Ultimate behaviour of adhesively bonded FRP lap joints

This paper deals with a numerical investigation on double-lap and symmetrical single-lap joints subjected to shear/bending moment and axial force. The analysis has been developed using the theoretical model proposed by the author in [Ascione F. Mechanical behaviour of FRP adhesive joints: a theoretical model; 2007].The mechanical behavior of the adhesive is modeled through two sets of independent interfacial springs capable of characterizing the normal and transversal interactions, respectively. The adherents are modeled following the hypotheses of the beam technical theory. The mathematical model is based on two fundamental hypotheses: the possibility to separate the shear-flexure problem from the extensional one; the total fracture energy is additionally broken down in a term relative to mode I of fracture (opening) and in a term relative to mode II of fracture (sliding).Five dimensionless parameters which influence the design problem of the joints are identified. Several examples of the ultimate domains of the interface between the adherents are also presented as well as comparisons with some results reported in literature.     

Key features of flexure hinges used as rotational joints

This article is supposed to serve as a guide for the design of flexure hinges that act as rotational joints. Firstly, flexure hinges with concentrated and distributed compliance are reviewed. They can be modeled by linear beam theories or by the theory of Elastica, respectively. Secondly, the transition between these limit cases is investigated by finite element methods (FEM). A planar symmetric flexure hinge with a circular notch serves as an exemplary geometry. By extending the notch the compliance is distributed. The deflection curves and the kinetics of desired and parasitic motions are chosen as key features to be studied. The corresponding results are compressed into a pseudo-rigid-body model (PRBM) approximation for a range of geometries. It turned out that the concentrated compliance matches best with an ideal rotational joint, but even for small displacements large stresses occur so that its range of operation is small. Distributing the compliance increases the range of operation, however stiffness within the task space decreases dramatically so that the design of a flexure hinge becomes a tradeoff between the two concurring goals large stiffness and large range of operation.     

Glued-in-rod timber joints: analytical model and finite element simulation

In this paper, a closed form analytical solution for glued-in-rod (GiR) joints is derived by solving the governing differential equations and accurately applying the boundary conditions in a cylindrical coordinate system for a GiR joint comprising of a rod, adhesive (glue) and timber. The results of the analytical model are compared with 3D continuum finite element simulations and it is shown that the closed-form solution developed can estimate the stress distribution in the adhesive and adherents (rod and timber) with good accuracy. Furthermore, the stiffness of GiR timber joints can be obtained from this analytical model. Closed-form solutions for pull–pull and pull–push test setup configurations are compared and it is shown that the maximum shear stress in the adhesive-adherent interface in a pull–push configuration is around 20% higher than that of the pull–pull counterpart. The typically (around 20%) lower strength of GiR joints in pull–push experiments compared to that of pull–pull tests can be attributed to this higher maximum shear stress which is predicted by the analytical model. A parametric study is carried out using the FE and analytical models and the effects of different variables on the distribution of stresses in the adhesive and adherents are studied.     

MOST EFFICIENT NLP TECHNIQUES IN OPTIMUM STRUCTURAL FRAME DESIGN

This paper identifies the two most efficient non-linear programming techniques for the solution of optimization problems of plane structural frames under multiple load systems. The problem is one of minimizing the mass or the frame subject to constraints on normal and shear stresses, the maximum transverse deflection and buckling load. In all, five different NLP techniques are tried, and it is found that the sequential linear and sequential convex programming techniques are the most efficient for the solution of the class of NLP problems under consideration     

Punch and crack problems in transversely isotropic bodies

An exact solution is proposed for the mixed boundary-value problem in a transversely isotropic half-space. Here, certain arbitrary shear tractions are prescribed inside a circular region, outside of which certain arbitrary tangential displacements are given. The normal stresses are supposed to be known all over the boundary. A particular case is considered, in detail, where normal stresses vanish all over the boundary with the shear tractions vanishing inside the circular region. A closed form expression is obtained for the tangential displacements inside the circular region directly through the displacements outside. As an example, a penny-shaped crack in an infinite transversely isotropic body is considered with arbitrary shear tractions acting on both sides of the crack. The formulae for the tangential displacements inside the circle and the shear stresses outside are obtained. Special cases where uniform shear and a concentrated tangential force arise are also discussed.     

