Similitude invariants and scaling laws for buckling experiments on anti-symmetrically laminated plates subjected to biaxial loading

Establishing the similitude between the model and prototype rigorously is a necessary step in designing an experiment efficiently. So far, to the best of the authors’ knowledge, no one has ever derived the similitude invariant for anti-symmetric cross- and angle-ply laminated plates subjected to biaxial loading before. This research paper is the first to establish the similitude invariant of anti-symmetric cross- and angle-ply laminated plates by applying the similitude transformation to the governing differential equations of buckling directly. Then the scaling laws for buckling loads of laminated plates subjected to biaxial loads are derived. But in reality, either due to the complexity of the scaling laws or to economize on costly experiments, it may not be feasible to construct the model conforming to the scaling laws completely, therefore partial similitude is investigated theoretically and approximate scaling laws are recommended. The buckling loads of the prototype predicted from the scaling laws are then compared with the available theoretical values. The complete similitude cases show exact agreement between results predicted from the scaling laws and the available analytical solutions. For partial similitude cases, the models distorted in stacking sequences, number of plies, and material properties are studied and the approximate scaling laws which yield good agreement are recommended.     

Some studies on free vibration of composite laminates

Free vibration analysis is carried out to study the vibration characteristics of composite laminates using the modified shear deformation, layered, composite plate theory and employing the Rayleigh-Ritz energy approach. The analysis is presented in a unified form so as to incorporate all different combinations of laminate boundary conditions and with full coverage with regard to the various design parameters of a laminated plate. A parametric study is made using a beam characteristic function as the admissible function for the numerical calculations.The numerical results presented here are for an example case of fully clamped boundary conditions and are compared with previously published results. The effect of parameters, such as the aspect ratio of plates, ply-angle, number of layers and also the thickness ratios of plies in laminates on the frequencies of the laminate, is systematically studied. It is found that for anti-symmetric angle-ply or cross-ply laminates unique numerical values of the thickness ratios exist which improve the vibration characteristics of such laminates. Numerical values of the non-dimensional frequencies and nodal patterns, using the thickness ratio distribution of the plies, are then obtained for clamped laminates, fabricated out of various commonly used composite materials, and are presented in the form of the design curves.     

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.     

具有局部分层的对称铺设层合板在非保守力作用下的屈曲分析

利用弹性非保守系统自激振动的拟固有频率变分原理,推导出复合材料矩形板受非保守随从力作用的变分方程,进而导出复合材料层合板含分层损伤的有限元基本方程及求解临界荷载力和固有频率的特征方程。用载荷增量法计算了不同边长比的复合材料矩形板在面内受随从力作用且含有分层的临界载荷,分析了不同角铺设方向对临界载荷的影响。分析表明,分层对临界载荷有一定影响,复合材料层合板的角铺设方向对临界载荷有较大影响。     

Post-buckling analysis of imperfect laminated plates under combined axial and thermal loads

A post-buckling analysis of composite laminated plates subjected to combined axial compression and uniform temperature loading is presented. The two cases of thermal post-buckling of initially compressed plates and of compression post-buckling of initially heated plates are considered. The initial geometrical imperfection of the plates is taken into account. The analysis uses a perturbation technique to determine buckling loads and post-buckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated plates. Typical results are presented in dimensionless graphical form.     

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.     

The effect of anisotropy on post-buckling behavior of laminated plates using semi-energy finite strip method

The effects of anisotropy (i.e. the ratio of longitudinal modulus of elasticity to transverse modulus of elasticity of a rectangular plate) on buckling and post-buckling performance of laminated plates under uniform end-shortening have been studied in this paper by implementing a Rayleigh–Ritz approach and a finite strip approach based on the concept of a rigorous post-buckling solution for composite plates and plate structures, namely the semi-energy approach. To validate the results, they are compared with those obtained from finite element method of analysis. The study of results has revealed that the buckling and post-buckling response of the laminates is significantly influenced by the changing of the anisotropy ratio.     

On the validity of the reduced bending stiffness method for laminated composite plate analysis

The ‘reduced bending stiffness’ (RBS) method has been used on occasions in the past as a means of simplifying the analysis of the flexural behavior of unsymmetrically laminated composite plates. However, the validity of the method has never been established. This paper makes direct comparisons between relatively simple, exact solutions for the static deflections, buckling loads and vibration frequencies of simply-supported plates and those arising from the RBS method. Extensive calculations are made for wide ranges of the physical parameters involved (aspect ratio, moduli ratio, lamination orientation angle, numbers of plies). The RBS method is found to yield sufficient accuracy for cross-ply plates, but errors up to 29% are obtained for angle-ply plates constructed with materials currently under development.     

