Modified mixed Ritz-DQ formulation for free vibration of thick rectangular and skew plates with general boundary conditions

Recently, the present authors proposed a simple mixed Ritz-differential quadrature (DQ) methodology for free and forced vibration, and buckling analysis of rectangular plates. In this technique, the Ritz method is first used to discretize the spatial partial derivatives with respect to a coordinate direction of the plate. The DQ method is then employed to analogize the resulting system of ordinary or partial differential equations. The mixed method was shown to work well for vibration and buckling problems of rectangular plates with simple boundary conditions. But, due to the use of conventional Ritz method in one coordinate direction of the plate, the geometric boundary conditions of the problem can only be satisfied in that direction. Therefore, the conventional mixed Ritz-DQ methodology may encounter difficulties when dealing with rectangular plates involving adjacent free edges and skew plates. To overcome this difficulty, this paper presents a modified mixed Ritz-DQ formulation in which all the natural boundary conditions are exactly implemented. The versatility, accuracy and efficiency of the proposed method for free vibration analysis of thick rectangular and skew plates are tested against other solution procedures. It is revealed that the proposed method can produce highly accurate solutions for the natural frequencies of thick rectangular plates involving adjacent free edges and skew plates using a small number Ritz terms and DQ sampling points.     

矩形平板自由和强迫振动以及屈曲的有限元-微分求积混合法(FE-DQM)

首次提出有限元法(FEM)和微分求积法(DQM)的组合应用,分析矩形平板的振动和屈曲问题.混合法综合了FEM几何适应性强,以及DQM的高精度和高效率.与已有文献的计算结果比较,验证了该方法的正确性.研究表明,使用少量的有限单元和不多的DQM样本点,就可以得到高精度的结果.由于该方法简单且具备进一步发展的潜力,被认为适用于这类问题的求解.     

Buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method

In this article, a combination of the finite element (FE) and differential quadrature (DQ) methods is used to solve the eigenvalue (buckling and free vibration) equations of rectangular thick plates resting on elastic foundations. The elastic foundation is described by the Pasternak (two-parameter) model. The three dimensional, linear and small strain theory of elasticity and energy principle are employed to derive the governing equations. The in-plane domain is discretized using two dimensional finite elements. The spatial derivatives of equations in the thickness direction are discretized in strong-form using DQM. Buckling and free vibration of rectangular thick plates of various thicknesses to width and aspect ratios with Pasternak elastic foundation are investigated using the proposed FE-DQ method. The results obtained by the mixed method have been verified by the few analytical solutions in the literature. It is concluded that the mixed FE-DQ method has good convergancy behavior; and acceptable accuracy can be obtained by the method with a reasonable degrees of freedom.     

A mixed differential quadrature and finite element free vibration and buckling analysis of thick beams on two-parameter elastic foundations

As a first endeavor, a mixed differential quadrature (DQ) and finite element (FE) method for boundary value structural problems in the context of free vibration and buckling analysis of thick beams supported on two-parameter elastic foundations is presented. The formulations are based on the two-dimensional theory of elasticity. The problem domain along axial direction is discretized using finite elements. The resulting system of equations and the related boundary conditions are discretized in the thickness direction and in strong-form using DQM. The method benefits from low computational efforts of the DQ in conjunction with the effectiveness of the FE method in general geometry and systematic boundary treatment resulting in highly accurate and fast convergence behavior solution. The boundary conditions at the top and bottom surface of the beams are implemented accurately. The presented formulations provide an effective analysis tool for beams free of shear locking. Comparisons are made with results from elasticity solutions as well as higher-order beam theory.     

A differential quadrature analysis of unsteady open channel flow

A rapid, convergent and accurate differential quadrature method (DQM) is employed for numerical simulation of unsteady open channel flow. To the best of authors’ knowledge, this is the first attempt to use the DQM in open channel hydraulics. The Saint-Venant equations and the related nonhomogenous, time dependent boundary conditions are discretized in spatial and temporal domain by DQ rules. The unknowns in the entire domain are computed by satisfying governing equations, boundary and initial conditions simultaneously. By employing DQM, accurate results can be obtained using dramatically less grid points in spatial and time domain. The stability of DQM solution is not sensitive to choosing time step or Courant number unlike other methods. Although numerical problems such as instability, oscillation and underestimation near critical depth can be seen by using other methods but DQM solution is smooth and accurate in this case. The results are sensitive to grid distribution in time domain. In light of this, Chebyshev–Gauss–Lobatto distribution performance is excellent. To validate the DQM solutions, the obtained results are compared with those of the characteristic method. In conclusion, DQM is a potential powerful method with minimum computational effort for unsteady flow simulation.     

