Free vibration of composite plates using the finite difference method

The finite difference method was used to solve differential equations of motion of free vibration of composite plates with different boundary conditions. The effects of shear deformation and rotary inertia on the natural frequencies of laminated composite plates are investigated in this paper. Four cases are studied: neglecting both shear deformation and rotary inertia, considering only rotary inertia, considering only shear deformation, and considering both. Solutions were obtained for symmetric and angle-ply laminated plates. The factors that affect natural frequencies of different composite plates, such as span-to-depth ratio, aspect ratio, angle-ply, and lamination sequence were also investigated. Results were found to agree well with exact and approximate solutions reported in literature. Shear deformation showed a considerable effect on the natural frequencies for composite plates, whereas the rotary inertia effect was found to be negligible.     

Transient dynamic response of composite circular cylindrical shells under radial impulse load and axial compressive loads

Free and forced vibration of composite circular cylindrical shells are investigated based on the first love's approximation theory using the first-order shear deformation shell theory. The boundary conditions (BCs) are considered as clamped-free edges. The dynamic response of the composite shells is studied under transverse impulse and axial compressive loads. The axial compressive load was less than critical buckling loads. The modal technique is used to develop the analytical solution of the composite cylindrical shell. The solution for the shell under the given loading conditions can be found using the convolution integrals. The effect of fiber orientation, axial load, and some of the geometric parameters on the time response of the shells has been shown. The results show that dynamic responses are governed primarily by natural period of the structure. The accuracy of the analysis has been examined by comparing results with those available in the literature and experiments.     

复合圆柱形壳在径向冲击荷载和轴压荷载作用下的瞬时动力响应

基于首次逼近理论,利用第一阶剪切变形理论对圆柱形壳的自由和强迫振动进行分析。边界条件（BCs）考虑为悬臂状态。分析复合壳体在横向冲击和轴向压力作用下的动力响应（轴压荷载小于临界屈曲荷载）,同时对复合柱形壳进行了建模分析。利用卷积积分对给定荷载状况下的壳体进行分析。揭示纤维方向、轴向荷载及一些几何参数对壳体时间响应的影响。结果表明：动力响应主要由结构的自振周期所控制。     

Free vibration of a paraboloidal shell of revolution including shear deformation and rotary inertia

The governing strain-displacement and curvature-displacement equations for paraboloidal shells including shear deformation and rotary inertia are solved for free vibration of closed shells. The finite element method is used to obtain three-dimensional frequency of vibration solutions for a variety of boundary conditions, free, fixed and simply supported. Assumptions concerning the circumferential vibrational behavior are incorporated that reduce the analysis to a single coordinate and the element shape function is formulated using the meridional coordinate. The results for frequency of vibration compare favorably with the available literature. Selected results for frequency of vibration are presented in tabular form for several shell parameters, including free, pinned and fixed boundary conditions. Representative mode shapes are plotted for a fixed boundary condition.     

Vibration Finite Element Analysis of Bimodular Laminates Using a Higher-Order Plate Theory

Abstract: In this work, a simple C° isoparametric higher-order plate element is developed to study the free vibration analysis of bimodulus laminated plates. The eqriations of motion for the higher-order plate theory are also derived variationally. The warping of cross section and transverse shear deformation are both considered. The natural frequencies and neutral surface locations are determined for benchmark problems. Compared with available analytical solutions, fast convergence and excellent agreement are observed for the present finite element formulation.     

夹层玻璃板PVB动力损伤检测的有限元格式

提出了一种夹层玻璃板的一阶剪切变形位移模式,由此建立了静力计算和自由振动的八结点有限等参单元公式.同时本文将“频率变化比”方法应用到夹层玻璃板PVB损伤检测中,用推导出的有限元公式建立了频率变化比法的分析过程.     

