Free vibration of composite rectangular plates with rectangular cutouts

A simple numerical method based on the Rayleigh principle is presented for predicting the natural frequencies of composite rectangular plates which exhibit special and general orthotropy. The method is illustrated for simply-supported rectangular plates having central rectangular cutouts and double square cutouts. The results are compared with the reported finite element and analytical results.     

Stress analysis of perforated composite plates

Thin-walled plates and panels of various constructions find wide use as primary structural elements in simple and complex configuration. In aerospace structures, panels with variously shaped cutout are often used. The understanding of the effects of cutout on the load bearing capacity and stress concentration of such plates is very important in designing of complex structures. An analytical investigation is used to study the stress analysis of plates with different central cutout. Particular emphasis is placed on flat square plates subjected to a uni-axial tension load. The results based on analytical solution are compared with the results obtained using finite element methods. The main objective of this study is to demonstrate the accuracy and simplicity of presented analytical solution for stress analysis of composite plates with central cutout. The effect of cutout geometry (circular, square, or special cutouts), material properties (isotropic and orthotropic), fiber angles, and cutout curvature are considered. The results presented herein, indicated that the presented method can be used to determine accurately the stresses and stress concentration in composite plates with special shape cutouts.     

Local fibre reinforcement of holes in composite multilayered plates

For various technical reasons, cutouts such as holes in thin-walled structures are inevitable and are of significant technical relevance. Unfortunately holes lead to an undesired stress concentration at the hole vicinity and a reduced strength of the structure. Therefore in practice a local reinforcement in the form of a ring is usually applied around the hole. The increasing requirements for modern structures in terms of low weight and high strength lead to the question of an optimal reinforcement design. The present paper addresses the new but well-approved techniques of the use of curved fibre format to determine the aforementioned optimal design of the reinforcement. The optimization of cutouts in laminated composite plates under bi-axial tensile loading conditions has been investigated using two approaches: the finite element and the Rayleigh-Ritz method. The used methodology implemented successfully theoretical results based on the complex potential theory. For the considered problems, the proposed methods were shown to successfully produce a constant objective function around the hole boundary under biaxial loading. The optimal reinforcement of holes in laminated composite plates illustrated that the optimum depends on the degree of orthotropy. Significant reduction of stress concentrations were demonstrated. The results obtained illustrate the necessity and usefulness of the applied optimization procedure.     

Global/local analysis of composite plates with cutouts

 The study focuses on the development of a simple and accurate global/local method for calculating the static response of stepped, simply-supported, isotropic and composite plates with circular and elliptical cutouts. The approach primarily involves two steps. In the first step a global approach, the Ritz method, is used to calculate the response of the structure. Displacement based Ritz functions for the plate without the cutout are augmented with a perturbation function, which is accurate for uniform thickness plates only, to account for the cutout. The Ritz solution does not accurately satisfy the natural boundary conditions at the cut-out boundary, nor does it accurately model the discontinuities caused by abrupt thickness changes. Therefore, a second step, local in nature is taken in which a small area in the vicinity of the hole and encompassing other points of singularities is discretized using a fine finite element mesh. The displacement boundary conditions for the local region are obtained from the global Ritz analysis. The chosen perturbation function is reliable for circular cutout in uniform plates, therefore elliptical cutouts were suitably transformed to circular shapes using conformal mapping. The methodology is then applied to the analysis of composite plates, and its usefulness successfully proved in such cases. The proposed approach resulted in accurate prediction of stresses, with considerable savings in CPU time and data storage for composite flat panels.     

Three-dimensional Finite Element Buckling Analysis of Honeycomb Sandwich Composite Shells with Cutouts

This paper investigates the buckling response of honeycomb sandwich composite shells with cutouts under axial compression. The Wilson's incompatible solid Finite Element (FE) is used around cutouts to obtain the detail stress distribution there. While to reduce the computational expense, a special multilayered relative degrees-of-freedom (DOF) shell FE is used to model the regions far from the cutouts. The efficiency and accuracy of this modeling scheme are illustrated by two benchmarks. Then parametric studies are carried out to reveal how the buckling response is influenced by the area, the shape and the orientation of cutouts.     

Boundary element method analysis of bending of inelastic plates with general boundary conditions

This paper presents an accurate boundary element method (BEM) formulation for the bending of inelastic Kirchhoff plates subjected to general boundary conditions. This approach is an extension of earlier work by the authors of this paper and other co-workers on elastic plate deformation where they had proposed a three-equation BEM scheme. Numerical results presented here include plates with cutouts and free edges. A rate type constitutive model is used here to describe nonelastic deformation behavior of the plate material.This research was performed while G.-S. Song was a visiting Scientist at Cornell University     

叠层复合材料方板多孔形状优化

采用一种无梯度仿生技术——基于等限制Tsai-Hill值准则的固定网格渐进优化方法(FGESO),研究了叠层复合材料方板在拉剪荷载时不同孔数、不同叠层构造条件下的最优孔形问题.在孔的周围不断把限制Tsai-Hill值小于删除标准的材料删除,直到稳定状态达到,然后提高删除标准继续迭代,直到达到指定的开孔面积.与传统渐进优化方法(ESO)的不同之处在于利用节点而不是单元的限制Tsai-Hill值来确定需要删除的材料,因此得到了比ESO更光滑的结果.例子证明了方法的普适性和有效性.还研究了两孔方板最优孔形的优化历程,结果反映了相邻开孔相互影响的一些规律.     

