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Generalized Differential Quadrature Finite Element Method for Cracked Composite Structures of Arbitrary Shape
Abstract
This paper investigates the dynamic behavior of moderately thick composite plates of arbitrary shape using the Generalized Differential Quadrature Finite Element Method (GDQFEM), when geometric dis-continuities through the thickness are present. In this study a five degrees of freedom structural model, which is also known as the First-order Shear Deformation Theory (FSDT), has been used. GDQFEM is an advanced version of the Generalized Differential Quadrature (GDQ) method which can discretize any derivative of a partial differential system of equations. When the structure under consideration shows an irregular shape, the GDQ method cannot be directly applied. On the contrary, GDQFEM can always be used by subdividing the whole domain into several sub-domains of irregular shape. Each irregular ele-ment is mapped on a parent regular domain where the standard GDQ procedure is carried out. The con-nections among all the GDQFEM elements are only enforced by inter-element compatibility conditions. The equations of motion are written in terms of displacements and solved starting from their strong for-mulation. The validity of the proposed numerical method is checked up by using Finite Element (FE) results. Comparisons in terms of natural frequencies and mode shapes for all the reported applications have been performed.
... All works via DQM yield excellent results due to the use of high order global basis functions and non-equally spaced sampling points that are roots of orthogonal polynomials [24]. The differential quadrature (DQ) procedure extended to analyze problems through subdomains is known as the differential quadrature element method (DQEM) [29] or the generalized differential quadrature finite element method (GDQFEM) [30,31]. The DQEM and GDQFEM also presented excellent results. ...
... Using the orthogonal polynomials on a quadrilateral domain [48], one can define DQ rules as Eqs. (31) to (34) on the domain with node distribution on the boundary of the element similar to Fig. 2 through similar procedure as Eqs. (26) to (28). ...
... Thus, the DQH rules for the first order derivatives of u(x, y) with respect to x and y at a discrete point xi = x(ξi, ηi), yi = y(ξi, ηi) (i = 1, 2,…, Nt) can be given as 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 (65) where Ain (1) and Bin (1) are respectively the weighting coefficients associate with the first order derivatives with respect to ξ and η, see Eq. (31). One can further write Eq. (65) in a single index notation form as (66) or , ...
A differential quadrature hierarchical finite element method (DQHFEM) using Fekete points was formulated for triangles and tetrahedrons and applied to structural vibration analyses. First, orthogonal polynomials on triangles and tetrahedrons that can be used as bases of the hierarchical finite element method (HFEM) were derived and simple formulas of transforming one dimensional non-uniform nodes to simplexes were presented. Then the non-uniform nodes were used as initial guesses to solve the Fekete points on simplexes through Newton–Raphson's method together with the orthogonal polynomials. New differential quadrature (DQ) rules on simplexes were formulated using the HFEM bases and the Fekete points. The numbers of nodes or bases on different edges and faces and inside the body of the new DQ elements do not relate with each other like the HFEM that can freely assign different numbers of bases on different edges and faces and inside the body. So the new DQ method was named as a differential quadrature hierarchical (DQH) method that uses either interpolation functions or orthogonal polynomials as bases inside the element. Its weak form was named as the DQHFEM. Besides the DQH method and its weak form, a simple method of generating high quality linear and high order triangular and tetrahedral meshes from a single NURBS patch was presented. Numerical tests of the DQHFEM through structural vibration analyses showed that high accuracy results can be obtained using only a few nodes even on curvilinear domains using the DQH bases on both physical and geometric fields. It was concluded that wide applications of the DQH method and the DQHFEM to science and engineering are possible and commercial codes based on them are deserved to be developed.
... In order to propose a universal method for vibration analysis of plate with irregularly shaped through cracks, a differential quadrature finite element method (DQFEM) proposed by Xing et al. [23,24] is used. The DQFEM has been widely used to model plate structures with complex geometries [25][26][27][28][29][30], which has been proved to be an efficient method using less element numbers and obtaining high accurate results. This method makes it possible to model cracks with irregular shapes simply by disconnecting the adjacent elements so that a crack emerges between the elements. ...
... The concept of finite element in finite element method is adopted into the DQFEM [25]. In order to deduce the equations of motion of the whole structure, we need to start from a single element. ...
... Substituting (33) into (30) and (25) and taking out the displacement vector, then the stiffness matrix and mass matrix of the whole system can be written as ...
A universal method combining the differential quadrature finite element method with the virtual spring technique for analyzing the free vibration of thin plate with irregular cracks is proposed. Translational and rotational springs are introduced to restrain the vertical displacement and orientation of the plate. The mass matrix and stiffness matrix for each element are deduced involving the effects of the virtual springs. The connection relationships between elements can be modified by setting the stiffness of the virtual springs. The vibration of two rectangular plates with three irregular shaped cracks and different boundary conditions are presented. The results are compared with those obtained by ANSYS, where the good agreement between the results validates the accuracy and efficiency of the present method.
... In particular, the present study considers a strong form pseudo-spectral technique implemented on arbitrarily shaped domains. In order words, the authors are decomposing the physical problem into quadrilateral domains and they are solving the mathematical problem upon each element with a strong form approach [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. As almost any numerical method the starting point is the functional approximation which might depend on the choice of basis functions and point collocation. ...
... The implementation of Eq. (13) is straightforward for partial differential systems of equations, nevertheless, only problems with regular domains can be solved. For this reason the authors extended the present pseudo-spectral approach to work with domains of arbitrary shape, such method has been termed Strong Formulation Finite Element Method (SFEM) [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. In order to achieve this, a mapping technique has been used which is able to transform a shape in Cartesian coordinates to a regular domain (of square shape). ...
... It has been assumed that no micro-couples can be applied to the solid but only in-plane pressures Q j , j ¼ 1; 2. The discrete system (14) cannot be solved directly without its discrete boundary conditions that can be easily carried out from Eq. (7). It is remarked that the implementation of the boundary conditions is the major drawback of the SFEM, while the implementation of the governing equation is quite straightforward [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. The solution is found by solving the standard linear problem ...
Cosserat theory of elasticity has been introduced for modelling micro-structured materials and structures. Micro-structured materials are able to re-distribute the stress resulting in lower stress peaks. Therefore, such effects are strongly underlined when composite structures have holes and discontinuities in which high stress gradients are generally observed in the classical theory of elasticity. However, general material configurations can be solved using numerical approaches, since exact solutions are only available for simple cases. The present paper deals with such problems using an advanced strong form pseudo-spectral method that uses domain decomposition to deal with geometric and material discontinuities.
... where [ ] T 0 1 2 3 , , , , N u u u u = u (37) (38) in which u 0 is the column vector of nodal displacements u 0 , x i are the nodes, x * i are the integral points in Cartesian coordinates. Other displacement vectors are defined in similar way as u 0 . ...
... In this section, free vibration analysis of two kind of arbitrary shaped composite plates are performed. The RBFEM results of this work are compared with the GDQFEM [37] and SRBFEM [30] ...
... (a) (b) (c) In Table 6, the first ten natural frequencies of the bipolar plates are compared with the results of the GDQFEM [37], the SRBFEM [30] and the commercial codes ABAQUS [37]. The radial basis function of the SRBFEM is the same Wendland RBF as the present work. ...
A layerwise shear deformation theory for composite laminated plates is discretized using a radial basis function finite element method (RBFEM). The RBFEM is the radial basis function (RBF) method in weak-form and is a partially mesh free method. Therefore, elements of complex shapes can be easily constructed. Compact-support Wendland function is used in the RBFEM. A layerwise theory based on a linear expansion of Mindlin’s first-order shear deformation theory in thickness direction is employed for static and dynamic analysis. The combination of the RBFEM with the layerwise theory allows an accurate and very flexible prediction of the field variables. Laminated composite and sandwich plates were analyzed. The RBFEM solutions were compared with various models in literatures and showed very good agreements with exact and other high accurate results in literatures based on similar layerwise theories. The analysis of composite plates based on the layerwise theory indicates that the RBFEM is an effective method for high accuracy analysis of large-scale problems.
... The differential quadrature (DQ) method is a highly effective approach for solving partial and ordinary differential equations. Its fundamental principle involves partitioning the domain of independent variables for smooth functions into discrete sample points according to specific rules and approximating the derivative of the function with respect to these independent variables as a weighted sum of their corresponding function values [11][12][13]. Therefore, the nth order derivative of function f (ϑ) with respect to ϑ at the ith point ϑ i is approximated as follows: ...
... Considering the boundary condition, Equation (13) can be written as ...
The dynamic characteristics of a structure serve as a crucial foundation for structural assessment, fault diagnosis, and ensuring structural safety. Therefore, it is imperative to investigate the impacts of uncertain parameters on the dynamic performance of structures. The dynamic characteristics of arches with uncertain parameters are analyzed in this paper. The uncertain parameters are regarded as non-probabilistic uncertainties and represented as interval variables. A model of an arch considering interval-uncertain parameters is built, and kinematic equations are established. The natural frequencies are obtained using the differential quadrature (DQ) method, and the relationships between natural frequency, radius, and central angle are also analyzed. On this basis, the Chebyshev polynomial surrogate (CPS) model is employed to solve the uncertain dynamic problem, and the natural frequencies are seen to be the objective functions of the CPS model. The accuracy verification of the model is achieved by comparison with the Monte Carlo simulation (MCS). Simulations are carried out considering different uncertainties, and the results show that the bounds of natural frequencies are influenced not only by the types of uncertain parameters, but also by their combinations.
... In terms of computational cost, GDQM is found to be one of the fastest numerical methods in the problems of structural dynamics [7] . Arbitrarily shaped structures are also treated rather easily by the GDQM through a mapping technique [7][8][9][10][11] . Moreover, any combinations of boundary conditions are easily handled in the solution process, moreover both linear and nonlinear problems may be solved. ...
... Linear aeroelastic analysis is first conducted for a set of fiber angles being According to the critical conditions given in Table ( 8 ), the maximum non-dimensional dynamic pressure occurs at < T 0 , T 1 > = < 0 • , 45 • > which corresponds to the smallest T 0 and the largest T 1 values. On the other hand, the smallest nondimensional dynamic pressure occurs at < T 0 , T 1 > = < 45 • , 0 • > which, on the contrary to the maximum case, corresponds to the largest T 0 and the smallest T 1 values. ...
Aeroelastic stability of tapered/skew variable stiffness composite cantilevered plates are considered in the current study at flutter and post-flutter regions. The variable stiffness behavior is obtained by altering the fiber angles continuously according to a selected curvilinear fiber path function in the composite laminates. Flutter speed, limit cycle oscillations (LCOs), and bifurcation diagrams of tapered/skewed plates are obtained at two different fiber path functions. Nonlinear structural model is utilized based on the virtual work principle. Fully nonlinear Green’s kinematic strain relations are used to account for the geometric nonlinearities and the first order shear deformation theory (FSDT) is employed to generalize the formulation for the case of moderately thick plates including transverse shear effects. One prominent target of the present study is to determine how the variable stiffness parameters affect the nonlinear behavior. Consequently, one may find the best fiber path with improved flutter and post-flutter characteristics for tapered/skew plates in supersonic flow. First order linear piston theory is used to model the aerodynamic loading. In order to get a reliable solution, the Generalized Differential Quadrature (GDQ) method is employed. Moreover, time integration of the equations of motion is carried out using the Newmark’s average acceleration technique. It will be shown that taperness/skewness as well as variable stiffness lamination parameters have significant effects on the aeroelastic stability margins. In addition, the post-critical behavior is found to be periodic or quasi-periodic at all the presented simulations with no specific route to chaos.
