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Abstract This paper was dealt with buckling and post-buckling behavior of short thin-walled Z-columns made of a carbon-epoxy laminate, subjected to eccentric compressive load. The buckling mode and the buckling load of real structures versus load eccentricity were discussed. The study involved both experimental tests were carried out on real laminated structures and numerical simulations by the finite element method. The buckling load of real structures was determined by approximation methods on the basis of experimental and numerical post-buckling equilibrium paths of the structure. Additionally, in numerical simulations, the bifurcation load value was determined by solving an eigen problem. In all cases, experimental findings and numerical results show high agreement. The study determined the quantitative influence of the direction and value of compressive load eccentricity on the buckling load and rigidity of the structure in the post-buckling state. [ABSTRACT FROM AUTHOR]

Abstract Repair of a composite laminate with delamination using micro-bolts is studied parametrically. The effects on compressive buckling load of delamination crack length, bolt material, bolt diameter and arrangement, and bolt–laminate clearance are considered. For 45- and 65-mm crack lengths, buckling load recovery rates using brass bolts are 90% and 73%, respectively. Using steel increases the rate by 6.6% compared to the intact specimen. Tests are performed fixing the number of steel bolts for bolt diameters of 0.6, 1.0, and 1.2 mm. The buckling load recovery rate using 1.2- versus 0.6-mm bolts increases by 6.8%. Finite element analysis confirms the effect of the bolt–laminate gap: 27% larger buckling load for no clearance than for a 0.1-mm gap. 18% higher buckling load, which is 97% of intact-specimen buckling load, is obtained for a 1.2-mm-diameter bolt repair by gluing bolts and holes and then tightening them compared with using bolts alone. [ABSTRACT FROM AUTHOR]

Abstract The present paper deals with the description of the stability loss and postbuckling for multilayered composite structures. Both, the experimental results and theoretical predictions are presented and discussed. The wide literature review is also pointed out considering the issue of both experimental and theoretical buckling analyses for composite rectangular plates and shells. A special attention is focused on the buckling and postbuckling behaviour od structures with cutouts and reinforcement. [ABSTRACT FROM AUTHOR]

Abstract In this work, a formulation for buckling and geometrically nonlinear analysis of functionally graded plates under thermo-mechanical loading is presented. The influence of high temperature environment on deflections due to mechanical loading as well the deflections produced by thermal loads acting alone or combined with mechanical loads is analysed. The implemented finite element model is based on a non-conforming triangular flat plate/shell element, with three nodes and eight degrees of freedom per node, associated with a higher order shear deformation theory. The solutions of some illustrative examples are evaluated involving the variation of volume fractions of constituent materials, temperature range and material combinations, which have significant impact on the deflections of plates. The numerical results obtained with the present model are presented and compared with alternative solutions. [ABSTRACT FROM AUTHOR]

Elastoplastic buckling behaviors of rectangular plates made from functionally graded materials (FGMs) are investigated in this paper. The elastoplastic material properties are assumed to vary smoothly through the thickness of the plates. The three dimensional material constitutive relation of FGMs is found by introducing the material homogenization method, named Tamura-Tomota-Ozawa(TTO) model, into J 2 deformation theory or J 2 flow theory. The uniform strain hypothesis helps to simplify the prebuckling state and derive the analytical expression of the position of the material elastoplastic interface. The buckling governing equations and the buckling critical condition of the structures are formulated under the framework of the classical plate theory. An iterative algorithm is designed to obtain the elastoplastic buckling critical load, a converging result between the prebuckling and the buckling critical internal forces. ABAQUS simulation well verifies the present theoretical predictions from J 2 flow theory, and is resorted to investigate the postbuckling behaviors of FGM plates. Discussions are addressed for the effects of the constituent distribution, the material plastic flow, the preloaded states of the plates, and the regions of buckling types are plotted as well. [ABSTRACT FROM AUTHOR]

In the present paper, the post-buckling behavior of a simply supported piezoelectric hybrid microplate subject to thermal, electrical and mechanical loads is studied. The size effect in the mechanical behavior of the microplate is captured by using the modified couple stress theory. The Mindlin plate theory is adopted to describe its deflection behavior with the von Karman’s geometric nonlinearity taken into account. Based on these assumptions and the principle of minimum potential energy, the equilibrium equations of the microplate and associated boundary conditions are derived. By applying the Galerkin method to the equilibrium equations, closed-form solutions for the critical thermal/mechanical buckling load and the load–displacement relation in the post-buckling stage are obtained. Furthermore, the effects of the material length scale parameter to thickness ratio, the applied electric field and in-plane boundary conditions on the buckling and post-buckling behavior of the piezoelectric hybrid microplate are discussed in detail. [ABSTRACT FROM AUTHOR]

Research is limited with respect to load-deflection behavior of slender concrete columns internally reinforced with Fiber Reinforced Polymer (FRP) spirals and longitudinal bars. An analytical buckling model based on numerical integration is presented to predict the load versus deflection performance of slender concrete columns reinforced with FRP spirals and longitudinal bars, subjected to eccentric loads. The model can be used to predict the behavior of slender concrete columns with various configurations including FRP and/or steel reinforcement, single or double spiral, and number of longitudinal bars. The longitudinal bars considered include steel, FRP, or hybrid reinforcement consisting of steel and FRP bars. The model is found to predict the experimental performance of slender concrete columns reinforced with Glass FRP longitudinal bars and spirals with satisfactory accuracy. The model is used to create interaction diagrams for FRP spiral-confined circular concrete columns with various slenderness ratios, reinforced with steel, FRP or hybrid reinforcement. [ABSTRACT FROM AUTHOR]

