Shallow Shell Theory Research Articles

Large amplitude free vibration of orthotropic shallow shells of complex shapes with variable thickness

The present formulation of the analysed problem is based on Donell’s nonlinear shallow shell theory, which adopts Kirchhoff’s hypothesis. Transverse shear deformations and rotary inertia of a shell are neglected. According to this theory, the non-linear strain-displacement relations at the shell midsurface has been proposed. The validity and reliability of the proposed approach has been illustrated and discussed, and then a few examples of either linear or non-linear dynamics of shells with variable thickness and complex shapes have been presented and

CHAOTIC VIBRATIONS IN FUNCTIONALLY GRADED DOUBLY CURVED SHELLS WITH INTERNAL RESONANCE

Chaotic vibrations of functionally graded doubly curved shells subjected to concentrated harmonic load are investigated. It is assumed that the shell is simply supported and the edges can move freely in in-plane directions. Donnell's nonlinear shallow shell theory is used and the governing partial differential equations are obtained in terms of shell's transverse displacement and Airy's stress function. By using Galerkin's procedure, the equations of motion are reduced to a set of infinite nonlinear ordinary differential equations with cubic and quadratic nonlinearities. A bifurcation analysis is carried out and the discretized equations are integrated at (i) fixed excitation frequencies and variable excitation amplitudes and (ii) fixed excitation amplitudes and variable excitation frequencies. In particular, Gear's backward differentiation formula (BDF) is used to obtain bifurcation diagrams, Poincaré maps and time histories. Furthermore, maximum Lyapunov exponent and Lyapunov spectrum are obtained to classify the rich dynamics. It is revealed that the shell may exhibit complex behavior including sub-harmonic, quasi-periodic and chaotic response when subjected to large harmonic excitations.

Free vibration analysis of orthotropic doubly-curved shallow shells based on the gradient elasticity

The comprehensive free vibration analysis of doubly-curved shallow shells which are made of an orthotropic material is presented. The size effect is taken into consideration using the gradient elasticity theory. Novozhilov’s linear shallow shell theory is used and it is assumed that the shell is simply supported. The governing equations of the doubly-curved shallow shells with consideration of the length scales are developed. Analytical solutions to the equations are proposed to obtain the frequencies of the shells. Numerical results are presented for various shell curvatures including spherical, cylindrical and hyperbolic paraboloidal shells and compared with those available in the literature.

An extension of Karman–Donnell's theory for non-shallow, long cylindrical shells undergoing large deflection

Conventionally cylindrical shells are often treated as shallow shells whose governing equation, according to Karman–Donnell's theory, can be approximated by those of thin plates. In this paper Karman–Donnell's theory for shallow shells is extended for long cylindrical shells undergoing large, nonlinear flexural deflection. The kinematic relations between the changes of curvature and the displacement are derived and the governing equations are established by considering the influence of the initial curvature of the cylindrical shells. In particular, the extended Karman–Donnell's theory is applied for the failure analysis of infinitely cylindrical shell under lateral pressure . A regional collapse mode is identified to occur in the shell with a longitudinal span proportional to (radius 3 /thickness) (1/2) and a transverse profile of dog bone shape. It is found that the buckling pressure of the shell is in proportion to (thickness/radius) 3 and converges to the classic solution given by Timoshenko and Gere (1961) . A comparison to the previous works indicates that ignoring the effect of the initial curvature will result in an overestimate of the buckling pressure for 33%. It shows that the initial curvature of long cylindrical shells has significant influence on the load carrying capacity and the extended Karman–Donnell's equations give very accurate predictions. ►The nonlinear kinematic relations of curvature change vs deflection for non-shallow, cylindrical shells are deduced.►The Karman–Donnell’s theory is extended by considering the effect of the initial curvature in circumferential direction. ►Post-buckling of infinitely long, cylindrical shells subjected to the lateral external pressure are analyzed.►A regional collapse mechanism is identified with a longitudinal span proportional to (radius 3 /thickness) (1/2) .►It shows that the initial curvature of long cylindrical shells has significant influence on the load carrying capacity.