... They utilized three-dimensional theory of elasticity to model the FGM plates in nearness of thermal environment. Kim [14] and Viola et al. [15] have also worked on vibration analysis of intact and cracked FGM rectangular plates under thermal environment using finite element method. Natarajan et al. [16] performed a detail study on vibration and buckling analysis of a functionally graded plate containing internal discontinuities in form of cracks using FEM and first-order shear deformation (FSDT) theory. ...
... (38)], it is seen that it is also affected by the surrounding fluid medium.5 Relation for central deflection of plateConsider a cracked orthotropic plate with all sides simply supported, subjected to a lateral uniformly distributed dynamic load (P z ) harmonically varying with time. For a plate in the absence of thermal moments (M T x = M T y = 0) and the presence of constant inplane forces (R T x and R T y ) due to thermal environment only [Eqs.(21 and 22)], the governing equation [Eq.(15)] becomes ...
Based on a non-classical plate theory, a nonlinear analytical model is proposed to analyze transverse vibration of thin partially cracked and submerged orthotropic plate in the presence of thermal environment. The governing equation for the cracked plate is derived using the Kirchhoff’s thin plate theory in conjunction with the strain gradient theory of elasticity. The effect of centrally located surface crack is deduced using appropriate crack compliance coefficients based on the simplified line spring model, whereas the effect of thermal environment is introduced using moments and in-plane forces. The influence of fluidic medium is incorporated in the governing equation in the form of fluid forces associated with its inertial effects. The equation has been solved by transforming the lateral deflection in terms of modal functions. The shift in primary resonance due to crack, length scale parameter and temperature has also been derived with central deflection. To demonstrate the accuracy of the present model, a few comparison studies are carried out with the published literature. The variation in fundamental frequency of the cracked plate is studied considering various parameters such as crack length, plate thickness, level of submergence, temperature and length scale parameter. It has been concluded that the frequency is affected by crack length, temperature and level of submergence. A comparison has also been made for the results obtained from the classical plate theory and Strain gradient theory. Furthermore, the variation in frequency response and peak amplitude of the cracked plate is studied using method of multiple scales to show the phenomenon of bending hardening or softening as affected by level of submergence, temperature, crack length and length scale parameter .
... The boundary characteristic orthogonal polynomials are employed in the RayleighRitz method to obtain the natural frequencies and associated mode shapes. Fantuzzi et al. [76,77,78] considered decomposition method for solving composite laminated arbitrary shape plates using differential quadrature method. ...
... It means that with vertical and horizontal lines, division points of all edges are connected. i.e. with three division in each edge, there are nine rectangular parts which should be solved separately and continuity conditions should apply among all subparts [76,77,78]. Continuity conditions include be equal in displacements, moments and forces. ...
Vibration analysis of composite laminated plates and shells using a spectral method
... The model is validated against the results from the commercial software ABAQUS based on the classical FEM and Generalized Differential Quadrature FEM (GDQFEM) based on the First-Order Shear Deformation Theory [17]. The presented model allows for the consideration of laminated composite plates of arbitrary shape. ...
... Free vibration analysis of the composite plate has been performed in order to validate the GLPT model against the GDQFEM solution from Ref. [17] and commercial software ABAQUS CAE. The first ten natural frequencies have been calculated for the intact plate, using two different mesh densities (app. ...
In the paper, free and forced vibrations of laminated composite plates having complex shape and embedded delaminations, have been considered. Computational model has been derived based on Reddy’s Generalized Laminated Plate Theory (GLPT), assuming layerwise linear variation of in-plane displacements and constant transverse displacement through the plate thickness. The assumed kinematics allows for the real cross sectional warping. The delamination openings in three orthogonal directions have been implemented using Heaviside step functions. Linear kinematics and Hooke’s constitutive law have been considered. Numerical solution has been obtained using the triangular 6- node layered finite elements, to approximately simulate the arbitrary geometry of the plate. To prevent layer overlapping in the delaminated zone, a novel node-to-node frictionless contact algorithm has been implemented in the computational model.
Delamination propagation has been predicted using a simple and efficient algorithm based on the Virtual Crack Closing Technique (VCCT), requiring the calculation of the virtually closed area in front of the delamination, the delamination openings behind the delamination and the reaction forces along the delamination front. All calculations have been performed using the original MATLAB code. The results have been compared with the existing data from the literature and the results from the commercial software. A variety of new results has been provided to serve as a benchmark for further investigations.
... Fantuzzi (2013) introduced the generalized differential quadrature finite element method. This method was used to investigate the free vibration of arbitrary shaped membranes (Fantuzzi et al. 2014) and cracked composite structures of arbitrary shape (Viola et al. 2013).They implemented the first-order shear deformation plate theories in their research. Although the presented results show the higher accuracy of this approach, the manner of applying continuity conditions between adjacent DQ elements and boundary conditions has not been explicitly proposed. ...
... The shear force and the total transverse force components are expressed by Viola et al. (2013) In order to introduce a refined approach in the differential quadrature element method, one can make some modifications in the aforementioned relations. So, the transverse displacement derivatives according to the following relationships are firstly considered as ...
The aim of the present study is to develop an elemental approach based on the differential quadrature method for free vibration analysis of cracked thin plate structures. For this purpose, the equations of motion are established using the classical plate theory. The well-known Generalized Differential Quadrature Method (GDQM) is utilized to discretize the governing equations on each computational subdomain or element. In this method, the differential terms of a quantity field at a specific computational point should be expressed in a series form of the related quantity at all other sampling points along the domain. However, the existence of any geometric discontinuity, such as a crack, in a computational domain causes some problems in the calculation of differential terms. In order to resolve this problem, the multi-block or elemental strategy is implemented to divide such geometry into several subdomains. By constructing the appropriate continuity conditions at each interface between adjacent elements and a crack tip, the whole discretized governing equations of the structure can be established. Therefore, the free vibration analysis of a cracked thin plate will be provided via the achieved eigenvalue problem. The obtained results show a good agreement in comparison with those found by finite element method.
... Tornabene et al. [32] examined the static bending of anisotropic doubly curved shells with arbitrary geometry and variable thickness resting on a Winkler-Pasternak using higher-order theory. Besides, Viola et al. [33] used the GDQ finite-element method for free vibration analysis of cracked composite structures. Tornabene et al. [34] developed a differential quadrature solution for the free vibrations study of shells and panels of revolution with a free-form meridian. ...
This article provides a new finite-element procedure based on Reddy’s third-order shear deformation plate theory (TSDT) to establish the motion equation of functionally graded porous (FGP) sandwich plates resting on Kerr foundation (KF). Although Reddy’s TSDT is attractive, it cannot be exploited as expected in finite-element analysis due to the difficulties in satisfying the zero shear stress boundary condition. In this study, the proposed element has four nodes, each with seven degrees of freedom (DOF). The performance of this element is confirmed by conducting various examples, showing its accuracy and range of applications. Then, some studies are performed to present the effects of input parameters on the vibration of FGP sandwich plates resting on KF.
... It has been demonstrated in many articles [68][69][70][71] that, if a non-uniform set of discretization points is chosen within the computational domain, GDQ provide higher performances with respect to other numerical methods. In particular, the advantages of GDQ technique employment are more evident when structures with very complex shapes and materials are investigated [72][73][74][75][76]. Since the GDQ method requires a very little number of DOFs, it has been successfully applied to CFD simulations [77,78], as well as fracture mechanics [79][80][81][82] whereas classical methods based on domain decomposition require a very high effort because of the need of a very refined discretization. ...
Laminated doubly-curved shells constituted by innovative materials have become a standard application in several engineering fields. However, this requires a proper formulation of such structures, since several very complex issues affecting such applications must be kept in mind, among all, the curvature effect and the coupling issues between two adjacent layers. In the present work a generalized formulation based on Higher Order Theories is proposed for the linear static analysis of curved structures with completely anisotropic materials in each layer of the lamination scheme. The shell model is described according to the Equivalent Single Layer approach and a mapping procedure based on a Non-Uniform Rational Basis Spline (NURBS) description of the shell edges is applied for the geometric description. Different thickness variations have been considered in the analyses. The fundamental equations of the static problem are derived from the minimum potential energy principle directly in the strong form and the effect of the Winkler-Pasternak support has been accounted for. The external surface load has been applied in each principal direction and arbitrary actions have been enforced to the external edges of the structure. The proposed approach has been validated with respect to the outcomes of some 3D Finite Element models and very good agreement is found between such simulations. Shells characterized by very complex shapes have been accounted for and accurate results have been found with a reduced number of degrees of freedom.
... For the study, hybrid composites are considered which is made from E-glass, Carbon, and Epoxy with different orientations to get the stress values of the plate under uniformly distributed transverse load. In other studies, such as [20,21,22], the resulting deflection in constructs was calculated using Matlab code to solve linear algebraic problems where the solution procedure was implemented using the generalized differential square (GDQ) technique in MATLAB code. The shear stresses and normal stresses subsequently evaluated with the present methodology agree well with the results get from semianalytical and numerical techniques. ...
The current review covers numerical analysis using Matlab program for multi-layer composite materials. Which involves studies are related to the intersection with fibers, between layers and piezoelectric layer or patch. It has been reported that using Matlab program has a great flexibility in analysis due to its library which includes various numerical methods. In addition to the ability of programming and developing the finite element technology to calculate the stress and strain in each layer based on different deformation methods such as (FSDT and HSDT), to obtain mechanical properties. It has been claimed that there is a deviation in results between Matlab and Ansys for the same 20-layer composite material. Using Matlab in dynamic analysis in various methods such as Newmark, Rayleigh damping, Timoshenko, and Euler-Bernoulli exhibit good agreement with natural frequencies and mode shapes. Moreover, Matlab is useful for the real-time process of data acquisition to deliver a digital model of a composite material coated with a piezoelectric plate and is an ideal material for sensing, detecting, and controlling vibration inhibition.
... Civalek [12][13][14] proposed a discrete singular convolution method, where the geometric transformation technique of four-node and eight-node had been applied one after another. Tornabene [15][16][17] proposed the generalized DQFEM for the dynamics analysis of composite plates. In this method, the whole plate was subdivided into several subdomains of irregular shapes. ...
This paper reports a modeling and experimental study on the free vibration characteristics of plates with curved edges. The model is established by the first-order shear deformation theory (FSDT) and Chebyshev differential quadrature method (CDQM). The one-to-one coordinate transformation technique is introduced into the CDQM to map the plate with curved edges into a square plate. The admissible displacement functions of the square plate are expanded by two-dimensional Chebyshev polynomials and discretized by Gauss-Lobatto sampling points. The boundary conditions are applied to the plate according to the projection matrix method. Experimental studies of six aluminum plates with different shapes are carried out to investigate the vibration characteristics and verify the validity of the proposed CDQM. Furthermore, the results of the present CDQM are also compared with those of the finite element method (FEM) and existing numerical approaches to examine its efficiency and accuracy. The results show that the current CDQM can rapidly and accurately compute the vibration characteristics of plates with curved edges under different boundary conditions.
