Analysis of Fiber-Reinforced Polymer Composite Grid Reinforced Concrete Beams

Transcription

1 ACI STRUCTURAL JOURNAL Title no. 100-S27 TECHNICAL PAPER Analysis of Fiber-Reinforced Polymer Composite Grid Reinforced Concrete Beams by Federico A. Tavarez, Lawrence C. Bank, and Michael E. Plesha This study focuses on the use of explicit finite element analysis tools to predict the behavior of fiber-reinforced polymer (FRP) composite grid reinforced concrete beams subjected to four-point bending. Predictions were obtained using LS-DYNA, an explicit finite element program widely used for the nonlinear transient analysis of structures. The composite grid was modeled in a discrete manner using beam and shell elements, connected to a concrete solid mesh. The load-deflection characteristics obtained from the simulations show good correlation with the experimental data. Also, a detailed finite element substructure model was developed to further analyze the stress state of the main longitudinal reinforcement at ultimate conditions. Based on this analysis, a procedure was proposed for the analysis of composite grid reinforced concrete beams that accounts for different failure modes. A comparison of the proposed approach with the experimental data indicated that the procedure provides a good lower bound for conservative predictions of load-carrying capacity. Keywords: beam; composite; concrete; fiber-reinforced polymer; reinforcement; shear; stress. INTRODUCTION In recent years, research on fiber-reinforced polymer (FRP) composite grids has demonstrated that these products may be as practical and cost-effective as reinforcements for concrete structures. 1-5 FRP grid reinforcement offers several advantages in comparison with conventional steel reinforcement and FRP reinforcing bars. FRP grids are prefabricated, noncorrosive, and lightweight systems suitable for assembly automation and ideal for reducing field installation and maintenance costs. Research on constructability issues and economics of FRP reinforcement cages for concrete members has shown the potential of these reinforcements to reduce life-cycle costs and significantly increase construction site productivity. 6 Three-dimensional FRP composite grids provide a mechanical anchorage within the concrete due to intersecting elements, and thus no bond is necessary for proper load transfer. This type of reinforcement provides integrated axial, flexural, and shear reinforcement, and can also provide a concrete member with the ability to fail in a pseudoductile manner. Continuing research is being conducted to fully understand the behavior of composite grid reinforced concrete to commercialize its use and gain confidence in its design for widespread structural applications. For instance, there is a need to predict the correct failure mode of composite grid reinforced concrete beams where there is significant flexural-shear cracking. 7 This type of information is critical for the development of design guidelines for FRP grid reinforced concrete members. Current flexural design methods for FRP reinforced concrete beams are analogous to the design of concrete beams using conventional reinforcement. 8 The geometrical shape, ductility, modulus of elasticity, and force transfer characteristics of FRP composite grids, however, are likely to be different than conventional steel or FRP bars. Therefore, the behavior of concrete beams with this type of reinforcement needs to be thoroughly investigated. OBJECTIVES The objectives of the present study were: 1) to investigate the ability of explicit finite element analysis tools to predict the behavior of composite grid reinforced concrete beams, including load-deflection characteristics and failure modes; 2) to evaluate the effect of the shear span-depth ratio in the failure mode of the beams and the stress state of the main flexural reinforcement at ultimate conditions; and 3) to develop an alternate procedure for the analysis of composite grid reinforced concrete beams considering multiple failure modes. RESEARCH SIGNIFICANCE The research work presented describes the use of advanced numerical simulation for the analysis of FRP reinforced concrete. These numerical simulations can be used effectively to understand the complex behavior and phenomena observed in the response of composite grid reinforced concrete beams. In particular, this effort provides a basis for the understanding of the interaction between the composite grid and the concrete when large flexural-shear cracks are present. As such, alternate analysis and design techniques can be developed based on the understanding obtained from numerical simulations to ensure the required capacity in FRP reinforced concrete structures. Background Several researchers have studied the viability of threedimensional FRP grids to reinforce concrete members. 