Problems For Shallow Shells Research Articles

Assessment of a new higher-order shear and normal deformation theory for the static response of functionally graded shallow shells

Abstract Static response of simply supported functionally graded (FG) shallow shells using a new higher-order shear and normal deformation theory is focused in this article. The effects of transverse strains and stresses on the bending response of FG shell are considered by the present theory. The current theory considers the impacts of transverse normal and shear deformations that meet the requirements for traction-free boundary conditions. The virtual work principle is applied to the mathematical formulation of the present theory. The simply supported doubly curved shallow shell problems under the static transverse load are analyzed using Navier’s solution technique. To verify the existing theory, the current results are, whenever possible, compared with those that have already been published. Additionally, a few benchmark results are presented in this article that will be helpful to researchers in the future.

Generalized Method of Finite Integral Transformations in Linear and Nonlinear Static Problems for Shallow Shells

Generalized Method of Finite Integral Transformations in Linear and Nonlinear Static Problems for Shallow Shells

Method of Solving Geometrically Nonlinear Bending Problems of Thin Shallow Shells of Complex Shape

A new numerical analytical method for solving geometrically nonlinear bending problems of thin shallow shells and plates of complex shape is given in the paper. The problem statement is performed within the framework of the classic geometrically nonlinear formulation. The parameter continuation method was used to linearize the nonlinear bending problem of shallow shells and plates. An increasing parameter t related to the external load, which characterizes the shell loading process, is introduced. For the variational formulation of the linearized problem, a functional in the Lagrange form, defined on the kinematically possible movement speeds, is constructed. To find the main unknowns of the problem of nonlinear bending of the shell (displacement, deformation, stress), the Cauchy problem was formulated by the parameter t for the system of ordinary differential equations, which was solved by the fourth order Runge-Kutta-Merson method with automatic step selection. The initial conditions are found from the solution of the problem of geometric linear deformation. The right-hand sides of the differential equations at fixed values of the parameter t, corresponding to the Runge-Kutta-Merson scheme, were found from the solution of the variational problem for the functional in the Lagrange form. Variational problems were solved using the Ritz method combined with the R-function method, which allows to accurately take into account the geometric information about the boundary value problem and provide an approximate solution in the form of a formula - a solution structure that exactly satisfies all (general structure) or part (partial structure) of boundary conditions. The test problem for the nonlinear bending of a square clamped plate under the action of a uniformly distributed load of different intensity is solved. The results for deflections and stresses obtained using the developed method are compared with the analytical solution and the solution obtained by the finite element method. The problem of bending of a clamped plate of complex shape is solved. The effect of the geometric shape on the stress-strain state is studied

The generalized finite integral transformations method in linear and nonlinear static problems for shallow shells

The generalized finite integral transformations method in linear and nonlinear static problems for shallow shells

Two-Layer Ferrocement Shells Stress-Strain State Modeling under the Fire Conditions

The article deals with the issues of strength calculation of two-layer ferrocement shells under fire conditions, taking into account the physical nonlinearity of the material. The change in the physicomechanical characteristics of materials under the action of temperature is taken into account. The solution of model problems for shallow shells of double curvature is presented.

THE METHOD OF SUCCESSIVE APPROXIMATIONS IN THE MATHEMATICAL THE-ORY OF SHALLOW SHELLS OF ARBITRARY THICKNESS


 
 
 The method of sequential approximations (MSA) in mathematical theory (MT) of transversal-isotropic shallow shells of arbitrary thickness is developed. MT takes into account all components of stress-strain state (SSS). SSS and boundary conditions are considered to be functions of three varia-bles. Three-dimensional problems are reduced to two- dimensional decompositions of all the compo-nents of the SSS into series in the transverse coordinate using Legendre polynomials and using the Reisner variational principle. The boundary conditions for stresses on the front surfaces of the shell are fulfilled precisely. Previous studies have shown the high efficiency of this MT. The boundary-value problem for a shallow shell is reduced to sequences of two boundary-value problems for the respective plates. One sequence describes symmetric deformation relative to the median plane, and the other sequence is skew symmetric. MSA makes it easier to find a common solution of differential equations (DE) for shallow shells. Highly accurate results for SSS are already in the first approxi-mation. MSA can be used when solving problems for shallow shells by other theories.
 
 

Complete systems method for linear and nonlinear problems of shallow shells theory

The approach to solving two-dimensional nonlinear (linear) boundary-value problems for shallow shells using the complete systems and quasilinearization Newton-Kantorovich-Raphson methods is developed. With the complete systems method in linear case, the original two-dimensional boundary-value problem is reduced to thise system of two interconnected one-dimensional problems, which is solved iteratively using the method similar to the Libman-Gauss-Seidel successive replacement method. The rational combination of the qusilinearization method and complete systems method makes it possible to construct a single generalized iterative process of solving the problem as a whole in nonlinear case. For the process, typical is rapid convergence (the number of iterations within the limits of one order) and a few number of approximating functions in the accepted presentation form (2−4).Possibilities of the approach proposed are illustrated by examples of solving mean-bending problems for shallow shells with rectangular planform.

