Stability analysis of dynamic thin shells

Donnell’s thin shell theory and basic equations based on the wave propagation method discussed in detail here, is used to investigate the natural frequencies of thin finite length circular cylindrical shells under various boundary conditions. Mode shapes are drawn to explain the circumferential mode numbern and axial mode numberm, and the natural frequencies are calculated numerically and compared with those of FEM (finite element method) to confirm the reliability of the analytical solution. The effects of relevant parameters on natural frequencies are discussed thoroughly. It is shown that for long thin shells the method is simple, accurate and effective.

A generalized displacement method has been previously presented for the analysis of thin plate‐shell structures with the use of bilinear 4‐node isoparametric shell elements. Following this approach, a procedure for the geometrically non‐linear analysis of thin plates and shells based on both updated and total Lagrangian formulations is developed. The results of some numerical examples are presented to show the versatility and effectiveness of the method.

A novel approximate theoretical solution is presented in this paper for the boundary disturbance of thin elastic shells of revolution with a nearly spherical middle surface and constant thickness. The indicated solution is a procedure of successive corrections, which was originally used by S. D. Poisson in handling non‐linear differential equations and introduced into thin shell analysis by E. F. Burmistrov.

We classify all fundamental electrically charged thin shells in general relativity, i.e., static spherically symmetric perfect fluid thin shells with a Minkowski spacetime interior and a Reissner-Nordstr\"om spacetime exterior, characterized by the spacetime mass and electric charge. The fundamental shell can exist in three states, nonextremal, extremal, and overcharged. The nonextremal state allows the shell to be located such that its radius can be outside its own gravitational radius, or can be inside its own Cauchy radius. The extremal state allows the shell to be located such that its radius can be outside its own gravitational radius, or can be inside it. The overcharged state allows the shell to be located anywhere. There is a further division, one has to specify the orientation of the shell, i.e., whether the normal out of the shell points toward increasing or decreasing radii. There is still a subdivision in the extremal state when the shell is at the gravitational radius, in that the shell can approach it from above or from below. The shell is assumed to be composed of an electrically charged perfect fluid, and the energy conditions are tested. Carter-Penrose diagrams are drawn for the shell spacetimes. There are fourteen cases in the classification of the fundamental shells, namely, nonextremal star shells, nonextremal tension shell black holes, nonextremal tension shell regular and nonregular black holes, nonextremal compact shell naked singularities, Majumdar-Papapetrou star shells, extremal tension shell singularities, extremal tension shell regular and nonregular black holes, Majumdar-Papapetrou compact shell naked singularities, Majumdar-Papapetrou shell quasiblack holes, extremal null shell quasinonblack holes, extremal null shell singularities, Majumdar-Papapetrou null shell singularities, overcharged star shells, and overcharged compact shell naked singularities.

An eighteen‐node, three‐dimensional, solid element with 54 degrees of freedom is presented for the finite element analysis of thin plates and shells. The element is based on the Hellinger‐Reissner principle with independent strain. The assumed independent strain is divided into higher and lower terms. The stiffness matrix associated with the higher order independent strain plays the role of stabilization matrix. A modified stress‐strain relation decoupling inplane and normal strain is used to incorporate thin shell behaviour. Numerical results demonstrate that, with a properly chosen set of assumed strain, this element is effectively free of locking even for very thin plates and shells.

This paper describes a new finite element scheme for the analysis of instability phenomena of arbitrary thin shells. A computationally efficient procedure is proposed for calculating the non‐linear stiffness and tangential stiffness matrices for a doubly‐curved quadrilateral element defined by co‐ordinate lines. The essential feature is the explicit addition of the non‐linear terms into the rigid‐body motion of the element. Thus the non‐linear and tangential element stiffness matrices can easily be generated by transforming the generalized element stiffness matrix for linear analysis, and the non‐linear terms of these matrices are separated into a number of component terms multiplied by the rigid‐body rotations. These component terms can be stored permanently and used to calculate efficiently the non‐linear and tangential stiffness matrices at each iteration. Illustrative examples are presented which confirm the validity of the present approach in the analysis of instability phenomena of thin plates and shells.

We construct new regular black hole solutions by matching the de Sitter solution and the Reissner-Nordstr\om solution with a timelike thin shell. The thin shell is assumed to have mass but no pressure and obeys an equation of motion derived from Israel's junction conditions. By investigating the equation of motion for the shell, we obtain stationary solutions of charged regular black holes and examine stability of the solutions. Stationary solutions are found in limited ranges of $0.87L\ensuremath{\le}m\ensuremath{\le}1.99L$, and they are stable against small radial displacement of the shell with fixed values of $m$, $M$, and $Q$ if $Mg0$, where $L$ is the de Sitter horizon radius, $m$ the black hole mass, $M$ the proper mass of the shell, and $Q$ the black hole charge. All the solutions obtained are highly charged in the sense of $Q/mg2\sqrt{3}\ensuremath{\approx}0.866$. By taking the massless limit of the shell in the present regular black hole solutions, we obtain the charged regular black hole with a massless shell obtained by Lemos and Zanchin and investigate stability of the solutions. It is found that Lemos and Zanchin's regular black hole solutions given by the massless limit of the present regular black hole solutions permit stable solutions, which are obtained by the limit of $M\ensuremath{\rightarrow}0$.

