Finite element free flexural vibration analysis of plates having various shapes and varying rigidities

Natural frequencies of skew, curved and tapered plates have been determined using the isoparametric quadratic plate bending element. Plates having linearly varying thickness in one direction and also those having parabolically varying thickness in orthogonal directions have been analysed. A simple stepped plate approach for a linearly tapered plate has also been considered to compare the results. Two approaches have been adopted for the solution of skew plates. In one case, the boundary conditions have been exactly satisfied by transforming the rotational displacements at the boundary nodes along and normal to the edge. In the other case, only the vertical deflection was locked. A curved plate has also been analysed by the first method. Results thus obtained from both cases have been compared.     

Fabrication of X-ray imaging zone plates by e-beam and X-ray lithography

X-ray imaging and microscopy techniques have been developed in worldwide due to their capabilities of large penetration power and high spatial resolution. Fresnel zone plates is considered to be one of the most convenient optic devices for X-ray imaging and microscopy system. The zone plates with aspect ratio of 7 and 13 have been fabricated by e-beam lithography combined with X-ray lithography in this paper. Firstly, the X-ray lithography mask of zone plates with outermost zone width of 100 nm was fabricated by e-beam lithography and gold electroplating techniques. Secondly, the zone plates with gold profile thickness of 700 and 1,300 nm were replicated by X-ray lithography and gold electroplating techniques. X-ray imaging and microscopy techniques were introduced to characterize the high-aspect-ratio zone plates’ inner structures. At the X-ray energy of 7.5 keV, the first-order focusing efficiency of zone plates with gold profile thickness of 700 nm is about 8.63%.     

Flexural failure analysis of anisotropic elliptic plates

An initial flexural failure analysis of laminated anisotropic elliptic clamped plates subjected to a uniform lateral pressure is presented. Two failure modes are distinguished: general lamina failure and interlaminar shear failure. For each of these modes dimensionless initial failure loads have been computed for a range of aspect ratios and fibre orientations. These results are complemented by a set of critical thickness ratios which serve to distinguish which mode of failure governs for a particular case and hence which failure loads apply. It is concluded that general lamina failure is the dominant failure mode for the practical range of thickness ratios, though the likelihood of interlaminar shear failure occurring is greater in CFRP plates than in GFRP plates.     

Nonlinear axisymmetric static and transient analysis of orthotropic thin tapered circular plates

This study deals with the geometrically nonlinear axisymmetric static and transient analysis of cylindrically orthotropic elastic thin tapered circular plates subjected to uniformly distributed and discrete central loads. Differential equations in terms of transverse displacement w and stress function ψ have been employed. The displacement w and stress function ψ are expanded in finite power series. The orthogonal point collocation method in space domain and Newmark-β scheme in time domain have been used. Step function dynamic loads are considered. Static and dynamic results have been presented for isotropic and orthotropic immovable clamped and simply supported plates with linearly varying thickness for three values of taper ratios and the effect of varying thickness has been investigated. A simple approximate method is used to predict the maximum dynamic response to step load from the results for static loads and is found to yield sufficiently accurate results.     

Fabrication of Fresnel zone plates with high aspect ratio by soft X-ray lithography

High focusing efficiency Fresnel zone plates for hard X-ray imaging is fabricated by electron beam lithography, soft X-ray lithography, and gold electroplating techniques. Using the electron beam lithography, Fresnel zone plates which has an outermost zone width of 100 nm and thickness of 250 nm has been fabricated. Fresnel zone plates with outermost zone width of 150 nm and thickness of 660 nm has been fabricated by using soft X-ray lithography.     

Large elastic deformations of clamped skewed plates by dynamic relaxation

This investigation is concerned with the nonlinear behaviour of clamped Isotropic skew plates of constant thickness subjected to a uniformly distributed transverse load. The recently developed numerical technique of dynamic relaxation has been adopted for the analysis. A detailed study of the large deflection behaviour of skew plates has been made by varying three parameters, viz. skew angle, load, and aspect ratio. Numerical results have been compared with the available solutions. Representative nondimensional solutions are presented in the form of graphs to elucidate the nonlinear effect due to large deflection at higher loads.     

