A Method for the Stress Analysis of Lap Joints

A theory is presented for the adhesive stresses in single and double lap joints under tensile loading, while subjected to thermal stress. The formulation includes the effects of bending, shearing, stretching and hygrothermal deformation in both the adherend and adhesive. All boundary conditions, including shear stress free surfaces, are satisfied. The method is general and therefore applicable to a range of material properties and joint configurations including metal-to-metal, metal-to-CFRP or CFRP-to-CFRP. The solution is numerical and is based on an equilibrium finite element approach. Through the use of an iterative procedure, the solution has been extended to cater for non-linear adhesive materials.     

Interlaminar Stress Distribution Within an Adhesive Layer in the Nonlinear Range

The solution for interlaminar stress distribution in an adhesive layer within a bonded structure is highly complex due to the predominantly inelastic nature of the polymeric adhesive. The present study attacks this problem using a nonlinear finite-element method and an empirical effective-stress-strain relationship derived experimentally for a typical epoxy adhesive loaded at a constant strain rate. The stress distribution in the critical zone of the interlaminar layer of a symmetric doubler model is presented.     

Interlaminar Stress Distribution Within an Adhesive Layer in the Nonlinear Range

The solution for interlaminar stress distribution in an adhesive layer within a bonded structure is highly complex due to the predominantly inelastic nature of the polymeric adhesive. The present study attacks this problem using a nonlinear finite-element method and an empirical effective-stress-strain relationship derived experimentally for a typical epoxy adhesive loaded at a constant strain rate. The stress distribution in the critical zone of the interlaminar layer of a symmetric doubler model is presented.     

Interfacial stresses in FRP-plated RC beams: Effect of adherend shear deformations

A recently popular method for retrofitting reinforced concrete (RC) beams is to bond fibre reinforced polymer (FRP) plates to their tensile faces. An important failure mode of such plated beams is the debonding of the FRP plates from the concrete due to high level of stress concentration in the adhesive at the ends of the FRP plate. This paper presents an improved solution for interfacial stresses in a concrete beam bonded with the FRP plate by including the effect of the adherend shear deformations. The analysis is based on the deformation compatibility approach where both the shear and normal stresses are assumed to be invariant across the adhesive layer thickness. In the present theoretical analysis, the adherend shear deformations are taken into account by assuming a parabolic shear stress through the thickness of both the concrete beam and the bonded plate. 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.     

A shear-lag model for functionally graded adhesive anchors

A shear-lag model for stress transfer through an adhesive layer of variable stiffness joining an anchor rod and the concrete is presented and the effect of such an inhomogeneous bondline on interfacial shear stress distribution in comparison with that of a homogeneous bondline anchor subjected to monotonic axial tension is investigated. A closed-form solution is presented for arbitrary distribution of shear stiffness of the bondline considering both bonded and debonded embedded-end conditions of the anchor. Subsequently, the specific cases of linear and constant distribution of stiffness are discussed in detail, and it is shown how the general solution can be simplified for these examples. For validation, the distribution of shear stress along the bondline for the aforementioned cases is compared with that of equivalent axisymmetric Finite Element (FE) models and the results are found to be in good agreement. The theoretical solution developed can be readily used to evaluate the pull-out performance of post-installed adhesive anchors. Variable stiffness adhesive interfaces deserve an interest in practical applications either to estimate the effect of loss of interface stiffness, due to degradation of the adhesive material, or to engineer the interface with optimum distribution of stiffness so as to maximize the structural performance of bonded systems.     

An Analytical Model of the Mechanical Behaviour of Elastic Adhesively Bonded Joints

Contact and adhesion between two elastic solids bonded by a thin elastic adhesive are considered. The peculiar case of a rigid flat cylindrical punch being adhered to an elastic layer is considered. In order to evaluate the influence of adhesive thickness on the mechanical behaviour of the bonded solids, an approximate solution of the contact problem is derived, using the Ritz method. Comparison of the analytical solution with FE calculations shows a good convergence towards the asymptotic value when the ratio of the punch radius to the layer thickness is greater than 4.

