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## Material Information- Title:
- Numerical Assessment of Reinforced Concrete Members Retrofitted with Fiber Reinforced Polymer for Resisting Blast Loading
- Creator:
- Long, Graham A
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2012
- Language:
- english
- Physical Description:
- 1 online resource (89 p.)
## Thesis/Dissertation Information- Degree:
- Master's ( M.E.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Civil Engineering
Civil and Coastal Engineering - Committee Chair:
- Krauthammer, Theodor
- Committee Members:
- Astarlioglu, Serdar
- Graduation Date:
- 5/5/2012
## Subjects- Subjects / Keywords:
- Beams ( jstor )
Blasts ( jstor ) Compressive stress ( jstor ) Engineering ( jstor ) Modeling ( jstor ) Moduli of elasticity ( jstor ) Reinforced concrete ( jstor ) Shear stress ( jstor ) Steels ( jstor ) Tensile stress ( jstor ) Civil and Coastal Engineering -- Dissertations, Academic -- UF assessment -- blast -- debonding -- fracture -- frp - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Civil Engineering thesis, M.E.
## Notes- Abstract:
- The use of fiber reinforced polymers for the retrofit of existing structures is a common practice in blast protection. The tools used to assess the behavior of structures have become more refined allowing for precise modeling of reinforced members retrofitted with fiber reinforced polymers (FRP). However, many of these tools require extensive computational power that is often time consuming. This study aims to adapt an existing expedient dynamic analysis assessment tool in order to account for the behavior of reinforced concrete members retrofitted with FRP for flexural enhancement. FRP retrofits have been shown to increase the capacity of the members under severe loading conditions. However, the strength associated with the FRP layer is not always able to fully develop due to the debonding behavior between the two surfaces. The model will seek to capture this behavior by employing fracture mechanics along the bonded interface. The results of the modeling tool will be validated by comparing them to those of finite element analysis programs as well as available experimental test data. An additional parametric study will be conducted to evaluate the models ability to capture the bond behavior when adjusting the properties associated with the FRP. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (M.E.)--University of Florida, 2012.
- Local:
- Adviser: Krauthammer, Theodor.
- Statement of Responsibility:
- by Graham A Long.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Copyright Long, Graham A. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Classification:
- LD1780 2012 ( lcc )
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PAGE 1 NUMERICAL ASSESSMENT OF RE INFORCED CONCRETE MEMBERS RETROFITTED WITH FIBER REINFORCED POLYMER FOR RESISTING BLAST LOADING By GRAHAM LONG A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERING UNIVERSITY OF FLORIDA 2012 1 PAGE 2 2012 Graham Long 2 PAGE 3 To my family and friends 3 PAGE 4 ACKNOWLEDGMENTS I would like to express my sincere gr atitude to Dr. Theodor Krauthammer for his advice and direction facilitating the completion of this research. I would also like to thank Dr. Serdar Astarlioglu for his support in the completion of this research. Finally, I would like to thank the Canadian Armed Forces and 1 ESU (1st Engineer Support Unit) for providing me with the opportunity to complete my graduate studies in structural engineering. 4 PAGE 5 TABLE OF CONTENTS page ACKNOWLEDG MENTS .................................................................................................. 4LIST OF TABLES ............................................................................................................ 7LIST OF FI GURES .......................................................................................................... 8LIST OF ABBR EVIATION S ........................................................................................... 10ABSTRACT ................................................................................................................... 15CHAPTER 1 INTRODUC TION ................................................................................................ 16Problem Stat ement ............................................................................................. 16Objective a nd Scope .......................................................................................... 17Research Sign ificanc e ........................................................................................ 182 BACKGROUND LITERATU RE REVI EW ........................................................... 19Dynamic Structural A nalysis Suite (DSAS) ......................................................... 19Blast Load ing ...................................................................................................... 19Materials ............................................................................................................. 21Concrete .................................................................................................. 21Compressive stress-strain cu rve ................................................... 21Tensile stress-strain cu rve ............................................................ 23Steel ......................................................................................................... 24Fiber Reinforced Polymers (FRP) ............................................................ 25Dynamic A nalysis ............................................................................................... 27Equivalent SDOF System ........................................................................ 28Equivalent mass ............................................................................ 28Equivalent loading func tion ........................................................... 29Resistance F unction ................................................................................ 30Numerical Int egration ............................................................................... 31Flexural Behavior ................................................................................................ 33Diagonal Shear Behavio r .................................................................................... 34Rate Effe cts ........................................................................................................ 35Fiber Reinforced Polymers ................................................................................. 37FRP Flexural Behavio r ............................................................................. 37FRP Shear B ehavior ................................................................................ 38FRP Rate E ffects ..................................................................................... 38FRP Size E ffects ...................................................................................... 39Debonding B ehavior ................................................................................ 39Shear stress .................................................................................. 40 5 PAGE 6 Normal st ress ................................................................................ 42Mixed mode in itiation .................................................................... 43Effective bond length ..................................................................... 45Ultimate bond st rength .................................................................. 46Load-Impulse (P-I ) Diagram s.............................................................................. 473 METHODOL OGY ............................................................................................... 58Structural Overview ............................................................................................ 58Debonding B ehavior ........................................................................................... 59DSAS Overall Algorith m ..................................................................................... 614 ANALYSI S .......................................................................................................... 65Material Model Validati on ................................................................................... 65Debonding Vali dation ......................................................................................... 66Dynamic lo ading ................................................................................................. 67Parametric Study ................................................................................................ 68FRP Ultimate Strength ............................................................................. 69FRP Elastic Modulus ................................................................................ 69FRP Thick ness ........................................................................................ 69FRP Widt h ............................................................................................... 705 SUMMARY AND CO NCLUSION S ..................................................................... 80Limitati ons .......................................................................................................... 80Future Development /R ecommendat ions ........................................................... 81General Conclusions .......................................................................................... 83REFERENCE LIST........................................................................................................ 85BIOGRAPHICAL SKETCH ............................................................................................ 89 6 PAGE 7 LIST OF TABLES Table page 4-1 Beam material properties for experimental test case .......................................... 784-2 DSAS inputs for materi al model veri fication ....................................................... 784-3 Peak results of unstrengthen static loading case. ............................................... 784-4 Peak results of FRP str engthened static l oading case ....................................... 794-5 Material data for param etric case studies ........................................................... 79 7 PAGE 8 LIST OF FIGURES Figure page 2-1 Free field pressure time vari ation ....................................................................... 482-2 Modified Hognestad stress-strain curve for concrete in compression. ................ 482-3 Confined concrete stress strain curve ................................................................ 492-4 Tensile stress strain curv e for reinforced concrete ............................................. 492-5 Steel Stress-stra in curve model. ......................................................................... 502-6 Deformation prof ile for a beam ........................................................................... 502-7 SDOF inelastic resistance model ........................................................................ 512-8 Stress strain diagram for reinforced concrete beam ........................................... 522-9 Shear reduct ion model ....................................................................................... 