... So far, scholars mainly utilize numerical methods to solve the vibration problems of the arbitrary-shaped plate and shell, for instance, finite element approach [1], generalized differential quadrature finite element method [2], general higher-order equivalent single layer theory [3], differential quadrature method [4,5], complex variable methods [6], and differential volume method [7]. Ahmad et al. [8] proposed a method to overcome the disadvantage of former ways when approximate geometric structure and the influence of shear displacement were ignored, by using the curved thick shell finite element method. ...
In this investigation, an improved Rayleigh–Ritz method is put forward to analyze the free vibration characteristics of arbitrary-shaped plates for the traditional Rayleigh–Ritz method which is difficult to solve. By expanding the domain of admissible functions out of the structural domain to form a rectangular domain, the admissible functions of arbitrary-shaped plates can be described conveniently by selecting the appropriate admissible functions. Adopting the spring model to simulate the general boundary conditions, the problems of vibration of the arbitrary plate domain can be solved perfectly. Then, a numerical method is introduced to figure out the structure strain energy, kinetic energy, and elastic potential energy of the boundary. Finally, comparing the result with the simulation results and reference examples, the accuracy and convergence of this method are testified. Therefore, an effective new method is proposed for the guidance of the related research and practical engineering problems.
... Their obtained results showed that the natural frequency decreases with increase in temperature gradient and crack length. Viola et al. [8], applied finite element method for the vibration analysis of thick composite plate with crack and extended it to plates of arbitrary shapes. The static solutions for cracked plate under influence of thermal environment are mostly done by different numerical techniques in literature, but an approximate analytical solution is only possible by means of the Line Spring Model (LSM). ...
Based on a non-classical plate theory, an analytical model is proposed for the first time to analyze free vibration problem of partially cracked thin isotropic submerged plate in the presence of thermal environment. The governing equation for the cracked plate is derived using the Kirchhoff"s thin plate theory and the modified couple stress theory. The crack terms are formulated using simplified line spring model whereas the effect of thermal environment is introduced using thermal moments and in-plane forces. The influence of fluidic medium is incorporated in governing equation in form fluids forces associated with inertial effects of its surrounding fluids. Applying the Galerkin method, the derived governing equation of motion is reformulated into well-known Duffing equation. The governing equation for cracked isotropic plate has also been solved to get central deflection which shows an important phenomenon of shift in primary resonance due to crack, temperature rise and internal material length scale parameter. To demonstrate the accuracy of the present model, few comparison studies are carried out with the published literature. The variation in natural frequency of the cracked plate with uniform rise in temperature is studied considering various parameters such as crack length, fluid level and internal material length scale parameter. Furthermore, the variation of the natural frequency with plate thickness is also established.
... The finite element method has been successfully applied to various engineering simulations and has a great advantage in dealing with continuous problems (Sarris and Papanastasious 2012;Birk and Behnke 2012;Erasmo et al. 2013;Zhang et al. 2017a, b). However, when it involves the discontinuous problems of cracked rock masses, FEM betrays huge limitation and deficiency. ...
Large cracks are important seepage channels inside fractured-vuggy reservoirs. Therefore, in this thesis, the calculation method of fully coupled modeling of the fractured saturated porous medium based on the extended finite element method (XFEM) is established to study the expanding regularity of cracks in fractured-vuggy reservoirs. Fully coupled governing equations are developed for hydro-mechanical analysis of deforming porous medium with fractures based on the stress balance equation, the seepage continuity equation and the effective stress principle. The final nonlinear fully coupled equations reflect not only the coupling effect of the physical quantity within the porous medium but also the coupling between the medium and the fracture. During the spatial dispersion of coupled equations based on XFEM, two kinds of additional displacement functions are introduced in the displacement model of the fracture area to reflect the strong discontinuity of the fracture surface. The pore pressure enhancement function is also applied to represent the weak discontinuous features of the normal pore pressure. The validity and efficiency of this model and calculation are verified through three calculating examples. The following crack propagation laws are obtained: (1) The larger the water flow rate is, the longer the crack propagation length is, and the larger the propagation width is (2) The greater the crack angle and the crack length, the easier it is to expand the crack. Besides, compared with dip angle, the crack length has a more sensitive influence to the crack propagation. (3) When multiple cracks exist, the larger the fracture spacing is, the easier the crack will expand.
... While modelling plates with voids using collocation methods, the domain has to be split into multiple patches and appropriate compatibility conditions have to be applied along the boundary [12]. The level set method [13] commonly used in IGA Galerkin methods, cannot be employed in collocation methods, as it cannot address the discontinuities without splitting the domain. ...
Modelling of plates with internal defects or cut outs is often an issue in the traditional isogeometric methods and hence extended isogeometric methods are developed. In order to reduce the computational cost involved in extended isogeometric methods based on weak form, a new extended isogeometric hybrid collocation–Galerkin method is proposed in the present paper. The natural frequencies of a homogeneous and functionally graded material plate with internal defects of varying gradient index, sizes and shapes are obtained using the proposed method. The obtained results are compared and found to be in good agreement with the reference results.
... The strong form finite element is implemented with a quadratic mapping with eight nodes per element as discussed in Viola et al. [2013b]. For the following numerical applications, the use of quadratic mapping is sufficient. ...
... The influence of holes on the free vibration behaviors of isotropic and orthotropic plates with simply-supported and clamped boundary conditions was investigated [33]. Viola et al. [42] carried out free vibration analyses for composite plates with various shapes containing elliptic holes and slits using the generalized differential quadrature finite element method. Fantuzzi et al. [12] derived a FE model to study the dynamics of multi-layered plates with discontinuities. ...
... The question of how the existence of a defect such as a crack can affect the vibrational behavior of plates has drawn research attention for quite some time [19][20][21][22][23][24]. The natural frequencies of various cracked composite plates were obtained by employing the so-called Generalized Differential Quadrature Finite Element Method (GDQFEM) [25,26]. Also, the GDQFEM was successfully used in the vibrational analysis of FGM sandwich shells with variable thicknesses [27]. ...
... In the present work, we employ the GDQ technique to solve the governing equations of the problem in a strong form, as increasingly used in the last years in the literature due to its great features of accuracy and reliability [44][45][46][47][48][49][50][51][52][53][54][55] . The main aspects of the method are illustrated in what follows for a 1D domain [ x 0 , x 1 ] for simplicity, and can be easily extended to a 2D domain (for more details see Tornabene et al. [56] ). ...
... This second feature can be seen as an issue or an advantage, because all boundary conditions must be written explicitly. Furthermore, the numerical solution is extremely accurate also at the inter-element boundaries and not only within the domains [13,14]. ...
The present study aims to show a novel numerical approach for investigating composite structures wherein inclusions and discontinuities are present. This numerical approach, termed Strong Formulation Finite Element Method (SFEM), implements a domain decomposition technique in which the governing partial differential system of equations is solved in a strong form. The provided numerical solutions are compared with the ones of the classic Finite Element Method (FEM). It is pointed out that the stress and strain components of the investigated model can be computed more accurately and with less degrees of freedom with respect to standard weak form procedures. The SFEM lies within the general framework of the so-called pseudo-spectral or collocation methods. The Differential Quadrature (DQ) method is one specific application of the previously cited ones and it is applied for discretizing all the partial differential equations that govern the physical problem. The main drawback of the DQ method is that it cannot be applied to irregular domains. In converting the differential problem into a system of algebraic equations, the derivative calculation is direct so that the problem can be solved in its strong form. However, such problem can be overcome by introducing a mapping transformation to convert the equations in the physical coordinate system into a computational space. It is important to note that the assemblage among the elements is given by compatibility conditions, which enforce the connection with displacements and stresses along the boundary edges. Several computational aspects and numerical applications will be presented for the aforementioned problems related to composite materials with discontinuities and inclusions.
... Analogously, the same scheme coupled with higher-order shear deformation theories was employed to achieve the numerical solution to similar problems [39][40][41][42][43][44][45][46][47][48][49][50][51]. Finally, it should be noted that the GDQ method can be used to solve the strong formulation when the reference domain is characterized by arbitrary geometries [52][53][54][55][56][57][58][59][60][61][62][63][64][65]. In general, the strong formulation needs a higher order of derivation in comparison with the corresponding weak form, which is introduced to weaken (or reduce) the order of differentiability [3]. ...
The main aim of the paper is to present a new numerical method to solve the weak formulation of the governing equations for the free vibrations of laminated composite shell structures with variable radii of curvature. For this purpose, the integral form of the stiffness matrix is computed numerically by means of the Generalized Integral Quadrature (GIQ) method. A two-dimensional structural model is introduced to analyze the mechanical behavior of doubly-curved shells. The displacement field is described according to the basic aspects of the general Higher-order Shear Deformation Theories (HSDTs), which allow to define several kinematic models as a function of the free parameter that stands for the order of expansion. Since an Equivalent Single Layer (ESL) approach is considered, the generalized displacements evaluated on the shell middle surface represent the unknown variables of the problem, which are approximated by using the Lagrange interpolating polynomials. The mechanical behavior of the structures is modeled through only one element that includes the double curvature in its formulation, which is transformed into a distorted domain by means of a mapping procedure based on the use of NURBS (Non-Uniform Rational B-Splines) curves, following the fundamentals of the well-known Isogeometric Analysis (IGA). For these reasons, the presented methodology is named Weak Formulation Isogeometric Analysis (WFIGA) in order to distinguish it from the corresponding approach based on the strong form of the governing equations (Strong Formulation Isogeometric Analysis or SFIGA), previously introduced by the authors. Several numerical applications are performed to test the current method. The results are validated for different boundary conditions and various lamination schemes through the comparison with the solutions available in the literature or obtained by a finite element commercial software.
The main aim of this book is to analyze the mathematical
fundamentals and the main features of the Generalized
Differential Quadrature (GDQ) and Generalized Integral
Quadrature (GIQ) techniques. Furthermore, another interesting
aim of the present book is to shown that from the two numerical
techniques mentioned above it is possible to derive two different
approaches such as the Strong and Weak Finite Element
Methods (SFEM and WFEM), that will be used to solve various
structural problems and arbitrarily shaped structures.
A general approach to the Differential Quadrature is
proposed. The weighting coefficients for different basis
functions and grid distributions are determined. Furthermore,
the expressions of the principal approximating polynomials
and grid distributions, available in the literature, are shown.
Besides the classic orthogonal polynomials, a new class of basis
functions, which depend on the radial distance between the
discretization points, is presented. They are known as Radial
Basis Functions (or RBFs). The general expressions for the
derivative evaluation can be utilized in the local form to reduce
the computational cost. From this concept the Local Generalized
Differential Quadrature (LGDQ) method is derived.
The Generalized Integral Quadrature (GIQ) technique can be
used employing several basis functions, without any restriction
on the point distributions for the given definition domain.
To better underline these concepts some classical numerical
integration schemes are reported, such as the trapezoidal rule or
the Simpson method. An alternative approach based on Taylor
series is also illustrated to approximate integrals. This technique
is named as Generalized Taylor-based Integral Quadrature (GTIQ)
method.
