02 1.15 55.6 73.1 54.1 84.3 1.03 0.87 2.598E-02 0.88 FRACTUREG1 2.27E-02 4.73E-03 4.80 165.9 239.2 158.5 244.2 1.05 0.98 1.893E-02 1.20 FRACTUREG2 2.43E-02 4.87E-03 4.98 162.5 229.9 161.7 246.1 1.00 0.93 1.804E-02 1.35 FRACTUREG3 1.42E-02 3.33E-03 4.27 208.8 269.2 207.8 292.6 1.00 0.92 1.548E-02 0.92 SLIDNGG4 1.23E-02 3.19E-03 3.86 217.5 250.8 206.9 286.6 1.05 0.88 1.648E-02 0.75 SLIDNGG5 3.42E-02 9.25E-03 3.70 113.6 178.4 108.0 177.0 1.05 1.01 1.531E-02 2.23 FRACTURE

θu(Exp)/θu(Theo)

Beam θu(Exp)/θu(Theo)

My(Exp)/My(Theo)

Mult(Exp)/Mult(Theo)

133

Sliding failure is unlikely to occur in lightly reinforced sections as mentioned previously. For example, let us consider a beam with large rotations and in which the reinforcing bars are about to fracture. In this case, the force in the ascending branch Pasc in Fig. 7 is much less than Psoft so that all of the compressive force can be assumed to be acting as Psoft and equal to the fracture strength of the reinforcing bars Prebars. From the analyses in Section 3.1, it can be shown that the maximum length of the softening zone required when all the bars are near fracture is given by

( )( )

−

++=−

αααααασ

αsincossin

cossincostan

mmc

w

PL

latb

rebarsreqsoft (14)

However, the length of the softening zone to cause sliding is given by

pk

slidecapsoft

sL

ε=− (15)

Hence if Lsof-req < Lsoft-cap for a given sslide, then sliding failure will never occur and bar fracture (or debonding which is unlikely) will cause failure instead. Having recognised that sliding failure does not always occur, an iterative procedure was used to determine the sliding capacity sslide of concrete confined by stirrups as this had been previously observed in spirally reinforced cylinder tests (Mander et al 1988) from Mattock’s test results. It was recognised that the degree of confinement influences the sliding capacity of the wedge and that the concrete cohesive component c is a function of fc so the confinement provided by the stirrups σlat was normalised by the compressive strength of the concrete fc. Hence stirrup confinement σlat can be written as

pc

ststy

c

lat

sbfA

f)(2σσ

= (16)

where sp is the spacing of stirrups, σy(st) is the yield stress of the stirrups, and it is assumed that 2 stirrup legs confine the concrete. By assuming that the experimental rotation θu is the rotation at which sliding occurs, then from the moment-rotation model the value of ssoft when θu is achieved is sslide. Initially all 31 tests were considered to have failed by sliding and the results are shown in Fig. 17. From the linear regression in Fig. 17, as a first approximation the sliding capacity sslide is

42.0f

76.2sc

latslide +

σ= (17)

where the coefficient of correlation is 0.20 and the units are in mm.

134

0.00

0.40

0.80

1.20

1.60

0.00 0.05 0.10 0.15 0.20 0.25σlat/fc

sslid

e(Ex

p)

Fig. 17 Derivation of sslide from all test rotations

This appears to be a reasonable initial approximation for the sliding capacity of the concrete wedge because previous research suggests the unconfined concrete sliding capacity is approx 0.4mm (Martinez et al.1984, and Mattock and Hawkins 1972). Now assuming that Eq.17 reflects the influence of confining pressure on the sliding capacity, the moment-rotation analysis approach was repeated and results omitted for tests in which bar fracture preceded sliding failure to produce a new equation for sslide. The procedure was repeated until only sliding failure occurred to give the results in Fig.18 and the following sliding capacity where the coefficient of correlation is 0.22 and the units are in mm.

61.0f

51.2sc

latslide +

σ= (18)

135

0.00

0.40

0.80

1.20

1.60

0.00 0.05 0.10 0.15 0.20 0.25σlat/fc

sslid

e(Ex

p)

Fig. 18 Variation in sslide with stirrup confinement

The theoretical rotation capacities based on the sliding capacities in Eq.18 are compared with the experimental rotations in Fig.19. As would be expected, the smaller rotations are associated with wedge sliding failure and the larger rotations with bar fracture. It is suggested that these rotation capacities are in reasonably good agreement bearing in mind the large scatters associated with rotation capacities (Panagotiokas and Fardis 2001) and also bearing in mind that the predominant material properties that affect rotation, that is the wedge sliding capacities sslide and the reinforcement bar bond characteristics τmax-δmax, have still to be quantified accurately.

136

0.00

0.01

0.02

0.03

0.04

0.05

0.00 0.01 0.02 0.03 0.04 0.05(Theo)

(Exp

)FRACTURE

SLIDING

Theoretical failure mode:

Fig.19 Comparison of experimental and theoretical rotations

4.4 Moment capacities Mattock (1964) noted that after yielding the cracks widened rapidly and this was also observed in the moment-rotation model. The experimental moment at yield is given in column 4 in Table 3 and the theoretical moment at yield from the moment-rotation model in column 6. The results are compared in column 8 with an average experimental to theoretical value of 1.02 with a standard deviation of 0.05. The correlation is very good as shown in Fig. 20. There was concern that using the ascending stress-strain relationship at its peak strain εpk as shown in Fig. 7 would overestimate the force in the ascending zone at low rotations when the strains may not have expected to reach their peak value. It can be seen that this has little effect if any on the moments at yield. The experimental moments at failure are given in column 5 in Table 3. The theoretical values from the moment-rotation model are given in column 7 where theoretical failure can occur through wedge sliding at sslide given by Eq. 18 or by bar fracture. The results are compared in column 9 in Table 3 where the mean is 0.94 with a standard deviation of 0.12 and also plotted in Fig. 21 which also shows good correlation.

137

0

100

200

300

400

0 100 200 300 400My(Theo) (kN-m)

My(

Exp)

(kN

-m)

Fig.20 Comparison between experimental and theoretical moment at yield

0

100

200

300

400

500

0 100 200 300 400 500Mult(Theo)

Mul

t(Exp

)

FRACTURESLIDING

Theoretical failure mode:

Fig. 21 Comparison between theoretical and experimental moment capacities

It has been shown that the rigid body rotation model reasonably predicts the rotation at failure and also the entire moment-rotation response considering the difficulty in quantifying experimentally the definitive rotation limit. It has now been shown that it also provides good estimates of the moment capacities.

138

Conclusions Previous published research by the authors has quantified the rotations within a crack using partial-interaction theory that allows for slip between the reinforcing bar and the concrete and which depends on the interface bond-slip characteristics as well as the yield and fracture strain of the reinforcing bar. Previous published research by the authors based on shear-friction theory has quantified the capacities of the concrete wedges in the compression zone and also the limit to the wedge compression force that occurs when the sliding capacity is reached. The present research links these two components together by using a novel rigid body displacement mechanism which can quantify not only the moment-rotation of the hinge but also the limits to the rotation due to either wedge sliding, reinforcing bar fracture or reinforcing bar debonding.

This new moment-rotation model was compared with 31 beam test results where hinge rotation was measured directly and was found to have good correlation with regard to moment capacity and rotation capacity as well as the moment-rotation behaviour up to failure. This new moment-rotation approach brings together partial-interaction theory, shear-friction theory and rigid body displacements to form a novel hinge model that allows for: any cross-sectional properties and geometries; concrete softening and flexural cracks; and reinforcing bar bond-slip characteristics, yield and fracture strains. This new moment-rotation model provides a tool for the design of reinforced concrete members for ductility such as for moment redistribution, column drift and the ability to absorb dynamic loads. 6. Acknowledgements This research was supported by an Australian Research Council Discovery Grant DP0663740 “Development of innovative fibre reinforced polymer plating techniques to retrofit existing reinforced concrete structures.”

139

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142

CHAPTER 6:

SYNOPSIS - THE DISCRETE ROTATION IN REINFORCED CONCRETE BEAMS

In Chapter 5, an iterative analysis technique was described to determine the moment-rotation behaviour of a reinforced concrete member. In this chapter, generic closed form mathematical expressions, that is a non-iterative technique, are presented to determine the moment-rotation response of a reinforced concrete member. Closed form mathematical moment-rotation expressions are also developed for the case where a softening wedge has not formed, that is the strain in the concrete is less than εpk. This was not the case in the iterative procedure outlined in Chapter 5 where it was always assumed that a softening wedge had formed and the slip of the softening wedge, ssoft, was used as the pivotal point in the analysis. Depending on the geometry and material properties of the beam, bar yield may or may not have occurred. If the slip of the reinforcement is less than that required to yield, then the load in the reinforcement Prebar, as a function of the slip is obtained from Equation 9 and the corresponding strain in the reinforcement is simply the force in the bar divided by the cross sectional area and Young’s modulus. From compatibility and equilibrium conditions and ignoring the tensile strength of the concrete, the force in the concrete is equal to the sum of the forces in the reinforcing bars. The corresponding compression force is simply the integration of the stress profile over the uncracked region of concrete. From equilibrium:

0bdf

Pεε3ε

bdfPε3

εε3εc

rr2pk

max,cc

r2pk2

max,cpk3

max,c =++−

Equation 29 out of the possible three roots for a cubic equation, the real solution yields the value of εc,max. From geometry the depth of the region of non softening concrete is therefore

( )rmax,c

max,casc εε

dεd

+=

Equation 30 where d is the effective depth of the section, εc,max is known from Equation 29 and εr is the known strain in the reinforcing bar. Now that εc,max and dasc are known, the corresponding moment is obtained by taking moments about the neutral axis of the compression force in concrete and the tension force in the reinforcement as

143

)dd(Pε4

ε32

εεbdf

)dd(Pydyεd

yεεd

yε2bfM

ascrpk

max,c

pk

max,c2ascc

ascr

d

0

2

pkasc

max,c

pkasc

max,cc

asc

−+

−=

−+

−

= ∫

Equation 31 and the rotation is

)dd(Δθ

asc

r

−=

Equation 32 If the slip of the reinforcement is greater than Δyield, but εc,max is less than εpk, then the load in the bar, as a function of the slip is simply determined from Equation 14 and the corresponding strain in the reinforcement is determined from the material stress strain relationship. The remaining procedure is identical to determine the moment and rotation. When the maximum strain in the concrete εc,max exceeds εpk in Figure M, then a softening wedge forms. In this case, the compression force in the concrete comprises two components: the first one due to the ascending branch of stress-strain curve where Pasc of depth dasc which can be estimated as discussed before; and the second component due to the softening of concrete Psoft with a corresponding depth dsoft associated with a wedge slip ssoft. The softening force Psoft for a given depth of softening dsoft is determined from Equation 23. For commonly accepted shear friction values Psoft is approximately 70%fc. For a given bar slip Δrebar in Figure L, derive Prebar from either Equation 9 or Equation 14. For equilibrium Prebar =Pcc= Pasc+Psoft Equation 33 The relative magnitudes of Psoft and Pasc are unknown at present since the depth of softening dsoft and the depth of the non softening region dasc are both unknown. The wedge displacement ssoft can be derived by integrating the slip strain across the wedge sliding plane where the slip-strain is given by

wedgepkslip εεε −= Equation 34 where εwedge is the axial strain in the wedge due to Psoft based on the ascending concrete stress-strain relationship as shown in Figure M. Hence the wedge slip is given by

144

αtanEdf2εLs

c

softcpksoftsoft ==

Equation 35 As the slip in the reinforcement Δrebar is compatible with the slip in the softening wedge ssoft

)ddd(Δ

αtandEdf2

ds

ascsoft

rebar

ascc

softc

asc

soft

−−==

Equation 36 For compatibility of strain and force equilibrium, the following quadratic equation can be obtained to solve for dasc

0bdfkPbdfEP16.4

667.0E

Δ707.0d

Ef77.2

csrebarasccc

rebar

c

rebar2asc

c

2c =−+

−−+

Equation 37 As dasc is known, then dsoft can be estimated as

bf707.0bdf667.0P

dc

asccrebarsoft

−=

Equation 38 The corresponding moment and rotation while the concrete undergoes softening are given by

crrebar2asccsoftascsofttcsoft hPbdf

85)d5.0d(bdf707.0M +++=

Equation 39

)ddd(Δ

θascsoft

rebarsoft −−

=

Equation 40 As the reinforcement slip is used as the variable in the above derivations, it is easy to impose the following limits to the reinforcement slip. Ultimately failure occurs when one of the following limits is reached: 1) Wedge sliding failure occurs at a slip of sslide which is experimentally determined. The corresponding reinforcement slip at wedge sliding from geometry is

asc

ascsoftslidesliderebar d

)ddd(sΔ

−−=−

Equation 41

145

2) Fracture of the reinforcing bar occurs at a reinforcement slip of ∆yp-ult from Equation 18. 3) Bar debonding occurs at a slip of δmax. In this chapter closed form mathematical expressions were developed for the moment-rotation response of reinforced concrete members. This chapter is important because it provides designers with a tool to evaluate the moment and rotation at failure only, whether that be through debonding or fracture of the plate or reinforcing bar, or through concrete failure. This is a less onerous analysis technique and will provide designers with the ability to evaluate the rotation capacity of a reinforced concrete member at failure only. This closed form approach to moment-rotation is not used subsequently in this thesis. Instead, the iterative moment-rotation technique described in Chapters 1 and 5 is used to develop the theoretical moment-rotation responses following.

146

Statement of Authorship The discrete rotation in reinforced concrete beams 1Mohamed Ali M.S., 2D.J. Oehlers, 3M. Haskett and 4M.C. Griffith 1Dr. Mohamed Ali,M.S. Senior Research Associate, School of Civil and Environmental Engineering University of Adelaide Wrote manuscript, developed closed form mathematical expressions SIGNED____________________ 2Professor Deric J. Oehlers School of Civil and Environmental Engineering University of Adelaide Supervised research, reviewed manuscript SIGNED____________________ 3Mr. Matthew Haskett PhD student School of Civil and Environmental Engineering University of Adelaide Developed discrete moment-rotation model, developed iterative moment-rotation solution technique, reviewed manuscript SIGNED____________________ 4Associate Professor Michael C. Griffith School of Civil and Environmental Engineering University of Adelaide Submitted ASCE Journal of Engineering Mechanics 15/10/09

147

Mohamed Ali, M.S., Oehlers, D.J., Haskett, M. & Griffith, M.C. (2009) The discrete rotation in reinforced concrete beams. ASCE Journal of engineering mechanics, submitted for print, October 2009.

A NOTE:


166

CHAPTER 7:

SYNOPSIS - INFLUENCE OF BOND ON THE HINGE ROTATION OF FRP PLATED BEAMS

In previous chapters, the rigid body rotation moment-rotation model was explained and limits to rotation through either concrete failure, of reinforcement debonding or fracture were presented. In this chapter, an example of an application of the rigid body rotation model is presented where the ductility of FRP plated members is assessed. Commonly, plated reinforced concrete members are considered to be brittle and non-ductile because the bond between the FRP and concrete is brittle. Experimentally, plated structures have shown ductility and hence to promote the acceptance of externally bonded (EB) and near surface mounted (NSM) fibre reinforced polymer (FRP) plates as a realistic method to improve flexural capacity we need to be able to quantify the influence of plating on ductility. Given the significant bond capacity of steel reinforcing bars, the retrofitting of EB or NSM plates almost always reduces the rotation capacity of the plated section. Ignoring other failure mechanisms like concrete sliding failure or fracture, EB plates typically debond at crack openings of approximately 0.5mm, NSM plates at crack openings approximately 3mm, and steel reinforcing bars at crack openings approximately 30mm. Hence, the rotations able to be achieved by EB plated members is less than that of NSM plated members and less than that of the unplated member. The range of rotations able to be achieved by EB, NSM and NSM plates with cover, are all shown in Figure P. An unplated 150mm deep slab is also shown for comparison.

167

σy ield=400MPa, Esh=2,000MPa, σf ract=600MPa, db=150mm

0

50

100

150

200

250

0.00 0.05 0.10 0.15 0.20

Rotation (radians)

Mom

ent (

kN-m

)

d=150mm,UNPLATED, 0.5% steel, 0%FRPd=150mm,PLATED, 0.5% steel, 0.25%FRPSOFTENINGYIELDEB DEBONDINGFRP FRACTURENSM DEBONDINGNSM COVER DEBONDINGSLIDING FAILURESTEEL DEBONDINGSTEEL FRACTURE

Figure P (Figure 9 in Chapter 7) – Moment-rotation response for unplated and plated 150mm

deep RC slab In Figure P yield of the reinforcing bar and the commencement of concrete softening occur at approximately the same rotation for the unplated slab. After yield, the moment continues to increase with strain hardening of the steel reinforcing bar until ultimately fracture occurs at a moment of 50kN-m. If it was assumed that bar fracture had not occurred and the reinforcing bar could continue to strain harden indefinitely then ultimately sliding failure would occur at a rotation of approximately 0.17 radians. This rotation corresponds to very large crack widths. For the plated 150mm deep slab, shown as the black line in Figure P, the rotation capacity of various plating types is assessed: EB, NSM and NSM with cover from Chapter 3. Each plating type has a different τ−δ relationship, and hence debonding, which occurs at δmax from the τ−δ relationship, commences at different rotations. EB plates have the most brittle bond and debond at crack widths less than 0.5mm (δmax≈0.25mm). This crack width corresponds to rotations less than 0.05 radians. NSM plates are more ductile and have greater slip capacities than EB arrangements and, therefore, allow the section to achieve greater rotation prior to debonding or fracture. NSM plates debond at crack widths up to 3mm, or rotations less than 0.4 radians in this example. Finally, NSM plates with cover have significant bond capacity and rotations up to 0.15 radians can be achieved. In this analysis, it is assumed that the FRP plates continue to attract load indefinitely and do not fracture. However, if fracture did occur at a stress of 1600MPa, a rotation of approximately 0.03 radians would be achieved as shown in Figure P. For lightly reinforced sections like 150mm deep slabs in Figure P, the probability of sliding failure is low due to the limit in the depth of softening dsoft that can be developed in a shallow section. Hence sliding failure is delayed in the unplated

NSM COVER

NSM

EB

SLIDING RANGE OF FRP RUPTURE

SLIDING

168

section, and occurs earlier in the plated section due to the increased percentage of reinforcement which directly increases the depth of the softening region. The unplated member fails by bar fracture before sliding in Figure P. As the section depth increases, the rotation at failure reduces. Since rotation is simply the geometric ratio of the crack separation to the crack height, larger section depths provide larger crack heights (the crack separation at failure is independent of the section depth), thus providing reduced rotations. The flexural capacity is increased due to the improved level arm and, hence, this reduction in ductility is offset by the increased flexural capacity. This is evident when comparing Figure P and Figure Q, where the rotations achieved by the 600mm deep section (Figure Q) are significantly less than those achieved by the 150mm deep section (Figure P).


