A three-parameter elastic foundation model for interface stresses in curved beams externally strengthened by a thin FRP plate

Externally bonding of fiber reinforced polymer (FRP) plates or sheets has become a popular method for strengthening reinforced concrete structures. Stresses along the FRP–concrete interface are of great importance to the effectiveness of this type of strengthening because high stress concentration along the FRP–concrete interface can lead to the FRP debonding from the concrete beam. In this study, we develop an analytical solution of interface stresses in a curved structural beam bonded with a thin plate. A novel three-parameter elastic foundation model is used to describe the behavior of the adhesive layer. This adhesive layer model is an extension of the two-parameter elastic foundation commonly used in existing studies. It assumes that the shear stress in the adhesive layer is constant through the thickness, and the interface normal stresses along two concrete/adhesive and adhesive/FRP interfaces are different. Closed-form solutions are obtained for these two interfacial normal stresses, shear stress within the adhesive layer, and beam forces. The validation of these solutions is confirmed by finite element analysis.     

FRP-to-concrete interfaces between two adjacent cracks: Theoretical model for debonding failure

External bonding of fibre reinforced polymer (FRP) composites has become a popular technique for strengthening concrete structures all over the world. The performance of the interface between FRP and concrete is one of the key factors affecting the behaviour of the strengthened structure. Existing laboratory research has shown that the majority of reinforced concrete (RC) beams strengthened with a bonded FRP soffit plate fail due to debonding of the plate from the concrete. Two types of debonding failures have been commonly observed: plate end debonding and intermediate crack induced debonding. In order to understand and develop methods to predict such debonding failures, the bond behaviour between concrete and FRP has been widely studied using simple shear tests on FRP plate/sheet-to-concrete bonded joints and a great deal of research is now available on the behaviour of these bonded joints. However, for intermediate crack induced debonding failures, the debonding behaviour can be significantly different from that observed in a simple shear test. Among other factors, the most significant difference may be that the FRP plate between two adjacent cracks is subject to tension at both cracks. This paper presents an analytical solution for the debonding process in an FRP-to-concrete bonded joint model where the FRP plate is subject to tension at both ends. A realistic bi-linear local bond-slip law is employed. Expressions for the interfacial shear stress distribution and the load–displacement response are derived for different loading stages. The debonding process is discussed in detail. Finally, results from the analytical solution are presented to illustrate how the bond length affects the behaviour of such bonded joints. While the emphasis of the paper is on FRP-to-concrete joints, the analytical solution is equally applicable to similar joints between thin plates of other materials (e.g. steel and aluminium) and concrete.     

Coupled mixed-mode cohesive zone modeling of interfacial debonding in simply supported plated beams

The development of predictive models for plate end debonding failures in beams strengthened with thin soffit plates is a topic of great practical relevance. After the early stress-based formulations, fracture mechanics approaches have become increasingly established. More recently, the cohesive zone (CZ) model has been successfully adopted as a bridge between the stress- and fracture mechanics-based treatments. However, the few studies of this nature propose complex formulations which can only be implemented numerically. To date, the only available analytical solution based on CZ modeling for the prediction of interfacial stresses/debonding in plated beams is limited to the determination of interfacial shear stresses and thus neglects the mixed-mode effects generated by the presence of interfacial normal stresses at the plate end. This paper presents a new analytical formulation based on the CZ modeling approach for the prediction of plate end debonding in plated beams. A key enhancement with respect to the previous solution is the use of a coupled mixed-mode CZ model, which enables a full account of mixed-mode effects at the plate end. The model describes the evolution of the interface after the end of the elastic regime, and predicts the value of the load at incipient debonding. The achievement of a closed-form solution for this quite complex case entails the introduction of a crucial simplifying assumption, as well as the ad hoc modeling of an effective cohesive interfacial response. The paper presents the analytical theory and compares its predictions with numerical and experimental results.     

Interfacial stress analysis of a thin plate bonded to a rigid substrate and subjected to inclined loading

The so-called peel test, in which a thin plate bonded to a substrate is subjected to an inclined pulling force, has been widely used to characterise the bond behaviour of adhesives. This paper presents an analytical solution for the interfacial normal and shear stresses in such a peel test to provide an improved understanding of its underlying mechanism. An approximate closed-form solution is also presented. The effect of the peel angle (i.e. the angle between the applied force and the substrate) on the interfacial stresses is discussed. Apart from being a widely used test for quantifying adhesive characteristics, the process of debonding in a peel test resembles that of intermediate flexural-shear or shear crack induced debonding in flexurally strengthened RC members, where a relative vertical displacement exists between the two sides of the crack, leading to an angle between the external plate and the concrete substrate. Therefore, the results of this study also offer some insight into the latter failure mode which is very important in the flexural strengthening design of RC members.     

