Elastic Conical Shells Research Articles

Water entry of an elastic conical shell

The axisymmetric problem of a conical shell impact onto an inviscid and incompressible liquid of infinite depth is studied. The shell is thin, and its deadrise angle is small. The problem is inertia dominated. Gravity, surface tension and viscous effects are not taken into account. The hydrodynamic loads acting on the shell and the shell displacements are determined at the same time. The model by Scolan (J. Sound Vib., vol. 277, issue 1–2, 2004, pp. 163–203) is used to find the flow and hydrodynamic pressure caused by the shell impact. This model is based on the Wagner theory of water impact, which was generalised to axisymmetric problems of hydroelastic slamming. Dry and wet modes of the conical shell, as well as the corresponding frequencies, are calculated. It is shown that a conical shell can be approximated by a circular plate only for a very small deadrise angle. Deflections and strains in the conical shell during the impact stage, when the wetted part of the shell increases at high rate, as well as the hydrodynamic loads, are determined and analysed.

DEFORMATION AND BUCKLING ANALYSIS OF SHALLOW CONICAL SHELLS BASED ON IMPROVED MODEL OF WIND LOAD USING ENVIRONMENT ANSYS

The process of deformation and buckling of shallow thin-walled elastic conical shells has been investigated for the case of significantly non-uniform stress-strain state due to the action of wind load based on improved model of pressure application schema to the surface of shallow shell and for hinged hedge of border. An improved model of wind load was based on data presented in terms [5, 6] and was a logical continuation of previous investigation of wind action on shallow conical shells based on model of first approach [3]. Deformation and buckling process investigation has been carried out using software ANSYS which effectivity was approved by the fact of being used by NASA for its aerospace projects. A model of shallow conical shell has been made using four-corner finite element SHELL 281 with 8 nodes that let us obtain not only symmetrical relatively to the axis of rotation buckling form but an asymmetrical too. Two types of computation have been made during numerical modeling – linear bifurcation computation with determination of linear pressure qcr value and corresponding to it buckling form, and computation of geometrically non-linear problem of deformation with determination of limit pressure qlim and corresponding buckling form. Obtained buckling forms have been compared to the deformed shape of shell surface when aerodynamic computations have been carried out using software ANSYS. An estimation analysis has been made for case of application of improved model of wind load in comparison to the previous investigation according to the values of baring capacity and buckling shape coherence during resolution of static tasks and comparison to the results of aerodynamic solution. An analysis of base parameter influence has been carried out for the model of first approach and current improved model according to the bearing capacity value and local extremums on schema of pressure intensity distribution of wind load. Specific moments of deformation process computations based on improved model using environment ANSYS have been mentioned and of further analysis on the basis of improved model with it specifics have been given too.

Liquid Induced Vibrations of Truncated Elastic Conical Shells with Elastic and Rigid Bottoms

A method of estimating natural modes and frequencies of vibrations for elastic shells of revolution conveying a liquid is proposed. The vibration modes of the liquid-filled elastic shells are presented as linear combinations of their own vibration modes without liquid. The explicit expression for fluid pressure is defined using Bernoulli’s integral and potential theory suppositions. Non-penetration, kinematic, and dynamic boundary conditions are applied at the shell walls and on a free liquid surface, respectively. The solution of the hydro-elasticity problem is found out using an effective technique based on coupled finite and boundary element methods. Computational vibration analysis of elastic truncated conical shells with different fixation conditions is accomplished. Sloshing and elastic walls frequencies and modes of liquid-filled truncated conical tanks are estimated. Both rigid and elastic bottoms of shells are considered. Some examples of numerical estimations are provided to testify the efficiency of the developed method

Optimal Control of an Axisymmetric Conical Shell with a Unilateral Elastic Foundation and Rigid Supports

AbstractWe study an optimal control problem for an elastic conical shell. The displacement vector u=(u,v) is then a function of one variable s∈(a,b), 0<a<b<∞. Here u is the meridional and v the normal displacement. The shell is assumed to be simply supported. Moreover, we consider several unilateral obstacles and an unilateral elastic foundation of Winkler type i.e. the reaction force is proportional of the positive part of the normal displacement. The state problem is formulated in a form of variational inequality for u. The design parameter e=(t,z,F) includes the variable thickness of the shell, the stiffness characteristics of the foundation and the friction coefficient. The existence of the optimal control will be explained. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An Alternative to F. Y. M. Wan’s Single Equation for an Elastic Right Circular Conical Shell