The Displacement Perspective During Ultimate Failure of Composite Laminates

This paper deals with the studies on the state of displacement of symmetric and anti-symmetric angle-ply and cross-ply laminated composite plates during its ultimate failure, subjected to transverse static load. First-order shear deformation theory (FSDT) is employed in conjunction with the finite element approach using eight-noded quadratic isoparametric element. The free vibration analyses of isotropic and laminated composite plates are carried out to ensure the overall validity of the present finite element formulation. The mid surface of the laminate is considered as the reference plane. The principal material directions in different laminae are oriented to produce a laminated structural element capable of resisting loads in several directions. The stiffness of a composite laminate is obtained from the properties of the constituent laminae. The affected stiffness of the failed lamina is discarded completely after the failure of weakest ply. The rigidity matrix of the laminate with remaining laminae is re-established. The re-evaluation process continues until the laminate fails completely. To investigate the displacement behaviour of laminates during the ultimate failure, parametric studies are carried out for different cases by varying the stacking sequences, fiber orientations, layer thicknesses, aspect ratios and the number of layers in the laminate. The comparison of results in terms of non-dimensional natural frequencies and ply-by-ply failure analyses obtained from the present investigation are made with those available in the reported literature.     

Postbuckling analysis of stiffened laminated composite panels,using a higher-order shear deformation theory

The investigation aims at: (i) constructing a modified higher-order shear deformation theory in which Kirchhoff's hypotheses are relaxed, to allow for shear deformations; (ii) validating the present 5-parameter-smeared-laminate theory by comparing the results with exact solutions; and (iii) applying the theory to a specific problem of the postbuckling behavior of a flat stiffened fiber-reinforced laminated composite plate under compression.The first part of this paper is devoted mainly to the derivation of the pertinent displacement field which obviates the need for shear correction factors. The present displacement field compares satisfactorily with the exact solutions for three layered cross-ply laminates. The distinctive feature of the present smeared laminate theory is that the through-the-thickness transverse shear stresses are calculated directly from the constitutive equations without involving any integration of the equilibrium equations.The second part of this paper demonstrates the applicability of the present modified higher-order shear deformation theory to the post-buckling analysis of stiffened laminated panels under compression. to accomplish this, the finite strip method is employed. A C ²-continuity requirement in the displacement field necessitates a modification of the conventional finite strip element technique by introducing higher-order polynomials in the direction normal to that of the stiffener axes. The finite strip formulation is validated by comparing the numerical solutions for buckling problems of the stiffened panels with some typical experimental results.     

Compressive strength of delaminated aerospace composites

An efficient analytical model is described which predicts the value of compressive strain below which buckle-driven propagation of delaminations in aerospace composites will not occur. An extension of this efficient strip model which accounts for propagation transverse to the direction of applied compression is derived. In order to provide validation for the strip model a number of laminates were artificially delaminated producing a range of thin anisotropic sub-laminates made up of 0°, ±45° and 90° plies that displayed varied buckling and delamination propagation phenomena. These laminates were subsequently subject to experimental compression testing and nonlinear finite element analysis (FEA) using cohesive elements. Comparison of strip model results with those from experiments indicates that the model can conservatively predict the strain at which propagation occurs to within 10 per cent of experimental values provided (i) the thin-film assumption made in the modelling methodology holds and (ii) full elastic coupling effects do not play a significant role in the post-buckling of the sub-laminate. With such provision, the model was more accurate and produced fewer non-conservative results than FEA. The accuracy and efficiency of the model make it well suited to application in optimum ply-stacking algorithms to maximize laminate strength.     

An exact approach to the elastic state of stress of shear deformable antisymmetric angle-ply laminated plates

The state space concept is used to develop Lévy-type solutions for anti-symmetric angle-ply laminated plates using the Yang, Norris and Stavsky (YNS) theory. The theory is a generalization of Mindlin's theory for isotropic plates applied to laminated anisotropic plates and includes shear deformation.

The solution is applicable to rectangular plates which are simply supported on two opposite edges and have arbitrary boundary conditions on the remaining ones. Numerical results are presented emphasizing the effects of length/thickness ratio, degree of anisotropy, number of layers, ply-angles and transverse shear correction coefficients on the elastic state of stress.     

Damage tolerance of fully orthotropic laminates in compression

The compression after impact (CAI) strength of fully orthotropic composite laminates with up to 21 plies is presented, as analysed by an existing strip model. Candidate layups, which can be symmetric, anti-symmetric or non-symmetric, are preselected to exhibit no elastic coupling response, with manufacturing rules applied. These criteria, along with the use of a simple surrogate sublaminate buckling model, were chosen to allow analysis of all feasible laminates in the design space without excessive computation time. Results indicate that although the inclusion of non-symmetric layups in the design space does not give benefits with respect to maximum achievable damage tolerance, these laminates can exhibit damage tolerance close to that of an anti-symmetric design for some ply counts, and better than symmetric solutions in most cases. It is also noted that in some instances increasing the number of plies in a laminate can actually reduce the highest achievable threshold load for damage tolerance, as a result of the large influence Poisson’s ratio has on sublaminate buckling. Average errors in the surrogate model were low in all cases, with maximum non-conservative errors less than 1%. The surrogate buckling model reduced computational time by over 99% when compared to the fully exhaustive search.     

