Bending and vibration of composite laminated plates

Hybrid-stress finite element method is applied for analysis of bending and vibration of composite laminated plates in this paper. Firstly, based on the modified complementary principle, a rectangular hybrid-stress plate bending element is presented which applies to analysis of laminates. Inside the element, different stress parameters are assumed according to different layers. The boundary displacements are determined by means of the assumption of YNS theory on the boundary of elements. The element formed in this way not only can take effects of transverse shear deformation and local warping into account, but also has less degrees of freedom. Then, problems of bending and vibration of laminates are solved by using this element, and the numerical results are compared with the exact solutions. This shows that the results obtained in the paper are very close to the exact results.     

Nonlinear bending behavior of 3D braided rectangular plates subjected to transverse loads in thermal environments

Nonlinear bending behavior of 3D braided rectangular plates subjected to transverse loads is investigated. A new micro-macro-mechanical model of unit cells is suggested. In this model, a 3D braided composite may be considered as a cell system and the geometry of each cell is deeply dependent on its position in the cross-section of the plate. The material properties of the epoxy are expressed as a linear function of temperature. Based on Reddy’s higher-order shear deformation plate theory and general von Kármán-type equations, analytical solutions for nonlinear bending behavior of simply supported 3D braided rectangular plates are obtained using mixed Galerkin-perturbation method. The numerical examples concern effects of geometric parameters, of fiber volume fraction, braiding angle and load boundary condition.     

基于高阶剪切变形理论的四边形求积元板单元及其应用

近年来由各类新型复合材料或功能梯度材料构成的板结构在工程领域得到了广泛应用,其显著特点是材料性能沿板厚变化.为合理考虑横向剪切应变,许多学者基于Reddy高阶剪切变形理论,构建了不同的有限元单元对该类板结构进行分析,但其中满足$C^{1}$连续条件的单元相对较少.本文基于Reddy高阶剪切变形理论,采用求积元方法,建立了$C^{1}$连续的四边形板单元.利用该单元对均质材料、复合材料、功能梯度材料构成的等厚度矩形板、变厚度矩形板及等厚度斜板的线弹性弯曲和自由振动问题进行了计算分析,并与现有文献中的相应计算结果进行了对比.研究表明:基于高阶剪切变形理论的四边形求积元板单元具有较高的计算效率和良好的适应性,文中各类材料构成的等变厚度矩形板及等厚度斜板均只需1个单元即可得到理想的计算结果.对于等/变厚度矩形板,可仅使用9$\times$9个积分点,而对于等厚度斜板,随着斜角的增大,所需积分点的数目逐渐增多至15$\times $15.该四边形求积元板单元可进一步用于新型复合材料板的非线性分析.      

NONLINEAR THREE-DIMENSIONAL ANALYSIS OF COMPOSITELAMINATED PLATES

ＮＯＮＬＩＮＥＡＲＴＨＲＥＥ－ＤＩＭＥＮＳＩＯＮＡＬＡＮＡＬＹＳＩＳＯＦＣＯＭＰＯＳＩＴＥＬＡＭＩＮＡＴＥＤＰＬＡＴＥＳ￥(江晓禹，张相周，陈百屏)ＪｉａｎｇＸｉａｏｙｕ；（ＳｏｕｔｈｗｅｓｔｅｒｎＪｉａｏｔｏｎｇＵｎｉｖｅｒｓｉｔｙ，Ｃｈｅｎｇｄｕ６...     

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The fiat shell element is constructed by combining a quadrilateral area co- ordinate method （QAC） based membrane element AGQ6- II, and a Timoshenko beam function （TBF） method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory （FSDT）, for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical re- sults show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.     

Analysis of composite laminated plates

In the static and dynamic analysis of composite laminates, a theory for the laminated plates is presented in this paper. Because the deflection W_b which is caused by the classical bending deformation and the deflection W₅ which is caused by the shear deformation are divided from the total deflection W in the theory, this makes it easy to solve the governing equations. In addition, this theory is convenient for the discussion and analysis of the effects of transverse shear deformations on bendings, vibrations and stabilities of laminated plates.     

