Shallow Spherical Caps Research Articles

Nonlinear Axisymmetric Static and Transient Response of Orthotropic Annular Shallow Spherical Caps

ABSTRACT This investigation deals with the nonlinear axisymmetric static and transient response of orthotropic annular shallow spherical caps with a free edge hole and with a rigid central mass, subjected to uniformly distributed load and a central load. The dynamic analogue of Marguerre equations, in terms of normal displacement w and stress function Ψ, are employed. An orthogonal point collocation method is used for spatial discretization and a Newmark-β scheme is used for time integration. Static load, step function load, and sinusoidal pulse load are considered. The influence of annular ratio on the response is studied for isotropic (β = 1) and orthotropic (β = 3) clamped caps, with a rise to thickness ratio of 1.5.

A finite element analysis of the punch stretching of elastic-plastic circular discs.

The Stretching of peripherally clamped, thin, circular discs over rigid, cyrindrical punches is studied by means of an Updated Lagrangian type finite element method. The analysis is based on the elastic-plastic membrane theory ; it accounts for finite strain and displacement and embodies the classical J 2 flow law and the J 2 deformation theory. The geometry of the punch face varied between a hemispherical and a relatively shallow spherical (flat-bottomed) cap. Three models were employed to account for friction at the interface of the punch and disk. These were a no-friction, a full sticking condition and Coulomb friction hypothesis. The disk fails by excessive thinning. The region of strain localization and the rate at which the local neck grows are influenced by the punch geometry and the interfacial frictional conditions. The Coulomb friction and full-sticking models promote the formation of a local neck, and this is further accelerated the flatter the base of the punch. The material property n-value also affects the local neck. The calculations support the hypothesis that the higher the n-value, the better the material distributes the strain. It is also demonstrated that the result is not much affected by the different material models (J 2 F and J 2 D).

Axisymmetric static and dynamic buckling of orthotropic shallow spherical caps with flexible supports

This investigation deals with the static and dynamic axisymmetric buckling of elastic orthotropic thin shallow spherical shells with elastically restrained edge for inplane and rotational displacements. Governing equations in terms of normal displacementw and stress function ψ have been employed. Orthogonal point collocation method is used for spatial discretisation and Newmark-β scheme is used for time-marching. The uniformly distributed static and step function conservative loadings normal to the underformed surface are considered. The present results are in good agreement with the available results. The influence of orthotropicity parameter β and the support stiffness parameters on the static and dynamic buckling loads has been investigated.

The apparent elastic modulus of the juxtarticular subchondral bone of the femoral head.

An experiment was undertaken to obtain approximate values for the intrinsic elastic modulus of subchondral bone. Shallow spherical caps, with uniform and incrementally controlled thickness, were machined from subchondral bone in the weight-bearing regions of 11 fresh-frozen normal femoral head autopsy specimens. Under application of polar point loads, the measured deflections were compared with a corresponding analytical shell solution, thus allowing back-calculation of the apparent modulus. Analogous tests were performed on similarly shaped specimens of stock Plexiglas of known modulus in order to estimate the precision of the testing method. The aggregate results for subchondral bone showed that its intrinsic stiffness correlated inversely with nominal shell thickness, but even the thinnest (1.0 mm thick) of these shells had an apparent modulus (mean = 1.372 GN/m2, SD = 414 MN/m2) well below that generally accepted for "pure" cortical bone (about 14 GN/m2). This stiffness deficit was very likely due to the presence of histologically evident marrow spaces. However, the low apparent modulus values measured in this study may not be fully representative of complex in vivo behavior, because in the testing of excised shells there is no radial compressive stress transfer to underlying cancellous bone.

Axisymmetric static and dynamic buckling of orthotropic shallow spherical cap with circular hole

Nonlinear axisymmetric static and dynamic buckling of clamped isotropic and cylindrically orthotropic elastic cap with central circular hole have been investigated. The governing equations are expressed in terms of normal displacement ω and stress function ψ. The orthogonal point collocation method has been used in the space domain and Newmark-β scheme in the time domain. The cases of shallow cap with free hole and with a hole plugged by rigid central mass are considered. Analysis has been carried out for uniformly distributed load and a ring load at the hole. Dynamic load is taken as a step function load. Detailed new results for static and dynamic buckling loads have been presented for the isotropic and orthotropic cases.