The major structural theories for the analysis of the
mechanical behavior of various structures are presented in depth
in the book. In particular, the strong and weak formulations
of the corresponding governing equations are discussed and
illustrated. Generally speaking, two formulations of the same
system of governing equations can be developed, which are
respectively the strong and weak (or variational) formulations.
Once the governing equations that rule a generic structural
problem are obtained, together with the corresponding boundary
conditions, a differential system is written. In particular, the
Strong Formulation (SF) of the governing equations is obtained.
The differentiability requirement, instead, is reduced through
a weighted integral statement if the corresponding Weak
Formulation (WF) of the governing equations is developed. Thus,
an equivalent integral formulation is derived, starting directly
from the previous one. In particular, the formulation in hand is
obtained by introducing a Lagrangian approximation of the
degrees of freedom of the problem.
The need of studying arbitrarily shaped domains or
characterized by mechanical and geometrical discontinuities
leads to the development of new numerical approaches that
divide the structure in finite elements. Then, the strong form or
the weak form of the fundamental equations are solved inside
each element. The fundamental aspects of this technique,
which the author defined respectively Strong Formulation Finite
Element Method (SFEM) and Weak Formulation Finite Element
Method (WFEM), are presented in the book.
A weak-form spectral element modeling and the corresponding experimental studies in this work are concerned to clarify free vibration mechanism for the plate of arbitrary shape having built-in complex hole and crack. To effectively address the discontinuity problem at irregular hole as well as crack of varying curvature, some quadrilateral plate elements with variable curvature edges are rigidly coupled together to form the investigated structural system. A flexible coordinate mapping strategy is here employed for satisfying the adaptability requirements of the introduced displacement functions to the sophisticated solution domain. Accordingly, the complete vibration equation of the physical model is solved based on the well-known first-order scheme of plate in conjunction with 2D spectral element functions, where the boundary constraints and compatibility relations in the form of energy quantification are imposed through the penalty function method. After that, modal experiments considering six unconstrained imperfect plates with pre-configured crack and hole, are incorporated with a series of acceptable solutions, provided by the available literature and some associated models of FEM, for disclosing the excellent predictive capability of the spectral element methodology in vibration performance. The reported eigenfrequencies and eigenmodes respecting arbitrarily shaped plate introducing intricate hole and curved crack are the first presented vibration solutions, and meaningful benchmarks for evaluating correctness of other numerical or analytical approaches.
The integration of generalized differential quadrature techniques and finite element methods has been developed during the past decade for engineering problems within classical continuum theories. Hence, the main objective of the present study is to propose a novel numerical strategy called the multi‐patch variational differential quadrature (VDQ) method to model the structural behavior of plate structures obeying the shear deformation plate theory within the strain gradient elasticity theory. The idea is to divide the two‐dimensional solution domain of the plate model into sub‐domains, called patches, and then to apply the VDQ method along with the finite element mapping technique for each patch. The formulation is presented in a weak form and due to the C1‐continuity requirements the corresponding compatibility conditions are applied through the patch interfaces. The Lagrange multiplier technique and the penalty method are implemented to apply the higher‐order compatibility conditions and boundary conditions, respectively. To show the efficiency of the proposed method, numerical results are provided for plate structures with both regular and irregular solution domains. The provided numerical examples demonstrate the applicability and accuracy of the method in predicting the bending and vibration behavior of plate structures following the higher‐order plate model.
Free vibration behavior of a composite shell/panel with and without a central square cutout has been studied using Multi-domain Generalized Differential Quadrature (GDQ) method. A physical domain is decomposed into several elements in such a way that all the elements have uniform thickness and material properties, as well as continuous loading and boundary conditions at their edges. The governing equations are derived based on the first-order shear deformation theory. They are formulated in a general form and can be converted to Donnell’s, Love’s, and Sanders’ theories. In addition, compatibility conditions are considered at the interface boundaries of adjacent elements as well as proper boundary conditions on other elements‘ edges. The GDQ method is employed to discretize these equations in both longitudinal and circumferential directions. By assembling these discretized relations, a system of algebraic equations will be generated. These equations can be solved through an eigenvalue solution to compute the natural frequencies of the whole shell/panel. Numerical results obtained by the presented method are compared with ABAQUS results and those available in the literature. After verifying the accuracy and precision of the proposed method, it is employed to study the effect of an opening on the vibrational behavior of composite shells and panels through a parametric study. The influence of the presence of a cutout and its size is investigated on shells/panels with various material properties, layup orientations, thicknesses, dimensions, and boundary conditions. The obtained results can be used as a benchmark for further researches.
In the present work, a finite element model in conjunction with higher-order shear deformation theory was developed to investigate the eigenfrequency responses of hybrid composite (banana-glass-epoxy) flat-panel structures. First of all, five different sets of banana-glass-epoxy hybrid composite plates were fabricated using an in-house fabrication facility. The test specimen for elastic property evaluation was prepared according to the ASTM standard. The necessary elastic properties were obtained by tensile test and utilized in numerical analysis. Furthermore, to develop the mathematical model of hybrid laminated composite, higher-order displacement kinematics were used that eliminates the consideration of shear correction factor. The model of plate was divided into several small parts using 2D nine noded elements. The elemental mass and stiffness of the panel were obtained and assembled to get the global mass and stiffness. The global values were substituted further into the governing equation derived from Hamilton principle. Furthermore, constraints conditions were imposed and the governing equation was solved to get the natural frequency. First, the numerical model’s sensitivity and accuracy were checked by stability and validation study. After that new numerical examples were solved and influences of different parameters including hybridization on frequency responses were studied and discussed in detail.
Due to corner stress singularities, free vibration analysis of thin sectorial plates with arbitrary boundary supports including a free vertex is a rather challenging problem. In this paper, a novel rotation-free quadrature element formulation based on Kirchhoff plate theory is presented to solve the problem. Different from all existing quadrature thin plate element formulations, rotational degrees of freedom are not used. Instead, a boundary point is modeled by two nodes separated by a very small distance and thus the C1 continuity is easily enforced. Explicit formulations with arbitrary number of nodes and nodes of any type are worked out to ease the p-refinement. Numerical results demonstrate that the proposed formulation is simple, straightforward and has excellent performance for solving thin sectorial plate with arbitrary combinations of boundary supports. Some tabulated data are believed new which may serve as a reference.
BackgroundA Evolution of material taken place for ages and in today's world even faster. The new concept faces a new set of challenges and is to execute in the real-world researchers need the material of new concepts. The journey that started from the base material of periodic table to composite materials has extended its steps towards functionally graded materials.
Review FactorThis paper reviews the effect of the appearance of a crack in component made of FGM. Wide applicability of FGM makes it more prone to crack during the service period. Therefore, it is necessary to evaluate the risk factors associated with cracks.ConclusionsFGM beam is most investigated structure made of FGM followed by plates and all. In most of cases, we observed that increase in any parameter of crack reduces the natural frequency of vibration on structure. It was also observed that theories implied also affect the results under the same parameter such as shear order deformation theory which provides greater value followed by Timoshenko, Euler Bernoulli, and Rayleigh in respective order.
An original work is presented for the free vibration analysis of thin functionally graded material (FGM) annular sectorial plates with arbitrary boundary supports using the weak form quadrature element method (QEM). Novel C¹ quadrature annular sectorial plate element is developed based on the physically neutral surface thin plate theory in cylindrical coordinate system. Three displacement functions are used to avoid the introduction of the mixed second order derivative degree of freedom at corner points. The derived explicit formulations with a varying node number can be applied for any type of node. A way to define the elastic spring constants is proposed to ensure the independent of the non-dimensional frequency parameter on the power-law exponent. The convergence and accuracy of the present model are validated by comparing the obtained results with those available in literatures. It is demonstrated that the proposed model is simple, straightforward and has good performance. A wide range of numerical results are reported to demonstrate the effect of the sector angle, ratio of inner/outer radii, and boundary conditions on the vibration behavior of the thin annular sectorial plate. Some tabulated results should be new which may serve as a reference for verifying numerical solutions obtained by other methods.
Communicated by Wei-Chau Xie.
In the context of variational differential quadrature finite element method (VDQFEM), the vibration and buckling of functionally graded graphene platelet reinforced composite (FG-GPLRC) plates with cutout are investigated in this study. The modified Halpin-Tsai model is used to compute the effective mechanical properties of nanocomposite plate. The mixed-type formulation of the higher-order shear deformation plate theory (HSDPT) is developed based on the Lagrange multiplier to model the mechanical behavior of FG-GPLRC plates. The variational-based differential quadrature element is developed to numerically solve the problem. Various comparative results are provided to check the validity of the proposed approach. Additionally, multiple numerical results are represented to examine the vibration and buckling behaviors of FG-GPLRC plates.
An analytical model is proposed to analyze the vibration and buckling problem of partially cracked thin orthotropic microplate in the presence of thermal environment. The differential governing equation for the cracked plate is derived using the classical plate theory in conjunction with the strain gradient theory of elasticity. The crack is modeled using appropriate crack compliance coefficients based on the simplified line spring model. The influence of thermal environment is incorporated in governing equation in form thermal moments and in-plane compressive forces. The governing equation for cracked plate has been solved analytically to get fundamental frequency and central deflection of plate. To demonstrate the accuracy of the present model, few comparison studies are carried out with the published literature. The stability and dynamic characteristics of the cracked plate are studied considering various parameters such as crack length, plate thickness, change in temperature, and internal length scale of microstructure. It has been concluded that the frequency and deflection are affected by crack length, temperature, and internal length scale of microstructure. Furthermore, to study the buckling behavior of cracked plate, the classical relations for critical buckling load and critical buckling temperature is also proposed considering the effect of crack length, temperature, and internal length scale of microstructure.
Purpose
To develop a new analytical model for vibration analysis of cracked-submerged orthotropic micro-plate affected by fibre orientation and thermal environment.
Methods
The proposed analytical model is based on Kirchhoff’s classical thin plate theory and the size effect is introduced using the modified couple stress theory. Effect of crack is deduced using appropriate crack compliance coefficients based on line spring model while the effect of thermal environment is introduced in terms of thermal in-plane moments and forces. The coupling of shear and normal stresses for fibre orientation is represented using the coefficient of mutual influence. The fluid forces associated with its inertial effects are added in the governing differential equation to incorporate the fluid–structure interaction effect.
Results
The results are presented for frequency response as affected by different fibre orientation, crack length, crack location, level of submergence, temperature variation and material length-scale parameter for simply supported boundary condition. Furthermore, to study the phenomenon of shifting of primary resonance in a cracked micro-plate, the classical relations for central deflection of plate is also proposed.
Conclusions
The results show that the fundamental frequency of micro-plate decreases by the presence of crack and thermal environment and this decrease in frequency is further intensified by the presence of surrounding fluid medium in present study. Another important conclusion is that with increase in temperature variation the reduction in frequency at 45° of fibre orientation is less when compared to 0 and 90° for both intact and cracked orthotropic plates.