Modeling of composite/concrete interface of RC beams strengthened with composite laminates

In this paper a finite element model for predicting shear and normal stresses in the adhesive layer of plated reinforced concrete beams has been developed. The numerical results carried out agree with those obtained in previous studies by other authors. It is found that shear stresses and high concentrations of peeling forces are present at the ends of the plates when the composite beam is loaded in flexure. These concentrations can produce premature failure of the strengthened beam because of debonding of the plate or cracking of the concrete cover along the level of internal steel reinforcement. The numerical simulation captures the actual interfacial stresses and, in particular, the maximum values of shear and normal stresses.     

Warping shear stresses in nonuniform torsion by BEM

 In this paper a boundary element method is developed for the nonuniform torsion of simply or multiply connected prismatic bars of arbitrary cross section. The bar is subjected to an arbitrarily distributed twisting moment, while its edges are restrained by the most general linear torsional boundary conditions. Since warping is prevented, beside the Saint–Venant torsional shear stresses, the warping normal and shear stresses are also computed. Three boundary value problems with respect to the variable along the beam angle of twist and to the primary and secondary warping functions are formulated and solved employing a BEM approach. Both the warping and the torsion constants using only boundary discretization together with the torsional shear stresses and the warping normal and shear stresses are computed. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The magnitude of the warping shear stresses due to restrained warping is investigated by numerical examples with great practical interest. Received: 13 November 2001 / Accepted: 2 October 2002     

Stress-function variational method for stress analysis of bonded joints under mechanical and thermal loads

High interfacial stresses near the ends of adherends are responsible for debonding failure of bonded joints used extensively in structural engineering and microelectronics packaging. This paper proposes a stress-function variational method for determination of the interfacial stresses in a single-sided strap joint subjected to mechanical and thermal loads. During the process, two interfacial shear and normal (peeling) stress functions are introduced, and the planar stresses of adherends of the joints are expressed in terms of the stress functions according to the static equilibrium equations. Two coupled governing ordinary differential equations (ODEs) of the stress functions are obtained through minimizing the complementary strain energy of the joints and solved explicitly in terms of eigenfunctions. The stress field of the joints based on this method can satisfy all the traction boundary conditions (BCs), especially the shear-free condition near the adherend ends. Compared to results based on finite element method (FEM) and other analytic methods in the literature, the present variational method is capable of predicting highly accurate interfacial stresses. Dependencies of the interfacial stresses upon the adherend geometries, moduli and temperature are examined. Results gained in this study are applicable to scaling analysis of joint strength and examination of solutions given by other methods. The present formalism can be extended conveniently to mechanical and thermomechanical stress analysis of other bonded structures such as adhesively bonded joints, composite joints, and recently developed flexible electronics, among others.     

An interface crack in an inhomogeneous stress field

The solutions for interface cracks of shear, opening and mixed modes (problems A, B, C) are obtained in elementary functions. In case C the crack surfaces partially overlap, slipping without friction. After removing the inhomogeneous stress field specified at the infinity the crack surfaces are loaded by normal and tangential stresses distributed according to a polynomial law. A detailed analysis is carried out for the solution corresponding to linear variation of these stresses. It develops that the strain-energy release rate varies slowly when going from problem C to problem A under compression-shear-bending of the piecewise-homogeneous plane, and from problem C to problem B when compression is replaced by tension. There is also a slow variation in the largest of moduli of the stress intensity factors with the elastic parameters. These results allow one to estimate the structure-toughness characteristics for inhomogeneous bodies by solving problems in simplified formulations.     

Interlaminar stresses in laminated composite plates, cylindrical/spherical shell panels damaged by low-velocity impact

Prediction of damage caused by low-velocity impact in laminated composite plate cylindrical/spherical shell panels is an important problem faced by designers using composites. Not only the in-plane stresses but also the interlaminar normal and shear stresses play a role in estimating the damage caused. The work reported here is an effort in getting better predictions of damage in composite plate cylindrical/spherical shell panels subjected to low-velocity impact.

The low-velocity impact problem is treated as a quasi-static problem. First, the in-plane stresses are calculated by 2-D nonlinear finite element analysis using a 48 degrees of freedom laminated composite shell element. The damage analysis is then carried out using a Tsai-Wu quadratic failure criterion and a maximum stress criteria. Interlaminar normal and shear stresses are predicted after taking into account the in-plane damage caused by low-velocity impact. The interlaminar stresses are obtained by integrating the 3-D equations of equilibrium through the thickness. The deformed geometry is taken into account in the third equation of equilibrium (in the thickness direction). After evaluating the formulation and the computer program developed for correctness, the interlaminar stresses are predicted for composite plates/shell panels which are damaged by low-velocity impact.     

