Closed-Form High-Order Analysis of RC Beams Strengthened with FRP Strips

A closed-form high-order analytical solution for the analysis of concrete beams strengthened with externally bonded fiber-reinforced plastic (FRP) strips is presented. The model is based on equilibrium and deformations compatibility requirements in and between all parts of the strengthened beam, i.e., the concrete beam, the FRP strip, and the adhesive layer. The governing equations representing the behavior of the strengthened beam, along with the appropriate boundary and the continuity conditions, are derived and solved with closed-form analytical solutions. Comparison of the closed-form high-order model with other simplified approaches, based on one- and two-parameter elastic foundation concepts, is included. It is shown that the current high-order model can be reduced, by omitting the appropriate terms, to the simplified theories. A numerical example of a typical RC beam strengthened with an externally bonded FRP strip is discussed with emphasis on the shear and peeling stress distributions at the edge of the FRP strip. Stress analysis results concerning the edge stresses determined by the high-order model are compared with those determined by the elastic foundation models and finite elements. Finally, a parametric study that characterizes the main parameters governing the magnitude and intensity of the edge stresses is performed. The paper is concluded with a summary and recommendations for the design of the strengthened beam.     

Delamination Failure of RC Beams Strengthened with FRP Strips—A Closed-Form High-Order and Fracture Mechanics Approach

An analytical model for the prediction of the interfacial delamination failure of reinforced concrete (RC) beams strengthened with externally bonded fiber-reinforced plastic strips (FRPs) is presented. The analysis is conducted through a comprehensive stress analysis of the strengthened member and a failure criterion based on fracture mechanics concepts. The stress analysis follows the closed-form high-order approach for the analysis of deformations, stresses, and stress resultants in the multilayered structure. The model is based on equilibrium and compatibility requirements in and between all constituents of the strengthened beam, i.e., the concrete beam, the FRP strip, and the adhesive layer. The governing equations of the bonded and the delaminated regions are derived, and along with a unique set of boundary and continuity conditions that model the cracking of the RC beam, they are solved in a closed form. The results provide the basis for the fracture analysis stage in which a criterion for the initiation and stable or unstable growth of the interfacial delaminations is derived. This criterion is based on the fracture mechanics concept of the elastic energy release rate and replaces the classical stress-based criteria. The energy release rate is evaluated through the path independent J-integral over the stress, deformation, and energy fields determined by the stress analysis. Three numerical examples concerning interfacial delamination triggered by cracking of the concrete and by the stress concentration at the edge of the FRP strip are presented. The emphasis is put on the development of the internal stress resultants in the RC beam and the FRP strip, the stresses at the adhesive layer, and the energy release rate with the growth of the delamination. The paper is concluded with a summary and some recommendations for the design of such strengthened beams.     

Analytical Solution for Fracture Analysis of CFRP Sheet–Strengthened Cracked Concrete Beams

Fiber-reinforced polymer (FRP) composite materials have been widely used in the field of retrofitting. Theoretical analysis of FRP plate- or sheet-strengthened cracked concrete beams is necessary for estimating service reliability of the structural members. In previous studies, the effect of a perfectly bonded FRP plate or sheet was equivalent to a cohesive force acting at the bottom of crack to delay the crack propagation in concrete and reduce the crack width. However, delamination between FRP and cracked beam is inevitable due to interfacial shear stress concentration at the bottom of crack. The intention of this paper is to present an analytical solution for fracture analysis of carbon FRP (CFRP) sheet–strengthened cracked concrete beams by considering both vertical crack propagation in concrete and interfacial debonding at CFRP-concrete interface. The interfacial debonding is modeled as the interfacial shear crack propagation in this paper. Four different stages are discussed after initial cracking state of the concrete. At the first stage, only fictitious crack propagation occurs in the concrete. At the second stage, macrocrack propagates in the concrete without interfacial debonding. At the third stage, both vertical macrocrack propagation in the concrete and horizontal shear crack propagation at the CFRP-concrete interface occur in the strengthened beam. The tensile stress in the CFRP sheet and interfacial shear stress along the span are formulated based on the deformation compatibility condition at the CFRP-concrete interface at this stage. Finally, macroshear crack propagates at the interface until the CFRP sheet is completely peeled out from the beam, and then the member is fractured. The applied load is determined as a function of the referred two crack lengths at different stages. At the beginning, the applied load increases to one peak value with the full propagation of fictitious crack at the first stage. At the third stage, the applied load is improved to another peak value due to the relatively high cohesive effect of the CFRP sheet. Then the two peak values are determined by the Lagrange multiplier method. The validity of the proposed analytical solution is verified with the experimental results and numerical simulations. It can be concluded that the proposed analytical solution can predict the load-bearing capacity of CFRP sheet-strengthened cracked concrete beams with reasonable accuracy.     

