Elastic large deflection analysis of isotropic rectangular Mindlin plates

Governing equations for the large deflection analysis of isotropic rectangular Mindlin plates are introduced and their solution using the DR algorithm is briefly outlined. Two computer programs, based respectively on interlacing and non-interlacing finite-differences, have been developed for the numerical solution of these equations. The programs have been verified by analysing a variety of thin and moderately thick plate problems for which alternative solutions are available. Sample results comparisons are presented in order to quantify the accuracy of the DR results. The non-interlacing finite-difference DR program is then used to compute new results for uniformly loaded square moderately thick plates with simply supported, clamped and combined simply supported and clamped edges.     

Rigid-plastic modelling of blast-loaded stiffened plates—Part II: Partial end fixity, rate effects and two-way stiffened plates

A new rigid-plastic analysis of stiffened plates subjected to uniformly distributed blast loads is developed. In this first part of a two-part paper, a uniform one-way stiffened plate with clamped ends is modelled as a singly symmetric beam, comprised of one stiffener and its tributary plating. Rigid-plastic analysis is then applied to this beam using an idealized piecewise linear bending moment-axial force capacity interaction relation or yield curve. Two solutions to the response are developed. The first solution is in closed form and is based on the solution of the resulting linearized differential equations. The second solution is obtained by approximating the response as a sequence of instantaneous mode responses, where the mode shapes are determined by an extremum principle which maximizes the rate of change of the kinetic energy. This latter solution may be extended to cases involving non-rigid boundaries and two-way stiffening and this is done in the second part of this paper. Here, the two solution methods are applied to several examples of one-way stiffened plates subjected to various blast-type pulses. Good agreement is obtained between the present results and those from elastic-plastic beam finite element and finite strip solutions.     

A DXDR large deflection analysis of uniformly loaded square, circular and elliptical orthotropic plates using non-uniform rectangular finite-differences

A finite-difference analysis of the large deflection response of uniformly loaded square, circular and elliptical clamped and simply-supported orthotropic plates is presented. Several types of non-uniform (graded) mesh are investigated and a mesh suited to the curved boundary of the orthotropic circular and elliptical plate is identified. The DXDR method-a variant of the DR (dynamic relaxation) method-is used to solve the finite-difference forms of the governing orthotropic plate equations. The DXDR method and irregular rectilinear mesh are combined along with the Cartesian coordinates to treat all types of boundaries and to analyze the large deformation of non-isotropic circular/elliptical plates. The results obtained from plate analyses demonstrate the potential of the non-uniform meshes employed and it is shown that they are in good agreement with other results for square, circular and elliptical isotropic and orthotropic clamped and simply-supported plates in both fixed and movable cases subjected to transverse pressure loading.     

Axisymmetric full-range analysis of transverse pressure-loaded circular plates

Incremental equations governing the axisymmetric full-range response of transverse pressure loaded, clamped and simply supported, circular plates are presented. The derivation and implementation of the full-section and layer constitutive equations is described in detail. The solution of the finite-difference approximations to the governing equations using the Dynamic Relaxation (DR) method is briefly outlined together with several refinements for improving computational efficiency. Both the full-section and layer analyses are used to compute plate deflections observed in several tests on uniformly loaded, clamped and simply supported, circular plates. It is observed that the layer analysis predicts the measured deflections more accurately than the full-section analysis. The analytical-experimental deflection correlation appears to be accurate for slender clamped plates and less accurate for stocky clamped and simply supported plates. It is suggested that various uncertainties about the plate test data may account, in part, for the lack of deflection correlation and that more carefully controlled plate tests should be undertaken in order to confirm this speculation.     

A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates

An exact closed-form frequency equation is presented for free vibration analysis of circular and annular moderately thick FG plates based on the Mindlin's first-order shear deformation plate theory. The edges of plate may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. The material properties change continuously through the thickness of the plate, which can vary according to a power-law distribution of the volume fraction of the constituents, whereas Poisson's ratio is set to be constant. The equilibrium equations which govern the dynamic stability of plate and its natural boundary conditions are derived by the Hamilton's principle. Several comparison studies with analytical and numerical techniques reported in literature and the finite element analysis are carried out to establish the high accuracy and superiority of the presented method. Also, these comparisons prove the numerical accuracy of solutions to calculate the in-plane and out-of-plane modes. The influences of the material property, graded index, thickness to outer radius ratios and boundary conditions on the in-plane and out-of-plane frequency parameters are also studied for different functionally graded circular and annular plates.     