In this paper, a new numerical method, named as the Free Element Collocation Method (FECM), is proposed for solving general engineering problems governed by the second order partial differential equations (PDEs). The method belongs to the group of the collocation method, but the spatial partial derivatives of physical quantities are computed based on the isoparametric elements as used in FEM. The key point of the method is that the isoparametric elements used can be freely formed by the nodes around the collocation node. To achieve a narrow bandwidth of the final system of equations, elements with a central node are recommended. For this purpose, a new 21-node quadratic element for 3D problems is constructed for the first time. Attributed to the use of isoparametric elements, FECM can result in more stable results than the traditional collocation method. In addition, the elements can be freely formed by local nodes, FECM has the advantage of mesh-free methods to fit complicated geometries of engineering problems. A number of numerical examples of 2D and 3D thermal and mechanical problems are given to demonstrate the correctness and efficiency of the proposed method.
The analysis of large amplitude vibrations of cracked plates is considered in this study. The problem is addressed via a Ritz approach based on the first-order shear deformation theory and von Kármán’s geometric nonlinearity assumptions. The trial functions are built as series of regular orthogonal polynomial products supplemented with special functions able to represent the crack behaviour (which motivates why the method is dubbed as eXtended Ritz); boundary functions are used to guarantee the fulfillment of the kinematic boundary conditions along the plate edges. Convergence and accuracy are assessed to validate the approach and show its efficiency and potential. Original results are then presented, which illustrate the influence of cracks on the stiffening effect of large amplitude vibrations. These results can also serve as benchmark for future solutions of the problem.
s
In this work a new strong-form numerical method, element differential method (EDM), is proposed to perform free and forced vibration analysis of elastodynamic problems. The present method establishes the global algebraic system equations directly based on the strong form of the equilibrium equations without using any variational principles or energy principles. In this method, the isoparametric elements with a node inside them are utilized to discrete the geometries. And the direct differentiation of their shape functions is used to characterize the geometry and physical variables. A novel collocation technique is then proposed to generate the system of equations, in which the dynamic equilibrium equations are collocated only at the internal nodes of elements, and the traction equilibrium equations are collocated at the interface and outer surface nodes. Compared with the standard finite element method, no integrals are involved to form the coefficients of the system and a reduced and lamped mass matrix can be directly obtained from the density properties of the target problem. Since the mass term only exists at the internal nodes of elements, the dynamic coupling of final system equations of the structure can be reduced, which will greatly save the computational resources. Numerical examples about free and forced 2D and 3D dynamic problems are given to demonstrate the correctness and efficiency of the proposed method.
Nonlinear air blast response of basalt composite plates is analysed by using a generalized differential quadrature (GDQ) method, which requires less solution time and decreases the complexity compared to finite element method. A test environment that contains a shock tube is designed and set to experiment on the transient response of blast loaded laminated plates. Experimental and numerical results show a good agreement in terms of displacement, strain, and acceleration versus time. The responses of glass/epoxy, Kevlar/epoxy, and carbon/epoxy composite plates are also investigated by using GDQ method and the results are compared with the basalt/epoxy composite plate and discussed.
The virtual spring technique is firstly introduced into the differential quadrature finite element method (DQFEM) to simulate the practical elastic restraints. The imposing procedures of the boundary conditions are simplified so that a certain kind of restraints can be easily achieved by merely setting different stiffness of the springs. The mapping technique is used to apply the DQFEM to irregular domain. The effects of different nodes collocation methods on the mapping results and vibration results are also discussed, through which one can conclude that the nodes distribution methods affect the accuracy of the mapping technique and the computing time. Especially, the uniformly distributed nodes are not the best selection for mapping process. The Guass Lobatto quadrature nodes are the good choice to obtain the better results in a relatively short time. Several numerical examples are carried out to demonstrate the validity and accuracy of the present solution by comparing with the results obtained by other researchers.
In this paper, a semi-analytical method is presented to study the transverse vibration of sector-like thin plate with various boundary conditions. A sector-like plate is generated from a sector plate whose circular edge is replaced by edges of other geometries. The expressions of boundary conditions for arbitrary point on the arc edge are given in a local rectangular coordinate system and transformed into the global polar coordinate system. Fourier expansion is performed on the expressions of the boundary conditions so that the boundary conditions are satisfied on the whole edge. Several examples are presented and the results are calculated numerically and compared with those in literature and obtained by FEM. The good agreement between the results validates the accuracy and applicability of this method for sector-like plates.
In this article, free flexural vibration and supersonic flutter analyses are studied for cantilevered trapezoidal plates composed of two homogeneous isotropic face sheets and an orthotropic honeycomb core. The plate is modeled based on the first-order shear deformation theory, and aerodynamic pressure of external flow with desired flow angle is estimated via the piston theory. For this goal, first applying the Hamilton's principle, the set of governing equations and boundary conditions are derived. Then, using a transformation of coordinates, the governing equations and boundary conditions are converted from the original coordinates into new computational ones. Finally, the differential quadrature method is employed and natural frequencies, corresponding mode shapes, and critical speed are numerically achieved. Accuracy of the proposed solution is confirmed by the finite element simulations and published experimental results. After the validation, effect of various parameters on the vibration and flutter characteristics of the plate are investigated. It is concluded that geometry of hexagonal cells in the honeycomb core has a weak effect on the natural frequencies and critical speed of the sandwich plate, whereas thickness of the honeycomb core has main influence on the natural frequencies and the critical speed. Besides, it is shown that the honeycomb core thickness has optimum values that lead to the most growth in the natural frequencies or critical speed. These optimum magnitudes can be taken into account by designers to increase the natural frequencies or expand flutter boundaries and make aircrafts safer in supersonic flights. It is also concluded that geometrical parameters of the hexagonal cells and thickness of the honeycomb core have no significant effect on the value of the critical flow angle.
In this paper, the transient behavior of an internally damaged laminated composite plate and shell structure under the influence of different mechanical loading types and constraint conditions has been analyzed numerically. For the numerical purpose, two well-known higher-order displacement kinematics are used to model the doubly curved shell panel in association with the finite-element steps. In addition, the internal delamination is modeled with the help of two sublaminate approaches including the intermittent continuity condition to obtain the necessary solutions. Further, the domain has been discretized with the assistance of a biquadratic nine-noded quadrilateral element. The panel motion equation is derived by integrating the total Lagrangian expression and solved to evaluate the time-dependent responses via an in-house computer code in association with Newmark's direct integration scheme. The stability of the found numerical solutions are checked through a convergence test and compared with established benchmark solutions. The performance of the developed numerical models are established by comparing the results with the subsequent experiments. Finally, the effect of internal debonding (size, position, and location) and other design parameters on the time-dependent deflections of the delaminated composite panel are examined including the geometries (spherical, cylindrical, elliptical, and hyperboloid) and discussed in detail.
As a useful tool for designing wings and tail fins of aircrafts, this paper presents an optimization for flutter characteristics of cantilevered functionally graded sandwich plates. The plate is composed of an isotropic homogeneous core and two functionally graded face sheets. The plate is modeled based on the first-order shear deformation theory. The aerodynamic pressure is estimated using supersonic piston theory and using Hamilton's principle, the set of governing equations and boundary conditions are then derived. Applying a transformation of coordinates, governing equations and boundary conditions are converted and solved numerically by differential quadrature method. Natural frequencies, damping ratio, corresponding mode shapes, critical aerodynamic pressure, and flutter frequency are calculated. In order to achieve an optimum design, particle swarm optimization is employed to find the best values of aspect ratio, thickness of the plate, thickness of the core, power law index, and angles of the plate which increase critical aerodynamic pressure. Some constrains on the angles of the plate and its mass and area (lift force) are also considered.
The authors are presenting a novel formulation based on the Differential Quadrature (DQ) method which is used to approximate derivatives and integrals. The resulting scheme has been termed strong and weak form finite elements (SFEM or WFEM), according to the numerical scheme employed in the computation. Such numerical methods are applied to solve some structural problems related to the mechanical behavior of plates and shells, made of isotropic or composite materials.
The main differences between these two approaches rely on the initial formulation – which is strong or weak (variational) – and the implementation of the boundary conditions, that for the former include the continuity of stresses and displacements, whereas in the latter can consider the continuity of the displacements or both.
The two methodologies consider also a mapping technique to transform an element of general shape described in Cartesian coordinates into the same element in the computational space. Such technique can be implemented by employing the classic Lagrangian-shaped elements with a fixed number of nodes along the element edges or blending functions which allow an “exact mapping” of the element. In particular, the authors are employing NURBS (Not-Uniform Rational B-Splines) for such nonlinear mapping in order to use the “exact” shape of CAD designs.
The free vibration behavior of quasi-isotropic carbon fiber laminated composite plates containing circular holes with free-clamped boundary conditions are numerically, analytically, and experimentally investigated. Finite element models based on classical plate theory (Kirchhoff) and the shear deformable theory (Mindlin) within the framework of equivalent single-layer and layer-wise concepts as well as the three-dimensional theory of elasticity are developed. These models are created using the finite element software, Abaqus, to determine the natural frequencies and the corresponding mode shapes. In addition, an analytical model based on Kirchhoff plate theory is developed. Using this approach, an equivalent bending-torsion beam model for cantilever laminated plates is extracted taking into account the reduction in local stiffness and mass induced by the center hole. Experimental vibration analyses are carried out using an optically-based vibration measurement tool to extract the frequency response functions and to measure the natural frequencies. Numerical and analytical natural frequency values are then compared with those obtained through experimental vibrational tests, and the accuracy of each finite element (FE) and analytical model type is assessed. It is shown that the natural frequencies obtained using the analytical and FE models are within 8% of the experimentally determined values.
Il presente manoscritto scaturisce dall’esperienza maturata nel corso di circa tredici anni di studio e di ricerca sulle strutture a guscio. Comprendono il periodo della tesi di laurea in “Scienza delle Costruzioni”, i tre anni del Dottorato di Ricerca in “Meccanica delle Strutture”, e alcuni anni di Assegni di Ricerca svolti dall’autore presso l’Alma Mater Studiorum - Università di Bologna.
Il titolo, Teoria delle Strutture a Guscio in Materiale Composito, illustra il tema trattato e la prospettiva seguita nella scrittura del volume. Il presente elaborato, nato dall’interesse di approfondire temi in parte affrontati nel corso di Scienza delle Costruzioni e nella redazione della tesi di Laurea e di Dottorato, si pone come obiettivo quello di analizzare il comportamento statico e dinamico dei gusci moderatamente spessi in materiale composito.
Il libro si articola in cinque capitoli, nei quali viene fornita nel dettaglio la teoria relativa alla statica e alla dinamica degli elementi strutturali analizzati e vengono presentati i risultati per i diversi problemi.
Partendo dalla Geometria Differenziale, fondamentale strumento per l’analisi delle strutture in esame, il primo capitolo presenta la Teoria delle Strutture a Guscio in Materiale Composito. Nella trattazione teorica si fa riferimento al campo di spostamento associato alla teoria di Reissner-Mindlin, nota nella letteratura scientifica anglosassone come “First-order Shear Deformation Theory” (FSDT). Una volta introdotte le equazioni di congruenza e le leggi di legame costitutivo, le equazioni indefinite di equilibrio e le condizioni naturali al contorno sono dedotte mediante il principio di Hamilton. Le equazioni del generico guscio a doppia curvatura, così ricavate e sintetizzate nello schema delle teorie fisiche, sono poi specializzate alle strutture di rivoluzione.