复合结构剪切板的屈曲与后屈曲研究

基于Ｒｅｉｓｓｎｅｒ剪切板理论，本文将现有理论模型推进了一步，导出了控制复合材料和构造矩形板平衡的广义Ｋａｒｍａｎ大挠度方程。此外，运用摄动技术，构造了一个单向压缩简支剪切矩形板的后屈曲渐近解，适用于面板和夹心材料都具正交各向异性的夹层板以及计及横向剪切变形对称铺设的层合板。文中给出了典型算例，并将当前结果与其它理论或数值解作了比较，显示出相当好的一致性。     

材料非线性对复合材料层合板热自由边界效应的影响

本文用准三维有限元法研究了材料非线性对复合材料层合板热自山边界效应的影响,给出了修正型Hahn-Tsai非线性应力-应变关系的三维形式。由本文非线性分析方法得到的层间应力与以往由线性分析方法得到的层间应力做了比较,结果表明:材料非线性能显著降低层间剪应力的集中程度,但对层间正应力影响不太明显。     

Nonlinear vibration of generally laminated anisotropic thick plates

Summary Nonlinear equations of motion of generally laminated anisotropic plates are derived by use of Hamilton's principle. The effects of transverse shear and rotatory inertia are included in the analysis. The equations of motion so obtained readily reduce to those obtained in a recent nonlinear theory of anisotropic plates including transverse shear and rotatory inertia and to the dynamic von Kármán equations of plates. Based on the Galarkin procedure and principle of harmonic balance approximate solutions to the governing equations of generally laminated rectangular plates are formulated for various boundary conditions. Including the effects of transverse shear and rotatory inertia numerical results for the ratio of nonlinear frequency to linear frequency of symmetric angle-ply and cross-ply laminates, unsymmetric angle-ply and cross-ply laminates and an arbitrarily laminated plate are presented graphically for various values of elastic properties, fiber orientation angle, number of layers, thickness-to-span ratio and aspect ratio. Present results are also compared with the existing data.

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Nichtlineare Schwingungen von anisotropen, dicken Verbundplatten

übersicht Die nichtlinearen Bewegungsgleichungen anisotroper, dicker Verbundplatten werden mit Hilfe des Prinzips von Hamilton hergeleitet. Dabei werden die Einflüsse der Schubspannungen und der Drehträgheit berücksichtigt. Mit dem Verfahren von Galerkin und der Methode der harmonischen Balance werden Näherungslösungen für Rechteckplatten bei verschiedenen Randbedingungen gewonnen. Numerische Ergebnisse für verschiedene Materialparameter sind graphisch dargestellt und werden mit bereits bekannten Ergebnissen verglichen.

     

LARGE AMPLITUDE FREE VIBRATIONS OF LAMINATED COMPOSITE PLATES

This paper deals with large amplitude free flexural vibrations of laminated composite plates using a 9-node Heterosis degenerated isoparametric quadrilateral element, including the effects of transverse shear and rotary inertia. The nonlinear dynamic equations of the plates are formulated in von Karman's sense. Amplitude-frequency relationships are obtained through dynamic response history using the Newmark numerical integration scheme. Detailed numerical results based on various parameters are presented for orthotropic laminated plates with different boundary conditions. The rectangular anti-symmetric cross-ply plates show the softening type of nonlinearity for initial small amplitudes. The displacement amplitudes decrease and nonlinear frequencies increase with the increment of time. Supported by the NNSFC (No. 19672033), the National Key Project on Basic Research and Applied Research (PD9521904) and the Doctoral Training Foundation of Education Commission of China(No. 98000304).     