Accurate bending analysis of rectangular plates with two adjacent edges free and the others clamped or simply supported based on new symplectic approach

The symplectic geometry approach is introduced for accurate bending analysis of rectangular thin plates with two adjacent edges free and the others clamped or simply supported. The basic equations for rectangular plates are first transferred into Hamilton canonical equations. Using the symplectic approach, the analytic solution of rectangular thin plate with two adjacent edges simply supported and the others slidingly supported is derived. Accurate bending solutions of title problems are then obtained using the superposition method. The approach used in this paper eliminates the need to pre-determine the deformation function and is hence more reasonable than conventional methods. Numerical results are presented to demonstrate the validity and efficiency of the approach as compared with those reported in other literatures.     

A semi-analytic approach for the nonlinear dynamic response of circular plates

This paper presents a new semi-analytic perturbation differential quadrature method for geometrically nonlinear vibration analysis of circular plates. The nonlinear governing equations are converted into a linear differential equation system by using Linstedt–Poincaré perturbation method. The solutions of nonlinear dynamic response and the nonlinear free vibration are then sought through the use of differential quadrature approximation in space domain and analytical series expansion in time domain. The present method is validated against analytical results using elliptic function in several examples for both clamped and simply supported circular plates, showing that it has excellent accuracy and convergence. Compared with numerical methods involving iterative time integration, the present method does not suffer from error accumulation and is able to give very accurate results over a long time interval.     

New benchmark solutions for free vibration of clamped rectangular thick plates and their variants

It is of significance to explore benchmark analytic free vibration solutions of rectangular thick plates without two parallel simply supported edges, because the classic analytic methods are usually invalid for the problems of this category. The main challenge is to find the solutions meeting both the governing higher order partial differential equations (PDEs) and boundary conditions of the plates, i.e., to analytically solve associated complex boundary value problems of PDEs. In this letter, we extend a novel symplectic superposition method to the free vibration problems of clamped rectangular thick plates, with the analytic frequency solutions obtained by a brief set of equations. It is found that the analytic solutions of clamped plates can simply reduce to their variants with any combinations of clamped and simply supported edges via an easy relaxation of boundary conditions. The new results yielded in this letter are not only useful for rapid design of thick plate structures but also provide reliable benchmarks for checking the validity of other new solution methods.     

Fundamental frequency analysis of microtubules under different boundary conditions using differential quadrature method

Microtubules (MTs) are a central part of the cytoskeleton in eukaryotic cells. The dynamic behaviors of MTs are of great interest in biomechanics. Many researchers have studied the vibration analysis of MTs by modeling them as an orthotropic cylindrical elastic shell and the exact solution to its displacements is investigated under simply supported boundary conditions. Other boundary conditions lead to some coupled equations, which there are no exact solution to them. Considering various boundary conditions requires implementing semi-analytic or numerical methods. In this study, the differential quadrature method (DQM) has been used to solve the nonlinear problem of seeking fundamental frequency. At first to verify the DQM results, this method has been applied to the equations of MTs under simply supported boundary condition. The coincidence of the exact solution results and the results of DQM shows the effectiveness and precision of this method. After verification, DQM has been used for the other boundary conditions. These boundary conditions are including clamped–clamped (CC), clamped–simply (CS), clamped–free (CF) and free–free (FF) constraints. Finally, the effect of edges boundary condition, radius of MTs and half wave numbers on the vibration behavior of MTs is considered.     