Per quanto riguarda le equazioni di legame elastico, una particolare attenzione viene riservata ai materiali compositi a causa del crescente sviluppo, cui si è assistito in questi ultimi anni in molti ambiti dell’ingegneria strutturale. L’interesse scientifico per questi materiali dalle elevate potenzialità applicative ha suggerito l’analisi statica e dinamica delle strutture a guscio in materiale composito. Una nuova classe di materiali compositi, recentemente introdotta in letteratura, viene anche presa in considerazione. Come ben noto, i materiali compositi laminati risultano affetti da inevitabili problemi di delaminazione dovuti alla presenza di interfacce in cui materiali diversi entrano a contatto. I “functionally graded materials” (FGMs) invece sono caratterizzati da una variazione continua delle proprietà meccaniche, quali ad esempio il modulo elastico, la densità del materiale, il coefficiente di Poisson, lungo una particolare direzione. Tale caratteristica è ottenuta facendo variare in maniera graduale, lungo una direzione preferenziale, la frazione in volume dei materiali costituenti attraverso opportuni processi produttivi. I FGMs risultano, pertanto, materiali non omogenei, tipicamente composti di un materiale metallico e uno ceramico.
Partendo dall’analisi dei gusci di traslazione e di rivoluzione a doppia curvatura, nel secondo capitolo vengono illustrate in dettaglio le equazioni fondamentali per le Principali Strutture a Guscio oggetto del presente volume. In questo capitolo si mostra come risulti possibile ricavare attraverso semplici relazioni geometriche le equazioni governanti il problema elastico dei gusci conici e cilindrici, delle piastre circolari e rettangolari e dei cilindri di traslazione a profilo generico dalle equazioni dei gusci di rivoluzione a doppia curvatura.
Nel terzo capitolo vengono presentate le Equazioni dell’Elasticità Tridimensionale in Coordinate Curvilinee Ortogonali, che costituiranno la base per un’adeguata ricostruzione dello stato tensionale e deformativo lungo lo spessore del guscio. La ricostruzione in parola si rende necessaria perché si sono trascurati determinati effetti al passaggio da una teoria tridimensionale a una bidimensionale al fine di diminuire il costo computazionale dell’analisi strutturale. Questo passaggio dalla teoria dell’elasticità tridimensionale a una teoria ingegneristica è reso possibile mediante l’introduzione opportune ipotesi e limita l’applicabilità delle teorie ingegneristiche all’interno di opportuni range di validità.
Le equazioni tridimensionali in coordinate curvilinee ortogonali vengono ricavate mediante il principio di Hamilton.
Il volume si completa con il capitolo quarto e quinto in cui vengono dedotte alcune importanti teorie strutturali bidimensionali (gusci sottili) e monodimensionali (archi e travi).
Il capito quarto si propone di derivare in maniera semplice e intuitiva la Teoria dei Gusci Sottili in Materiale Composito a partire dalla teoria dei gusci moderatamente spessi sviluppata nel primo capitolo. In particolare vengono illustrate la Teoria di Kirchhoff-Love e la Teoria Membranale. Infine, il capitolo quinto espone la Teoria degli Archi e delle Travi in Materiale Composito. In particolare, le equazioni della Teoria di Timoshenko e della Teoria di Eulero-Bernoulli, per le travi ad asse curvilineo e non, vengono dedotte direttamente dalle equazioni dei gusci di traslazione a singola curvatura e delle piastre.
Over the years the Differential Quadrature (DQ) method has distinguished because of its high accuracy, straightforward implementation and general ap- plication to a variety of problems. There has been an increase in this topic by several researchers who experienced significant development in the last years.
DQ is essentially a generalization of the popular Gaussian Quadrature (GQ) used for numerical integration functions. GQ approximates a finite in- tegral as a weighted sum of integrand values at selected points in a problem domain whereas DQ approximate the derivatives of a smooth function at a point as a weighted sum of function values at selected nodes. A direct appli- cation of this elegant methodology is to solve ordinary and partial differential equations. Furthermore in recent years the DQ formulation has been gener- alized in the weighting coefficients computations to let the approach to be more flexible and accurate. As a result it has been indicated as Generalized Differential Quadrature (GDQ) method.
However the applicability of GDQ in its original form is still limited. It has been proven to fail for problems with strong material discontinuities as well as problems involving singularities and irregularities. On the other hand the very well-known Finite Element (FE) method could overcome these issues because it subdivides the computational domain into a certain number of elements in which the solution is calculated. Recently, some researchers have been studying a numerical technique which could use the advantages of the GDQ method and the advantages of FE method. This methodology has got different names among each research group, it will be indicated here as Generalized Differential Quadrature Finite Element Method (GDQFEM).
The purpose of this PhD Thesis is to introduce the limitations of the di- rect GDQ method and more importantly the implementation technique of the GDQFEM. Moreover, in order to show the accuracy, stability and flexibility of the current methodology some numerical examples are shown. The examples are related to the mechanics of civil and mechanical engineering structures such as membranes, state plane structures and flat plates. The static and dynamic behaviour of these structures are proposed in the following chapters. Numerical comparisons with literature and FE analyses are reported and very good agreement is observed in all the computations.
Il presente volume scaturisce dall’esperienza maturata nel corso di circa nove anni di studio e di ricerca sulle strutture a guscio e sul metodo di Quadratura Differenziale. Comprendono il periodo della tesi di laurea in “Scienza delle Costruzioni”, i tre anni del Dottorato di Ricerca in “Meccanica delle Strutture”, e alcuni anni di Assegni di Ricerca svolti dall’autore presso l’Alma Mater Studiorum - Università di Bologna. Il titolo, Meccanica delle Strutture a Guscio in Materiale Composito, illustra il tema trattato e la prospettiva seguita nella scrittura del volume. Il presente elaborato si pone come obiettivo quello di analizzare il comportamento statico e dinamico dei gusci moderatamente spessi in materiale composito attraverso l’applicazione del Metodo Generalizzato di Quadratura Differenziale (GDQ Method). Una particolare attenzione viene riservata oltre che ai compositi fibrosi e laminati anche ai “functionally graded materials” (FGMs). Essi risultano materiali non omogenei, caratterizzati da una variazione continua delle proprietà meccaniche lungo una particolare direzione. La soluzione numerica GDQ viene confrontata con i risultati presenti in letteratura e con quelli forniti e ricavati mediante l’utilizzo di diversi programmi di calcolo strutturale basati sul metodo agli elementi finiti (FEM).
This work presents the static and dynamic analyses of laminated doubly-curved shells and panels of revolution resting on Winkler-Pasternak elastic foundations using the Generalized Differential Quadrature (GDQ) method. The analyses are worked out considering the First-order Shear Deformation Theory (FSDT) for the above mentioned moderately thick structural elements. The effect of the shell curvatures is included from the beginning of the theory formulation in the kinematic model. The solutions are given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution struc-tures. The discretization of the system by means of the GDQ technique leads to a standard linear problem for the static analysis and to a standard linear eigenvalue problem for the dynamic analysis. Comparisons between the present for-mulation and the Reissner-Mindlin theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs. Very good agreement is observed. Finally, new results are presented in order to invest-tigate the effects of the Winkler modulus, the Pasternak modulus and the inertia of the elastic foundation on the behav-ior of laminated shells of revolution.
This study deals with a mixed static and dynamic optimization of four-parameter functionally graded material (FGM) doubly curved shells and panels. The two constituent functionally graded shell consists of ceramic and metal, and the volume fraction profile of each lamina varies through the thickness of the shell according to a generalized power-law distribution. The Generalized Differential Quadrature (GDQ) method is applied to determine the static and dynamic responses for various FGM shell and panel structures. The mechanical model is based on the so-called First-order Shear Deformation Theory (FSDT). Three different optimization schemes and methodologies are implemented. The Particle Swarm Optimization, Monte Carlo and Genetic Algorithm approaches have been applied to define the optimum volume fraction profile for optimizing the first natural frequency and the maximum static deflection of the considered shell structure. The optimization aim is in fact to reach the frequency and the static deflection targets defined by the designer of the structure: the complete four-dimensional search space is considered for the optimization process. The optimized material profile obtained with the three methodologies is presented as a result of the optimization problem solved for each shell or panel structure.
A two-dimensional theory of flexural motions of isotropic, elastic plates is deduced from the three-dimensional equations of elasticity. The theory includes the effects of rotatory inertia and shear in the same manner as Timoshenko’s one-dimensional theory of bars. Velocities of straight-crested waves are computed and found to agree with those obtained from the three-dimensional theory. A uniqueness theorem reveals that three edge conditions are required.
Composite materials consist of two or more materials which together produce desirable properties that may not be achieved with any of the constituents alone. Fiber-reinforced composite materials, for example, consist of high strength and high modulus fibers in a matrix material. Reinforced steel bars embedded in concrete provide an example of fiber-reinforced composites. In these composites, fibers are the principal loadcarrying members, and the matrix material keeps the fibers together, acts as a load-transfer medium between fibers, and protects fibers from being exposed to the environment (e.g., moisture, humidity, etc.).
Smart structures that contain embedded piezoelectric patches are loaded by both mechanical and electrical fields. Traditional plate and shell theories were developed to analyze structures subject to mechanical loads. However, these often fail when tasked with the evaluation of both electrical and mechanical fields and loads. In recent years more advanced models have been developed that overcome these limitations. Plates and Shells for Smart Structures offers a complete guide and reference to smart structures under both mechanical and electrical loads, starting with the basic principles and working right up to the most advanced models. It provides an overview of classical plate and shell theories for piezoelectric elasticity and demonstrates their limitations in static and dynamic analysis with a number of example problems. This book also provides both analytical and finite element solutions, thus enabling the reader to compare strong and weak solutions to the problems. Key features: compares a large variety of classical and modern approaches to plates and shells, such as Kirchhoff-Love , Reissner-Mindlin assumptions and higher order, layer-wise and mixed theories introduces theories able to consider electromechanical couplings as well as those that provide appropriate interface continuity conditions for both electrical and mechanical variables considers both static and dynamic analysis accompanied by a companion website hosting dedicated software MUL2 that is used to obtain the numerical solutions in the book, allowing the reader to reproduce the examples given as well as solve problems of their own The models currently used have a wide range of applications in civil, automotive, marine and aerospace engineering. Researchers of smart structures, and structural analysts in industry, will find all they need to know in this concise reference. Graduate and postgraduate students of mechanical, civil and aerospace engineering can also use this book in their studies.
This article presents the Ritz method for the vibration analysis of sandwich plates having an orthotropic core and laminated facings. The planform of the plate may take on any arbitrary shape. On the basis of the Mindlin plate theory and the Ritz method, the governing eigenvalue equation for determining the natural frequencies was derived. The Ritz method was automated and made computationally effective for general-shaped plates with any boundary conditions by (1) adopting the product of polynomial functions and boundary equations that were raised to appropriate powers and (2) applying Green's theorem to transform the integration over the general-shaped domain into a closed line integration. The Ritz formulation and software were verified by the close agreement with vibration frequencies obtained by previous researchers for a wide range of subset plate problems involving isotropic, laminated, and sandwich plates of various shapes. Moreover, sample natural frequencies of sandwich plates with laminated facings are presented for some quadrilateral plate shapes. These frequencies should be useful as reference results to researchers who are developing new methods or software for vibration analysis of sandwich plates.