复合材料层合厚板锯齿理论及有限元分析

对于较厚复合材料弯曲问题,已有锯齿型厚板理论最大误差超过35%。为了合理地分析较厚复合材料弯曲问题,发展了准确高效的锯齿型厚板理论。此理论位移变量个数独立于层合板层数,其面内位移不含有横向位移一阶导数,构造有限元时仅需C0插值函数,故称此理论为C0型锯齿厚板理论。基于发展的锯齿理论,构造了六节点三角形单元并推导了复合材料层合/夹层板弯曲问题有限元列式。为验证C0型锯齿厚板理论性能,分析了复合材料层合/夹层厚板弯曲问题,并与已有C1型锯齿理论对比。结果表明,本文的C0型锯齿厚板理论最大误差15%,比已有锯齿型厚板理论准确高效。     

Stochastic nonlinear bending response of laminated composite plates with system randomness under lateral pressure and thermal loading

In this paper, the effect of sensitivity of randomness in system parameters on the nonlinear transverse central deflection response of laminated composite plates subjected to transverse uniform lateral pressure and thermal loading is examined. System parameters such as the lamina material properties, expansion of thermal coefficients, lamina plate thickness and lateral load are modelled as basic random variables. A higher order shear deformation theory in the von-Karman sense is used to model the system behavior of the laminated plate. A direct iterative-based C ⁰ nonlinear finite element method in conjunction with the first-order perturbation technique developed by the authors is extended for thermal problem to obtain the second-order response statistics, i.e., mean and variance of the nonlinear transverse deflection of the plate. Typical numerical results of composite plates with temperature independent and dependent material properties subjected to uniform temperature and combination of uniform and transverse temperature are obtained for various combinations of geometric parameters, uniform lateral pressures, staking sequences and boundary conditions. The results have been compared with those available in the literature and an independent Monte Carlo simulation.     

The effect of initial geometric imperfection on the nonlinear resonance of functionally graded carbon nanotube-reinforced composite rectangular plates

The purpose of the present study is to examine the impact of initial geometric imperfection on the nonlinear dynamical characteristics of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) rectangular plates under a harmonic excitation transverse load. The considered plate is assumed to be made of matrix and single-walled carbon nanotubes (SWCNTs). The rule of mixture is employed to calculate the effective material properties of the plate. Within the framework of the parabolic shear deformation plate theory with taking the influence of transverse shear deformation and rotary inertia into account, Hamilton’s principle is utilized to derive the geometrically nonlinear mathematical formulation including the governing equations and corresponding boundary conditions of initially imperfect FG-CNTRC plates. Afterwards, with the aid of an efficient multistep numerical solution methodology, the frequency-amplitude and forcing-amplitude curves of initially imperfect FG-CNTRC rectangular plates with various edge conditions are provided, demonstrating the influence of initial imperfection, geometrical parameters, and edge conditions. It is displayed that an increase in the initial geometric imperfection intensifies the softening-type behavior of system, while no softening behavior can be found in the frequency-amplitude curve of a perfect plate.     

Mechanical behavior of functionally graded material plates under transverse load—Part I: Analysis

An elastic, rectangular, and simply supported, functionally graded material (FGM) plate of medium thickness subjected to transverse loading has been investigated. The Poisson’s ratios of the FGM plates are assumed to be constant, but their Young’s moduli vary continuously throughout the thickness direction according to the volume fraction of constituents defined by power-law, sigmoid, or exponential function. Based on the classical plate theory and Fourier series expansion, the series solutions of power-law FGM (simply called P-FGM), sigmoid FGM (S-FGM), and exponential FGM (E-FGM) plates are obtained. The analytical solutions of P-, S- and E-FGM plates are proved by the numerical results of finite element method. The closed-form solutions illustrated by Fourier series expression are given in Part I of this paper. The closed-form and finite element solutions are compared and discussed in Part II of this paper. Results reveal that the formulations of the solutions of FGM plates and homogeneous plates are similar, except the bending stiffness of plates. The bending stiffness of a homogeneous plate is Eh³/12(1 − ν²), while the expressions of the bending stiffness of FGM plates are more complicated combination of material properties.     