Inplane vibration analysis of polar orthotropic annular plates with linearly varying thickness

The natural frequencies of inplane vibrations of polar orthotropic annular plates with linearly varying thickness have been analysed. A semi-analytical method of analysis has been used where the radial and tangential displacements are expanded in the circumferential direction as Fourier series and the radial behaviour is solved using the finite element method. For the sake of comparison, results have been obtained starting with two different kinds of displacement functions. The frequencies have been studied with respect to various boundary conditions. aspect ratios, thickness ratios, ratios of moduli, and two fibre directions.     

Vibration of triangular plates of variable thickness

The problem of the natural frequencies and mode shapes of cantilevered triangular plates with variable thickness and arbitrary planform is solved using the finite element technique. This is done for various combinations of four non-dimensional geometric parameters, namely, the aspect ratio, the two thickness ratios along the two coordinate directions and the sweepback angle. The frequencies for the various cases are tabulated and a few typical mode shapes have been presented graphically.     

Triangular element for analysis of perforated plates under inplane and transverse loads

A C⁰-type triangular element formulation in orthogonal curvilinear co-ordinates has been developed, based on assumptions of transverse inextensibility and constant shear angle through thickness for analysis of perforated plates subjected to inplane and transverse loads. The assumed quadratic displacement potential energy approach is utilized in obtaining an element stiffness matrix and consistent load vector, which are numerically integrated. Numerical results have been obtained using a straight-sided triangular version, which behaves like a subparametric element, for stretching and bending analyses of perforated plates.     

Optimal plastic design of axisymmetric solid plates with a maximum thickness constraint

A systematic method for deriving least-weight solutions for plastically designed, axially symmetric solid plates of prescribed maximum thickness is presented. It is shown that the solution in general consists of regions having (a) stiffeners of maximum depth but infinitesimal width running in one direction only, with a plate of vanishing thickness in between the stiffeners or (b) a “smooth” solid plate. It follows that the majority of existing papers on least-weight solid plates, based on smooth thickness variation throughout, have failed to locate the global optimum. The method is illustrated with examples of circular plates. The weight of the optimal solution is compared with that of intuitively selected designs.     

The quadratic MITC plate and MITC shell elements in plate bending

The analysis of plates can be achieved using the quadratic MITC plate or MITC shell elements. The plate elements have a strong mathematical basis and have been shown to be optimal in their convergence behavior, theoretically and numerically. The shell elements have not (yet) been analyzed mathematically in depth for their rates of convergence, with the plate/shell thickness varying, but have been shown numerically to perform well. Since the shell elements are general and can be used for linear and nonlinear analyses of plates and shells, it is important to identify the differences in the performance of these elements when compared to the plate elements. We briefly review the quadratic quadrilateral and triangular MITC plate and shell elements and study their performances in linear plate analyses.     

Natural vibrations of shear deformable cantilevered skewed trapezoidal and triangular thick plates

In a recent companion paper, the efficacy of higher-order shear deformable, C⁰ continuous, Lagrangian isoparametric plate finite element analyses has been demonstrated on cantilevered skewed (parallelogram) thick plates. The present work extends the method to include skewed thick plates having trapezoidal and triangular planforms, and is the first known vibrational study of such plates. Extensive and accurate nondimensional frequency tables and graphical charts are presented for a series of trapezoidal plates showing the effect of aspect ratio, chord ratio, thickness ratio, and skew angle. The need for the present higher-order shear deformable plate finite element method for skewed trapezoidal plate vibrations increases as the skew angle increases and as the aspect ratio, chord ratio, and thickness ratio decreases. Some theoretical and experimental data hitherto published for delta and skewed triangular cantilevered plates are compared with results obtained using the present finite element method. No published theoretical results for cantilevered skewed trapezoidal and triangular thick plates are known to exist.     

复杂圆板形零件厚度的快速精密测量

复杂圆板形零件的快速和精密测量在工业生产中具有重要的意义,使用两个非接触式的激光位移传感器可实现圆板形零件厚度的精密和快速测量,影响测量精度的因素包括传感器的精度、传感器激光的同轴度、板的偏斜度、圆板上孔洞和沟槽的分布、测量环境温度的变化,通过合理设计测量装置的结构,减少两个传感器的激光的同轴度偏差和板的偏斜度,增加恒温装置消除环境温度对测量结果的影响,为了去除由圆板上的孔洞和沟槽所带来的含有粗大误差的数据点,综合采用了阈值法和差分法相结合的数据处理方法,并用去除了粗大误差的测量值序列的平均值作为测量结果,实验证明这种测量方法具有精度高和速度快的特点.     