The original direct differential quadrature (DQ) method has been known to fail for problems with strong nonlinearity and material discontinuity as well as for problems involving singularity, irregularity, and multiple scales. But now researchers in applied mathematics, computational mechanics, and engineering have developed a range of innovative DQ-based methods to overcome these shortcomings. Advanced Differential Quadrature Methods explores new DQ methods and uses these methods to solve problems beyond the capabilities of the direct DQ method.
After a basic introduction to the direct DQ method, the book presents a number of DQ methods, including complex DQ, triangular DQ, multi-scale DQ, variable order DQ, multi-domain DQ, and localized DQ. It also provides a mathematical compendium that summarizes Gauss elimination, the Runge–Kutta method, complex analysis, and more. The final chapter contains three codes written in the FORTRAN language, enabling readers to quickly acquire hands-on experience with DQ methods.
Focusing on leading-edge DQ methods, this book helps readers understand the majority of journal papers on the subject. In addition to gaining insight into the dynamic changes that have recently occurred in the field, readers will quickly master the use of DQ methods to solve complex problems.
This paper investigates the static analysis of doubly-curved laminated composite shells and panels. A theoretical
formulation of 2D Higher-order Shear Deformation Theory (HSDT) is developed. The middle surface
of shells and panels is described by means of the differential geometry tool. The adopted HSDT is
based on a generalized nine-parameter kinematic hypothesis suitable to represent, in a unified form,
most of the displacement fields already presented in literature. A three-dimensional stress recovery procedure
based on the equilibrium equations will be shown. Strains and stresses are corrected after the
recovery to satisfy the top and bottom boundary conditions of the laminated composite shell or panel.
The numerical problems connected with the static analysis of doubly-curved shells and panels are solved
using the Generalized Differential Quadrature (GDQ) technique. All displacements, strains and stresses
are worked out and plotted through the thickness of the following six types of laminated shell structures:
rectangular and annular plates, cylindrical and spherical panels as well as a catenoidal shell and an elliptic
paraboloid. Several lamination schemes, loadings and boundary conditions are considered. The GDQ
results are compared with those obtained in literature with semi-analytical methods and the ones computed
by using the finite element method.
The aim of this study is to clarify the discrepancy regarding the critical flow speed of straight pipes conveying fluids that appears to be present in the literature by using the Generalized Differential Quadrature method. It is well known that for a given “mass of the fluid” to the “mass of the pipe” ratio, straight pipes conveying fluid are unstable by a flutter mode via Hopf bifurcation for a certain value of the fluid speed, i.e. the critical flow speed. However, there seems to be lack of consensus if for a given mass ratio the system might lose stability for different values of the critical flow speed or only for a single speed value. In this paper an attempt to answer to this question is given by solving the governing equation following first the practical aspect related to the engineering problem and than by simply considering the mathematics of the problem. The
Generalized differential quadrature method is used as a numerical technique to resolve this problem. The differential governing equation is transformed into a discrete system of algebraic equations. The stability of the system is thus reduced to an eigenvalue problem. The relationship between the eigenvalue branches and the corresponding unstable flutter modes are shown via Argand diagram. The transfer of flutter-type instability from one eigenvalue branch to another is thoroughly investigated and discussed. The critical mass ratios, at which the transfer of the eigenvalue branches related to flutter take place, are determined.
This paper is focused on the Generalized Differential Quadrature (GDQ) Method to study the free vibration of conical shell structures. The treatment is conducted within the theory of linear elasticity, when the material behaviour is assumed to be homogeneous and isotropic. The governing equations of motion are expressed as functions of five kinematic parameters. Numerical solutions are obtained.
Differential Quadrature (DQ) is a high-order numerical scheme, yielding very accurate results by use of very small number of nodal points. But it requires the functions to be determined highly differentiable. In the presence of material discontinuity in an elastic medium, direct application of DQ would yield poor results, and this issue has been addressed through a numerical example in this paper. After that, a multi-domain DQ approach has been proposed to solve the discontinuity difficulty. The approach is characterized by being first-order accurate at the interfaces of two different materials, but high-order accurate elsewhere. Numerical examples are given to demonstrate the effectiveness of the method.
In this paper, a new numerical method, the differential quadrature element method has been developed for two-dimensional analysis of bending problems of Reissner-Mindlin plates. The basic idea of the differential quadrature element method is to divide the whole variable domain into several subdomains (elements) and to apply the differential quadrature method for each element. The detailed formulations for the differential quadrature element method and compatibility conditions between elements are presented. The convergent characteristics and accuracy of the differential quadrature element method are carefully investigated for the solution of th two-dimensional bending problems of Reissner-Mindlin plates. Finally, the differential quadrature element method is applied to analyze several bending problems of two-dimensional Reissner-Mindlin plates with different discontinuities including the discontinuous loading conditions, the mixed boundaries, and the plates with cutout. The accuracy and applicability of this method have been examined by comparing the differential quadrature element method solutions with the existing solutions obtained by other numerical methods and the finite element method solutions generated using ANSYS 5.3.
This work applies the famous Ritz method to analyze the free vibrations of rectangular plates with internal cracks or slits. To retain the important and useful feature of the Ritz method providing the upper bounds on exact natural frequencies, the paper proposes a new set of admissible functions that are able to properly describe the stress singularity behaviors near the tips of the crack and meet the discontinuous behaviors of the exact solutions across the crack. The validity of the proposed set of functions is confirmed through comprehensive convergence studies on the frequencies of simply supported square plates with horizontal center cracks having different lengths. The convergent frequencies show excellent agreement with published accurate results obtained by an integration equation technique, and are more accurate than those obtained by a previously published approach using the Ritz method combined with a domain decomposition technique. Finally, the present solution is employed to obtain accurate natural frequencies and mode shapes for simply supported and completely free square plates with internal cracks having various locations, lengths, and angular orientations. Most of the configurations considered here have not been analyzed in the previously published literature. The present results are novel, and are the first published vibration data for completely free rectangular plates with internal cracks and for plates with internal cracks, which are not parallel to the boundaries.
A new numerical approach for solving warping torsion problems is proposed. The approach uses the differential quadrature element method (DQEM) to discretize the differential equations defined on each element. The resulting overall discrete equations can be solved using the solvers of the linear algebra. Numerical results of the DQEM warping torsion model are presented.
Nonlinear free vibration analysis of thin-to-moderately thick laminated composite skew plates is presented based on the first order shear deformation theory (FSDT) using differential quadrature method (DQM). The geometrical nonlinearity is modeled using Green’s strain and von Karman assumptions in conjunction with the FSDT of plates. After transforming and discretizing the governing equations, which includes the effects of rotary inertia, direct iteration technique as well as harmonic balance method is used to solve the resulting discretized system of equations. The effects of skew angle, thickness-to-length ratio, aspect ratio and also the impact due to different types of boundary conditions on the convergence and accuracy of the method are studied. The resulted solutions are compared to those from other numerical methods to show the accuracy of the method. Numerical solutions for nonlinear frequency laminated skew plates under different geometrical parameters and mixed boundary conditions are presented.
The vibration responses of orthotropic plates on nonlinear elastic foundations are numerically modeled using the differential quadrature method. The differential quadrature technique is utilized to transform partial differential equations into a discrete eigenvalue problem. Numerical results and those from literature closely correspond to each other. Numerical results demonstrate that elastically restrained stiffness, plate aspect ratio and foundation stiffness significantly impact the dynamic behavior of orthotropic plates.
In this paper, a novel numerical solution technique, the moving least squares differential quadrature (MLSDQ) method is employed to study the free vibration problems of generally laminated composite plates based on the first order shear deformation theory. The weighting coefficients used in MLSDQ approximation are obtained through a fast computation of the MLS shape functions and their partial derivatives. By using this method, the governing differential equations are transformed into sets of linear homogeneous algebraic equations in terms of the displacement components at each discrete point. Boundary conditions are implemented through discrete grid points by constraining displacements, bending moments and rotations of the plate. Combining these algebraic equations yields a typical eigenvalue problem, which can be solved by a standard eigenvalue solver. Detailed formulations and implementations of this method are presented. Convergence and comparison studies are carried out to verify the reliability and accuracy of the numerical solutions. The applicability, efficiency, and simplicity of the present method are all demonstrated through solving several sample problems.
The differential quadrature element method (DQEM) and differential quadrature finite difference method (DQFDM) have been proposed by the author. The differential quadrature, generic differential quadrature and mapping technique are used in developing DQEM and DQFDM. The two novel numerical methods are effective for solving generic continuum mechanics problems. In this paper, these two numerical methods are used to solve the composite two-dimensional elasticity problems. Numerical procedures are summarized and presented. Numerical results are also presented. They demonstrate the DQEM and DQFDM.
The dynamic response of shear-deformable axisymmetric orthotropic circular plate structures is solved by using the DQEM to the spacial discretization and EDQ to the temporal discretization. In the DQEM discretization, DQ is used to define the discrete element model. Discrete dynamic equilibrium equations defined at interior nodes in all elements, transition conditions defined on the inter-element boundary of two adjacent elements and boundary conditions at the structural boundary form a dynamic equation system at a specified time stage. The dynamic equilibrium equation system is solved by the direct time integration schemes of time-element by time-element method and stages by stages method which are developed by using EDQ and DQ. Numerical results obtained by the developed numerical algorithms are presented. They demonstrate the developed numerical solution procedure.
Based on the same concept as generalized differential quadrature (GDQ), the method of Fourier expansion-based differential quadrature (FDQ) was developed and then applied to solve the Helmholtz eigenvalue problems with periodic and non-periodic boundary conditions. In FDQ, the solution of a partial differential equation is approximated by a Fourier series expansion. The details of the FDQ method and its implementation to sample problems are shown in this paper. It was found that the FDQ results are very accurate for the Helmholtz eigenvalue problems even though very few grid points are used. ©1997 John Wiley & Sons, Ltd.
A new numerical approach for solving the problem of a beam resting on an elastic foundation is proposed. The approach uses the differential quadrature (DQ) to discretize the governing differential equations defined on all elements, the transition conditions defined on the interelement boundaries of two adjacent elements, and the boundary conditions of the beam. By assembling all the discrete relation equations, a global linear algebraic system can be obtained. Numerical results of the solutions of beams resting on elastic foundations obtained by the DQEM are presented.
An investigation on the vibrational behavior of cracked rectangular plates is reported. Vibration analysis is carried out for plates with a crack (1) emanating from an edge or (2) centrally located. A highly computationally efficient and accurate domain decomposition method is presented. To establish this discrete model, the cracked plate domain is assumed to be an assemblage of small subdomains with the appropriate shape functions formed according to the boundary conditions. The complete coupling process leads to a governing eigenvalue equation that can be solved to obtain the vibration frequencies. These results, where possible, are compared with the work of other investigators. With these results, the effects of various geometric parameters on the vibration response of cracked plates can be examined. Some concrete conclusions can be deduced from a careful examination of these results.