Abstract Cylindrical lattice structures are types of lightweight structures and can utilize the orthotropy of fiber-reinforced composite materials. Maximization of the buckling loads is important because they dominate the strength of the structures. On the basis of the fact that a hyperboloidal shape deviation increases the buckling loads of cylindrical homogeneous shells, the changes in the buckling loads of cylindrical lattice structures with a hyperboloidal shape deviation are discussed. Further, the buckling loads of both conventional cylindrical lattice structures and shape-deviated hyperboloidal lattice structures with respect to the compressive, bending, and torsional loads are calculated using a finite element method. The results show that the shape deviation decreases the buckling loads, while the change in mass is negligible. The effects of the shape deviation on the buckling loads differ between homogeneous shell structures and lattice structures. [ABSTRACT FROM AUTHOR]

A general analytical procedure is proposed to design tapered laminated composite structures. It is based on: 1) the introduction of two variables representing a laminate configuration, 2) the analytical determination of the lower bound of the number of layers in each of the segments constituting the construction, 3) the definition of the reference sublaminate which configuration is transferred to all segments/parts of the construction, 4) the gradual building of configurations for symmetric balanced laminates having odd or even number of plies. The solution is verified by the comparison with results existing in the literature for both inner and outer tapers. A multi-panel composite structure is considered to demonstrate the applicability of the proposed method. As usual, in a such class of problems, the whole construction is divided into segments/panels. The individual segments are subjected to the simultaneous action of in-plane, tensile or compressive and shear loads. The present results demonstrate the simplicity and effectiveness of the method and non-uniqueness of solutions. The paper intends to clarify the physical sense of the discussed problem. [ABSTRACT FROM AUTHOR]

Grid-stiffened composite structures not only allow for significant structural weight reduction but also are competitive in terms of structural stability and damage tolerance compared with conventional stiffened candidates. As the development of Automated Fibre Placement (AFP) technology matures, a unitized construction of skin and stiffeners is easily manufacturable. In this paper, a curved stiffener layout is optimized to enhance the structural buckling resistance. A linear variation of stiffener angles is used, resulting in the formation of a locally rhombic lattice pattern. Due to the spatial variation of angle and spacing induced by the use of curved stiffeners, analytic solutions for the responses are not generally applicable. Thus, global and local buckling loads are calculated based on finite element models by a previously-developed global/local coupled strategy. Since the stiffeners are not explicitly modelled in the finite element calculations, a fixed mesh is used for gradient-based optimization. Both parametric design and optimization are performed in order to find the optimal curved grid pattern, whose practical performances are assessed by post-buckling analysis. A comparison between the performances of structures with curved stiffeners, with straight stiffeners, and variable-stiffness skins with curved fibres, demonstrates the potential of curved stiffener configurations in improving the structural efficiency. [ABSTRACT FROM AUTHOR]

A solution of the buckling problem for a shear loaded composite anisogrid lattice plate with the clamped edges based on the Galerkin method is presented in this paper. The lattice plate is modelled as an equivalent continuous orthotropic plate with effective stiffness parameters. The deflection of buckled plate is presented in the form of a double series containing clamped–clamped beam functions. The critical in-plane shear load is found solving the generalised eigenvalue problem for a homogeneous system of algebraic equations in which the unknowns are the coefficients of the double series. Based on this solution, the effects of the plate dimensions and parameters of lattice structure on the value of critical load are investigated and analysed. Results of these analyses are successfully verified using the finite element method. An approximate analytical expression providing fast and reliable way of calculation of the critical buckling load is obtained for the lattice plate composed of the ribs made of the same composite material and with the same size of their cross-sections. [ABSTRACT FROM AUTHOR]

As a first attempt, optimum stacking sequences of thick laminated composite plate is obtained to maximize its buckling load via employing the finite element (FE), genetic algorithms (GAs) and particle swarm optimization (PSO) methods. The higher-order shear deformation theory (HSDT) is used to obtain governing equations of the plate. The finite element method (FEM) is employed to solve these equations and to gain buckling load of the plate. The FEM solution is linked with a developed optimization algorithm which is a mix of genetic algorithms (GAs) and particle swarm optimization (PSO) techniques. Some examples are solved to show applicability and usefulness of the proposed hybrid method for maximizing buckling load of the plate via finding optimum stacking sequences of the plate. Also, influences of different parameters on the optimum stacking sequences of the thick plate are studied. [ABSTRACT FROM AUTHOR]

Stiffened cylindrical shells are widely used in modern engineering structures, especially in aircraft, oil-transmitting pipeline, spacecraft industry and ocean engineering. Due to the increasing demands for high structures performances, this paper presents the effect of stiffeners on the nonlinear buckling of cylindrical shells with functionally graded (FG) coatings under torsional load. The cylindrical shell is reinforced with external axial stiffeners. The material properties are assumed to vary continuously through the thickness direction. Equilibrium and stability equations for cylindrical shells are derived by using the classical shell theory (CST) with von-Karman nonlinear kinematic relations. Using analytical approach, Galerkin procedure, and the Airy stress function, the resulting equations are employed to obtain the closed-form expression for the critical buckling loads and load–deflection relation. The influences of the number of stiffeners, volume fraction exponent, the thickness of the metal layer, and geometric parameters on the nonlinear buckling behavior of the cylindrical shell with FG coatings are examined in details. [ABSTRACT FROM AUTHOR]