Vibration analysis of laminated functionally graded shallow shells with clamped cutout of the complex form by the Ritz method and the R-functions theory

Abstract The R-functions theory and Ritz approach are applied for analysis of free vibrations of laminated functionally graded shallow shells with different types of curvatures and complex planforms. Shallow shells are considered as sandwich shells of different types: a) face sheets of the shallow shells are made of a functionally graded material (FGM) and their cores are made of an isotropic material; b) face sheets of the shallow shells are isotropic, but the core is made of FGM. It is assumed that FGM layers are made of a mixture of metal and ceramics and effective material properties of layers are varied accordingly to Voigt’s rule. Formulation of the problem is carried out using the first-order (Timoshenko’s type) refined theory of shallow shells. Different types of boundary conditions, including clamped, simply supported, free edge and their combinations, are studied. The proposed method and the created computer code have been examined on test problems for shallow shells with rectangular planforms. In order to demonstrate the possibility of the developed approach, novel results for laminated FGM shallow shells with cut of the complex form are presented. Effects of different material distributions, mechanical properties of the constituent materials, lamination scheme, boundary conditions and geometrical parameters on natural frequencies are shown and analyzed.

Numerical methods for thermally stressed shallow shell equations

We develop efficient and accurate numerical methods to solve a class of shallow shell problems of the von Karman type. The governing equations form a fourth-order coupled system of nonlinear biharnomic equations for the transverse deflection and Airy's stress function. A second-order finite difference discretization with three iterative methods (Picard, Newton and Trust-Region Dogleg) are proposed for the numerical solution of the nonlinear PDE system. Three simple boundary conditions and two application-motivated mixed boundary conditions are considered. Along with the nonlinearity of the system, boundary singularities that appear when mixed boundary conditions are specified are the main numerical challenges. Two approaches that use either a transition function or local corrections are developed to deal with these boundary singularities. All the proposed numerical methods are validated using carefully designed numerical tests, where expected orders of accuracy and rates of convergence are observed. A rough run-time performance comparison is also conducted to illustrate the efficiency of our methods. As an application of the methods, a snap-through thermal buckling problem is considered. The critical thermal loads of shell buckling with various boundary conditions are numerically calculated, and snap-through bifurcation curves are also obtained using our numerical methods together with a pseudo-arclength continuation method. Our results are consistent with previous studies.

Linear and nonlinear free vibration analysis of laminated functionally graded shallow shells with complex plan form and different boundary conditions

Linear and geometrically nonlinear vibrations of the three-layered functionally graded shallow shells with a complex form of the base are studied. It is assumed that outer and inner layers are made of functionally graded materials (FGM) or an isotropic material (metal or ceramic). The first-order shear deformation theory of shallow shells (FSDT) is employed. Effective material properties of layers are varied along the thickness according to a power law. To calculate different mechanical characteristics for different types of lamination schemes, analytical expressions are obtained. The linear problem is solved by combining the Ritz and the R-functions method. Linearization of the nonlinear problem is carried out by a novel original approach. Namely, the initial problem is reduced to solving a sequence of linear problems including those vibrations related to linear, special type of the elasticity problem and the nonlinear system of ordinary differential equations. Validation of the proposed method and the developed software has been examined on test problems for shallow shells with rectangular plan form and different boundary conditions. In order to demonstrate possibilities of the proposed approach, new results for laminated FGM shallow shells of the complex form of the base are presented. Effects of different material distributions, lamination schemes, curvatures, boundary conditions, and geometrical parameters on natural frequencies and backbone curves are reported and analyzed.

Solvability of geometrically nonlinear boundary-value problems for shallow shells of Timoshenko type with pivotally supported edges

We study solvability of a geometrically nonlinear, physically linear boundary-value problems for elastic shallow homogeneous isotropic shells with pivotally supported edges in the framework of S. P. Timoshenko’s shear model. The purpose of work is the proof of the theorem on existence of solutions. Research method consists in reducing the original system of equilibrium equations to one nonlinear differential equation for the deflection. The method is based on integral representations for displacements, which are built with the help of the general solutions of the nonhomogeneous Cauchy-Riemann equation. The solvability of equation relative to deflection is established with the use of principle of contraction mappings.

Investigating geometrically nonlinear vibrations of laminated shallow shells with layers of variable thickness via the R-functions theory

A novel numerical/analytical approach to study geometrically nonlinear vibrations of shells with variable thickness of layers is proposed. It enables investigation of shallow shells with complex forms and different boundary conditions. The proposed method combines application of the R-functions theory, variational Ritz’s method, as well as hybrid Bubnov–Galerkin method and the fourth-order Runge–Kutta method. Mainly two approaches, classical and first-order shear deformation theories of shells are used. An original scheme of discretization regarding time reduces the initial problem to the solution of a sequence of linear problems including those related to linear vibrations with a special type of elasticity, as well as problems governed by non-linear system of ordinary differential equations. The proposed method is validated by the investigation of test problems for shallow shells with rectangular planform and applied to new vibration problems for shallow shells with complex planforms and variable thickness of layers.