A refined non-conforming triangular plate/shell element for linear and geometrically nonlinear analysis of plates and shells is developed in this paper based on the refined non-conforming element method (RNEM). A conforming triangle membrane element with drilling degrees of freedom in Cartesian coordinates and the refined non-conforming triangular plate-bending element RT9, in which Kirchhoff kinematic assumption was adopted, are used to construct the present element. The displacement continuity condition along the interelement boundary is satisfied in an average sense for plate analysis, and the coupled displacement continuity requirement at the interelement is satisfied in an average sense, thereby improving the performance of the element for shell analysis. Selectively reduced integration with stabilization scheme is employed in this paper to avoid membrane locking. Numerical examples demonstrate that the present element behaves quite satisfactorily either for the linear analysis of plate bending problems and plane problems or for the geometrically nonlinear analysis of thin plates and shells with large displacement, moderate rotation but small strain.

Exploring thin-shell dynamics in regular charged black hole through T-duality

The dual reciprocity method (DRM) and the radial integration method (RIM) are techniques used in BEM to transform domain integrals into boundary integrals. They are suitable for boundary element formulations where a complete fundamental solution is either unavailable or very complex, because in these cases one or more terms can remain as domain integrals in order to use a simpler fundamental solution. Thus, a large number of problems can be solved with the knowledge of a few number of fundamental solutions and additional terms as inertia or non-linear effects, can be treated as body forces and taken to the boundary. In both methods, the remaining terms are approximated through a finite series expansion involving proposed approximating functions and coefficients to be determined. This expansion is substituted in the generated domain integrals that are, subsequently, tr ansformed into boundary integrals. In this paper, a boundary element formulation for thin shallow shells with no domain discretization is presented. The domain integrals due to the curvature of the shells are tran sformed into boundary integrals using the radial integration method. Two approximation functions are used in this work. The first is the radial ba sis function that has been used extensively in the DRM given by:

Computational generation of free-form shells in architectural design and civil engineering

Shells are used in many modern structures because of their efficiency and economy, their ability to retain their form, and because of other features stemming from their reaction to certain loads. Their shape allows certain membrane stress systems to develop parallel to their tangential plane and become prime carriers of the deformation. Indeed the analysis of many thin shells is solely based on the membrane theory of shells which neglects their bending rigidity. On the other hand, bending becomes important in the presence of rapidly changing loads (e.g. concentrated loads, line loads etc.), near edge constraints, near discontinuities in the shell geometry and in nonlinear deformations. The development of a general membrane and bending theory as well as related numerical implementations are subjects of intense research efforts which aim at providing deeper understanding of the mechanics of load carrying. The literature on this subject has been growing at a rapid pace over the last decades.

FREE SURFACE EFFECTS ON THE DYNAMICS OF CYLINDRICAL SHELLS PARTIALLY FILLED WITH LIQUID

In this paper, a new solution method is proposed for determining the natural frequency of a given mode for a finite-length circular cylindrical thin shell with a circumferential part-through crack. The governing equation of the cracked cylindrical shell is derived by integrating the line-spring model with the classical thin shell theory. The proposed method calculates the natural frequency from an initial trial to satisfy both the governing equations and appropriate boundary conditions through an optimization process. The initial trial is proposed to satisfy the governing equations by using the beam modal function to determine the modal wavenumbers and mode shapes of cylindrical shells in the axial direction, assuming the flexural mode shapes of cylindrical shells in the axial direction to be of the same form as that of a flexural vibration beam with the same boundary conditions. Four representative sets of boundary conditions are considered: simply supported (SS-SS), clamped-clamped (C-C), clamped-simply supported (C-SS), and clamped-free (C-F). Compared with the finite element (FE) method, the proposed solution method is verified to provide an accurate and efficient way to calculate the dynamic characteristics of both intact and cracked cylindrical shells.

At present the Semiloof element is probably one of the most efficient available for the solution of thin shells of arbitrary geometry. Experience in static situations indicates that accurate results can be obtained for non‐trivial geometric and loading configurations with relatively coarse meshes. Variable thickness shells or discontinuous thicknesses can be accommodated and no difficulties are encountered in modelling sharp corners or multiple junctions in structures. The element is basically non‐conforming but some measure of C1 continuity is provided by the introduction of normal rotation variables at Loof nodes on the element boundary. This paper examines the element behaviour when applied to elastic instability and vibration situations. An eigenvalue solution scheme based on Sturm sequences is presented which does not require the usual elimination of a percentage of the total nodal variables by static condensation. Finally the method is assessed by the solution of several numerical examples.