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The paper proposes a mixed strain- and stress-based topology optimization method for designing the ideal geometry of carbon fibers in composite laminates subjected to either applied tractions or prescribed displacements. On the basis of standard micromechanical approaches, analytical elastic solutions for a single cell, assumed to be a Representative Volume Element (RVE), are ad hoc constructed by involving anisotropy induced by fiber orientation and volume fraction, also taking into account inter-laminar stresses and strains. The analytical solutions are then implemented in a Finite Element (FE) custom-made topology optimization-based procedure rewritten to have as output the best curves the reinforcing fibers have to draw in any composite laminate layer to maximize the overall panel stiffness or to minimize the elastic energy. To verify the effectiveness of the proposed strategy, different structures undergoing either in-plane or out-plane boundary conditions have been selected and theoretically investigated, determining the optimal fibers' maps and showing the related results in comparison to standard sequences of alternate fibers disposition for the same composites. Two optimized panels were at the end actually produced using an innovative Automated Fiber Placement (AFP) machine and consolidating the materials by means of autoclave curing processes, in this way replicating the fiber paths obtained from theoretical outcomes. As a control, two corresponding composite structures were also built without employing the fiber optimization strategy. The panels have been tested in laboratory and the theoretical results have been compared with the experimental findings, showing a very good agreement with our predictions and confirming the capability of the proposed algorithm in suggesting the arrangement of the fibers to obtain enhanced mechanical performances. It is felt that the hybrid analytical-FE topology optimization strategy, in conjunction with the possibilities offered by AFP devices, could pave the way for a new generation of ultra-lightweight composites for aerospace, automotive and many industrial applications.

This paper presents a Timoshenko beam finite element for nonlinear analysis of planar masonry arches. Considering small displacement and strain assumption, the element governing equations are defined according to a force-based formulation that adopts three different parametrizations of the axis planar curve, permitting the exact description of the element geometry for arbitrarily curved arches. Specific quadrature techniques are illustrated to perform numerical integration over the curved axis. A two-scale arch-to-beam homogenization procedure reproduces the nonlinear response of periodic masonry materials, where an equivalent Timoshenko straight beam describes the behavior of the reference Unit Cell made of a single linear elastic brick and a nonlinear mortar layer. Formation of the hinges characterizing the collapse mechanism of the arch is detected taking advantage of the quadrature rule along the axis and a fracture energy based regularization technique is employed to avoid damage localization. The proposed curved beam model is implemented in a standard FE analysis code and is used to perform several numerical applications. After validating the proposed formulation through benchmarking tests under linear elastic material response, the numerical simulation of six experimental tests is shown, concerning masonry arches characterized by different shapes and undergoing in-plane bending. The numerical results are validated through experimental outcomes and other FE models.

Aim of the present paper is to develop an efficient multiscale procedure for studying the mechanical response of structural elements made of elastoplastic or viscoplastic composite materials. The micro and the macro scales are considered separated. At the microscale a PieceWise Uniform Transformation Field Analysis (PWUTFA) homogenization technique is adopted to derive the overall response of a periodic composite. Thus, a Unit Cell (UC) containing all the properties of the heterogeneous material is analyzed and divided in subsets; in each one the inelastic strain is considered uniform, i.e. constant, and represents the history variable of the analysis. Elastoplastic and elasto-viscoplastic models with isotropic hardening are adopted in order to describe the nonlinear response of the constituents. A new numerical procedure is developed in order to solve the evolutive problem in all the subsets simultaneously adopting a predictor-corrector technique. The corrector phase is solved by means of a modified Newton-Raphson iterative procedure. Furthermore, the tangent consistent with the algorithm is computed and adopted in the multiscale computations. Numerical applications are carried out in order to assess the efficiency of the proposed multiscale approach.

The paper presents a study concerning the local bond behavior of FRCM strengthening systems externally applied to structural supports. The aim of the study is to analyze the main parameters influencing the interaction between the reinforcement and the mortar layer and their role. To this purpose, a theoretical approach is performed in order to analyze the main phases characterizing the bond behavior of FRCM-strengthening systems. In particular, the attention has been devoted to the upper mortar layer by analyzing its influence on the bond behavior of FRCM-strengthening systems.