532-11 Section forces for FRP reinforced conc rete beam .............................................. 552-14 Typical respons e functi ons ................................................................................. 573-1 Test beam fr om study ........................................................................................ 623-2 Comparison of debonding proce ss. .................................................................... 623-3 DSAS debonding algori thm ................................................................................ 633-4 Overall DSAS algorithm ...................................................................................... 644-1 Validation of the mate rial model of DSAS. .......................................................... 714-2 Load-displacement curve for validati on of the debonding model in DSAS. ........ 724-3 Moment-curvature model for validation of the debonding model in D SAS. ......... 734-4 Time-displacement response to blast load ing case ........................................... 734-5 Pressure-impulse curve under uniform blas t loading. ......................................... 744-6 Effect of FRP Yield Str ength. .............................................................................. 754-7 Effect of FRP Modul us of Elas ticity .................................................................... 764-8 Effect of FR P thickne ss. ..................................................................................... 77 8 PAGE 9 4-9 Effect of FRP width. ............................................................................................ 78 9 PAGE 10 LIST OF ABBREVIATIONS Aci Area of concrete at each layer Asi Area of steel at each layer bc Concrete width bf FRP width c Neutral axis depth C Damping coefficient dsi Depth of steel layer D Nominal diameter of hoops D Nominal diameter of longitu dinal compression reinforcement Ead Adhesive modulus of elasticity Ec Elastic modulus of concrete Enhc Concrete enhancement factor in compression Enhct Concrete enhancement factor in tension Enhs Steel enhancement factor Ef FRP modulus of elasticity Es Steel modulus of elasticity F Force fc Compressive strength of concrete fci Stress of concrete layer Fci Concrete layer forces Fe Equivalent Force ft Tensile strength fs Steel stress 10 PAGE 11 fsi Stress of steel layer Fsi Steel layer forces fu Ultimate steel stress fy Steel yield stress fy Yield stress of the hoops/stirrups Gf FRP interfacial fracture energy Gn Normal interfacial fracture energy Gs Shear interfacial fracture energy GT Total interfacial fracture energy h Concrete section depth H Average core dimension of the confined concrete compression area is Impulse K Stiffness Ke Resistance factor KL Load factor Km Mass factor l Length Le Effective bond length Lb Bond length M Mass mci Concrete layer moment Me Equivalent mass Mfl Ultimate moment capac ity due only to flexure 11 PAGE 12 Mn Moment at mid-span msi Steel layer moment Mt Total mass Mu Ultimate moment c apacity due only to shear Pso Incident pressure Pu Bond strength tad Thickness of adhesive layer tf Thickness of FRP to Time of positive Phase R Resistance function Rm Maximum resistance Re Equivalent resistance function s Hoop spacing sn Normal stress slip snf Slip at maximum normal stress ss Shear stress slip ssf Slip at final debonding sso Slip at maximum shear stress SRF Shear reduction factor u1 Axial displacement of member u2 Axial displacement of debonded section u Displacement u Velocity 12 PAGE 13 u Acceleration w Uniformly distributed load w1 Vertical displacement of member w2 Vertical displacement of debonded section Y1 Distance from debonded interfac e to member neutral axis Y2 Distance from debonded interface to debonded section neutral axis x Position zi Depth of concrete layer Newmark-Beta constant l Effective bond length factor w Bond width ratio factor Increment space Deflection Total Strain c Concrete Strain s Steel strain sh Strain hardening in steel f Final strain in steel Strain rate s Quasi-static strain rate Web reinforcement ratio Shape function bl Blast loading decay coefficient 13 PAGE 14 n Normal shear stress Interfacial shear stress max Max interfacial shear stress 14 PAGE 15 15 Abstract of Thesis Pres ented to the Graduate School of the University of Florida in Partial Fulf illment of the Requirements for t he Degree of Master of Engineering NUMERICAL ASSESSMENT OF RE INFORCED CONCRETE MEMBERS RETROFITTED WITH FRP FOR RESISTING BLAST LOADING By Graham Long May 2012 Chair: Theodore Krauthammer Major: Civil Engineering The use of fiber reinforced polymers for t he retrofit of existi ng structures is a common practice in blast protection. The tools used to assess the behavior of structures have become more refined allowing for pr ecise modeling of reinforced members retrofitted with fiber reinforced polymers (FRP ). However, many of these tools require extensive computational power t hat is often time consuming. This study aims to adapt an existing expedient dynamic analysis assessment tool in order to account for the behavior of reinforced concrete members retrof itted with FRP for flexural enhancement. FRP retrofits have been shown to increase t he capacity of the members under severe loading conditions. However, the strength associated with the FRP layer is not always able to fully develop due to the debonding behavior between the two surfaces. The model will seek to capture this behavior by employing fracture mechanics along the bonded interface. The results of the modeli ng tool will be validated by comparing them to those of finite element analysis programs as well as available experimental test data. An additional parametric study will be c onducted to evaluate the models ability to capture the bond behavior when adjusting t he properties associ ated with the FRP. PAGE 16 CHAPTER 1 INTRODUCTION Problem Statement The vulnerability of many existing structures has been demonstrated by the various incidents over the last half century. Many of these structures were not designed to withstand severe loading due to an explosiv e loading incidents. As a result, there has been a requirement to increase the blast re sistance for other critical infrastructure systems in order to preserve life and oper ations. While extensive studies were conducted on the various means for retrofitti ng existing structures under normal service and seismically-induced loading, it is only wit hin the last two decades that the research has expanded to include severe short durati on loading cases. The choice on which method to use often depends on a variety of fact ors. While different facilities will have different governing factors, t he factor that continually is expressed is the cost to effectiveness ratio. For this reason, t he use of externally bonded fiber reinforced polymer (FRP) strips or mats have been ident ified as a suitable option due to its high strength to weight ratio. In addition, the ease of application allows for the retrofit to be completed in a timely manner. The employment of FRP alon g the tension face allows for an initial increase in beam flexural strength while r educing the deflection. Howeve r, the effectiveness of the FRP retrofit is limited by t he bond strength. In most ca ses, the FRP will debond from the beam before the it r eaches its ultimate strength. It is therefore impo rtant to model the bond and the debonding behavior in order to accurately determine the effectiveness of a retrofit. 16 PAGE 17 Extensive design calculations are often re quired for determining the full effect of these severe load cases on structural member s and the means to counteract the effect. The use of an advanced finite element meth od is computationally expensive and may not be suitable in time sensitive situations. Conversely, the use of design guidelines provides methods that do not require ext ensive analysis to design against specific threats but may not apply to all cases. In addition, these guidelines are often conservative and may produce results that may not be cost efficient due to conservative limits. Therefore, an approach that combines a high level of accuracy as well as utilizes a simplified analysis method to minimize com putational time is required to produce accurate results when time is limited. Dy namic Structural Analysis Suite (DSAS) is a computer program that has been under dev elopment for many years and has had extensive research completed on reinforced concrete structural members. However, retrofitting these structures with new composite materials has not been incorporated into that research and further work needs to be done to develop an expedient algorithm to account for these new methods. Objective and Scope The objective of this research project is to develop an algorithm that is both computationally expedient and sufficiently accurate for the purpose of analyzing reinforced concrete beams retrofitted with ex ternally bonded fiber reinforced polymer (FRP) strips subjected to severe short dy namic loading. The research project will be completed in the following steps: Develop an algorithm to evaluate t he flexural and deformation responses of reinforced concrete members retrofitted with externally bonded fiber reinforced polymer strips on the tensi on face of the member and im plement this algorithm as a new module in DSAS. 17 PAGE 18 Validate the algorithm model in and the results obtained in DSAS to available experimental data. Complete a parametric study using D SAS to fully understand the effect of FRP layer on reinforced concrete members subj ected to severe short duration load cases, by comparing their behavior wit h and without the retrofit. Various thicknesses and strengths of the FRP layer will also be examined to determine the effect on debonding and flexural enhancement. Research Significance The algorithm to be developed in this st udy will be incorporated into the computer code, DSAS, in order further develop the resi stance function of the components. DSAS will then have the capability to assess the response of reinforced concrete retrofitted with externally bonded fiber reinforced polymer strips. 