Analysis of Scaling and Instability in FRP-Concrete Shear Debonding for Beam-Strengthening Applications

The debonding mode of failure, which is observed in girders strengthened using externally attached fiber-reinforced polymer (FRP) sheets, is studied in this paper. A numerical analysis of the direct-shear response of FRP attached to concrete substrate is performed to study the initiation, formation, and propagation of an interfacial crack between the two adherents. The material response of the bimaterial interface, which includes postpeak softening, is incorporated into the numerical model. The load response obtained numerically is shown to be in close agreement with that determined experimentally from direct shear tests on concrete blocks strengthened with FRP sheets. An instability in the load response is predicted close to failure and the arc-length method is used to obtain the entire load response past the displacement-limit point. The instability in the load response is shown to be a result of snapback, where both the load and the displacement decrease simultaneously. The effect of the bonded length on the stress transfer between the FRP and concrete and on the ultimate failure is also analyzed. It is shown that there is a scaling in the load capacity when the bonded length does not allow for the establishment of the full stress-transfer zone associated with interface crack growth. From the results of the numerical analysis, a fundamental understanding of interfacial crack propagation and instability at failure in concrete members strengthened using externally bonded FRP is developed. Using a simple energy based formulation; it is shown that in strengthened girders, the instability at complete debonding of FRP from concrete translates into an explosive failure associated with a sudden release of energy.     

Interfacial Stresses in Beams and Slabs Bondedwith Thin Plate

A new popular method for retrofitting reinforced concrete beams and slabs is to bond fiber-reinforced plastic (FRP) plates to the soffit. An important failure mode for such strengthened members is the debonding of the FRP plate from the member due to high interfacial stresses near the plate ends. Accurate predictions of the interfacial stresses are a prerequisite for designing against debonding failures. In this paper, a theoretical interfacial stress analysis is presented for simply supported beams and slabs bonded with a thin FRP composite or steel plate and subjected to a uniformly distributed load in combination with a uniform bending moment. The analysis leads to an exact closed-form solution, in which a plane stress model is used for beams and a plane strain model is used for slabs. The salient features of the new analysis include the consideration of nonuniform stress distributions in and the satisfaction of the stress boundary conditions at the ends of the adhesive layer. Numerical results from the present analysis are presented both to demonstrate the advantages of the present solution over existing ones and to illustrate the main characteristics of interfacial stress distributions in beams and slabs.     

In-Plane Nonlinear Buckling Analysis of Deep Circular Arches Incorporating Transverse Stresses

Large discrepancies exist among current classical theories for the in-plane buckling of arches that are subjected to a constant-directed radial load uniformly distributed around the arch axis. Discrepancies also exist between the classical solutions and nonlinear finite-element results. A new theory is developed in this paper for the nonlinear analysis of circular arches in which the nonlinear strain-displacement relationship is based on finite displacement theory. In the resulting variational equilibrium equation, the energy terms due to both nonlinear shear and transverse stresses are included. This paper also derives a set of linearized equations for the elastic in-plane buckling of arches, and presents a detailed analysis of the buckling of deep circular arches under constant-directed uniform radial loading including the effects of shear and transverse stresses, and of the prebuckling deformations. The solutions of the new theory agree very well with nonlinear finite-element results. Various assumptions often used by other researchers, in particular the assumption of inextensibility of the arch axis, are examined. The discrepancies among the current theories are clarified in the paper.     