Debonding of FRP-plated reinforced concrete beam,a bond-slip analysis. I. Theoretical formulation

External bonding of FRP plates or sheets has emerged as a popular method for strengthening reinforced concrete. Debonding along the FRP–concrete interface can lead to premature failure of the structure. In this study, a bond-slip model is established to study the interface debonding induced by a flexural crack in a FRP-plated concrete beam. The reinforced concrete beam and FRP plate are modeled as two linearly elastic Euler–Bernoulli beams bonded together through a thin layer of FRP–concrete interface. The interface layer is essentially modeled as a large fracture processing zone of which the stress–deformation relationship is described by a nonlinear bond-slip model. Three different bond-slip models (bi-linear, triangular and linear-damaging) are used. By dividing the debonding process into several stages, governing equations of interfacial shear and normal stresses are obtained. Closed-form solutions are then obtained for the interfacial shear and normal stresses and the deflection of the beam in each stage of debonding. In such a way, the proposed model unifies the whole debonding process, including elastic deformation, debonding initiation and growth, into one model. With such a superior feature, the proposed model provides an efficient and effective analytical tool to study FRP–concrete interface debonding.     

Cohesive zone modeling of interfacial stresses in plated beams

The elastic analysis of interfacial stresses in plated beams has been the subject of several investigations. These studies provided both first-order and higher-order solutions for the distributions of interfacial shear and normal stresses close to the plate end in the elastic range. The notable attention devoted to this topic was driven by the need to develop predictive models for plate end debonding mechanisms, as the early models of this type adopted debonding criteria based on interfacial stresses. Currently, approaches based on fracture mechanics are becoming increasingly established. Cohesive zone modeling bridges the gap between the stress- and energy-based approaches. While several cohesive zone analyses of bonded joints subjected to mode-II loading are available, limited studies have been conducted on cohesive zone modeling of interfacial stresses in plated beams. Moreover, the few available studies present complex formulations for which no closed-form solutions can be found. This paper presents an analytical cohesive zone model for the determination of interfacial stresses in plated beams. A first-order analysis is conducted, leading to closed-form solutions for the interfacial shear stresses. The mode-II cohesive law is taken as bilinear, as this simple shape is able to capture the essential properties of the interface. A closed-form expression for the debonding load is proposed, and the comparison between cohesive zone modeling and linear-elastic fracture mechanics predictions is discussed. Analytical predictions are also compared with results of a numerical finite element model where the interface is described with zero-thickness contact elements, using the node-to-segment strategy and incorporating decohesion and contact within a unified framework.     

Analytical solution for a strained reinforcement layer bonded to lip-shaped crack under remote mode Ⅲ uniform load and concentrated load

In this paper,the analytical solution of stress field for a strained reinforcement layer bonded to a lip-shaped crack under a remote mode Ⅲ uniform load and a concentrated load is obtained explicitly in the series form by using the technical of conformal mapping and the method of analytic continuation.The effects of material combinations,bond of interface and geometric configurations on interfacial stresses generated by eigenstrain,remote load and concentrated load are studied.The results show that the stress concentration and interfacial stresses can be reduced by rational material combinations and geometric configurations designs for different load forms.     

Analytical solution for a strained reinforcement layer bonded to lip-shaped crack under remote mode III uniform load and concentrated load

In this paper, the analytical solution of stress field for a strained reinforcement layer bonded to a lip-shaped crack under a remote mode III uniform load and a concentrated load is obtained explicitly in the series form by using the technical of conformal mapping and the method of analytic continuation. The effects of material combinations, bond of interface and geometric configurations on interfaciai stresses generated by eigenstrain, remote load and concentrated load are studied. The results show that the stress concentration and interfaciai stresses can be reduced by rational material combinations and geometric configurations designs for different load forms.     

Improved theoretical solution for interfacial stresses in concrete beams strengthened with FRP plate

In this paper, an improved theoretical interfacial stress analysis is presented for simply supported concrete beam bonded with a FRP plate. The adherend shear deformations have been included in the present theoretical analyses by assuming a linear shear stress through the thickness of the adherends, while all existing solutions neglect this effect. Remarkable effect of shear deformations of adherends has been noted in the results. Indeed, the resulting interfacial stresses concentrations are considerably smaller than those obtained by other models which neglect adherent shear deformations. It is shown that both the normal and shear stresses at the interface are influenced by the material and geometry parameters of the composite beam. This research is helpful for the understanding on mechanical behavior of the interface and design of the FRP–RC hybrid structures.     

