Assessment of zigzag theory for static loading, buckling, free and forced response of composite and sandwich beams

An efficient zigzag one-dimensional (1D) theory of laminated beams is assessed by comparison of analytical solutions of simply supported beam with exact 2D elasticity solutions for static patch load, natural frequencies, harmonic transverse load with sinusoidal longitudinal variation and buckling under axial load. These results for a highly inhomogeneous test beam, symmetric composite beam, anti-symmetric composite beam and sandwich beam are also compared with first order (FSDT) and third order (TOT) shear deformation theories. The effect of the laminate lay-up and the thickness-to-span ratio on the accuracy is assessed. The zigzag theory yields much more accurate results than the TOT and FSDT.     

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.     

Vibrations of multiple delaminated beams

The free vibrations of beams with two enveloping delaminations have been solved analytically without resorting to numerical approximation. The multiple delaminated beam is analyzed as several interconnected Euler–Bernoulli beams. The differential stretching and the bending–extension coupling are considered in the formulation. The influence of the sizes and locations of the delaminations on the primary and secondary frequencies and mode shapes of a beam are investigated. For clamped–clamped beams, the primary frequency shows a high sensitivity for the long delamination but a low sensitivity for the second short delamination, while for the secondary frequency, the sensitivity is high for both delaminations. For cantilever beams, the primary and secondary frequencies show a high sensitivity for the long delamination but low sensitivity for the second short delamination. Results are compared with the analytical and experimental data reported in the literature to verify the validity of the present model.     

Buckling analysis of trapezoidal composite sandwich plate subjected to in-plane compression

The results of the stability analysis of simply supported layered isosceles trapezoidal plate subjected to axial in-plane compression are presented. The layup configuration is confined to symmetric laminates. The solution has been obtained by means of the Galerkin orthogonalisation method combined with the proposed method of the coordinate system transformation. The results of the analytical solution are compared with the verifying FEM calculation. The computed results in the form of graphs of the buckling force as a function of material and geometrical parameters of the panel are included.     

Glass/Epoxy复合材料叠层板面内剪切连续损伤模型

为确定S2玻璃纤维/环氧树脂(S2-Glass/Epoxy) 叠层复合材料面内剪切应力-应变关系,对S2-Glass/Epoxy 叠层复合材料面内剪切拉伸载荷下的弹、塑性连续损伤本构模型及应用进行了研究。基于平面应力状态下的连续损伤力学模型,通过典型面内剪切拉伸实验,分别建立了忽略塑性应变和考虑塑性应变的两种连续损伤力学(CDM)模型,并确定相关参数。通过ABAQUS/Explicit 用户子程序VUMAT接口,分别采用两种CDM模型对S2-Glass/Epoxy 叠层复合材料面内剪切拉伸实验进行有限元数值计算,与实验结果对比,验证模型可靠性,并分析单元类型对有限元计算结果的影响。研究结果表明: 忽略塑性应变的CDM模型可以很好地预测复合材料面内剪切失效强度,但不能较好地预测其非线性力学响应; 考虑塑性应变,将塑性硬化与损伤耦合后的CDM模型则能较好的预测复合材料非线性力学响应和面内剪切失效强度; 该平面应力状态下建立的CDM模型可用于壳单元进行复合材料有限元数值计算,横向剪切作用导致传统壳单元数值计算的载荷位移曲线略低于平面应力单元计算结果; 减缩积分算法有利于提高有限元数值计算结果的准确性。     

Buckling response of laminated composite stiffened plates subjected to partial in-plane edge loading

This article presents the buckling analysis of laminated composite stiffened plates subjected to partial in-plane edge loading. The finite element method is used to carry out the analysis. The eight-noded isoparametric degenerated shell element with C⁰ continuity and first-order shear deformation and a compatible three-noded curved beam element are used to model the plate skin and the stiffeners, respectively. The eigen value analysis is carried out to track the buckling load. The convergence study is performed for some specific problems and the results are compared with the available results in the literature. It is observed that the convergence of results is very fast for this finite element model. Effect of different parameters like orientation of fibers, number of layers, and loading types are considered in the present investigation. It is also observed that all these parameters have significant effect on the buckling response of the composite stiffened plate.     