Numerical simulation reveals a strong effect of the adhesive compressibility, for a thin adhesive. This effect is assessed quantitatively from a double asymptotic expansion of the analytical solution. Lastly, the rupture stress is evaluated as a function of the adhesive thickness, from the knowledge of the solution of the contact problem: the asymptotic value for a very thin adhesive is determined independently.     

An Analytical Model of the Mechanical Behaviour of Elastic Adhesively Bonded Joints

Contact and adhesion between two elastic solids bonded by a thin elastic adhesive are considered. The peculiar case of a rigid flat cylindrical punch being adhered to an elastic layer is considered. In order to evaluate the influence of adhesive thickness on the mechanical behaviour of the bonded solids, an approximate solution of the contact problem is derived, using the Ritz method. Comparison of the analytical solution with FE calculations shows a good convergence towards the asymptotic value when the ratio of the punch radius to the layer thickness is greater than 4.

Numerical simulation reveals a strong effect of the adhesive compressibility, for a thin adhesive. This effect is assessed quantitatively from a double asymptotic expansion of the analytical solution. Lastly, the rupture stress is evaluated as a function of the adhesive thickness, from the knowledge of the solution of the contact problem: the asymptotic value for a very thin adhesive is determined independently.     

Load transfer in adhesive double-sided patch joints

In this work, the application of adhesively bonded joints to connect two structural elements with a double-sided patch is studied. On the basis of the shear lag model, a simple closed-form solution was obtained. The analytical solutions can be used to predict the shear stress in the adhesive and the load transfer between the structural elements and the external patches. The load and shear stress distributions in the adhesively bonded region are presented. For verification of the analytical model, finite element analyses were employed to calculate the load transfer and shear stress for the double-sided patch joint under static tensile loadings. Good agreement was found between the theoretical predictions and numerical results. To obtain a better understanding of the joints, the effects of adhesive thickness, adhesive shear modulus and patch Young's modulus on the load transfer and shear stress distributions were investigated. The results show that the maximum shear stress occurs at the edge of the adhesive. The maximum value of the shear stress increases as the adhesive shear modulus and patch Young's modulus increase and as the adhesive thickness decreases. A more gradual load transfer can be achieved by increasing the adhesive thickness and decreasing the adhesive shear modulus. The simple analytical solution presented in this paper has the advantages of avoiding the numerical difficulties and giving explicit relationship between the stress state and joint parameters. Moreover, from the designer's point of view a closed-form and easy-to-use solution is preferred.     

Shear stress distribution in adhesive layers of a double-lap joint with void or bond separation

In this work, stress distribution in adhesive layers of a double-lap joint subjected to tension and suffering from a void or a partial debond at the adhesive–adherend interface is examined. For symmetric voids, the deduced equilibrium equations are decoupled for better application of boundary conditions at the extreme ends of each adhesive layer. The proposed method of solution has resulted in better estimates on peak shear stress developed in the adhesive layers. The results based on analytical solution are compared with those of finite element findings. Very good agreement is observed between the two. The major difference between stresses stemming from debonds and voids occurs at the edge of the large size defects. For small central defects, it is hardly discernible by the stresses to distinguish the type of defect. Moreover, there appears to be an optimum length to thickness ratio for each adhesive layer which produces minimum peak interfacial shear stress. This value seems to be a function of defect size and location. A double-lap joint shows to experience smaller interfacial shear stresses due to a single void or debond in comparison with a single-lap joint with a similar defect. The peak interfacial shear stress in a double-lap joint suffering from symmetric voids or debonds is still lower than that of a single-lap joint with a single defect of the same size and location.     

A New Coupled Solution for Interfacial Stresses Analysis in a Repaired Beam with an Adhesively Bonded Prestressed FRP Plate

This paper focuses on a new coupling solution for determining the elastic interfacial shear and normal stresses in an adhesive joint between a strengthening plate and a simply supported beam. The mismatch of the curvatures in the beam and plate is considered by including both the effect of the adherend shear deformations and the prestressed laminates model. This new method leads to the coupling of governing differential equations for the interfacial shear and normal stresses. Most of the other solutions in the literature assume that the beam and plate have an equal curvature to uncouple this effect. In this paper, however, a solution is presented to calculate the interfacial stresses of beams strengthened with a prestressed composite plate having a new rigidity model coupled with the shear lag effect, which are neglected by the previous studies. It is found that the present method can predict accurately stresses in the interior and near the ends of the adhesive layer, where the stress fields can be significantly influenced by the edge effects. A parametric study was carried out to show how the stress concentration and distribution are influenced by the dimensions of the adherends and the material properties of the strengthened beam.     