Effect of End Tapering on Crack-Induced FRP Debonding from the Concrete Substrate

The technique of bonding fiber-reinforced plastic (FRP) plates on the tensile side of concrete members has been proved to be an effective method for structural strengthening. For a RC beam strengthened with FRP plates, failure may occur by concrete cover separation near the plate end, or crack-induced debonding initiated by an opening crack away from the plate end. Experimental investigations have shown that tapering of the FRP plate at its ends is effective in reducing the stress concentrations and increasing the loading for plate end failure to occur. However, with reduced averaged thickness of the tapered plate, crack-induced debonding is also easier to occur. To optimize plate tapering in practical designs, an approach to analyze crack-induced debonding of tapered FRP plates is required. In the present investigation, FRP debonding is first studied with the finite-element method. In the analysis, a three-parameter model is employed for the shear slip relation between concrete and the FRP plate. Based on the findings from FEM analysis, simplifying assumptions are derived and an analytical model is developed for calculating the stresses in the FRP plate and along the concrete-to-FRP interface. The analytical stress distributions show good agreement with those from the FEM analysis. Using the analytical model, the effect of FRP tapering is quantitatively assessed. Also, the effects of various parameters on the ultimate failure load are simulated.     

Cohesive Interface Modeling of Debonding Failure in FRP Strengthened Beams

A theoretical model that incorporates the concept of the cohesive interface approach for the debonding analysis of reinforced concrete beams strengthened with externally bonded fiber reinforced polymer (FRP) strips is presented. The cohesive interface concept is adopted for modeling of the debonding process near the critical adhesive-concrete interface, whereas the adhesive layer itself is modeled as a two-dimensional elastic medium. Thus, the stress and deformation fields within the adhesive layer, the coupling between the shear and normal stresses and, especially, their influence on the tractions across the cohesive interface are taken into account. The nonlinear relations between the tractions and the displacement jumps across the cohesive interface are derived using a potential function and account for the peeling effects and for the coupling between the shear-slip and the peeling-separation laws. Numerical results that examine the capabilities of the model, provide insight into the stability characteristics of the debonding mechanism, and highlight some aspects of the debonding problem are presented. A summary and conclusions close the paper.     

On Edge Stresses Control in Strengthened RC Beams with FRP Strips: Adhesive Layer Profile Effect

A high-order analytical model for the analysis of reinforced concrete (RC) beams strengthened with fiber-reinforced plastic (FRP) strips bonded with adhesive layer of variable thickness is presented. The model is based on the closed-form high-order approach, and it provides the means for the analysis of beams retrofitted with generally curved FRP strips and adhesive layers of arbitrary profile. The analysis is comprehensive and includes the local and overall response of the structure. An emphasis is put on the stress concentration that occurs at the edge of the FRP strip and in many cases leads to brittle and sudden failure of the strengthened member. The field equations and the boundary and continuity conditions are derived using the variational principle of virtual work along with the kinematic relations of small deformations. The governing equations of the generally curved FRP strip include large curvatures and are introduced via coordinate transformation from its local curvilinear system into the global Cartesian one. The derived model is used for the investigation of various adhesive profiles and their influence on the shear and vertical normal stresses at the edges of the FRP strip. The results focus on the stress concentrations involved and reveal that proper design and application of the adhesive profile can significantly reduce the edge stresses, thus preventing the brittle mode of failure. The paper is concluded with a summary and recommendations for the analysis, design, and application of the strengthening process.     