18 PAGE 19 CHAPTER 2 BACKGROUND LITERA TURE REVIEW This chapter will provide a review of t he concepts and studies associated with the behavior of reinforced concrete members retrof itted with FRP. The topics will include a look at the structural anal ysis process employed by the Dynamic Structural Analysis Suite as well as the characteristics and behav iors associated with externally bonded FRP materials. The material is based on a re view of recent studies and publications in the field of protective structur es and fiber reinforced polymer. Dynamic Structural Analysis Suite (DSAS) DSAS is a multifunctional program that is capable of modeling the response of a wide range of structural co mponents under both st atic and dynamic loads (Chee et al. 2008; Tran et al. 2009; Morency et al. 2010). It has the capability to perform timehistory and load-impulse (P-I) analyses by empl oying accurate resistance functions that can be derived for the various structural elem ents in its library. This program utilizes a combination of advanced fully-nonlinear and Physics-based structural behavioral models for expedient structur al behavior computations. Blast Loading Blast loading is a shockwave caused by a sudden release of energy from an explosive device. In the case of an ai r blast at ground level, the shockwave is simultaneously reflected off the ground and t he wave expands out in a hemispherical configuration. The loadi ng produced from this sho ckwave can be assumed to be uniformly distributed over the entire structur al member, except when the point of initiation is extremely close to the structure. In whic h case, advanced computational fluid dynamic (CFD) simulations are requir ed to determine the complicated interaction 19 PAGE 20 between the blast wave and a structure. The loading time hist ory can be illustrated using the free field pressure -time variation obtained from TM 5-855-1 (1986) by means of Krauthammer (2008) and is illustrated in Figure 21. Loading due blast is characterized by an initial peak overpressure that decays over time based on a decay coefficient. The loading will consist of two phases, positive and negative. Loading due to a blast and impact is characterized by the peak overpressure, the duration, and the impulse. In order to det ermine the pressure on the structure as a function of time, the modified Friedlander equation wi ll be used in order to minimize the computational demand instead of other more precise equations. blt o soe t t PtP/1)( (2-1) The impulse can be obtained by the area enclosed by the load-time curve. ot sdttPi0)( (2-2) Where is the incident over pressure. soPot is the positive phase duration. bl is the decay coefficient. si is the positive impulse. The negative phase loading is much smalle r than the positive phase loading and is often omitted in dynamic analysis. In addition the positive phase is often idealized as a triangular load for simplified calculations. 20 PAGE 21 Materials Concrete Concrete is a non-homogenous material composed of cement and aggregates mixed with water. The combined effects of incorporating the materials results in a nonlinear material behavior as the concrete develops cracks. Concrete is much stronger in compression than in tension. Compressive stress-strain curve Concrete compressive strength is typi cally evaluated using uniaxial compression tests on plain concrete cylinders. The m odified Hognestad stress-strain curve, as shown in Figure 2-2, is commonly used to represent unconfined concrete in compression. It consists of a second degree parabola for the ascending branch followed by a linear descending branch. The ascending branch, for 0 c 2 0 0 "2 cc ccff (2-3) The descending branch, for 0038.00thanlessandc 0" ccd ccEff (2-4) where is the concrete compressive stress. cfc cff' "9.0 is the maximum concrete compressive stress. (2-5) cf' is the uniaxial concrete compressive strength under standar d test cylinder. c is the concrete strain. c cE f" 08.1 is the concrete strain at ma ximum compressive stress. (2-6) 21 PAGE 22 Elastic modulus is given in ACI Committee 318 (2008) by Equation 2-7 and 2-8. cE c ccfwE '335.1for (2-7) 3 3/160 /90 ftlbwftlbc c cf E '57000 for normal weighted concrete with in psi. (2-8) cf'Descending elastic modulus, is given by Equation 2-7. cdE ksior f Ec cd500 0038.0 "15.00 (2-9) For confined concrete, Krauthammer et al. (1988) outlines a material model that incorporates the effects due to the expandi ng concrete being restrained by lateral reinforcement. Figure 2-3 illustrates the material model. Due to the dependence on the core dimensions, the values calculated wi ll be readjusted as the neutral axis changes during analysis. The ascending branch, for 0 c 0 0 0 0 02 1 c c c c c c cfK E K E f (2-10) The descending branch, for K c 3.0 0 1 8.010 0 c c cZ fKf (2-11) The steady state, for cK 3.0 c cfKf 3.0 (2-12) c yrf f s h 734.0 1005.00024.00 (2-13) 22 PAGE 23 c y rf f D D h s K 245.010091.01 (2-14) 002.0 1000 002.03 4 3 5.0 c c rf f s h Z (2-15) H is the average core dimension of the confined concrete compression zone, measured to the outside of the stirrups. r is the confining steel volume to the confined concrete core volume per unit length of the element in compression zone. is the longitudinal compression reinforcement ratio. yf is the yield stress of the hoops. is the spacing of the hoops. sD is the nominal diameter of the hoops. D is the nominal diameter of the longitudinal compression reinforcement. Tensile stress-strain curve Concrete has a low tensile strength compar ed to its compressive strength and its strength is based on either the splitting tensile strength or th e modulus of rupture. The behavior of concrete in tension is a two phase model. Initially, the model is linear up to the failure point at which point cracking occurs. At this point, there is a significant drop in strength but the concrete retains some residual strength and the concrete model follows a curved softening path. Hsu (1993) de scribed the tensile stressstrain curve of concrete as follows and is illustrated in Figure 2-4: Ascending branch, for crc cccEf (2-16) 23 PAGE 24 c cf E'47000 (2-17) Descending branch, for crc 4.0 c cr crcff (2-18) c crf f'75.3 (2-19) crfis the concrete cracking stress of plai n concrete in pounds per square inch. cr is the cracking strain. Steel The steel reinforcement is strong in t ension and is used to add ductility to the concrete member. There are different steel stress-strain models available that can encompass elastic and perfectly plastic behavior, as well as strain hardening. The model outline in Park and Pauley (1975) outlines all three of these behavior characteristics and is illustrated in Figure 25 follows the three stages according to the following procedure. for ys yssEf (2-20) for shys ysff (2-21) for shs 21302 60 2 60 2 r m m ffshs shs shs ys (2-22) 24 PAGE 25 2 215 160130 r r r f f my u (2-23) sE is the steel modulus of elasticity. s is the steel strain. sh is the strain at the comm encement of steel hardening. su is the ultimate steel strain. f is the final steel strain. sf is the steel stress. uf is the ultimate steel stress. yf is the steel yielding stress. Fiber Reinforced Polymers (FRP) FRP strengthened members has exhibited significantly higher load-carrying capacity and stiffness. Depend ing on the system type, thickne ss, fiber angle orientation and geometry, load capacity increases can r ange between 1.5 to 5 times that of an unstrengthen member (Kachlakev et al. 2000). The most common FRP utilizes unidirectional fibers oriented along the longit udinal length. Multi-direction weaved mats can also be used to provide strength in two directions. Moreover, the use of unidirectional plies can also be layered in order to achieve two-directional strength. FRP material is strong in tension, but is weak in compression. The compression strength is based on t he epoxy. While not as strong as steel, it can improve the members strength without subst antial structural weight in creases. Most studies have assumed FRP to be linearly elastic up to failure. This assumption will also be 25 PAGE 26 incorporated into this study. However, individual product information would be needed to predict behavior of a reinforced structur e due to the various different manufacturing techniques and processes. Environmental conditions and long-term lo ading can significantly weaken the material properties. The materials are subjected to creep and fatigue when experiencing long loading periods reduci ng the effectiveness of the bond behavior between the fibers and the epoxy matrix. Furt hermore, moisture c ontent, UV radiation temperature effects, and chemical reacti ons can further degrade t he strength of the member. Fibers are the load carrying components of the composite materials and occupy a large volume of the overall la minate. Fibers are available in various materials such as glass, carbon, aramid and boron. Glass fi bers tend to be more ductile than Carbon fibers but will have a lower ultimate tensile strength. Because these products are proprietary materials, they tend to have specific characteristics that can alter the predicted behavior. Therefore, it is important to obtain a ll the material data sheets for the purpose obtaining the best results. Epoxy and resin materials can serve multip le purposes when applied to FRP. A primer is used to prepare any surface before attaching the FRP material. This allows for better adhesion and reduces the risk of poor bonds. Adhesives attach to components together and are used when attachi ng FRP strips. Saturants impregnate the fiber matrix and are used when performing a wet-layup reinforcing of fibers. Epoxies have low tensile strength when com pared to fibers but have good ductility. 26 PAGE 27 Thick epoxy layers will be subject to exce ssive deformation when loaded reducing the ability of the fiber to transmit its tensile strength to the member. Dynamic Analysis The dynamic analysis of either a mult i-degree of freedom system or a single degree of freedom system can be represented by the sa me equation of motion: )()( tFuRuCuM (2-24) M is the mass acting on inertia of the system. Cis the damping coefficient. )( uRis the stiffness function. is the forcing function. )( tFuis the displacement. uis the velocity. uis the acceleration. The resistance function R(u) will often be replaced by Ku for the purposes of dynamic analysis when the system being evaluated is linearly elastic. The resistance function will be discussed furt her in a later section. Rigorous analysis is often co mputationally expensive and is only truly feasible when the resistance and load functions are represented by convenient mathematical functions. As such, an approximate method is often used for analysis of the structure when only a localized evaluation of deflection is required. Th is requires the idealization of the structure and the loading. Equivalent values for the mass ( Me), the resistance function ( Re(u) ), and the forcing function ( Fe(t) ) are developed using shape functions as described in Biggs (1964) and Krauthammer et al. (1988). 