Linear and non-linear 3-D analysis of hybrid composite laminates

In this paper, linear and non-linear 3-D solutions are presented for hybrid composite laminates subjected to uniform transverse loadings. The perturbation method and a variational principle are used to obtain solutions which satisfy the linear and non-linear 3-D differential equations of equilibrium, the strain/displacement relations, the stress/strain relations, the boundary conditions and the continuity conditions at layer interfaces. The distributions of displacements and stresses in the plates are shown. The effect of different non-linearities on laminated plates is considered, and the importance of transverse shear stresses and transverse normal stress is analysed.     

Sandwich shell finite element for dynamic explicit analysis

The contribution of this paper consists of new development of transverse shear stresses through the thickness and finding an expression for the critical time step for explicit time integration of layered shells. This work presents the finite element (FE) formulation and implementation of a higher‐order shear deformable shell element for dynamic explicit analysis of composite and sandwich shells. The formulation is developed using a displacement‐based third‐order shear deformation shell theory. Using the differential equilibrium equations and the interlayer requirements, special treatment is developed for the transverse shear, resulting in a continuous, piecewise quartic distribution of the transverse shear stresses through the shell thickness. Expressions are developed for the critical time step of the explicit time integration for orthotropic homogeneous and layered shells based on the developed third‐order formulation. To assess the performance of the present shell element, it is implemented in the general non‐linear explicit dynamic FE code DYNA3D. Several problems are solved and results are presented and compared to other theoretical and numerical results. Copyright © 2002 John Wiley & Sons, Ltd.     

An experimental–numerical hybrid technique for three dimensional stress analysis

The paper describes a numerical method for determining the stress distribution in the interior of a three-dimensional body using experimentally determined surface stresses, and the interior displacements from surface displacements. The normal and shear stresses inside the body are obtained by solving Laplace's equation in terms of sum of normal stresses together with the three-dimensional compatibility equations in terms of stresses, using the finite difference technique, when the stresses on the surface of the body are known. On the other hand if surface displacements are known (from which strain components could be determined) then displacement components in the interior of a body can be determined by solving Laplace's equation in terms of sum of normal strains together with the three-dimensional equilibrium equations in terms of displacements. It is shown that axi-symmetric problems can also be solved in an identical way by transforming the equations into cylindrical co-ordinates. The application of the method has been illustrated through several examples.     

An efficient FE model and Least Square Error method for accurate calculation of transverse shear stresses in composites and sandwich laminates

Accurate evaluation of transverse stresses in laminated composites and sandwich plates using 2D FE models involves cumbersome post-processing techniques. In this paper a simple and efficient method has been proposed for accurate evaluation of through-the-thickness distribution of transverse stresses in composites and sandwich laminates by using a displacement based C⁰ FE model (2D) derived from Refined Higher Order Shear Deformation Theory (RHSDT) and a Least Square Error (LSE) method. The C⁰ FE model satisfies the inter-laminar shear stress continuity conditions at the layer interfaces and zero transverse shear stress conditions at the top and bottom of the plate. In this model the first derivatives of transverse displacement have been treated as independent variables to circumvent the problem of C¹ continuity associated with the above plate theory (RHSDT). The LSE method is applied to the 3D equilibrium equations of the plate problem at the post-processing stage, after in-plane stresses are calculated by using the above FE model based on RHSDT. Thus the proposed method is quite simple and elegant compared to the usual method of integrating the 3D equilibrium equations at the post-processing stage for calculation of transverse stresses in a composite laminate. In the proposed method, the first two equations of equilibrium are utilized to compute the transverse shear stress variation through the thickness of a laminated plate whereas the third equation of equilibrium gives the normal stress variation. Accuracy of the proposed method is demonstrated in the numerical examples through comparison of the present results with those obtained from different models based on higher order shear deformation theory (HSDT) and 3D elasticity solutions.     

Stress analysis of steel beams strengthened with a bonded hygrothermal aged composite plate

In this paper, the problem of interfacial stresses in steel beams strengthened with bonded hygrothermal aged composite laminates is analyzed using linear elastic theory. The analysis is based on the deformation compatibility approach developed by Tounsi (Int. J. Solids Struct. 43:4154–4174, 2006) where both the shear and normal stresses are assumed to be invariant across the adhesive layer thickness. The adopted model takes into account the adherend shear deformations by assuming a linear shear stress through the depth of the steel beam. This solution is intended for application to beams made of all kinds of materials bonded with a thin composite plate. For steel I-beam section, a geometrical coefficient ξ is determined to show the effect of the adherend shear deformations. This research is helpful for the understanding on mechanical behaviour of the interface and design of such structures.     