The buckling response and load carrying capacity of thin-walled open cross-section profiles made of Fibre Metal Laminates, subjected to static axial compression loading are considered. These include thin-walled Z-shape and channel cross-section profiles adopting a 3/2 FML lay-up design, made of 3 aluminium layers. The objective of the investigation is the comparison of standard thickness Fibre Reinforced Plastic layers versus thin-ply material technology. Whilst thin ply designs differ only by the layer thickness, they offer an exponential increase in stacking sequence design freedoms, allowing detrimental coupling effects to be eliminated. The benefit of different hybrid materials are also considered. The comparisons involve semi-analytical and finite element methods, which are validated against experimental investigations. [ABSTRACT FROM AUTHOR]

The purpose of this paper was to study the effect of I-shape stiffener stiffness on the buckling and post-buckling, up to collapse behavior of stiffened composite panel under the uniform uniaxial compression load. Six stiffened composite panel configurations, which were orthogonally designed with two skin configurations and three I-shape stiffener configurations, were took into account in the experimental investigation. Three specimens for each configuration (18 specimens in total) were manufactured for test. Before tests, the key dimensions of I-stiffener and skin were measured. Furthermore, all the specimens were also inspected visually and detected by the ultrasonic C-scan system. No initial delamination or debonding area was found on all the specimens according to their C-scan images. The experimental results show that the equivalent compression stiffness of I-shape stiffener has great influence on the buckling load and failure mode of stiffened panel, but a little effect on the failure load of stiffened panel. The skin thickness has great impact on the buckling load and the final failure load. [ABSTRACT FROM AUTHOR]

This paper presents an analytical solution for the buckling and free vibration analysis of laminated beams by using a refined and generalized shear deformation theory which includes the thickness expansion. The eigenvalue equation is derived by employing the Rayleigh quotient, and the Ritz method is used to approximate the displacement field. The functions used in the Ritz method are chosen as either a pure polynomial series or a hybrid polynomial-trigonometric series. The hybrid series is used due to the superior convergence and accuracy compared to conventional pure polynomial series for certain boundary conditions. The boundary conditions are taken into account using the penalty method. Convergence of the results is analyzed, and numerical results of the present theory are compared with other theories for validation. Nondimensional natural frequencies and critical buckling loads are obtained for a variety of stacking sequences. The effect of the normal deformation on the fundamental frequencies and critical buckling loads is also analyzed. [ABSTRACT FROM AUTHOR]

Buckling analysis of nanobeam is significantly important for design of nano-electro-mechanical systems (NEMS), especially for those under non-uniform temperature induced commonly by the operation of NEMS. In this work, such issue is investigated with the aids of size-dependent model, i.e. nonlocal thermoelasticity, and size effect of heat conduction on buckling property is considered for the first time. Euler–Bernoulli beam is adopted and reformulated within nonlocal theory. Analytical solution to critical load for various boundary conditions, e.g. SS, CF, CS and CC, is obtained using eigenvalue method. And the temperature distribution is determined using nonlocal heat conductive law. The effect of elastic and thermal nonlocal parameter on the critical load is systematically discussed. It is observed that classical models over-predict the critical load, which may be dangerous in practical applications. The present work is expected to be helpful for design of nanobeam-based devices and systems. [ABSTRACT FROM AUTHOR]

Unstiffened cylindrical shells are very sensitive to geometric imperfections such as production-related deviations from the ideal geometry (conventional imperfections) as well as imperfect boundary conditions, material and wall thickness imperfections (non-conventional imperfections). The load carrying capability of unstiffened shells is reduced significantly by those imperfections. The NASA SP-8007 design guideline from 1968 is currently used for shell design. This guideline provides knock-down factors for the buckling loads and is based on experimental data of isotropic and orthotropic test specimen. Therefore the structural behavior of composite material is not considered adequately. Based on the single-perturbation load approach (SPLA), this paper introduces a new physical based approach, to determine a lower bound for the buckling load of unstiffened composite cylindrical shells with respect to conventional and non-conventional imperfections, the constant single-buckle imperfection (CSBI) principle. The CSBI principle is based on the theory of the metastability of the geometric imperfect cylindrical shell which is introduced within this paper. The results indicate that the CSBI principle has the potential to provide an improved shell design in order to reduce weight and cost of thin-walled shells. [ABSTRACT FROM AUTHOR]

The present study is concerned with an analysis of load and frequency interaction effects in the dynamic buckling of sandwich plates and shells with a soft, transversely flexible core. The analysis is based of a general geometrically nonlinear soft-core sandwich model presented and a corresponding extended Galerkin solution strategy presented earlier. In the present study, the formalism is adopted and transformed into an eigenvalue problem. Assuming large transverse average displacements but small amplitudes for the vibration problem, a partially decoupled problem is obtained where in a first step the static buckling and face wrinkling problems can be addressed without consideration of the dynamic problem. In a second step, the dynamic eigenvalue problem can be solved, using the static solution as input parameters. The model is applied in parametric studies on the vibration response of plane and doubly curved sandwich panels with rectangular projection. In all cases, strong interaction effects between the external static preload and the natural frequencies are observed. The development of overall buckling or local face wrinkling instabilities enabled due to the transverse flexibility of the core result in distinct drops of the first natural frequency. [ABSTRACT FROM AUTHOR]