Impact of thermal loads on interfacial debonding in FRP strengthened beams

The effect of thermal loads on the debonding mechanisms in beams strengthened with externally bonded composite materials is analytically investigated. The analytical approach adopts a high-order stress analysis model and a fracture mechanics model that uses the concept of the energy release rate through the thermo-mechanical form of the J-integral. The two models are combined to synthesize the relation between the energy release rate, the mechanical loads, the thermal loads, and the interfacial crack length simulating the thermo-mechanical debonding process. The model is supported through comparison with experimental results taken from the literature. The comparison quantifies and explains various phenomena observed in the experiments and mainly the non-monotonic dependency of the debonding failure load on the temperature. The impact of the temperature on the interfacial stresses and on the stability of the debonding process is also studied. Finally, the effect of an uniform thermal load on the debonding behavior of a strengthened beam is studied revealing the impact of the thermal load on the debonding stability and strength characteristics.     

Modeling of mixed-mode debonding in the peel test applied to superficial reinforcements

This paper focuses on modeling of the interface between a rigid substrate and a thin elastic adherend subjected to mixed-mode loading in the peel test configuration. The context in which the investigation is situated is the study of bond between fiber-reinforced polymer (FRP) sheets and quasi-brittle substrates, where FRP sheets are used as a strengthening system for existing structures. The problem is approached both analytically and numerically. The analytical model is based on the linear-elastic fracture mechanics energy approach. In the numerical model, the interface is discretized with zero-thickness contact elements which account for both debonding and contact within a unified framework, using the node-to-segment contact strategy. Uncoupled cohesive interface constitutive laws are adopted in the normal and tangential directions. The formulation is implemented and tested using the finite element code FEAP. The models are able to predict the response of the bonded joint as a function of the main parameters, which are identified through dimensional analysis. The main objective is to compute the debonding load and the effective bond length of the adherend, i.e., the value of bond length beyond which a further increase has no effect on the debonding load, as functions of the peel angle. The detailed distributions of interfacial shear and normal stresses are also found. Numerical results and analytical predictions are shown to be in excellent agreement.     

Interfacial stress analysis and prediction of debonding for a thin plate bonded to a curved substrate

This paper focuses on the analytical and numerical modeling of the interface between a rigid substrate with simple constant curvature and a thin bonded plate. The interfacial behavior is modeled by independent cohesive laws in the normal and tangential directions, coupled with a mixed-mode fracture criterion. The newly developed analytical model determines the interfacial shear and normal stress distributions as functions of the substrate curvature, during the various behavioral stages of the interface prior to the initiation of debonding. The model is also able to predict the debonding load and the effective bond length. In the numerical model the interface is modeled by zero-thickness node-to-segment contact elements, in which both the geometrical relationships between the nodes of the discretized problem and the interface constitutive laws are suitably defined. Numerical results and comparisons between the predictions of the two models are presented.     

THE ANALYSIS OF INTERLAMINAR STRESSES FOR COMPOSITE LAMINATED SHALLOW SHELLS WITH INTERFACIAL DAMAGE

Based on the general six-degrees-of-freedom plate theory towards the accurate stress analysis and nonlinear theory of shallow shells, considering the damage effect of the interlaminar interface and using the variation principle, the three-dimensional non-linear equilibrium differential equations of the laminated shallow shells with interfacial damage are derived. Then, considering a simply supported laminated shallow shell with damage and under normal load, an analytical solution is presented by using finite difference method to obtain the interlaminar stresses. Numerical results show, the stiffness of the shell is weakened, greater absolute values of displacements as well as smaller interlaminar stresses are obtained by interfacial damage. When the interfacial damage is further increased, delamination occurs obviously under normal pulling load and pure shear slip occurs under normal pressure load. The portion of the load undertaken by the two sides of the interface is more different. Different mechanical behaviors are shown in both sides of the interface, and the discontinuation of stresses and displacements takes place in the interface.     