Mechanical Buckling Analysis of Composite Panels with/without Cutouts

A simplified analytical solution suitable for simple stacking sequences was developed using the Euler buckling theory,the structure’ s equations of equilibrium and laminate panel mathematical formulation.Comparing these results with numerical results reveals the accuracy of the method and even more,allows us to validate the numerical analysis.Therefore,two important results are obtained:a simplified analytical solution for the buckling problem and validation of the numerical results.Another important and novel finding is related to the influence of the angle ply orientation and of the cutouts,on the buckling load.Under symmetrical boundary conditions and loading case,rectangular panels with elliptical cutouts,give better results for 90° oriented plies than for 0 oriented ones.With a compression load applied in the X direction,and with material properties 10 times better in X direction than in Y direction,the best results are obtained when the load is aligned with the Y direction associated to the material reference frame.Moreover,panels with cutouts seem to behave better than panels without cutouts under certainply orientation angles.     

Effective widths of compression-loaded of perforated cross-ply laminated composites

In this study the influence of eccentric circular cutouts on the prebuckling and postbuckling stiffness, and effective widths of compression loaded laminated composite plates are presented. The effective-widths concept is derived based on nonlinear finite element analysis for the plates with and without cutout. Several behavioral trends are described that initially appear to be nonintuitive. The results demonstrate a complex interaction between cutout size and the degree of plate orthotropy that affects the axial stiffness and effective width of plate subjected to compression loads. Also these investigations show that the cutout dimension have a more considerable effect on prebuckling stiffness compare to postbuckling one. It will show that the stiffness ratio of the postbuckling over prebuckling is increased by cutout size. Considering the effective-width ratios concept provide a simple means for incorporating the postbuckling strength and stiffness of plate subcomponents into the design of stiffened structures.     

Behaviour of pipes with a circular hole subjected to compressive loading

In the fabrication of offshore structures, cutouts are often made on pipes to allow access to the interior of the pipes for maintenance and servicing. Thus, there is a necessity to investigate the effects of cutouts on the buckling load of pipes. Both the finite element and experimental techniques have been employed to investigate several series of pipes with and without cutouts. The results obtained were in reasonably good agreement with those obtained by the semiempirical solution.     

A strength-based multiple cutout optimization in composite plates using fixed grid finite element method

The design of interior cutouts in laminated composite panels is of great importance in aerospace, automobile and structural engineering. Based on the Tsai–Hill failure criterion of the first ply, this paper presents a newly developed Fixed (FG) Grid Evolutionary Structural Optimization (ESO) method to explore shape optimization of multiple cutouts in composite structures. Different design cases with varying number of cutouts, ply orientations and lay-up configurations are taken into account in this study. The examples demonstrate that the optimal boundaries produced by FG ESO are much smoother than those by traditional ESO. The results show the remarkable effects of different opening numbers and various lay-up configurations on resulting optimal shapes. The paper also provides an in-depth observation in the interactive influence of the adjacent cutouts on the optimal shapes.     

Experimental and computed natural frequencies of square pultruded GRP plates: effects of anisotropy, hole size ratio and edge support conditions

Experiments have been carried out to determine the free vibration frequencies and mode shapes of 3.2 mm thick, pultruded GRP, square plates with six combinations of clamped (C), simply supported (S) and free (F) edge supports. Comparison of experimental and theoretical/numerical frequencies confirms that thin homogeneous orthotropic/anisotropic plate theory provides a reasonable model for predicting the free vibration response of pultruded GRP plates. Additional vibration experiments were carried out on plates with central circular cutouts. The hole size ratios were varied from about 0.1 to 0.4 for three combinations of clamped (C) and simply supported (S) edge conditions. Finite-element (FE) frequency and mode shape predictions based on orthotropic plate theory were again shown to be in reasonable agreement with the experimental frequencies and modes.     

圆柱壳非圆大开孔的应力集中研究

本文以薄圆柱壳大开孔应力集中问题的理论解为基础,利用保角映射方法对圆柱壳非圆大开孔的应力集中问题进行研究。在给出映射函数的基础上,对承受不同外部载荷下开椭圆大孔圆柱壳孔边应力集中系数进行了数值计算,并与相应的圆柱壳小开孔理论解进行了比较。结果表明:大开孔解答更具有合理性。     

Failure Behavior and Strength of Composite I-Section Beam with Double Cutouts and Stiffener Reinforcement

