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Euromech-Colloquium Nr. 165 The shell-theory development has changed its emphasis during the last two decades. Nonlinear problems have become its main motive. But the analysis was until recently predominantly devoted to shells designed for strength and stiffness. Nonlinearity is here relevant to buckling, to intensively vary able stress states. These are (with exception of some limit cases) covered by the quasi-shallow shell theory. The emphasis of the nonlinear analysis begins to shift further - to shells which are designed for and actually capable of large elastic displacements. These shells, used in industry for over a century, have been recently termedj1exible shells. The European Mechanics Colloquium 165. was concerned with the theory of elastic shells in connection with its applications to these shells. The Colloquium was intended to discuss: 1. The formulations of the nonlinear shell theory, different in the generality of kine­ matic hypothesis, and in the choice of dependent variables. 2. The specialization of the shell theory for the class of shells and the respective elastic stress states assuring flexibility. 3. Possibilities to deal with the complications of the buckling analysis of flexible shells, caused by the precritial perturbations of their shape and stress state. 4. Methods of solution appropriate for the nonlinear flexible-shell problems. 5. Applications of the theory. There were 71 participants the sessions were presided over (in that order) by E. Reissner, J. G. Simmonds, W. T. Koiter, R. C. Tennyson, F. A. Emmerling, E. Rarnm, E. L. Axelrad.

Inhaltsverzeichnis

Frontmatter

The Nonlinear Thermodynamical Theory of Shells: Descent from 3-Dimensions without Thickness Expansions

Abstract
A glance at the current engineering literature on shells reveals two strong, interacting trends: the inexorable rise and spread of the finite element method and the pressures of economics that are demanding light, efficient structures for automobiles, air- and spacecraft.
James G. Simmonds

On the Derivation of the Differential Equations of Linear Shallow Shell Theory

Abstract
Given that derivations of linear shallow shell theory depend on (1) geometric shallowness assumptions and (2) static shallowness assumptions it is the purpose of the present discussion to verify the less obvious validity of the second set of assumptions through the application of an iterative procedure. The result of this justification includes the observation that additional steps of the iterative procedure will be meaningful only in conjunction with simultaneous iterative improvements in regard to the geometric shallowness assumption.
E. Reissner

Geometrically Nonlinear Theory and Incremental Analysis of Thin Shells

Abstract
Within the last years, a wide variety of complicated geometrically and physically nonlinear shell problems was approximately solved using mainly the Finite Element Method (FEM). Complex engineering structures need large systems of nonlinear equations for incremental load paths. Own investigations, using the SHEBA-element with 63 DOF’s /4/, /8/, have shown that the limit capacity even of large and fast scalar computers (in our case a Cyber 76/73 configuration) may be exceeded in the case of critical and postcritical calculations.
E. Stein, W. Wagner, K.-H. Lambertz

Flexible Shells

Summary
The nonlinear intrinsic equations of elastic thin shells are simplified by the assumption widely tested in the linear theory as the basis of the Novozhilov’s complex equations. The obtained equations directly specialize themselves for the class of shells designed for large elastic displacements. Another aim of this contribution is to further the vector form of nonlinear shell-theory. This form combines the graphic staticgeometric duality with the short-cut way between the invariant and physical-component presentation.
E. L. Axelrad

Seismic Behavior of Liquid Filled Shells

Summary
The elastic behavior of vertical-axis cylindrical liquid storage tanks subject to horizontal seismically induced motions of the base is considered. Tanks without as well as with a dome are examined. The effect of initial geometric imperfections is studied, as is the influence of circumferential reinforcing rings on dynamic behavior of the shell-liquid system.
W. A. Nash, S. H. Shaaban, L. Watawala, S. C. Lee

On Geometrically Non-Linear Theories for Thin Elastic Shells

Summary
The present paper deals with geometrically non-linear first approximation Kirchhoff-Love type theories for thin elastic shells undergoing small strains accompanied by moderate, large or unrestricted rotations. All theories will be given in an entirely Lagrangian description. We shall start our considerations with a general theory valid for small strains and arbitrary, unrestricted rotations. Then, this general theory will be simplified for shell problems in which the shell material elements undergo large rotations according to the classification scheme given below. Three variants will be derived which admit large rotations about tangents to the shell middle surface and either large, moderate or small rotations about the normal. Finally, the general shell equations will be simplified for shells undergoing moderate rotations about tangents to the shell middle surface and either moderate or small rotations about the normal. All theories presented here are derivable from variational principles.
R. Schmidt

Buckling and Post-Buckling of Shells for Unrestricted and Moderate Rotations

Summary
In the first part of this paper a Lagrangean nonlinear theory of thin elastic shells for unrestricted rotations is considered, where the boundary conditions are non-rational functions of the shell deformations. Using energy considerations the equations of critical equilibrium are derived, which define the general eigenvalue problem of shell buckling. Furthermore post-buckling equations are obtained by application of the static perturbation technique.
If the rotations of the shell elements can be restricted to be moderate, essential simplifications in the prebuckling, buckling and post-buckling equations are achieved, which is shown in the second part.
H. Stumpf

On Entirely Lagrangian Displacemental Form of Non-Linear Shell Equations

Summary
Equations of equilibrium and corresponding four geometric and static boundary conditions are derived for an entirely Lagrangian non-linear theory of thin shells. In case of a linearly elastic material and conservative external forces all shell relations are exactly derivable as stationarity conditions of the Hu — Washizu free functional. The set of equations is consistently reduced in the case of the geometrically non-linear theory of thin elastic shells undergoing large/small rotations.
W. Pietraszkiewicz