In 1970, F. Y. M. Wan derived a single, complex-valued ordinary differential equation for an elastically isotropic right circular conical shell (“On the Equations of the Linear Theory of Elastic Conical Shells,” Studies Appl. Math., 49, pp. 69–83). The unknown was the nth Fourier component of a complex combination of the midsurface normal displacement and its static-geometric dual, a stress function. However, an attempt to formally replace the Fourier index n by a partial derivative in the circumferential angle θ results in a partial differential equation, which is eighth order in θ. The present paper takes as unknowns the traces of the bending strain and stress resultant tensors, respectively, and derives static-geometric dual partial differential equations of fourth order in both the axial and circumferential variables. Because of the explicit appearance of Poisson ratios of bending and stretching, these two equations cannot be combined into a single complex-valued equation. Reduced equations for beamlike (axisymmetric and lateral) deformations are also derived.

Dynamic analysis of conical shells conveying fluid

A formulation, based on the semi-analytical finite element method, is proposed for elastic conical shells conveying fluids. The structural equations are based on the shell element proposed by Ramasamy and Ganesan [Finite element analysis of fluid-filled isotropic cylindrical shells with constrained viscoelastic damping, Computers & Structures 70 (1998) 363–376] while the fluid model is based on velocity potential formulation used by Jayaraj et al. [A semi-analytical coupled finite element formulation for composite shells conveying fluids, Journal of Sound and Vibration 258(2) (2002) 287–307]. Dynamic pressure acting on the walls is derived from Bernoulli's equation. By imposing the requirement that the normal component of velocity of the solid and fluid are equal leads to fluid–structure coupling. The computer code developed has been validated using results available in the literature for cylindrical shells conveying fluid. The study has been carried out for conical shells of different cone angles and for boundary condition like clamped–clamped, simply supported and clamped free. In general, instability occurs at a critical fluid velocity corresponding to the shell circumferential mode with the lowest natural frequency. Critical fluid velocities are lower than that of equivalent cylindrical shells. This result holds good for all boundary conditions.

Axisymmetric Instabilities for Elastic Conical Shells under Compressive End Loadings

We deal with the problem of equilibrium and buckling of nonlinear elastic axisymmetric shells, whose referential shape is a truncated circular cone, subject to compressive end loadings. It is a non-trivial generalization of the cylindrical case, already studied by Cohen and Pastrone; in this particular case, suitable constitutive restrictions, which are not needed for cylinders, must be satisfied to allow the conical shell to sustain end loadings, even small, without bending. In particular, we consider thin Kirchhoff shells and prove, by means of the bifurcation theory of Poincaré, the non-uniqueness of solutions of the boundary value problem associated with the equilibrium equations, the assigned end loadings and the geometrical constraints: i.e., the axisymmetry and the inextensibility along meridians. The critical loads are determined as well as the bifurcation points. If the material is hyperelastic the equations of equilibrium are derived from a variational principle and, for some special form of the strain energy function, a Hamiltonian formulation can be provided. The possibility of a non-convex strain energy function is briefly discussed.

Buckling of a conical thin shell with variable thickness under a dynamic loading

This study considers the buckling of an elastic truncated conical shell having a meridional thickness expressed by an arbitrary function, subject to a uniform external pressure, which is a power function of time. At first, the fundamental relations and Donnell type dynamic buckling equation of an elastic conical shell with variable thickness have been obtained. Then, employing Galerkin's method, those equations have been reduced to a time-dependent differential equation with variable coefficients. Finally, applying the Ritz type variational method, the critical static and dynamic loads, the corresponding wave numbers, dynamic factor and critical stress impulse have been found analytically. Using the results, thus obtained, the effects of the thickness variations with a power or an exponential function, the variation of the semi-vertex angle and the variation of the power of time in the external pressure expression are studied through pertinent computations. It is observed that these factors have appreciable effects on the critical parameters of the problem in the heading.

The dynamic stability of a nonhomogeneous orthotropic elastic truncated conical shell under a time dependent external pressure

In this research, the dynamic stability of an orthotropic elastic conical shell, with elasticity moduli and density varying in the thickness direction, subject to a uniform external pressure which is a power function of time, has been studied. After giving the fundamental relations, the dynamic stability and compatibility equations of a nonhomogeneous elastic orthotropic conical shell, subject to a uniform external pressure, have been derived. Applying Galerkin's method, these equations have been transformed to a pair of time dependent differential equations with variable coefficients. These differential equations are solved using the method given by Sachenkov and Baktieva (1978). Thus, general formulas have been obtained for the dynamic and static critical external pressures and the pertinent wave numbers, critical time, critical pressure impulse and dynamic factor. Finally, carrying out some computations, the effects of the nonhomogeneity, the loading speed, the variation of the semi-vertex angle and the power of time in the external pressure expression on the critical parameters have been studied.