Parametric instability analysis of moderately thick FGM cylindrical panels using FSM

In the current paper, it is the first time that a dynamic stability analysis of moderately thick cylindrical panels made from FG materials is conducted by employing finite strip formulations based on a Reddy-type third order shear deformation theory (HSDT). Two versions of finite strip methods (FSM), namely semi-analytical and B-spline methods are developed. The mechanical properties of FGM are assumed to change in thickness direction according to a power-law function. The temperature effects are ignored. The strain terms are expressed in terms of the Koiter–Sanders theory of shallow shells. In order to demonstrate the capabilities of the developed methods in predicting parametric behavior of the subject structures, some representative results are obtained and compared with those in the literature wherever available.

Nonlinear Stability Analysis for FGM Shallow Conical Shells Subjected to Thermal and Mechanical United Loads

Based on the Von Karman large deflection theory of shallow shell, the problem of nonlinear stability of symmetrically FGM shallow conical shells under the action of mechanical and thermal loads is investigated. The nonlinear governing equation under the united action of mechanical and thermal loads of FGM shallow conical shells whose physical parameters change with the power rate is derived and solved by the modified iteration method. The nonlinear characteristic relation of load, deflection and temperature is obtained. The extremum buckling principle is employed to determine the critical buckling load. The influences of gradient constants, geometric parameters and temperature differences on buckling are discussed as well.

Some features of the natural flexural vibrations of circular rings with initial imperfections

The effect of small initial deviations from an ideal circular shape on the frequencies and modes of the natural oscillations of a thin ring under conditions of plane deformation is investigated. The analysis is based on the equations of motion of the shallow shell theory when the length of the shell tends to infinity. A new approach is proposed to the construction of a finite-dimensional model of a ring in which it is assumed that initial imperfections lead to the interaction of flexural and radial vibrations. A number of assertions are formulated and proved regarding the specific features which accompany the vibrations of an imperfect ring that apparently also have to be taken into account when investigating the dynamic characteristics of thin circular cylindrical shells with initial imperfections.

Influence of physical and geometrical system parameters uncertainties on the nonlinear oscillations of cylindrical shells

This work investigates the influence of physical and geometrical system parameters uncertainties and excitation noise on the nonlinear vibrations and stability of simply-supported cylindrical shells. These parameters are composed of both deterministic and random terms. Donnell's non-linear shallow shell theory is used to study the non-linear vibrations of the shell. To discretize the partial differential equations of motion, first, a general expression for the transversal displacement is obtained by a perturbation procedure which identifies all modes that couple with the linear modes through the quadratic and cubic nonlinearities. Then, a particular solution is selected which ensures the convergence of the response up to very large deflections. Finally, the in-plane displacements are obtained as a function of the transversal displacement by solving the in-plane equations analytically and imposing the necessary boundary, continuity and symmetry conditions. Substituting the obtained modal expansions into the equation of motion and applying the Galerkin's method, a discrete system in time domain is obtained. Several numerical strategies are used to study the nonlinear behavior of the shell considering the uncertainties in the physical and geometrical system parameters. Special attention is given to the influence of the uncertainties on the parametric instability and escape boundaries.

The analysis of interlaminar stresses for composite laminated shallow shells with interfacial damage

Based on the general six-degrees-of-freedom plate theory towards the accurate stress analysis and nonlinear theory of shallow shells, considering the damage effect of the interlaminar interface and using the variation principle, the three-dimensional non-linear equilibrium differential equations of the laminated shallow shells with interfacial damage are derived. Then, considering a simply supported laminated shallow shell with damage and under normal load, an analytical solution is presented by using finite difference method to obtain the interlaminar stresses. Numerical results show, the stiffness of the shell is weakened, greater absolute values of displacements as well as smaller interlaminar stresses are obtained by interfacial damage. When the interfacial damage is further increased, delamination occurs obviously under normal pulling load and pure shear slip occurs under normal pressure load. The portion of the load undertaken by the two sides of the interface is more different. Different mechanical behaviors are shown in both sides of the interface, and the discontinuation of stresses and displacements takes place in the interface.