Vibration and buckling analysis of plates of abruptly varying stiffness

The application of B-spline functions and the Rayleigh-Ritz procedure to analyze vibration and buckling of plates with abruptly varying stiffnesses is presented. Numerical examples of stepped thickness plates and perforated plates are presented and the results are discussed in comparison with those of other approximate methods.To demonstrate the versatility of the present method, vibration and buckling of skew plates of stepped thickness are also studied for various skew angles, ratios of thickness and ratios of width.     

Solutions of plates on a heterogeneous elastic foundation

In this paper the solution of the problem of interaction of a plate with a two parameter elastic foundation is presented. Vertical displacements of rectangular plates having free edges resting on a heterogeneous elastic foundation have been analysed by using of finite element method. The heterogeneity of the foundation caused by the orthotropy has been taken into account when forming the equations of equilibrium and variable thickness of the foundation by approximating it by Haar's functions. The obtained results have been compared with the results for a homogeneous foundation.     

Fabrication of intermediate mask for deep x-ray lithography

 This paper presents the fabrication of intermediate x-ray mask for deep x-ray lithography. In order to have working mask with absorbers thickness larger than 10 μm, the intermediate mask should have absorbers of 0.7 μm in thickness. To demonstrate intermediate mask fabrication, x-ray zone plates are fabricated on the 1.2 μm low-stress silicon-rich silicon nitride (SiN_x) membrane with the tri-layer Chromium-Tungsten-Chromium (Cr–W–Cr) as the x-ray absorbers. The chromium layers both 200 angstroms are used as adhesion and for stress relief. The SiN_x film is deposited with low pressure chemical vapor deposition (LPCVD) and the free standing membrane are formed by KOH silicon backside etching. With the e-beam lithography and reactive ion etching, width of 0.8 μm of outmost zone of the x-ray zone plates has been achieved on the membrane. The scanning electron microscopy (SEM) images of the x-ray zone plates and pictures of intermediate masks are demonstrated. Received: 25 August 1997/Accepted: 3 September 1997     

Application of the collocation method to vibration analysis of rectangular mindlin plates

The flexural vibration analysis of rectangular Mindlin plates using the collocation method is described. The results obtained by the present method are compared with published results for plates with uniform thickness and two opposite edges simply supported. The comparison shows that the method yields very good results with a relatively small number of collocation points, and that estimates for the higher modes can be obtained without any difficulties. Furthermore, the method is applied to plates with linearly varying thickness, and new findings are presented for the frequencies of plates.     

Application of a three-dimensional mixed finite element model to the flexure of sandwich plate

The bending analysis of sandwich plates consisting of very stiff face sheets and a comparatively flexible core material offers challenge due to large variation in the magnitude of stress and strain components in the face and in the core regions of the plate. Similarly, the displacement fields do vary in zigzag manner at the layer interface of stiff face sheet and the soft core, thereby making the transverse strains highly discontinuous at such layer interfaces. All these behavioural aspects indicate that only an individual layerwise model can appropriately analyze sandwich plates. A layerwise (three-dimensional), mixed, 18-node finite element (FE) model developed by Ramtekkar et al. [Mech. Adv. Mater. Struct. 9 (2002) 133] has been employed for the accurate evaluation of transverse stresses in sandwich laminates. The FE model consists of six degrees-of-freedom (three displacement components and three transverse stress components τ_xz, τ_yz, σ_z, where z is the thickness direction) per node which ensures the through thickness continuity of transverse stress and displacement fields. Results obtained by using the FE model have shown excellent agreement with the available elasticity solutions for sandwich plates. Additional results on the variation of transverse strains have also been presented to highlight the magnitude of discontinuity in these quantities due to difference in properties of the face and the core materials of sandwich plates.     

Thermal stress analysis of laminated composite plates and shells

In this paper, Semiloof shell finite element formulation has been extended to thermal stress analysis of laminated plates and shells. The accuracy of the formulation has been verified using sample problems available in the literature. Thermal stresses in cross-ply and angle-ply laminated plates and shells subjected to thermal gradients across the thickness are presented for different boundary conditions, taking into account the temperature dependence of the material properties. The behaviour of laminates under thermal load is found to be different from that under mechanical loads in certain respects.