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.     

双轴受压反对称角铺设复合材料层合板在固支边界下的后屈曲和模态跃迁

为有效分析双轴受压反对称角铺设复合材料层压板在固支边界下的后屈曲性能, 由渐近修正几何非线性理论推导其双耦合四阶偏微分方程(即应变协调方程和稳定性控制方程), 通过双Fourier级数将耦合非线性控制偏微分方程转换为系列非线性常微分方程, 从而获得相对简单的求解方法。使用广义Galerkin方法求解与角交铺设复合层合板相关的边界值问题, 研究了模态跃迁前后不同复杂程度的后屈曲模式。对四层固支边界复合层合板的数值模拟结果表明: 该解析法与有限元方法在主后屈曲区域的线性屈曲荷载计算结果吻合良好; 有限元方法在解靠近二次分岔点时失去收敛性, 而解析方法可深入后屈曲区域, 准确捕捉模态跃迁现象; 对于反对称角铺设层合板, 可仅用纯对称模态来定性预测主后屈曲分支、二次分岔荷载及远程跃迁路径。      

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.     

Nonlinear analysis of bimodular composite plates under compression

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.     

Energy-release-rate evaluation for delamination growth prediction in a multi-plate model of a laminate composite

In this paper, numerical methods for the evaluation of the energy-release-rate along a delamination periphery under conditions of local buckling of the delaminate, as well as global buckling of the entire laminate, are presented. A multi-plate model, using independent Reissner-Mindlin plate models for each of the delaminated and undelaminated plies, with Reissner-Mindlin constraints for relating the degrees of freedom of the delaminated plates to those of the undelaminated plate at the crack front, is used to model the laminate with embedded delaminations. Explicit expressions, in terms of finite element nodal or Gauss-point variables, are derived for the pointwise energy release rate in terms of the J-integral and the Equivalent Domain Integral in the context of a typical multi-plate model for characterising the delamination growth. A finite element method with a 3-noded quasi-conforming shell element, and an automated post-buckling solution capability, is used for conducting the numerical analyses in this paper. Using these numerical results, mechanisms of multiple buckling modes and their effect on the propagation of embedded delaminations in plates, are studied.This work has been supported by a grant from FAA to the Center of Excellence for Computational Modeling of Aircraft Structures, Georgia Institute of Technology. The first author also acknowledges the support by the INDO-US Science and Technology Fellowship under the auspices of US Agency for International Development.     

Buckling Characteristics of Symmetrically and Antisymmetrically Laminated Composite Plates with Central Cutout

A numerical and experimental study was carried out to determine the effects of anti-symmetric laminate configuration, cutout and length/thickness ratio on the buckling behavior of E/glass-epoxy composite plates. The buckling loads were presented for symmetrically and anti-symmetrically laminated plates subjected to axial compression load. The study included two different laminate configurations ([90/45/−45/0]_as and [90/45/−45/0]_s), two different cutout shapes (circular and semi-circular), two different length/thickness ratios (L/t = 75 and 37.5) and three boundary conditions (clamped–clamped [CC], clamped–pinned [CP] and pinned–pinned [PP]). Firstly, the buckling loads of eight-ply E/glass-epoxy rectangular plates were determined experimentally. Then, the buckling loads of the laminated composites were calculated by ANSYS finite-element computer code. The changing in buckling load of the composites due to the presence of cutout and changing of length/thickness ratio was calculated. Finally, the experimental test results were compared to the buckling loads of plates obtained from the finite element analysis.     

Effect of in-plane boundary restraints on optimal design of skew laminates under buckling loads

Symmetrically laminated cross-ply and angle-ply skew plates subject to uniaxial buckling loads and various combinations of in-plane boundary restraints are studied using a shear deformable theory. For this purpose a finite element code is developed and applied to a couple of verification problems. The formulation of the parabolic iso-parametric plate element is briefly given and numerical results obtained for the verification problems related to stability analysis and stress diffusion are presented. The effect of in-plane restraints on the non-uniform distribution of in-plane stresses is studied by means of contour graphs. Next the buckling loads are maximized with respect to layer thicknesses in the case of cross-ply laminates and with respect to fiber orientations in the case of angle-ply laminates. The optimization results show that the exclusion of the in-plane restraints, which arise in several engineering applications, may lead to errors in the stability analysis and consequently in the design of laminated plates against buckling.