3,5,9,10 One specific type of three-dimensional FRP reinforcement is constructed from commercially manufactured pultruded FRP profiles (also referred to as FRP grating cages). Figure 1 shows a schematic of the structural members present in a concrete beam reinforced with the three-dimensional FRP reinforcement investigated in this study. A pilot experimental and analytical study was conducted by Bank, Frostig, and Shapira 3 to investigate the feasibility of developing three-dimensional pultruded FRP grating cages to reinforce concrete beams. Failure of all beams tested occurred due to rupture of the FRP main longitudinal reinforcement in the shear span of the beam. Experimental results also revealed that most of the deflection at high loads appeared to occur due to localized rotations at large flexural crack widths ACI Structural Journal, V. 100, No. 2, March-April MS No received March 27, 2002, and reviewed under Institute publication policies. Copyright 2003, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion will be published in the January-February 2004 ACI Structural Journal if received by September 1, ACI Structural Journal/March-April 2003

2 Federico A. Tavarez is a graduate student in the Department of Engineering Physics at the University of Wisconsin-Madison. He received his BS in civil engineering from the University of Puerto Rico-Mayagüez and his MSCE from the University of Wisconsin. His research interests include finite element analysis, the use of composite materials for structural applications, and the use of discrete element methods for modeling concrete damage and fragmentation under impact. ACI member Lawrence C. Bank is a professor in the Department of Civil and Environmental Engineering at the University of Wisconsin-Madison. He received his PhD in civil engineering and engineering mechanics from Columbia University in He is a member of ACI Committee 440, Fiber Reinforced Polymer Reinforcement. His research interests include FRP reinforcement systems for structures, progressive failure of materials and structural systems, and durability of FRP materials. Michael E. Plesha is a professor in the Engineering Mechanics and Astronautics Program in the Department of Engineering Physics at the University of Wisconsin- Madison. He received his PhD from Northwestern University in His research interests include finite element analysis, discrete element analysis, dynamics of geologic media, constitutive modeling of geologic discontinuity behavior, soil structure interaction modeling, and continuum modeling of jointed saturated rock masses. developed in the shear span near the load points. The study concluded that further research was needed to obtain a better understanding of the stress state in the longitudinal reinforcement at failure to predict the correct capacity and failure modes of the beams. Further experimental tests on concrete beams reinforced with three-dimensional FRP composite grids were conducted to investigate the behavior and performance of the grids when used to reinforce beams that develop significant flexural-shear cracking. 7 Different composite grid configurations were designed to study the influence of the FRP grid components (longitudinal bars, vertical bars, and transverse bars) on the load-deflection behavior and failure modes. Even though failure modes of the beams were different depending upon the characteristics of the composite grid, all beams failed in their shear spans. Failure modes included splitting and rupture of the main longitudinal bars and shear-out failure of the vertical bars. Research results concluded that the design of concrete beams with composite grid reinforcements must account for failure of the main bars in the shear span. A second phase of this experimental research was performed by Ozel and Bank 5 to investigate the capacity and failure modes of composite grid reinforced concrete beams with different shear span-to-effective depth ratios. Three different shear spandepth ratios (a/d) were investigated, with values of 3, 4.5, and 6, respectively. 11 The data obtained from this recently completed experimental study was compared with the finite element results obtained in the present study. Experimental studies have shown that due to the development of large cracks in the FRP-reinforced concrete beams, most of the deformation takes place at a relatively small number of cracks between rigid bodies. 12 A schematic of this behavior is shown in Fig. 2. As a result, beams with relatively small shear span-depth ratios typically fail due to rupture of the main FRP longitudinal reinforcement at large flexural-shear cracks, even though they are over-reinforced according to conventional flexural design procedures. 