Inelastic buckling of skew and rhombic thin thickness-tapered plates with and without intermediate supports using the element-free Galerkin method

In this paper, skew and rhombic isotropic plates subjected to in-plane loadings are analyzed using the element-free Galerkin method. Inelasticity effect is included in the buckling analysis while plates are thin thickness-tapered type. The governing differential equation for a plate in plastic range of response is numerically solved using the Galerkin method. The shape functions are constructed using the moving least squares (MLS) approximation and the essential boundary conditions are introduced into the formulation through the use of the Lagrange multiplier method and the orthogonal transformation techniques. The Stowell theory for the plastic buckling of flat skew plates with variable thicknesses is used. The inelastic analysis is based on the Ramberg–Osgood representation of the stress–strain curve which is used in the deformation theory of plasticity. Using this method the initial inelastic local buckling of skew plates with or without intermediate line supports is studied. Stiffness and geometric matrices are formulated by weak form of the Galerkin method. Finally, the inelastic local buckling loads of these plates are obtained and the results are compared with known solutions in the literature.     

扇形板的富里哀—贝塞尔级数解

本文以加补充项的富里哀—贝塞尔双重级数的位移模式,对扇形弹性薄板在各种边界件条下的弯曲和振动问题,提出了一种应用范围比较广的、便于计算的、解析形式的解法.作为算例,文中给出了各种角度的径向边界简支或固定的扇形板在均布荷载或集中荷载作用下产生的挠度和弯矩的分布曲线,并与有关文献的数值结果进行了比较.本文推广了加补充项的富氏级数法的应用范围,并计算出各种角度的径向边界简支的扇形板的自振频率和节线的图表.     

Free vibration analysis of circular plates by differential transformation method

This study analyses the free vibrations of circular thin plates for simply supported, clamped and free boundary conditions. The solution method used is differential transform method (DTM), which is a semi-numerical-analytical solution technique that can be applied to various types of differential equations. By using DTM, the governing differential equations are reduced to recurrence relations and its related boundary/regularity conditions are transformed into a set of algebraic equations. The frequency equations are obtained for the possible combinations of the outer edge boundary conditions and the regularity conditions at the center of the circular plate. Numerical results for the dimensionless natural frequencies are presented and then compared to the Bessel function solution and the numerical solutions that appear in literature. It is observed that DTM is a robust and powerful tool for eigenvalue analysis of circular thin plates.     

An efficient mixed methodology for free vibration and buckling analysis of orthotropic rectangular plates

In this paper, a simple and efficient mixed Ritz-differential quadrature (DQ) method is presented for free vibration and buckling analysis of orthotropic rectangular plates. The mixed scheme combines the simplicity of the Ritz method and high accuracy and efficiency of the DQ method. The accuracy of the proposed method is demonstrated by comparing the calculated results with those available in the literature. It is shown that highly accurate results can be obtained using a small number of Ritz terms and DQ sample points. The proposed method is suitable for the problem considered due to its simplicity and potential for further development.     

Buckling mode localization in rib-stiffened plates with misplaced stiffeners – Kantorovich approach

Buckling mode localization in rib-stiffened plates with randomly misplaced stiffeners is studied in this paper. The method of Kantorovich on reducing a partial differential equation to a system of ordinary differential equations is employed to obtain the deflection surface of the rib-stiffened plates under axial compressive load. The edges of the plates normal to the stiffeners can be either simply supported or clamped. The solutions of the deflection surface are then expressed in the form of transfer matrices. The expressions of the solutions obtained for the case of one edge simply supported and one edge clamped and the case of two edges clamped are similar to those for the case of two edges simply supported. When the two edges are simply supported, the method of Kantorovich yields the exact results. Localization factors, which characterize the average exponential rates of growth or decay of amplitudes of deflection, are determined using the method of transfer matrix. The method of Kantorovich is a general approximate method, which is applicable for various support conditions.     

A mixed method for free and forced vibration of rectangular plates

This paper presents a very first combined application of Ritz method and differential quadrature (DQ) method to vibration problem of rectangular plates. In this study, the spatial partial derivatives with respect to a coordinate direction are first discretized using the Ritz method. The resulting system of partial differential equations and the related boundary conditions are then discretized in strong form using the DQ method. The mixed method combines the simplicity of the Ritz method and high accuracy and efficiency of the DQ method. The results are obtained for various types of boundary conditions. Comparisons are made with existing analytical and numerical solutions in the literature. Numerical results prove that the present method is very suitable for the problem considered due to its simplicity, efficiency, and high accuracy.     

