Application of the R-Functions Theory to Problems of Nonlinear Dynamics of Laminated Composite Shallow Shells and Plates: Review

A review of studies performed using the R-functions theory to solve problems of nonlinear dynamics of plates and shallow shells is presented. The systematization of results and studies for the problems of free and parametric vibrations and for problems of static and dynamic stability is fulfilled. Expansion of the developed original method of discretization for nonlinear movement equations on new classes of nonlinear problems is shown. These problems include researches of vibrations of antisymmetric laminated cylindrical and spherical panels; laminated composite shallow shells with variable thicknesss of the layers; functionally graded (FG) shallow shells and others. The basic issues that arise when using RFM are described. The future prospects of using the theory of R-functions for solving problems of nonlinear dynamics of plates and shallow shells with complex form are formulated. First of all this is an algorithms development and creation of the associated software to apply multi-modes approximations; improvement of approximation tools for nonlinear problems; investigation of the cracked functionally graded shallow shells; FG panels under thermal environments; parametric vibrations, static and dynamical stability of the multilayered and FG plates and shells

Geometrically non-linear vibration and meshless discretization of the composite laminated shallow shells with complex shape

To study the geometrically non-linear vibrations of the composite laminated shallow shells with complex plan form the approach, based on meshless discretization, is proposed. Non-linear equations of motion for shallow shells based on the first order shear deformation shell theories are considered. The discretization of the motion equations is carried out by method based on expansion of the unknown functions in series for which eigenvectors of the linear vibration obtained by RFM (R-functions method) are employed as basic functions. The factors of these series are functions (generalizing coordinates) depending on time. Due to applying the basic variational principle in mechanics by Ostrogradsky-Hamilton the corresponding system of the ordinary differential equations by Euler is obtained The non-linear ordinary differential equations are derived in terms of amplitudes of the vibration modes. The offered method is expounded for multi-modal approximation of unknown functions. Backbone curves of the spherical shallow shell with complex plan form are obtained using only the first vibration mode by the Bubnov-Galerkin method. The effects of lamination schemes on the behavior are discussed

Application of the R-Functions Method for Nonlinear Bending of Orthotropic Shallow Shells on an Elastic Foundation

Geometrically nonlinear behavior of orthotropic shallow shells subjected to the transverse load and resting on Winkler’s foundation is investigated. On base of the R-function theory and variational methods problem's solution for shells with complex plan form is proposed. The algorithm to finding upper and lower critical loads is developed. The stress-strain state of shallow shells with the complex planform is investigated including different boundary conditions, properties of material and elastic foundation

Investigation of Geometrically Nonlinear Vibrations of Laminated Shallow Shells with Layers of Variable Thickness by Meshless Approach

Geometrically nonlinear vibrations of laminated shallow shells with layers of variable thickness are studied. Nonlinear equations of motion for shells based on the first order shear deformation and classical shells theories are considered. In order to solve this problem we use the numerically-analytical method proposed in work [1]. Accordingly to this approach the initial problem is reduced to consequences of some linear problems including linear vibrations problem, special elasticity ones and nonlinear system of ordinary differential equations in time. The linear problems are solved by the variational Ritz’ method and Bubnov-Galerkin procedure combined with the R-functions theory [2]. To construct the basic functions that satisfy all boundary conditions in case of simply-supported shells we propose new solutions structures. The proposed method is used to solve both test problems and new ones

Multi-modal geometrical non-linear free vibrations of composite laminated plates with the complex shape

Geometrically non-linear free vibrations of the composite laminated plates are investigated using new multi modal approach to discretization of motion equations . The non-linear governing equations for laminated plates are derived by Hamilton’s principle using first-order shear deformation theory. Due to proposed algorithm of the discretization all unknown functions except of transverse displacement are eliminated and governing equations are reduced to system of ordinary differential equations in time by the Bubnov-Galerkin procedure. The expansion of all unknown functions in the truncated Fourier series is performed using the eigenfunctions of the linear vibration problems and solutions of the sequence of elasticity problems. All auxiliary problems are solved by RFM (R-functions method)