5,7,13,14 Due to the aforementioned behavior for beams reinforced with composite grids, especially those that exhibit significant flexural-shear cracking, it is postulated that the longitudinal bars in the member are subjected to a uniform tensile stress distribution, plus a nonuniform stress distribution due to localized rotations at large cracks, which can be of great importance in determining the ultimate flexural strength of the beam. The present study investigates the stress-state at the flexuralshear cracks in the main longitudinal bars, using explicit finite element tools to simulate this behavior and determine the conditions that will cause failure in the beam. ACI Structural Journal/March-April 2003 Fig. 1 Structural members in composite grid reinforced concrete beam. Fig. 2 Deformation due to rotation of rigid bodies. Numerical analysis of FRP composite grid reinforced beams Implicit finite element methods are usually desirable for the analysis of quasistatic problems. Their efficiency and accuracy, however, depend on mesh topology and severity of nonlinearities. In the problem at hand, it would be very difficult to model the nonlinearities and progressive damage/ failure using an implicit method, and thus an explicit method was chosen to perform the analysis. 15 Using an explicit finite element method, especially to model a quasistatic experiment as the one presented herein, can result in long run times due to the large number of time steps that are required. Because the time step depends on the smallest element size, efficiency is compromised by mesh refinement. The three-dimensional finite element mesh for this study was developed in HyperMesh 16 and consisted of brick elements to represent the concrete, shell elements to represent the bottom longitudinal reinforcement, and beam elements to represent the top reinforcement, stirrups, and cross rods. Figure 3 shows a schematic of the mesh used for the models developed. Beams with span lengths of 2300, 3050, and 3800 mm were modeled corresponding to shear span-depth ratios of 3, 4.5, and 6, respectively. These models are referred to herein as short beam, medium beam, and long beam, respectively. The cross-sectional properties were identical for the three models. As will be seen later, the longitudinal bars play an important role in the overall behavior of the system, and therefore they were modeled with greater detail than the rest of the reinforcement. The concrete representation consisted of 8-node solid elements with dimensions 25 x 25 x 12.5 mm (shortest dimension parallel to the width of the beam), with one-point integration. The mesh discretization was established so that the reinforcement nodes coincided with the concrete nodes. The reinforcement mesh was connected to the concrete mesh by shared nodes between the concrete and the 251

3 Fig. 3 Finite element model for composite grid reinforced concrete beam. Fig. 4 Short beam model at several stages in simulation. reinforcement. As such, a perfect bond is assumed between the concrete and the composite grid. The two-node Hughes-Liu beam element formulation with 2 x 2 Gauss integration was used for modeling the top longitudinal bars, stirrups, and cross rods in the finite element models. In this study, each model contains two top longitudinal bars with heights of 25 mm and thicknesses of 4 mm. The models also have four cross rods and three vertical members at each stirrup location, as shown in Fig. 3. The vertical members have a width of 38 mm and a thickness of 6.4 mm. The cross rod elements have a circular cross-sectional area with a diameter of 12.7 mm. To model the bottom longitudinal reinforcement, the four-node Belytschko- Lin-Tsay shell element formulation was used, as shown in Fig. 3, with two through-the-thickness integration points. Boundary conditions and event simulation time To simulate simply supported conditions, the beam was supported on two rigid plates made of solid elements. The finite element simulations were displacement controlled, which is usually the control method for plastic and nonlinear behavior. That is, a displacement was prescribed on the rigid loading plates located on top of the beam. The prescribed displacement was linear, going from zero displacement at t = 0.0 s to 60, 75, and 90 mm at t = 1.0 s for the short, medium, and long beams, respectively. The corresponding applied load due to the prescribed displacement was then determined by monitoring the vertical reaction forces at the concrete nodes in contact with the support elements. The algorithm CONTACT_AUTOMATIC_SINGLE_ SURFACE in LS-DYNA was used to model the contact 252 ACI Structural Journal/March-April 2003

4 between the supports, load bars, and the concrete beam. This algorithm automatically generates slave and master surfaces and uses a penalty method where normal interface springs are used to resist interpenetration between element surfaces. The interface stiffness is computed as a function of the bulk modulus, volume, and face area of the elements on the contact surface. The finite element analysis was performed to represent quasistatic experimental testing. As the time over which the load is applied approaches the period of the lowest natural frequency of vibration of the structural system, inertial forces become more important in the response. Therefore, the load application time was chosen to be long enough so that inertial effects would be negligible. The flexural frequency of vibration was computed analytically for the three beams using conventional formulas for vibration theory. 