Natural flexural vibration of a simply supported rectangular plate with a symmetrically located crack parallel to one edge is considered. The problem is analyzed by means of finite Fourier transformation of discontinuous functions. The unknowns of the problem are the discontinuities of the displacement and of the slope across the crack. The singularities of the moments at the tips are “built” in the solution. The conditional equations are obtained by satisfying the boundary conditions at the crack's edge. This requires differentiation of Fourier series representing a discontinuous function. The characteristic equation in form of an infinite determinant is obtained. Numerical data for certain geometries as well as comparison with existing results are included.
A method is presented for accurately determining the natural frequencies of plates having V-notches along their edges. It is based on the Ritz method and utilizes two sets of admissible functions simultaneously, which are (1) algebraic polynomials from a mathematically complete set of functions, and (2) corner functions duplicating the boundary conditions along the edges of the notch, and describing the stress singularities at its sharp vertex exactly. The method is demonstrated for free, square plates with a single V-notch. The effects of corner functions on the convergence of solutions are shown through comprehensive convergence studies. The corner functions accelerate convergence of results significantly. Accurate numerical results for free vibration frequencies and nodal patterns are tabulated for V-notched square plates having notch angle α=5° or 30° at different locations and with various notch depths. These are the first known frequency and nodal pattern results available in the published literature for rectangular plates with V-notches.
The differential quadrature element method (DQEM) has been proposed. The element weighting coefficient matrices are generated by the differential quadrature (DQ) or generic differential quadrature (GDQ). By using the DQ or GDQ technique and the mapping procedure the governing differential or partial differential equations, the transition conditions of two adjacent elements and the boundary conditions can be discretized. A global algebraic equation system can be obtained by assembling all of the discretized equations. This method can convert a generic engineering or scientific problem having an arbitrary domain configuration into a computer algorithm. The DQEM irregular element torsion analysis model is developed.
A new numerical technique, the differential quadrature element method (DQEM), has been developed for solving the free vibration of the discontinuous Mindlin plate in this paper. By the DQEM, the complex plate domain is decomposed into small simple continuous subdomains (elements) and the differential quadrature method (DQM) is applied to each continuous subdomain to solve the problems. The detailed formulations for the DQEM and the connection conditions between each element are presented. Several numerical examples are analyzed to demonstrate the accuracy and applicability of this new method to the free vibration analysis of the Mindlin plate with various discontinuities which are not solvable directly using the differential quadrature method.
In this paper, a highly accurate and rapidly converging hybrid approach is presented for the Quadrature Element Method (QEM) solution of plate free vibration problems. The hybrid QEM essentially consists of a collocation method in conjunction with a Galerkin finite element technique, to combine the high accuracy of the Differential Quadrature Method (DQM) for the efficient solution of differential equations with the generality of the finite element formulation. This results in superior accuracy with fewer degrees of freedom than conventional FEM or FDM. A series of numerical tests is conducted to assess the performance of the quadrature plate element in free vibration problems. Anisotropic and stepped thickness plates are investigated as well as mixed boundary conditions and point supports at the edges. In all cases, the results obtained are quite accurate.
Free vibration analysis of composite cylindrical shells with different boundary conditions is presented in this paper using differential quadrature method (DQM). Equations of motion are derived based on first order shear deformation theory taking the effects of shear deformation and rotary inertia terms into account. By applying the differential quadrature formulation and the required modified relationships for implementing the different boundary conditions, equations of motion of a circular cylindrical shell are transformed into a set of algebraic equations. By solving this algebraic system natural frequencies of circular cylindrical shells made of fibrous composite materials with different fibre angles are evaluated. The results thus obtained are then compared with some available results and a good agreement is observed. In all the cases studied here efficiency, ease and usefulness of the DQM are well illustrated.
In this paper a differential quadrature method is applied for static, free vibration, and stability analysis of skewed and trapezoidal composite thin plates. To mechanize the modelling procedure, a general transformation scheme is employed to transfer the variation of the variables in the computational to the physical domain and vice versa. Examples are shown to show the accuracy and convergence of the solutions under different geometrical parameters and boundary conditions. The accuracy is demonstrated by comparing the results with those of other numerical methods.
A new numerical approach for solving the vibration of nonuniform shear deformable axisymmetric orthotropic circular plate problems is proposed. The approach uses the differential quadrature (DQ) to discretize the governing equations defined on each element, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of the plate. It is a differential quadrature element method (DQEM) analysis model. Numerical results of sample problems solved by this DQEM analysis model are presented. They prove that the developed DQEM analysis model of shear deformable axisymmetric orthotropic circular plates is efficient.
The Quadrature Element Method (QEM) is a new concept in numerical methods which was introduced only recently. A quadrilateral quadrature element is developed in the paper and applied to the solution of two dimensional potential problems governed by Poisson or Laplace equations. The results of three examples are in good agreement with the available solutions. It is shown that the present quadrilateral quadrature element is very efficient and effective. Most significantly, it can be employed to solve differential equations of physical problems with irregular geometry and complex boundary conditions.
A new numerical approach for solving steady-state heat conduction problems by using the irregular elements of the differential quadrature element method (DQEM) is proposed. The mapping technique is used to transform the governing partial differential equation, the natural transition condition of two adjacent elements and the Newmann boundary condition defined on the irregular physical element into the parent space. The differential quadrature technique is used to discretize the transformed relation equations defined on the regular element in the parent space. Various techniques for selecting and implementing the constraint conditions, at element corners, are proposed. A global algebraic equation system can be obtained by assembling all of the discretized relation equations. Numerical procedures are summarized and the related computer code is developed. Numerical results are presented. They demonstrate the developed DQEM steady-state heat conduction analysis model.
A 2D Unconstrained Third Order Shear Deformation Theory (UTSDT) is presented for the evaluation of tangential and normal stresses in moderately thick functionally graded cylindrical shells subjected to mechanical loadings. Eight types of graded materials are investigated. The functionally graded material consists of ceramic and metallic constituents. A four parameter power law function is used. The UTSDT allows the presence of a finite transverse shear stress at the top and bottom surfaces of the graded cylindrical shell. In addition, the initial curvature effect included in the formulation leads to the generalization of the present theory (GUTSDT). The Generalized Differential Quadrature (GDQ) method is used to discretize the derivatives in the governing equations, the external boundary conditions and the compatibility conditions. Transverse and normal stresses are also calculated by integrating the three dimensional equations of equilibrium in the thickness direction. In this way, the six components of the stress tensor at a point of the cylindrical shell or panel can be given. The initial curvature effect and the role of the power law functions are shown for a wide range of functionally cylindrical shells under various loading and boundary conditions. Finally, numerical examples of the available literature are worked out.
In this paper, generalized differential quadrature techniques are applied to the computation of the in-plane free vibrations of thin and thick nonuniform circular arches in undamaged and damaged configurations, when various boundary conditions are considered. Structural damage is represented by one crack in different positions and with various damage levels. The crack present in a structural member can be considered as a local stiffness reduction at the fracturing section, which changes the dynamic behaviour of the structure. Much effort has been devoted to dealing with in-plane free vibration analysis of circular arches, but only a few researchers have studied cracked circular arch structures. The present analysis refers to the complete in-plane equations of motion of non-uniform circular arches, in terms of displacements and rotation. Shearing and axial deformations as well as rotary inertia are taken into account. For given geometric and boundary conditions, the presence of a crack will cause displacements and rotations of sections along the arch greater than the corresponding values resulting in an uncracked structure. In order to evaluate the effect of cracks, a cracked section is modelled as an elastic hinge with rotational constant which has to simulate the local flexibility caused by the cracked section itself. A crack will produce discontinuities in slope of the elastic curve of the arch at the fractured cross-section. It should be noted that in our investigation of the in-plane dynamic response variation of damaged arches with variable cross-section, the localized cracks will always be considered as open.
The main aim of this paper is to provide a general framework for the formulation and the dynamic analysis computations of moderately thick laminated doubly-curved shells and panels. A 2D higher-order shear deformation theory is also proposed and the differential geometry is used to define the arbitrary shape of the middle surface of shells and panels with different curvatures. A generalized nine-parameter displacement field suitable to represent in a unified form most of the kinematical hypothesis presented in literature has been introduced.
Formal comparison among various theories have been performed in order to show the differences between the well-known First-order Shear Deformation Theory (FSDT) and several Higher-order Shear Deformation Theories (HSDTs). The 2D free vibration shell problems have been solved numerically using the Generalized Differential Quadrature (GDQ) technique. The GDQ rule has been also used to perform the numerical evaluation of the derivatives of the quantities involved by the differential geometry to completely describe the reference surfaces of doubly-curved shell structures. Numerical investigations concerning four types of shell structures have been carried out. GDQ results are compared with those presented in literature and the ones obtained using commercial programs such as Abaqus. Very good agreement is observed.
In this paper, the generalized differential quadrature (GDQ) method is applied to solve classical and nonclassical nonconservative stability problems. Various cantilever beams subjected to slope-dependent forces are considered. First, the governing differential equation for a nonuniform column subjected to an arbitrary distribution of compressive subtangential follower forces is obtained. The effect of the variability of the mechanical properties along the beam length is also considered. Then, the application of the GDQ procedure leads to a discrete system of algebraic equations from which the system critical loads can be obtained by solving an associated eigenvalue problem. A parametrical study for different levels of nonconservativeness of the applied load is carried out for some classical benchmark cases such as Beck's, Leipholz's and Hauger's column problems. Finally, applications to geometrically and mechanically tapered beams subjected to nonpotential subtangential follower forces are investigated as nonclassical cases.It has been proved that the method can efficiently solve structural nonconservative elastic problems and, more in general, problems governed by a nonsymmetric system of algebraic equations.
Basing on the First-order Shear Deformation Theory (FSDT), this paper focuses on the dynamic behaviour of moderately thick functionally graded parabolic panels and shells of revolution. A generalization of the power-law distribution presented in literature is proposed. Two different four-parameter power-law distributions are considered for the ceramic volume fraction. Some symmetric and asymmetric material profiles through the functionally graded shell thickness are illustrated by varying the four parameters of power-law distributions. The governing equations of motion are expressed as functions of five kinematic parameters. For the discretization of the system equations the Generalized Differential Quadrature (GDQ) method has been used. Numerical results concerning four types of parabolic shell structures illustrate the influence of the parameters of the power-law distribution on the mechanical behaviour of shell structures considered.
a b s t r a c t In this paper, the generalized differential quadrature (GDQ) method is applied to study the dynamic behavior of functionally graded materials (FGMs) and laminated doubly curved shells and panels of revolution with a free-form meridian. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature a generalization of the Reissner–Mindlin theory, proposed by Toorani and Lakis, is adopted. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bé zier curves are used to define the meridian curve of the revolution structures. Firstly, the differential quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Secondly, the discretization of the system by means of the GDQ technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the Reissner–Mindlin and the Toorani–Lakis theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed. Finally, different lamination schemes are considered to expand the combination of the two functionally graded four-parameter power-law distributions adopted. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the lamina thickness direction. A two-constituent functionally graded lamina consists of ceramic and metal those are graded through the lamina thickness. A parametric study is performed to illustrate the influence of the parameters on the mechanical behavior of shell and panel structures considered.