This work is carried out to study the influence of double cutouts and stiffener reinforcements on the performance of I-section Carbon Fibre/Epoxy composites beam, including buckling, post-buckling behavior and the ultimate failure. The cantilever I-section beam with two diamond-shaped cutouts in the web and three longitudinal L-shaped stiffeners bonded to one side is subjected to a shear load at free end. Both numerical modelling and Experiment of I-section CFRP beam are performed. In numerical analysis, Tsai-Wu failure criterion is utilized to detect the first-ply-failure load in nonlinear analysis by predicting the load-deflection response. Good agreements are obtained from comparison between the numerical simulations and test results. For the double-hole beam web, the two cutouts show close surface deformation amplitude, which indicates that the stiffeners make the force transformation more effective. Comparing to the numerical result of corresponding beam with single cutout and stiffener reinforcement, the longitudinal stiffeners can not only play a significant role in improving the structural stability (increase about 30%), but also take effects to improve the deformation compatibility of structure. Local buckling happened within the sub-webs partioned by the stiffener and the buckling load is different but close. With post-buckling regime, the two areas show similar deformation characteristic, while the sub-web close to fixed end bears more shear load than the sub-web close to loading end with the increase of normal deformation of structure. The catastrophic failure load is approximate 75.6% higher comparing to buckling load. Results illustrate that the tensile fracture of the fiber is the immediate cause of the ultimate failure of the structure.     

Localized collapse of shells of revolution using a local-global strategy

The mode of collapse in many metallic shells is, oftentimes, one of localized collapse, wherein severe plastic deformation and high displacement gradients are localized within some region. The combined material and geometric non-linearity emanating from this local region is the predominant cause of instability in shells of revolution. In such a situation a local-global strategy is very useful for an efficient analysis. This strategy consists of employing two dimensional degenerated isoparametric shell elements with non-linear capability in a well defined 'local' zone where all the non-linearities are expected to be localized and which also contains some form of local deviation from axisymmetry such as a circumferential imperfection, cutouts, cracks, etc. Linear elastic ring type elements are employed in the remaining axisymmetric zone and the two zones are linked through a transformation between the degrees of freedom involved. The solution of the non-linear problem is achieved by appropriate condensation procedures to reduce the number of active degrees of freedom and the load incrementation is achieved by the well known 'arc length' iterative procedure. Numerical exampies are presented to demonstrate that this strategy is very efficient and accurate for problems with well defined non-linear local zones. The capability of this strategy for treating problems with local material discontinuities such as cracks, cutouts, etc. as an efficient alternative to a complete two dimensional discretization is pointed out.     

The effect of cutout positioning on the dynamic characteristics of a curved composite panel

An investigation of the effects of eccentrically located square cutouts on the natural frequencies and mode shapes of circular, cylindrical composite panels was conducted. A video holographic interferometry technique was used to determine experimentally the first five natural frequencies and mode shapes of the panels. Numerically, predictions for these same data were obtained using a finite element method. A clamped-free-clamped-free boundary condition was selected for this study. Cutouts were eccentrically located in both the circumferential and axial directions. The experimental and numerical results for both the natural frequencies and mode shapes correlated well. The per cent differences between the experimental and numerical frequencies averaged around 7% and were in all cases less than 10%. It was observed that, in general, panels with eccentrically located cutouts experienced only small frequency deviations from those with centered cutouts. The amount of variation observed tended to increase with cutout size.     

Multiscale finite element method for a locally nonperiodic heterogeneous medium

A generalization of the mathematical homogenization theory to account for locally nonperiodic solutions is presented. Such nonperiodicity may arise either due to the rapidly varying microstructure (e.g.: graded materials, microcracks) or because the macroscopic solution is not smooth and may have significant variation within a microstructure. In the portion of the problem domain where the material is formed by a spatial repetition of the base cell and the macroscopic solution is smooth, a double scale asymptotic expansion and solution periodicity are assumed, and consequently, mathematical homogenization theory is employed to uncouple the microscopic problem from the global solution. For the rest of the problem domain it is assumed that the periodic solution does not exist (cutouts, cracks, free edges in composites, etc.) and the approximation space is decomposed into macroscopic and microscopic fields. Compatibility between the two regions is explicitly enforced. The proposed method is applied to resolve the structure of the microscopic fields in the single ply composite plates with a centered hole and with a centered crack and in the [0/90] _s laminated plate. Numerical results are compared to the reference solution, an engineering global-local approach, and the direct extraction from the mathematical homogenization method.     

Cutout reinforcements for shear loaded laminate and sandwich composite panels

This paper presents the numerical and experimental studies of shear loaded laminated and sandwich carbon/epoxy composite panels with cutouts and reinforcements aiming at reducing the cutout stress concentration and increasing the buckling stability of the panels. The effect of different cutout sizes and the design and materials of cutout reinforcements on the stress and buckling behaviour of the panels are evaluated. For the sandwich panels with a range of cutout size and a constant weight, an optimal ratio of the core to the face thickness has been studied for the maximum buckling stability. The finite element method and an analytical method are employed to perform parametric studies. In both constant stress and constant displacement shear loading conditions, the results are in very good agreement with those obtained from experiment for selected cutout reinforcement cases. Conclusions are drawn on the cutout reinforcement design and improvement of stress concentration and buckling behaviour of shear loaded laminated and sandwich composite panels with cutouts.