17 Accordingly, it was determined that having a load application time of 1.0 s was sufficiently long so that inertial effects are negligible and the analysis can be used to represent a quasistatic experiment. For the finite element simulations presented in this study, the CPU run time varied approximately from 22 to 65 h (depending on the length of the beam) for 1.0 s of load application time on a 600 MHz PC with 512 MB RAM. Material models Material Type 72 (MAT_CONCRETE_DAMAGE) in LS-DYNA was chosen for the concrete representation in the present study. This material model has been used successfully for predicting the response of standard uniaxial, biaxial, and triaxial concrete tests in both tension and compression. The formulation has also been used successfully to model the behavior of standard reinforced concrete dividing walls subjected to blast loads. 18 This concrete model is a plasticitybased formulation with three independent failure surfaces (yield, maximum, and residual) that change shape depending on the hydrostatic pressure of the element. Tensile and compressive meridians are defined for each surface, describing the deviatoric part of the stress state, which governs failure in the element. Detailed information about this concrete material model can be found in Malvar et al. 18 The values used in the input file corresponded to a 34.5 MPa concrete compressive strength with a 0.19 Poisson s ratio and a tensile strength of 3.4 MPa. The softening parameters in the model were chosen to be 15, 50, and 0.01 for uniaxial tension, triaxial tension, and compression, respectively. 19 The longitudinal bars were modeled using an orthotropic material model (MAT_ENHANCED_COMPOSITE_DAMAGE), which is material Type 54 in LS-DYNA. Properties used for this model are shown in Table 1. Because the longitudinal bars were drilled with holes for cross rod connections, the tensile strength in the longitudinal direction of the FRP bars was taken from experimental tensile tests conducted on notched bar specimens with a 12.7 mm hole to account for stress concentration effects at the cross rod locations. The tensile properties in the transverse direction were taken from tests on unnotched specimens. 11 Values for shear and compressive properties were chosen based on data in the literature. The composite material model uses the Chang/Chang failure criteria. 20 The remaining reinforcement (top longitudinal bars, stirrups, and cross rods) was modeled using two-noded beam elements using a linear elastic material model (MAT_ELASTIC) with the same properties used for the longitudinal direction in the bottom FRP longitudinal bars. A rigid material model Fig. 5 Experimental and finite element load-deflection results for short, medium, and long beams. Fig. 6 Typical failure of composite grid reinforced concrete beam (Ozel and Bank 5 ). Table 1 Material properties of FRP bottom bars E x 26.7 GPa X t MPa E y 14.6 GPa Y t MPa G xy 3.6 GPa S c 6.9 MPa ν xy 0.26 X c MPa β 0.5 Y c MPa (MAT_RIGID) was used to model the supports and the loading plates. FINITE ELEMENT RESULTS AND DISCUSSION Graphical representations of the finite element model for the short beam at several stages in the simulation are shown in Fig. 4. The lighter areas in the model represent damage (high effective plastic strain) in the concrete material model. As expected, there is considerable damage in the shear span of the concrete beam. Figure 4 also shows the behavior of the composite grid inside the concrete beam. All displacements in the simulation graphics were amplified using a factor of 5 to enable viewing. Actual deflection values are given in Fig. 5, which shows the applied load versus midspan deflection behavior for the short, medium, and long beams for the experimental and LS-DYNA results, respectively. The jumps in the LS-DYNA curves in the figure represent the progressive tensile and shear failure in the concrete elements. As shown in this figure, the ultimate load value from the finite element model agrees well with the experimental result. The model slightly over-predicts the stiffness of the beam, however, and under-predicts the ultimate deflection. The significant drop in load seen in the load-deflection curves produced in LS-DYNA is caused by failure in the ACI Structural Journal/March-April

5 Fig. 7 Medium beam model at several stages in simulation. Fig. 8 Long beam model at several stages in simulation. longitudinal bars, as seen in Fig. 4. The deformed shape seen in this figure indicates a peculiar behavior throughout the length of the beam. It appears to indicate that after a certain level of damage in the shear span of the model, localized rotations occur in the beam near the load points. These rotations create a stress concentration that causes the longitudinal bars to fail at those locations. This deflection behavior was also observed in the experimental tests. Figure 6 shows a typical failure in the longitudinal bars from the experiments conducted on these beams. 