Linear Theory Of Thin Shells Research Articles

Numerical modeling of natural vibrations of coaxial shells partially filled with fluid, taking into account the effects on the free surface

This work is devoted to the numerical analysis of vertical elastic coaxial cylindrical shells completely or partially filled with a quiescent compressible fluid with account for sloshing of its free surface. The problem is solved in the axisymmetric formulation using a semi-analytical version of the finite element method. The fluid medium is described by the wave equation, which together with the conditions prescribed at its boundaries is reduced to the weak form by the Bubnov - Galerkin method. A mathematical formulation of the dynamic problem for thin-walled structures is developed using the variational principle of virtual displacements and linear theory of thin shells based on Kirchhoff - Love hypotheses. The fluid pressure on the walls of the structure is calculated according to Bernoulli's equation. Sloshing modes caused by gravitational effects on free surface of liquid medium are excluded from the resulting system of equations through the use of the iteration dynamic condensation method. The numerical model has been verified by comparing it with the known data for the case of a single shell partially filled with fluid. The influence of the fluid level on the lowest natural frequencies of the system vibrations at different variants of kinematic boundary conditions for shells (rigid clamping at both edges, cantilever support) and different widths of the annular gap between them has been evaluated. It has been found that for the examined configurations, the level of the fluid in the annular channel has a stronger influence on the frequency spectrum compared to the level of fluid in the cavity of the inner shell due to the fact that vibration frequencies change in a wider range.

An analysis of annular plate in curvilinear non-orthogonal coordinates with the help of equations of a shell theory

The complete system of equations of a linear theory of thin shells in curvilinear non-orthogonal coordinates proposed in the paper was taken as the basis of the investigation. Earlier, this system was used for static analysis of a long developable helicoid. In the article, this system is applied for the determination of stress-strain state of annular and circular plates under action of the external axisymmetric uniform load acting both in the plane of the plate and out-of-their plane. Presented results for annular plate given in the non-orthogonal coordinates ex-pand a number of problems that can be solved analytically. They can be used as the first terms of series of expansion of displacements of degrees of the small parameter if a small parameter method is applied for examining a long tangential developable helicoid.

Complicated features and their solution in analysis of thin shell and plate structures

Introduction. It is instructive to assemble a list of applications from a historical point of view, and to take as a connecting theme the way in which the introduction of the thin shell as a structural form made an important contribution to the development of several branches of engineering. The following is a brief list, which is by no means complete and complicated features and their solutions for analysis of such structures. Solution technique, methods. The linear theory of thin elastic shells with arbitrary shape of the middle surface is derived on the basis of Kirchhoff’s assumptions that were used in the development of the plate bending theory introduced in Part I. These assumptions are formulated for the linear theory of thin shells of an arbitrary shape. The problem of the study of this article is to identify simple way of solution to analyze complicated features of thin shell structure by introducing modern and new programmable theories and aspects. Especially the intersecting line of connecting thin shell structures. Results. It is possible to successfully model explicitly a panel profile that can be used for optimization studies for use as possible future test studies. It has also been shown, that if test data exists, a numerical solution can be very accurately modeled to match the test data by modifying the material properties of the model. Discussion. The article should encourage structural engineers to solve complicated features in thin shell structures and design for construction of such structure which are rarely constructing in country like Nepal due to lack of skilled manpower.

Calculation of thin isotropic shells beyond the elastic limit by the method of elastic solutions

The paper focuses on the model of calculation of thin isotropic shells beyond the elastic limit. The determination of the stress-strain state of thin shells is based on the small elastic-plastic deformations theory and the elastic solutions method. In the present work the building of the solution based on the equilibrium equations and geometric relations of linear theory of thin shells in curved coordinate system α and β, and the relations between deformations and forces based on the Hirchhoff-Lave hypothesis and the small elastic-plastic deformations theory are presented. Internal forces tensor is presented in the form of its expansion to the elasticity tensor and the additional terms tensor expressed the physical nonlinearity of the problem. The functions expressed the physical nonlinearity of the material are determined. The relations that allow to determine the range of elastic-plastic deformations on the surface of the present shell and their changing in shell thickness are presented. The examples of the calculation demonstrate the convergence of elastic-plastic deformations method and the range of elastic-plastic deformations in thickness in the spherical shell. Spherical shells with the angle of half-life regarding 90 degree vertical symmetry axis under the action of equally distributed ring loads are observed.