Bio-inspired helicoidal carbon fiber reinforced polymer composite (CFRPC) laminates are one of the novel composite structures that have potential applications in many engineering fields. The optimization design of helicoidal layups with linear, nonlinear Recursive, Fibonacci, Exponential and their combination-form arrangements are carried out using ABAQUS and ASA algorithm on Isight platform. Two cases of the compressive postbuckling under uniaxial compression and the thermal postbuckling under uniform temperature rise of helicoidal CFRPC laminated plates stacked with optimal layups are performed and the secondary buckling and mode jumping in the postbuckling region are examined for the first time. Numerical investigations are carried out for 16-ply symmetrical helicoidal laminated plates with various width-to-thickness ratios under different boundary conditions and are compared with quasi-isotropic counterparts. The results show that the bio-inspired helicoidal CFRPC laminated plates can enhance the buckling load and the buckling temperature as well as the compressive and the thermal postbuckling strength. In some cases, the secondary buckling and mode jumping may occur for rectangular helicoidal laminated plate in the postbuckling region. [ABSTRACT FROM AUTHOR]

Practical applications such as airplane wings are usually subjected to combined thermal and mechanical loads, and they hence are prone to buckling failure. Preceding works on the buckling of advanced materials, e.g., functionally graded materials, under combined thermal and mechanical loads are rather rare in literature. In this paper, we report new numerical results of thermal-mechanical buckling of functionally graded rectangular and skew plates (FGPs) under combined thermal and mechanical loads. The numerical responses of buckling are computed using isogeometric analysis (IGA) based on the first-order shear deformation plate theory (FSDT) without shear-locking effect. We present formulations and then provide validation of numerical results computed by the proposed formulation against reference existing solutions. Parametric study is also performed to explore insight into the effects of various numerical aspect ratios such as gradient index, plate aspect ratio, loading type, skew angle, and boundary condition, etc. on mechanical response of FGPs. The stability diagrams are also presented. [ABSTRACT FROM AUTHOR]

The free vibration and buckling behaviors of foam-filled composite corrugated sandwich plates under thermal loading are investigated theoretically. A refined shear deformation theory is extended incorporating two different combinations of hyperbolic and parabolic shear shape functions. Equivalent thermoelastic properties of the foam-filled corrugation are obtained using the method of homogenization based on the Gibbs free energy. Based on hyperbolic-polynomial variation of all displacements across the thickness of both face sheets and sandwich core, the shear plate theory accounts for both transverse shear and thickness stretching effects. The theoretical predictions are validated against existing results as well as finite element simulations. The effects of geometric and material parameters on natural frequency and critical temperature change for buckling are systematically investigated. The proposed theory is not only accurate but also simple in predicting the free vibration and thermal buckling responses of composite sandwich plates with foam-filled corrugated cores. [ABSTRACT FROM AUTHOR]

This paper presents a unified nonlinear analytical solution of bending, buckling and vibration for the temperature-dependent functionally graded (FG) rectangular plates subjected to thermal load. Geometric nonlinearity resulted from mid-plane stretching is considered. Material properties of FG plates are assumed to vary with temperatures and the volume fractions of the constituents. Newly proposed higher order shear deformation theories in present available literature are sorted out and given a unified application. Three types of mathematical models, P-FG, S-FG and E-FG models, describing effective material properties of functionally graded materials (FGMs) are discussed. Finally, based on the unified nonlinear analytical solution, influences of material heterogeneity, thermal load, and plate geometry on bending, buckling and vibration of FG plates are studied. Outcomes reveal that the nonlinear solution exhibits better accuracy in calculation of shear stress in bending responses. The thermal load plays an important role in determining the bending, buckling and vibration of the FG plate. In addition, the characteristics of these three types of mathematical models to simulate effective material properties of FGMs are numerically compared and discussed. [ABSTRACT FROM AUTHOR]

An approximate analytical solution of the buckling problem formulated for a composite sandwich cylindrical shell subjected to a uniform external lateral pressure is presented in the paper. Both ends of the shell of finite length are fully clamped. The problem is solved using the Galerkin method. The analytical formula for the critical load has been obtained and verified by comparison with a finite-element solution. Based on this formula buckling analyses of sandwich pipes with different types of core materials normally used in underwater applications have been performed. Applications of the formula to the solutions of design problems have been demonstrated for the clamped–clamped sandwich pipes considering constraints imposed on the value of critical pressure. [ABSTRACT FROM AUTHOR]

In this paper, a re-look into the upper and lower bound solution to the buckling of composite columns is presented in details and this includes the complete derivation of the expressions for the kinematical compatibility conditions. Included in this contribution are also the results of the current finite element simulation together with the results of the experiments. Moreover, further investigations are conducted to test whether the proposed three methods of calculating the effective flexural stiffness can be employed in cooperation with the developed mathematical model to form the desired upper and lower bounds for the buckling loads of both the perfect and single-delaminated fiber-reinforced composite beams for the various parameters that are studied. In these analyses, single-delaminated fiber-reinforced composite beams which consist of random ply-orientations and stacking sequences are employed to put up a stringent test on the generalness and applicability of these three methods. It is observed that for all the cases that were examined, these three means of evaluating the effective bending modulus of a fiber-reinforced composite beam pass are validated. [ABSTRACT FROM AUTHOR]

Morphing aircraft aim to increase the performance of aircraft over multiple flight conditions, by enabling shape changes in flight in order to optimise their aerodynamic properties for the current conditions. The skin of a morphing aircraft is a critical component. It must be compliant in degrees of freedom that are required for actuation, to minimise the actuation loads. However, it must also carry structural loads, and therefore be stiff in load bearing degrees of freedom. This leads to a requirement for extremely anisotropic material systems. A common solution is the use of a corrugated panel. However, previous work on corrugations has not addressed the problem of compressive buckling loads. This work analyses the performance of corrugated panels under buckling loads, and optimises corrugation patterns for the objectives of weight, buckling performance, and actuation compliance. Simplified analytical models that derive properties equivalent to conventional plates are used to obtain approximate estimates of the buckling loads. Furthermore a new mode of buckling, that occurs entirely in-plane and is unique to panels with extreme anisotropy is analysed. The simple models allow optimisation to be performed, and both a single-objective and a multi-objective approach are demonstrated. The results are compared to Finite Element Analysis. [ABSTRACT FROM AUTHOR]