Stress Analysis of Adhesively-bonded Joints Under In-plane Shear Loading

A closed-form stress analysis of an adhesively-bonded lap joint subjected to spatially-varying in-plane shear loading is presented. The solution, while similar to Volkersen's treatment of tension loaded lap joints, is inherently two-dimensional and, in general, predicts a multi-component adhesive shear stress state. A finite difference numerical solution of the derived governing differential equation is used to verify the accuracy of the closed-form solution for a joint of semi-infinite geometry. The stress analysis of a finite-sized doubler is also presented. This analysis predicts the adhesive stresses at the doubler boundaries, and can be performed independently from the complex stress state that would exist due to a patched crack or hole located within the interior of the doubler. The analytical treatment of lap joints under combined tension and shear loading is now simplified since superposition principles allow the stress states predicted by separate shear and tension cases to be added together. Applications and joint geometries are discussed.     

Viscoelastic Analysis of an Adhesive Tubular Joint

The investigations so far available with regard to stress analysis of adhesive joints assume that the adhesive is elastic. In the present analysis the time dependent properties of the adhesive are taken into account by assuming that the adhesive is viscoelastic. The viscoelastic analysis of a tubular joint has been attempted using a prony series fitting for the relaxation modulus of two adhesives. The long term redistribution of the stresses in the adhesive is evaluated using the finite element method.

Viscoelastic analysis of an adhesive tubular joint has been performed for the first time, using the finite element method with a prony series fitting for the relaxation modulus of the adhesive. For a typical epoxy it has been found that not only the elastic stresses are different at different levels but also the viscoelastic response shows considerable variation from one level to another. As large a reduction as 57 % is noticed in the normal stress and an even larger reduction of 62% is noticed in the shear stress over three decades of time.

The technique used in this report for the solution of the long term response can be employed for a study of the short-term response also.     

Strength of adhesive joints with adherend yielding: I. Analytical model

A sandwich element can be isolated in all two-dimensional adhesive joints, thereby simplifying the analysis of strain and stress. An adhesive sandwich model has been developed that accommodates arbitrary loading, a bilinear adherend stress-strain response, and any form of nonlinear adhesive behavior. The model accounts for both the bending deformation and the shear deformation of the adherends. Stress and strain distributions in the adhesive were obtained by solving a system of six differential equations using a finite-difference method. For a sample adhesive sandwich, the adhesive strains and stresses from the new model were compared with those of other models. Finally, the model was coupled with an analytical solution for the detached section of an adhesive joint in peel. The stress and strain distributions in the adhesive and the root curvature of the peel adherend were then compared with finite element results. An accompanying article in this issue uses the model with experimental peel data to investigate the suitability of various adhesive failure criteria.     

Studying the thermoviscoelastic response of the Redux 312 adhesive—Part 2: Modeling

The aim of this work is to propose a numerical model of a bonded joint that takes into account the thermoviscoelastic response of the adhesive to compute the stress distribution in the adhesive as well as in the adherends. The adhesive is assumed to obey a generalized Maxwell model. The effect of temperature is taken into account by considering that the branches of the model change with temperature. The numerical solution is obtained iteratively. The particular case of the Redux 312 adhesive is then considered as an example. The procedure is applied in various cases exhibiting an increasing complexity. Some significant differences are observed with the usual solution obtained within the framework of elasticity if the effects of both temperature and time are taken into account.     