A global/local third-order Hermitian displacement field with damaged interfaces and transverse extensibility: analytical formulation

A third-order Hermitian zig-zag plate theory is presented as development of the classical cubic zig-zag displacement field. In addition to the capabilities of the previous model ((i) transverse shear flexibility, (ii) through-the-thickness continuity of the transverse shear stresses, (iii) traction-free condition on the two external surfaces of the laminate and (iv) possibility to study damaged interfaces), the Hermitian model offers some interesting improvements ((i) through-the-thickness linear varying transverse displacement, (ii) evaluation of the normal transverse deformability in general and of the corresponding normal stress in particular, (iii) traction equilibrium condition on the external surfaces and (iv) use of the displacements and transverse shear stresses of the external surfaces as degrees of freedom of the plate model). By means of the virtual work principle of the three-dimensional linear elasticity theory, the two-dimensional equations of motion and boundary conditions are obtained. Some numerical results are finally presented to show the particular nature of the through-the-thickness Hermitian shape functions and to test the model performances in evaluating the transverse normal stress.     

De Saint-Venant flexure-torsion problem handled by Line Element-less Method (LEM)

In this paper, the De Saint-Venant flexure-torsion problem is developed via a technique by means of a novel complex potential function analytic in all the domain whose real and imaginary parts are related to the shear stresses. The latter feature makes the complex analysis enforceable for the shear problem. Taking full advantage of the double-ended Laurent series involving harmonic polynomials, a novel element-free weak form procedure, labelled Line Element-less Method (LEM), is introduced, imposing that the square of the net flux across the border is minimized with respect to expansion coefficients. Numerical implementation of the LEM results in systems of linear algebraic equations involving positive-definite and symmetric matrices solving only contour integrals. Some numerical applications are reported to assess not only the efficiency and accuracy of the method to handle shear stress problems but also the robustness in the sense that exact solutions when available are captured straight away.     

Indentation of functionally graded beams and its application to low-velocity impact response

The problem of contact between a rigid cylindrical indenter and a functionally graded (FG) beam is studied. The elastic modulus of the material varies in an exponential fashion across the thickness of the beam. For the sake of comparison indentation of a homogeneous beam is also considered. In the case of FG beams indentation of both soft and hard sides of the beam are studied. Results are presented for contact force–contact length relations and contact stresses in the three types of beams. Maximum normal strains and stresses and maximum transverse shear stresses are plotted as a function of strain energy (work done by the indenter) in the beam. The results are extended to low-velocity impact problems. It is seen that for a given impact energy in low-velocity impacts, the maximum stresses and strains are significantly lower in FG beams when the impact occurs on the softer side of the beam.     

Delamination in composite plates: influence of shear deformability on interfacial debonding

An elastic interface model is introduced to investigate the effects of in-plane and out-plane shear stresses on interfacial debonding in laminated composite plates by means of the energy release rate concept. This is done by utilising an improved laminated plate model in which the Reissner–Mindlin kinematics type for each layers is coupled with an adhesion mechanism modelled by means of a linear interface model, acting in the opening and sliding failure mode directions. The problem is faced through an analytical solution procedure. Increasing the stiffnesses of the interface leads to restoring displacement continuity at the interface between layers and to recovering energy release rate components through the work performed by the singular stress field at the crack tip. In view of the great importance of shear deformation in laminated composite plates the effect of shear stresses on the mechanism of delamination are investigated pointing out new features which emerge from the interaction of normal and shear stresses acting on the transverse section near the crack tip. Several examples of mixed mode delamination schemes used in experimental applications are examined, showing the influence of transverse shear stresses in coupling with normal stresses on energy release rates determination.     

Out-of-plane behavior of unreinforced masonry walls strengthened with FRP strips

The out-of-plane behavior of unreinforced masonry walls strengthened with externally bonded fiber reinforced polymer (FRP) strips is analytically studied. The analytical model uses variational principles, equilibrium requirements, and compatibility conditions between the structural components (masonry units, mortar joints, FRP strips, and adhesive layers) and assumes one-way flexural action of the strengthened wall. The masonry units and the mortar joints are modeled as Timoshenko’s beams. The FRP strips are modeled using the lamination and the first-order shear deformation theories, and the adhesive layers are modeled as 2D linear elastic continua. The model accounts for cracking of the mortar joints and for the development of debonding zones near the cracked joints. Numerical and parametric studies that reveal the capabilities of the model, throw light on the interaction between the variables, and quantitatively explain some aspects of the behavior of the strengthened wall are also presented.