0

200

400

600

800

1000

1200

0.00 0.05 0.10 0.15Rotation (radians)

Mom

ent (

kN-m

)

d=600mm,UNPLATED, 1.5% steel, 0%FRP

d=600mm,PLATED, 1.5% steel, 0.75%FRP

SOFTENING

YIELD

EB DEBONDING

FRP FRACTURE

NSM DEBONDING

NSM COVER DEBONDING

SLIDING FAILURE

STEEL DEBONDING

STEEL FRACTURE

Figure Q (Figure 10 in Chapter 7) – M-θ for 600mm deep section

For unplated sections in general, after yielding of the steel reinforcing bars the depth of the softening region is relatively constant and, hence, so too is the height of the crack. Therefore, by increasing the section depth four times, as is the case in this example when comparing the 150mm deep slab and 600mm deep beams analysed in Figure P and Figure Q, the height of the crack is also increased by a similar amount. For a given rotation, this larger crack size of the deeper beam directly causes a greater slip in the reinforcement, where the slip of the reinforcement can be calculated according to the relationship between rotation and the distance from the reinforcement to the crack tip. In the case of an FRP plate, the displacement is given by δFRP=θsofthcr, and for a steel reinforcing bar δst =θsoft(hcr-c), where c is the cover to the bar. Therefore, all other things being constant, if the depth of the section is quadrupled then the slip required for a given rotation is also quadrupled. For this reason, deeper sections cannot achieve the same rotation capacity at failure when all other things are constant.

NSM COVER NSM EB

FRACTURE

SLIDING

SLIDING

169

Deeper sections are also more likely to experience concrete sliding failure. Unlike shallow sections where the depth of softening is limited, the only limit to the depth of softening is caused by the cross sectional area of reinforcement. Hence sliding failure is more probable. If a sliding capacity of 0.4mm is assumed (from Equation 28), then sliding failure in the unplated section occurs almost immediately after yield of the reinforcing bars for the unplated 600mm deep beam in Figure Q. If the influence of confinement provided by the stirrups is considered, and the sliding capacity of concrete is assumed to be approximately 1.5mm, then sliding failure is delayed significantly and rotations in excess of 0.1 radians can be achieved. The rotation at failure is very sensitive to the sliding capacity of concrete, especially in deeper sections where sliding is more probable.

170

Statement of Authorship Influence of bond on the hinge rotation of FRP plated beams, 1Haskett, M., 2Mohamed Ali M.S., 3Oehlers D.J., and 4Wu, C. 1Haskett, M., Postgraduate student School of Civil, Environmental and Mining Engineering Adelaide University Wrote manuscript, performed all analyses, developed theory SIGNED____________________ 2M. S. Mohamed Ali, Dr. Senior Research Associate, School of Civil, Environmental and Mining Engineering Adelaide University 3Deric J. Oehlers, Professor School of Civil, Environmental and Mining Engineering Adelaide University Reviewed manuscript, supervised research SIGNED____________________ 4Chengqing Wu, Dr. Lecturer School of Civil, Environmental and Mining Engineering Adelaide University Advances in Structural Engineering, Vol.12(6), pp. 833-843

171

JOURNAL PAPER – INFLUENCE OF BOND ON THE HINGE ROTATION OF FRP PLATED BEAMS

Haskett, M., Mohamed Ali M.S., Oehlers D.J., and Wu, C. Abstract Fibre reinforced polymer (FRP) plate reinforcement is a brittle material which has a brittle interfacial bond with concrete. This can lead to the misconception that all FRP retrofitting techniques provide brittle members and, hence, limited rotational capacity which has severe limitations for structural applications. This paper shows that the FRP reinforcement behaviour is but one of three components that govern the rotational capacity of plated reinforced concrete beam hinges. It is shown that FRP retrofitted beams and slabs can achieve ductile behaviour and provide rotational capacity and, furthermore, that the rotational capacity of FRP plated members depends very importantly on the interface bond characteristics. Keywords FRP, IC debonding, bond, member ductility, RC beam, rotation, softening. 1. Introduction Structural engineers have long recognised the great importance of member ductility and its associated mechanism of moment redistribution in designing safe structures (Baker 1956, Mattock 1967, Wood 1968, Priestley and Park 1987, Fantilli et al 1998, Panagiotakos and Fardis 2001, Oehlers 2006, and Oehlers et al 2006). An FRP plated beam is illustrated in Fig. 1. For ease of understanding, the beam has been divided between the non-hinge region and the hinge region. The non-hinge region is defined as where the concrete stress in the stress-profile σnon-hinge always lies within the first branch of the concrete stress-strain relationship in Fig. 2. In contrast, the hinge region is defined as where some of the concrete in the stress profile σhinge in Fig. 1 is softening, that is it lies in the second branch in Fig. 2.

non-hinge region hinge region

σnon-hinge

non-hinge region

non-softening stress profile

softening stress profile

concrete softening wedgesteel reinforcement FRP reinforcementσhinge

Figure 1 - Rotational components of FRP plated RC beam

0

σstart

σsoft

εstart εslide

σ

ε

first branch

second branchresidual strength

slidingfailuresslide

idealised

Figure 2 - Idealised concrete material properties

172

The rotation within the non-hinge region in Fig. 1 can be obtained from numerical models based on standard procedures of equilibrium and compatibility (Oehlers et al 2007a, Oehlers et al 2005 and Mohamed Ali et al 2007). The rotation within the hinge region can be visualised as a rigid body rotation across the primary flexural crack shown as θ in Fig. 3 (Oehlers et al 2007a, Oehlers et al 2007b). To allow the crack to open up in the hinge region and facilitate this rigid body rotation, the reinforcement (whether it is FRP plates or steel reinforcing bars) slips by an amount δ relative to the adjacent concrete and the concrete softens within a wedge shaped region shown shaded. Hence, the softening capacity of the concrete, and ultimate slip capacity of the reinforcement (FRP plates and steel reinforcing bars) are both limiting factors to the rotational capacity of the hinge or softening region. It can also be seen in Fig. 3 how the rotation within the softening zone interacts with the rotation of the primary flexural crack and that these rotations should not be added, but rather either can limit the rotation of the hinge.

δrebar

θ

θrigid body rotation

concrete softening wedge primary

crack face

rigid body rotation

rigid body rotation

θsoft-limitθfract-debond-limit

δFRP

hrebar

hcr=hFRP

c

α

Figure 3 - Hinge rotation limits

2. Softening rotation Shear-friction theory is now a well established area of research (Mattock and Hawkins 1972, Hsu et al 1987, Walraven et al 1987, Mattock et al 1988, Duthinh 1999, and Veccchio and Lai 2004). Research based on shear-friction theory has quantified the rotational capacity of the softening zone (Oehlers et al 2007b). Although not the subject of this paper, the outcome from this shear-friction research is illustrated in Fig. 4 where it can be seen that the softening wedge shown shaded resists a force Psoft shown in the force profile Phigh, which shows the distribution of internal forces within the section immediately at the crack face and has an equivalent stress σsoft shown in the stress profile σhigh, which similarly shows the distribution of internal stresses within the section immediately at the crack face. Furthermore, the failure of the wedge is caused by sliding as shown along the wedge-beam interface, that is when the slip ssoft exceeds a critical value sslide. The analysis in Fig. 4 uses the idealised concrete stress-strain relationship in Fig. 2 where the first branch is considered to be a material property and the second br anch is considered to be an equivalent stress-strain relationship that represents the shear-friction sliding mechanism of the concrete softening wedge in Fig. 4. Research has shown (Oehlers et al 2007c) that a lower bound to the second branch in Fig. 2 is the following equivalent or residual stress:

173

latcsoft f σσ 3.471.0 += (1)

where fc is the concrete cylinder compressive strength and σlat the concrete confining stress. This equivalent or residual stress σsoft can be maintained until uncontrolled sliding across the shear-friction failure plane at sslide, magnitudes of which have been determined experimentally (Mattock and Hawkins 1972, Mattock et al. 1988 and Mattock 1988). Hence the equivalent stress σsoft can be used in a standard equilibrium and compatibility analysis (Oehlers et al 2007a) such as that shown on the left hand side of Fig. 4 to give the moment-curvature relationship for specific values of the depth of the wedge dsoft in Fig. 4. As a note, assuming that the stress-strain relationship of the ascending or first branch of concrete in Fig. 2 is as given by Hognestad (1951), then the average compressive stress over the ascending region is 0.67fc, whereas, the average compressive stress of the softening region in Eq. 1 for unconfined concrete is slightly greater at 0.71fc until sliding failure occurs. This shows that the presence of softening does not cause any significant reduction in moment as the curvature and the depth of the softening region increases, that is until sliding occurs at sslide at the equivalent strain shown as εslide in Fig. 2. The analysis in Fig. 4 that incorporates shear-friction theory gives dsoft. Shear-friction theory also defines the length of the hinge Lsoft in terms of dsoft as the angle of the softening wedge α in Figs. 3 and 4 is approximately 26o (Oehlers et al 2007b). Integration of the curvature along the length of the hinge gives the rotation for that specific value of dsoft. By gradually increasing the curvature, and hence the depth of the softening region, dsoft, the moment-rotation relationship of the hinge can be obtained. It is important to note that the moment-rotation analysis depicted in Fig. 4 does not give the rotation limit to softening. The softening limit occurs when the slip of the wedge ssoft exceeds sslide. By assuming that the wedge is a rigid body, the slip ssoft can be obtained by integrating the strain distribution along the length of the shear-friction wedge. The strain is constant where the shear friction wedge forms, and corresponds to the strain at the commencement of softening, εstart in Figs. 2 and 4. Hence, the slip of the wedge is given by dsoftεstart/sinα.

highest χ edge

χstart

εstart σstart

εstart

σstart

σhighεhigh

dsoft

Phigh

Prising

σsoft

Prebarχsoft

lowest χ edge

Psoft

PFRP

Lsoft

ssoft

εstart

εlow σlow

softening wedge

α

neutral axis

Figure 4 - Softening rotation limit

174

The fracture or the debonding of the reinforcement can also limit the rigid body rotation across the primary flexural crack. The rotation δ/h of the primary flexural crack in Fig. 3 is simply limited by the slip capacity ∆ult (Oehlers et al 2007c) of the reinforcement due to intermediate crack (IC) debonding or fracture which is the subject of this paper. In the case of an un-plated RC member, the limit to the rotation across the primary flexural crack is either caused by fracture of the reinforcing bar or localised debonding of the bar around the crack front. In the case of an FRP plated RC member, the rotation across the primary flexural crack can be limited by fracture or localised debonding of the steel reinforcing bar around the crack front, or, and more likely, by fracture or debonding of the FRP reinforcement. In this paper, IC debonding theory is first used to quantify the slip at fracture or debonding of FRP plates and steel reinforcement. This is then used in a moment-rotation analysis to illustrate the limits that this imposes on the rotation of unplated and FRP plated RC beams. The influence of bond on the rotational capacity of FRP plated RC members is then discussed where it is shown that the bond-slip characteristics of the reinforcement control the ability of the primary flexural crack to rotate. It is also shown that for shallow RC beams, the ultimate slip capacity of the reinforcement often limits rotation, and for deeper RC beams, the rotation is more likely to be limited by the concrete softening component.

3. IC resistance slip capacities The bond between steel reinforcement or FRP reinforcement and concrete interface can be idealised as a uni-linear variation with a maximum shear capacity of τmax at zero slip and a minimum shear capacity of zero at a slip δmax (Haskett et al 2008a, Mohamed Ali et al. 2006). The rotational limit across the primary crack has been shown to be dependent on the ultimate slip capacity, ∆ult, of the reinforcement. Mathematically, the ultimate slip capacity is the lesser of the slip at which IC debonding commences, δmax, or the slip at which the axial reinforcement fractures, δfract as shown in Eq. 2.

{ }max,min δδ fractult =∆ (2) The value of δmax in Eq. 2 corresponds to the slip in the local bond stress-slip (τ−δ) relationship beyond which bond is not transferred across the reinforcement-concrete interface. In the case of externally bonded (EB), near surface mounted (NSM) and NSM With Cover (Oehlers et al. 2007d) plating techniques shown in Fig. 5, the value of δmax is given by Eq. 3 (Oehlers et al. 2007d) where: ϕf is the aspect ratio df/bf in Fig. 5 (Seracino et al. 2007) that reflects the increased slip at the commencement of IC debonding of NSM relative to EB plating arrangements; c refers to the cover provided to an embedded NSM strip; and d is the depth of the NSM strip. In the case of deformed steel reinforcing bars, a reasonable magnitude of δmax is 15mm (Haskett et al. 2008a) as observed from experimental results (Eligehausen et al. 1983).

55.0576.0

max 078.0802.0976.0

+

+=

ddc

f

f

ϕϕ

δ (Units N and mm) (3)

175

The parameter ϕf is determined according to the definition of the IC debonding failure plane, shown in Fig. 5 for both EB and NSM techniques where ϕf=df/bf. In Fig. 5, tb and td are assumed to be 1mm because experimentally it has been observed that approximately 1mm of concrete remains adhered to the FRP strip when IC debonding occurs.

Figure 5 – IC debonding failure perimeters and aspect ratios

FRP plates and low ductility steel reinforcing bars also are prone to fracture prior to debonding or pulling out. When this occurs, the ultimate slip capacity is reduced and the rotational capacity across the crack front is also compromised. This limit depends on the maximum interface bond strength. For EB, NSM and NSM With Cover plates, τmax is given by Eq. (4) (Oehlers et al. 2007d), and for deformed steel reinforcing bars it is approximately 2.5√fc (CEB 1992) where fc is the cylinder compressive strength of the concrete.

65.06.0526.0

max 976.0

+

=d

dcfcfϕτ (MPa) (4)

For linear-elastic materials such as FRP plates when fracture occurs prior to the commencement of debonding, that is δ<δmax, fracture occurs at the following slip which is the slip in a beam with a single flexural crack and, hence, in general a lower bound to the slip in a beam (Haskett et al 2008a)

( ){ }1max cos1 afract λδδ −= (5) where:

EALper

max

max2

δτ

λ = (6)

and

L per Lper

A p tb

td

tb

td

A p d f

d f b f

b f

failure plane EB

failure plane NSM

concrete

Lper

failure plane of

td

A p

d f

bf

c

d

bar diameter: db

176

λ

τελ

=per

fract

LEA

a max1

arcsin

(7)

in which EA is the axial rigidity of the reinforcement and εfract is the fracture strain of the linear-elastic reinforcement. For an elasto-plastic material like steel, the slip at the commencement of IC debonding or pull-out is the same as for a linear-elastic material and is given by δmax from the local τ−δ relationship, which is approximately 15mm for deformed reinforcing bars. However, the slip at fracture of a steel reinforcing bar is not as per Eqs. 5-7 because of strain hardening. When a steel bar yields, strain hardening commences with a reduced strain hardening modulus, Esh as shown in Fig. 6. The reduced strain hardening modulus post yield leads to a rapid increase in slip of the steel reinforcement for a given increase in stress. Hence, a lower bound to the slip at fracture of the steel reinforcing bar is a function of both the elastic Modulus (Es) and the strain hardening modulus (Esh). The slip at fracture of a steel reinforcing bar is given by Eq. 8, which is the sum of the slip at yield and the increase in slip from yield to fracture (for a complete discussion of the derivation of the slip at fracture refer to Haskett et al. 2008b)

[ ]{ } [ ]{ }shfractelelfract aa λδλδδ cos1cos1 maxmax −+−= (8) where

el

byel

el

df

aλ

τλ

= max4arcsin

(9)

sbel Edmax

max2 4δ

τλ = (10)

( )

sh

yfractbsh

fract

fd

aλ

τσλ

−

= max4arcsin

(11)

shbsh Edmax

max2 4δ

τλ = (12)

In Eqs. 8-12, δmax is the local slip value beyond which no bond stress is transferred across the bar-concrete interface as stated previously, db is the bar diameter (mm), Es is the elastic Modulus (MPa), fy is the yield stress of the bar (MPa), and σfract is the fracture stress of the bar (MPa) as shown in Fig. 6.

177

σfract

fracture

strain

stressEsh

Es

εfractεy

fy

Figure 6 – Idealised steel stress-strain properties

Hence, the slip capacity of a deformed reinforcing bar is the minimum of the slip at which IC debonding commences, δmax, or the slip at which the reinforcing bar fractures, that is δfract in Eq. 8. Mathematical equations have been presented that relate the slip of a linear-elastic or elasto-plastic reinforcement to the force in that reinforcement. It has also been shown that there are three factors that can limit the rotation of a plated RC member. Concrete softening limits the rotation when sliding failure occurs across the shear-friction sliding plane, which occurs when the reinforcement neither fractures nor debonds prior to sliding. Alternatively, the reinforcing bar can pull-out (debond) or fracture prior to concrete softening or the FRP plate debonding or fracturing. Failure of the steel reinforcing bars is unlikely to occur given the significant slip capacity of deformed steel reinforcing bars relative to FRP plates. However, with the increased use of low ductility steel, bar fracture is a realistic limit to rotation (Gilbert and Sakka 2007). Finally, the FRP plate can debond or fracture prior to the softening sliding limit being reached or the reinforcing bar achieving its ultimate slip capacity. This is especially probable because the slip at debonding of EB and NSM FRP plates, as given by Eq. 3, is often low. In the following section, the methodology of the development of moment-rotation expressions is presented. These moment-rotation expressions are subsequently used to demonstrate the various rotation limits that occur in plated and unplated RC beams of various depths. 4. Softening -Moment rotation analysis A continuous reinforced concrete beam is shown in Fig. 7. Rotation θ can occur about any of the four zones that are shown of length Lsoft. For ease of analysis, it can be assumed that Lsoft is much less than the span of the beam so that the moment over the length Lsoft can be considered to be constant.

(Lsoft)left (Lsoft)right

(dsoft)left (dsoft)right

(Lsoft)middle(Lsoft)middle

(dsoft)middle

θleftθright

θmiddle θmiddle

flexural crack face

Figure 7 - Rotational zones in a beam

178

At each zone in Fig. 7, the rotation θ, also shown in Fig. 3, can be derived from a sectional moment-curvature analysis as shown in Fig. 4. The rotation θ in Fig. 3 is obtained by integrating the curvature of the softening region over the length of the softening region, Lsoft. The limit to this rotation θ in Fig. 3 occurs either when the wedge slides or the reinforcement debonds or fractures. In the following section, moment-rotation relationships for various RC sections are presented, where it can be seen that bond characteristics influence the rotational capacity of both plated and unplated RC members. 5. Influence of bond on rotational ductilities The bond characteristics and material properties play a critical role in determining the rotational capacity of an RC member. To illustrate the effect of various parameters that influence the rotation, 3 different RC member sections are analysed, with moment-rotation (M-θ) figures presented for both unplated and plated sections. Examples of the moment-rotation curves are given in Figs. 9 to 11. These curves were derived by ignoring all failure modes, and then the point at which one particular failure would have occurred, say debonding of an EB plate when the slip of that plate exceeded 0.2mm, was inserted, assuming prior failure had not occurred. The RC sections analysed are shown in Fig. 8. For all sections analysed the stress-strain properties of the steel reinforcing bars are assumed to be: Es=200,000MPa, σyield=400MPa, Esh=2,000MPa, σfract=600MPa. Each RC section was analysed using the standard sectional method of analysis illustrated in Fig. 4. A curvature was fixed and the neutral axis depth was guessed. From the resulting strain profile, the tensile and compressive forces were calculated. If the tensile and compressive forces were not equal, the neutral axis was changed until the forces balanced. This iterative process was adopted for each sectional analysis.

b=1000mm

d=150mm

Ast=0.5%Unplated, or EB or NSM FRP plate - Area FRP=0.5Ast

d=350 or 600mm

b=300mm

Ast=1.5%Unplated, or EB or NSM FRP plate - Area FRP=0.5Ast

Figure 8 - Reinforced concrete sections: typical slab and beam

179

5.1 Slab of depth 150 mm: The M−θ distributions of the unplated and plated slab in Fig. 8 are shown in Fig. 9. The positions where concrete softening, yield of the reinforcing bar, wedge sliding failure, steel debonding and fracture occur are all shown. On the M−θ relationship for the plated section, the approximate positions where FRP fracture and FRP debonding occur for EB, NSM and NSM With Cover plating techniques are also shown. In the sliding analysis, it was considered that sliding commences at a slip of 0.4mm. The presence of stirrups or other confining pressures such as FRP wrapping can significantly increase this value (Oehlers et al 2007c) and this is discussed in the analysis of the RC beams where it can be seen in Fig 10 that sliding occurs in deep beams immediately after the commencement of softening if an ultimate slip capacity of 0.4mm is assumed.