Modeling of corrugated laminates

This work compares two modeling approaches for corrugated laminates. Both models use a unit cell and a generalized plane-strain approach in order to determine a substitute plate model. The first model is based on a linear thin-shell theory and the second model uses a self-programmed finite element program with planar elements. The comparisons show the differences of the models as well as the limits of the shell theory approach. Furthermore, the realizable anisotropy of corrugated laminates which is important for morphing wing applications is investigated.

A new energy harvester using a cross-ply cylindrical membrane shell integrated with PVDF layers

In this paper, we proposed a smart cylindrical membrane shell panel (SCMSP) model for vibration-based energy harvester. The SCMSP is made of an orthotropic elastic core covered by outer PVDF layers with transverse polarization vector. Electrodynamics governing equations of motion are derived by applying extended Hamilton’s principle. The governing equations are based on Donnell’s linear thin shell theory. The SCMSP displacement fields are expanded by means of double Fourier series satisfying immovable edges with free rotation boundary conditions and coupled system of linear partial differential equations are obtained. The discretized linear ordinary differential equations of motion are obtained using Galerkin method. The output power is taken as an indicating criterion for the generator. A parametric study for MEMS applications is conducted to predict the power generated due to radial harmonic ambient vibration. Optimal resistance value is also obtained for the particular electrode distribution that gives maximum output power. A low vibration amplitude (5 Pa), and a low-frequency (471.79 Hz) vibration source is targeted for the resonance operation, in which the output power of 0.4111 μW and peak-to-peak voltage of 0.2952 V are predicted.

Vibration analysis of bi-layered FGM cylindrical shells

In the present study, a vibration frequency analysis of a bi-layered cylindrical shell composed of two independent functionally graded layers is presented. The thickness of the shell layers is assumed to be equal and constant. Material properties of the constituents of bi-layered functionally graded cylindrical shell are assumed to vary smoothly and continuously through the thickness of the layers of the shell and are controlled by volume fraction power law distribution. The expressions for strain–displacement and curvature–displacement relationships are utilized from Love’s first approximation linear thin shell theory. The versatile Rayleigh–Ritz approach is employed to formulate the frequency equations in the form of eigenvalue problem. Influence of material distribution in the two functionally graded layers of the cylindrical shells is investigated on shell natural frequencies for various shell parameters with simply supported end conditions. To check the validity, accuracy and efficiency of the present methodology, results obtained are compared with those available in the literature.

Dynamic stiffness analysis of toroidal shells

The free vibration of a toroidal shell is studied using the dynamic stiffness method. The dynamic stiffness method eliminates both spatial discretization error and mesh generation. Moreover, with a finite number of degrees of freedom, the dynamic stiffness method can predict an infinite number of natural frequencies. The dynamic behavior of the toroidal shell is modeled by DMV (Donnell-Mushtari-Vlasov) linear thin shell theory in the present paper. However, the procedure can be adapted to be used with any other linear thin shell theory without difficulty. Since a close form solution of toroidal shell using DMV theory is not (yet) possible, in order to obtain the desired dynamic stiffness matrix, a finite number of Fourier's series terms are taken in the circumferential direction and the unknown longitudinal displacements are then solved from the reduced governing equations exactly. The solution obtained from the dynamic stiffness method can be regarded as semi-analytical due to the Fourier approximation. With the dynamic stiffness matrices in hands, a toroidal shell with different boundary conditions and connections (to other toroidal shells) can be analyzed. This paper presents the procedure and assumption made in order to obtain the dynamic stiffness matrix of a toroidal shell in harmonic oscillation. Also some numerical examples will be given and discussed.