NONLINEAR FLUTTER OF LAMINATED PLATES WITH IN-PLANE FORCE AND TRANSVERSE SHEAR EFFECTS*

Based on a simplified higher-order shear deformation plate theory (SDPT) and von Karman large deformation assumption, a high-precision higher-order triangular-plate element that can be used to deal with transverse shear effects is developed for the nonlinear flutter analysis of composite laminates. The element presents no shear-locking problem due to the assumption that the total transverse displacement of the plate is expressed as the sum of the displacement due to bending and that due to shear deformation. Quasi-steady aerodynamic theory is employed for the flutter analysis. Newmark numerical time integration method is applied to solve the nonlinear governing equation in time domain. Results show that the in-plane force on the plate will increase the maximum plate displacement but will not influence the maximum plate motion speed. However, the aerodynamic pressure will increase both the maximum displacement and velocity of the plate. The transverse shear will have profound influence on the flutter boundary for a thick plate and under certain conditions it will change the plate motion from buckled but dynamically stable to a limit-cycle oscillation.     

Differential quadrature method for bending of orthotropic plates with finite deformation and transverse shear effects

Based on the Reddy ‘s theory of plates with the effect of higher-order shear deformations, the governing equations for bending of orthotropic plates with finite deformations were established. The differential quadrature ( DQ ) method of nonlinear analysis to the problem was presented. New DQ approach, presented by Wang and Bert ( DQWB), is extended to handle the multiple boundary conditions of plates. The techniques were also further extended to simplify nonlinear computations. The numerical convergence and comparison of solutions were studied. The results show that the DQ method presented is very reliable and valid. Moreover, the influences of geometric and material parameters as well as the transverse shear deformations on nonlinear bending were investigated. Numerical results show the influence of the shear deformation on the static bending of orthotropic moderately thick plate is significant.     

A state space approach for exact analysis of composite laminates and functionally graded materials

A class of problems of composite laminates and functionally graded materials (FGM) under extension, twisting, and bending is formulated in the state space setting. A solution approach for exact analysis of the deformation and stress fields in the media is developed. Exact solutions for torsion of cross-ply laminates and certain FGM are derived, which satisfy exactly the equations of anisotropic elasticity, the end conditions, the traction-free boundary conditions on the bounding planes of the rectangular section, and the interfacial continuity conditions in multilayered composite laminates, regardless of the number of layers. The solutions serve as useful benchmarks for numerical modeling and material characterization of composite laminates and FGM.     

Geometrically nonlinear large deformation analysis of triangular CNT-reinforced composite plates

A first known investigation on the geometrically nonlinear large deformation behavior of triangular carbon nanotube (CNT) reinforced functionally graded composite plates under transversely distributed loads is investigated. The analysis is carried out using the element-free IMLS-Ritz method. In this study, the first-order shear deformation theory (FSDT) and von Kármán assumption are employed to account for transverse shear strains, rotary inertia and moderate rotations. A convergence study is conducted by varying the supporting size and number of nodes. The effects of transverse shear deformation, CNT distribution and CNT volume fraction on the nonlinear bending characteristics under different boundary conditions are examined.     

Thermal bending analysis of shear-deformable laminated anisotropic plates by the GDQ method

The generalized differential quadrature (GDQ) method was used to determine the inter-laminar stresses and deflections in a laminated rectangular anisotropy plate under thermal bending involving the effect of shear deformation. We obtained the non-dimensional stresses and transverse center deflection in cross-ply and angle-ply anti-symmetric, anisotropic laminates subjected to thermal load with sinusoidal temperature distribution. We found that the shear deformation has significant effects on the stresses and deflections for laminated anisotropic plate with moderately side-to-thickness ratio under thermal load and bending state.     

ANALYSIS OF COMPOSITE LAMINATE BEAMS USING COUPLING CROSS-SECTION FINITE ELEMENT METHOD

Beams and plates manufactured from laminates of composite materials have distinct advantages in a significant number of applications. However, the anisotropy arising from these materials adds a significant degree of complexity, and thus time, to the stress and deformation analyses of such components, even using numerical approaches such as finite elements. The analysis of composite laminate beams subjected to uniform extension, bending, and/or twisting loads was performed by a novel implementation of the usual finite element method. Due to the symmetric features of the deformations, only a thin slice of the beam to be analysed needs to be modelled. Conventional three-dimensional solid finite elements were used for the structural discretization. The accurate deformation relationships were formulated and implemented through the coupling of nodal translational degrees of freedom in the numerical analysis. A sample solution for a rectangular composite laminate beam is presented to show the validity and accuracy of the proposed method.