STRENGTH OF ADHESIVE JOINTS WITH ADHEREND YIELDING: I. ANALYTICAL MODEL

A sandwich element can be isolated in all two-dimensional adhesive joints, thereby simplifying the analysis of strain and stress. An adhesive sandwich model has been developed that accommodates arbitrary loading, a bilinear adherend stress-strain response, and any form of nonlinear adhesive behavior. The model accounts for both the bending deformation and the shear deformation of the adherends. Stress and strain distributions in the adhesive were obtained by solving a system of six differential equations using a finite-difference method. For a sample adhesive sandwich, the adhesive strains and stresses from the new model were compared with those of other models. Finally, the model was coupled with an analytical solution for the detached section of an adhesive joint in peel. The stress and strain distributions in the adhesive and the root curvature of the peel adherend were then compared with finite element results. An accompanying article in this issue uses the model with experimental peel data to investigate the suitability of various adhesive failure criteria.     

A closed-form solution to statically indeterminate adhesive joint problems—exemplified on ELS-specimens

A beam/adhesive-layer model is developed. For this model a closed-form solution method applicable to arbitrary boundary conditions is presented. This enables the solution of a large number of practical problems which may be statically indeterminate. The stress state in the adhesive layer and the adherends of the beam/adhesive-layer model is also scrutinized. The method is exemplified in an analysis of the end-loaded split (ELS) specimen, commonly used to determine fracture energies of adhesive layers. The effect of the flexibility of the adhesive layer on the energy release rate and the critical crack length for stable crack growth is examined. Both symmetric and unsymmetric ELS-specimens are studied.     

Elastic analysis and engineering design formulae for bonded joints

The general elastic plane strain problem of adhesively bonded structures which consist of two different adherends is considered. To facilitate a truly general approach the adhesive joint is modelled as an adherend-adhesive sandwich with any combination of tensile, shear and moment loading being applied at the ends of both adherends. A full elastic analysis is presented which calculates the adhesive shear and tensile stresses in the overlap region, this analysis has been validated for a range of load cases using a finite element program. Basic design approaches are outlined and explicit expressions are developed which enable the simple evaluation of the stress distributions in the adhesive overlap. Simplified two parameter design formulae are also produced which accurately describe the peak stresses at the ends of the adhesive overlap in both the transverse and longitudinal shear directions. In all of the analyses the adherends are assumed to behave as linear elastic cylindrically bent plates with the adhesive forming an elastic interlayer between them. In the simplified analyses only one component of adhesive stress is considered, while in the full elastic analysis two components of stress are considered with a consequent increase in the complexity of the required solution method, but also an increase in accuracy over the simplified analyses for a wider range of joint configurations.     

Stress analysis of the adhesive resin layer in a reinforced pin-loaded joint used in glass structures

A reinforced pin-loaded joint used to assemble elements in a tempered glass structure consists of a steel bolt and a steel ring glued to a glass plate through an adhesive resin layer. The stiffness of a typical resin material is generally much lower than the stiffness of steel or glass. This fact leads us to make the assumption that the stress field in the adhesive resin layer is essentially due to the relative rigid displacements of the steel ring with respect to the glass plate. On the basis of this assumption, an analytical solution is obtained for the stresses in the adhesive resin layer. This solution is compared with and validated by the numerical results obtained by the finite element method.     

Elastic Analysis of Adhesive Butt Joints

The investigation of adhesive butt joints using the finite element method of solution indicates a variation of stresses through the thickness of the adhesive in contrast to the existing analyses which assume zero stress variation. Both plane stress and axisymmetric cases have been investigated and the effect of adhesive modulus and thickness on the stress distribution have been considered. It has been shown that there is significant variation of normal and shear stresses across the adhesive thickness especially at the adhesive-air interface.     

The Time-Dependent Mechanical Behaviour of an Interlaminar Layer in a Structural Bonded Model

The main problems involved in the adhesive bonding of structural systems stem from the low stiffness and strength of the interlaminar adhesive layer (IAL), its non-linear time-dependent mechanical behaviour, and its sensitivity to temperature and humidity. The present article, which is a continuation of previous papers on this subject, focuses on the time-dependency of the state of stress and strain within an adhesive layer under mechanical loading up to the non-linear range.

The prediction of the interlaminar stresses and strains was obtained by an iterative numerical procedure, using the finite element method. The solution is based on a non-linear, time-dependent effective loading function derived from an empirical stress-strain relationship of the adhesive material and from a postulated function describing the relationship between the stress and the strain tensors of each element, on the one hand, and their respective effective values. The findings indicate the expected trend of stress decrease and strain increase with time.