In practice, voltage would be stored in actuators due to applied mechanical loads or voltage specified as input to them. The effect of stored voltage in actuators on buckling analysis has not been studied so far. This article addresses fully coupled electromechanical buckling analysis of active laminated composite thin to thick plates using partial hierarchical Rayleigh–Ritz solution. The formulation is derived from variational principle with consideration for shear deformation plate theory. Another major contribution is to postpone the initiation of buckling of a laminated plate by increasing its flexural stiffness using feedback piezoelectric control. [ABSTRACT FROM AUTHOR]

In this paper, the thermal effect on buckling and free vibration characteristics of functionally graded (FG) size-dependent Timoshenko nanobeams subjected to an in-plane thermal loading are investigated by presenting a Navier type solution for the first time. Material properties of FG nanobeam are supposed to be temperature-dependent and vary continuously along the thickness according to the power-law form. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived based on Timoshenko beam theory through Hamilton’s principle and they are solved applying analytical solution. According to the numerical results, it is revealed that the proposed modeling can provide accurate frequency results of the FG nanobeams as compared to some cases in the literature. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as thermal effect, material distribution profile, small scale effects, beam thickness and mode number on the critical buckling temperature and normalized natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the thermal buckling and vibration behavior of a FG nanobeams is significantly influenced by these effects. [ABSTRACT FROM AUTHOR]

This paper presents a method for the determination of the buckling loads of symmetrically laminated composite plates under uniaxial compressive load with explicit consideration of bending–twisting-coupling (BTC). The plates under consideration are assumed to be thin and balanced with a symmetric stacking sequence and consist of layers with ply orientations of {0°/±45°/90°}, but otherwise arbitrary lamination schemes. The characteristic laminate quantities are described in a non-dimensional notation in order to capture all combinations of material and stacking sequences that are physically possible. Plates with rectangular planforms in the framework of Classical Laminate Theory (CLT) under several different sets of boundary conditions are considered, and the buckling problem of such composite laminated plates is solved in an approximate manner by using the classical Rayleigh–Ritz method. Several combinations of the boundary conditions ‘simply supported’ and ‘clamped’ are analyzed, and the obtained results are compared with finite-element simulations for verification purposes. Based on the computed data, two different design equations for sufficiently long plates are developed which allow for a rapid and closed-form analytical assessment of the buckling behavior of symmetrically layered composite plates under explicit consideration of bending–twisting coupling. It is found that the closed-form analytical analysis equations are simple in their application and yet still work with very satisfying accuracy which makes them very suitable for use in practical engineering work. [ABSTRACT FROM AUTHOR]

Cutouts are always located in aerospace composite structures to meet specified requirements on the service and functions. The presence of cutout, especially large-sized cutout, would result in a significant stress concentration, and consequently a complicated failure mechanism. This paper presents an experimental and numerical study on the post-buckling damage process of high-performance carbon fiber reinforced polymer (CFRP), T800/X850, multidirectional composite laminated specimen. The simply supported specimen is with a central large-sized elliptical cutout and subjected to shearing load. In the numerical analysis, a damage model incorporating both advanced strain softening schemes and cross-coupling effect between the failure mechanisms is constructed for the damage simulation. Regularizations are implemented for coupling the damage variables to address the deficiencies of the existing model and access the real physical damage behavior of laminated composites. The strain invariant failure theory is employed for evaluation of delamination. To assess the objectivity of the present numerical model, the numerical predictions are compared with experimental data. The predicted damage and propagation patterns show good agreement with experiment results. [ABSTRACT FROM AUTHOR]

Semi-analytical models for the linear buckling analysis of unstiffened laminated composite cylinders and cones with flexible boundary conditions are presented. The Classical Laminated Plate Theory and the First-order Shear Deformation Theory are used in conjunction with the Donnell’s non-linear equations to derive the buckling equations. Axial, torsion and pressure loads can be applied individually or combined in the proposed models. The stiffness matrices are integrated analytically and for the conical shells an approximation is proposed to overcome non-integrable expressions. Comparisons with the literature show that the classical base functions available for axial compression cannot capture the buckling modes for non-orthotropic laminates. For torsion loads these classical shape functions do not catch the buckling modes even when applying the assumption of pure orthotropy, and it is shown how the proposed models correlate well with experimental data from the literature and finite element results. The use of elastic constraints at the boundaries allows the simulation of different boundary conditions in a versatile way and it is shown how those constants can be adjusted in order to change from one type of boundary condition to another. [ABSTRACT FROM AUTHOR]

Buckling strength is the most important design criterion for laminated composite structures subjected to compressive loading. The aim of the present work is to introduce an explicit procedure in designing laminated composite plates for maximum buckling load subjected to strength and stiffness constraints using the polar formalism, which relates stiffness to the critical buckling load in order to simplify the calculation and optimization of the buckling load. The polar representation of plane elasticity is effectively used in this work in order to introduce a unified optimization method. Design for maximum flexural stiffness as well as laminate effective elastic modulus was developed in order to achieve the optimal orientation of the layers giving the maximum buckling strength. Three optimization procedures were proposed. The first one is unconstrained optimization and the two other cases are constrained optimization subjected to the flexural stiffness and the elastic modulus constraints. The quasi-homogeneous laminates, which have been used as a special class of laminate stacks, were proposed as the general solution for the optimal stacking sequence of the layers. [ABSTRACT FROM AUTHOR]