点简支正交各向异性矩形薄板弯曲的精确解

本文利用迭加原理,给出了点简支正交各向异性短形薄板弯曲问题的封闭的级数式解答.简支点的位置和横向载荷的分布均可任意.用本文的级数解给出的算例与以往的数值解是十分一致的.     

3D free vibration analysis of laminated cylindrical shell integrated piezoelectric layers using the differential quadrature method

This paper addresses the free vibration problem of multilayered shells with embedded piezoelectric layers. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. The shell has arbitrary end boundary conditions. For the simply supported boundary conditions closed-form solution is given by making the use of Fourier series expansion. Applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free end conditions. Natural frequencies of the hybrid laminated shell are presented by solving the eigenfrequency equation which can be obtained by using edges boundary condition in this state equation. Accuracy and convergence of the present approach is verified by comparing the natural frequencies with the results obtained in the literatures. Finally, the effect of edges conditions, mid-radius to thickness ratio, length to mid-radius ratio and the piezoelectric thickness on vibration behaviour of shell are investigated.     

Analysis of Mindlin micro plates with a modified couple stress theory and a meshless method

A modified couple stress theory and a meshless method is used to study the bending of simply supported micro isotropic plates according to the first-order shear deformation plate theory, also known as the Mindlin plate theory. The modified couples tress theory involves only one length scale parameter and thus simplifies the theory, since experimentally it is easier to determine the single scale parameter. The equations governing bending of the first-order shear deformation theory are implemented using a meshless method based on collocation with radial basis functions. The numerical method is easy to implement, and it provides accurate results that are in excellent agreement with the analytical solutions.     

A hybrid DQ/LMQRBF-DQ approach for numerical solution of Poisson-type and Burger’s equations in irregular domain

Differential quadrature (DQ) is an efficient and accurate numerical method for solving partial differential equations (PDEs). However, it can only be used in regular domains in its conventional form. Local multiquadric radial basis function-based differential quadrature (LMQRBF-DQ) is a mesh free method being applicable to irregular geometry and allowing simple imposition of any complex boundary condition. Implementation of the latter numerical scheme imposes high computational cost due to the necessity of numerous matrix inversions. It also suffers from sensitivity to shape parameter(s). This paper presents a new method through coupling the conventional DQ and LMQRBF-DQ to solve PDEs. For this purpose, the computational domain is divided into a few rectangular shapes and some irregular shapes. In such a domain decomposition process, a high percentage of the computational domain will be covered by regular shapes thus taking advantage of conventional DQM eliminating the need to implement Local RBF-DQ over the entire domain but only on a portion of it. By this method, we have the advantages of DQ like simplicity, high accuracy, and low computational cost and the advantages of LMQRBF-DQ like mesh free and Dirac’s delta function properties. We demonstrate the effectiveness of the proposed methodology using Poisson and Burgers’ equations.     

A two-variable first-order shear deformation theory considering in-plane rotation for bending,buckling and free vibration analyses of isotropic plates

This paper presents a two-variable first-order shear deformation theory considering in-plane rotation for bending, buckling and free vibration analyses of isotropic plates. In recent studies, a simple first-order shear deformation theory (S-FSDT) was developed and extended. It has only two variables by separating the deflection into bending and shear parts while the conventional first-order shear deformation theory (FSDT) has three variables. However, the S-FSDT provides incorrect predictions for the transverse shear forces on the insides and the twisting moments at the boundaries except simply supported plates since it does not consider in-plane rotation. The present theory also has two variables but considers in-plane rotation such that it is able to correctly predict the responses of plates with any boundary conditions. Analytical solutions are obtained for rectangular plates with two opposite edges that are simply supported, with the other edges having arbitrary boundary conditions. Numerical results of deflections, stress resultants, buckling loads and natural frequencies are presented with the FSDT, the S-FSDT and the present theory. Comparative studies demonstrate the effects of in-plane rotation and the accuracy of the present theory in predicting the bending, buckling and free vibration responses of isotropic plates.