Semi-analytical finite strip method based on the shallow shell theory in buckling analysis of cold-formed sections

This paper presents a development of the finite strip method based on the shallow shell theory of Marguerre. Most of existing semi-analytical finite strips are based on the plate theories of Kirchhoff and of Mindlin. As it is showed in the paper, the plate theories can be considered as the particular cases of the shallow shell theory. Numerical results show the efficiency of the finite strip based on the shallow shell theory in the buckling analysis of prismatic structures which have curved corners.

An analytical study of the non-linear vibrations of cylindrical shells

The primary resonance response of simply supported circular cylindrical shells is investigated using the perturbation method. Donnell's non-linear shallow-shell theory is used to derive the governing partial differential equations of motion. The Galerkin technique is then employed to transform the equations of motion into a set of temporal ordinary differential equations. Considering only the primary resonance case, the method of multiple scales is used to study the periodic solutions and their stability. The necessary and sufficient conditions for appearance of the so-called companion mode are also discussed. To this end, a range of the possible multi-mode solution is obtained for response and excitation amplitudes and also excitation frequency as a function of damping, geometry and material properties of the shell. Parametric studies are performed to illustrate the effect of different values of thickness, length and material composition on the possibility of the companion mode participation in primary resonance response.

Vibration of doubly curved shallow shells with arbitrary boundaries

The first comprehensive study of shallow shell vibrations subjected to as many as 21 possible boundary conditions is presented. Thin shallow shell theory is used. Relatively accurate results for natural frequencies of doubly-curved shallow shells have been obtained. These can be used for benchmarking by researchers as well as reference data for practicing engineers. The Ritz method is used to solve for natural vibrations of these shells with arbitrary boundary conditions. Natural frequencies are presented for various shell curvatures including spherical, cylindrical and hyperbolic paraboloidal shells.

Buckling of FGM cylindrical shells subjected to pure bending load

In this paper, buckling behaviors of composite cylindrical shells made from functionally graded materials (FGMs) subjected to pure bending load were investigated. The material properties were assumed to be graded along the thickness. The non-uniform bending force on the shell section was considered in the buckling government equation of FGM cylindrical shells based on the Donnell shallow shell theory. The prebuckling deformation of the FGM cylindrical shells was neglected and the buckling mode was assumed to occur non-uniformly in local district along the shell circumferential direction. The eigenvalue method was used to obtain the buckling critical condition. The theoretical results were in excellent agreement with those of ABAQUS code. Results show that the inhomogenity of the materials is significant for buckling of FGM cylindrical shells.

Stability of cylindrical shells with initial imperfections under the action of external pressure

We use the equations of nonlinear theory of shallow shells to solve the problem of stability of thin elastic isotropic cylindrical shells, with small initial shape imperfections, that are under the action of external uniform pressure. The problem solution is constructed by the Rayleigh-Ritz method with the approximation of the shell midsurface displacement by double functional sums in trigonometric and beam functions. The system of nonlinear algebraic equations is solved by using the methods of continuation with respect to a close-to-best parameter. For the initial imperfections of the shells, we use their normalized deflections from the limit points of overcritical branches of the loading trajectories. We consider various cases of the shell fixation and support under loading by lateral and hydrostatic uniform pressure. We also construct the range of values of the critical pressure, which, with the maximal deviation of the shell shape from the cylindrical shape up to 30%, covers practically all known experimental data.

Effect of inplane edge constraints on natural frequencies of simply supported doubly curved shallow shells

Reasonably accurate natural frequencies are presented for simply supported shallow shells on rectangular planform subjected to 55 possible combinations of edge constraints. Thin shallow shell theory is used. Accurate natural frequencies of doubly curved shallow shells are presented and can be used for benchmarking by researchers as well as reference data for practicing engineers. Natural frequencies are presented for various shell curvatures including spherical, cylindrical and hyperbolic paraboloidal shells.