This paper proposes a novel numerical optimization procedure with mixed integer and continuous design variables for optimal design of laminated composite plates subjected to buckling loads. In the present optimization problem, the objective function is to maximize the buckling load factor. The design variables are fibre orientation angles and thickness of layers, in which the fibre orientation angles are integer variables and thickness are continuous variables. The constraints include the limitation of variables and the total thickness of the plate. For analyzing the buckling behavior of laminated composite plates, a recently proposed smoothed finite element method named the cell-based smoothed discrete shear gap method (CS-DSG3) is employed. For solving the current optimization problems which contain both integer and continuous variables, an improved different evolution algorithm, named mixed-variable different evolution (mDE) is proposed. In the mDE, the mutation and selection phases of the original DE are replaced by an adaptive mutation mechanism and an elitist selection technique, respectively. These improvements not only help balance effectively the global and local search abilities of the DE, but also help deal with integer and continuous design variables. The reliability and effectiveness of the proposed optimization procedure are investigated through some numerical examples for optimal design of laminated composite plates with 2, 3, 4 and 10 layers subjected to buckling loads. Additionally, the influence of different loading and boundary conditions on the optimal solution is also investigated. [ABSTRACT FROM AUTHOR]

Current paper is the second part of Kolakowski Z, Mania RJ paper entitled: “ Influence of the coupling matrix B on the interactive buckling of FML-FGM columns with closed cross-section under axial compression ”, Composite Structures, in press. The problem of non-linear multi-mode buckling of thin-walled columns with open cross-sections made of Functionally Graded Materials ( FGMs ) and Fibre Metal Laminate ( FML ) was discussed. FGM layers are a thermal barrier, whereas the FML composite provides rigidity and strength. This new hybrid thin-walled structure is very interesting but there are no interactive buckling solutions. The non-linear problem of buckling was solved with the analytical-numerical method ( ANM ) based on Koiter’s theory. An interaction of global buckling mode with many different local buckling modes was discussed. A plate model and the classical laminate theory are used to define government relations. The differential equilibrium equations were obtained with a variational method. Each column was made of an FML sublayer, a single AL-TiC-type FGM layer and/or a TiC ceramic layer. The FML sublayer consisted of two outer aluminium layers and even carbon-epoxy prepreg interlayers. The layup configuration of the FML/FGM composite is non-symmetric, so the coupling matrix B is non-trivial. Simply supported columns are under mechanical compression only. [ABSTRACT FROM AUTHOR]

A semi-analytical model to predict the non-linear behavior of unstiffened cylinders and cones considering initial geometric imperfections and various loads and boundary conditions is presented. The formulation is developed using the Classical Laminated Plate Theory (CLPT) and Donnell’s equations, solving for the complete displacement field. The non-linear static problem is solved using a modified Newton–Raphson algorithm with line-search. A numerical integration scheme for the non-linear matrices is proposed and details regarding the implementation of the proposed method are given. Two methods to include measured imperfections into the analyses are presented and for one method the effect of using different approximation levels for the imperfection field on the non-linear response is investigated, and a minimum approximation accuracy that should be used is determined. The semi-analytical results are verified using finite elements and previous models from the literature. The implemented routines are distributed on-line and are based on a matrix notation simply applicable to other problems. [ABSTRACT FROM AUTHOR]

This paper presents a unified Lévy-type solution procedure for the buckling analysis of both thin and thick composite plates under biaxial loads. The plates are simply-supported at two opposite edges, while the two remaining sides are subjected to any combination of simply-supported, clamped and free conditions. The problem is formulated in the context of a variable-kinematic approach, offering the advantage of automatically handling theories of various order. Both layerwise and equivalent single layer theories are considered. The governing equilibrium equations are derived analytically from the Principle of Virtual Displacements (PVD), and are solved exactly referring to the Lévy-type procedure. The accuracy of the predictions is demonstrated by comparison with results available in literature, including exact 3D solutions. A comprehensive set of benchmark results is provided for plates subjected to different loading and boundary conditions and characterized by various width-to-thickness ratios. [ABSTRACT FROM AUTHOR]

Abstract The critical buckling load of FRP pultruded columns of different shapes under static and dynamic load are presented. FRP columns with cross sections of sheet pile, square tube, I-section and T-section have been studied and the shape effect on the instability is discussed. The static buckling load calculated from the numerical simulations was then compared with the results from theoretical values and ratios reported for the different cross sections. The comparison shows that the outcomes of the FE analysis were more conservative than the theoretical values some cases. The simulation of dynamic buckling loads showed a considerable sensitivity to the model’s boundary conditions. The simulation of the horizontal loading suggested that local buckling tends to occur close to the base of the column. Based on the results, it is suggested that full numerical analysis including dynamic and horizontal loading needs to be conducted for the safe design of thin-wall FRP elements. [ABSTRACT FROM AUTHOR]

This paper is a discussion of “Optimization for buckling loads of grid stiffened composite panels”. That paper presented an improved smear method to study the buckling behavior of stiffened composite cylindrical shells. Although this is a simple and efficient procedure but there is a basic error in superimposing the stiffness matrices of the shell and the stiffeners and calculating the equivalent stiffness parameters. This error is shown by two simple examples. [ABSTRACT FROM AUTHOR]