Local Interface Shear Fracture of Bonded Steel Joints with Various Bondline Thicknesses

Interfacial fracture is a critical issue for extensive applications of adhesively bonded structures to a variety of modern industries. Extensive global experimental tests have been conducted to measure the global behavior of adhesively bonded joint, such as ultimate load capacity and toughness. Recently, several studies have also been employed to characterize the local interfacial traction–separation laws. However, very few tests have investigated the dependency of the local interfacial constitutive laws on the adhesive thickness, particularly, under Mode-II loading conditions. In this work, six typical adhesive thicknesses (from 0.1 mm to 1.0 mm) are prepared for the bonded joints with a configuration of end notched flexure (ENF) specimen to realize the Mode-II fracture loading (shear fracture). With a recently developed analytical model, the global energy release rates of the ENF specimens are experimentally measured. Meanwhile, with the image analysis technique, the local slips between the two adherends are obtained. Finally, based on the J-integral theory, the local interfacial constitutive laws at different bondline thicknesses are obtained. Several experimental findings are reported in this work. This work may provide valuable baseline experimental data for the input in cohesive zone model (CZM) based analytical and numerical simulations.     

三层薄板的热屈曲测量及特征反演

The paper presents a new study on a tri-layer thin plate.Shadow moiré implemented with an advanced phase unwrapping technique is employed to obtain actual flexural deformation of a real-life plate sample subjected to thermal loads.An analytical model is re-formulated to provide the plate with global closed-form solutions of the plate deflection as well as the interfacial stress and strain.With the measurements and the solutions available,an inverse iterative approach is developed to evaluate and maximize th...     

三层薄板的热屈曲测量及特征反演

The paper presents a new study on a tri-layer thin plate. Shadow moiré implemented with an advanced phase unwrapping technique is employed to obtain actual flexural deformation of a real-life plate sample subjected to thermal loads. An analytical model is re-formulated to provide the plate with global closed-form solutions of the plate deflection as well as the interfacial stress and strain. With the measurements and the solutions available, an inverse iterative approach is developed to evaluate and maximize the correlation between the measured and the predicted thermal flexure, leading to ascertained materials' constitutive and thermal behaviour. The inverse search algorithm starts with estimated ranges of material property parameters and progressively updates them to finally approach the respective true values. The established model solutions along with the optimized material properties matrix enable the prediction of the interfacial stresses/strains for the specific plate sample under evaluation.     

Analytical solution for a reinforcement layer bonded to an elliptic hole under a remote uniform load

A general solution to a reinforced elliptic hole embedded in an infinite matrix subjected to a remote uniform load is provided in this paper. Investigations on the present elasticity problem are rather tedious due to the presence of material inhomogeneities and complex geometric configurations. Based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique, the general expressions of the displacement and stresses in a reinforcement layer and the matrix are derived explicitly in a series form. Some numerical results are provided to investigate the effects of the material combinations and geometric configurations on the interfacial stresses. The results show that there exists an optimum design of a reinforcement layer such that both the magnitude of stress concentration and the interfacial stresses could be fairly reduced.     

A rigid triangular inclusion partially bonded in an elastic matrix

The two-dimensional problem of a rigid rounded-off angle triangular inclusion partially bonded in an infinite elastic plate is studied. The unbonded part of the inclusion boundary forms an interfacial crack. Based on the complex variable method for curvilinear boundaries, the problem is reduced to a non-homogeneous Hilbert problem and the stress and displacement fields in the plate are obtained in closed form. Special attention is paid in the investigation of the stress field in the vicinity of the crack tip. It is found that the stresses present an oscillatory singularity and the general equations for the local stresses are derived. The singular stress field is coupled with the maximum circumferential stress and the minimum strain energy density criteria to study the fracture characteristics of the composite plate. Results are given for the complex stress intensity factors, the local stresses, the crack extension angles and the critical applied loads for unstable crack growth from its more vulnerable tip or two types of interfacial cracks along the inclusion boundary.     

Linear viscoelastic analysis of straight and curved thin-walled laminated composite beams

This paper is devoted to study the behavior, in the range of linear viscoelasticity, of shear flexible thin-walled beam members constructed with composite laminated fiber-reinforced plastics. This work appeals to the correspondence principle in order to incorporate in unified model the motion equations of a curved or straight shear-flexible thin-walled beam member developed by the authors, together with the micromechanics and macromechanics of the reinforced plastic panels. Then, the analysis is performed in the Laplace or Carson domains. That is, the expressions describing the micromechanics and macromechanics of a plastic laminated composites and motion equations of the structural member are transformed into the Laplace or Carson domains where the relaxation components of the beam structure (straight or curved) are obtained. The resulting equations are numerically solved by means of finite element approaches defined in the Laplace or Carson domains. The finite element results are adjusted with a polynomial fitting. Then the creep behavior is obtained by means of a numerical technique for the inverse Laplace transform. Predictions of the present methodology are compared with experimental data and other approaches. New studies are performed focusing attention in the flexural–torsional behavior of shear flexible thin-walled straight composite beams as well as for thin-walled curved beams and frames.