27 PAGE 28 Equivalent SDOF System SDOF analyses are often used due to their computational efficiency. However, many of the structural members that need to be analyzed are continuous such as beams and slabs. In order to analyze them as a single degree of freedom system, the member will need to be transformed into an equi valent system. The process outlined in Biggs (1964) allows for the determination of the transverse displace ment of the member at various points along the mem ber by use of a shape function ( x) derived by the application of static loads. This assumes that t he calculated equivalent SDOF displaced shape is equal to that of the real struct ure at the same time. However, the stresses and forces are not directly equivalent to those of the real st ructure. Since this method accounts for only the elastic and plastic domains with ideal boundary conditions, alternate means are required to obt ain a more accurate representation of an equivalent system. Krauthammer et al. (1988) modifies this model by taking into account the transition phases between the two domains and is valid for all boundary conditions. Figure 2-6 illustrates the deformation profile that can be represented by the shape function. Equivalent mass The equivalent mass of the structure is der ived by balancing the kinetic energy of the moving parts of the beam for both the re al and the equivalent system. Equating the two systems will provide the following equivalent mass relationship. L edxxxmM0 2)()( (2-25) From which, the equivalent mass factor can be derived from a ratio of the equivalent mass to the total mass. 28 PAGE 29 temMMK / (2-26) Both of these will be affected at each time step in dynamic analysis due to the change in the shape profile of the member Krauthammer et al. (1988) uses the following linear interpolation procedure to ca lculate the equivalent mass factor at each time step. However, the equivalent mass will remain constant during hinge formation, since there is only a minor effect on the inelastic deformed shape function. )()1( )1( i ii mi im mimuu uu KK KK (2-27) The use of the DSAS program eliminates the need for the equivalent mass factor by calculating an equivalent mass function for each step based on displacements. Using finite elements, an equi valent mass is gener ated with the following relationship: Nnodes j i mid i j i j i ed d M M2 (2-28) Where i is the load increment and j is the node location along the beam. Equivalent loading function The equivalent forcing function is derived in a similar way as the equivalent mass. It is derive by balancing the external work for both the real and eq uivalent systems. Equating the two systems will provide the following equivalent load ing relationship. (2-29) L i i i extxFdxtxtxpF0)(),(),(),( From which, the equivalent loading factor can be derived from a ratio of the equivalent load to the total load. teLFFK / (2-30) 29 PAGE 30 The same approach was used for computing the equivalent load factor at each time step as was used for the equivalent mass factor. )()1( )1( i ii Li iL LiLuu uu KK KK (2-31) The use of the DSAS program eliminates the need for the equivalent loading factor by calculating an equivalent loading function for each step based on displacements. Applying the same approach as the equivalent mass, an equivalent resistance function is generated using the following function: i e Nnodes j i mid i j i j i eR d d f F (2-32) From which an equivalent load ing function can be developed. )( )( )( ),( tw uw uF tuFe e (2-33) Where w is the static load that would result in the control displacement. Resistance Function The resistance function represents the restor ing force a member exhibits to return to its initial condition when subjected to an external load. Biggs (1964) suggests that, for most structures, a simp lification can be made using a bilinear function to compute the resistance factor using the maximum plas tic-limit load as the maximum resistance Rm for the function. In addi tion, the resistance factor KR must always equal the load factor KL. However, a majority of the case s that will be discussed require further development of the resist ance function since the me mbers being evaluated have nonlinear characteristics. 30 PAGE 31 For dynamic analysis cases, the effect of load reversal needs to be considered in the resistance function model. The Krautha mmer et al. (1988) model expands on the Sozen (1974) model by including all materi al and support nonlinearities in a piecewise, multilinear curve. Figure 2-7 illustrates the modified resistance-displacement model compared to the bilinear model. If the maximum dynamic displacement does not exceed the yield point at A or A, the behavio r will remain elastic and will oscillate about zero displacement. However, if the maxi mum dynamic displacement exceeds the yield point, plastic deformation will occur and the member will have a residual displacement once it comes to rest. If point C is exceeded, the member will have failed in flexure. Otherwise, the member will unload according to the path outlined in the model. While the resistance of beams and one-way slabs can be analyzed using the method previously describe, two way sl abs need to be analyzed by calculating the resistance in both directions and super positi oning the resistance functions to obtain the total system resistance model. In addition, if the section is not symmetrical, a different resistance model will be r equired for load reversal. Numerical Integration Due to the nonlinear characteristics of the members that will be evaluated, a closed form solution may not be possible. As su ch, a numerical solution that is valid for a wide range of cases is required. The approach used to numerically integrate the equation of motion is chosen in order to mi nimize computational demands while still providing accuracy. Either an implicit or explicit time step can be used to evaluate the system. The implicit method evaluates the equation of moti on at the next time step while an explicit method evaluates it at the current time step. Wh ile an implicit method is computationally expensive for a MDOF syst em, it is feasible fo r a SDOF system. As 31 PAGE 32 such, a modified Newmark-Beta method will be used for the purpose of this research. The Equations 2-34 and 2-35 are used to compute the velocities and displacements for this method. ttt tttuu t uu 2 (2-34) 2 22 1 tutu tuuutt t tttt (2-35) The is taken as 1/6 corresponding to t he linear acceleration method as proposed in Tedesco (1999) instead of 1/4 corresponding to the cons tant average acceleration method originally proposed by Newmark (1962). The followi ng steps are used in order to solve for the equation of motion for the system. Use the known velocity and displacement at time t to calculate the acceleration at time t Estimate the acceleration at the next time step. Compute both the velocity and the displa cement at the next time step using the estimated acceleration value. Compute the acceleration at the next ti me step using the equation of motion and the computed velocity and displacement. Compare the computed acceleration to the estimated acceleration. If the convergence tolerance is achieved, conti nue on to the next time step. Otherwise, repeat the process using the calculated acceleration as the new estimate in step 2 of the process. The level of accuracy and stability of t he final result must be considered when selecting a value of time step. Using a time step that is less than a tenth of the natural period of the system will usual allow for a fa st convergence. However, the time step must also be small enough in order to accoun t for the loading of the member. In order to further improve the efficiency of the of the evaluation, a sma ller time step can be 32 PAGE 33 used during the loading phase of the system and a larger time step can be used during the free vibration phase. Flexural Behavior Flexural behavior of reinforced concrete has been studied extensively since most structural design is based on flexural failure (MacGregor 2009; Park and Gamble 2000). The use of a Moment Curvature relationship is the primary method for representing flexural behavior and its function is generated using strain com patibility and equilibrium. Flexural Behavior in reinforced concre te is based on three assumptions. Sections perpendicular to the axis of bending that are plane before bending will remain plane after bending. The strain in reinforcement is equal to the strain in the concrete at the same level. The stresses in the concrete and reinforc ement can be computed from the strains by using stress-strain curves for concrete and steel. For most cases, tension in concrete is ignored. However, in blast loading cases, the tension is included since the strain rate may have a significant effect on the tension strength of the concrete. The stra in rate effects will be further discussed in a later section. Figure 2-8 shows a stress strain diagram of a reinforced concrete beam cross section under flexural loading. By dividing the section into layers, a more accurate representation of the behavior can be achi eved instead of using the Whitney stress block which is the primary design method under normal loading conditions. The moment-curvature diagram can be obtained by incrementing the curvat ure from zero to failure and calculating the stra in and stresses in each layer. An iterative process would be used to ensure that equilibrium and compatib ility are satisfied for each layer. The stress-strain values for each material will be defined by individual predefined material models. Both confined concrete and unconfined concrete will need to be considered. 33 PAGE 34 Diagonal Shear Behavior When a member fails in flexure, it is a combination of both the flexural behavior and the diagonal shear behavior. It occurs when flexural stress and shear stress act together such that there is a large enough stress to create cracks perpendicular to the principle tensile stress along the member to prevent brittle fa ilure. As such, the use of web reinforcement is required. Due to t he diagonal shear influence on the deflection of beams, the use of a shear reduction factor is required in the analysi s of the member. Krauthammer et al. (1988) modified a shear reduction factor orig inally proposed by Krauthammer et al. (1979) to account for deep and slender beams. )(fluMMSRF (2-36) Krauthammer et al. (1979) used the relati onships in Equations 2-37, 2-38, and 239 to describe the minimum SRF ratio as a function of the tensile longitudinal reinforcement ratio, 0.1 :%65.00 m fl uM M (2-37) )0065.0(6.360.1 :%88.1%65.0 m fl uM M (2-38) 6.0 :%88.2%88.1 m fl uM M (2-39) The modified minimum moment capacity ra tio with web reinforcement as proposed by Krauthammer et al. (1988) identified by point 2P in Figure 2-9 is given Equation 2-40. tan 0.1 m fl u m fl u m fl uM M M M M M (2-40) 34 PAGE 35 Where the angle of compressive strut at ultimate is calculated for a deep rectangular beam by )5.2/1( da08.4)/(72.2* da (2-41) and a slender rectangular beam )7/5.2( daby 22.7)/(06.3* da (2-42) cyff /* (2-43) Where is the web reinforcement ratio. The computed moments are then multiplied by the SR F and the curvature is divided by the SRF (Krauthammer 1988). This model is included in the DSAS program. Rate Effects Various studies have found that the strength and modulus of elasticity of concrete and steel increase significantly when subjected to high loading rates such as impact and blast Shanna (1991). Of the two techniques used in analysis, one is a dynamic enhancement factor based on straining rate to increase the material properties used in the derivation of moment-curvature, diagonal s hear and direct shear relationships. The other applies the enhancement factor directly to the resistance function by multiplying the shear and flexure capac ity of the section by the enhancement factor. The use of either the strain rate, stre ss rate, or the loading rate can be used to develop the enhancement factor. However, si nce the strain rate is the controlling parameter for the development of the moment-curvature re lationship, it will be used as the independent parameter. Based on the DSAS program the following models are used for the material enhancement factors. The use of Soroushian and Obaseki (1986) 35 PAGE 36 model has been recommended by both Shanna (1991) and by Krauthammer et al. (2002). The enhancement factor for steel mate rial is provided by Equation 2-45. )( )( )( )(staticf dynamicf staticf dynamicf Enhu u y y s (2-44) y y y sf f f Enh )log( 05.065.02.11.3 (2-45) However, the Soroushian and Obaseki ( 1989) model underestimates the dynamic increase factor for concrete in compression under high strain rates. Therefore, the CEB-FIP (1990) model is used. The enhancem ent factor for concrete material is provided by Equation 2-46 and 2-47. )( )( staticf dynamicf Enhc c c (2-46) For 1sec30026.1/s cEnh (2-47) For 1sec303/1/s cEnh (2-48) Where (quasi-static strain rate) 16sec1030 xs)2156.6(10 (2-49) )1450/95/(1cf (2-50) For concrete in tension, the model proposed by Ross et al. (1989) was used and is given by Equation 2-51. 