0

50

100

150

200

250

0.00 0.05 0.10 0.15 0.20

Rotation (radians)

Mom

ent (

kN-m

)


Figure 9 –M−θ for unplated and plated 150mm deep RC slab sections

It can be seen in Fig. 9, that the presence of plates significantly increases the ultimate strength of the section as would be expected and that the unplated member fails by bar fracture before sliding as commonly occurs in slabs. Figure 9 also suggests that the range of rotation that can be achieved is highly correlated to the type of plating arrangement. As an example, wet lay up or EB plates (0.1mm to 1.2mm thick and 25 to 100mm wide) debond at approximately the same rotation as when the steel yields. Standard NSM plating arrangements (1.2 or 2.4 mm thick and 10 or 20mm deep) have greater slip capacity than EB arrangements and, therefore, allow the section to achieve greater rotation prior to debonding or fracture. In this example, NSM plating arrangements debond after the steel reinforcing bars yield, and can provide rotations up to 0.03 radians. This rotation is slightly beyond the rotation where the FRP material fractures (assuming a fracture stress of approximately1600 MPa). If a material with increased ultimate strength was used, and the NSM With C over technique was used, then rotations ranging from 0.03 to 0.08

NSM COVER NSM

EB

SLIDING RANGE OF FRP

SLIDING

180

radians can be achieved, assuming that plate debonding is the limiting factor to rotation. Wedge sliding failure of the RC member is also a possibility, and occurs when the wedge of softened concrete has slid more than its sliding capacity of approximately 0.4mm for unconfined and uncracked concrete. In this example, sliding failure of the plated section occurs in the range of rotations achievable with a standard NSM plating arrangement. An interesting observation is that in the plated M−θ relationship sliding appears a possibility, whereas in the unplated M−θ relationship sliding occurs well beyond the rotation at which reinforcing bars theoretically debond or fracture. This is because the sliding of the softened wedge of concrete is directly proportional to the length of the softening region and, hence, the depth of the softening zone; both of which are functions of the tensile force in the reinforcement. For lightly reinforced shallow sections like a slab, the depth of the softening region is limited, thereby, limiting the length of the softening zone and, therefore, the slip of the wedge. As the tensile force increases, as is the case when the section is strengthened with FRP plates, the depth of the softening region is increased substantially, increasing the probability of sliding failure of the wedge. Steel fracture or debonding were not evident in the M−θ of the plated section. The presence of the FRP plate, which can produce significant forces well in excess of those that can be achieved in steel bars, causes the depth of the neutral axis, and, hence, the softening region to shift down and closer to the FRP plate. As the depth of the neutral axis increases (i.e. the section becomes over-reinforced or the neutral axis depth factor ku increases) the strain in the steel reinforcing bar also reduces, removing the probability of debonding or fracture of the steel bar. Similar behaviour is also experienced in the plated beam section, where it is observed that reinforcing bars are less likely to control rotation of plated members. 5.2 Beam of depth 600 mm: The 600mm deep beam, as shown in Fig. 8, was also analysed as both an unplated and plated section. M−θ relationships for the unplated and plated section are shown in Fig. 10. As shown in Fig. 10, it can be observed that the range of rotations achievable for the 600mm deep section is noticeably smaller than that which can be achieved by the 150mm deep slab. As an example, a rotation of 0.17 radians can be achieved at steel reinforcing bar debonding in the 150mm deep unplated section, whereas, for the 600mm deep unplated RC section a rotation of only 0.045 radians can be achieved at debonding of the steel reinforcing bar.

181


0

200

400

600

800

1000

1200


Mom

ent (

kN-m

)

d=600mm,UNPLATED, 1.5% steel, 0%FRP

d=600mm,PLATED, 1.5% steel, 0.75%FRP

SOFTENING

YIELD

EB DEBONDING

FRP FRACTURE

NSM DEBONDING

NSM COVER DEBONDING

SLIDING FAILURE

STEEL DEBONDING

STEEL FRACTURE

Figure 10 - M−θ for unplated and plated 600mm deep beam sections

For unplated sections in general after yielding of steel bars, the depth of the softening region is relatively constant and, hence, so too is the height of the crack. Therefore, by increasing the section depth four times, as is the case in this example when comparing the 150mm deep slab and 600mm deep beams analysed, the height of the crack is also increased by a similar amount. For a given rotation, this larger crack size of the deeper beam directly causes a greater slip in the reinforcement, where the slip of the reinforcement can be calculated according to the relationship between rotation and the distance from the reinforcement to the crack tip. In the case of an FRP plate, the displacement is given by δFRP=θsofthcr in Fig. 3, and for a steel reinforcing bar δst =θsoft(hcr-c), where c is the cover to the bar. Therefore, all other things being constant, if the depth of the section is quadrupled then the slip required for a given rotation is also quadrupled. Figs. 9 and 10 also indicate that deeper beams are more prone to sliding failure than shallow slabs. As discussed previously, this is because the length and depth of the softening region are relatively small for a shallow member like a slab, while for a deeper section these parameters are not limited by section depth and can become larger. If a sliding capacity of the wedge of 0.4mm is assumed, as per the slab, then sliding failure in the 600mm deep beam occurs almost immediately after the commencement of softening. As curvature increases, the depth of the softening region increases for the unplated section but more rapidly for the plated section because the force in the FRP plate continues to increase for increasing curvatures. This increase in the depth of the softening region causes the length of the softening zone to increase and the wedge to slide more than the unconfined wedge slip capacity (0.4mm) relatively quickly. This suggests that deep beams have limited rotational capacity due to softening.

NSM COVER NSM EB

FRACTURE

SLIDING

SLIDING

182

An ultimate wedge slip capacity of 0.4mm was determined experimentally in tests using unconfined concrete. In the case of slabs, this value appears reasonable since slabs do not contain stirrups; however, in the case of beams which contain stirrups, the stirrups create a confining pressure on the concrete which increases the ultimate slip capacity of the wedge (Oehlers et al 2007d, Martinez et al 1984). The increase in the ultimate slip capacity of the wedge is often expressed in terms of the effective confinement pressure relative to the unconfined concrete strength. Tests by Martinez et al. (1984) showed that when this ratio approaches 0.3, an ultimate wedge slip capacity at failure in excess of 7mm can be achieved. If in this example, 12 mm diameter stirrups at 200mm centres with a yield strength 500MPa were used as shear reinforcement, the ultimate slip capacity of the wedge can increase to approximately 1.5mm (Oehlers et al. 2007d). Using this value enables the unplated RC beam to rotate 50 times more at sliding failure, as shown in Fig. 10, than when using 0.4mm of ultimate wedge slip capacity. The increased rotation capacity of the plated RC beam when assuming a wedge slip capacity of 1.5mm is not as significant because the presence of FRP plates increases the rate at which the depth of the softening region increases compared to the unplated RC beam. In this example, the sliding rotation at failure of the plated beam is increased 10 times when this increased wedge slip capacity is used. It should be noted that this paper is not specifically addressing the ultimate slip capacity of the wedge or the effect of stirrups in confining concrete, but rather to highlight that the rotation capacity is particularly sensitive to the slip value used, and to additionally highlight the importance that stirrups provide in confining concrete. It appears intuitively that the deeper sections are more likely to fail through the wedge sliding since both the depth and length of the softening zone, dsoft, and Lsoft respectively, are able to achieve greater magnitudes for deeper beams. This suggests that shallow sections, for example the slab shown in Fig. 8, are more likely to fail by an alternative mechanism, like FRP debonding or fracture. Shallow sections, however, have smaller crack heights, where this suggests that for a fixed ultimate slip capacity (∆ult) which is a function of reinforcement and material properties only, the rotation limit of the reinforcement (whether it is a steel bar or FRP plate) should be greater. Thus, sliding is less probable in shallow sections, but the rotation limit of shallow sections due to rupture or debonding of the reinforcement is also greater since the crack height, hcr, of shallow sections is smaller. Hence, it is difficult to visualise which mechanism limits the rotation of plated structures without observing the interaction between the various mechanisms that can cause failure. Figure 10 also shows the range of rotations than can be achieved with various plating arrangements. Once again, as was the case for the slab, the EB plating technique debonds well before fracture of the plates, suggesting poor material use. The EB plate also debonds prior to the commencement of sliding of the wedge. Typical NSM plates debond after the commencement of concrete sliding. The increased slip capacity at debonding of NSM relative to EB plates means that NSM RC members do have significant rotational capacity. Fracture of the plates occurs, as shown in Fig. 10, at rotations achievable with standard NSM plating arrangements. Once again, NSM With Cover plates possess the greatest rotation capacity, where they can achieve in excess of twice the rotation compared to the standard NSM plating technique, assuming other forms of failure such as sliding, steel fracture or debonding

183

are precluded. From the analyses, it can also been seen that the plated sections did not display steel debonding and fracture occurs at rotations only achievable with the NSM With Cover plating technique. As the curvature increases, the presence of FRP causes the depth of the neutral axis (and hence the depth of the softening zone) to continue to shift lower, limiting the slip of the steel reinforcing bar and hence, preventing debonding. 5.3 Beam of depth 350 mm: To further highlight the relationship between beam depth and the predominant mode of failure, a 350mm deep unplated and plated RC beam in Fig. 8 was analysed. The moment-rotation relationships for the unplated and plated beam are shown in Fig. 11.


0

50

100

150

200

250

300

350

400


Mom

ent (

kN-m

)


Figure 11 - M−θ for unplated and plated 350mm deep beam sections

In Fig. 11, similar relationships between rotation and plating technique can be observed. In this example, it can be seen that if an ultimate wedge sliding capacity of 1.5mm is assumed, which is achievable for concrete confined by stirrups, wedge sliding failure occurs in the plated section at rotations that are only achievable with the NSM With Cover plating technique. This indicates the advantage of this technique in strengthening existing RC members. All other observations presented previously in the discussion of the 600mm deep plated and unplated sections are still valid. Importantly, these three analyses indicate that for sections with identical percentage steel, and hence constant neutral axis depth ku factors, the section depth plays a critical role in determining the rotational capacity of the section. This is generally not reflected in codes worldwide, which often consider a beams rotational capacity to be independent of beam depth and only a function of its ku factor or percentage reinforcement.

NSM COVER NSM EB

FRACTURE

SLIDING

1.5 mm SLIDING CAPACITY

0.4 mm SLIDING CAPACITY

184

6. Conclusions It has been shown that three failure mechanisms can limit the rotation of unplated and plated RC members: concrete wedge sliding; FRP plate or steel reinforcing bar debonding; and FRP plate or steel reinforcing bar fracture. In shallow RC sections like slabs, failure of the reinforcement generally limits rotation. Conversely in deeper RC sections, wedge sliding failure was shown to limit rotation. It was also shown that EB plates often debond soon after concrete softening, thereby limiting the rotational capacity of EB RC members. NSM plated sections debond later, enabling the plated structure to rotate significantly more. Furthermore, the NSM With C over plating technique was shown to provide the greatest rotational capacity and the most efficient material use as FRP fracture often precedes debonding. The different ranges of rotation able to be achieved by the various plating arrangements is directly related to the bond characteristics, as the rotation at debonding is directly related to the value of δmax in the local bond stress-slip relationship, and the propensity to fracture prior to debonding is also highly correlated to the local bond characteristics. It was also shown that FRP plated deep RC sections suffer plate debonding or fracture at rotations significantly less than what occurs with equivalent shallow plated sections.

185

References Baker, A. L. L., 1956. Ultimate Load Theory Applied to the Design of Reinforced and

Prestressed Concrete Frames, Concrete Publications Ltd., London, 91. CEB. (1992). "CEB-FIP Model Code 90." London. Duthinh, D., 1999. Sensitivity of shear strength of reinforced concrete and prestressed

concrete beams to shear friction and concrete softening according to modified compression field theory, ACI Structural Journal, Vol. 96, No. 4, July-August, 496-508.

Eligehausen, R., Popov, E. P., and Bertero, V. V. (1983). "Local bond stress-slip relationship of deformed bars under generalized excitations." UCB/EERC-83/23, Earthquake Engineering Research Center, University of California, Berkeley.

Fantilli, A.P., Ferretti, D., Iori, I., and Vallini, P., 1998. Flexural deformability of reinforced concrete beams. Journal of Structural Engineering (ASCE), 1041-1049.

Gilbert, R. I., and Sakka, Z. I. (2007). "Effect of reinforcement type on the ductility of suspended reinforced concrete slabs." Journal of Structural Engineering, 133(6), 834-843.

Haskett, M., Oehlers, D.J., and Mohamed Ali M.S. (2008a) “Local and global bond characteristics of steel reinforcing bars”, Engineering Structures. Vol 30(2), 376-383.

Haskett, M., Oehlers, D.J., Mohamed Ali and Wu, C. (2008b) “Yield penetration hinge rotation in reinforced concrete beams”. Submitted ASCE Structural Journal.

Hognestad, E., (1951). “A study of combined bending and axial load in reinforced concrete members”, University of Illinois, Eng. Expt Station Bull, No 399.

Hsu, T.T.C., Mau, S.T., and Chen, B., 1987. Theory on Shear Transfer Strength in Reinforced Concrete, Structural Journal American Concrete Institute, Vol. 84, No.2, March-April, 149-160.

Martinez, S., Nilson, A.H., and Slate, F.O.. (1984). Spirally reinforced high-strength concrete columns. ACI Journal, Sept.-Oct., 431-441.

Mattock, A.H., 1967. Discussion of Rotational Capacity of Reinforced Concrete Beams, by W.D.G.

Mattock, A.H., and Hawkins, N.M. (1972). “Shear transfer in reinforced concrete recent research”. Precast Concrete Institute Journal, March- April, 55-75.

Mattock, A.H., Mau, S.T., and Hsu, T.T. (1988). Comments on “Influence of Concrete Strength and Load History on the Shear Friction Capacity of Concrete Members”, PCI Journal, Jan-Feb., pp 165- 169.

Mattock, A. (1988). “Reader Comments”, PCI Journal, Vol.33, No.1, Jan- Feb., 165-166.

Mohamed Ali, M. S., Oehlers, D. J., and Seracino, R. (2006). "Vertical shear interaction model between external FRP transverse plates and internal steel stirrups." Engineering Structures, 28(3), 381-389.

Mohamed Ali, M.S., Oehlers, D.J. and Griffith, M.C. (2007) “Simulation of plastic hinges in FRP plated RC beams”. Submitted ASCE Composites for Construction, Accepted 16/12/07.

Oehlers, D.J. (2006) “Ductility of FRP plated flexural members”. Cement and Concrete Composites, 28, pp 898-905.

Oehlers, D.J., Campbell, L., Haskett, M, Antram, P., and Byrne, R., (2006) “Moment redistribution in RC beams retrofitted by longitudinal plating”. Advances in Structural Engineering, Vol.9 No.2 April, pp 257-264.

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Oehlers, D.J., Liu, I.S.T. and Seracino, R. (2005) “The gradual formation of hinges throughout reinforced concrete beams.” Mechanics Based Design of Structures and Machines, 33, pp375-400.

Oehlers, D.J., Mohamed Ali, M.S., Griffith, M.C. and Ozbakkaloglu, T., (2007a) “Fundamental issues that govern the rotation of FRP retrofitted RC columns and beams: the intractable plastic hinge ductility problem”, The Inaugural Asia-Pacific Conference on FRP in Structures APFIS 2007, Hong Kong.

Oehlers, D.J., Mohammed Ali, M.S., and Griffith, M.C (2007b). “Concrete component of the rotational ductility of reinforced concrete flexural members”. Accepted Advances in Structural Engineering.

Oehlers, D.J., Mohammed Ali, M.S., Griffith, M.C., and Ozbakkaloglu, T., (2007c) “Residual strength of actively confined concrete”. Submitted ACI Structural Journal 15/11/07

Oehlers, D.J., Haskett, M., Wu, C, Seracino, R. (2007d) “Embedding NSM FRP plates for improved IC debonding resistance”. Submitted Journal of Composites for Construction, ASCE, 20/11/07

Panagiotakos, T.B. and Fardis, M.N., 2001. Deformations of Reinforced Concrete Members at Yielding and Ultimate, ACI Structural Journal, Vol. 98, No. 2, 135-148.

Priestley, M.J.N., Park, R., 1987. Strength and Ductility of Concrete Bridge Columns under Seismic Loading, ACI Structural Journal, Title No. 84-S8, 61-76, Jan.-Feb.

Seracino, R., Raizal Saifulnaz M.R., and Oehlers, D.J. (2007) “Generic debonding resistance of EB and NSM plate-to-concrete joints”. ASCE Composites for Construction. 11 (1), Jan-Feb, 62-70.

Vecchio, F.J. and Lai, D., 2004. Crack shear-slip in reinforced concrete elements, Journal of Advanced Concrete Technology,Vol.2, no.3,October, 289-300.

Walraven, J., Frenay, J., and Pruijssers, A., 1987. Influence of Concrete Strength and Load History on the Shear Friction Capacity of Concrete Members, PCI Journal, V.32, No.1, Jan-Feb., 66-84.

Wood, R.H., 1968. Some controversial and curious developments in the plastic theory of structures, In Engineering Plasticity; Heyman, J., Leckie, F.A. Eds; Cambridge University Press: Cambridge, 665-691.

187

CHAPTER 8:

SYNOPSIS - ANALYSIS OF MOMENT REDISTRIBUTION IN FRP PLATED RC BEAMS

In Chapters 5 and 6, the rigid body rotation mechanism was described and methods presented to determine the moment-rotation response of a reinforced concrete section. In this and the following chapters, an application of the use of this moment-rotation response is presented where the moment and rotation at failure from the RBR model is used to determine the moment redistribution capacity of a member. Ductility is a critical component in reinforced concrete design. It provides a member with the ability to redistribute moment, to give prior warning of failure, and absorb energy during earthquakes and dynamic loadings. The RBR mechanism can be used to quantify the ductility of a reinforced concrete member in terms of the moment redistribution capacity at failure, where moment redistribution is simply the ability of a member to maintain a given moment whilst simultaneously rotating to allow the other regions of the member to attain more moment. Most national standards quantify the moment redistribution capacity of a reinforced concrete section using the commonly accepted neutral axis (ku) approach. This approach requires the neutral axis depth to be determined from a standard linear strain based analysis. For this neutral axis depth, the curvature at failure can be determined where failure is always assumed to be caused by concrete crushing. Since the strain at concrete crushing is fixed, sections with low ku factors have a greater curvature at failure. The corresponding rotation at failure is simply the curvature at failure integrated over the empirically derived plastic hinge length, which is often assumed to be proportional to the section depth. Hence, sections with low ku factors are considered to have larger rotation capacities and hence are more ductile than sections which have larger ku factors. This simple approach has provided engineers with the ability to approximate the ductility and redistribution capacity of reinforced concrete members, but is limited in its accuracy because it relies on empirically derived plastic hinge length expressions, requires concrete crushing to be the singular mode of failure, and can not accommodate the partial interaction behaviour between the reinforcement and adjacent concrete. In this chapter, an alternative procedure to determine moment redistribution is presented which can accommodate any failure mode, allows for the partial interaction behaviour between the reinforcement and concrete and does not require a hinge length approximation. Consider a continuous beam of span L of flexural rigidity EI which is subjected to a uniformly distributed load, w, where hinges are first forming at the supports as shown in Figure R, where in this example the hogging or -ve region is redistributing moment to the sagging or +ve region.