板壳结构非线性屈曲分析的修正拉格朗日法

本文采用退化曲壳有限单元,推导了板壳结构非线性有限元分析的、修正的拉格朗日法(Updated Lagrauge简称U.L.法,下同),并编制了非线性有限元程序.利用本文理论方法既可分析板壳的太变形问题,同时也可考虑材料进入非线性后的太应变问题.通过对一些板壳屈曲问题的分析对比,证明了本文理论方法的正确性和有效性.     

Analysis and optimization of laminated composite circular cylindrical shell subjected to compressive axial and transverse transient dynamic loads

Optimization is one of the important stages in the design process. In this paper the genetic algorithms method is applied for weight and transient dynamic response and two constraints including critical buckling loads and principle strains optimization of laminated composite cylindrical shells. The multi-objective function seeks the minimum structural weight and transient dynamic response. Nine design variables including material properties (fibre and matrix), volume fraction of fibre, fibre orientation and thickness of each layer are considered. In analytical solution, vibration of composite circular cylindrical shells are investigated based on the first-order shear deformation shell theory. The boundary conditions are assumed to be fully simply support. The dynamic response of the composite shells is studied under transverse impulse and axial compressive loads. The modal technique is used to develop the analytical solution of the composite cylindrical shell. The solution for the shell under the given loading conditions can be found using the convolution integrals. An example of simply supported laminated composite cylindrical shells is given to demonstrate the optimality of the solution obtained by the genetic algorithms technique. Results are shown that the weight coefficient of multi-objective function and the type of the constraints have considerable effect on the optimum weight and dynamic response.     

Tension buckling and parametric instability characteristics of doubly curved panels with circular cutout subjected to nonuniform tensile edge loading

This paper deals with the study of tensile buckling, vibration, and parametric instability behaviour of doubly curved panels with central circular cutout subjected to uniaxial in-plane partially distributed tensile edge loadings using finite element method. First order shear deformation theory is used to model the curved panels, to consider the effects of the transverse shear deformation and rotary inertia. The vibration analysis for this problem shows that for certain parameters of the tensile loading, the frequencies of the panel initially rise with the load, but begin to decrease with increasing tension, showing the onset of tension buckling. The parametric instability behaviour under tensile periodic edge loading with different load parameters shows that instability regions are influenced by the cutout size, load width and its location.     

Dynamic analysis of functionally graded truncated conical shells subjected to asymmetric moving loads

The dynamic behavior of functionally graded (FG) truncated conical shells subjected to asymmetric internal ring-shaped moving loads is studied. The material properties are assumed to have continuous variations in the shell thickness direction. The equations of motion are derived based on the first-order shear deformation theory (FSDT) using Hamilton׳s principle. The finite element method (FEM) together with Newmark׳s time integration scheme is employed to discretize the equations of motion in the spatial and temporal domain, respectively. The formulation and method of solution are validated by studying their convergence behavior and carrying out the comparison studies in the limit cases with existing solutions in the literature. Then, the influences of material graded index, radius-to-length ratio, semi-vertex angle, thickness, boundary conditions and moving load velocity on the dynamic behavior of the FG truncated conical shells are studied. In addition, the difference between the responses of the FG shells under symmetric and asymmetric loadings is compared.     

Geometrically exact displacement-based shell theory

The deformed geometry often is the most important information for applications of highly flexible plates/shells, and a geometrically exact shell theory should be displacement-based in order to directly and exactly describe any greatly deformed geometry. The main challenges of modeling a shell undergoing large deformation are how to describe its deformed reference plane and its differential element's large rotations and how to derive objective strains in terms of global displacements and rotations that contain both elastic straining and rigid-body movement. This paper presents a truly geometrically exact displacement-based shell theory without singularity problems. The theory fully accounts for geometric nonlinearities, all possible initial curvatures, and extensionality by using Jaumann strains and stresses, exact coordinate transformation, and orthogonal virtual rotations. Moreover, transverse shear deformations are accounted for by using a high-order shear deformation theory. The derived fully nonlinear strain–displacement relations enable geometrically exact forward analysis (obtaining the deformed geometry under a set of known loads) and inverse analysis (obtaining the required loads for a desired deformed geometry). Several numerical examples are used to demonstrate the accuracy and capabilities of the geometrically exact shell theory. Moreover, different theoretical and numerical problems of other geometrically nonlinear shell theories are shown to be mainly caused by the use of Mindlin plate theory to account for transverse shears, Green–Lagrange strains to account for geometric nonlinearities, and/or Euler and Rodrigues parameters to model large rotations.     