086.2 10))log7(0164.0( eEnhct (2-51) 36 PAGE 37 Fiber Reinforced Polymers Fiber Reinforced Polymers are capable of enhancing the overall strength of a beam through composite action. The FRP is attached using an adhesive epoxy and the bond strength of the epoxy is an important component in the failure mechanism of the beam. The use of FRP in reinforcing a member can result in the following failure modes illustrated in Figure 2-10: Rupture of the FRP-EB after yieldi ng of the tension steel reinforcement. Secondary concrete crushing after yielding of tension steel reinforcement. Primary concrete crushing in compression before yielding of the reinforcing steel. Shear/tension delamination of the concrete cover. Debonding of FRP from t he concrete substrate. The most common types of failure observed in various tests are the debonding of the FRP or the concrete cover delamination. This occurs primarily when the FRP reinforcement is insufficiently anchored into the member. FRP Flexural Behavior The use of FRP enhances the flexural str ength, but can reduce the ductility of the member limiting its deflection. Due to the limited deflection, th e strengthening of the member can delay the onset of the cracking and diffuses t he crack pattern over the length of the member (Buyle-Bodin et al. 2002). The assumptions made for a reinforced concrete member are also applied when FRP is used. In addition, the shear deformation within the adhesive layer is negl ected based on the premise that the adhesive layer is very thin. Also, no rela tive slip exists between the FRP and the concrete substrate. While these assumptions do not accurately reflect the behavior of 37 PAGE 38 FRP, they are necessary for computational efficiency. In addition, the degree of inaccuracy does not have a significant effect on the flexural strengt h of the member. The process for determining the moment-cur vature of the reinforced member is similar to that of a normal reinforced concrete member. However, the total strains in the member are adjusted to account for the strain of the FRP on the bottom layer. Due to the very small thickness of the FRP layer, it will be assumed to act in the same location of the bottom layer of the conc rete. As such, the strain of the FRP will be the same as the strain acting on the bottom layer of conc rete. Until debonding occurs, the FRP layer will be considered another laye r of tensile reinforcement. FRP Shear Behavior The use of FRP applied to the bottom layer of the concrete does not typically add to the shear strength of the member (Kac hlakev et al. 2000). Diagonal cracks still develop at similar loading leve ls. However, the FRP allows the ability to maintain the integrity of the member in the presence of a shear crack. Shear failure will then be accompanied by the transverse rupture of the FRP composite. If there is a vertical displacement, the peeling of the FRP memb rane will need to be considered and will be discussed in more detail in the debonding section of this review. FRP Rate Effects The strain-rate effect for a FRP plate when compared to those of concrete and steel was found to be negligible according to J ohnson et al. (2005). Therefore, it will not be considered in the analysis of this study. Ho wever, rate effects will have an effect on the bond strength of the FRP-to-concrete in terface. This is attributed to the dependence of bond strength on the concrete s hear strength. Therefore, the rate 38 PAGE 39 enhancement factor previously discussed will need to be applied to the concrete model for tension. FRP Size Effects The amount of FRP applied to base of t he member will need to be considered when calculating the capacity. Higher FRP re inforcement ratios ha ve lower deflection capacities and higher stiffness. However, increasing the th ickness of the FRP by using multiple plys does not always lead to a hi gher member capacity. The studies performed by Pham et al. (2004) and Maalaj et al. (2005) show that the effectiveness of the FRP is reduced as the relative stiffness of FRP to st eel increases. Maalaj et al. (2005) further illustrates that the peak interfacial shear stresses seems to increase with increasing FRP reinforcement. As a result, debonding will occur earlier in the loading process. Debonding Behavior Debonding will occur well belo w the rupture strain of the FRP, leading to an underutilization of the strength of the reinforcement (Smi th et al. 2010; Rougier et al. 2006). Debonding of the FRP from the concre te often occurs when the surface has not properly been prepared. If the FRP is properly applied, various tests have shown that the failure will not be at the concrete adhesive interface, but within the concrete cover just above the adhesive layer. This is due to the shear strength of the adhesive layer being higher than that of the concrete cover. The amo unt of the concrete cover that delaminates will be dependent on the material properties within the concrete. Debonding occurs at locations of high stress concentrations such as formed cracks or at the ends of the FRP plate and is directly re lated to the shear stress experience in the interfacial layer. Depending upon the location of failure, a specific set of criteria will need to be satisfied. The common types of debonding failure include 39 PAGE 40 plate-end debonding, midspan debonding, and shear-span debonding. Plate-end debonding depends largely on the interfacial shear and the normal stress concentration at the cut-off points of the FRP plate. Once debonding o ccurs, the crack will propagate towards the center of the member. The Mid-span debonding starts at regions of max moment and usually propagates towards the nearest plate end. The higher interface stresses are developed by the opening of a major crack and rely primarily on shear stress concentrations. Shear span and in termediate shear crack debonding are affected by crack widening as well as relative vertical displacement. The vertical displacement causes a peeling effect on t he FRP and influences the initiation of the bond failure. Therefore, both normal and shear stresses influence the debonding process. Once debonding has occurred, the behavior of the member will revert back to that of the unstrengthened member (Pham 2004). Shear stress Shear Stress develops between the laye rs of a beam when subjected to a difference in axial loading. In the case of a beam under flexural loading, the axial loading in each section of the beam is based on the different strain values derived from the curvature of the beam. Figure 2-11 shows the develop ment of the shear stress between the layers. Given an adhesive laye r thickness, Equations 2-52 and 2-53 defines the shear stress withi n the adhesive layer assuming the strain varies linearly across its depth (Li et al. 2009). )( 112uu ta a (2-52) aaaG (2-53) 40 PAGE 41 If the adhesive has a thickness approaching zero, the shear stress along the concrete interface will be similar to the shear stress along the FRP interface. Assuming a uniform bond stress distribution, an aver age bond stress can be evaluated based on Equation 254(Aiel lo and Leone 2008). bf frpLb T (2-54) The local shear stress for the interfac e is a primary concern when trying to understand the debonding failure between the FRP reinforcement and the concrete. The use of pull tests are used to derive bond-slip models that represent the shear stresses that result from the horizontal slip of the member These models focus solely on determining the shear stresses. The bilin ear model has been used in most studies and provides a relatively accurate approximat ion of the behavior for the interface. The primary factors affecting the bond-slip behavior are concrete strength, bond length, FRP axial stiffness, FRP-to-concrete width ratio, adhesive stiffness and adhes ive strength. Lu et al. (2005) developed both a precise, simplified, and a bilinear model for determining bond-slip and strength using meso-scale finite element results with available test data. The models proposed by Lu et al. (2005) provided better results over all the previous models for calcul ating both the bond str ength and the strain distributions in the FRP. All three of t he models closely resembled the test data with the precise model only slightly more accu rate. Therefore, the use of the less complicated computational models could be used without a significant loss of accuracy. For the purpose of this study, Lu et al. (2005) bilinear model will be used. The bilinear model is illustrated in Figure 212 and described using the following equations: 41 PAGE 42 so ss smax if (2-55) sosss sosf ssfss ss max if (2-56) sfssosss 0 if (2-57) sfsss The final slip is calculated using Equation 2-58 max/2 f sfGs (2-58) The max shear stress and corresponding slip are given by Equations 2-59 and 2-60. twf 50.1max (2-59) tw sof s 0195.0 (2-60) The interfacial fracture energy can be expressed as Equation 2-61. tw ff G2308.0 (2-61) The FRP-to-concrete width ratio factor can be derived by the following formula proposed by Lu et al. (2005) cf cf wbb bb /25.1 /25.2 (2-62) fb is the width of FRP plate. cb is width of concrete member. tf is the tensile strength. Normal stress Cracks that experience both vertical and horizontal displacements will be subjected to both peeling and pulling forc es. The peeling force will generate normal stresses acting perpendicular to the interface of the member increasing the ease of 42 PAGE 43 debonding. However, the peeling effect is only significant near the crack and decreases as it move further along the beam. Theref ore, debonding initiation will be significantly affected but the development of the ultimate l oad will be less sensitive to the effect of peeling (Pan and Leung 2007). The use of t he triangular model described in Wang and Zhang (2008) will be used to illustrate the open traction-separation of the FRP-toconcrete interface. This m odel illustrated in Figure 2-13 will allow the simplification of the calculation without losing much accuracy in the results. The effect of normal stress on the debond ing process becomes negligible once the crack has extended beyond a certain poi nt (Wang and Zhang 2008). The normal stress and max slip will be will be determined using 12ww t Ead ad n 2-63 nn nfGs /2 2-64 nGis the normal interfacial fracture energy depe nding upon if the fracture occurs in the concrete cover or the adhesive layer. Mixed mode initiation While flexural debonding is largely dependent on shear stresses, it is does not account for vertical crack displacement from shear cracks, which could result in significant error. Therefore, in order to accommodate all debonding scenarios, normal stresses need to be included with the shear st resses in the analysis. For the purposes of this study, a mixed-mode nonlinear bond st ress-slip model will be used. The normal and the shear-stress slip models described in the previous sections will be included in the analysis. 