Aim of this work is the evaluation of FRCM capacity in repairing pre-damaged clay brick walls under out of plane loads. Full scale experimental tests have been performed damaging clay brick walls subjected to out of plane loads; then, tests are performed on damaged walls repaired with innovative composite grids with inorganic matrix (FRCM). The boundary constraints of the walls (two adjacent edges constrained and the other two free) allowed to simulate a non-uniform out of plane behaviour of the wall subjected to a pointwise normal load applied at the free corner. The aim of this study was to assess the potentiality of FRCM to recover stiffness after significant damage and to improve the lateral (out-of-plane) capacity of the repaired wall. Experimental evidences demonstrated that the externally bonded strengthening was not strongly affected by debonding and it was able to prevent a flexural failure; the collapse of the wall occurred at almost double load with respect to the unreinforced configuration, despite previous significant damage, and that the failure was governed by shear sliding. Future developments include experimental tests of strengthened walls not previously damaged, to investigate on the effect of pre-damage on out of plane strengthening capacity of FRCM.

A new homogenization technique, based on the Transformation Field Analysis, able to determine the overall behavior of viscoplastic heterogeneous materials, is proposed. This method is derived developing a variational formulation of the compatibility and evolution equations governing two classical elastic–viscoplastic models, i.e. Perzyna and Peric models. A representation form for the stress field and the plastic multiplier is defined on the representative volume element (RVE) of the composite material, resulting in a significant reduction of the history variables ruling the evolution problem. A numerical procedure is developed integrating the evolution equation by a backward-Euler algorithm and solving the single time step developing a specific form of the predictor corrector technique. The accuracy of the proposed homogenization method is assessed through some numerical examples for elastic–viscoplastic composites. The applications aim to verify the effectiveness of the presented technique in modeling the elastic–viscoplastic composite behavior adopting a reduced number of history variables. The homogenization results are compared with the ones obtained by nonlinear finite element micromechanical analyses.

In the present work, a new model of the FRP-concrete or masonry interface, which accounts for the coupling occurring between the degradation of the cohesive material and the FRP detachment, is presented; in particular, a coupled interface-body nonlocal damage model is proposed. A nonlocal damage and plasticity model is developed for the quasi-brittle material. For the interface, a model which accounts for the mode I, mode II and mixed mode of damage and for the unilateral contact and friction effects is developed. Two different ways of performing the coupling between the body damage and the interface damage are proposed and compared. Some numerical applications are carried out in order to assess the performances of the proposed model in reproducing the mechanical behavior of the masonry elements strengthened with external FRP reinforcements.

The present paper discusses simple compatibility, equilibrium, and constitutive equations for a non-prismatic planar beam. Specifically, the proposed model is based on standard Timoshenko kinematics (i.e., planar cross-section remain planar in consequence of a deformation, but can rotate with respect to the beam center-line). An initial discussion of a 2D elastic problem highlights that the boundary equilibrium deeply influences the cross-section stress distribution and all unknown fields are represented with respect to global Cartesian coordinates. A simple beam model (i.e. a set of Ordinary Differential Equations (ODEs)) is derived, describing accurately the effects of non-prismatic geometry on the beam behavior and motivating equation’s terms with both physical and mathematical arguments. Finally, several analytical and numerical solutions are compared with results existing in literature. The main conclusions can be summarized as follows. (i) The stress distribution within the cross-section is not trivial as in prismatic beams, in particular the shear stress distribution depends on all generalized stresses and on the beam geometry. (ii) The derivation of simplified constitutive relations highlights a strong dependence of each generalized deformation on all the generalized stresses. (iii) Axial and shear-bending problems are strictly coupled. (iv) The beam model is naturally expressed as an explicit system of six first order ODEs. (v) The ODEs solution can be obtained through the iterative integration of the right hand side term of each equation. (vi) The proposed simple model predicts the real behavior of non-prismatic beams with a good accuracy, reasonable for the most of practical applications.

This paper presents two micromechanical and a multiscale finite element models for the analysis of masonry walls under out-of-plane instability effects. A two-dimensional modeling of the wall is considered in all approaches, assuming a cylindrical bending. The micromechanical analyses are performed considering elastic beams to model the bricks and either nonlinear beams or interfaces to model the mortar layers. The beam finite elements rely on the force-based formulation and account for large displacements by making use of the corotational approach. This latter is properly formulated and extended to interface elements to include nonlinear geometry effects. The multiscale model is defined by applying a two-scale beam-to-beam homogenization procedure, developed for masonry elements with periodic brick arrangements. Hence, a Unit Cell made of a single linear elastic brick and a single nonlinear mortar layer is introduced at the microscopic level, which is linked to the macroscale level through a semi-analytic homogenization technique. In all models, a damage formulation with friction plasticity governs the mortar constitutive relationship. Computational details on model implementation and solution procedures are given for all approaches. Correlation studies are performed to assess the proposed numerical micromechanical and multiscale procedures. In particular, the behavior of an unreinforced wall tested under a compressive eccentric load is reproduced and advantages and disadvantages of the proposed approaches are discussed.