Geometrical aspects of acoustic radiation from a shallow spherical cap

Geometrical properties are used to obtain the acoustic field from harmonic and pulse excitation of a shallow spherical cap in a baffle. The axial field from harmonic oscillation is obtained by observing the equivalence with the field from oscillation of a suitably chosen full sphere. This defines a convenient Legendre polynomial expansion of the velocity on the cap and leads to an axial field in the form of a series of products of spherical Bessel and Hankel functions. Each term corresponds to a Bessel function variation of the potential in the focal plane. The full solution for a uniform pulse excitation is obtained from the geometry of intersecting spheres. The simplicity and finite duration of the pulse solution suggests its use as convolution kernel for general time dependence of the excitation.

Dynamic Snap-Through of Shallow Arches and Spherical Caps

Dynamic Snap-Through of Shallow Arches and Spherical Caps

Dynamic buckling of isotropic and orthotropic shallow spherical cap with circular hole

The axisymmetric dynamic behaviour of clamped isotropic and orthotropic spherical cap with central circular hole under a uniform step pressure of infinite duration is investigated. The critical loads are determined by using the rapidly converging Chebyshev series in a space domain and a Houbolt numerical integration scheme in time domain. Three different cases of openings are investigated and detailed numerical results have been obtained for the isotropic case and two different cases of orthotrophy, which may be useful for design engineers.

Dynamic buckling of shells: Evaluation of various methods

The problem of dynamic stability is substantially more complex than the buckling analysis of a shell subjected to static loads. Even at this date suitable criteria for dynamic buckling of shells, which are both logically sound and practically applicable, are not easily available. Thus, a variety of analyses are available to the user, encompassing various degrees of complexity, and involving a range of simplifying assumptions. The purpose of this paper is to compare and evaluate some of these solutions by applying them to a specific problem. A shallow spherical cap, subjected to an axisymmetric, uniform-pressure, step loading, is used as the structural example. The predictions, by various methods, of the dynamic buckling of this shell into unsymmetric modes, are then investigated and compared. The approximate methods used by Akkas are compared to the more rigorous and general solutions of the KSHEL, STARS, DYNASOR, and SATANS computer programs, and the various simplifying assumptions utilized are evaluated. Also included in the comparisons, are the predictions of the relatively simple “dynamic buckling model” approach of Budiansky and Hutchinson. The approaches utilized by the more complex programs [KSHEL (spatial integration, modal superposition, perturbation approach), DYNASOR (finite elements, time integration of non-linear dynamic equilibrium equations), SATANS (finite differences, pseudo load method, time integration), STARS (spatial and time integration, non-linear equilibrium or perturbation approaches)] will in turn be compared in terms of accuracy, idealization complexity, ease of use, and user expertise and experience required for analysis. The comparisons show that the more approximate methods underpredict the dynamic buckling loads for this problem. In addition, some basic assumptions of the simpler solutions are found to be invalid.

Direct dynamic analysis of shells of revolution using high-precision finite elements

For the transient dynamic analysis of structural systems, the direct numerical integration of the equations of motion may be regarded as an alternative to the mode superposition method for linear problems and a necessity for nonlinear problems. When compared to a modal superposition solution, the direct integration approach is attractive in that the eigenvalue problem is avoided. Depending on the amount of information required from the dynamic analysis, e.g. frequencies, frequencies and mode shapes, and/or a complete time history, a direct integration scheme may prove to be more efficient than a modal superposition solution for some linear problems as well. The purpose of this study is to develop and demonstrate a direct integration algorithm which is compatible with an existing high-precision rotational shell finite element. Excellent comparative efficiency for static problems was achieved with this element by the incorporation of the exact geometry, the utilization of high-order interpolation polynomials and, yet, the retention of only a minimum number of nodal variables in the global formulation. Likewise, accurate and efficient results for the free vibration analysis of rotational shells were facilitated by the inclusion of a consistent mass matrix and the utilization of a rationally justified kinematic condensation procedure. The approach to the direct integration stage is strongly tempered by the established characteristics of this element which enable a given shell to be modeled accurately in the spatial domain with a comparatively coarse discretization. The equations of motion for a shell of revolution under conservative loading are derived from Hamilton's variational principle and specialized for the discretization of a rotational shell into curved shell elements. Degrees of freedom in excess of those required to establish minimum (C°) continuity at the nodal circles are eliminated through kinematic condensation. Some guidance as to the proper order of the polynominal approximations for a dynamic analysis is provided by earlier free vibration studies. Whereas the condensation is exact for static problems, it is only approximate for dynamic response and it was found that the accuracy of the eigenvalues obtained for the reduced problem decreases with increasing order of the condensed functions. This tendency is counted by the desirability of using sufficiently high-order interpolations so as to permit accurate stress computations, both at the nodal circles and between nodes since a coarse discretization is necessary to realize maximum efficiency. It was found that cubic polynomials were generally satisfactory from both standpoints, except in localized regions of high stress gradients where quintic polynomials were employed. The finite element discretizations for the direct integration studies were selected on this basis. For the high-precision finite element at hand, the efficiencies achieved in the space domain are demonstrated by the ability to achieve precise solutions with relatively coarse discretization patterns. The resulting comparatively large elements are not subject to accurate representation by diagonal mass matrices so that an implicit, consistent mass approach is followed. Efficiency in the time domain as well rests on the successful modeling of rotational shells subject to dynamic loading using coarse discretitations in space and large increments in time. Computational efficiency and accuracy are demonstrated for various problems documented in the literature, including a shallow spherical cap subject to a step pulse and a hyperboloidal shell under a simulated dynamic wind pressure.