11 As shown in this figure, there is considerable damage in the shear span of the member. Large shear cracks develop in the beam, causing the member to deform in the same fashion as the one seen in the finite element model. Figure 7 shows the medium beam model at several stages in the simulation. The figure also shows the behavior of the main longitudinal bars. Comparing this simulation with the one obtained for the short beam, it can be seen that the shear damage is not as significant as in the previous simulation. The deflected shape seen in the longitudinal bars shows that this model does not have the abrupt changes in rotation that were observed in the short beam, which would imply that this model does not exhibit significant flexural-shear damage. For this model, the finite element analysis slightly over-predicted both the stiffness and the ultimate load value obtained from the experiment. On the other hand, the ultimate deflection was under-predicted. Failure in this model was also caused by rupture of the longitudinal bars at a location near the load points. In the experimental test, failure was caused by a combination of rupture in the longitudinal bars as well as concrete crushing in the compression zone. This compressive failure was located near the load points, however, and could have been initiated by cracks formed due to stress concentrations produced by the rigid loading plates. 11 Figure 8 shows the results for the long beam model. Comparing this simulation with the two previous ones, it can be seen that this model exhibits the least shear damage, as expected. As a result, the longitudinal bars exhibit a parabolic shape, which would be the behavior predicted using conventional moment-curvature methods based on the curvature of the member. Once again, the stiffness of the beam was slightly over-predicted. However, the ultimate load 254 ACI Structural Journal/March-April 2003

6 Table 2 Summary of experimental and finite element results Fig. 9 (a) Tensile force in longitudinal bars; and (b) internal moment in longitudinal bars. value compares well with the experimental result. Failure in the model was caused by rupture of the longitudinal bars. Failure in the experimental test was caused by a compression failure at a location near one of the load application bars, followed by rupture of the main longitudinal bars. Figure 5 also shows the time at total failure for each beam, which can be related to the simulation stages given in Fig. 4, 7, and 8 for the short, medium, and long beam, respectively. To investigate the stress state of a single longitudinal bar at ultimate conditions, the tensile force and the internal moment of the longitudinal bars at the failed location for the three finite element models was determined, as shown in Fig. 9(a) and (b). It is interesting to note that for the short beam model, the tensile force at failure was approximately 51.6 kn, while for the medium beam model and the long beam model the tensile force at failure was approximately 76.5 kn. On the other hand, the internal moment in the short beam model was approximately 734 N-m, while the internal moment was approximately 339 N-m for both the short beam model and the long beam model. It is clear that the shear damage in the short beam model causes a considerable localized effect in the stress state of the longitudinal bars, which is important to consider for design purposes. According to Fig. 9(a), the total axial load in the longitudinal bars for the short beam model produces a uniform stress of 130 MPa, which is not enough to fail the element in tension at this location. However, the ultimate internal moment produces a tensile stress at the bottom of the longitudinal bars of 141 MPa. The sum of these two components produces a tensile stress of 271 MPa. When this value is entered in the Chang/Chang failure criterion for the tensile longitudinal direction, the strength is exceeded and the elements fail. Using conventional over-reinforced beam analysis formulas, the tensile force in the longitudinal bars at midspan would be obtained by dividing the ultimate moment obtained from the experimental test by the internal moment arm. This would imply that there is a uniform tensile force in each longitudinal bar of 88.1 kn. This tensile force is never achieved in the finite element simulation due to considerable shear damage in the concrete elements. As a result of this shear damage in the concrete, the curvature at the center of the beam is not large enough to produce a tensile force in the bars of this magnitude (88.1 kn). The internal moment in the longitudinal bars shown in Fig. 9(b), however, continues to develop, resulting in a total failure load comparable to the experimental result. As mentioned before, the force in the bars according to the simulation was approximately 51.6 kn, which is approximately half the load predicted using conventional methods. Therefore, the use of conventional beam analysis formulas to analyze this composite grid reinforced beam would not only erroneously predict the force in the longitudinal bars, but it would also predict a concrete Beam Experimental Total load capacity, kn Flexural analysis Finite element analysis Tensile force in each main bar, kn Flexural analysis Finite element analysis Short Medium Long compression failure mode, which was not the failure mode observed from the experimental