Buckling analysis of rectangular composite plates with rectangular cutout subjected to linearly varying in-plane loading using fem

A numerical study is carried out using finite element method, to examine the effects of square and rectangular cutout on the buckling behavior of a sixteen ply quasi-isotropic graphite/epoxy symmetrically laminated rectangular composite plate [0°/+45°/-45°/90°]_2s, subjected to various linearly varying in-plane compressive loads. Further, this paper addresses the effects of size of square/rectangular cutout, orientation of square/rectangular cutout, plate aspect ratio(a/b), plate length/thickness ratio(a/t), boundary conditions on the buckling bahaviour of symmetrically laminated rectangular composite plates subjected to various linearly varying in-plane compressive loading. It is observed that the various linearly varying in-plane loads and boundary conditions have a substantial influence on buckling strength of rectangular composite plate with square/rectangular cutout.     

Postbuckling strengths of composite laminate with various shaped cutouts under in-plane shear

The aim of present investigation is to study the buckling and postbuckling response and strengths under positive and negative in-plane shear loads of simply-supported composite laminate with various shaped cutouts (i.e., circular, square, diamond, elliptical-vertical and elliptical-horizontal) of various sizes using finite-element method. The FEM formulation is based on the first order shear deformation theory which incorporates geometric nonlinearity using von Karman’s assumptions. The 3-D Tsai-Hill criterion is used to predict the failure of a lamina while the onset of delamination is predicted by the interlaminar failure criterion. The effect of cutout shape, size and direction of shear load on buckling and postbuckling responses, failure loads and failure characteristics of quasi-isotropic [i.e., (+45/−45/0/90)_2s] laminate has been discussed. In addition, the effect of composite lay-up [i.e., (+45/−45/0/90)_2s, (45/−45)_4s and (0/90)_4s] has also been reported. It is observed that the cutout shape has considerable effect on the buckling and postbucking behaviour of the quasi-isotropic laminate with large size cutout. It is also observed that the direction of shear load and composite lay-up have substantial influence on strength and failure characteristics of the laminate.     

Stability and failure of composite laminates with various shaped cutouts under combined in-plane loads

The objective of this paper is to study stability and failure of a composite laminate with a centrally placed cutout of various shapes (i.e., circular, square, diamond, elliptical-vertical and elliptical-horizontal) under combined action of uni-axial compression and in-plane shear loads. The FEM formulation based on the first order shear deformation theory and von Karman’s assumptions has been utilized. Newton–Raphson method is used to solve nonlinear algebraic equations. Failure of a lamina is predicted by the 3-D Tsai–Hill criterion whereas the onset of delamination is predicted by the interlaminar failure criterion. The effects of cutout shape, direction of shear load and composite lay-up on buckling and postbuckling responses, failure loads and failure characteristics of the laminate has been discussed. An efficient utilization of material strength is observed in the case of laminate with circular cutout as compared to the laminate with other shaped cutouts. In addition, it is also concluded that although the buckling strength of the (0/90)_4s laminate is lower than that of the (+45/?45/0/90)_2s and (45/?45)_4s laminates, but its strength is increased in the advanced stage of postbuckling deformation.     

Optimization of shear-deformable laminated plates under buckling and strength criteria

A shear deformable laminated theory is used to study the optimal design of rectangular plates under biaxial compressive loads. Such loads lead to plate failure by buckling or material failure which corresponds to the violation of the selected strength criterion. The minimum of the two loads (buckling load or material failure load) determines the critical failure load for a given set of problem parameters. At the optimum values of the ply angles, buckling or both failure criteria may be operational depending on the laminate thickness. The present study investigates the effect of laminate thickness on the optimal design and gives numerical results for symmetrically laminated angle-ply plates.     

Buckling analysis of composite laminates under end shortening by higher‐order shear deformable finite strips

A higher‐order shear deformable finite strip is developed and employed in the buckling analysis of laminated composite plates when subjected to uniform end shortening. This enables the transverse shear deformation to be accurately incorporated. The permitted laminate material properties are quite general, encompassing anisotropy and full coupling between in‐plane and out‐of‐plane behaviour. Results with respect to the number of plies, thickness of laminate and ratios of E₁₁/E₂₂ are presented for unsymmetric cross‐ply and angle‐ply lay‐ups and for laminates with arbitrary lay‐up arrangements. Copyright © 2002 John Wiley & Sons, Ltd.     