Free vibration analysis of a toroidal shell

The free vibration of a toroidal shell (curved pipe) is studied using DMV (Donnell-Mushtari-Vlasov) thin shell theory, Fourier's series and Galerkin's method. Some behaviors of the vibration of cylindrical shell, as a special case of toroidal shell, are discussed. Also, a 90-degree-bend curved pipe is studied numerically. Results obtained by the present method are compared to those obtained by finite element and other methods. DMV linear thin shell theory is employed for simplicity. However, the method applies to any other shell theories without difficulties.

The role of bending in the finite element analysis of thin shells

The role of bending in the finite element analysis of problems in the linear theory of thin shells is examined through an in-depth study of the behavior of a vehicle finite element. The chosen element is the very simple combined constant strain and constant bending moment flat triangle. It emerges that this vehicle element is a superb performer and the examination of its behaviour reveals that there are two quite different roles for the bending freedoms. One role concerns inextensional bending movements which extend over the whole model. The other role concerns local rotational movements which accompany the curvature changes of inextensional bending and of edge effect. Extensive numerical comparisons of demonstration inextensional bending movements are made against solutions obtained from the classical theory for shells which are very deep and which have strongly negative Gaussian curvature. Comparison of edge effect concerns a circular cylindrical thin shell. The paper concludes by giving details of a rudimentary matrix procedure which is currently under development and which is intended for use in the preliminary assessment of thin shell finite element models.

Linear theory of thin shells on the basis of the concept of added vectors

Linear theory of thin shells on the basis of the concept of added vectors

Thin shell analysis of non-circular pipe bends

The equations of Novozhilov's linear thin shell theory are applied to the in-plane bending and pressurising of non-circular pipe bends with lateral symmetry. Neglecting end effects, results are given for deflection and stress factors with wall thicknesses up to 0.3 of the minimum radius of the non-circular shape. Methods for determining the decay constants of end effects and for simulating built-in end conditions are presented for both circular and non-circular pipe bends but no results are given.

Some intrinsic considerations on the non linear theory of thin shells

Equations of compatibility and equilibrium are deduced in the case of finite deflection in anintrinsic form and referred to the undeformed state. The integrability relations, the needed auxiliary relations of algebra and the equations of static equivalence are directly applied. No use of a variational principle and only a minimal reference to the usual hypotheses of shell geometry is made. Physical interpretation is easier in this intrinsic representation.

Efect of normal pressure on intrinsic equations in the consistent second approximation theory of thin elastic shells

A linear thin shell theory of the lowest order second approximation is formulated in the case of the application of normal surface loads. A set of intrinsic equations are presented in terms of three stress resultants and three changes of curvature, where the 3-dimensional effect is shown to be necessary. A comparison is made with the refined interior equations derived by John.

The bending and pressurizing of pipe bends with flanged tangents

Pipe bends with flanged tangents are analyzed, by linear thin shell theory, for the stresses and flexibilities by pure inplane bending and pressurization, assuming that they are not interactive. Results are presented for flexibilities, decay constants and stresses. Zero tangent lengths, that is, flanges directly on the bend, substantially reduce the stresses and bend deflections due to applied moments, but tangents longer than one pipe circumference effectively allow 90° bends to act as if they are unterminated. With pressurization, flanges on 90° bends only marginally reduce stresses and bend deflections compared to those for an unterminated bend.