tests. The curves for the medium beam model and the long beam model, shown in Fig. 9, show that for both cases, the beam shear span-depth ratio was sufficiently large so that the stress state in the longitudinal bars would not be greatly affected by the shear damage produced in the beam. As such, the ultimate axial force obtained in the longitudinal bars for both models was close to the ultimate axial load that would be predicted by using conventional methods. In summary, Table 2 presents the ultimate load capacity for the three models, including experimental results, conventional flexural analysis results, and finite element results. As shown in this table, conventional flexural analysis under-predicts the actual ultimate load carried by the beams and a better ultimate load prediction was obtained using finite element analysis. The tensile load in the bars was computed (analytically) by dividing the experimental moment capacity by the internal moment arm computed by using strain compatibility. Although the finite element results over-predicted the ultimate load for the medium and long beams, the simulations provided a better understanding of the complex phenomena involved in the behavior of the beams, depending on their shear span-depth ratio. The results for tensile load in the bars reported in this table suggest that composite grid reinforced concrete beams with values of shear span-depth ratio greater than 4.5 can be analyzed by using the current flexural theory. It is important to mention that the concrete material model parameters that govern the post-failure behavior of the material played a key role in the finite element results for the three finite element models. In the concrete material formulation, the elements fail in an isotropic fashion and, therefore, once an element fails in tension, it cannot transfer further shear. Because the concrete elements are connected to the reinforcement mesh, this behavior causes the beam to fail prematurely as a result of tensile failure in the concrete. Therefore, the parameters that govern the post-failure behavior in the concrete material model were chosen so that when an element fails in tension, the element still has the capability to transfer shear forces and the stresses will gradually decrease to zero. Because the failed elements can still transfer tensile stresses, however, the modifications caused an increase in the stiffness of the beam. In real concrete behavior, when a crack opens, there is no tension transfer between the concrete at that location, causing the member to lose stiffness as cracking progresses. Regarding shear transfer, factors such as aggregate interlock and dowel action would contribute to transfer shear forces in a concrete beam, and tensile failure in the concrete would not affect the response as directly as in the finite element model. ACI Structural Journal/March-April

7 Stress analysis of FRP bars As discussed previously, failure modes observed in experimental tests performed on composite grid reinforced concrete beams suggest that the longitudinal bars are subjected to a uniform tensile stress plus a nonuniform bending stress due to localized rotations at locations of large cracks. This section presents a simple analysis procedure to determine the stress conditions at which the longitudinal bars fail. As a result of this analysis, a procedure is presented to analyze/design a composite grid reinforced concrete beam, considering a nonuniform stress state in the longitudinal bars. A more detailed finite element model of a section of the longitudinal bars was developed in HyperMesh 16 using shell elements, as shown in Fig. 10. A height of 50.8 mm was specified for the bar model, with a thickness of 4.1 mm. The length of the bar and the diameter of the hole were 152 and 12.7 mm, respectively. The material formulation and properties were the same as the ones used for the longitudinal bars in the concrete beam models, with the exception that now the unnotched tensile strength of the material (X t = 521 MPa) was used as an input parameter because the hole was incorporated in the model. The finite element model was first loaded in tension to establish the tensile strength of the notched bar. The load was applied by prescribing a displacement at the end of the bar. Figure 10 shows the simulation results for the model at three stages, including elastic deformation and ultimate failure. As expected, a stress concentration developed on the boundary of the hole causing failure in the web of the model, followed by ultimate failure of the cross section. A tensile strength of 274 MPa was obtained for the model. A value of 267 MPa was obtained from experimental tests conducted on notched bars (tensile strength used in Table 2), demonstrating good agreement between experimental and finite element results. A similar procedure was performed to establish the strength of the bar in pure bending. That is, displacements were prescribed at the end nodes to induce bending in the model. Figure 11 shows the simulation results for the model at three stages, showing elastic bending and