Steel Reinforced Polymers (SRP) have been recently introduced and adopted as strengthening systems for existing constructions. Although the interest for this kind of strengthening materials is increasing also for masonry structures, in particular for historic constructions, only few studies specifically pertained to masonry elements strengthened with SRPs are available in the literature. In this context, a twofold purpose is at the basis of the present paper. The first one consists of experimentally investigating the bond behavior of masonry elements strengthened with SRPs using different types of adhesives and strips density. The second purpose aims at developing simple formulations for evaluating the bond resistance of masonry elements strengthened with SRPs applied with different types of adhesive systems.

The effects of mortars joints in masonry substrate reinforced with FRP is investigated from the numerical point of view. The analysis has been conducted by means of a new interface model specifically developed to reproduce the debonding process occurring between an elastic thin body in adhesion with a cohesive support material, as concrete or masonry. The model accounts for mode I and mode II of failure, considering the effect of the in-plane deformation of the interface, i.e. the possible elongation or confinement of the material constituting the interface. Numerical results are compared with experimental evidences showing the good performances of the proposed model in investigating the transferring phenomena in heterogeneous interfaces and, in particular, in studying the influence of the presence of mortar joints in the masonry texture in the debonding process.

A new macro-element based on the equivalent frame approach is presented to analyze the nonlinear structural response of masonry panels under monotonic lateral loadings. A nonlinear elastic response is assumed for the masonry material and the sectional response of the beam is derived performing analytical integration. A two-node equilibrated force-based (FB) beam finite element (FE) is formulated. The FE is composed of a central flexible part, characterized by a no-tension constitutive relationship, and a lumped nonlinear shear hinge arranged in series, in order to capture the main flexural and shear nonlinear mechanisms characterizing the masonry panel response. Some applications on experimental prototypes are presented, showing a very satisfactory agreement between the numerical results and the experimental outcomes.

A recently proposed cohesive-zone model which effectively combines damage, friction and mechanical interlocking has been revisited and further validated by numerically simulating the pull-out test, from a concrete block, of a ribbed steel bar in the post-yield deformation range. The simulated response is in good agreement with experimental measurements of the bond slip characteristics in the post-yield range of deformed bars reported in the literature. This study highlights the main features of the model: with physically justified and relatively simple arguments, and within the sound framework of thermodynamics with internal variables, the model effectively separates the three main sources of energy dissipation, i.e. loss of adhesion, friction along flat interfaces and mechanical interlocking. This study provides further evidence that the proposed approach allows easier and physically clearer procedures for the determination of the model parameters of such three elementary mechanical behaviours, and makes possible their interpretation and measurement as separate material property, as a viable alternative to lumping these parameters into single values of the fracture energy. In particular, the proposed approach allows to consider a single value of the adhesion energy for modes I and II.

The nonlinear structural analysis represents a powerful tool for the study of the seismic response of new and existing constructions. In particular, nonlinear static analysis, also denoted as pushover analysis, is widely used for RC and steel structures and, recently, new seismic codes extended the use of this type of analysis to the case of masonry structures. The reliability of the results deduced from nonlinear analyses is not always ensured since the numerical results can depend on the modeling strategy and on the adopted solution techniques. Moreover, in the case of masonry structures strengthened with fiber reinforced polymers (FRP), particular regards has to be addressed not only to the masonry material model but also to the modeling of FRP-system, taking into account the complex phenomena characterizing the interaction between the FRP and the masonry supports. Aim of the present paper is to discuss some strategies, in the framework of the nonlinear finite element analysis, regarding the elements choice and the models to adopt for reliable nonlinear analyses of masonry structures reinforced with FRP strips. Specific structural problems are described and solved, discussing the role and the reliability of the adopted strategies. Finally, the seismic analysis of a masonry façade and the design of suitable FRP strengthening systems are performed. © 2012 Elsevier Ltd. All rights reserved.