Dynamic response of a submerged hemispherical shell to earthquake motions

AbstractThe dynamic response to ground motion of hemispherical shells in a fluid, medium is studied numerically. In the analysis, linear thin shell theory is used and the fluid is assumed to be compressible and inviscid. The effect of the duration of the ground motion on the dynamic response is studied using two forcing functions, one with a very short duration and the other in the form of a Heaviside function. As special cases, dynamic responses of the shell in vacuo and of a rigid hemisphere in the fluid medium are investigated. The results are valid also for a ring‐stiffened complete spherical shell accelerating in an acoustic medium.

Forced axisymmetric motions of viscoelastic cylindrical shells

The isothermal response of a viscoelastic cylindrical shell, of finite length, to arbitary axisymmetric surface forces, initial conditions, and boundary conditions is considered within the linear theory of thin shells. The problem is formulated with the effects of shear deformation and rotatory inertia included; the viscoelastic properties are assumed to be isotropic and homogeneous. The response is first found formally in terms of a causal Green's function. It is then shown that when Poisson's ratio is constant, the causal Green's function can be expanded in a series of orthonormal spatial eigenfunctions of an associated elastic shell eigenvalue problem. The resulting solution for the general problem is an eigenfunction series with Laplace transformed time-dependent coefficients. The general solution is applied to predicting the motion of a uniform, simply-supported cylindrical shell, initially quiescent, which is subjected to a step pressure moving with constant velocity. For this example, the relaxation function of the shell material in uniaxial extension is taken to be that of a standard linear solid. The motions predicted by simpler shell models, namely, shells with bending only and without bending, are also considered for comparison. Here, the absolute values of the Fourier coefficients in the shell displacement series go to zero faster than the inverse of the first or second power of positive integers when bending is excluded or included, respectively. Numerical results are presented for a moderately long and relatively thick, nearly elastic, cylindrical shell.

The echo signal of a finite spherical pulse from a fluid filled spherical shell: PMM vol. 40, n≗ 4, 1976, pp. 648–654

A method is proposed for determining the echo signal of a centrosymmetrical pressure pulse from a fluid filled elastic spherical shell immersed in an infinite fluid medium. The medium surrounding the shell and the fluid in it are considered to conform to the theory of perfect compressible fluid. Motion of the shell is defined in accordance with Timoshenko's linear theory of thin shells. The problem is solved with the use of Fourier's integral transformation in terms of time and Watson's transformation in terms of polar distance. Mechanisms of formation of the various echo signal components are described. The determination of the reflected component of echo signal and of its components radiated by creeping and peripheral waves from empty spherical shells was considered in [1]. The effect of the echo signal of waves which pass either a fluid or an elastic cylinder was analyzed in [2, 3] by the method of Watson's transformation. Here, the echo signal from a fluid filled spherical shell is investigated, taking into account besides the indicated echo signal components also the effect of waves which propagate partly in the shell, as peripheral waves, and partly as waves that pass through the fluid filler.

A Cauchy system in the linear theory of thin shells of revolution

The differential equations of the linear theory of thin shells of revolution as derived by A. Kalnins in terms of eight fundamental state variables from the Reissner-Meissner equations, and subject to appropriate boundary conditions, are reduced to a system of integro-differential equations of the Riccati type subject to initial conditions. The reduction is meaningful on theoretical grounds since it offers an alternative description of a boundary value problem in terms of a Cauchy system, and valuable from a numerical point of view. on account of the inherent stability of the resulting sequential algorithms. A discussion on rigorous and computational aspects closes the presentation.

A longitudinal crack in a cylindrical shell under internal pressure

The study relates to the strength of a cracked shell. Linear thin shell theory is employed to obtain solutions for internal pressure and uniform circumferential bending. A shallow shell approximation is used, and shown valid providing crack length and shell thickness are small enough in comparison with shell radius. Initial formulation as a boundary value problem is shown equivalent to two coupled singular integral equations. These were solved numerically using a computer. Stresses are found to display singularities as inverse square-root of distance from a crack tip, as with the flat plate. The main results are graphs of normal and bending stress singularity strengths against a curvature/crack-length dimensionless parameter.