ultimate failure caused by flexural failure at the tension flange. As shown in this figure, the width of the top flange was modified to prevent buckling in the flange (which was present in the original model). Because buckling would not be present in a longitudinal bar due to concrete confinement, it was decided to modify the finite element model to avoid this behavior. To maintain an equivalent cross-sectional area, the thickness of the flange was increased. A maximum pure bending moment of 2.92 kn-m was obtained for the model. Knowing the maximum force that the bar can withstand in pure tension and pure bending, the model was then loaded at different values of tension and moment to cause failure. This procedure was performed several times to develop a tensionmoment interaction diagram for the bar, as shown in Fig. 12. The discrete points shown in the figure are combinations of tensile force and moment values that caused failure in the finite element model. This interaction diagram can be used to predict what combination of tensile force and moment would cause failure in the FRP longitudinal bar. Considerations for design The strength design philosophy states that the flexural capacity of a reinforced concrete member must exceed the flexural demand. The design capacity of a member refers Fig. 10 Failure on FRP bar subjected to pure tension. Fig. 11 Failure on FRP bar subjected to pure bending. to the nominal strength of the member multiplied by a strengthreduction factor φ, as shown in the following equation φm n For FRP reinforced concrete beams, a compression failure is the preferred mode of failure, and, therefore, the beam should be over-reinforced. As such, conventional formulas are used to ensure that the selected cross-sectional area of the longitudinal bars is sufficiently large to have concrete compression failure before FRP rupture. Considering a concrete compression failure, the capacity of the beam is computed using the following 8 Experimental tests have shown, however, that there is a critical value of shear V s crit in a beam where localized rotations due to large flexural-shear cracks begin to occur. The ultimate moment in the beam is assumed to be related to this shear-critical value and it is determined according to the following equation where n is the number of longitudinal bars. Once the beam has reached the shear-critical value, it is assumed (conservatively) that the tensile force t, which is the force in each bar at the shear-critical stage, remains constant and any additional load is carried by localized internal moment m in the longitudinal bars. Furthermore, it is assumed that at this stage the concrete is still in its elastic range, and, therefore, the internal moment arm i e can be determined by equilibrium and elastic strain compatibility. The tensile force t in Eq. (5) is computed M u M n = A f f f d a 2 -- a = A f f f β 1 f c b β f f E f ε 1 d a = cu a M n = n ( t i e + m) (1) (2) (3) (4) (5) 256 ACI Structural Journal/March-April 2003

8 Table 3 Summary of results for three beams using proposed approach Beam Experimental ultimate shear, kn Theoretical shear critical, kn Equation for moment capacity Total load capacity, kn Experimental Analytical P n = M n /a s Tension in each main bar, kn Short M n = t i e + m Medium M n = A f f f (d a/ 2) Long M n = A f f f (d a/2) according to the following equation for a simply supported beam in four-point bending t = V crit s a s ni e (6) where a s is the shear span of the member. The obtained value for the tension t in each bar is then entered in Eq. (7), which is the equation for the interaction diagram, to determine the ultimate internal moment m in Eq. (5) that causes the bar to fail. In this equation, t max and m max are known properties of the notched composite bar. t m = m max for t > 0; m > 0 t max (7) The aforementioned procedure is a very simplified analysis to determine the capacity of a composite grid reinforced concrete beam, and, as can be seen, it depends considerably on the shearcritical value V s crit established for the beam. This value is somewhat difficult to determine. Based on experimental data, a value given by Eq. (8) (analogous to Eq. (9-1) of ACI 440.1R-01) can be considered to be a lower bound for FRP reinforced beams with shear reinforcement. V crit s = 7ρ f E f 1 90β 1 f c 6 -- f c bd where f c is the specified compressive strength of the concrete in MPa. In summary, the ultimate moment capacity in the beam is determined according to one of the following equations M n A f f f d a 2 -- crit = for Vult < V s crit M n = n ( t i e + m) for V ult > V s (8) (9) (10) According to Eq. (9), if the ultimate shear force computed analytically based on conventional theory does not exceed the shear-critical value V s crit, the moment capacity can be computed from flexural analysis. On the other hand, if the computed ultimate shear force is greater than V s crit, Eq. (10) is used. Table 3 presents a summary showing the load capacity for the three beams obtained experimentally and analytically using the present approach. As shown in this table, the