43 PAGE 44 The slip for the of the normal and shear stresses can be derived using a coordinate system and ar e represented by (W ang and Zhang, 2008): 12wwsn (2-65) 222111wYuwYuss (2-66) ns is the normal stress slip. ss is the shear stress slip. 21,uuare the axial displace ments of the member and the debonded section 21, ww are the vertical displacements of the member and the debonded section are the distances from debonding interface to neutral axis of each member 21, YYFor the purposes of this study, the no rmal and shear stresses will be treated as independent since there is little experimenta l data outlining their interaction. This assumption will allow for the simplification of the analysis and the level of error should be minimal. Debonding will occur when the total fracture energy of the system is reached. By combining the fracture energi es of both the normal and shear stress slip models, the total fracture energy developed at each stage of the evaluation can be determined. snTGGG (2-67) n s nn ndssGn0 (2-68) s s s sdssGs0 (2-69) Since the failure will likely occur in the c oncrete cover, the fractural energy will be based on the concrete material properties. Using a simple linear debonding criteria 44 PAGE 45 developed by Hutchinson and Suo (1991), fu ll debonding will occur when the following equation is satisfied. 1 sc s nc nG G G G (2-70) As previously discussed, the effect on debonding due to normal stresses will become insignificant after the crack propagates past a certain point. At that point, debonding will be determined sole ly by shear stresses. Effective bond length Despite the FRP being applied to the entire member, only a small section closest to the loading is used to determine the debonding process. Any bond length that extends beyond the effective bond length do es not add a significant value to the strength of anchorage. As the crack propagates further away from the original loading, the resistance in the FRP at a point further away becomes active against bond-slip. Yuan et al. (2004) developed an effective l ength model for a bilinear bond-slip model that will be used in this study. a a aLe2 21 2 21 1tan tan ln 2 1 (2-71) fftEs0 max 1 (2-72) ff ftEss )(0 max 2 (2-73) f fs ss a0 299.0arcsin 1 (2-74) 45 PAGE 46 The factor of 0.99 was used instead of the original val ue of 0.97 proposed by Yuan et al. (2004) in order to achieve a closer agreement to model proposed by Chen and Teng (2001). This percentage in dicates the level of bond strength that is achieved in an infinitely long bonded joint with the higher percentage providing the more stringent definition. The effective bond length defined by Yuan et al. (2004) can be applied to plate end debonding. However, during beam flexur e, intermediate crack induced debonding can occur and would require a different anchorage length. Teng et al. (2003) identifies the intermediate crack induced effective lengt h in Equation 2-75. They further explain that the anchorage bond length should be twice the size of the effective bond length. c ff ef tE L (2-75) Ultimate bond strength The ultimate bond strength is the maximu m strength that can be sustained by the shear interface up to a maximum limit w hen the bond length is equal to the effective bond length. If the load is greater than t he ultimate bond strengt h, the crack will continue to propagate until such time that t he load can be sustained by the remaining bonded zone. Equation 2-76 obtained from Lu et al. (2005) pr edicts the ultimate bond strength. ffffluGtEbP 2 (2-76) where e b lL L 2 sin (2-77) 46 PAGE 47 If the bond length is greater than the effe ctive length, the bond length factor,l ,will be equal to one as no further bond strengt h is gained from additional anchorage. Load-Impulse (P-I) Diagrams P-I Diagrams provide the means of deter mining the level of damage on a structure or member. The points of t he curve represent a pre-defined threshold response criteria based on the equivalent peak load and impulse required to achieve the specific behavior. Any combination of pressure and impulse that is below the curve is safe. The curve is broken down into three separat e regimes as illustrated in Figure 2-14. The impulsive regime is characterized by loads of short duration where the maximum response is not reached until t he load duration is complete. The dynamic regime is representative of the maximum response being reached close to the end of the loading of the member. Finally, the quasi-static regime indicates when a member has reached the maximum response before the loadi ng has been removed. Krauthammer (2008) outlines various methods for developing the curv e. Simple problems allow for the use of closed form solutions, while other cases ma ke use of the energy balance method. However, for complex problems such as those to be discussed during this research, a numerical solution must be employed. Blasko et al. (2007) outlines the method that has been employed in the DSAS program and whic h will be applied to this study. 47 PAGE 48 Figure 2-1. Free field pressure time variation. [Reprinted with permission from Krauthammer, T. 2008. Moder n Protective Structures (Page 68, Figure 3-1). CRC Press, Boca Raton, FL.] Figure 2-2. Modified Hognestad stress-strain curve for concrete in compression. 48 PAGE 49 Figure 2-3. Confined concrete stress strain curve. [Reprinted with permission from Krauthammer, T. 1988. A computati onal method for evaluating modular prefabricated structural element for rapi d construction of facilities, barriers, and revetments to resist modern convent ional weapons effects. Technical report (Page 11, Figure 3). Tyndall Air Force Base, FL.] Figure 2-4. Tensile stress strain curve for reinforced concrete. 49 PAGE 50 Figure 2-5. Steel Stre ss-strain curve model. Figure 2-6. Deformation profile for a beam. 50 PAGE 51 Figure 2-7. SDOF inelastic resistance model. [Reprinted with permission from Krauthammer, T. 1988. A computati onal method for evaluating modular prefabricated structural element for rapi d construction of facilities, barriers, and revetments to resist modern convent ional weapons effects. Technical report (Page 103, Figure 38). Ty ndall Air Force Base, FL.] 51 PAGE 52 Figure 2-8. Stress strain diagr am for reinforced concrete beam. 52 PAGE 53 Figure 2-9. Shear reduction model. [Reprinted with permission from Krauthammer, T. 1988. A computational method for evaluat ing modular prefabr icated structural element for rapid construction of facili ties, barriers, and revetments to resist modern conventional weapons effects. Technical report (Page 37, Figure 12). Tyndall Air Force Base, FL.] 53 PAGE 54 Figure 2-10. Failure modes of FRP rein forced concrete members. [Reprinted with permission from Teng, J.G. 2003. Interm ediate crack-induced debonding in RC beams and slabs.(Page 448, Figure 1) C onstruction and Building Materials.] 54 PAGE 55 Figure 2-11. Section forces for FRP reinforced concrete beam. 55 PAGE 56 Figure 2-12. Local bond-slip bilinear model. [Repr inted with permission from Yuan, H. 2004. Full-range behavior of FRP-to-concrete bonded joints. (Page 555, Figure 3) Engineering Structures.] Figure 2-13. Normal stress traction-separation model. [Reprinted with permission from Wang, J. 2008. Nonlinear fracture me chanics of flexural-shear crack induced debonding of FRP strengthened concrete beams. (Page 2920, Figure 3) International Journal of Solids and Structures.] 56 PAGE 57 57 Figure 2-14. Typical response functions. A) Shock spectrum. B) P-I diagram. [Reprinted with permission from Krautha mmer, T. 2008. Modern Protective Structures (Page 328, Figure 8-2). CRC Press, Boca Raton, FL.] PAGE 58 CHAPTER 3 METHODOLOGY The emphasis of this chapter outlines the development of an algorithm for a reinforced concrete beam retrofitted on the tension face with FRP. This algorithm will be implemented into the non-linear dynamic analysis procedure in DSAS and will account for debonding behavior during beam deflection due to flexural loading. A basic overview of the concepts that will be employ ed in the new model will be discussed and reviewed. Structural Overview The structure of interest in this study is a reinforced concrete beam that has been retrofitted on the tension side with FRP. T he initial evaluation of the beam will be done under four-point-static loading to valid ate the debonding model. The beams analyzed will be based on the experimental beam descri bed in Ross et al. (1994) and a graphic depiction of the beam is shown in Figure 3-1. The beam is reinforced with US# 4 rebar in tension and US# 3 rebar for the compression steel and the stirrups. Once the static behavior has been confirmed, the beams will be loaded dynamically using a uniform blast pressure. Us ing a uniform blast pr essure will limit the risk of discontinuities that can increase the chances of developing flexural shear cracks that would be linked to peeling effects on the FRP layer. Failure assessments associated with dynamic load ing will be based on values gener ated by pressure impulse diagrams. For most loading cases, when the beams have been properly prepared, the beams have failed in the concrete cover and not in the epoxy. This is due to most epoxies having a much higher tensile strength than that of the concrete. Predicting epoxy failure 58 PAGE 59 will not be feasible since workmanship fa ctors cannot be easily accounted for in analysis. Therefore, it was assumed that the debondin g behavior will occur in the concrete cover. However, the experimen tal results witnessed del amination within the Debonding Behavior The debonding behavior associated with FRP in the tension face is evaluated using the non-linear SDOF dynamic analysis software and incorporates the various concepts discussed in Chapter 2. The debondi ng failure will limit the ability of the FRP layer to fully develop until its ultimate strengt h as illustrated in Figure 3-2. Due to the limitations of evaluatin g the vertical shear deformations in the current version of DSAS, the debonding algorithm was based on only the interfacial shear. The basis of the analysis is the modification of the moment cu rvature diagram. Utilizing the strain based on the curvature of the beam, the tensile stress is determined for the FRP layer. The tensile stress is then transferred to the concrete producing interfacial shear resulting in slippage along the interface as a result of c oncrete failure. The bond will remain elastic up until reaching the maximum shear stress of the material and then will transfer to the elastic-softening stage. Upon reaching the fr acture energy of the concrete, debonding will be initiated and the residual shear stress will allow the crack to propagate further along the interface. If the ultimate bond st rength defined by exceeds that of the imposed shear stresses, the remainder of the section will not debond. Until the bond fails, the FRP layer will act as an additional layer of tensile reinforcement for the purpose of section analysis. The DSAS Algo rithm for debonding is illustrated in Figure 3-3. The initiation of the debonding behavior will be based on achieving the fracture energy required for failure of the material along the bond interface. The fracture energy 59 PAGE 60 will be determined by utilizing a simplified bili near bond-slip model. From the model, the properties were developed for the debonding init iation utilizing the following formulas. max/2 f sfGs (3-1) twf 50.1max (3-2) tw sof s 0195.0 (3-3) tw ff G2308.0 (3-4) Using the bond-slip properties, both the effective bond length and the bond strength are calculated. Due to the si mplified analysis model used by the DSAS program, the size of each section to be ev aluated is greater than the effective bond length. Therefore, the bond strength will not be less t han the ultimate bond strength. a a aLe2 21 2 21 1tan tan ln 2 1 (3-5) fftEs0 max 1 (3-6) ff ftEss )(0 max 2 (3-7) f fs ss a0 299.0arcsin 1 (3-8) ffffluGtEbP 2 (3-9) The debonding module receives a calculated strain from the moment-curvature analysis based on one point within the section. Therefore, t he shear stress distribution is taken as uniformly distributed across the entire bond length. The adhesive layer 60 PAGE 61 thickness is taken as zero and the shear st ress is based on the tensile forces generated in the FRP layer. bf frpLb T (3-10) The debonding failure is then based on co mparing the FRP generated interfacial shear and the bond strength. The interfacial shear stress is averaged over the entire bond length, where as the bond strength is only averaged over the effective bond length since only the effective bond length is active against the debonding behavior. If the interfacial shear stress the bond strength, the section will be considered to have failed and the FRP layer will be removed from the moment-curvature. DSAS Overall Algorithm The behavior of the bond affects the analysis of the section and, in turn, the moment-curvature diagram. Therefore t he overall algorithm will summon the debonding behavior during the generation of the Moment curvature stage as outlined in Figure 34. If the FRP layer has debonded, it will rema in debonded for all subsequent time steps during the dynamic analysis. 