188

EI

w (kN/m)

hinge either sideof centre-line

L/2(a) beam

centre-linesupports

centre-linesupports

L/2

(b) moment Mh=2/3Ms

Mst (wL2/8)Mh

(c) static deformation

yθh(w) θh(w)w (kN/m)

(d) restraint deformation

yθh(Mh) θh(Mh)

MhMh

Ms=1/3Mst

Figure R (Figure 6 in Chapter 8) - Hinge at supports: continuous beam with UDL

The elastic moment distribution is shown in Figure R(b), where the support moment Mh is twice the midspan moment Ms and the static moment Mst is wL2/8. The rotation at the supports due to the static moment θh(w) is shown in Figure R(c) and that due to the support moments Mh is shown in Figure R(d). When the moment distribution is elastic, the rotation at the supports in Figure R(c) and (d) sum to zero, and hence the support hinges do not require any rotation capacity. Once the moment distribution deviates from the elastic case in Figure R(b), some rotational capacity is required. Ultimately, failure occurs when the rotation required for a given amount of moment redistribution, that is the algebraic sum of the rotations in Figure R(c) and (d), exceeds the rotation capacity of the section from the rigid body rotation (RBR) model. The behaviour of the beam can be visualised as the sum of the static deformation due to the applied load w, and a deformation due to the applied restraint moment Mh. These two rotations are shown in Figure R(c) and (d) respectively. These rotations at the support are obtained through integration of the curvature distribution, where various boundary conditions are required to solve for the constants of integration. An example of a boundary condition is that the rotation at midspan is zero by symmetry. The rotation at the supports due to the applied load w in Figure R(c) is given by

EI24wL3

)w(h−

=θ

Equation 42 and the rotation at the support due to the applied restraint moment Mh in Figure R (d)is

EI2LMh

)Mh(h =θ

189

Equation 43 Hence, the total rotation at a support is the difference between these two rotations, where Mst is the static moment, wL2/8.

EI2LM

EI3LM

EI2LM

EI3L

8wL

EI2LM

EI24wL hsth

2h

3

h −=−=−=θ

Equation 44 Let us define the moment redistribution factor KMR as the moment redistributed as a proportion of the elastic moment such that KMR varies from 0→1 so that 100KMR is the commonly used percentage moment redistributed. Then for a continuous member subjected to a uniformly distributed load such that the hogging or -ve (support) moment is two-thirds the static moment prior to redistribution and for redistribution from hogging region (support) to sagging region (midspan).

st

hst)sh:UDL(MR

M32

MM32

K−

=→

Equation 45 Rearranging Equation 45 in terms of Mh, and substituting for Mh in Equation 44 yields

LKEI3MMR

hst

θ=

Equation 46 Rearranging Equation 45 to make Mst the subject, and substituting for Mst in Equation 46 yields

LMEI2EI2K

hh

h)sh:UDL(MR +θ

θ=→

Equation 47 where the subscript h refers to the hogging region and subscript s the sagging region. It can be seen that the redistribution capacity of the hogging region is a function of the flexural rigidity but it is not directly proportional to the flexural rigidity as it occurs in both the numerator and the denominator. The flexural rigidity of the cracked section as opposed to the uncracked section would underestimate the moment redistribution capacity. Hence, if we know the moment and rotation at failure, the redistribution capacity of the section can easily be determined for the loading situation. A similar approach can also be used to determine the moment redistribution relationship for redistribution from the sagging to hogging region and an identical expression is achieved.

190

The ductility of three plating types, “wet lay-up” EB plates, “pultruded” EB plates, and NSM plates are compared in terms of their moment redistribution capacities. The moment-rotation response obtained from the rigid body rotation analysis is converted to a moment-moment redistribution relationship according to Equation 47, and is shown in Figure S. In Figure S, the moment redistribution at the start of debonding is shown as a grey circle, which coincides with a plate slip δmax, and the moment redistribution at the end of debonding is shown as a black circle. If it is assumed that debonding can propagate for a distance d/2 away from the crack face without a drop in plate axial load, that is debonding is being restricted to approximately the hinge region, then at the end of debonding the additional crack opening through localised debonding is simply the integration of the strain at debonding over the debonded length d/2. The extra displacement capacity (and hence rotation) provided by localised debonding is therefore greater when the plating arrangement achieves larger debonding strains. Hence, in Figure S the “wet lay-up” EB plate, which is thinner than the “pultruded” EB plate, but has the same bond characteristics, achieves a larger debonding strain and therefore can provide a greater increase in redistribution capacity from the start to the end of debonding. In the case of the “wet lay-up” EB plate the increase in moment redistribution through localised debonding is approximately 200%.

0.040.05

0.05

0.65

0.31

0.040.50

0.10 0.18

0.19

0

20

40

60

80

100

120

140

0.0 0.2 0.4 0.6 0.8KMR

Mom

ent (

kN-m

)

UnplatedEB-pultrudedEB- "wet lay-up"NSMYieldingBar FractureStart debondingEnd debondingPlate Fracture

Figure S (Figure 12 in Chapter 8) – Comparison of moment redistribution capacities From Figure S, the EB pultruded plate has significantly less redistribution capacity than the unplated section, and only a minimal increase in flexural strength. The EB wet lay-up arrangement is twice as ductile as the pultruded EB plates, with a corresponding 15% increase in flexural capacity. The NSM FRP arrangement is the most ductile due to its superior bond performance and also displays the greatest increase in flexural capacity through plating. However, at failure, it is still not as ductile as the unplated RC section.

Unplated

NSM

EB pultruded

EB “wet lay-up”

191

Statement of Authorship Analysis of moment redistribution in FRP plated RC beams Matthew Haskett1, Deric John Oehlers2, Mohamed Ali M.S.3, and Chengqing Wu4 1Mr. Matthew Haskett PhD student School of Civil, Environmental and Mining Engineering University of Adelaide Wrote manuscript, developed all theory, performed all analyses SIGNED____________________ 2Professor Deric J. Oehlers School of Civil, Environmental and Mining Engineering University of Adelaide Supervised research and reviewed manuscript SIGNED____________________ 3Dr. Mohamed Ali M.S. Senior Research Associate, School of Civil, Environmental and Mining Engineering University of Adelaide 4Dr. Chengqing Wu Senior Lecturer, School of Civil, Environmental and Mining Engineering School of Civil, Environmental and Mining Engineering University of Adelaide Accepted Composites in Construction, ASCE, 4/12/09

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A Haskett, M., Oehlers, D.J., Mohamed Ali, M.S. & Wu, C. (2010) Analysis of Moment Redistribution in Fiber-Reinforced Polymer Plated RC Beams. Journal of Composites for Construction, v. 14(4), pp. 424-433

A NOTE:



A http://dx.doi.org/10.1061/(ASCE)CC.1943-5614.0000098

A

http://dx.doi.org/10.1061/(ASCE)CC.1943-5614.0000098

216

CHAPTER 9: SYNOPSIS - MOMENT REDISTRIBUTION IN REINFORCED

CONCRETE MEMBERS In Chapter 8, a relationship between moment redistribution, moment and rotation at failure, span length and flexural rigidity was developed for a continuous member. The ability of a section to redistribute moment allows the load capacity of the beam to be increased. The proportional increase in applied load (∆w) due to moment redistribution is given by

1K11wΔ

MR

−

−

=

Equation 48 For example, if the engineer wishes to redistribute the support moment by 25% (and the rotation capacity is sufficient) then from Equation 48, with KMR = 0.25, the increase in the applied load is Δw = 33%. Various parameters influence ductility and moment redistribution. From the rigid body rotation approach, it can be demonstrated that the moment-redistribution capacity increases with increasing bar diameter, concrete confinement and bar fracture strain and reduces with increasing bond strength. The effect of bar diameter on the slab moment redistribution capacity is shown in Figure T where it can be seen that the bar diameter has virtually no effect on the moment-redistribution capacity at the start of debonding as this depends on δmax. However, increasing the bar diameter has a significant effect on increasing the moment redistribution factor at bar fracture (grey triangles). This is because the load at fracture is constant, hence, the depth of the concrete mobilised to resist this compressive load, dasc+dsoft, is also constant and so consequently is the height of the crack hcr and height to the rebar hrebar. Hence, from Chapter 4 where it was shown that ∆fract increases for increasing bar diameter, the increase in rotation (and moment redistribution) at fracture is proportional to the increase in the slip at bar fracture. The redistribution at sliding failure is also shown in Figure T for both a sliding capacity of 0.8mm and 1.2mm. In this case for a 400mm deep section, a sliding capacity in excess of 0.8mm is required for fracture to occur prior to concrete failure.

217

Figure T (Figure 13 in Chapter 9) KMR response with varying bar diameter

Bond stress also influences moment redistribution. The peak bond stress (τmax) of deformed reinforcing bars was shown to be approximately 2.5√f c in Chapter 2, but it was shown in Chapter 4 that using a peak bond stress of 1.25√f c provides a better approximation of the empirical rotations, and better models the bond behaviour in beams. Expressing the bond stress as kb√fc, where kb varies from 0.75 to 2.75, the influence of bond stress on moment redistribution is shown in Figure U. From Figure U, increased bond strength reduces the moment redistribution capacity at fracture but increases the moment redistribution capacity at debonding.

218

Figure U (Figure 14 in Chapter 9) - KMR response with interface bond strength for 400 mm beam The intermediate crack debonding capacity was shown to be a function of the interfacial fracture energy of the bond relationship in Chapter 4 (Equation 13). Increasing the magnitude of the peak bond stress τmax (whilst maintaining δmax at 15mm) increases the interfacial fracture energy Gf of the local bond stress slip relationship. This directly provides an increased IC debonding load at failure. The larger load at debonding failure directly means that for equilibrium the depth of concrete in compression (dsoft+dasc) must also increase, thereby reducing hcr. Thus, since the debonding slip is fixed at δmax=15mm, the rotation at debonding failure must also increase in response to the increased IC debonding load due the corresponding decrease in the depth of the flexural crack. However, it must be emphasised further that debonding is unlikely to control failure as the reinforcement slip capacity δmax is very large.

An opposite effect is observed when comparing moment-redistribution capacity at bar fracture in Figure U: the moment-redistribution capacity reduces for increasing bond strength. Increasing the bond strength τmax increases the shear stress that can be transferred across the bar-concrete interface, providing a reduction in the slip at which bar fracture occurs. As an example, increasing the bond stress from 0.75√fc to 2.75√fc results in the slip at bar fracture reducing by a factor of 5.8 from 12.8mm to 2.2mm for a 12mm bar. This reduction in slip at fracture correspondingly reduces the rotation at fracture by a similar factor. The load and moment at fracture are unaffected by the reduced slip at fracture for higher bond stresses because the fracture stress is constant (600MPa) in this analysis and, hence, the height of the crack (at fracture) is unaffected by increased bond stress. Thus, in Figure U the moment redistribution at fracture is reducing for increasing bond stress kb.

219

The influence of fracture strain of the reinforcing bars is clear: larger fracture strains allow greater rotation and hence moment redistribution at failure. The moment-redistribution capacities obtained using the present approach and ignoring debonding are compared in Figure V to those suggested in codes worldwide for beams of varying depth. The ku factor used in Figure V was obtained using the Australian rectangular stress block approach and gamma factors and is defined as the depth of the neutral axis divided by the effective depth of the section. It can be seen in Figure V that the depth of the member is very important, with the moment-redistribution capacity reducing with increased member depth. Furthermore, the variation consists of two components. At low values of ku, bar fracture controls the moment-redistribution capacity. With increasing ku the moment-redistribution capacities rise slowly which is followed by sliding failure associated with rapidly reducing moment-redistribution capacities. Furthermore, the transition between these failure modes occurs at reduced ku values for increasing beam depths.

Figure V (Figure 17 in Chapter 9) - KMR response with neutral axis depth factor for 400 mm beam As the ku factor increases (by increasing the neutral axis depth through an increase in the number of bars in the section rather than the bar diameter), the total tensile force developed for a given slip is much greater. Since the slip at bar fracture is constant, the increased number of bars means that the depth of the softening zone increases and, hence, rotation at fracture also increases with increasing ku. For this reason, when fracture occurs, increasing the ku factor increases the redistribution capacity. When sliding failure occurs, increasing the ku factor reduces the redistribution capacity for all section depths. Increasing the steel area reduces the crack height hcr. The reduction in hcr at the onset of sliding failure is associated with a reduction in rotation and hence reducing moment redistribution for increasing ku values. Hence the intercept of these two failure modes occurs at a peak redistribution as shown in Figure V. It can be seen that the code values have the same dual trend but importantly the code values should be seen as just part of a family of curves which depends to a large extent on the depth of the member.

220

Statement of Authorship Moment redistribution in reinforced concrete members By, Deric John Oehlers1, M atthew H askett2, M ohamed A li M .S.3 and. M ichael C . Griffith4

1Professor Deric J. Oehlers School of Civil and Environmental Engineering University of Adelaide Supervised research, reviewed manuscript SIGNED____________________ 2Mr. Matthew Haskett PhD student School of Civil and Environmental Engineering University of Adelaide Wrote manuscript, performed all analyses, developed model SIGNED____________________ 3Dr. Mohamed Ali M.S. Senior Research Associate, School of Civil and Environmental Engineering University of Adelaide 4Associate Professor Michael C. Griffith School of Civil and Environmental Engineering University of Adelaide Accepted Proceedings, ICE, Structures and Buildings, 18/6/09

221

A Oehlers, D.J., Haskett, M., Mohamed Ali, M.S. & Griffith, M.C. (2010) Moment redistribution in reinforced concrete beams. Proceedings of the Institution of Civil Engineers: Structures and Buildings, v. 163(SB3), pp. 165-176

A NOTE:



A http://dx.doi.org/10.1680/stbu.2010.163.3.165

A

243

CHAPTER 10:

SYNOPSIS - DESIGN FOR MOMENT REDISTRIBUTION IN RC BEAMS RETROFITTED WITH STEEL PLATES

In Chapter 8, it was shown that the ability of reinforced concrete member to redistribute moment is proportional to the flexural rigidity and span length of the member, which can both be easily obtained, and the moment and rotation capacity of the hinge. In this chapter, the moment redistribution capacity of reinforced concrete sections with externally bonded and near surface mounted steel plates are analysed. In this chapter, the structural mechanics approach for moment redistribution presented in Chapter 8 is extended to assess the moment redistribution capacity of a propped cantilever with a point load at midspan. When considering moment redistribution from midspan to the support, the loading arrangement can be idealised as a cantilever of length L/2 in Figure W(c) with shear P-RR, where RR is the reaction at the right end support, and a cantilever of length L/2 with an applied moment Ms at position L/2 as shown in Figure W(d).

EI

P

hinge at midspan

L(a) beam

L

(b) elasticmoment

(c) shear deformation

θs(P)

P-RR

(d) applied moment deformation

θs(Ms)Ms

P

L/2 L/2

5PL/32 5PL/32

6PL/32

RR

Figure W (Figure 1 in Chapter 10)– Hinge at midspan: propped cantilever with point load

The rotation at midspan due to the shear force P-RR in Figure W(c) is

EI4LM

EI8PL s

2

)P(s −=θ

Equation 49 where the reaction RR is 2Ms/L, and the flexural rigidity of the member is EI. The rotation at midspan due to the applied moment Ms in Figure W (d) is

244

EI2LMs

)Ms(s =θ

Equation 50 The total rotation at midspan θs is therefore

EI4

LMEI2

LMEI8

PL ss2

s −−=θ

Equation 51 When the moment distribution is elastic, then the midspan moment Ms is 5PL/32 as shown in Figure W(b). Replacing Ms in Equation 3 with 5PL/32, the elastic rotation at midspan is

EI128PLθ

2

)elastic(s =

Equation 52 Thus, when the midspan rotation is not equal to PL2/128EI, the elastic moment distribution is not achieved and, hence, moment redistribution is occurring. The total hinge rotation at midspan is

EI32LM24LM15

EI128PL

EI4LM

EI2LM

EI8PL sst

2ss

2

s−

=−−−=θ

Equation 53 where the static moment Mst is PL/4. The moment redistribution factor KMR is defined as the moment redistributed as a proportion of the elastic moment. As the elastic moment at midspan is 5PL/32, or 5/8 the static moment Mst, then

st

sst)hs:P(MR

M85

MM85

K−

=→

Equation 54 where the subscript P:s→h refers to moment redistribution for a point load P and redistribution from the sagging region (s) to (→) the hogging region (h). Rearranging Equation 54 with Ms the subject, the midspan moment in terms of the static moment and moment redistribution is

( )MRsts K1M85M −=

Equation 55 Rearranging Equation 53 in terms of Mst and substituting into Equation 55 and solving for KMR gives

245

LM3EI4EI4

Kss

s)hs:P(MR +θ

θ=→

Equation 56 A similar approach can also be used to express the moment redistribution as a function of the span length, flexural rigidity, and moment and rotation at failure when redistribution is occurring from the hogging to sagging region. In this case:

LMEI3EI3K

hh

h)sh:P(MR +θ

θ=→

Equation 57 Hence, knowing the moment and rotation at failure from the RBR model, the moment redistribution capacity of a propped cantilever with a point load can be determined. Partial-interaction equations quantifying steel reinforcing bar load for a given displacement were presented in Chapter 4, and used successfully in modelling the behaviour of deformed steel reinforcing bars. These basic equations are generic and can accommodate the presence of yielding, the reduced strain hardening modulus post yield, and any shape of reinforcement. It was shown for a reinforcing bar prior to yield that the relationship between displacement, ∆steel, and axial load, Psteel, is

δ∆−δ

λ

τ=

max

steelmax

el

permaxsteel arccossin

LP

Equation 58 where τmax is the peak bond stress, δmax the slip at which no more bond stress can be transferred across the bar-concrete interface and obtained from the local bond stress-slip relationship, and Psteel and ∆steel are the load and displacement of any steel reinforcement. Equation 58 is appropriate for a range of displacements (∆steel) from zero to the slip at which yield occurs (∆yield)

τ

λ−δ=∆

maxper

steelyelmaxyield L

Afarcsincos1

Equation 59 For NSM and EB steel plates, τmax and δmax are given by

( ) 6.0cfmax f078.0802.0 ϕ+=τ (units of N and mm)

Equation 60

f

526.0f

max 078.0802.0976.0

ϕ+ϕ

=δ (units of N and mm)

Equation 61

246

and where

( )steelsmax

permax2el AE

Lδ

τ=λ

Equation 62 in which Lper and ϕf are both functions of the plate geometry from Figure H. The load at debonding of a deformed steel reinforcing bar is assumed to be simply the yield load Asteelfy plus the increase in load through strain hardening:

( ) ( )steelshpermaxmaxysteelbarReIC AELδτfAP += Equation 63 Equation 63 assumes that the entire interfacial fracture energy is available to strain harden the reinforcing bar. This assumption is suitable when considering deformed steel reinforcing bars since they have significant interfacial fracture energy, and the majority of the interfacial fracture energy is available to strain harden the bar. When EB or NSM steel plates are used, the assumption that the entire interfacial fracture energy is available to strain harden the plate is less appropriate. For EB plates debonding may occur prior to yield, rendering the assumption that the entire interfacial fracture energy is available to strain harden the plate inappropriate. Hence, the load at which debonding of EB and NSM steel plates occurs is not given by Equation 63. Figure X(a) shows the distribution of slip and bond along a steel plate from the crack face on the left hand side and Figure X (b) shows the local bond stress/slip relationship which is assumed to be linear although the following argument applies to any bond-slip shape. When the yield load, Py is applied at the loaded end, the slip at the loaded end is ∆yield, as shown by the feint black line in the “slip distribution”. Over the active length of embedment, Lyield in Figure X (a), the slip reduces along line B-E from ∆yield at the loaded end to zero at position Lyield. Furthermore, the bond stress distribution varies along line I-K from τyield at the loaded end and increasing to τmax at position Lyield. Hence, the range of bond stresses when load Py is applied is τmax to τyield, and the range of slips ∆yield to zero. Thus, the range of the bond-slip relationship required to yield the reinforcement is shown as the un-hatched area in the bond stress-slip relationship in Figure X (b), where the slip varies from ∆yield to zero.