equation used to determine the flexural capacity depends on the ultimate shear obtained for each beam. As seen in this procedure, the only difficulty in applying these formulas is the fact that an equation needs to be determined Fig. 12 Tension-moment interaction diagram for longitudinal bar. to compute the maximum moment that the bar can carry as a function of the tensile force acting in the bar. If a specific bar is always used, however, this difficulty is eliminated, and if the flexural demand is not exceeded, a higher capacity can be obtained by increasing the number of longitudinal bars in the section. According to the results obtained for the three beams analyzed herein, the proposed procedure will under-predict the capacity of the composite grid reinforced concrete beam, but it will provide a good lower bound for a conservative design. Furthermore, it will ensure that the longitudinal bars will not fail prematurely as a result of the development of large flexural-shear cracks in the member, and thus the member will be able to meet and exceed the flexural demand for which it was designed. CONCLUSIONS Based on the explicit finite element results and comparison with experimental data, the following conclusions can be made: 1. Failure in the FRP longitudinal bars occurs due to a combination of a uniform tensile stress plus a nonuniform stress caused by localized rotations at large flexural-shear cracks. Therefore, this failure mode has to be accounted for in the analysis and design of composite grid reinforced concrete beams, especially those that exhibit significant flexuralshear cracking; 2. The shear span for the medium beam and the long beam studied was sufficiently large so that the stress state in the longitudinal bars was not considerably affected by shear damage in the beam. Therefore, the particular failure mode observed by the short beam model is only characteristic of ACI Structural Journal/March-April

9 beams with a low shear span-depth ratio. Moreover, according to the proposed analysis for such systems, both the medium beam and the long beam could be designed using conventional flexural theory because the shear-critical value was never reached for these beam lengths; 3. Numerical simulations can be used effectively to understand the complex behavior and phenomena observed in the response of composite grid reinforced concrete beams and, therefore, can be used as a complement to experimental testing to account for multiple failure modes in the design of composite grid reinforced concrete beams; and 4. The proposed method of analysis for composite grid reinforced concrete beams considering multiple failure modes will under-predict the capacity of the reinforced concrete beam, but it will provide a good lower bound for a conservative design. These design considerations will ensure that the longitudinal bars will not fail prematurely (or catastrophically) as a result of the development of large flexural-shear cracks in the member, and thus the member can develop a pseudoductile failure by concrete crushing, which is more desirable than a sudden FRP rupture. ACKNOWLEDGMENTS This work was supported by the National Science Foundation under Grant. No. CMS Javier Malvar and Karagozian & Case are thanked for providing information regarding the concrete material formulation used in LS-DYNA. Jim Day, Todd Slavik, and Khanh Bui of Livermore Software Technology Corporation (LSTC) are also acknowledged for their assistance in using the finite element software, as well as Strongwell Chatfield, MN, for producing the custom composite grids. NOTATION a = depth of equivalent rectangular stress block a s = length of shear span in reinforced concrete beam b = width of rectangular cross section d = distance from extreme compression fiber to centroid of tension reinforcement E f = modulus of elasticity for FRP bar E x = modulus of elasticity in longitudinal direction of FRP grid material E y = modulus of elasticity in transverse direction of FRP grid material G xy = shear modulus of FRP grid members f c = specified compressive strength of concrete f f = stress in FRP reinforcement in tension i e = internal moment arm in the elastic range M n = nominal moment capacity m = internal moment in longitudinal FRP grid bars n = number of longitudinal FRP grid bars S c = shear strength of FRP grid material t = tensile force in a longitudinal bar at the shear critical stage V crit s = critical shear resistance provided by concrete in FRP grid reinforced concrete V ult = ultimate shear force in reinforced concrete beam X c = longitudinal compressive strength of FRP grid material X t = longitudinal tensile strength of FRP grid material Y c = transverse compressive strength of FRP grid material Y t = transverse tensile strength of FRP grid material β = weighting factor for shear term in Chang/Chang failure criterion β 1 = ratio of the depth of Whitney s stress block to depth to neutral axis ε cu = concrete ultimate strain ρ f = FRP reinforcement ratio ν xy = Poisson s ratio of FRP grid material REFERENCES 1. 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