Experimental Investigation and Fracture Analysis of Debonding between Concrete and FRP Sheets

The last few years have witnessed a wide use of externally bonded fiber reinforced polymer (FRP) sheets for strengthening existing reinforced and prestressed concrete structures. The success of this strengthening method relies on the effectiveness of the load-transfer between the concrete and the FRP. Understanding the stress transfer and the failure of the concrete–FRP interface is essential for assessing the structural performance of strengthened beams and for evaluating the strength gain. This paper describes an experimental investigation of the interfacial bond behavior between concrete and FRP. The strain distributions in concrete and FRP are determined using an optical technique known as digital image correlation. The results confirm that the debonding process can be described in terms of crack propagation through the interface between concrete and FRP. The data obtained from the analysis of digital images was used to determine the interfacial material behavior for the concrete–FRP interface (stress versus relative displacement response) and the fracture parameter GF (fracture energy). The instability in the test response at failure is shown to be the result of snapback, which corresponds with the elastic unloading of the FRP as the load carrying ability of the interface decreases with increasing slip.     

Three-Parameter Model for Debonding of FRP Plate from Concrete Substrate

Concrete beams retrofitted with bonded fiber reinforced plastic (FRP) plates often fail by debonding of the plate from the concrete surface. To predict the failure load in design, a proper debonding model is required. As debonding is a nonlinear process involving material softening, it can be analyzed once the interfacial shear (τ) versus sliding (s) relationship is known. Recent experimental results indicate that the simplest τ-s relationship should involve three parameters: the maximum shear stress for debonding to initiate, the initial residual stress right after debonding occurs, and a parameter governing the reduction of shear stress with sliding. In this paper, a FRP debonding model based on these three parameters is developed. The applicability of the model is verified through comparison with experimental results. Through a systematic parametric study, the effect of various material and geometric properties on the debonding process is investigated. Implications to the design of FRP strengthened members are highlighted.     

Experimental Study of Interfacial Shear Stresses in FRP-Strengthened RC Beams

This paper presents the results of an experimental study on the distribution of shear stresses along the interface between concrete and the carbon fiber-reinforced polymer (FRP) in 29 plate-strengthened beams, where the primary test variables are: Clear cover, plate length, plate thickness (area), and compressive strength of concrete. FRP strain measurement was accomplished using either the photographic technique of digital image correlation or a series of electrical-resistance strain gages, both providing similar results. The distribution of shear stresses is found to be smoother than predicted by several analytical expressions available in the literature. Another substantial observation is the existence of a second region of peak stress, occurring near the center of the shear span in all of the beams with longer plate lengths, which the authors believe is associated with the singular application of shear corresponding to the point load, as well as the transition from elastic to plastic behavior occurring in the rebar. Because the overall nature of the stress distribution is sufficiently smooth, it is very reasonable to approximate it as a constant stress.     

Energy-Based Modeling Approach for Debonding of FRP Plate from Concrete Substrate

For concrete beams and slabs strengthened with bonded fiber reinforced plastic (FRP) plates, plate debonding from the concrete substrate is a common failure mode. In this paper, the debonding process is modeled as the propagation of a crack along the concrete/adhesive interface, with frictional shear stress acting behind the crack tip. Crack propagation is taken to occur when the net energy release of the system equals the interfacial fracture energy. The analysis is first performed for the special case with constant shear stress along the debonded interface, and then for the general case with slip softening in the debonded zone. From the results, a direct correspondence between energy-based and strength-based analyses can be established for arbitrary softening behavior along the interface. Specifically, through the proper definition of an effective interfacial shear strength, the conventional strength-based approach can be employed to give the same results as the much more complicated energy-based analysis. Also, based on the relation between the effective shear strength and other material parameters, it is possible to explain the very high interfacial shear stresses observed in experimental measurements. As an application example, distribution of plate stress and interfacial shear stress for the linear softening case is derived. The model results are found to be in good agreement with experimental measurements, showing that the simple linear softening model can describe the debonding process in real material systems.     