基于大挠度理论的减振器阀片精确变形模拟和研究

为获得筒式减振器环形阀片弯曲变形的大挠度表达式与半径的关系,基于圆薄板大挠度理论,提出了薄板变形问题表达式内在统一的表示,结合针对环板外边缘挠度的大挠曲变形解析式与小挠曲变形方程,推导出环形阀片大挠曲变形与半径相关的混合解法解析式与变形修正系数。利用ABAQUS有限元软件对环形阀片进行了仿真分析,验证了混合解法解析式的精确度。分析了基于混合解法的叠加阀片弯曲变形,对叠加阀片的等效厚度与弯曲变形刚度进行了研究,得出的解析式与结论可用于减振器阀片的设计、调校与仿真分析。     

The deflection coefficient of a rectangular plate reinforced in the middle

In this paper, we investigated the deflection coefficients of rectangular plates that are reinforced in the middle. For reinforced plates with three different aspect ratios and two types of boundary conditions (simply supported and clamped), we derived the deflection coefficients with respect to the elastic modulus ratio and the relative length of the inner plate using the least-squares method. We performed a finite element analysis of the models, and calculated the deflection coefficients of reinforced plates in terms of the deflection coefficients of simple (nonreinforced) plates. The results can be extended to various types of reinforced rectangular plates.     

Analysis of rectangular stiffened plates under uniform lateral load based on FSDT and element-free Galerkin method

This paper presents an element-free Galerkin (EFG) method for the static analysis of concentrically and eccentrically stiffened plates based on first-order shear deformable theory (FSDT). The stiffened plates are regarded as composite structures of plates and beams. Imposing displacement compatible conditions between the plate and the stiffener, the displacement fields of the stiffener can be expressed in terms of the mid-surface displacement of the plate. The strain energy of the plate and stiffener can be superimposed to obtain the stiffness matrix of the stiffed plate. Because there are no elements used in the meshless model of the plate, the stiffeners need not to be placed along the meshes, as is done in the finite element methods. The stiffeners can be placed at any location, and will not lead to the re-meshing of the plate. The validity of the EFG method is demonstrated by considering several concentrically and eccentrically stiffened plate problems. The present results show good agreement with the existing analytical and finite element solutions. The influences of support size (denoted by a scaling factor d_max) and order of the complete basis functions (N_c) on the numerical accuracy are also investigated. It is found that larger support size and higher order of basis function will furnish better convergence results.     

考虑阀片大挠曲变形的减振器建模与参数辨识

为建立减振器参数化数学模型,基于圆薄板大挠曲理论和弹性力学原理,推导出了阀片大挠曲变形与半径的关系式。建立外半径不全相等时环形叠加阀片的力学模型,提出用大挠曲变形解析方法与有限元分析相结合的等效模型求解方式推导出该环形叠加阀片挠曲变形的解析式。根据活塞阀系叠加阀片与复原弹簧同时变形的串联原理,建立了该双筒式减振器在复原与压缩行程中阻尼特性的数学模型。采用NSGA-Ⅱ算法对模型中难以测量的参数进行辨识,辨识后的减振器外特性仿真结果与试验数据基本吻合,说明所建立的参数化模型比较可靠,对减振器的性能调试具有指导意义。     

Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers

This paper employs an analytical method to analyze vibration of piezoelectric coupled thick annular functionally graded plates (FGPs) subjected to different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Reddy's third-order shear deformation theory (TSDT). The properties of host plate are graded in the thickness direction according to a volume fraction power-law distribution. The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study closed-form expressions for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. The present analysis is validated by comparing results with those in the literature and then natural frequencies of the piezoelectric coupled annular FG plate are presented in tabular and graphical forms for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric, power index and boundary conditions.     