On the influence of laminate stacking on buckling of composite cylindrical shells subjected to axial compression

The buckling loads of laminated cylinders can strongly depend on the position of the differently oriented layers within the shell. This paper deals with two different laminated orthotropic cylinders with opposite stacking sequence of the laminate layers. Cylinders of this construction had been thoroughly tested within a BRITE EURAM project. Analytical and semi-analytical methods have been used to predict the buckling loads, and the results are reported in this paper as well as test results for comparison. An explanation of the striking influence of stacking sequence is given. With some more examples the findings are verified. It is suggested that the presented results can be used for benchmarking purpose.     

Post-buckling behaviour of thermoplastic matrix composite laminates subjected to pure shear

The present study focuses on the determination of the buckling load and post-buckling behaviour of simply supported laminated composite rectangular panels loaded in shear. The nonlinear structural response is studied with a non-linear finite element approach. In order to determine the accuracy of the procedure, several tests have been performed comparing the finite element solutions for isotropic and laminated composite rectangular panels with existing ones, adopting different sequences of lamination and different length to width ratios. The analysis then considers the behaviour of laminates produced with innovative thermoplastic matrix composites developed in the frame of a national research program.     

Study of the effect of in-plane and interlaminar stresses on damping in fluid-filled laminated composite cylinders

The modal strain energy method is used in conjunction with a three-dimensional finite element analysis in the characterization of the effects of in-plane and interlaminar stresses in fluid-filled composite laminate cylindrical shells. A semi-analytical, 8-noded isoparametric finite element, which includes both the symmetric and antisymmetric modes in the circumferential direction, is used in the analysis. The effects of fiber angle, contained fluid height, size parameter of the shell, and stacking sequence on the contribution of in-plane and interlaminar stresses to the overall system damping in fluid-filled, composite laminate cylindrical shells are studied.     

Control of mechanical deformation of a laminate by electrical load to piezoelectric actuator considering the effects of damping and transverse shear

In this paper, we treat the control of dynamic deformation of a laminate by applying electrical load to piezoelectric actuators. Dynamic behavior of the laminate is analyzed considering the effect of damping due to interlaminar shear and the effect of transverse shear. The analytical model is a rectangular laminate composed of fiber-reinforced laminae and piezoelectric layers. The model is assumed to be a symmetric cross-ply laminate with all edges simply supported and to be subjected to unavoidable mechanical load and to electrical loads to piezoelectric actuators. Behavior of the laminate is analyzed based on the first-order shear deformation theory. The effect of damping due to interlaminar shear is incorporated into our analysis by introducing the interlaminar shear stresses which satisfy the Newton’s law of viscosity. The following quantities are obtained: (1) natural frequencies of the laminate, (2) weight functions for the deflection and rotations and (3) transient deflection due to loads varying arbitrarily with time. Moreover, the methods to control the deflection due to mechanical load by applying electrical voltage to the piezoelectric actuator are shown.     

In situ study of short-beam shear tests for composite materials

The shear behaviour of a unidirectional E-glass/epoxy composite was studied with four-point short-beam bending tests carried out in the interior of a scanning electron microscope. The damage process of the composite was followed during the tests and the local matrix shear strain was measured. A relationship was established experimentally between the beam shear stress and the local interface shear strain. Finite element calculations gave the correct stress distribution in the beam.     

Interaction of laminate damage and adhesive disbonding in composite scarf joints subjected to combined in-plane loading and impact

Impact tests were carried out on composite laminates and composite scarf repairs, while both were subjected to in-plane loading with tensile pre-strain levels up to 5000 microstrain. The results show that pre-straining of the composite laminates has no noticeable influence on the size of the delamination area for the given impact energy of 8 J, which represents a typical barely-visible impact on thin-skin composite structures. For composite scarf joints, however, resulting damage has been found to be a combination of adhesive disbonding and matrix cracking (delamination and intraply cracking) in the composite laminate. The size of this mixed type of damage increases significantly with increasing pre-strain levels. A finite element model was developed to investigate the interaction between adhesive disbonding and composite delamination. The computational results reveal that both delamination and adhesive disbonding are dominated by the mode II fracture. Since the critical mode II fracture energy release rate for composite laminates (G_IIC = 1.08 kJ/m²) is much less than that pertinent to the adhesive (G_IIC = 3.73 kJ/m²), delamination tends to occur first in the composite laminates, which then shield the growth of disbonding in the adhesive.     