247

yield

max

yield

max

PIC

yield

max

Pyield

max

maxyield

yield

Loaded end

Slip distribution

Bond stressdistribution

Lyield

LIC

(a) Bond behaviour (b) Bond slip relationship

A

C

FE

B

HG

I J

K L

LIC-LyieldCrack face

Figure X (Figure 8 in Chapter 10) – Slip and bond stress distribution for various loads

If we now prescribe an increase in load above Py to cause strain hardening and then debonding at PIC, then the slip at the loaded end increases from Δyield to max that is from point B to A in Figure X (a). Over the active length of embedment, LIC in Figure X (a), the slip reduces from max at the loaded end to zero at position LIC along A-F. The bond stress distribution now varies from zero at the loaded end at G to max at position LIC at L. The original distributions of slip B-E and bond stress I-K are still repeated but have been shifted a distance LIC-Lyield away from the loaded end to C-F and J-L. Thus, the increase in load in the reinforcement, that is PIC-Py, is due to the presence of bond stresses ranging from zero at the loaded end to yield over the distance LIC-Lyield, which is shown as the area G-H-J in Figure X(a), over which length the slip distribution varies from max to yield shown as area A-B-C. This is also equivalent to the “unused energy” shown in Figure X(b) as the hatched area which was not required to yield the reinforcement but has been used to strain harden the reinforcement. Consequently, the ‘unused energy” area in Figure X(b) is the fracture energy available to strain harden up to debonding. That is, the term √τmaxδmax in Equation 63 should be replaced as in the following equation to accurately estimate the debonding force for reinforcement with small bond fracture energy such as for EB plates. The debonding load, considering the energy required to yield the reinforcement, which we will refer to as PIC(EB/NSMplates), is therefore:

steelshperyieldmaxyieldysteel)NSMplates/EB(IC AELΔδτfAP Equation 64 If yield occurs prior to debonding, then the debonding strain

248

( )y

shsteel

peryieldmaxyieldICsteelysteelspermaxmax EA

LAfAEL ε+

∆−δτ=ε=>>δτ

Equation 65 where εIC is the strain at debonding. If sufficient anchorage length is provided, then plate debonding can occur over some length Ldeb with no reduction in axial load. Thus, if we assume that debonding can occur over an arbitrary debonding length, Ldeb, without a reduction in load then the slip at the end of debonding, Δdeb(end), is simply the slip at the start of debonding (δmax) plus the additional slip due to the integration of strain over the debonded length Ldeb. Conservatively estimating Ldeb as half the section depth the plate slip at the end of debonding is

debICmax)end(deb Lε+δ=∆ Equation 66 This additional slip capacity through localised debonding of the steel plates adjacent to the crack front is a critical component of the ductility of steel plates. Yield of the plate provides larger debonding strains (and hence additional slip capacity due to localised debonding) than what would be achieved by FRP plates for example. The redistribution capacity (KMR) at the start and end of debonding of EB steel plates is shown in Figure Y. Various EB plate dimensions were assessed, where the cross sectional area of the plate was maintained at 30mm2, but the dimensions were varied from 1.2mm to 7mm in 1.2mm increments as shown. Changing the plate width whilst maintaining the same cross sectional area changes the bond stress-slip relationship according to Figure H and Equation 60 and Equation 61. Increased plate thicknesses provide slightly larger values of δmax from the local bond stress-slip relationship. Hence, at the start of debonding (which occurs at a slip of δmax) increased plate thickness provides slightly increased ductility. Plate dimensions influence moment redistribution at the end of debonding significantly. At the end of debonding, which considers the additional slip by integrating the debonding strain over an arbitrary debonding length, increased plate thickness reduces the ductility of the EB section. Increased plate thickness reduces the debonding strain due to the reduced contact area of the plate with the concrete. In Figure Y, two redistributions are shown: the grey square refers to redistribution from the sagging to hogging region (s->h) and the grey triangle from the hogging to sagging region (h->s). Approximately twice the redistribution can be accommodated when the hogging region is redistributing moment compared to redistribution from sagging region, and this is purely due to structural mechanics.

249

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.2 2.4 3.6 4.8 6.0 7.2tp (mm)

KM

R

plate debond (start)plate debond (end) s->hplate debond (end) h->s

Figure Y (Figure 9 in Chapter 10) – Influence of plate dimension on MR of EB plates

In Figure Y, the linear reducing relationship between moment redistribution at the end of debonding and plate thickness indicates that ultimately at large plate thicknesses (and small plate widths since the cross sectional area is constant) plate debonding occurs before plate yield. A dimensionless number, Rpl, exists which is the ratio of the peak plate stress if it behaves perfectly elastically and the yield stress. Designers should choose plate properties such that the dimensionless number Rpl is greater than 1, which ensures yield occurs before debonding and hence provides greater ductility. The larger the value of the parameter Rpl the greater the debonding strain and hence the ductility.

y

pl

spermaxmax

y

ICpl f

AEL

fR

δτ

=σ

=

Equation 67 From Equation 67, the yield stress clearly influences ductility; plates with lower yield stress yield earlier and hence provide more ductility through greater strain hardening capacity. The redistribution behaviour of NSM steel plates also displays similar properties. NSM plates have significantly better bond capacity and hence ductility than EB plates. Debonding commences at larger crack widths, and NSM plates can also achieve greater increases in strain hardening due to increased contact with the concrete (Lper) and greater interfacial fracture energy. When comparing EB and NSM plates the behaviours follow the same trends, only the redistribution capacities of NSM plates are significantly greater than EB plates.

250

Statement of Authorship Design for moment redistribution in RC beams retrofitted with steel plates Matthew Haskett1, Deric John Oehlers2 and Mohamed Ali M.S.3 1Mr. Matthew Haskett PhD student School of Civil, Environmental and Mining Engineering University of Adelaide Wrote manuscript, performed all analyses, developed theory SIGNED____________________ 2Professor Deric J. Oehlers School of Civil, Environmental and Mining Engineering University of Adelaide Supervised research, reviewed manuscript SIGNED____________________ 3Dr. Mohamed Ali M.S. Senior Research Associate, School of Civil, Environmental and Mining Engineering University of Adelaide Advances in Structural Engineering, Vol. 13(2), pp. 379-392

251

JOURNAL PAPER – DESIGN FOR MOMENT REDISTRIBUTION IN RC BEAMS RETROFITTED WITH STEEL PLATES

Matthew Haskett, Deric John Oehlers and Mohamed Ali M.S. Abstract It is now common practice to retrofit reinforced concrete members by adhesively bonding steel or fibre reinforced polymer plates to their surfaces. However, tests have shown that these plated RC structures tend to have less member ductility, or rotational capacity, than the unplated structure because of premature plate debonding. In this paper, structural mechanics approaches are described for both: quantifying the moment rotation capacity, or member ductility, of steel plated RC flexural members; and quantifying the moment redistribution capacity from the moment rotation capacity. It is shown how the moment redistribution structural mechanics model can be used to design for member ductility directly and, furthermore, it is applied to both externally bonded and near surface mounted steel plates. As would be expected, it is shown that steel plating produces more ductile members than fibre reinforced polymer plating. Keywords: Reinforced concrete; beams; ductility; moment rotation; moment redistribution; near surface mounted; externally bonded; steel plates. 1 - Introduction In general, reinforced concrete flexural members are designed directly for member strength and stiffness and indirectly for member ductility by ensuring that: the steel reinforcing bar yields before the concrete crushes; the steel reinforcement material is ductile so that bar fracture does not precede concrete crushing; and the bond between the reinforcement and the concrete is strong enough to ensure concrete crushing occurs prior to reinforcing bar failure. However, these criteria cannot be ensured with plated members because of the weak bond and consequently the tried and well tested indirect approaches for ensuring member ductility cannot be used directly for plated structures. In this paper, a generic structural mechanics model (Oehlers et al. 2008, Haskett et al. 2009a) for quantifying the moment rotation of any reinforced concrete hinge is described as well as a structural mechanics model for moment redistribution in a propped cantilever. These models are then used to quantify the moment redistribution capacity of an RC flexural member retrofitted with externally bonded (EB) steel plates, and subsequently with near surface mounted (NSM) steel plates. It is shown that steel yielding can provide significant ductility benefits that are not attainable when fibre reinforced polymer (FRP) plates are used. By way of examples, the ductility of EB and NSM steel plated sections are assessed, and a dimensionless number Rpl is introduced which indicates whether EB or NSM plates yield before they debond. 2 – Moment redistribution analysis 2.1 - Redistribution from midspan to support Moment redistribution is an important concept in structural engineering, and is a measure of the ability of the section to maintain a constant moment whilst simultaneously rotating and redistributing moment to another region of the member. Typically, the redistribution capacity of unplated RC sections has been quantified in

252

terms of the ku factor, which is the ratio of the depth of the neutral axis to the effective depth of the section. Sections with lower ku factors, which are achieved by under-reinforcing a section, provide greater curvature at failure and, hence, since rotation is the integration of curvature over the hinge length, they achieve greater rotation (and moment redistribution) at failure. It can, however, also be shown (Scholz 1993, Haskett et al. 2009b) from a structural mechanics approach that moment redistribution is a function of the moment (Mcap) and rotation (θcap) at failure, the flexural rigidity (EI) and the span of the member (L). In a previous paper (Haskett et al. 2009b), the redistribution capacity of a continuous member subjected to a uniformly distributed load was quantified in terms of the aforementioned parameters. In this paper a propped cantilever is considered with a point load at midspan, as in one span of the two span continuous beam in Figure 1(a). Let us first consider moment redistribution from midspan to support. The loading arrangement can be idealised as a cantilever of length L/2 in Figure 1(c) with shear P-RR, where RR is the reaction at the right end support, and a cantilever of length L/2 with an applied moment Ms at position L/2 as shown in Figure 1(d).

EI

P

hinge at midspan

L(a) beam

L

(b) elasticmoment

(c) shear deformation

θs(P)

P-RR


θs(Ms)Ms

P

L/2 L/2

5PL/32 5PL/32

6PL/32

RR

Figure 1– Hinge at midspan: propped cantilever with point load

The rotation at midspan due to the shear force P-RR in Figure 1(c) is

EI4LM

EI8PL s

2

)P(s −=θ Equation 1

where the reaction RR is 2Ms/L, and the flexural rigidity of the member is EI. The rotation at midspan due to the applied moment Ms in Figure 1(d) is

EI2LMs

)Ms(s =θ Equation 2

253

The total rotation at midspan θs is therefore

EI4

LMEI2

LMEI8

PL ss2

s −−=θ Equation 3

When the moment distribution is elastic, then the midspan moment Ms is 5PL/32 as shown in Figure 1(b). Replacing Ms in Equation 3 with 5PL/32, the elastic rotation at midspan is

EI128PL2

)elastic(s =θ Equation 4

Thus, when the midspan rotation is not equal to PL2/128EI, the elastic moment distribution is not achieved and, hence, moment redistribution is occurring. The total hinge rotation at midspan is

EI32LM24LM15

EI128PL

EI4LM

EI2LM

EI8PL sst

2ss

2

s−

=−−−=θ Equation 5

where the static moment Mst is PL/4. The moment redistribution factor KMR is defined as the moment redistributed as a proportion of the elastic moment. As the elastic moment at midspan is 5PL/32, or 5/8 the static moment Mst, then

st

sst)hs:P(MR

M85

MM85

K−

=→ Equation 6

where the subscript P:s→h refers to moment redistribution for a point load P and redistribution from the sagging region (s) to (→) the hogging region (h). Rearranging Equation 6 with Ms the subject, the midspan moment in terms of the static moment and moment redistribution is

( )MRsts K1M85M −= Equation 7

Rearranging Equation 5 in terms of Mst and substituting into Equation 7 and solving for KMR gives

LM3EI4EI4

Kss

s)hs:P(MR +θ

θ=→ Equation 8

2.2 - Redistribution from support to midspan The analysis of a propped cantilever with concentrated point loads at midspan and moment redistribution from support to midspan is simpler than the converse in Section 2.1 since symmetry about the hinge at the central support exists and consequently the slope at the central support must remain at zero. The idealisation is

254

shown in Figure 2 where the elastic moment distribution is shown in Figure 2(b), and under elastic conditions there is no rotation at the hinge.

EI

P

hinge at support

L

(a) beam

L

(b) elasticmoment

(c) load deformation

θh(P)


θh(Mh)

L/2 P L/2

A B C

P

5PL/325PL/32

6PL/32

Mh

L/2

Figure 2- Hinge at support: propped cantilever with point load

The rotation at the central support (hogging region) due to the applied load P in Figure 2(c) is given by

EI16PL2

)P(h =θ Equation 9

where the subscript ‘h’ refers to the hogging region. The rotation at the central support due to the applied moment Mh in Figure 2(d) is given by

EI3LM h

)Mh(h =θ Equation 10

Under elastic conditions where no moment redistribution is occurring, the slope at the central support is zero by symmetry. Thus, when rotation occurs at the central support moment redistribution is occurring. The total hinge rotation is therefore

EI3LM

EI16PL h

2

h −=θ Equation 11

As before, the static moment is PL/4 and moment redistribution is

st

hst)sh:P(MR

M43

MM43

K−

=→ Equation 12

255

where the subscript P:h→s refers to moment redistribution with a point load (P), from the hogging (h) to (→) sagging (s) region. The elastic moment at the support is 6PL/32 from Figure 2(b), or 3/4 the total static moment Mst. Rearranging Equation 12 with Mh the subject, the support moment in terms of moment redistribution and static moment

( )MRsth K1M43M −= Equation 13

Rearranging Equation 11in terms of Mst and substituting into Equation 13 and solving for KMR gives

LMEI3EI3K

hh

h)sh:P(MR +θ

θ=→ Equation 14

Thus if the moment and rotation at failure are known or can be calculated, then the redistribution capacity of a section is able to be determined. A rigid body rotation mechanism (Oehlers et al. 2008, Haskett et al. 2009a) has been proposed which models the behaviour of the section using partial interaction (PI) and shear friction (SF) theory, which, importantly, provides a moment-rotation response from the commencement of loading to failure. Thus, knowing the moment and rotation capacity of the section at failure, Mh or Ms and θh or θs respectively in Equation 8 and Equation 14, then the redistribution capacity can be determined. In the following section, the rigid body rotation (RBR) mechanism is explained, which allows the moment and rotation at failure to be determined independent of hinge length and independent of failure mode. 3 – Generic moment-rotation (rigid body rotation) model Member ductility, or rotational capacity, of unplated reinforced concrete flexural members is commonly determined from a sectional moment curvature analysis in which the curvature at failure is determined and is integrated over an empirical hinge length to obtain the rotation capacity. The difficulty in this approach has always been in determining the length of the hinge. Various hinge length expressions have been developed (Baker 1956, Sawyer 1964, Corley 1966, Mattock 1967, Priestley and Park 1987) which provide good correlation from the experimental set from which they were obtained but significant scatter when used outside their experimental set (Panagiotakos and Fardis 2001). This hinge length approach was developed for normal flexural members where concrete crushing controlled failure and consequently cannot be used directly or should be used with extreme care when the reinforcement fractures or debonds prior to concrete crushing. In response to this difficulty, a new method has been proposed (Haskett et al 2009c, 2009a and 2009b; Oehlers et al 2008 and 2009), a rigid body rotation (RBR) approach, which does not assume or require a hinge length to determine a hinge’s rotational capacity or ductility. This method differs from the standard strain approach, as instead a displacement profile, rather than a strain profile, is assumed and the axial tensile and compressive forces are determined according to this displacement profile. It is a partial interaction (PI) approach and, hence, assumes that there is slip of the reinforcement relative to the surrounding concrete. Consequently, an essential requirement in the analysis technique is to be able to quantify the axial load in the reinforcement for a given slip.

256

To ensure the viability of this approach, mathematical expressions for the load-slip response (P−∆) for all forms of reinforcement, including FRP and steel reinforcing bars and plates is required, and these expressions for deformed steel reinforcing bars and steel plates are presented in Section 4. Partial interaction theory has been used previously in the analysis of FRP plates, where closed form mathematical solutions have been developed (Yuan et al. 2004, Mohamed Ali et al. 2006) enabling the load-slip (P−∆) response to be quantified mathematically. Thus, even when brittle materials such as FRP are used, or when brittle bond occurs such as with adhesively applied EB and NSM plates, provided the P−∆ response of the reinforcement can be determined, the RBR approach can be used determine the moment and rotation at failure. Until now, this partial interaction technique has not been required to analyse RC sections because steel reinforcing bars have significant bond capacity and, therefore, it is reasonable to assume that the force in the reinforcing bars is not limited by its bond transfer capacity and, consequently, failure can only be caused by concrete crushing. However, the increased use of brittle steel reinforcing bars and brittle plating techniques (EB) has required the development of this RBR approach. Visual representation of the RBR model is shown in Figure 3, and a complete description of the RBR model is presented elsewhere (Haskett et al. 2009a, Haskett et al. 2009b, Oehlers et al. 2008). In brief, the hinge is idealised in Figure 3(a) as a single crack where rotation θ is directly controlled by the slip between the axial reinforcement and concrete. The limit to the bar or plate slip is at debonding or fracture. Furthermore, the softening of the concrete is represented by a wedge where the wedge force is limited by the sliding capacity of the wedge, ssoft. The analysis is shown in Figure 3(b) where the force in the reinforcement Prebar and Pplate is related to the reinforcement slip Δrebar and Δplate through partial interaction theory. The compressive force in the concrete is that in the ascending branch Pasc and that in the wedge Psoft where the latter can be obtained from shear-friction theory and the former from standard procedures.