Influence of stiffener location on the stability of stiffened plates under compression and in-plane bending

The elastic behaviour of stiffened plates under non-uniform edge compression is investigated. As a first stage, an energy formulation, in which the structure is modelled as assembled plate and beam elements, is presented. A Sequential Quadratic Programming (SQP) algorithm is then used to evaluate the buckling load and the associated buckling mode for given plate/stiffener geometric properties. Results are presented showing the influence of the stiffener location on the stability of the structure under combined compression and bending. Then, a strategy is presented for optimum location of the stiffener. The proposed methodology can be used to develop an improved design procedure for efficient design of stiffened plates under this type of loading.     

Axisymmetric free vibration of thick annular plates

This paper presents a numerical analysis of the axisymmetric free vibration of moderately thick annular plates using the differential quadrature method (DQM). The plates are described by Mindlin’s first-order shear-deformation theory. The first five axisymmetric natural frequencies are presented for uniform annular plates, of various radii and thickness ratios, with nine possible combinations of free, clamped and simply supported boundary conditions at the inner and outer edges of the plates. The accuracy of the method is established by comparing the DQM results with some exact and finite element numerical solutions and, therefore, the present DQM results could serve as a benchmark for future reference. The convergence characteristics of the method for thick plate eigenvalue problems are investigated and the versatility and simplicity of the method is established.     

Buckling analysis of circular and annular composite plates reinforced with carbon nanotubes using FEM

The critical compressive load in the buckling of circular and annular composite plates reinforced with carbon nanotubes (CNTs) is calculated using finite element method. The developed model is based on the third-order shear deformation theory for moderately thick laminated plates. Effects of CNTs orientation angles and thickness-to-inner radius ratio on the buckling of composite plates are discussed. The results are compared with those obtained by analytical method based on classical plate theory. The finite element method shows lower values for critical buckling load because of the elimination of shear strain in the classical plate theory.     

Nonlinear study of large deflection of simply supported piezoelectric layered-plate under initial tension

The nonlinear problem of large deflection of a simply supported piezoelectric layered plate under initial tension is studied. The approach follows von Karman’s plate theory for large deflection for a symmetrically layered isotropic case including a piezoelectric layer. The nonlinear governing equations are solved using a finite difference method, by taking the associated linear analytical solution as an initial guess in the numerical iteration procedure. The results for a nearly monolithic plate under a very low applied voltage are found to correlate well with available solutions for a single-layered case under pure mechanical loading and thus the present approach is validated. For three-layered plates made of typical silicon based materials, various initial tension and lateral pressure are considered, and different applied voltages up to a moderate magnitude are implemented. No edge effect was observed, in contrast to the cases of clamped plates in literature. In additions, varying the layer moduli seems to have an insignificant effect upon the structural responses of the layered plate. On the other hand, the piezoelectric effect tends to be apparent only in a low pretension condition. For a relatively large pretension, the effect of initial tension becomes dominant, yielding nearly unique solutions for the structural responses, regardless of the magnitudes of the applied voltage and the lateral pressure.     

Analytical and experimental studies for deformation of circular plates subjected to blast loading

In this paper, experimental responses of the clamped mild steel, copper, and aluminum circular plates subjected to blast loading are investigated. Extensive experimental results concerning variations of central deflection are presented. Approximate energy analysis is used for the analytical modeling of the mid-point deflection thickness ratio of the circular plates. The essence of the model is to describe the deformation profile, which depends on the distribution of impulsive load. By using the present modeling, one can show how the midpoint deflection varies with the variations of the important parameters. The results obtained from the present model show very good agreement with the experimental data.     