A dynamical model for nonlinear viscoelastic corrugated circular plates with shallow sinusoidal corrugations

An integrated mathematic model and an efficient algorithm on the dynamical behavior of homogeneous viscoelastic corrugated circular plates with shallow sinusoidal corrugations are suggested. Based on the nonlinear bending theory of thin shallow shells, a set of integro-partial differential equations governing the motion of the plates is established from extended Hamilton’s principle. The material behavior is given in terms of the Boltzmann superposition principle. The variational method is applied following an assumed spatial mode to simplify the governing equations to a nonlinear integro-differential variation of the Duffing equation in the temporal domain, which is further reduced to an autonomic system with four coupled first-order ordinary differential equation by introducing an auxiliary variable. These measurements make the numerical simulation performs easily. The classical tools of nonlinear dynamics, such as Poincare map, phase portrait, Lyapunov exponent, and bifurcation diagrams, are illustrated. The influences of geometric and physical parameters of the plate on its dynamic characteristics are examined. The present mathematic model can easily be used to the similar problems related to other dynamical system for viscoelastic thin plates and shallow shells.

Boundary Condition Effects on the Parametric Stability of Moderately Thick Laminated Cylindrical Panels

A Reddy type, third order shear deformation theory of shells is applied to the development of two versions of finite strip method (FSM), namely semi-analytical and spline methods, to predict the parametric stability and instability regions in the case of cylindrical moderately thick composite laminated panels. The structures are assumed to be under harmonic in-plane loads in the context of the so-called parametric loading. The linear strain terms are expressed in terms of the Koiter-Sanders theory of shallow shells. In order to demonstrate the capabilities of the developed methods in predicting parametric behavior of the subject structures, some representative results are obtained and compared with those in the literature wherever available.

Fundamental Frequency Optimization of Laminated Cylindrical Panels by Elitist-Genetic Algorithm

In this paper, fundamental frequency optimization of symmetrically laminated cylindrical panels is investigated by the Elitist-Genetic algorithm (E-GA). The number of layers, the fiber orientation angles, edge conditions, length/width (a/b) and length/R (a/R) ratios are considered as design variables. The classical shallow shell theory (Donnell’s formulation) is applied to calculate the natural frequencies of laminated cylindrical curved panels. To improve the speed of the optimization process, the elitist strategy is used in the Genetic algorithm and the fitness function is computed with a semi-analytical finite strip method (FSM). A program based on Maple is used for this purpose. To check the validity, the obtained results are also compared with some other stacking sequences.

A theoretical analysis of FGM doubly curved shallow thin shell based on physical neutral surface

In this paper, a new theory about functionally gradient materials (FGM) is constructed from the concept of physical neutral surface and classical double curved shallow thin shell theory (CDSST). In this theory, there is no stretching-b ending coupling effect in constitutive equations, governing equations and boundary conditions have the simple forms as those of CDSST about homogeneous isotropic materials, so the solution procedure is as easy as homogeneous isotropic shell. Using this new theory, linear bending and vibration solutions are presented using analytical method and a nonlinear bending approximate solution is given out using Galerkin variational method. This new theory about FGM has more merits in engineering applications, because it is easier and simpler than any of geometric middle surface FGM double curved shallow shell theories.

Nonlinear strain gradient elastic thin shallow shells

Abstract The governing equilibrium equations for strain gradient elastic thin shallow shells are derived, considering nonlinear strains and linear constitutive strain gradient elastic relations. Adopting Kirchhoff’s theory of thin shallow structures, the equilibrium equations, along with the boundary conditions, are formulated through a variational procedure. It turns out that new terms are introduced, indicating the importance of the cross-section area in bending of thin plates. Those terms are missing from the existing strain gradient shallow thin shell theories. Those terms highly increase the stiffness of the structures. When the curvature of the shallow shell becomes zero, the governing equilibrium for the plates is derived.