∆plate

∆rebar

primaryflexuralcrack

compressionwedgerigid

bodyrotation

θ

α α

θθ

Pasc

Psoft

Prebar

Pplate

softening region

non-softeningregion

crack width

ssoft∆asc

ssoftεsoft εsoft

A

A

(a) rigid body displacements (b) moment-rotation analysis Figure 3– Rigid body displacement and rotation analysis

257

Without presenting the entire methodology for the analysis technique, which is presented in detail elsewhere (Oehlers et al. 2009), using the RBR approach it is possible to develop a moment-rotation response for any section up to and including failure. This moment-rotation response can accommodate all failure modes such as concrete failure (referred to as sliding rather than crushing in this methodology), bar failure (fracture or debonding) or plate failure (debonding or fracture). An example of a M-θ response is shown in Figure 4, where the section analysed is shown in Figure 5 and where EB plates, 1.2mmx20mm wide, of yield stress 400MPa, and total cross sectional area 20% the total cross sectional area of steel have been applied to the soffit of the section. The moments and rotations at which plate yield and debonding, and reinforcing bar yield and debonding have also been shown in Figure 4. In Figure 4 it is clear that plate yield, shown as the grey diamond, occurs at a moment of approximately 600kN-m. Reinforcing bar yield and plate debonding both occur later at a moment of approximately 800 kN-m, shown as the grey square and triangle respectively. Continuing the analysis by assuming that plate debonding (and hence failure) had not occurred at a moment of approximately 800kN-m, bar debonding finally occurs at a moment in excess of 900kN-m, as shown by the black circle in Figure 4.

0

200

400

600

800

1000

0.E+00 2.E-02 4.E-02 6.E-02 8.E-02θ (rad)

M (k

N-m

)

600mm deep sectionReinforcing bar yieldPlate debondingPlate yieldBar debonding

Figure 4 – M-θ response for plated section, with failure modes

Mcap , θcap

b=300mm

d=600mm

c=25mm

Ast=2%=3450mm2

db=12mm, fy=400MPa, ffract=600MPa, Esh=2,000MPa

fc=30MPa

EB or NSM platevarious tp and bp

258

Figure 5– Base RC section to be retrofitted From the M−θ response in Figure 4, we can determine the moment and rotation at failure, shown as Mcap and θcap respectively. In this example as shown in Figure 4, and discussed earlier, failure is caused by plate debonding at a moment of approximately 800kN-m. Knowing the moment and rotation at failure, the moment redistribution capacity at failure can be determined. For a propped cantilever subjected to a point load at midspan, the redistribution capacity is given by Equation 8 or Equation 14, depending on which region is redistributing moment. In the following section, the existing mathematical closed form expressions for the load-slip (P−∆) relationships of steel reinforcing bars and EB and NSM steel plates are presented as they are essential for the RBR approach. 4 – Existing closed form P−∆ equations Partial-interaction equations quantifying steel reinforcing bar load for a given displacement (Prebar and ∆rebar in Figure 3) have been developed previously (Haskett et al. 2009c), and used successfully in modelling the behaviour of deformed steel reinforcing bars. These basic equations are generic and can accommodate the presence of yielding, the reduced strain hardening modulus post yield, and any shape of reinforcement. It was shown for a reinforcing bar (Haskett et al. 2009c) that prior to yielding the relationship between displacement, ∆steel, and axial load, Psteel, is

δ∆−δ

λ

τ=

max

steelmax

el

permaxsteel arccossin

LP Equation 15

where τmax is the peak bond stress, δmax the slip at which no more bond stress can be transferred across the bar-concrete interface and obtained from the local bond stress-slip relationship, and Psteel and ∆steel are the load and displacement of any steel reinforcement in Figure 3 where

( )steelsmax

permax2el AE

Lδ

τ=λ Equation 16

Equation 15 is appropriate for a range of displacements (∆steel) from zero to the slip at which yield occurs (∆yield), where the slip at which yield occurs is given by (Haskett et al. 2009c)

τ

λ−δ=∆

maxper

steelyelmaxyield L

Afarcsincos1 Equation 17

The length Lper is the concrete failure plane surrounding the cross-section of the reinforcement and shown in Figure 6. Lper is a critical parameter in any debonding analysis and governs the total force that can be transferred across the reinforcement-concrete interface. From Figure 6 it can be seen that the failure plane (Lper) of EB plates is significantly less than that of NSM plates or reinforcing bars. This introduces

259

limits to the total force that can be developed in EB plates and is discussed in detail later.

LperLper

Ap of EB plate1 mm

1 mm

1 mm

1 mm

Ap of NSM plate

df

df

bf

bf

failure plane EB plate

failure plane NSM plate

concrete surface

concreteelement

bp tp

tp bp

Lper

1 mmdp

Abar of reinforcing bar

failure plane of reinforcing bar

(a) externally bonded plate (b) near surface mounted plate (c) reinforcing barFigure 6– Failure planes of axial reinforcement

For deformed steel reinforcing bars, τmax is approximately 1.25√f c and δmax approximately 15mm (Haskett et al. 2008c). For EB and NSM plates (Seracino et al. 2007),

( ) 6.0cfmax f078.0802.0 ϕ+=τ (units of N and mm) Equation 18

f

526.0f

max 078.0802.0976.0

ϕ+ϕ

=δ (units of N and mm) Equation 19

where the aspect ratio, ϕf in Equation 18 and Equation 19 is the ratio of the depth of the failure plane perpendicular to concrete face (df) to the width of the failure plane parallel to concrete face (bf), where the parameters df and bf are shown in Figure 6. An example of the relative magnitudes of δmax and τmax is shown in Figure 7, which is approximately to scale. It can be seen that the peak bond stress, τmax, is approximately the same for EB plates, NSM plates, and deformed reinforcing bars. For example, for 30MPa concrete it is approximately 7 MPa. However, the significant difference between the three types of axial reinforcement is the range of slip (δ) over which bond stress can be transferred. In Figure 7, it can be seen that the slip capacity of reinforcing bars, δmax (Rebar), is significantly greater than that of EB (δmax (EB)), and NSM (δmax (NSM)) plates. For deformed steel reinforcing bars bond stress can be transferred up to a slip of approximately 15mm (where δmax (Rebar) in Figure 7 is approximately 15 mm), whereas bond stresses can only be transferred for slips up to approximately 2mm for NSM plates (where δmax (NSM) in Figure 7 is approximately 2 mm), and less than 0.5mm for EB plates (where δmax (EB) in Figure 7 is approximately 0.5 mm). Hence, EB plates debond earliest, followed by NSM plates and finally, at huge crack widths, deformed steel reinforcing bars.

260

The interfacial fracture energy of deformed reinforcing bars is shown hatched in Figure 7, and is representative of the total force that can be developed in the reinforcement prior to debonding. It has been shown (Seracino et al. 2007) that the interfacial fracture energy governs the total force that can be transferred across the reinforcement-concrete interface, where ultimately debonding of the reinforcement occurs when the slip (∆rebar or ∆plate) exceeds the slip at which debonding commences, δmax, from the local bond stress-slip relationship. Since the interfacial fracture energy of deformed reinforcing bars is very large, as shown in Figure 7, the load at which deformed reinforcing bars debond is significantly greater than EB or NSM plates, all other things considered.

τmax

δmax (Rebar)

τ (MPa)

0δ

Deformed bars

NSM plateEB plate

Interfacialfracture energy(deformed bars)

δmax (NSM)

δmax (EB)

Figure 7– Local bond stress-slip (τ−δ) relationship

The load at which debonding occurs is given by

( ) ( )steelshpermaxmaxysteel1MethodIC AELfAP δτ+= Equation 20 The second component of Equation 20 reflects the increase in strength due to strain hardening of the reinforcement. Equation 20 also assumes that the slip at yield is negligible, and that no interfacial fracture energy is “used” to yield the reinforcement. This is a reasonable assumption for deformed bars given their large interfacial fracture energy and the small slips at which bar yield occurs. For example, only approximately 2% of the total interfacial fracture energy is used to yield a 12mm diameter bar of yield stress 400MPa. Hence, when considering deformed steel bars, it can be assumed that the entire interfacial fracture energy remains to strain harden the bar, introducing minimal error. When EB or NSM plates are used, the assumption that the entire interfacial fracture energy is available to strain harden the plate is less appropriate. For EB plates ∆yield/δmax can approach 1, or indeed yield may not occur, rendering the assumption that the entire interfacial fracture energy is available to strain harden the plate inappropriate. Hence, the load at which debonding of EB and NSM plates occurs requires a slight modification to Equation 20.

261

Figure 8(a) shows the distribution of slip and bond along a reinforcing bar from the crack face on the left hand side and Fig. 8(b) shows the local bond stress/slip relationship which is assumed to be linear although the following argument applies to any bond-slip shape. When the yield load, Py is applied at the loaded end, the slip at the loaded end is ∆yield, as shown by the feint black line in the “slip distribution”. Over the active length of embedment, Lyield in Figure 8(a), the slip reduces along line B-E from ∆yield at the loaded end to zero at position Lyield. Furthermore, the bond stress distribution varies along line I-K from τyield at the loaded end and increasing to τmax at position Lyield. Hence, the range of bond stresses when load Py is applied is τmax to τyield, and the range of slips ∆yield to zero. Thus, the range of the bond-slip relationship required to yield the reinforcement is shown as the un-hatched area in the bond stress-slip relationship in Figure 8(b), where the slip varies from ∆yield to zero.

δ

τ

∆yield

δmax

τyield

τmax

PIC

∆yield

δmax

Pyield

τ

δ

τmax

δmax∆yield

τyield

Loaded end

Slip distribution

Bond stressdistribution

Lyield

LIC

(a) Bond behaviour (b) Bond slip relationship

A

C

FE

B

HG

I J

K L

LIC-LyieldCrack face

Figure 8 – Slip and bond stress distribution for various loads If we now prescribe an increase in load above Py to cause strain hardening and then debonding at PIC, then the slip at the loaded end increases from Δyield to δmax that is from point B to A in Fig. 8(a). Over the active length of embedment, LIC in Figure 8(a), the slip reduces from δmax at the loaded end to zero at position LIC that is along A-F. The bond stress distribution now varies from zero at the loaded end at G to τmax at position LIC at L. The original distributions of slip B-E and bond stress I-K are still repeated but have been shifted a distance LIC-Lyield away from the loaded end to C-F and J-L. Thus, the increase in load in the reinforcement, that is PIC-Py, is due to the presence of bond stresses ranging from zero at the loaded end to τyield over the distance LIC-Lyield, which is shown as the area G-H-J in Fig. 8(a), over which length the slip distribution varies from δmax to ∆yield shown as area A-B-C. This is also equivalent to the “unused energy” shown in Fig. 8(b) as the hatched area which was not required to yield the reinforcement but has been used to strain harden the reinforcement. Consequently, the

262

‘unused energy” area in Figure 8(b) is the fracture energy available to strain harden up to debonding. That is, the term √τ maxδmax in Equation 20 should be replaced as in the following equation to accurately estimate the debonding force for reinforcement with small bond fracture energy such as for EB plates in Figure 7. The debonding load, considering the energy required to yield the reinforcement, which we will refer to as Method 2, is therefore:

( ) ( )steelshperyieldmaxyieldysteel)2Method(IC AELfAP ∆−δτ+= Equation 21 Thus, knowing the load at which debonding occurs we can determine the strain at plate or bar debonding which we need later to determine the debonding slip as in Equation 26. If yield occurs prior to debonding, then

( )y

shsteel

peryieldmaxyieldICsteelysteelspermaxmax EA

LAfAEL ε+

∆−δτ=ε=>>δτ

Equation 22 where εIC is the strain at debonding, which is required to determine the slip at the end of plate debonding and used in Equation 26. Alternatively, if yield does not precede debonding, which can occur with brittle bond conditions, such as EB plates then

ssteel

permaxmax

s

ICICsteelysteelspermaxmax EA

L

EAfAEL

δτ=

σ=ε=><δτ

Equation 23 After yield occurs, the relationship between load and slip is governed by the strain hardening modulus of the steel. For slips greater than ∆yield, and less than δmax, and assuming for simplicity that the entire interfacial fracture energy is available post yield, the relationship between load and slip is given by (Haskett et al. 2009c)

steelymax

yieldsteelmax

sh

permaxsteel Afarccossin

LP +

δ

∆+∆−δ

λ

τ=

Equation 24 where:

( )steelshmax

permax2sh AE

L

δ

τ=λ Equation 25

Finally, from previously discussed theory (Haskett et al. 2009b), if sufficient anchorage length is provided, then plate debonding can occur over some length Ldeb with no reduction in axial load. Thus, if we assume that debonding can occur over an arbitrary debonding length, Ldeb, without a reduction in load then the slip at the end of debonding, Δdeb(end), is simply the slip at the start of debonding (δmax) plus the additional slip due to the integration of strain over the debonded length Ldeb. Ldeb can

263

be conservatively estimated as half the section depth (Haskett et al. 2009b). Hence, the plate slip at the end of debonding is

debICmax)end(deb Lε+δ=∆ Equation 26 Thus, we know the slip of the reinforcement at the start of debonding, δmax, the load at which debonding commences, PIC as given by Equation 21, and the slip of the reinforcement at the end of debonding as given by Equation 26. Knowing these values, we can determine the moment and rotation at the start and end of debonding, and hence the moment redistribution at the start and end of debonding. In the following section the redistribution capacity of various EB and NSM sections is discussed, where it is shown that steel plated RC sections can achieve significant amounts of moment redistribution. 5 – Moment redistribution capacity of plated sections 5.1 Externally bonded steel plates The majority of plate debonding research has considered the use of EB and NSM fibre reinforced polymer (FRP) plates (Oehlers 2006; Chen et al 2007; Mohamed Ali et al 2008). The use of FRP has been considered to be preferable to steel plates because FRP behaves perfectly linear-elastically up to plate fracture or debonding, whichever comes first. Hence, the plate continues to attract load up to failure with constant loading stiffness, maximising the benefit in plating by providing the greatest increase in flexural capacity. Conversely, steel plates have often been ignored because of the difficulty in modelling them due to their elasto-plastic behaviour, and the axial load limit that can be developed in the plate (relative to FRP) due to the presence of yielding. However, steel plates can yield prior to debonding, which provides greater debonding strains and hence increased ductility due to localised debonding. Consider a base RC section, as shown in Figure 5, that is being retrofitted to increase its flexural capacity, whilst at the same time having some minimum ductility requirements. To initially increase the flexural capacity of the RC section, consider applying EB steel plates with 20% cross sectional area of the total cross sectional area of the steel reinforcing bars, which, in this example, is 690mm2 of EB steel plate. The steel plates have properties: yield stress 400MPa, elastic modulus 200,000MPa, and strain hardening modulus 2,000MPa post yield. These properties are used throughout this paper unless noted otherwise. A fracture stress or strain has not been assumed in this analysis, but can be accommodated in the RBR methodology if necessary. To assess the influence of plate dimensions, the cross sectional area of each plate was maintained constant at 30mm2, whilst the thickness of the plate was varied from 1.2mm to 7mm, in 1.2mm increments. The breadth of the plate was also therefore varied, meaning that the failure plane, Lper in Figure 6, which is the contact width over which the bond is transferred, reduced as the plate thickness increased. This reduces the total force that can be transferred across the plate-concrete interface, and ultimately may mean that the plate may debond prior to yield. The aspect ratio of the plate, which is the ratio of df and bf as defined in Figure 6 and is a function of plate thickness and width, influences both τmax and δmax. Therefore, changing the plate width to maintain the same cross sectional area for various plate thicknesses also changes the bond stress-slip relationship. As the plate width reduces

264

for increasing plate thickness the increase in peak bond stress (τmax) is negligible. Conversely, peak slip (δmax) increases noticeably for increased plate thickness. The peak bond stress, τmax, and the peak slip after which no more bond stress can be transferred across the plate-concrete interface, δmax, are given by Equation 18 and Equation 19 respectively. The influence of plate dimension on moment redistribution at the start and end of plate debonding is shown in Figure 9, where in this paper moment redistribution is calculated from both Equation 8 and Equation 14, and in the legend s→h refers to redistribution from the sagging to hogging region, and h→s redistribution from the hogging to sagging region. The moment and rotation at failure are determined from the RBR model described in Section 3. The salient values from the analysis are also tabulated in Table 1.

0.0

0.1

0.2

0.3

0.4

0.5

0.0 1.2 2.4 3.6 4.8 6.0 7.2tp (mm)

KM

R

plate debond (start)plate debond (end) s->hplate debond (end) h->s

Figure 9– Influence of plate dimensions on MR of EB plates – Apl=30mm2

Figure 9 shows that as the plate thickness increases (corresponding to reduced plate width) the moment redistribution capacity at the start of plate debonding gradual increases. This very minimal increase in MR at the start of plate debonding (Figure 9 and rows 10 and 11 of Table 1) is attributed to the increased slip at the start of plate debonding δmax due to the improved aspect ratio of thicker, less wide plates. For clarity, at the start of debonding redistribution from the sagging to hogging region (Equation 8) has been shown, since the redistribution capacity at the start of debonding when considering redistribution from the hogging to sagging region is very similar. In Figure 9, it is clear that as the plate thickness increases the moment redistribution at the end of debonding continues to reduce. This inverse relationship between ductility and plate thickness was also observed in an early experimental investigation on the ductility of plated RC concrete beams (Sharif et al. 1994). Ultimately, at a plate thickness of 7.2mm in this example, there is a significant reduction in moment redistribution such that the redistribution at the start and end of debonding are similar. This suggests that the debonding strain continues to reduce as the plate becomes thicker and less wide. Knowing that the IC debonding strain is a function of τmax, δmax, Apl, Es, Esh, εy, and Lper, the significant reduction in debonding strain as the plate width reduces is primarily due to the reduced failure plane, Lper, row 5 in Table 1.

265

Indeed, as the plate width reduces (plate thickness increases), thereby reducing Lper, the strain at debonding continually reduces until ultimately at a plate thickness of 7.2mm debonding occurs prior to plate yield. This is also shown in Table 1 row 6, where for plate thickness 7.2mm the plate does not yield and also in row 7 where the stress at debonding is less than the yield stress fy. The absence of yielding results in limited additional ductility provided by localised debonding adjacent to the crack front. It can be seen in Table 1 that when yield occurs the slip at the end of debonding (row 9) is significantly greater than the slip at the start of debonding (row 4) and that up to 95% of the total slip at the end of debonding is caused by localised debonding around the crack front. The strain at the commencement of debonding is shown in Table 1 row 8. It can also be seen in Figure 9, and Table 1 when comparing rows 12 and 13, that approximately twice the redistribution can be accommodated when the hogging region is redistributing moment compared to redistribution from sagging region. The influence of redistributing region on moment redistribution capacity is purely due to structural mechanics.