Dynamics of a conical shell containing supersonic gas flow

Unsteady dynamics of an elastic conical shell with a circular cross-section are considered when the smaller base is clamped and the larger base is free. Inside the shell gas flows supersonically from the smaller end to the larger end. The shell is also transversally loaded. All processes are assumed to axisymmetric. Flutter is investigated and for a stable situation the evolution of the bending of the cone generators under the action of a given load is analysed. The behaviour of the shell is described by the equations of technical theory in mixed form, the influence of the flux being taken into account via a piston model. A non-local approximation to the solution by a special system of orthogonal polynomials is used. Its properties are discussed and numerical results are given.

Stress and strain determination in a nonlinearly elastic conical shell subjected to aerodynamic loading and heating

Stress and strain determination in a nonlinearly elastic conical shell subjected to aerodynamic loading and heating

Buckling of Axially Compressed Conical Shells

The buckling of a truncated elastic conical shell subjected to an axial compression is a classical problem in shell structures. The paper reinvestigates the buckling of an axially compressed truncated conical shell with rigid bulkheads. Two improvements are achieved. First, the condition that the total horizontal displacement must vanish due to rigid bulkhead and axisymmetry is treated as a constraint. This constraint is incorporated into the system through the use of the Lagrange multiplier; then the variational method is used to derive a complete set of boundary conditions for conical shells. Second, the stability is evaluated in the deformed state using the asymptotic solutions of the pair of Donnell-type equations for axisymmetric configuration. The results indicate that the buckling strength of conical shells depends mainly on the condition of the smaller end. In addition to the vertex angle, the distance ratio plays, at least, an equally important role.

An asymptotic solution of the non-linear equations of conical shells

An asymptotic solution to the Donnell type non-linear equations of elastic conical shells is presented. Closed form solutions for the displacements and stress resultants including eight constants of integration are obtained, by considering the lateral displacement to be of the order of the thickness t, and omitting terms of order up to (t/r) 1 2 in comparison to unity. A numerical example for a conical frustum under axial load is given. The results indicate that buckling occurs near the smaller end of the frustum, as observed in an experiment.

On the Equations of the Linear Theory of Elastic Conical Shells

In a recent article [19], it was shown that the equations of the linear theory of elastic shells of revolution under arbitrary loads, after a harmonic analysis in the polar angle of the base plane of the shell, can be reduced to two simultaneous fourth order ordinary differential equations for a stress function and a displacement function. The exact reduction procedure of [19] was applied to a circular cylindrical shell. Also mentioned was the possibility of a simplified procedure based on the fact that there is an inherent error in shell theory of the order h/R, where h is the shell thickness and R is a representative magnitude of the two principal radii of curvature. In the present paper, the exact and the simplified procedure of [19] will be applied to a conical shell frustrum. The simplified procedure leads to two differential equations as well as to auxiliary equations for the calculation of stress resultants and couples which differ from those of the exact procedure only in terms which are O(h/R). The differential equations obtained here are very similar in form to those of shallow shell theory [6,8] as well as to those of the Donnell type theory [15]. They do in fact reduce to the shallow shell equations when the slope angle /!? between a meridian and the base plane is very small. On the other hand, for /3 = z/2, they reduce to the “correct” equations for circular cylindrical shells of [16,19] while the results of [15] reduce only to the Donnell equation. Aside from their usefulness in obtaining the solution of specific problems, our results also delimit the range of applicability of the Donnell type equations for conical shells previously obtained in [5,15].

An asymptotic solution of conical shells of constant thickness.

Constant thickness elastic conical shells subject to lateral loads, deriving asymptotic solution

On the solution of conical shells of linearly varying thickness subjected to lateral normal loads

the solution of elastic conical shells of linearly varying thickness is hinged on an eight degree characteristic equation. In this paper, a method of solving this equation is presented which results in a general asymptotic solution. The particulat solution due to a lateral normal load which is constant along meridians and has a sinusoidal distribution in the circumferential direction is given. Included is also an illustrative example of a semicircular cone which is simply supported along two generators with one end fixed and the other end free.