Francesco Tornabene

Vibration analysis of spherical structural elements using the GDQ method

This paper deals with the dynamical behaviour of hemispherical domes and spherical shell panels. The First-order Shear Deformation Theory (FSDT) is used to analyze the above moderately thick structural elements. 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, written in terms of internal resultants, are expressed as functions of five kinematic parameters, by using the constitutive and the congruence relationships. The boundary conditions considered are clamped (C), simply supported (S) and free (F) edge. Numerical solutions have been computed by means of the technique known as the Generalized Differential Quadrature (GDQ) Method. These results, which are based upon the FSDT, are compared with the ones obtained using commercial programs such as Abaqus, Ansys, Femap/Nastran, Straus, Pro/Engineer, which also elaborate a three-dimensional analysis. The effect of different grid point distributions on the convergence, the stability and the accuracy of the GDQ procedure is investigated. The convergence rate of the natural frequencies is shown to be fast and the stability of the numerical methodology is very good. The accuracy of the method is sensitive to the number of sampling points used, to their distribution and to the boundary conditions.

Four-parameter functionally graded cracked plates of arbitrary shape: A GDQFEM solution for free vibrations.

ABSTRACT The dynamic behavior of moderately thick FGM plates with geometric discontinuities and arbitrarily curved boundaries is investigated. The Generalized Differential Quadrature Finite Element Method (GDQFEM) is proposed as a numerical approach. The irregular physical domain in Cartesian coordinates is transformed into a regular domain in natural coordinates. Several types of cracked FGM plates are investigated. It appears that GDQFEM is analogous to the well-known Finite Element Method (FEM). With reference to the proposed technique the governing FSDT equations are solved in their strong form and the connections between the elements are imposed with the inter-element compatibility conditions. The results show excellent agreement with other numerical solutions obtained by FEM.

Mixed Static and Dynamic Optimization of Four-Parameter Functionally Graded Completely Doubly Curved and Degenerate Shells and Panels Using GDQ Method

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.

General higher-order layer-wise theory for free vibrations of doubly-curved laminated composite shells and panels

ABSTRACT The present article illustrates a general formulation for a higher-order layer-wise theory related to the analysis of the free vibrations of thick doubly-curved laminated composite shells and panels. The theoretical framework relates to the dynamic analysis of shell structures by using a general displacement field based on the Carrera Unified Formulation (CUF), including the stretching effect for each layer. The order of the expansion along the thickness direction is taken as a free parameter. The starting point of the present general higher-order layer-wise formulation is to propose a kinematic assumption, with an arbitrary number of degrees of freedom. The main aim of this work is to determine the explicit fundamental operators that can be used for the layer-wise (LW) approach. These fundamental operators are obtained for the first time by the author and are related to motion equations of doubly-curved shells described in an orthogonal curvilinear co-ordinate system. The free vibration shell and panel problems are computationally solved using the generalized differential quadrature (GDQ) and generalized integral quadrature (GIQ) techniques. The numerical results are compared with recent papers in the literature and commercial finite element codes.

Analysis of Sandwich Plates by Generalized Differential Quadrature Method

We combine a layer-wise formulation and a generalized differential quadrature technique for predicting the static deformations and free vibration behaviour of sandwich plates. Through numerical experiments, the capability and efficiency of this strong-form technique for static and vibration problems are demonstrated, and the numerical accuracy and convergence are thoughtfully examined.

The strong formulation finite element method: stability and accuracy

The Strong Formulation Finite Element Method (SFEM) is a numerical solution technique for solving arbitrarily shaped structural systems. This method uses a hybrid scheme given by the Differential Quadrature Method (DQM) and the Finite Element Method (FEM). The SFEM takes the best from DQM and FEM giving a highly accurate strong formulation based technique with the adaptability of finite elements. The present study investigates the stability and accuracy of SFEM when applied to 1D and 2D structural components, such as rods, beams, membranes and plates using analytical and semi-analytical well-known solutions. The numerical results show that the present approach can be very accurate using a small number of grid points and elements, when it is compared to standard FEM.