61 PAGE 62 Figure 3-1. Test beam from study. Figure 3-2. Comparison of debonding process. 62 PAGE 63 Figure 3-3. DSAS debonding algorithm. 63 PAGE 64 Figure 3-4. Overall DSAS algorithm. 64 PAGE 65 CHAPTER 4 ANALYSIS The objective of the validation of the research is to confirm the methodology presented in Chapter 3. The validati on of the methodology was conducted by comparing results obtained from the DSAS analysis and compared to experimental test results performed under Ross et al (1994). The first step wa s to the validation of the material model followed by the validati on of the debonding process. Once the debonding process was evaluated, the debonding process was analyzed utilizing dynamic loading cases. Furt her analysis was conducted to determine if the algorithm accounted for expected performance changes associated with the FRPs strength, modulus of elasticity, thickness of the la yer, and the width of the FRP layer. Material Model Validation The material models used were based on established DSAS models. The concrete material is based on the Modified Hognestad Parabola discussed in Chapter 2. The steel model was based on the three stage hardening model outlined in Chapter 2. The material properties defined by the exper iment are included in Table 4-1. DSAS input values beyond those in the experiment ar e included in Table 4-2. Verification of the modified code for DSAS was complet ed by comparing the values for the unretrofitted beam with the existing DSAS vers ion 3.2.1. The inclusion of the FRP module was found not to affect the current version of the program as the same test results were obtained for both the existing ve rsion and the modified version. Figure 4 1 compares the material model from the experiments conducted by Ross et al. (1994) with those computed in DSAS. The load-displa cement plot of the DSAS model closely follows that of the experiment under four point loading. The peak displacements and 65 PAGE 66 forces are captured in Table 4-3. The di fference between the two peak displacements is slightly over ten percent at the point of failure. The DSAS model fails earlier than the experimental value. This could be attribut ed to the strain values associated with the reinforcing steel. There was no information pr ovided on the failure strain of the steel. Therefore, the value used in the DSAS program coul d have been underestimated which in turn would have led to the premature failure of the member and does not have an impact on the programs analysis. Therefore, the DSAS material model is able to accurately capture the behavior of reinforced concrete beams under static loading case. Debonding Validation The debonding validation was carried out us ing the experimental results from Ross et al. (1994). The experimental test produced debonding between the epoxy FRP interface which is contrary to the pr imary assumption that debonding will occur in the concrete cover. However, the model was set up to account for t hat variable as long as the bond strength of the epo xy was known. It cannot predict if debonding will occur in the epoxy layer prior to the actual failure of the bond. The comparison of the DSAS model and the experimental data can be viewed in Figure 4 2. The comparison of the load-displa cement graphs indicated that the DSAS debonding model closely followed the path of the ex perimental values. However, the DSAS model failed to yield at the same point as the Experimental values. It continued to see an increase in strength prior to the debonding of the FRP layer. Furthermore, the experimental value had a quick drop-off fr om the peak load where as the DSAS model declined from the peak value al ong a more gradual incline. However, the difference between the two values is less than ten per cent and is included in Table 4-4. 66 PAGE 67 The DSAS program evaluates sections along the beam and in an attempt to maintain the efficiency of the program; the mesh size is lim ited to increments that are larger than effective bond length resulting in an inability to predict the transferring of loads through crack propagation. Furthermore the model bases the failure of the bond on an average uniform shear stress due to the strain value provided by the momentcurvature analysis. This allows for a simp lified solution but limits the accuracy of the member assessment. The shear stress distribut ion is not uniform along the interface. It has a peak shear stress near the crack and then dissipates in a non-linear fashion further away from the crack. Therefore, the non uniform may have shear stresses that exceed the ultimate bond strength while uniform shear stresses average them out possibly resulting in no debonding. However, under these static loading conditions, the DSAS model performed well and was able to closely capture the debonding behav ior. Dynamic loading The primary purpose of the DSAS program is to evaluate the dynamic loading cases of various different components. Ther efore it is important to evaluate the member under blast loading cases. The beam was loaded with a uniform blast pressure to evaluate the effects associated wit h the FRP retrofit under dynamic loading. Figure 4-4 illustrates the time displacement of both the retr ofitted and the unretrofitted beam that had been subjected to a peak load of 50 psi that decayed linearly to zero after 0.01s. Neither of the beams fail ed under the loading, however, the beam enhanced with the FRP layer was able to reduce the maximum deflection of the beam from 1.55967 in to 1.120034 in. This illustrates that the FRP layer is capable of reducing the ductility of the beam. Increasing the stiffness of the beam could lead to crushing of the concrete while the FRP laye r remains bonded to the tension face of the 67 PAGE 68 beam. After the concrete crushes, the beam is subjected to a greater risk of brittle failure if the FRP ruptures or debonds from the tension face. The use of pressure-impulse diagrams a llows for the development of a failure envelop that simplifies the a ssessment of a structure under various loading cases. Figure 4-5 illustrates a pressure-impulse diagram demonstrating the ultimate failure criteria for both cases. The beam with t he FRP was able to sustain greater loading conditions that the unretrofitted beam. Under quasi-static loading conditions, the FRP retrofitted beam was able to withstand conditions nearly double that of the unstrengthened member. However, the differ ence in resistance was much less under the impulsive regime. This is due to the loading occurring for a short duration with the load often being removed before the member undergoes any significant deformation. The behavior of the beam is then determined by its natur al frequency and damping. Moreover, the loading in the impulsive regime has a greater chance of developing direct shear failure of which FRP has minimal resistance against. Parametric Study The following sections discuss the various aspects associated with the behavior of the FRP layer and its effect s on the debonding behavio r. The input for the parametric study can be found in Table 4-5. This study will focus on the effect of the ultimate strength, the elastic modulus, t he thickness and the width of t he reinforcing strip. There are a wide range of FRP materials and co ver a wide range of material property combinations. This study only served to identify key components and isolate them for the purpose of identifying thei r effect on the bond behavior. 68 PAGE 69 FRP Ultimate Strength The ultimate strength of FRP has no effect on the debonding process. The strength of the beam is limited by the debonding process. However, weak FRP layers that are connected to the beam with a strong bond ma y experience rupture in the fibers before debonding can occur. This is proce ss is rare and occurs when anchor systems are employed to prevent debonding. Otherwise, failure will occur due to debonding as demonstrated by Figure 4-6. The pressure-im pulse diagram distinctly showed a failure of a weaker FRP while the remaining FRP layers with higher strengths grouped together. This grouping can also be used to ill ustrate the failure criteria for debonding. FRP Elastic Modulus The behavior of the elastic modulus is shown in Figure 4-7. A high elastic modulus will increase the stiffne ss of the member but will lead to early failure of the FRP layer. Using an FRP layer that has a low Elastic Modulus will have more ductile behavior allowing the further deformation of the FRP layer and increase the behavior of the material in the impulsive domain. Incr easing the elastic modulus will also increase the stiffness in the impulsive domain but will still result in failure of the member due to the beam being overly rigid. The beam will be able to resist initial large loading scenarios but will FRP Thickness The thickness of the FRP layer was in creased by adding additional plies to the bottom of the existing layer. Initially, t he increased thickness enhances the strength of the beam. However, there is a limit on t he amount of thickness that can be added before there is a reversal of behavior. Figur e 4-8 shows that there is an initial larger increase in strength by adding a second layer to the FRP. However, increasing the 69 PAGE 70 number of layers further limits the advantage provided by the excess material and would not be a cost beneficial. In addition, increas ing the number of layers will increase the stiffness of the FRP strip resu lting in greater shear forces along the interface leading to possible debonding of the layers. The pressu re-impulse diagram showed the thickness peaking at 0.0350 in. The subse quent layer resulted in a small reversal suggesting that the peak thickness would be close to 0.0350in. The last thickness was four times the original thickness and produced a composite strength much less than the thinnest layer. FRP Width The width of the FRP layer had the greates t impact on the stiff ness of the beam. Increasing the width provided the best results for enhancing the strength of beam. This is directly attributed to the gr eater dispersion of the interfac ial shear stresses that result in debonding. In addition to reducing the shear concentration, the increased area provides greater tensile st rength to the beam. Figure 4-9 illustrates differences between the strips. 