Experimental, numerical and analytical results for bending and buckling of rectangular orthotropic plates

Solid unstiffened, sandwich and hat-stiffened rectangular orthotropic fiber reinforced plastic (FRP) plates were tested for buckling by in-plane compression and for stresses and deflections under uniform out-of-plane pressure. The solid unstiffened and hat-stiffened plates were 154 × 77 cm (1 × w) (72 × 36 in), while the sandwich plates were 102 × 77 cm (1 × w) (48 × 36 in). Balsa core was used in the sandwich plates and in the hat-stiffeners. The two short edges of the unstiffened and sandwich plates were clamped, while the two long edges were simply supported. The two long edges of the hat-stiffened plates were free, while the short edges were clamped. The buckling load, as well as stresses and deflections from the tests, were then compared to those from finite element analysis (FEA) and analytic solutions. There was reasonably good agreement between FEA, analytic, and experimental buckling stresses for the unstiffened solid plates. There was reasonable agreement in buckling stresses between FEA and experimental results for the hat-stiffened plate. There was poor agreement between FEA, analytic, and experimental elastic buckling results for the sandwich plates because they failed in local buckling prior to global buckling. Under out-of-plane uniform pressure, FEA and analytic solutions of the stresses and deflections for the unstiffened solid plates agreed well with experimental results. There was poor agreement between FEA and experimental results for stresses and deflections of the hat-stiffened or sandwich plates. Experimental error could be traced, in part,to plate fabrication, the method of applying out-of-plane pressure, edge support, and instrumentation accuracy.     

Dynamic pulse-buckling behavior of ‘quasi-ductile'' carbon/epoxy and E-glass/epoxy laminated composite beams

Dynamic pulse-buckling response of carbon/epoxy and E-glass/epoxy laminated composite beams with [(±67.5)_n]_s ply sequence, subject to axial impact was investigated experimentally and numerically. The laminated beams deformed like ductile metals, retaining a residual deformed shape after being axially impacted, exhibiting no obvious delamination. The ‘crest' deflection of the beams was found to be linearly proportional to the impact energy. The numerical investigation showed that the beams' top and bottom surfaces experienced stresses (transverse stress component) in excess of the tensile strength limits of the matrices.     

Efficient modelling of delamination buckling in composite cylindrical shells under axial compression

Composite cylindrical shells and panels are widely used in aerospace structures. These are often subjected to defects and damage from both in-service and manufacturing events. Delamination is the most important of these defects. This paper deals with the computational modelling of delamination in isotropic and laminated composite cylindrical shells. The use of three-dimensional finite elements for predicting the delamination buckling of these structures is computationally expensive. Here combined double-layer and single-layer of shell elements are employed to study the effect of delamination on the global load-carrying capacity of such systems under axial compressive load. It is shown that through-the-thickness delamination can be modelled and analysed effectively without requiring a great deal of computing time and memory. A parametric study is carried out to study the influence of the delamination size, orientation and through-the-width position of a series of laminated cylinders. The effect of material properties is also investigated. Some of the results are compared with the corresponding analytical results. It is shown that ignoring the contact between the delaminated layers can result in wrong estimations of the critical buckling loads in cylindrical shells under compressive load.     

Numerical analysis of fracture in ceramic coatings subjected to thermal loading

In this paper the general purpose finite element code ANSYS has been employed to analyse fracture in ceramic coatings subjected to thermal loading. An approach is developed in which hypothetical material properties have been considered as material data for coupled (thermal and structure) finite element analysis. These properties were chosen by assumed changes in some functional properties of ZrO₂-G.G. coatings. The aim was to evaluate the stress intensity factors in different coatings. Furthermore, to demonstrate the influence of crack length and coating geometry on the stress intensity in coatings, finite element analyses were carried out for various cases. The normalized stress intensity factors were obtained. The results showed that the shorter the crack length and the thinner the coating, the sounder the coatings. Furthermore, coatings representing a wide range of thermal and mechanical properties have a close normalized stress intensity factor values. It is also concluded that the finite element technique can be used to optimize the design and the processing of ceramic coatings.     