Inter-laminar stress recovery procedure for doubly-curved, singly-curved, revolution shells with variable radii of curvature and plates using generalized higher-order theories and the local GDQ method

ABSTRACT The stress and strain recovery procedure already applied for solving doubly-curved structures with variable radii of curvature has been considered in this article using an equivalent single layer approach based on a general higher-order formulation, in which the thickness functions of the in-plane displacement parameters are defined independently from the ones through the shell thickness. The theoretical model considers composite structures in such a way that employs the differential geometry for the description of doubly-curved, singly-curved, revolution with variable radii of curvature and degenerate shells. Furthermore, the structures at hand can be laminated composites made of a general stacking sequence of orthotropic generically oriented plies. The governing static equilibrium equations are solved in their strong form using the local generalized differential quadrature (GDQ) method. Moreover the generalized integral quadrature (GIQ) is exploited for the evaluation of the stress resultants of the model under study. Several numerical applications are presented and the local GDQ results are compared with finite element method (FEM) commercial codes.

Boundary conditions in 2D numerical and 3D exact models for cylindrical bending analysis of functionally graded structures

The cylindrical bending condition for structural models is very common in the literature because it allows an incisive and simple verification of the proposed plate and shell models. In the present paper, 2D numerical approaches (the Generalized Differential Quadrature (GDQ) and the finite element (FE) methods) are compared with an exact 3D shell solution in the case of free vibrations of functionally graded material (FGM) plates and shells. The first 18 vibration modes carried out through the 3D exact model are compared with the frequencies obtained via the 2D numerical models. All the 18 frequencies obtained via the 3D exact model are computed when the structures have simply supported boundary conditions for all the edges. If the same boundary conditions are used in the 2D numerical models, some modes are missed. Some of these missed modes can be obtained modifying the boundary conditions imposing free edges through the direction perpendicular to the direction of cylindrical bending. However, some modes cannot be calculated via the 2D numerical models even when the boundary conditions are modified because the cylindrical bending requirements cannot be imposed for numerical solutions in the curvilinear edges by definition. These features are investigated in the present paper for different geometries (plates, cylinders, and cylindrical shells), types of FGM law, lamination sequences, and thickness ratios.

Stress and Strain Recovery of Laminated Composite Doubly-Curved Shells and Panels Using Higher-Order Formulations

The present paper investigates the static behaviour of doubly-curved laminated composite shells and panels. A two dimensional Higher-order Equivalent Single Layer approach, based on the Carrera Unified Formulation (CUF), is proposed. The differential geometry is used for the geometric description of shells and panels. The numerical solution is calculated using the generalized differential quadrature method. The through-the-thickness strains and stresses are computed using a three dimensional stress recovery procedure based on the shell equilibrium equations. Sandwich panels are considered with soft cores. The numerical results are compared with the ones obtained with a finite element code. The proposed higher-order formulations can be used for solving elastic problems involved in the first stage of any scientific procedure of analysis and design of masonry structures.

Linear Static Behavior of Damaged Laminated Composite Plates and Shells

A mathematical scheme is proposed here to model a damaged mechanical configuration for laminated and sandwich structures. In particular, two kinds of functions defined in the reference domain of plates and shells are introduced to weaken their mechanical properties in terms of engineering constants: a two-dimensional Gaussian function and an ellipse shaped function. By varying the geometric parameters of these distributions, several damaged configurations are analyzed and investigated through a set of parametric studies. The effect of a progressive damage is studied in terms of displacement profiles and through-the-thickness variations of stress, strain, and displacement components. To this end, a posteriori recovery procedure based on the three-dimensional equilibrium equations for shell structures in orthogonal curvilinear coordinates is introduced. The theoretical framework for the two-dimensional shell model is based on a unified formulation able to study and compare several Higher-order Shear Deformation Theories (HSDTs), including Murakami’s function for the so-called zig-zag effect. Thus, various higher-order models are used and compared also to investigate the differences which can arise from the choice of the order of the kinematic expansion. Their ability to deal with several damaged configurations is analyzed as well. The paper can be placed also in the field of numerical analysis, since the solution to the static problem at issue is achieved by means of the Generalized Differential Quadrature (GDQ) method, whose accuracy and stability are proven by a set of convergence analyses and by the comparison with the results obtained through a commercial finite element software.