Table 1 – Influence of plate parameters on ductility 1 tp 1.2 2.4 3.6 4.8 6.0 7.2 2 bp 25.0 12.5 8.3 6.3 5.0 4.2 3 τmax 6.2 6.2 6.2 6.2 6.3 6.3 4 δmax 0.21 0.30 0.35 0.40 0.43 0.46 5 Lper 29.0 16.5 12.3 10.3 9.0 8.2 6 ∆yield 0.08 0.16 0.23 0.30 0.38 NA 7 σIC 431.1 420.6 414.4 409.6 404.4 396.3 8 εIC 1.75E-02 1.23E-02 9.21E-03 6.79E-03 4.20E-03 1.98E-03 9 ∆db (end) 5.47 3.99 3.12 2.43 1.69 1.05 10 KMR (start) s->h 0.01 0.01 0.02 0.02 0.02 0.02 11 KMR (start) h->s 0.02 0.03 0.04 0.04 0.04 0.05 12 KMR (end) s->h 0.24 0.18 0.14 0.11 0.08 0.05 13 KMR (end) h->s 0.41 0.33 0.27 0.22 0.16 0.10 14 Rpl 1.27 1.12 1.06 1.03 1.01 0.99 The dimensionless number Rpl in Table 1 row 14 is the ratio of the peak stress attainable in the reinforcement σIC, when it behaves perfectly linearly elastically and the yield stress fy; this parameter indicates whether the plate debonds prior to yielding. It is particularly important for EB plates where the interfacial fracture energy is significantly less than for NSM plates or deformed reinforcing bars such that for certain plate dimensions yield may not occur. This is shown in Figure 7 where the interfacial fracture energy of EB plates is approximately 20% that of NSM plates, and 2-5% that of deformed reinforcing bars. The dimensionless parameter Rpl is

y

pl

spermaxmax

y

ICpl f

AEL

fR

δτ

=σ

= Equation 27

266

Designers should choose suitable plating dimensions and techniques and a suitable yield stress such that Rpl>1. This ensures that yield occurs prior to plate debonding, and therefore ensures that the steel plate can achieve the benefits of strain hardening, the most critical being the significant increase in strain post yield. This ensures that a degree of ductility is present in the plating arrangement, allowing the rotation at the end of debonding to increase. In Table 1 the parameter Rpl is shown in row 14, where for wider plates, and hence plates that have a greater bond transfer capacity due to their greater contact width with the concrete, greater debonding strains can be achieved, and hence the ratio of IC debonding stress to yield stress is greater. To investigate the influence of yield stress on the behaviour of steel plates the EB plating arrangement shown in Figure 5 was once again analysed. As before, 20% cross sectional area of steel plate was provided, with the plate dimensions maintained constant at 1.2mm thick and 25mm wide. The yield stress of the plate was varied from 200MPa to 700MPa, in 100MPa increments. The influence of yield stress on ductility is shown in Figure 10 where for clarity the moment redistribution at the end of debonding is only shown. Once again in Figure 10 the redistribution capacity from hogging to sagging (h→s) is approximately twice the redistribution capacity of the same section from sagging to hogging (s→h).

0.270.24

0.47 0.460.41

0.05

0.28

0.10

0.21

0.0

0.1

0.2

0.3

0.4

0.5

200 300 400 500 600 700 fy (MPa)

KM

R

s->hh->s

Figure 10– Influence of yield stress on ductility at end of plate debonding:

Apl=30mm2

From Figure 10 it is clear that yield stress significantly influences the moment redistribution capacity of EB plated sections, where for sections with yield stress greater than 500MPa only limited redistribution is available. When the parameter, Rpl=1, this indicates that yield and debonding occur at exactly the same load, indicating that the debonding strain is the yield strain. For this plate configuration, and solving Equation 27 for fy with Rpl=1, debonding and yield occur simultaneously for fy=502MPa. This is reflected in Figure 10, where for yield stress less than 500MPa significant redistribution is achieved. In the following section, the behaviour of near surface mounted (NSM) steel plates is analysed. It will be shown that the dimensionless parameter Rpl is less appropriate for NSM plates since for most plating arrangements yield occurs prior to debonding, due to the improved bond performance of NSM compared to EB plates.

267

5.2 – Near surface mounted steel plates NSM steel plates have significantly better bond capacity than EB plates, and as such have much greater redistribution capacity. The improved redistribution capacity is evident at both the start and end of debonding. The slip at which NSM plates commence debonding is in the order of 5 times greater than that of a similar EB plate, and the strains achieved by NSM plates at debonding can similarly be up to five times greater than that achieved by EB plates of the same dimensions. Due to the significant bond stresses that are developed between the NSM plate and the surrounding concrete the dimensionless parameter Rpl introduced previously is not necessary since yield will almost always occur prior to debonding for almost all plate dimensions and yield stresses. In addition, since the slip at yield of NSM plates is generally much less than the peak slip (δmax), the error between Equation 20 and Equation 21 is minimal. To highlight the flexibility of this RBR approach to predicting moment redistribution we will now consider the section shown in Figure 5 with NSM steel plates of dimension 10mmx10mm, total cross sectional area 20% of the cross sectional area of the steel reinforcing bars and varying yield stress. The moment redistribution capacity of the various yield stress plates is shown in Figure 11. In Figure 11, it can be seen that significant redistribution capacity can be achieved by the thick NSM plates, and these redistribution capacities are greater than that of common EB plating arrangements. In this example the 700MPa plate does not yield prior to debonding, and that significantly less moment redistribution is available than when a plate with lower yield stress is used. Generally though for most yield stresses, at the end of debonding significant redistribution capacity is present, of which some can be used in design.

0.0

0.2

0.4

0.6

200 300 400 500 600 700fy (MPa)

KM

R

plate debond (start) s->h plate debond (end) s->hplate debond (start) h->s plate debond (end) h->s

Figure 11 – Moment redistribution for 10x10mm NSM plates of varying fy

6 – Conclusions In this paper, mathematical equations have been presented that quantify moment redistribution in propped cantilevers in terms of member parameters such as span length and flexural rigidity and sectional parameters such as moment and rotation at failure. A rigid body rotation model has also been presented which can be used to determine the moment and rotation at failure for any reinforced concrete section with any failure mode. Mathematical equations have been presented that model the behaviour of externally bonded and near surface mounted steel plates up to and

268

beyond yield, and the moment redistribution capacity of various EB and NSM plating arrangements has been quantified. A dimensionless parameter Rpl was introduced, which is particularly relevant to EB steel plates, and provides an indication of the ductility of a steel plated section. In this paper, it is shown that a designer must consider both ductility and strength requirements for a retrofitted section before deciding on the most suitable plating arrangement.

269

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Haskett, M., Oehlers, D.J., Mohammed Ali, M.S., Wu, C., (2009c). “Yield penetration hinge rotation in reinforced concrete beams”, ASCE Structural Journal, Vol. 135, Issue 2, Feb., 130-138

Haskett, M., Oehlers, D.J., Mohammed Ali, M.S., (2008c). “Local and global bond characteristics of deformed steel reinforcing bars”, Engineering St ructures, 30(2), February, 2008, p 376-383

Haskett, M.H., Oehlers, D. J., Mohamed Ali M.S., and Wu, C., (2009b). “Analysis of moment redistribution in FRP plated RC beams”, Submitted ASCE J ournal of Composites for Construction, 16/03/09.

Mattock, A.H. (1967), “Discussion of Rotational Capacity of Reinforced Concrete Beams”, by W.D.G. Corley, Journal of the Structural Division of ASCE, Vol. 93, No. 2, pp. 519-522.

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Notation: bf - width of failure plane (for EB and NSM plates) bp - plate width b - width of concrete c - cover provided to reinforcing bars db - diameter of reinforcing bar df - depth of failure plane (for EB and NSM plates) d - total depth of concrete fc- compressive strength of concrete ffract - fracture stress of steel fy - yield stress of steel ku - depth of the neutral axis as proportion of effective depth of section ssoft - sliding capacity of softening wedge of concrete tp- plate thickness α - angle softening wedge forms δmax - slip beyond which no more bond stress is transferred ∆ - displacement of loaded end of axial reinforcement ∆deb(end) - slip at the end of debonding which allows for localised debonding adjacent to crack front ∆plate - displacement of a plate ∆rebar - displacement of a reinforcing bar ∆steel - displacement of any form of steel reinforcement (plate or bar) ∆yield - slip at yield of any form of steel reinforcement εIC - debonding strain εy - yield strain ϕf - aspect ratio of EB or NSM plate λel - elastic parameter λsh - plastic parameter θ - rotation θcap - rotation capacity θh - rotation at support θh(Ms) - rotation at support due to applied moment θh(P) - rotation at support due to shear force θs - rotation at support θs(elastic) - elastic rotation at midspan θs(Ms) - rotation at midspan due to applied moment θs(P) - rotation at midspan due to shear force σIC - debonding stress τmax - maximum bond stress τyield - bond stress transfer capacity corresponding to a slip ∆yield A - cross sectional area of axial reinforcement Ast - cross sectional area of steel EI - flexural rigidity Esh - strain hardening modulus of steel Es –elastic Young’s modulus of steel KMR - percent moment redistribution, expressed from 0-1

272

KMR(P:h→s) - moment redistribution from hogging to sagging region for propped cantilever with point load KMR(P:s→h) - moment redistribution from sagging to hogging region for propped cantilever with point load L - span length Ldeb - length over which a plate can debond with no reduction in load LIC - embedment length required to develop IC debonding load Lper - failure plane of reinforcement Lyield - embedment length required to yield reinforcement Mcap - moment capacity Mh - moment in hogging (support) region Ms - moment in sagging (midspan) region Mst - static moment P - point load Pasc - force in the non-softening region of concrete PIC - debonding load Pplate - axial force in a plate Prebar - axial force in a reinforcing bar Psoft - force in the softening region of concrete Psteel - force in steel reinforcement (plate or bar) Py - load at yield Rpl - dimensionless number that indicates if yield occurs before debonding RR - external support reaction in a propped cantilever

273

CHAPTER 11:

SYNOPSIS - DESIGN FOR MOMENT REDISTRIBUTION IN FRP PLATED RC MEMBERS

In this chapter, the concept of moment redistribution, which was explained in Chapters 8, 9, and 10, is extended to model moment redistribution in real beams. An iterative procedure for moment redistribution is explained, which is dependent on the shape of the moment-rotation response of the hinge. The moment-rotation behaviour of the non-hinging region is also a critical component in assessing the redistribution capacity of a section. In the classical moment-curvature approach (which is akin to a moment-rotation response), strain hardening is often ignored and it is assumed that once yield occurs there is infinite increase in ductility (curvature) whilst the flexural capacity remains constant. We will refer to this as being a “perfectly elasto-plastic” moment-rotation response, shown as O-A-B in Figure Z. In the rigid body moment rotation response approach, we consider strain hardening so after yield of the reinforcing bars there is an increase in moment with a corresponding increase in rotation. This response is shown as O-A-C in Figure Z.

B (Myield, θcap)

D

H

Myield

Mcap

θcap

C (Mcap, θcap)

θyield

MI

O θH

IMH

θS

SMS

A (Myield, θyield)

E( θtotal>θcap)F( θtotal<θcap)

κ1

θΙ

M

θ

Figure Z (Figure 4 in Chapter 11) – Various moment-rotation responses (not to scale) Firstly, consider the case when the same section is used in both the sagging and hogging regions of a member, and where each region has a perfectly elasto-plastic moment-rotation response O-A-B. Up to yield at Myield (point A in Figure Z), the moment distribution remains elastic such that the moment in the sagging region is half that of the hogging region. That is, the moment in the hogging region is Myield and the sagging region Myield/2. Since the hogging region achieves its flexural capacity first, it needs to maintain moment and rotate to allow the sagging region to achieve its

274

flexural capacity Myield. Thus the hogging regions rotates from θyield at point A towards θcap at B, maintaining its moment at Myield and redistributing moment to the sagging region whilst it rotates. If the hogging hinge has sufficient rotation capacity to allow the sagging region to attain Myield, then full redistribution has occurred and the total static moment is 2Myield. Importantly, in this situation where a perfectly elasto-plastic moment-rotation response exists, the region having the moment redistributed to it does not need to rotate to allow full moment redistribution to occur. Alternatively, the moment rotation response can reflect the presence of strain hardening as in O-A-C in Figure Z, where both the sagging and hogging region have the same moment-rotation response. We can firstly assume that maximum moment redistribution can occur, such that the moment and rotation of the plastic hinge is at C in Figure Z. According to this moment-rotation relationship, the moment redistribution capacity of the hinge is determined from Equation 47. According to this moment redistribution capacity, the corresponding moment in the non-hinging region of the beam is given by Equation 68, where Mhog is Mcap in Figure Z, and KMR is from Equation 47, where Mcap and θcap are both known from Figure Z.

( ) ( ))K1(2

K21MM

MR

MRhogredistsag −

+=

Equation 68 If at this moment, yield of the reinforcing bars has not occurred (i.e. (Msag)redist<Myield) then the region having moment redistributed to it is still elastic and hence has not rotated. This absence of rotation means that the plastic hinge does not need to rotate any more to accommodate the rotation in the non-plastic region of the beam, and full redistribution can occur. If the non-plastic region has yielded, for example somewhere along A-C in Figure Z, then rotation is also occurring in this region. This rotation is also induced in the primary hinging region. Hence, the sum of the rotations in the non hinge and hinge regions must equal the rotation capacity of the hinging region, θcap. The issue is that the moment in the non-hinging region is given by Equation 68 (the rotation is simply from the moment-rotation response), and is a function of the moment and rotation of the hinging region. Hence, an iterative procedure is required to solve for moment redistribution. The iterative procedure is simple: Assume a hinge moment and rotation, for example point H in Figure Z. For this moment and rotation determine the redistribution capacity of the hinge from Equation 47. According to this redistribution capacity determine the corresponding moment in the non-hinging region from Equation 68. According to this moment determine the corresponding rotation in the “non-hinging region” from its moment-rotation response. Determine the sum of the rotations in the “hinging” and “non-hinging” regions: θ1-total. If θ1-total is greater than θcap from the RBR model, then the initial guess of the moment and rotation in the “hinging” region was too high and needs to be reduced, to say Point S in Figure Z. For this moment and rotation the procedure is repeated, until the sum of the “hinging” and “non-hinging” rotations is equal the rotation capacity θcap from the RBR model.

275

An example of moment redistribution for a section with an elasto-plastic moment rotation response like O-A-C in Figure Z, is shown graphically in Figure AA. On the ordinate the percentage of the total primary hinge rotation is graphed, and the abscissa the total static moment is graphed.

0%

20%

40%

60%

80%

100%

50 100 150 200Static moment (kNm)

% o

f tot

al s

tatic

mom

ent o

r %

of t

otal

prim

ary

hing

e ro

tatio

nHogging region - Moment Sagging region - MomentSagging region - Rotation Hogging yieldSagging yield

Figure AA (Figure 11 in Chapter 11) - Relative rotation and moment contributions for both

regions for a given static moment In Figure AA, it can be seen that initially the distribution of moment is almost elastic and very little moment redistribution occurs. Under elastic conditions, the hogging moment is approximately 2/3 the static moment and the sagging region 1/3 the static moment. As the static moment continues to increase, the hogging (primary hinging) region continues to redistribute moment to the sagging region. At reinforcing bar yield in the sagging region, point A in Figure Z and shown as the black triangle in Figure AA, significant rotation starts to occur in the sagging region. Ultimately at failure at a total static moment of 180kN-m, 80% of the rotation in the “primary” hinge is caused by rotation in the “secondary” hinge, and the moment in the secondary hinge is also greater than that in the primary hinge. The influence of cross sectional area of FRP on moment redistribution in FRP plated members is assessed using this approach, where it is assumed that the non-hinge region remains elastic, that is along O-D in Figure Z. From this analysis technique, sections with greater cross sectional areas of FRP have greater redistribution capacity. As the cross sectional area of FRP increases, more concrete is engaged to ensure longitudinal equilibrium, and hence the crack height reduces and the depth of softening increases. Thus, all other things being constant such as plating style and failure mechanism, a larger cross sectional area of FRP will provide greater redistribution and moment capacity. This is in direct contrast with the ku approach which suggests under-reinforced sections are more ductile. However, it needs to be remembered that as the area of reinforcement increases the probability of wedge sliding failure increases which will reduce the rotation as with the ku approach. As the cross sectional area of FRP increases, a larger depth of softening region is mobilised causing the wedge slip ssoft to increase. Ultimately there is a limit to the amount of

276

FRP that can be provided, at which point uncontrollable wedge sliding precedes debonding.

277

Statement of Authorship Design for moment redistribution in FRP plated RC beams Deric John Oehlers1, Matthew Haskett2, and Mohamed Ali M.S.3 1Professor Deric J. Oehlers School of Civil, Environmental and Mining Engineering University of Adelaide Supervised research, reviewed manuscript SIGNED____________________ 2Mr. Matthew Haskett PhD student School of Civil, Environmental and Mining Engineering University of Adelaide Wrote manuscript, developed theory, performed all analyses SIGNED____________________ 3Dr. Mohamed Ali M.S. Senior Research Associate, School of Civil, Environmental and Mining Engineering University of Adelaide Submitted, ASCE Composites in Construction, 16/3/09

278

A NOTE:


Oehlers, D.J., Haskett, M. & Mohamed Ali, M.S. (2009) Design for moment redistribution in FRP plated beams. ASCE Journal of composites in construction, submitted for print, March 2009.

302

CHAPTER 12:

SYNOPSIS - EVALUATING THE SHEAR-FRICTION RESISTANCE ACROSS SLIDING PLANES IN CONCRETE

In previous chapters, the rigid body rotation moment-rotation mechanism was discussed, iterative and closed form solutions for developing discrete moment-rotation responses were presented, and an application of the discrete moment-rotation response was presented where the moment and rotation at failure from the RBR approach was used to predict allowable amounts of moment redistribution in unplated and plated members. Until now, the behaviour and modelling of the concrete in the softening wedge has not been studied extensively and, in terms of our moment redistribution analyses, it was always assumed that the axial reinforcement failed by debonding or fracture prior to sliding of the concrete softening wedge. In this chapter, an approach for determining the shear friction properties of initially uncracked concrete is presented, which ultimately will be incorporated into the RBR model to more accurately simulate the behaviour of the softening region of concrete. Shear friction or aggregate interlock refers to the transfer of shear stresses across sliding planes in concrete through the projection of aggregate particles from one sliding plane into another. These projections bear against the opposing sliding plane and this bearing facilitates the transfer of shear stresses. Under increasing levels of confinement, the magnitude of the shear stress able to be transferred across the sliding plane increases. In this chapter, a mathematical approach is developed to derive the shear friction parameters of initially uncracked concrete. These parameters are the shear stress (τN) and crack separation (hcr), which are expressed in terms of the displacement (∆) of and normal stress (σN) across the sliding plane. This is slightly different to a previous shear friction approach used by Walraven to model initially cracked sliding planes, where he expressed the shear stress (τN) and normal stress (σN) in terms of the displacement (∆) and crack separation (hcr) of the sliding plane. Consider a concrete cylinder of height Lprism and diameter dprism subjected to a uniform hydrostatic confinement (σlat), with an axial stress in the cylinder σsoft, such that the axial stress σsoft is somewhere along the softening branch A-B in Figure DD. A softening wedge forms as shown in Figure BB(a), where the angle the wedge forms (α) is known and is a function of various known shear friction parameters. Across this sliding plane normal stresses of magnitude σN and shear stresses of magnitude τN exist.

303

σsoft

σlat

α

sliding plane

σsoft

σlat

τNσN

wedge

(a) Equilibrium

α

wedge sliding plane

dprism

wedge

Lprism

(b) Cylinder

cone

εcone

δaxial

localisedcrushing

δaxial

εreal

σsoft

∆sm

(δlat)sm

(c) Smooth sliding

εrealcylinder

∆

(δlat)sm+ (δlat)agg

(d) Aggregate interlock

same onthis side

hcr

Figure BB (Figure 6 in Chapter 12) – Equilibrium of a wedge and cylinder deformations

From cylinder equilibrium in Figure BB, the shear stress and normal stress are a function of the geometry of the wedge and the softening (σsoft) and lateral stresses (σlat):

αασ−αασ=τ cossin2cossin latsoftN Equation 69

ασ+ασ=σ 2lat

2softN cos2sin

Equation 70 Compatibility of the confined cylinder is also required. The deformation of the cylinder if the wedge surface is smooth is shown in Figure BB(c). The cylinder bearing surface moves δaxial relative to the cylinder mid-height so that the tips of the cone crush locally as shown to allow this movement. This movement of the cones towards mid-height causes the wedge to move laterally (δlat)sm over a radius dprism/2 as shown, which is one component of the cylinder dilation. This is the movement associated with a smooth interface and is shown in Figure 6(c). The opening up, or separation, of the initially uncracked sliding interface, which is, for want of a better term, referred to as an aggregate interlock effect, causes the deformations shown in Figure BB(d). These deformations are enlarged and shown in Figure CC(a), and the resolution of this aggregate interlock deformation is shown in Figure CC (c).