Nonlinear free vibration of tapered Mindlin plates with edges elastically restrained against rotation using DQM

Large amplitude free vibration analyses of tapered Mindlin rectangular plates with elastically restrained against rotation edges are investigated using different differential quadrature method (DQM). The governing equations are based on the first-order shear deformation plate theory in conjunction with Green's strain and von Karman assumption. The spatial derivatives are discretized using DQM and the harmonic balance method is used to transform the resulting differential equations into frequency domain. A direct iterative method is used to solve the nonlinear eigenvalue system of equations. The convergence of the method is shown and their accuracy is demonstrated by comparing the results with those of the limiting cases, i.e. nonlinear free vibration analysis of plates with classical boundary conditions and also linear free vibration analysis of tapered plates. The effects of the elastic restraint coefficient at the edges and the geometrical parameters on the ratio of the nonlinear natural frequency to linear natural frequency of plates with linearly and bi-linearly varying thickness are studied.     

A Four-Variable First-Order Shear Deformation Theory Considering the Variation of In-plane Rotation of Functionally Graded Plates

This paper presents a four-variable first-order shear deformation theory considering in-plane rotation of functionally graded plates. In recent studies, a simple first-order shear deformation theory was developed and extended to functionally graded plates. It has only four variables, separating the deflection into bending and shear parts, while the conventional first-order shear deformation theory has five variables. However, this simple first-order shear deformation theory only provides good predictions for simply supported plates since it does not consider in-plane rotation varying through the thickness of the plates. The present theory also has four variables, but considers the variation of in-plane rotation such that it is able to correctly predict the responses of the plates with any boundary conditions. Analytical solutions are obtained for rectangular plates with various boundary conditions. Comparative studies demonstrate the effects of in-plane rotation and the accuracy of the present theory in predicting the responses of functionally graded plates.     

Free vibration analysis of variable thickness thin and moderately thick plates with elastically restrained edges by DQM

A differential quadrature (DQ) procedure is developed for free vibration analysis of variable thickness moderately thick plates with edges elastically restrained against translation and rotation. The governing equations are based on the first order shear deformation theory and the rotary inertia effects are considered. Comparisons with known thin plate and uniform thickness Mindlin plate solutions are carried out to verify the applicability and accuracy of the analysis. Plates with linear or nonlinear varying thickness in one or two directions can be considered. It is demonstrated that using this DQ procedure, classical boundary conditions such as simply supported, clamped and free edges for variable thickness, thin as well as moderately thick plates can be simulated without any numerical difficulties. The effects of different combinations of constraints at edges, aspect ratio, thickness-to-length ratio, and stiffness parameter values on accuracy and convergence behaviors of the plates are presented. Some new results are presented for bi-linearly variable thickness plates with elastically restrained edges.     

Stability analysis of cross-ply laminated shells of revolution using a curved axisymmetric shell finite element

A curved axisymmetric shell finite element based on a consistent first-order shear deformable shell theory is developed for the linear stability analysis of cross-ply laminated shells of revolution under compressive loads. Finite element analysis results are presented for isotropic, orthotropic and cross-ply laminated shells of revolution in comparison with the analytical and numerical results found in the literature. These comparisons demonstrate the applicability and the high performance of the element in stability analysis of thin and moderately thick cross-ply laminated composite shells of revolution under compressive loads.     