Micromechanical modelling of oval particulates subjected to bi-axial compression

One of the questions that still remain unanswered among researchers dealing with granular materials is how far the particle shape affects the micro-macroscopic features of granular assemblies under mechanical loading. The latest advances made with particle instrumentation allow us to capture realistic particle shapes and size distribution of powders to a fair degree of accuracy at different length scales. Industrial applications often require information on the micromechanical behaviour of granular assemblies having different particle shapes and varying surface characteristics, which still remains largely unanswered. Traditionally, simulations based on discrete element method (DEM) idealise the shape of individual particles as either circular or spherical. In the present investigation, we analyse the influence of particle shape on the shear deformation characteristics of two dimensional granular assemblies using DEM. We prepared the assemblies having nearly an identical initial packing fraction (dense), but with different basic shapes of the individual particles: (a) oval and (b) circular for comparison purposes. The granular assemblies were subjected to bi-axial compression test. We present the evolution of macroscopic strength parameters and microscopic structural/topological parameters during mechanical loading. We show that the micromechanical properties of granular systems are significantly influenced by the shape of the individual particles constituting the granular assemblies.     

Classes of exact solutions for buckling of multi-step non-uniform columns with an arbitrary number of cracks subjected to concentrated and distributed axial loads

The governing differential equation for buckling of a multi-step non-uniform column subjected to combined concentrated and distributed axial loads, each step of which has an arbitrary number of cracks with or without spring supports, is expressed in the form of bending moment. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks in the column. In this paper, the distribution of flexural stiffness of a non-uniform column is arbitrary, and the distribution of axial forces acting on the column is expressed as a functional relation with the distribution of flexural stiffness and vice versa. The governing equation for buckling of a one-step non-uniform column is reduced to a differential equation of the second-order without the first-order derivative by means of functional transformation. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for six important cases. The exact buckling solutions of one-step non-uniform columns are thus found. Then a new approach that combines the exact buckling solution of a one-step column and the transfer matrix method is presented to establish the eigenvalue equation for buckling of a multi-step non-uniform column with spring supports. The main advantage of the proposed method is that the eigenvalue equation for buckling of a non-uniform column with an arbitrary number of cracks, any kind of two end supports and various spring supports at intermediate points can be conveniently determined from a second order determinant. Due to the decrease in the determinant’s order as compared with previously developed procedures the computational time required by the present method can be reduced significantly. A numerical example is given to examine the accuracy of the proposed method and to investigate the effect of cracks on buckling of a multi-step non-uniform column.     

Non-linear buckling and postbuckling of shear deformable anisotropic laminated cylindrical shell subjected to varying external pressure loads

Non-linear buckling and postbuckling of a moderately thick anisotropic laminated cylindrical shell of finite length subjected to lateral pressure, hydrostatic pressure and external liquid pressure has been presented in the paper. The material of each layer of the shell is assumed to be linearly elastic, anisotropic and fiber-reinforced. The governing equations are based on a higher order shear deformation shell theory with von Kármán–Donnell-type of kinematic non-linearity and including the extension/twist, extension/flexural and flexural/twist couplings. The non-linear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, moderately thick, anisotropic laminated cylindrical shells with different values of shell parameters and stacking sequence. The results confirm that there exists a circumferential stress along with an associate shear stress when the shell is subjected to external pressure.     

Buckling of filament-wound composite cylinders subjected to hydrostatic pressure for underwater vehicle applications

The buckling and failure characteristics of moderately thick-walled filament-wound carbon–epoxy composite cylinders under external hydrostatic pressure were investigated through finite element analysis and testing for underwater vehicle applications. The winding angles were [±30/90]_FW, [±45/90]_FW and [±60/90]_FW. ACOS, an in-house finite element program, successfully predicted the buckling pressure of filament-wound composite cylinders with 2 ∼ 23% deviation from the test results. The analysis and test results showed that the cylinders do not recover the initial buckling pressure after buckling and that this leads directly to the collapse. Major failure modes in the test were dominated by the helical winding angles.