Analysis of Geometrically Nonlinear Vibrations of Functionally Graded Shallow Shells of a Complex Shape

Geometrically nonlinear vibrations of functionally graded shallow shells of complex planform are studied. The paper deals with a power-law distribution of the volume fraction of ceramics and metal through the thickness. The analysis is performed with the use of the R-functions theory and variational Ritz method. Moreover, the Bubnov-Galerkin and the Runge-Kutta methods are employed. A novel approach of discretization of the equation of motion with respect to time is proposed. According to the developed approach, the eigenfunctions of the linear vibration problem and some auxiliary functions are appropriately matched to fit unknown functions of the input nonlinear problem. Application of the R-functions theory on every step has allowed the extension of the proposed approach to study shallow shells with an arbitrary shape and different kinds of boundary conditions. Numerical realization of the proposed method is performed only for one-mode approximation with respect to time. Simultaneously, the developed method is validated by investigating test problems for shallow shells with rectangular and elliptical planforms, and then applied to new kinds of dynamic problems for shallow shells having complex planforms

Application of R-Functions Theory to Study Parametric Vibrations and Dynamical Stability of Laminated Plates

The problem of nonlinear parametric vibrations and stability analysis of the symmetric laminated plates is considered. The proposed method is based on multimode approximation of displacements and solving series auxiliary linear tasks. The main feature of the work is the application of the R-functions theory, which allows investigating parametric vibrations of plates with complex shape and different boundary conditions

Investigation of the stress-strain state of the laminated shallow shells by R-functions method combined with spline-approximation

The bending behavior of the laminated shallow shells under static loading has been studied using the R functions theory together with the spline-approximation. Formulation is based on the first order shear deformation theory. Due to usage of the R-functions theory the laminated shallow shells with complex shape and different types of the boundary conditions can be investigated. Application of the spline-approximation allows getting reliable and validated results for non concave domains and domains with holes. The proposed method is implemented in the appropriate software in framework of the mathematical package MAPLE. The analysis of influence of certain factors (curvature, packing of layers, geometrical parameters, boundary conditions) on a stress-strain state is carried out for shallow shells with cut-outs. The comparison of obtained results with those already known from literature and results obtained by using ANSYS are also presented

Geometrical analysis of vibrations of functionally graded shell panels using the R-functions theory

An approach for investigation of geometrically nonlinear vibrations of functionally graded shallow shells and plates with complex planform is proposed. It combines the application of the R-functions theory (RFM), variational Ritz’s method, the procedure by Bubnov-Galerkin and Runge-Kutta method. The presented method is developed in the framework of the first–order shear deformation shallow shell theory (FSDT). Shell panels under consideration are made from a mixture of ceramics and metal. Power law of volume fraction distribution of materials through thickness is chosen. Investigation of nonlinear vibrations of functionally graded shallow shells and plates with arbitrary planform and different types of boundary conditions is carried out. Test problems and numerical results have been presented for one-mode approximation in time. Effect of volume fraction exponent, geometry of a shape and boundary conditions on the natural frequencies is brought out

Applicatin of R-functions Theory to Nonlinear Vibration Problems of Laminated Shallow Shells with Cutouts

In present work an effective method to research geometrically nonlinear free vibrations of elements of thinwalled constructions that can be modeled as laminated shallow shells with complex planform is applied. The proposed method is based on joint use of R-functions theory, variational methods and Bubnov-Galerkin procedure. It allows reducing an initial nonlinear system of motion equations of a shallow shell to the Cauchy problem. The mathematical formulation of the problem is performed in a framework of the refined first-order theory. The appropriate software is created within POLE-RL program system for polynomial results and using C+ + programs for splines. New problems of linear and nonlinear vibrations of laminated shallow shells with cutouts are solved. To confirm reliability of the obtained results their comparison with the ones obtained using spline-approximation and known in literature is provided. Effect o f boundary condition on cutout is studied