304

α

B

A

∆

(a) Crack behaviour

(b) Smooth analysis

(c) Rough analysis

hcr

C α

A

B

∆

∆sinα

∆cosα

hcr

C B

hcr/cosα

αhcrsinα

zero gap

Figure CC (Figure 7 in Chapter 12) – Resolutions of deformations

The movement across a sliding plane is shown in Figure CC(a). Prior to movement, points A and B are adjacent to each other. However, sliding across a smooth interface causes the displacement Δ shown enlarged with its components in Figure CC(b). Furthermore, the separation across the rough surface hcr causes the additional displacements in Figure CC(c). The lateral and axial strains are a function of the lateral and axial displacements respectively. The axial and lateral strains as a function of the cylinder dimensions and the displacement (∆) and crack separation (hcr):

realprism

cr

prismaxial L

sinh2L

cos2ε+

α+

α∆=ε

Equation 71

realcprism

cr

prismlat cosd

h2d

sin2εν+

α+

α∆=ε−

Equation 72 The idealised axial stress-strain relationship of confined concrete used in the analysis procedure is shown in Figure DD.

305

Ec

fco (unconfined)fcc [confined]

ascending branch softening branch

εpeak

[εcc] - confined(εco) - unconfined

B

A

sliding

εslide

εaxial

[εsoft]

σaxial

[σsoft]

εreal

σresidual

ε

σ

CD

EF

G

Teng et al.(2007)

H

O

Figure DD (Figure 5 in Chapter 12) – Idealised concrete axial stress-strain relationship

The ascending branch O-A is a material property, and usually determined directly from tests. The descending softening branch A-B is governed by shear friction behaviour over the sliding planes, and is where the shear friction parameters of initially uncracked concrete are extracted. For a given softening stress, say σsoft at point C in Figure DD, the axial (εsoft) and lateral strains (εlat) are also known. The strain for a softening stress σsoft on the descending portion of the stress-strain relationship can also be determined from previously developed mathematical relationships. Hence, for a given softening stress, the axial and lateral strains are all known. Thus, for a given lateral confinement σlat and a given softening stress σsoft, we have 4 equations (Equations 69-72) with 4 unknowns: τN, σN, hcr and ∆. Hence, we can solve for these four unknowns for varying softening stresses along A-B in Figure DD. The shear stress (τN) and normal stress (σN) across the sliding plane are solved from Equations 69 and 70 for a fixed/constant lateral confinement σlat and a given softening stress σsoft. The crack separation and displacement are a function of the cylinder dimensions and the axial and lateral strains. Rearranging Equations 71 and 72 for hcr and ∆:

( )α

α∆−ε−ε=

sin2cos2L

h realsoftprismcr

Equation 73

306

( )

α−

ααα

ε−ε−εν−ε−

=∆

sind2

dsin2

sincosdL

prismprism

prism

realsoftprismrealclat

Equation 74 Using this approach, the shear friction parameters for 50MPa concrete are shown in Figure EE. The shear stress is shown on the positive ordinate, the crack separation on the negative ordinate, and the displacement on the abscissa. Varying levels of confinement (σN) are considered across the sliding plane, as shown in the legend. The confinement varies from 30%fco to 55%fco in Figure EE. Mathematically, the normal stress across the crack face is a function of both the softening stress σsoft and the lateral confinement σlat. From Equation 70, the minimum normal stress across the crack plane occurs when no lateral confinement is provided and the minimum softening stress σresidual in Figure DD. According to this, the minimum normal stress is σresidualsin2α. The residual stress is approximately 70%fco for commonly accepted shear friction values, hence for α=38˚, the minimum normal stress across the sliding interface is approximately 24%fco. This is reflected in Figure EE where the minimum normal stress graphed is 30%fco.

fco=50MPa

-25

-15

-5

5

15

25

0 0.5 1 1.5 2 2.5∆ (mm)

30%fco

35%fco

40%fco

45%fco

50%fco

55%fco

UNCRACKEDENVELOPE

Psoft

τN (MPa)

Uncracked envelope

1.5

0.5

hcr (mm)2.5

σresidual

30%fco

35%fco

40%fco

45%fco

50%fco

55%fco

σresidual

0

A

BC

D

E

18.3

1.28

UNCRACKED ENVELOPE

Figure EE (Figure 8 in Chapter 12) – Shear friction parameters for initially uncracked concrete: fco

50MPa up to ssoft=70%σresidual From Figure EE, it can be seen that the shear stress is reducing for increasing displacements, with a corresponding increase in the crack separation. This is the case for all normal confining stresses. For a given normal stress, each marker in Figure EE corresponds to a different softening stress σsoft. For example, for a constant normal stress

307

of 30%fco, point A in Figure EE corresponds to a softening stress σsoft of 113%σresidual, point B 100%σresidual and point C 74%σresidual. For a normal stress 45%fco, point D corresponds to a softening stress of 100%σresidual. A failure envelope for the shear friction parameters is also shown in Figure EE, which has been obtained from Mattock’s shear stress-normal stress relationship in Figure FF. For example, Point E in Figure EE is the maximum shear stress able to be transferred for a normal stress of 40%fco. This shear stress has been determined from Figure FF point E, where the shear stress is 0.366fco, or for 50MPa concrete 18.3MPa.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4σN/fco

Vu/f

co

Figure FF (Figure 2 in Chapter 12) – Normalised shear transfer as a function of normal stress for

initially cracked planes In Figure EE the “Uncracked envelope” failure plane corresponds closely with the achievement of the residual stress σresidual in Figure DD. This indicates that sliding failure coincides with the attainment of the residual stress σresidual in Figure DD. This further supports the failure envelope in Figure EE which was obtained from Mattock’s bounds.

initially uncracked sliding planes

initially cracked sliding planes

E 0.36

308

Statement of Authorship Evaluating the shear friction resistance across sliding planes in concrete 1Matthew Haskett, 2Deric John Oehlers, 3Mohamed Ali M.S. and 4Surjit Kumar Sharma 1Mr. Matthew Haskett PhD student School of Civil, Environmental and Mining Engineering University of Adelaide Wrote manuscript, performed all analyses SIGNED____________________ 2Professor Deric J. Oehlers School of Civil, Environmental and Mining Engineering University of Adelaide Supervised research, reviewed manuscript SIGNED____________________ 3Dr. Mohamed Ali M.S. Senior Research Associate, School of Civil, Environmental and Mining Engineering University of Adelaide 4Dr. Surjit Kumar Sharma Technical Officer Bridges and Structures Central Road Research Institute New Delhi INDIA Developed theory and analysis technique, reviewed manuscript SIGNED____________________ Submitted ASCE Materials, 11/6/09

309

Haskett, M., Oehlers, D.J., Mohamed Ali, M.S. & Sharma, S.K. (2011) Evaluating the shear-friction resistance across sliding planes in concrete. Engineering structures, v. 33(4), pp. 1357-1364

A NOTE:



A http://dx.doi.org/10.1016/j.engstruct.2011.01.013

A

331

CHAPTER 13:

SYNOPSIS - THE SHEAR-FRICTION AGGREGATE INTERLOCK RESISTANCE ACROSS SLIDING PLANES IN CONCRETE

In the previous chapter, a method was presented for determining the shear friction behaviour of hydrostatically confined concrete cylinders, and an example of the shear friction parameters (τN, hcr, σN and ∆) of initially uncracked 50MPa concrete was presented graphically. Importantly, the shear friction parameters were presented for a constant normal stress, which allowed a failure envelope to be developed using Mattock’s shear friction research. In this chapter, the approach presented in Chapter 12 for extracting the shear friction parameters of initially uncracked hydrostatically confined concrete is used to develop mathematical expressions for shear stress and crack separation in terms of normal confining stress and displacement for various concrete strengths. Using the analysis procedure described in Chapter 12, the shear stress-slip relationship for 30MPa concrete with varying normal confinements is shown in Figure GG.

y = -4.35x + 12.84

y = -1.86x + 16.68

0

5

10

15

20

0 0.5 1 1.5 2 2.5∆ (mm)

τ N (M

Pa) 30% fco

35% fco

40% fco

45% fco

50% fco

30%fco

35%fco

40%fco

45%fco

50%fco

Figure GG (Figure 5 in Chapter 13) – Influence of normal confining stress on shear stress

From the regressions analyses in Figure GG, it can be seen that higher confining stresses provide greater shear stress transfer capacity for a given displacement. Higher confining stresses also provide a lesser reduction in shear stress with increasing displacement. That is, the slope of the regression analyses (mτ) in Figure GG is reducing for increasing normal confinements. Similarly, higher confining stresses provide greater shear stress at zero displacement. Thus, the slope and the ordinate intercept (cτ) of the shear stress-displacement relationship in Figure GG are both a function of the normal confining stress σN/fco. Hence, we can express the shear stress-slip relationship as

ττN cΔmτ +=

30%fco: τN = -4.35∆+12.84

50%fco: τN = -1.86∆+16.68

332

Equation 75 where mτ and cτ are both dependent on the normal confining stress across the sliding plane and are determined from linear regression analyses. The relationship between the slope of the shear stress relationship (mτ) and the normal confining stress (σN/fco) is shown in Figure HH.

y = 0.56x-1.76

y = -12.67x + 8.24

0.00

1.00

2.00

3.00

4.00

5.00

0.00 0.25 0.50 0.75 1.00σN/fco

-mτ (N

/mm

3 )

Figure HH (Figure 6 in Chapter 13) – Influence of confinement on -mτ

For confinements less than 50%fco, the variation in mτ with normal stress (σN/fco) from Figure HH is

24.8fσ

67.12mco

Nτ −

=

Equation 76 For confinements up to fco, the variation in mτ with normal stress (σN/fco) from Figure HH is

76.1

co

Nτ f

σ56.0m

−

−=

Equation 77 The influence of normal confining stress on the peak shear stress capacity, cτ is shown in Figure II.

-m τ = -12.67(σN/fco)+8.24

-m τ = 0.56(σN/fco)-1.76

333

y = 23.16x0.48

y = 18.91x + 7.31

0.0

5.0

10.0

15.0

20.0

25.0

0.00 0.20 0.40 0.60 0.80 1.00 1.20σN/fco

c τ (M

Pa)

Figure II (Figure 7 in Chapter 13) – Influence of confinement on cτ

Unlike Figure HH, where the linear and power regressions were different, the linear and power regressions in Figure II are very similar. Hence, for confinements up to fco, the parameter cτ in terms of the normal confining stress

48.0

co

Nτ f

σ16.23c

=

Equation 78 Substituting Equations 76 and 78 into Equation 75, the shear stress-slip relationship for initially uncracked 30MPa concrete for normal confining stresses less than 50%fco is given by

48.0

co

N

co

NN f

σ16.23Δ24.8

fσ

67.12τ

+

−

=

Equation 79 Substituting Equations 77 and 78 into Equation 75, the shear stress-slip relationship for initially uncracked 30MPa concrete for normal confining stresses less than fco is

48.0

co

N76.1

co

NN f

σ16.23Δ

fσ

56.0τ

+

−=

−

Equation 80 The relationship between crack separation and displacement and normal confining stress for 30MPas concrete was also quantified using an identical approach.

79.1fσ

08.3Δ56.5fσ

09.8hco

N

co

Ncr −

+

+

−=

Equation 81

c τ = 18.91(σN/fco)+7.31 cτ = 23.16(σN/fco)0.48

334

These expressions are both very accurate with low standard deviation of the error in modelling the “experimentally derived” shear friction parameters. The influence of compressive strength of concrete on the shear friction parameters was also studied, where the concrete strength varied from 30MPa to 70MPa. The influence of concrete strength on the shear stress and crack separation are:

( )( )

( )50.0

co

Nco

0.2f01.0

co

NcoN f

σ74.0f80.0Δ

fσ

90.0f05.0τco

−+

+−=

−

Equation 82

( ) ( )

−+

+−+

+−

−= 10.3f04.0

fσ

12.5f06.0Δ95.8f11.0fσ

22.13f17.0h coco

Ncoco

co

Ncocr

Equation 83 Equation 82 provides very good agreement with the experimental shear friction parameters for various displacements, concrete strengths and normal confining stresses, but the crack separation expression Equation 83 is less accurate. The shear friction parameters of initially cracked concrete were also investigated. Walraven and Reinhardt developed mathematical expressions for the shear friction properties of initially cracked concrete planes for a constant crack separation. This prevents the development of a failure envelope for initially cracked concrete since the normal stress is constantly changing. Using Walraven and Reinhardt’s previous expressions as a source of “experimental data”, various regression analyses were performed. The shear stress and crack separation of initially cracked concrete in terms of the displacement, concrete strength and normal confining stress:

( ) ( )

( ) ( )06.0f01.0fσ

86.2f34.2

Δ22.0f02.0fσ

40.0f33.0τ

coco

Nco

coco

NcoN

−+

−+

−−+

+−=

Equation 84

( ) ( )

−

+

+−+

−=

−

1.0fσ

02.0

Δ)662.0f003.0fσ

024.1f004.0h

024.1f003.0

co

N

coco

Ncocr

co

Equation 85 These expressions model the shear stress and crack separation behaviour of initially cracked concrete relatively accurately. The relationship between shear and normal stress at failure for both initially cracked and uncracked concrete is known from Mattock’s research, Figure FF in Chapter 12. Hence, from Chapter 12 for initially

335

uncracked concrete, the relationship between shear and normal stress, for normal stresses greater than 17%fco:

coNu f25.0σ29.0V += Equation 86

Substituting the shear stress relationships for initially uncracked concrete (Equation 82) into Equation 86 and solving for ∆, the slip at failure:

( )

( )( )0.2f01.0

co

Nco

50.0

co

NcocoN

sliding co

fσ

90.0f05.0

fσ

74.0f80.0f25.0σ29.0Δ −

+−

−−+

=

Equation 87 Repeating the same procedure for initially cracked concrete:

( ) ( )

( )

−−

+−

−−

−−+

=

22.0f02.0fσ

40.0f33.0

06.0f01.0fσ

86.2f34.2f11.0σ73.0Δ

coco

Nco

coco

NcocoN

sliding

Equation 88 Thus, for a given compressive strength of concrete and knowing the normal stress across the sliding plane, the displacement at the commencement of sliding failure for a given normal stress can be determined. As expected, the shear stress at failure for initially uncracked concrete is greater than the shear stress for initially cracked concrete. Also, as expected, higher strength concrete fails at a greater shear stress. A generic expression for sliding failure has also been developed for concrete of any strength, which is the first step in accurately modelling wedge failure in the rigid body rotation model.

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Statement of Authorship The shear-friction aggregate-interlock resistance across sliding planes in concrete 1Matthew H askett, 2Deric Joh n O ehlers, 3Mohamed A li M .S. and 4Surjit K umar Sharma 1Mr. Matthew Haskett PhD student School of Civil, Environmental and Mining Engineering University of Adelaide Performed all analyses, wrote manuscript, developed all mathematical expressions SIGNED____________________ 2Professor Deric J. Oehlers School of Civil, Environmental and Mining Engineering University of Adelaide Supervised research, reviewed manuscript SIGNED____________________ 3Dr. Mohamed Ali M.S. Senior Research Associate, School of Civil, Environmental and Mining Engineering University of Adelaide 4Dr. Surjit Kumar Sharma Technical Officer Bridges and Structures Central Road Research Institute New Delhi INDIA Accepted, Magazine of Concrete Research, 16/2/2010

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Haskett, M., Oehlers, D.J., Mohamed Ali, M.S. & Sharma, S.K. (2010) The shear friction aggregate interlock resistance across sliding planes in concrete. Magazine of concrete research, v. 62(12), pp. 907-924.

A NOTE:



A http://dx.doi.org/10.1680/macr.2010.62.12.907

A

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CHAPTER 14:

CONCLUSIONS AND RECOMMENDATIONS In this thesis, a discrete moment-rotation model is presented that can model the moment-rotation response of any reinforced concrete member from the commencement of loading to failure. This model utilises shear friction theory to model the behaviour of concrete, partial interaction theory to model the behaviour of the reinforcement and relates the concrete and reinforcement behaviour through a rigid body rotation. Previously published pull tests are analysed to determine the local bond stress-slip relationship for deformed steel reinforcing bars. This is a structural mechanics approach to determine bond properties, and can accommodate any embedment length. This is especially important in the derivation of the bond properties of deformed steel reinforcing bars, because experimentally short embedment length is generally provided to prevent yielding of the steel. A partial interaction numerical model is also derived to compare the theoretical P−∆ response for a given bond (τ−δ) relationship with the experimental response. A similar approach is also adopted to determine the influence of embedment on the bond characteristics of NSM FRP plates. Partial interaction theory is subsequently used to derive mathematical expressions for the load-slip (P-∆) behaviour of steel reinforcing bars. This approach has the ability to accurately model the elasto-plastic behaviour of steel. Closed form mathematical expressions are derived for the rigid body rotation technique to provide designers with the ability to determine the moment and rotation of any reinforced concrete member at failure only. An example of the application of the discrete moment-rotation approach is presented. Moment redistribution expressions are derived for various loading arrangements, where moment redistribution is proportional to the moment and rotation at failure (from the discrete rotation model), the flexural rigidity and span length. Importantly, this approach to moment redistribution can allow for failure mechanisms like plate debonding and fracture that the existing neutral axis parameter ku approach cannot accommodate. Various examples of moment redistribution are presented, where it was shown that both EB and NSM plating arrangements do have some redistribution capacity. The redistribution capacity of reinforced concrete sections with EB and NSM steel plates was also assessed, where it was shown that localised debonding around the crack front can provide significant additional ductility. A practical example of moment redistribution is also presented, where it is shown that the moment-rotation response of both regions of the member is important when considering moment redistribution in practice. A method for extracting the shear friction properties of hydrostatically confined initially uncracked concrete cylinders is also presented, and this approach is used to quantify mathematically the generic shear friction parameters of initially uncracked concrete. A failure bound is also developed for these shear friction parameters, which

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is the first step in predicting the sliding capacity of the concrete in the softening wedge in the moment-rotation model. Further research is still required to develop this discrete rotation model. At present, the model does not consider the presence of stirrups, and therefore does not reflect the “pseudo-confined’ nature of the softening region of concrete. When the stirrup confinement is considered, it is expected that the theoretical moment-rotation response will display reducing flexural capacity after the peak moment is achieved. At present, the moment continues to increase until failure. The behaviour of stirrups can be modelled in this approach by considering the confinement the stirrups provide to the softening wedge sliding plane. If the stirrup material properties are known, and the separation of the softening wedge is also known, then from partial interaction theory the force in the stirrups can be determined and the corresponding normal confinement of the softening wedge is also known. The shear stress across the sliding plane can subsequently be determined from shear friction theory, and therefore the softening force in the wedge Psoft can be determined. This model also has the ability to consider the influence of FRP confinement. The analysis technique is similar to that described above: separation of the softening wedge induces an axial stress in the FRP wrap that can be determined from partial interaction theory. This axial stress further confines the softening wedge and allows an increased wedge softening force Psoft, which can be determined from shear friction theory. The increased normal confinement provided by stirrups or FRP confinement, both increases the softening force, Psoft, but also delays sliding failure.

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The discrete rigid body rotation of reinforced concrete ...€¦ · In Chapter 5the rigid body rotation model is described in detail and an iterative solution , technique presented