Shallow Caps with a Localized Pressure Distribution Centered at the Apex

Abstract
Under favourable loading conditions, dome-shaped thin elastic shells of revolution are known to exhibit a predominantly inextensional bending deformation in the form of a finite axisymmetric dimple centered at the pole. For example, it has been shown in [1,2] that polar dimpling is possible when a spherical shell is subject to an axisymmetric normal pressure distribution which is directed inward near a pole and outward in an adjacent region(3). To a good approximation, the dimple base radius, which characterizes the location of the dimple base and therefore the dimple size, was shown to depend on the external loading in a simple way. To bring out the essential idea behind the asymptotic method for constructing the simple solution, results for a spherical cap with a clamped edge were first presented in [1] for a quadratically varying pressure distribution along the shell meridian. Analogous and more general results were reported for a complete spherical shell in [2] for a meridionally sinusoidal pressure distribution.
Frederic Y. M. Wan

On the Buckling and Postbuckling of Spherical Shells

Summary
For the axisymmetric buckling problem of a complete spherical shell we are able to give a complete imperfection sensitivity analysis for the most complicated case to occur, namely that of a double eigenvalue, if we restrict our analysis to nonlinear terms of quadratic order, which is correct mathematically as the bifurcation system turns out to be two determinate. As we make a classical bifurcation analysis our results are strictly local and therefore are of restricted practical importance in two respects. Firstly we do not take care of the fact that we have closely spaced eigenvalues. We comment on this in chapter 7. Secondly we are not able to show that the experimentally observed single dimple solution (Fig. 8) is one of the stable solutions found from our bifurcation equations up to terms of third order.
R. Scheidl, H. Troger

Implicit Relaxation Applied to Postbuckling Analysis of Cylindrical Shells

Summary
The paper deals with a viscous approach, where an artificial damping is used in order to arrive at points of equilibrium in the pre- or postbuckling range of shell structures.
The artificial damping is based on a creep type procedure with the time playing a purely ficticious role. The required numerical stability is obtained by using an implicit operator in time and a variable viscosity and time step. The viscosity and the time step are combined as a variable damping parameter which depends on the unbalanced forces of the structure.
The procedure allows to find states of static equilibrium with the load applied all at once. Incrementation is not necessary unless the path has to be traced in small steps. With a perturbation different branches of the nonlinear solution can be obtained.
Examples are given for elastic thin walled cylindrical shells in the postbuckling range. The used finite elements are based on a functional with displacements, axial forces and bending moments as nodal variables.
B. Kröplin

Nonlinear Bending of Curved Tubes

Abstract
The determination of the nonlinear deformation and of the collapse load of elastic curved tubes subjected to bending loads is investigated. The precritical deformation of the tubes is determined on the basis of the semi-membrane theory. The stability analysis is done with the aid of the hypothesis of local buckling. The collapse loads are compared with the results of a bending experiment with Hostaphan-tubes.
F. A. Emmerling

Nonlinear Finite Element Analysis of Shells under Pressure Loads Using Degenerated Elements

Abstract
The paper briefly describes the concept of degeneration for the derivation of shell elements. Then the procedure to describe arbitrarily large displacements and rotations is explained. The problem of displacement dependent pressure loads is discussed in detail. A clear classification of the load definition allows to identify when a load is conservative and when it is not. Some remarks to the solution procedures in nonlinear analyses are added and selected numerical examples are given.
Ekkehard Ramm

Contribution on the Numerical Analysis of Thin Shell Problems

Summary
This contribution reviews our results on the numerical analysis of thin shell problems. Essentially, it includes theorems of existence (and uniqueness) for a solution in the linear and non linear static case as well as the study of the free vibration problem; next some results from the mathematical study of the approximation are given; the implementation of these methods is illustrated by numerical simulation of arch dam problems. Finally some open problems are pointed out.
M. Bernadou

A Total Lagrangian Finite Element Formulation for the Geometrically Nonlinear Analysis of Shells

Abstract
The analysis of structures subjected to large displacements by means of the finite element method has attracted the attention of many researchers in recent years and different publications have been reported in the literature [1]–[8].
J. Oliver, E. Oñate

Large Deformations of Elastic Conical Shells

Summary
Axisymmetric conical shells under axial forces are investigated. A geometrically nonlinear approach leads to the REISSNER-MEISSNER equations, which allow the calculation of large deformations. These two nonlinear second order equations have been integrated by a matrix method, suggested by E.L. AXELRAD. Another effective solution, suitable for a small computer, uses the RUNGE-KUTTA-integration combined with an iteration program for the unknown boundary values. Useful results for the practical design of conical springs have been published, which give slight corrections to the famous ALMEN-LASZLO-paper of 1936. While conical springs are rather flat and thick, the presented theory can be used for steep and thin shells. Then the “spring-characteristics”, the force-deflection curves, become very complicated and their stability must be discussed. A simple criterion of stability is derived here out of the DIRICHLET definition. Presented as a problem of catastrophe theory, interesting curves in the parameter plane are obtained.
W. Hübner

The Mechanics of Drape

Summary
The special difficulties of modelling the complex deformations of textile fabrics are discussed in the context of flexible shell theory, and the particular requirements of analysis methods in this area are described. Examples of previous attempts to model particular problems of complex fabric deformations are used to illustrate the advantages and disadvantages of the methods used. Finally, an outline is given of a computational approach which is under development at Leeds at the moment.
D. W. Lloyd
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