Effects of boundary conditions on buckling and postbuckling responses of composite laminate with various shaped cutouts

The objective of the present work is to study the effects of flexural boundary conditions on buckling and postbuckling behavior of axially compressed quasi-isotropic laminate, (+45/−45/0/90)_2s with various shaped cutouts (i.e., circular, square, diamond, elliptical–vertical and elliptical–horizontal) of various sizes using the finite element method. The FEM formulation is based on first order shear deformation theory and von Karman’s assumptions are used to incorporate geometric nonlinearity. The 3-D Tsai-Hill criterion is used to predict the failure of a lamina while the onset of delamination is predicted by the interlaminar failure criterion. It is observed that the laminates clamped and simply supported on all edges have the highest and lowest buckling and postbuckling strength, respectively, irrespective of cutouts shape and size. It is also noted that a fully clamped laminate with large size elliptical–vertical cutout can take higher compressive buckling load than the laminate without cutout for same boundary condition.     

Detection of stiffness reductions in laminated composite plates from their dynamic response using the microgenetic algorithm

In this study, we investigate a method to detect damage in a laminated composite structure by analyzing its dynamic response to impact loads. The combined finite element method (FEM) with seven degrees of freedom (DOF) and the advanced microgenetic algorithm described in this paper may allow us not only to detect the damaged elements but also to find their locations and the extent of damage. A high order shear deformation theory (HSDT) is used to predict the structural behavior and to detect damage of laminated composite plates. The effects of noise associated with the uncertainty of measurements due to the complex nature of composites are considered for [0/90]_s and [±45]_s layup sequences. The results indicate that the new method is computationally efficient in characterizing damage for complex structures such as laminated composites.     

Magnetoelectric performance of cylindrical Ni–lead zirconate titanate–Ni laminated composite synthesized by electroless deposition

The cylindrical Ni–lead zirconate titanate (PZT)–Ni laminated composites with various magnetostrictive–piezoelectric phase thickness ratios were synthesized by electroless deposition. The influences of the bias magnetic field (H _dc) and the ac magnetic field frequency (f) on magnetoelectric (ME) effect are discussed. It is seen that the ME voltage coefficient depends strongly on H _dc and f. The ME voltage coefficient and electromechanical resonance frequency increase as the magnetostrictive–piezoelectric phase thickness ratio increases. The calculated resonant frequency increases with the magnetostrictive–piezoelectric phase thickness ratio, which agrees well with the experimental results. The maximum ME voltage coefficient of the cylindrical Ni–PZT–Ni laminated composite is 3.256 V cm⁻¹ Oe⁻¹, which is much higher than that of the plate laminated composite with the same magnetostrictive–piezoelectric phase thickness ratio. Electroless deposition is an efficient method to prepare ME laminated composites with complex structures. Proper resonant frequency and stronger ME effect can be obtained by optimizing the structure.     

Strength prediction in composites with stress concentrations: classical Weibull and critical failure volume methods with micromechanical considerations

Application of Weibull statistics to tensile strength prediction in laminated composites with open holes is revisited. Quasi-isotropic carbon fiber laminates with two stacking sequences [45/0/−45/90]_s and [0/45/90/−45]_s with three different hole sizes of 2.54, 6.35 and 12.7 mm were considered for analysis and experimental examination. The first laminate showed 20% lower strength for smaller and 10% for the larger hole sizes. A novel critical failure volume (CFV) method with minimum scaling length constraint as well as the traditional Weibull integral method were applied. The strength prediction was based on the state of stress in the 0^° ply by taking into account the redistribution of stress due to matrix damage in the form of splitting, delamination and matrix cracking of off axis plies. The state of matrix damage precipitating failure was recorded by using X-radiography and examined by a sectioning technique. The measured extent of damage was then included in a 3D stress analysis procedure by using a mesh independent crack modeling method to account for fiber direction stress redistribution. The CFV method gave results within one standard deviation from experimentally observed strength values for both laminates and all three hole sizes. The Weibull integral method underpredicted the strength in all cases from as much as 20–30% for smaller hole sizes to 8% for the large holes. The accuracy of failure predictions using CFV is attributed to the introduction of a minimum scaling length. This length has a physical meaning of the width of a process zone of formation of fiber macro-crack as a result of single fiber break interaction. Direct measurement or rigorous evaluation of this parameter is, however, difficult. Consistent with referenced micromechanical studies, its value was assigned equal to six times the Rosen’s ineffective length.     