Interfacial Behavior and Debonding Failures in FRP-Strengthened Concrete Slabs

It has been demonstrated, through laboratory investigations and various field projects, that the external bonding of fiber- reinforced polymer (FRP) laminates is an effective technique for the structural enhancement of reinforced concrete slabs. In such applications, failure is generally governed by debonding of the FRP laminate. Nevertheless, numerical simulations to date of FRP-strengthened slabs have usually been based on the assumption of full bond between the concrete and FRP. In this study, the interfacial behavior between the FRP laminates and the concrete substrate is accounted for by introducing appropriate bond-slip models for the interface in a nonlinear finite-element analysis of FRP-strengthened two-way slabs. The numerical model is capable of simulating slabs strengthened in shear or in flexure; it can be applied to arbitrary FRP configurations, and can also accommodate both passive as well as prestressed FRP strengthening schemes. Results are presented in terms of load-deflection relationships, ultimate load capacities, failure modes, and interfacial slip and stress distributions. When compared to test results reported in the literature, the analysis is shown to lead to excellent predictions in that, for the entire set of FRP-strengthened specimens considered, the average of the numerical-to-experimental load capacity ratios is 0.966, with a standard deviation of 0.066. Furthermore, in all cases when FRP debonding was observed experimentally, the analysis correctly predicted the mode of failure.     

Stress Transfer and Fracture Propagation in Different Kinds of Adhesive Joints

To effectively and efficiently utilize fiber-reinforced plastic (FRP) laminates (plates or sheets) in strengthening civil infrastructures, a design strategy integrating the properties of FRP reinforcement and composite structural behavior needs to be adopted. The interfacial stress transfer behavior including debonding should be considered to be one of the most important effects on the composite structural behavior. In this paper, two kinds of nonlinear interfacial constitutive laws describing the pre- and postinterfacial microdebonding behavior are introduced to solve the nonlinear interfacial stress transfer and fracture propagation problems for different kinds of adhesive joints in FRP/steel-strengthened concrete or steel structures. Expressions for the maximum transferable load, interfacial shear stress distribution, and initiation and propagation of interfacial cracks (debonding) are derived analytically. In addition, numerical simulations are performed to discuss the factors influencing the interfacial behavior and the theoretical derivations are validated by finite-element analysis.     

Seismic Response of Interior RC Beam-Column Joints Upgraded with FRP Sheets. II: Analysis and Parametric Study

In this paper a procedure for analytical prediction of joint shear strength of interior beam-column joints, strengthened with externally bonded fiber-reinforced polymer (FRP) sheets, has been presented. The procedure is based on the formulation available in the literature. To implement the available formulation for shear capacity prediction a computer program has been developed. Using this program shear capacity of the joint and joint shear stress variation at various stages of loading have been predicted and compared with experimental observations; presented in Part I of this study. Predictions show good agreement with experimental test results. The formulation is further extended to predict diagonal tensile stresses in the joint. The effectiveness of FRP quantity on joint shear strength and on various strains has been studied on parametric basis. It is observed that even a low quantity of FRP can enhance shear capacity of the joint significantly and its effectiveness can be further increased if debonding is suppressed (e.g., through mechanical anchorages). Effect of column axial load on shear strength of the joint has also been studied. It is observed that axial load increases the confinement of the joint core, which in turn increases the shear capacity of the joint.     

Rational Approach to Shear Design in Fiber-Reinforced Polymer-Prestressed Concrete Structures

The use of plasticity-based shear design methods for fiber-reinforced polymer (FRP) reinforced and prestressed concrete, as they are used at present, is inappropriate in the long term. In particular, the use of a plasticity-based truss model for shear behavior seems to be unsound, as reliance is placed on a predominantly elastic zone to redistribute stresses. A better approach to shear design would be to employ a model incorporating force equilibrium and compatibility of strains so that the elastic properties of the FRP could be included rationally. This would help to develop a real understanding and form a basis on which new guides and codes could be founded. In tandem with a more rational analytical approach, new configurations and types of FRP reinforcement need to be developed and researched so that these materials can be used more efficiently. An analytical approach to investigate the shear response of FRP-reinforced and -prestressed concrete has been developed, based on equilibrium and compatibility across a shear discontinuity. The analytical model presented here was developed in conjunction with an experimental program. Correlation between the analytical and experimental results is good and more accurate than the current guideline provisions for concrete beams containing FRP reinforcement.     