Accurate dynamic response of laminated composites and sandwich shells using higher order zigzag theory

Forced vibration response of laminated composite and sandwich shell is studied by using a 2D FE (finite element) model based on higher order zigzag theory (HOZT). This is the first finite element implementation of the HOZT to solve the forced vibration problem of shells incorporating all three radii of curvatures including the effect of cross curvature in the formulation using Sanders' approximations. The proposed finite element model satisfies the inter-laminar shear stress continuity at each layer interface in addition to higher order theory features, hence most suitable to model sandwich shells along with composite shells. The C₀ finite element formulation has been done to overcome the problem of C₁ continuity associated with the HOZT. The present model can also analyze shells with cross curvature like hypar shells besides normal curvature shells like cylindrical, spherical shells etc. The numerical studies show that the present 2D FE model is more accurate than existing FE models based on first and higher order theories for predicting results close to those obtained by 3D elasticity solutions for laminated composite and sandwich shallow shells. Many new results are presented by varying different parameters which should be useful for future research.     

Analysis and shape optimisation of variable thickness prismatic folded plates and curved shells — Part 1: Finite strip formulation

This paper deals with the linear elastic analysis of prismatic folded plate and shell structures supported on diaphragms at two opposite edges with the other two edges arbitrarily restrained. The analysis is carried out using curved, variable thickness, Mindlin-Reissner finite strips. The theoretical formulation is presented for a family of C (0) strips and the accuracy and relative performance of the strips are examined for curved situations. Some variable thickness and elastically supported plates are considered and the interesting phenomenon of the occurence of boundary layers in the twisting moments and shear forces is highlighted for a common boundary condition. Other examples analysed include box girders and cylindrical shells. In all cases transverse shear deformation effects are included and the contributions to the strain energy from membrane, bending and transverse shear behaviour noted. In a companion paper these accurate and inexpensive finite strips are used for structural shape optimisation.     

Uncertainty propagation in dynamics of composite plates: A semi-analytical non-sampling-based approach

In this study, the influences of spatially varying stochastic properties on free vibration analysis of composite plates were investigated via development of a new approach named the deterministic-stochastic Galerkin-based semi-analytical method. The material properties including tensile modulus, shear modulus, and density of the plate were assumed to be spatially varying and uncertain. Gaussian fields with first-order Markov kernels were utilized to define the aforementioned material properties. The stochastic fields were decomposed via application of the Karhunen-Loeve theorem. A first-order shear deformation theory was assumed, following which the displacement field was defined using admissible trigonometric modes to derive the potential and kinetic energies. The stochastic equations of motion of the plate were obtained using the variational principle. The deterministic-stochastic Galerkin-based method was utilized to find the probability space of natural frequencies, and the corresponding mode shapes of the plate were determined using a polynomial chaos approach. The proposed method significantly reduced the size of the mathematical models of the structure, which is very useful for enhancing the computational efficiency of stochastic simulations. The methodology was verified using a stochastic finite element method and the available results in literature. The sensitivity of natural frequencies and corresponding mode shapes due to the uncertainty of material properties was investigated, and the results indicated that the higher-order modes are more sensitive to uncertainty propagation in spatially varying properties.     

Dynamic stability analysis of composite skew plates subjected to periodic in-plane load

Here, the dynamic stability characteristics of simply supported laminated composite skew plates subjected to a periodic in-plane load are investigated using the finite element approach. The formulation includes the effects of transverse shear deformation, in-plane and rotary inertia. The boundaries of the instability regions are obtained using the Bolotin's method and are represented in the non-dimensional load amplitude-excitation frequency plane. The principal and second instability regions are identified for different parameters such as skew angle, thickness-to-span ratio, fiber orientation and static in-plane load.     

预应力纤维金属层板圆柱壳受侧向脉冲压力时的动力响应

进行承受初始轴向和内部压力的纤维金属层板柱壳的动力响应分析。基于Love一次近似理论,利用一次剪切变形理论建立壳体平衡方程与拉力-位移关系。采用伽辽金法求解屈曲平衡方程、壳体自由及受迫振动问题。研究纤维金属层板参数如:材料属性、板层数、纤维方位、预应力对动力性能的影响。结果表明,各种板层数与预应力情况下,纤维金属层板对结构自振频率及动力性能均具有重要影响。