Optimal Design of Fibre-reinforced Laminated Plates Accounting for Manufacturing Uncertainty

A procedure to design symmetrically laminated plates under buckling loads for minimum weight with manufacturing uncertainty (tolerance) in the ply angle and plate thickness, which are the design variables, is described. A minimum buckling load capacity is the design constraint implemented. It is assumed that the probability of any tolerance value occurring within the tolerance band, compared with any other, is equal, and thus the approach is a worst case scenario approach. The effects of bending–twisting coupling are neglected in implementing the procedure, and the Downhill Simplex method is used as the search technique, but the methodology is flexible and allows any appropriate problem formulation and search algorithm to be substituted. Two different tolerance scenarios are used for the purposes of illustrating the methodology, and plates with varying aspect ratios and loading ratios are optimally designed and compared. The results demonstrate the importance of carrying out design optimisation of composite structures with the effects of manufacturing tolerances included.     

Thermal and mechanical fatigue analysis of CFRP laminates

[0°/90°]_s and [±45°]_s CFRP laminated plates were analysed using a finite element formulation for their fatigue behaviour. A fatigue criterion which is based on the laminate interlaminar stresses and the basic lamina fatigue parameters was used. Thermal effects were included in the formulation. In particular, initial thermal stresses resulting from the curing of the laminate were also included in the analysis. The results showed that both laminates had predicted S-N behaviour similar to that from experiments of past investigators. Also, the fatigue behaviour for the [±45°]_s laminate between room temperature and the curing temperature were found to be the same. However, in the case of the [0°/90°]_s laminate the fatigue strength at high temperatures was found to be lower than that at low temperatures.     

Optimal design of hybrid laminates with discrete ply angles for maximum buckling load and minimum cost

The design of hybrid symmetric laminated plates consisting of high-stiffness surface and low-stiffness core layers is presented. In the first problem the maximization of buckling load is carried out over a discrete set of ply angles. In the second problem the minimum number of high-stiffness plies is determined for a given buckling load to minimize the material cost. Boolean variables are introduced to specify stacking sequence. Solution of the linear optimization problem yields an optimal stacking sequence. The effect of hybridization is investigated for various problem parameters such as the aspect ratio of the laminate and the number of plies. The optimal designs are obtained with upper bound constraints on the effect of bending-twisting coupling stiffnesses. Results are given for hybrid graphite-epoxy/glass-epoxy laminates under both uniaxial and biaxial loadings.     

Computational approach of piezoelectric shells by the GDQ method

A piezoelectric laminated cylindrical shell with shear rotations effect under the electromechanical loads and four sides simply supported boundary condition was studied by using the two-dimensional generalized differential quadrature (GDQ) computational method. The typical hybrid composite shells with 3-layered cross-ply [90°/0°/90°] graphite–epoxy laminate and bounded PVDF layers are considered under the sinusoidal pressure loads and electric potentials on the shell. The governing partial differential equation with first-order shear deformation theory in terms of mid-surface displacements and shear rotations can be expressed in series equations by the GDQ formulation. Thus we obtain the GDQ numerical solutions of non-dimensional displacement and stresses at center position of laminated piezoelectric shells. Displacement is generally affected by the thickness of laminated piezoelectric shells under the action of mechanical load. Stresses are generally affected by the thickness and the length of laminated piezoelectric shells under the actions of mechanical load and electric potential.     

Analysis of critical buckling load of laminated composites plate with different boundary conditions using FEM and analytical methods

In this study, the critical buckling load of fiber reinforced composite plate was calculated by analytical and finite element methods. The critical buckling loads and composite deformations were obtained on the basis of plate dimensions ratio (L_x/L_y). The deformation behaviour of the plate is shown for modes i = 1,2 and different values of orientation angles [0/θ]₂. As a result of the analysis, for the composite plates with symmetric and antisymmetric layup the critical buckling loads were obtained for different support conditions. It was observed that difference between results increased depending on θ angle, although critical buckling load values obtained by using numerical and finite element methods were nearly same in [0/θ]₂ orientation angles.     