This paper studies a closed-form solution for the nonlinear postbuckling behavior of long unsymmetrical rotationally-restrained laminated composite plates subjected to shear load. The analysis method described can be applied to the postbuckling behavior of stiffened plates under limitations of the local buckling mode, in which the portion of panel with two stiffeners is modeled as a thin plate with two edges restrained with torsion springs. Nondimensional parameters are introduced in order to cover a wide class of material properties and laminate configurations. An appropriate mode shape function is proposed for the postbuckling process, under the assumption that the buckling mode remains unchanged as the deflection increases. Substituting the shape function into the governing equations, an precise closed-form solution for the load–deflection relationship is derived using the Galerkin method. To conclude, stiffened composite plates with representative unsymmetrical laminate configurations are used as analytical models for a comparison of the closed-form solution. The finite element simulation demonstrates the closed-form solution’s ability to predict postbuckling behavior with high computational efficiency. [ABSTRACT FROM AUTHOR]

Abstract The study investigates the effect of eccentric load on the stability and postcritical states of thin-walled CFRP channel section columns under compression. Test specimens are subjected to compression on a testing machine provided with a fixture for applying eccentric compressive loads. Loading forces, deflection and strains of the column walls and web are measured. The experiments also involve examination of the operating performance of a structure undergoing buckling and determination of its postcritical equilibrium paths describing the relationship between load and deflection. Based on experimental results, numerical models of composite structures are designed and verified by the FEM. The scope of the numerical analysis involves performing a linear analysis of stability, which allows for determining the buckling mode depending on the amplitude of compressive load eccentricity and corresponding critical loads. The next stage of the analysis involves performing a nonlinear analysis of the structures with implemented geometric imperfections reflecting the lowest buckling modes. Based on obtained results, postcritical equilibrium paths of the developed FEM models are determined. Obtained equilibrium paths are then compared with experimental characteristics of real structures. The numerical results and experimental findings show a satisfactory agreement. [ABSTRACT FROM AUTHOR]

Abstract Pultruded glass fiber reinforced polymer (GFRP) sections face a critical issue of premature web crippling failure when loaded in the transverse direction. However, limited research on the web crippling behavior of asymmetric sections like channel section was reported in the literature. In addition, the effect of specimen length on the web crippling behavior has not been explicitly investigated. This paper presents an experimental study on the web crippling behavior of pultruded GFRP channel sections under transverse bearing load. Four channel sections with various dimensions were tested under two transverse loading conditions, i.e. interior-two-flange (ITF) and exterior-two-flange (ETF). For each section, two specimen lengths were adopted to study the effect of specimen length on the web crippling behavior. Therefore, three variables including sectional dimensions, specimen length and loading conditions were considered in the experimental program. Two failure modes, namely web-flange junction failure and web buckling failure, were observed. The load-displacement curves and web crippling capacities of all specimens were reported and compared. Finally, design equations were proposed considering the failure mechanisms. The predicted web crippling capacities generally agreed well with the experimental results. [ABSTRACT FROM AUTHOR]

Abstract Wind turbine blades present different types of failure mechanisms and modes which are associated with specific loading conditions. Trailing edge failure mode has been documented in full-scale blade tests as one of the failure types observed in blades on service. Trailing edge failure is characterized by failure of the trailing edge adhesive joint and the buckling of the trailing edge sandwich panels. This failure is governed by the contribution of edgewise, flapwise and torsion moments, with edgewise moments being the main driver. This paper describes a blade sub-component test setup suitable for studying trailing edge failure on static and fatigue load conditions, which is an improvement in the experimental verification of a trailing edge blade design. The test setup and design drivers are described and studied via FE models. Static and fatigue test results are reported for a full-scale blade section sub-component obtained from a 34 [m] wind turbine blade. Moreover, experimental results are discussed and compared with FE models to describe and study the trailing edge failure mechanism. [ABSTRACT FROM AUTHOR]

Abstract In this paper, aiming at better understanding the damage tolerance of T-stiffened composite panels, impact damage and the influences on compression behaviors were studied with experiment methods using single T-stiffened panels. Low velocity impact (LVI) was conducted on the stiffener flange tip from the smooth side. Serious impact damage, including stiffener/skin interface debonding, fibre breakage and delamination, was observed. After the impact test, compression experiment was performed to characterize both the buckling behavior and compression failure mechanism. The compression results revealed that the buckling was promoted by the impact damage and induced bending moments which drove delamination inside the skin to propagate beneath the stiffener and led to the failure of the stiffened panel. Consequently, the impact damage decreased the buckling load and failure load of the single T-stiffened panel by 9.62% and 34.1%, respectively. [ABSTRACT FROM AUTHOR]

The feasibility of probabilistic design methods to overcome conservatism associated with deterministic design methods for unstiffened cylindrical composite shells has been shown, but studies are frequently based on assumptions regarding relevant uncertainties due to the lack of available data. The reliability based calibration method suggested here relies on extensive measurements regarding the statistical characteristics of thickness, fibre volume fraction, fibre orientation, material stiffness and geometric imperfections of 11 previously tested cylinders. Furthermore, the stiffness of the potting of tested cylinders is estimated using Finite Element Analysis and the introduction of a pre-stress state through the potting process investigated. Following a sensitivity study, Bayesian inference is used to update the information of relevant uncertain variables. Monte Carlo simulation is employed to compute the distribution functions of the buckling load, and these are used to calibrate structural safety factors at a chosen reliability level. A multiplicative error model is established to compute safety factors covering the model uncertainty inherent to the approach. In combining these two safety factors in a Bayesian sense, a transparent, updatable safety-factor concept is developed, leading to less conservative design loads than currently considered. [ABSTRACT FROM AUTHOR]