A multi-scale model for the structural analysis of the in-plane response of masonry panels, characterized by periodic arrangement of bricks and mortar, is presented. The model is based on the use of two scales: at the macroscopic level the Cosserat micropolar continuum is adopted, while at the microscopic scale the classical Cauchy medium is employed. A nonlinear constitutive law is introduced at the microscopic level, which includes damage, friction, crushing and unilateral contact effects for the mortar joints. The nonlinear homogenization is performed employing the Transformation Field Analysis (TFA) technique, properly extended to the macroscopic Cosserat continuum. A numerical procedure is developed and implemented in a Finite Element (FE) code in order to analyze some interesting structural problems. In particular, four numerical applications are presented: the first one analyzes the response of the masonry Representative Volume Element (RVE) subjected to a cyclic loading history; in the other three applications, a comparison between the numerically evaluated response and the micromechanical or experimental one is performed for some masonry panels.

The aim of the present paper is the development of a simple procedure for the analysis of the bond behavior of fiber-reinforced polymer (FRP) sheets or plates externally applied to masonry supports for the strengthening or repair of masonry constructions. The procedure allows evaluation of the bond strength and the fracture energy developed during the debonding process through simple formulas based on a few parameters, evaluated either by standard tests performed on the materials making up the support and the strengthening system or by theoretical considerations. A brief discussion on the main experimental evidence and the theoretical models provided by the literature is also reported in this paper. The comparison between the theoretical results obtained by applying the proposed procedure and the experimental data deduced from literature is carried out.

The paper deals with the problem of the determination of the in-plane behavior of periodic masonry material. The macromechanical equivalent Cosserat medium, which naturally accounts for the absolute size of the constituents, is derived by a rational homogenization procedure based on the Transformation Field Analysis. The micromechanical analysis is developed considering a Cauchy model for masonry components. In particular, a linear elastic constitutive relationship is considered for the blocks, while a nonlinear constitutive law is adopted for the mortar joints, accounting for the damage and friction phenomena occurring during the loading history. Some numerical applications are performed on a Representative Volume Element characterized by a selected commonly used texture, without per- forming at this stage structural analyses. A comparison between the results obtained adopting the proposed procedure and a nonlinear micromechanical Finite Element Analysis is presented. Moreover, the substantial differences in the nonlinear behavior of the homogenized Cosserat material model with respect to the classical Cauchy one, are illustrated.

Fiber Reinforced Plastic (FRP) can be used for strengthening concrete or masonry constructions. One of the main problem in the use of FRP is the possible detachment of the reinforcement from the support material. This paper deals with the modeling of the FRP-concrete or masonry damage interface, accounting for the coupling occurring between the degradation of the cohesive material and the FRP detachment. To this end, a damage model is considered for the quasi-brittle material. In order to prevent strain localization and strong mesh sensitivity of the solution, an integral-type of nonlocal model based on the weighted spatial averaging of a strain-like quantity is developed. Regarding the interface, the damage is governed by the relative displacement occurring at bond. A suitable interface model which accounts for the mode I, mode II and mixed mode of damage is developed. The coupling between the body damage and the interface damage is performed computing the body damage on the bond surface. Numerical examples are presented.

The present work deals with the buckling and post-buckling behaviour of bimodular laminated composite plates. The fiber-governed bimodular constitutive model is adopted for the analysis and the geometrical nonlinearities are accounted for by using the consistent small strain and moderate rotation theory for a shear deformable Mindlin plate. Governing equations of the problem are developed and the finite element formulation is given using lagrangian C°-elements. Some numerical examples are developed to analyze the buckling and post-buckling behaviour of laminated bimodular plates.

International audience; The present paper deals with the derivation of a higher order theory of interface models. In particular, it isstudied the problem of two bodies joined by an adhesive interphase for which ‘‘soft’’ and ‘‘hard’’ linearelastic constitutive laws are considered. For the adhesive, interface models are determined by usingtwo different methods. The first method is based on the matched asymptotic expansion technique, whichadopts the strong formulation of classical continuum mechanics equations (compatibility, constitutiveand equilibrium equations). The second method adopts a suitable variational (weak) formulation, basedon the minimization of the potential energy. First and higher order interface models are derived for softand hard adhesives. In particular, it is shown that the two approaches, strong and weak formulations, leadto the same asymptotic equations governing the limit behavior of the adhesive as its thickness vanishes.The governing equations derived at zero order are then put in comparison with the ones accounting forthe first order of the asymptotic expansion, thus remarking the influence of the higher order terms and ofthe higher order derivatives on the interface response. Moreover, it is shown how the elastic properties ofthe adhesive enter the higher order terms. The effects taken into account by the latter ones could play animportant role in the nonlinear response of the interface, herein not investigated. Finally, two simpleapplications are developed in order to illustrate the differences among the interface theories at thedifferent orders.