A semi-analytical approach for the geometrically nonlinear analysis of trapezoidal plates

This paper presents a semi-analytical approach for the geometrically nonlinear analysis of skew and trapezoidal plates subjected to out-of-plane loads. The thin elastic plate theory with nonlinear von Kármán strains is used for the nonlinear large deflection analysis of the plate. The solution of the governing nonlinear partial differential equations with variable coefficients is reduced to an iterative solution of nonlinear ordinary differential equations using the multi-term extended Kantorovich method. The geometry of the trapezoidal plate is mapped into a rectangular computational domain. Parallelogram (skew) plates are considered as a particular case of the general trapezoidal ones. The capabilities and convergence of the method are numerically examined through comparison with other semi-analytical and numerical methods and with finite element analyses. The applicability of the approach to the nonlinear large deflection analysis of skew and trapezoidal plates is demonstrated through various numerical examples. The numerical study focuses on combinations of geometry, loading and boundary conditions that are beyond the applicability of other semi-analytical methods.     

求解直边简支环扇板振动特性的积分方程方法

将傅里叶－贝塞尔级数引入积分方程方法，推导出一种研究沿直边简支环扇形板结构振动特性的简捷，高效的积分方程方法，根据积分方程和傅里叶一贝塞尔级数理论，首先采用由第一、二类贝塞尔函数组成的完备正交函数系构造直边简支环扇板的格林函数，然后由叠加原理将环扇形板的自由振动问题转化为积分方程特征值问题；进而将积分方程形式的特征值问题转化为无穷阶正定对称矩阵的标准特征值问题，计算结果表明，该方法不仅运算简捷，精度高，适用性强，而且为分析更为复杂的环扇板的振动问题提供可靠的基础。     

Dynamic relaxation method for nonlinear buckling analysis of moderately thick FG cylindrical panels with various boundary conditions

Using new approach proposed by Dynamic relaxation (DR) method, buckling analysis of moderately thick Functionally graded (FG) cylindrical panels subjected to axial compression is investigated for various boundary conditions. The mechanical properties of FG panel are assumed to vary continuously along the thickness direction by the simple rule of mixture and Mori-Tanaka model. The incremental form of nonlinear formulations are derived based on First-order shear deformation theory (FSDT) and large deflection von Karman equations. The DR method combined with the finite difference discretization technique is employed to solve the incremental form of equilibrium equations. The critical mechanical buckling load is determined based on compressive load-displacement curve by adding the incremental displacements in each load step to the displacements obtained from the previous ones. A detailed parametric study is carried out to investigate the influences of the boundary conditions, rule of mixture, grading index, radius-to-thickness ratio, length-to-radius ratio and panel angle on the mechanical buckling load. The results reveal that with increase of grading index the effect of radius-to-thickness ratio on the buckling load decreases. It is also observed that effect of distribution rules on the buckling load is dependent to the type of boundary conditions.

     

Generic nonlinear behavior of antisymmetric angle-ply laminated plates

A generic presentation of the large deflection, postbuckling and nonlinear flexural vibration behavior of antisymmetric angle-ply laminated plates is investigated. A transformation between on-axis and off-axis stiffnesses is presented such that all the stiffnesses, including bending-stretching coupling, in the nonlinear governing equations of angle-ply laminates can be expressed in terms of bounded global constants in orthotropic plates. The results presented are applicable to a large variety of composite materials and show that a higher generalized rigidity ratio will lead to smaller deflection of plates, higher buckling loads and lower postbuckling deflection, and higher relative large-amplitude natural frequencies. The information presented should be helpful to anyone seeking a better understanding of the correlation between composite material properties and nonlinear behavior of laminated plates.     

Large amplitude vibrations of thin elastic plates by the method of conformal transformation

A unified method for investigating large amplitude vibrations of thin elastic plates of any shape under clamped edge boundary conditions is presented, based on Von Karman governing equations generalised to the dynamical case. The conformal mapping technique is introduced and the domain is conformally transformed on to the unit circle. The deflection function is chosen beforehand in conformity with the prescribed boundary conditions and the stress function is solved taking only the first term of the mapping function. The transformed differential equations are solved by the Galerkin procedure to obtain the second order nonlinear differential equation for the unknown time function. The time equation is readily solved in terms of Jacobian elliptic functions. Frequency of linear and nonlinear oscillations as well as static nonlinear case are analysed for plates of circular, and regular polygonal shape. Results obtained are compared with other known results. From the comparative study of different results it is observed that the first term approximation of the mapping function yields fairly accurate results with less computational effort.