Buckling Analysis of Angle-ply Composite and Sandwich Plates by Combination of Geometric Stiffness Matrix

Buckling response of angle-ply laminated composite and sandwich plates are analyzed using the global-local higher order theory with combination of geometric stiffness matrix in this paper. This global-local theory completely fulfills the free surface conditions and the displacement and stress continuity conditions at interfaces. Moreover, the number of unknowns in this theory is independent of the number of layers in the laminate. Based on this global-local theory, a three-noded triangular element satisfying C¹ continuity conditions has also been proposed. The bending part of this element is constructed from the concept of DKT element. In order to improve the accuracy of the analysis, a method of modified geometric stiffness matrix has been introduced. Numerical results show that the present theory not only computes accurately the buckling response of general laminated composite plates but also predicts the critical buckling loads of soft-core sandwiches. However, the global higher-order theories as well as first order theories might encounter some difficulties and overestimate the critical buckling loads for soft-core sandwich plates.     

Stability analysis of delaminated composite beams

This study describes the dynamic stability of composite cantilever beams subjected to periodic axial loading with delaminations at pre-set locations. A computer code based on the finite element method is developed to calculate the natural frequencies, critical buckling loads and dynamic instability regions of the woven and laminated composite beams with different stacking sequences ([0]₄, [0/90]_s and [90]₄), corresponding to this peculiar delamination case. The results of the developed code for the natural frequencies are compared with the natural frequencies obtained experimentally and numerically with commercial FEA (ANSYS). The critical buckling loads are also compared with the ones obtained from ANSYS simulations.     

Optimum design of laminated composite under axial compressive load

In the present study optimal design of composite laminates, with and without rectangular cut-out, is carried out for maximizing the buckling load. Optimization study is carried out for obtaining the maximum buckling load with design variables as ply thickness, cut-out size and orientation of cut-out with respect to laminate. Buckling load is evaluated using a ‘simple higher order shear deformation theory’ based on four unknown displacements u, v, w _b and w _s . A C¹ continuous shear flexible finite element based on HSDT model is developed using Hermite cubic polynomial. It is observed that for thick anti-symmetric laminates, the non-dimensional buckling load decreases with increase in aspect ratio and increase in fibre orientation angle. There is a decrease in the non-dimensional buckling load of symmetric laminate in the presence of cut-out.     

Effect of cobalt substitution on magnetic and transport properties of Nd0·5Sr0·5Mn1?xCoxO3 (x = 0·1, 0·3 and 0·5)

The magnetic and transport properties of the compounds Nd_0·5Sr_0·5Mn_1-x_{{\rm 1}-{x}}Co_x_{{x}}O₃ (x = 0·1, 0·3 and 0·5), synthesized by citrate–gel route have been investigated. The spin transition in cobaltates at low temperatures affects the magnetic as well as transport properties. The irreversibility behaviour between the zero-field cooled (ZFC) and field cooled (FC) magnetization as a function of temperature becomes stronger with increasing Co content. This is understood on the basis of glassy behaviour, which becomes more robust with increasing Co substitution. The non-saturating M–H behaviour indicates strong magnetic inhomogeneities which may cause the magnetic phase separation at the nanoscopic length scale. The double exchange interaction is stronger between Mn^3 +–O^2 −–Mn^4 + as compared to Co^3 +–O^2 −–Co^4 + pairs. Co-substitution suppresses the double exchange which will lead to cluster/spin glass like behaviour as well as semiconducting features due to localization of charge carriers (mobile e_g{e}_{\rm g} electrons).     

High‐order layerwise finite element for the damped free‐vibration response of thick composite and sandwich composite plates

A high‐order layerwise finite element methodology is presented, which enables prediction of the damped dynamic characteristics of thick composite and sandwich composite plates. The through‐thickness displacement field in each discrete layer of the laminate includes quadratic and cubic polynomial distributions of the in‐plane displacements, in addition to the linear approximations assumed by linear layerwise theories. Stiffness, mass and damping matrices are formulated from ply to structural level. Interlaminar shear stress compatibility conditions are imposed on the discrete layer matrices, leading to both size reduction and prediction of interlaminar shear stresses at the laminate interfaces. The C¹ continuous finite element implemented yields an element damping matrix in addition to element stiffness and mass matrices. Application cases include thick [0/90/0], [±θ]_S and [±θ] composite plates with interlaminar damping layers and sandwich plates with composite faces and foam core. In the latter case, modal frequencies and damping were also experimentally determined and compared with the finite element predictions. Copyright © 2008 John Wiley & Sons, Ltd.