Front and Side View Image Correlation Measurements on FRP to Concrete Pull-Off Bond Tests

Understanding the transfer of force by bond between externally bonded fiber-reinforced polymer (FRP) reinforcement and concrete is an important step in formulating good models for predicting debonding failures observed in externally bonded reinforcement strengthened systems. In this paper, a 3D optical displacement measurement system was used to capture the full-field displacements from the front and side view in pull-off bond specimens. The experiments were carried using six specimens with carbon FRP (CFRP) strips having different axial stiffnesses but a constant bond length to the concrete substrate. Using the optical measurements, it was possible to obtain the in-plane displacement or slip and the out-of-plane displacement or separation between the CFRP strip and the concrete. It was demonstrated, that the usual assumption of pure shear stresses in such pull-off tests is not true and that the bond behavior is a two-dimensional problem involving shear and peeling stresses. The bond behavior in CFRP strip to concrete pull-off tests was characterized by three stages: (1) the initiation of the first crack; (2) the initiation of debonding; and (3) failure by complete debonding. Based on the test results it was found that there was a dependency between the maximum bond shear stress, the maximum fracture energy of the FRP-concrete interface, and the stiffness of the FRP. However, the slip values after initiation of debonding (Stage 2) were independent of the FRP stiffness. The measured anchorage force and anchorage length were in good agreement with predictions from existing code equations.     

Development of the Nonlinear Bond Stress–Slip Model of Fiber Reinforced Plastics Sheet–Concrete Interfaces with a Simple Method

A new analytical method for defining the nonlinear bond stress–slip models of fiber reinforced plastics (FRP) sheet–concrete interfaces through pullout bond test is proposed. With this method, it is not necessary to attach many strain gauges on the FRP sheets for obtaining the strain distributions in FRP as well as the local bond stresses and slips. Instead, the local interfacial bond stress-slip models can be simply derived from the relationships between the pullout forces and loaded end slips. Based on a series of pullout tests, the bond stress–slip models of FRP sheet–concrete interfaces, in which different FRP stiffness, FRP materials (carbon FRP, aramid FRP, and glass FRP), and adhesives are used, have been derived. Only two parameters, the interfacial fracture energy and interfacial ductility index, which can take into account the effects of all interfacial components, are necessary in these models. Comparisons between analytical results and experimental ones show good accordance, indicating the reliability of the proposed method and the proposed bond stress–slip models.     

Investigation of Bond in Concrete Structures Strengthened with Near Surface Mounted Carbon Fiber Reinforced Polymer Strips

Fiber reinforced polymer (FRP) materials are currently produced in different configurations and are widely used for the strengthening and retrofitting of concrete structures and bridges. Recently, considerable research has been directed to characterize the use of FRP bars and strips as near surface mounted reinforcement, primarily for strengthening applications. Nevertheless, in-depth understanding of the bond mechanism is still a challenging issue. This paper presents both experimental and analytical investigations undertaken to evaluate bond characteristics of near surface mounted carbon FRP (CFRP) strips. A total of nine concrete beams, strengthened with near surface mounted CFRP strips were constructed and tested under monotonic static loading. Different embedment lengths were used to evaluate the development length needed for effective use of near surface mounted CFRP strips. A closed-form analytical solution is proposed to predict the interfacial shear stresses. The model is validated by comparing the predicted values with test results as well as nonlinear finite element modeling. A quantitative criterion governing the debonding failure of near surface mounted CFRP strips is established. The influence of various parameters including internal steel reinforcement ratio, concrete compressive strength, and groove width is discussed.