70 PAGE 71 Figure 4-1. Validation of t he material model of DSAS. 71 PAGE 72 Figure 4-2. Load-displacement curve for validation of the debonding model in DSAS. 72 PAGE 73 Figure 4-3. Moment-curvature model for validation of the debonding model in DSAS. Figure 4-4. Time-displacement response to blast loading case. 73 PAGE 74 Figure 4-5. Pressure-impulse curve under uniform blast loading. 74 PAGE 75 Figure 4-6. Effect of FRP Yield Strength. 75 PAGE 76 Figure4-7. Effect of FRP Modulus of Elasticity. 76 PAGE 77 Figure 4-8. Effect of FRP thickness. 77 PAGE 78 Figure 4-9. Effect of FRP width. Table 4-1. Beam material proper ties for experimental test case. Ec (psi) fc (psi) c u f y (psi) Es (psi) ffr p (psi) Efr p (psi) 5000000 7950 0.00245 0.003 60000 29000000 320000 20000000 Table 4-2. DSAS inputs for material model verification. sh u f fu(psi) damping 0.00207 0.018 0.018 70000 0.05 Table 4-3. Peak results of unstrengthen static loading case. dexp (in) dDSAS (in) Difference (%) Pexp (lb) PDSAS (lb) Difference (%) 3.72264 3.271156 12.13 10352.4 10280.3 0.7 78 PAGE 79 Table 4-4. Peak results of FR P strengthened static loading case. dexp (in) dDSAS (in) Difference (%) Pexp (lb) PDSAS (lb) Difference (%) 2.14152 2.320851 8.37 22377.5 2347.07 4.89 Table 4-5. Material data for parametric case studies. Test ffr p (psi) Efr p (psi) tf (in) bf (in) 1 320000 20000000 0.0175 8 2 160000 20000000 0.0175 8 3 480000 20000000 0.0175 8 4 640000 20000000 0.0175 8 5 320000 30000000 0.0175 8 6 320000 40000000 0.0175 8 7 320000 20000000 0.0350 8 8 320000 20000000 0.0525 8 9 320000 20000000 0.0700 8 10 320000 20000000 0.0175 4 11 320000 20000000 0.0175 2 79 PAGE 80 CHAPTER 5 SUMMARY AND CONCLUSIONS This chapter outlines the various observa tions made as part of the validation for the debonding behavior as it pertains to the modification of the DSAS program. The limitations observed as part of the analysis will be address as well as recommendations and future research opportunities for the fu rther development of the retrofit model. Limitations Various limitations were encountered during the validation process of this study. The results are based primarily on experimental static loading cases and as there are limited severe dynamic loading cases associated with beams information to perform adequate comparisons to validate the dynam ic behavior of the debonding model in DSAS. Previous dynamic enhancement factor s are addressed in Chapter 2, but a lack of experimental dynamic tests lim ited the ability to evaluate if high strain rates in the epoxy and the FRP layers have a signifi cant impact on the debonding behavior when compared to steel and concrete. However, since most debonding behavior occurs in concrete, it may not be possible to accurate ly separate strain rate s associated with the FRP and epoxy from the concrete. Most models for bending incorporate a mi xed mode of failure. DSAS currently lacks the ability to properly evaluate the effe cts of discontinuities in the displacement due to flexural cracks that may result in peeling effects. Therefore, the DSAS debonding module is limited to the ev aluation of debonding based solely on the interfacial shear. While the deviations wil l be minimal under pure bending cases with uniform loading, it will limit that programs ab ility to accurately predict the behavior of more severe loading cases. Both close pr oximity blast loading and impact loading will 80 PAGE 81 increase the risk of flexural shear cracking re sulting in peeling effects. Including the peeling effects in the debonding initiation criteria through mixed mode evaluation will reduce the need for additional shea r stress. Therefore, not including the peeling effects will lead to predicted beam strengths that will be higher than the actual loading case. The behavior of debonding is based on two processes, the initiation of the debonding and the crack propagati on. The DSAS model adequately captures the initiation of the debonding but is limited on tracking the cra ck propagation. The size of the mesh generated for the DSAS program limits the ability to track this behavior. While the model for the individual se ction analysis produced good result s, they did not transfer to the overall beam analysis. Future Development /Recommendations The algorithm developed in this research to account for FRP debonding along the tension face provides an additional capability fo r the use in analyzing the retrofitting of reinforced concrete beams. However, the ev aluation of the behavior is limited to simple beam cases and further development is needed to accurately represent the behavior under various other support and loading conditions. The first recommendation is to perform addi tional experimental testing to validate the approach addressed in this study. There are limited test results addressing severe dynamic loading cases on reinforced concrete beams retrofitted with externally bonded FRP. Many of the existi ng beam tests are conducted using semi-static loading cases and many of the dynamic cases are conducted on slabs and walls. The main issue that would be evaluated during the tests would be to evaluate how the debonding crack propagates under severe loading cases. Furthe rmore, future experimental tests would allow closer evaluation of strain rate effe cts on both the FRP layer and the epoxy resin. 81 PAGE 82 The second recommendation is to expand the debonding model out to account for fixed end support beams. The used of fi xed end supports will cause negative moment situations at the supports that could lead to buckling of the FRP strips forcing premature debonding at the ends and reduc ing the capacity. Ha rdened epoxy has small compression strength and acts as a solid elem ent. Even if the tension on the FRP layer is not enough to initiate fracture damage due to shear stresses along the interface, a buckling action could result in a peeling effect initiating crack propagation starting at the end of the beam and progressing inward. The third recommendation is to include the addition of an anchor system in the FRP reinforcing layer. The first issue will account for the inclusi on of anchors along the supports. The used of anchors along the suppor ts will allow for the development of a tension membrane. This wi ll allow the increased enhancem ent of the member even after the FRP debonds from the face of the concrete. Fu rthermore, the inclusion of anchors along the edges will force debonding to initiate in the cent er of the member compared to at the plate ends under norma l bonding conditions. The second issue will account for the inclusion of an anchoring system along the face of the beam. The bonded region will only sustain up to an ultima te strength defined by the effective bond length. If the shear stresses along the in terfaces exceed that amount, the debonding crack will continue to propagate unt il either the shear stresses are less than the ultimate strength or the FRP completely debonds from the bottom of the beam. The inclusion of anchors along the base of the beam will limit the ability for the crack to propagate beyond a given distance and will prevent total failu re of the FRP layer. Future research should evaluate the types of anchors and t he placement of the anchors. 82 PAGE 83 The final recommendation is to research the effectiveness of FRP when applied as shear reinforcement. The use of FRP along t he tension face is effective at increasing the strength of the member in flexure. However, the de velopment of flexural shear cracks will allow for debonding to occur at lesse r loading cases. Applying exterior bonded FRP along the shear plain could reduce the size of the diagonal cracking in the member. Furthermore, wrapping the FRP up the sides of the beam will add increased anchor strength by increasing the bonded area that needs to fail in order for debonding to initiate and propagate along the tension face. General Conclusions A new assessment module was developed for DSAS to address the strengthening of reinforced concrete members with FRP on the tension face. The inclusion of this module has the ability to enhance the capability of the program to analyze members that have been subjected to FRP retrofits. Fu rther research is required to accurately capture the behavior of retrofits under diffe rent load cases and bou ndary supports. The current model is capable of capturing the debonding behavior based solely on the interfacial shear which is suitable for beams that do not experience any discontinuities due to flexural shear cracks. This study has shown that the inclusi on of FRP along the tension face has the ability to initially increase the strength of the member. The FRP provides the beam the ability to sustain higher loads and reduces the maximum displacem ent. However, the strength increase is limited by the bond str ength at the FRP-concrete interface. Furthermore, employment of FRP over re inforces the members and increases the possibility of concrete crushing along the comp ression face. Failure of the FRP, either 83 PAGE 84 84 by debonding or rupture, after the concrete is damaged could result in brittle failure of the member. The choice of FRP and its properties wil l determine the behavior of the beam. Overly stiff members will suit able under quasi-static loading. 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Structural Dynamics: Theory and Applications Addison-Wesley Longman Inc: Menlo Park, CA. Teng, J.G., Smith, S.T., Y ao, J., Chen, J.F. (2003). Intermedi ate crack-induced debonding in RC beams and slabs. Construction and Building Materials 17(6-7), 447-462. Tran, P.T., Krauthammer, T., Astarlioglu, S. (2009). E ffect of short-duration-highimpulse variable axial and transverse loads on reinforced concrete column. Report CIPPS-TR-001-2009 Center for Infrastructu re Protection and Physical Security, University of Florida, FL. Wang, J., Zhang, C. (2008). Nonlinear frac ture mechanics of fl exural-shear crack induced debonding of FRP strengthened concrete beams. International Journal of Solids and Structures 45(10), 2916-2936. Yuan, H., Teng, J.G., Seraci no, R., Wu, Z.S., Yao, J. (2004). Full-range behavior of FRP-to-concrete bonded joints. Engineering Structures 26(5), 553-565. PAGE 89 BIOGRAPHICAL SKETCH Graham Long was born in The Pas, Mani toba, Canada. In 2000, he joined the Canadian Forces and attended the Royal M ilitary College of Canada under a military sponsored officer program. Upon graduation in May 2004, he earned hi s Bachelor of Engineering degree in Civil Engineering and obtained his officer s commission and begun serving as a military engineer. In 2006, he was deployed to Af ghanistan as a part of a construction engineering support unit. Upon his return, he was posted to Canadian Forces Base Suffield as a member of the base infrastr ucture engineering branch. In 2010, he was awarded a postgraduate scholarship from th e Canadian Forces to pursue a masters degree in civil engineering at the University of Florida, specializing in the field of protective structures. Upon completion of his masters pr ogram, Graham will be assigned to the Engineering Support Unit in Moncton, New Brunswick, Canada. 89 |