Free vibration of shallow shells using the Rayleigh—Ritz method and penalty parameters

In the present work, the Rayleigh—Ritz method (RRM) and the penalty function method (PFM) are used to solve vibration problems of shallow shells of rectangular planform with spherical, cylindrical, and hyperbolic paraboloidal geometries with classical boundary conditions. Problems with more complicated constraints can also be solved using the method presented in this work, which consists in developing the stiffness and mass matrices of a completely free shallow shell using the RRM and then all constraints of the shell are defined by the PFM through the use of either artificial stiffness or artificial inertia with either positive or negative values.

A mixed finite element method for a shallow shell problem with a stress function

In this paper we consider the problem of a deflected mode of a shallow shell. The stress function and the normal component of the displacement of the median surface of the shell are unknown functions. We propose a mixed variational statement of the problem, where the second derivatives of the stress function and the normal component of the displacement of the median surface are additional unknowns. This enables us to construct the finite element approximation of the initial problem. We prove the existence of a unique solution of the approximating problem and estimate the rate of convergence of the discrete solution.

INVESTIGATION OF STATIC AND DYNAMIC BEHAVIOR OF ANISOTROPIC INHOMOGENEOUS SHALLOW SHELLS BY SPLINE APPROXIMATION METHOD

The present report proposes an efficient approach to solving within the framework of the classic and refined models the stress‐strain problems of shallow shells as well as the problems on free vibrations. In accordance with the approach the initial system of partial differential equations is reduced to one‐dimensional problems by using approximation of the solution in terms of basic splines in one coordinate. The boundary‐value problems obtained and eigenvalue boundary‐value problems for systems of ordinary differential high‐order equations are solved by the stable numerical method of discrete ortogonalization. Santrauka Nagrinėjamas efektyvus lėkštųjų kevalų įtempių, deformacijų ir laisvųjų svyravimų nustatymo algoritmas, pagrįstas klasikiniais ir tobulesniais skaičiavimo modeliais. Algoritme dalinių diferencialinių lygčių sistema yra transformuojama į vienmatį uždavinį, sprendžiamą naudojant pagrindinių splainų aproksimaciją į vieną koordinatę. Gaunamas kraštinis uždavinys, kuris sprendžiamas kaip kraštinis savųjų reikšmių nustatymo uždavinys. Uždavinio sąlygas atitinka įprastų aukštesnės eilės diferencialinių lygčių sistema, kuriai spręsti taikomi patikimi diskrečiosios ortogonalizacijos skaitiniai metodai.

Solving stationary problems for shallow shells by a generalized Kantorovich–Vlasov method

A generalized Kantorovich–Vlasov method is used to solve stationary problems for shallow shells with rectangular planform and arbitrary boundary conditions. The efficiency of the approach is illustrated by examples

Hadamard well-posedness for a class of nonlinear shallow shell problems

This paper is concerned with the nonlinear shallow shell model introduced in 1966 by W.T. Koiter in [On the nonlinear theory of thin elastic shells. III, Nederl. Akad. Wetensch. Proc. Ser. B 69 (1966) 33–54, Section 11] and later studied in [M. Bernadou, J.T. Oden, An existence theorem for a class of nonlinear shallow shell problems, J. Math. Pures Appl. (9) 60(3) (1981) 285–308]. We consider a version of this model which is based upon the intrinsic shell modeling techniques introduced by Michel Delfour and Jean-Paul Zolésio. We show existence and uniqueness of both regular and weak solutions to the dynamical model and that the solutions are continuous with respect to the initial data. While existence and uniqueness of regular solutions to nonlinear dynamic shell equations has been known, full Hadamard well-posedness of weak solutions, as shown in this paper, is a new result which solves an old open problem in the field.

Uniqueness and continuous dependence on the initial data for a class of non-linear shallow shell problems

This note is concerned with the non-linear shallow shell model introduced in 1966 by W.T. Koiter, and later studied by M. Bernadou and J.T. Oden. We show the uniqueness of the solution to the dynamical model and that this solution is continuous with respect to the initial data. To cite this article: J. Cagnol et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Solution of vibration problems for shallow shells of arbitrary form by the R-function method

An effective method for the free vibration of arbitrary plan-form shallow shells is proposed. The algorithm is based on the R-function theory and variational Ritz method. The effectiveness of the method offered is illustrated by examples of shallow shells of a complex plan form at different boundary conditions. Three types of curvatures are considered, which are the spherical, circularly cylindrical and of hyperbolic paraboloid shape. A number of the test problems were solved to check the veracity of the proposed method. The comparison between the obtained results and those available in the literature, was carried out.

Coupled Thermoelasticity Problems of Shallow Shells

The derived differential equations govern a coupled linear thermoelastic behaviour in the frame of the Kirchhoff-Love model with the neglected effect of shell element rotary inertia. Then an abstract Cauchy problem for a coupled system by two differential equations in the Hilbert space is defined. It generalizes a series of coupled thermoelastic problems including those considered previously (Kirchhoff-Love and Timoshenko-type shells). All considerations are valid for a wide class of initial and boundary value problems.