The vibration behaviors of functionally graded carbon nanotube (FG-CNT) reinforced composite rectangular plates subjected to in-plane loads are studied in this paper. Three types of carbon nanotube (CNT) distributions are researched in this investigation. Reddy’s third-order shear deformation theory (HSDT) is applied to evaluate the displacement fields of the plate. The equation of motion of the structural system is formulated with use of the Hamilton’s principle. The plates investigated in this study are simply-supported on two opposite edges and therefore, in order to solve the coupling set of equations of motion, the state-space Levy method is applied. Based on the Levy solution, the natural frequencies and vibration mode shapes of FG-CNT reinforced composite plates subjected to in-plane loads with different boundary conditions are presented for different thickness-length ratios and aspect ratios. Moreover, the critical in-plane loads for the buckling of different FG-CNT plates are calculated and compared. [ABSTRACT FROM AUTHOR]

Experimental studies and numerical simulations are performed to investigate the structural response of adhesively bonded CFRP cylindrical splice-joint panels subjected to axial compression. Splicer joints used in aerospace composite structures need to sustain the design ultimate loads in the presence of flaws due to manufacturing (debonds), in order to demonstrate compliance with flightworthiness. CFRP cylindrical splice-joint specimens with and without debonds are subjected to compression load. Experimental data reveals that the load-carrying capacity of splicer joint decreases in the presence of debonds. Numerical simulations are carried out in line with experimental observations using finite element analysis. Good correlations with experimental results are obtained both in terms of buckling load capacity and load–displacement responses. Non-destructive methods, pulsed thermography and acoustic emission, are employed to detect and monitor the response of debond respectively. Parametric studies are also performed using finite element analysis to assess the influence of bi-directional loading and the effect of the physical gap provided between basic panels in the splicer joints. [ABSTRACT FROM AUTHOR]

In this work a new device has been developed to estimate compression-after-impact (CAI) strength. This device allows the testing of laminates thinner than those recommended by CAI test standards. The proposed device is composed of a support structure, with a set of vertical ribs that stabilize the specimen during the test, increasing the buckling load. A numerical analysis was made to ensure that global buckling does not occur in the laminate during the CAI test, and that there is no interference with the damage area. Laminate specimens were tested with the proposed device and the ASTM device. For specimens 4.416 mm thick (thickness according to ASTM D7137 standard), the test results were similar with both devices. For thinner laminates, higher CAI strength was estimated with the proposed device than with the ASTM device, showing that the global buckling was delayed. [ABSTRACT FROM AUTHOR]

The problems of a multi-mode buckling approach which is based on Koiter’s theory of a hybrid column are presented in this paper. An interaction of global buckling modes with the local ones is discussed. There are many different local and global buckling modes. Their selected combinations are dangerous and cause a reduction in the load-carrying capacity. All walls of the hybrid column were plane and made of many layers. The outer layers were thermal barriers and made of a TiC ceramic layer or an AL-TiC-type FGM. The inner layers were composed of aluminium layers and a few carbon-epoxy laminate layers. The classical laminate theory is used to define the ABD matrix which described the relations between applied loads and the associated deformations. The layup configuration of the hybrid column is general, so the coupling submatrix B is non-trivial. This submatrix has a significant impact on the value of local buckling load, whereas its effect on the value of global buckling load can be neglected. The main topic discussed in this paper is whether and how individual elements of the B submatrix can change the load-carrying capacity of a hybrid column. A detailed discussion is conducted for simple supported columns with opened cross-sections subjected to mechanical loads only. Thermal effects are neglected. [ABSTRACT FROM AUTHOR]

Thin-walled composite beams of complex sectional contour are widely used along with the traditional elements of reinforced structures. Based on the generalized theory of elasticity of thin-walled beams, the mathematical model of flexural–torsional deformation of open-profile thin-walled composite beams of non-uniform cross section under thermo-mechanical loading has been developed. To confirm the reliability of developed approach to determination of the spatial displacements of thin-walled composite beams of non-uniform cross section, several experimental studies were conducted. Based on the analysis of the range of typical profiles, for the further analysis we selected several compound cross sections for which the analytical dependences were obtained regarding displacements of their characteristic points. Our studies can serve as a base for development of the practical recommendations for the optimal designing and manufacturing of thin-walled anisotropic profiles, as well as proposing the measures to reduce the value of deformations caused by the production process. [ABSTRACT FROM AUTHOR]

The part 1 [1] of this two-part paper presented an experimental study concerning the web-crippling phenomenon on GFRP pultruded I-shaped profiles and showed that they are very sensitive to transverse compressive loads (lower mechanical properties in the transverse direction) and web-crippling failure. This part 2 reports numerical and analytical results aiming at a better understanding of the web-crippling phenomenon and providing enough data to complement the observations made in the experimental campaign. The scope of these studies was to predict and replicate the behaviour exhibited by the four different GFRP profiles tested (I100, I120, I200 and I400) when subjected to different transverse loading configurations (ETF and ITF) and bearing lengths (varying from 15 to 100 mm). Shell finite element (SFE) models with the Tsai Hill failure criterion were developed in ABAQUS, aiming at simulating the experimental tests performed in part 1. This procedure enabled the comparison of the numerical results with the experimental data, as well as the observation of the stress fields and damage evolution. In general, these models presented good agreement with the test results. An analytical formula was also developed to predict the web-crippling load of GFRP profiles and presented very good agreement with the test results. [ABSTRACT FROM AUTHOR]