On the buckling of shallow spherical caps subjected to uniform external pressure

On the buckling of shallow spherical caps subjected to uniform external pressure

Nonlinear Axisymmetric Buckled States of Shallow Spherical Caps

Nonlinear axisymmetric buckled states of a thin, elastic spherical cap satisfying edge conditions for which the spherical shape remains one possible state of equilibrium when the cap is subjected to a uniform radial load are studied by constructive methods. The approach used is based upon the Lyapunov–Schmidt method and shows that buckled states exist near every eigenvalue of the linearized problem.

Snap-through buckling of eccentrically stiffened shallow spherical caps

The problem of snap-through buckling of a clamped, eccentrically stiffened shallow spherical cap is considered under quasi-statically applied uniform pressure and a special case of dynamically applied uniform pressure. This dynamic case is the constant load infinite duration case (step time-function) and it represents an extreme case of blast loading-large decay time, small decay rate.The analysis is based on the nonlinear shallow shell equations under the assumption of axisymmetric deformations and linear stress-strain laws. The eccentric stiff eners are disposed orthogonally along directions of principal curvature in such a way that the smeared mass, and extensional and flexural stiffnesses are constant. The stiffeners are also taken to be one-sided with constant eccentricity, and the stiffener-shell connection is assumed to be monolithic.The method developed in an earlier paper is employed. In this method, critical pressures are associated with characteristics of the total potential surface in the configuration space of the generalized coordinates.In addition, buckling of the complete thin eccentrically stiffened spherical shell under uniform quasi-statically applied pressure is considered, and these results are used to check the numerical answers. The complete spherical shell is stiffened in the same manner as the shallow cap.The results are presented in graphical form as load parameter vs initial rise parameter. Geometric configurations corresponding to isotropic, lightly stiffened, moderately stiffened and heavily stiffened geometries are considered. By lightly stiffened geometry one means that most of the extensional stiffness is provided by the thin shell. A computer program was written to solve for critical pressures. The Georgia Tech Univac 1108 high speed digital computer was used for this purpose.

有限要素モード重畳法による回転殻の動的非線型解析 : その 2 : 浅い周辺固定球殻の動的挫屈解析

The object of the present paper is to formulate the the efficient method for the nonlinear dynamic analyses of rotational shells by the combined use of the finite element method and the mode superposition method. The effects of the geometrical imperfections are taken into consideration. In the Part 2, a shallow spherical cap having a clamped edge subjected to the lateral pressure is analysed for λ=5, 6, 7, 8 and 9 and the results are compared to be in fair agreement with those given by other authers. The shell subjected to the simultaneous dynamical action of a uniform pressure and a concentrated load on the apex with various ratio is investigated and the results show the large amount of reducction of the dynamic buckling load due to a slight concentrated load. The interaction formula for the dynamic buckling of the lateral pressure and the concetrated load is discussed.

Non‐linear dynamic analysis of axisymmetric shells

AbstractIncremental equations of motion are derived from a Lagrangian variational formulation for the large displacement elastic‐plastic and elastic‐viscoplastic dynamic analysis of deformable bodies. The material constitutive behaviour is described in terms of the symmetric Piola–Kirchhoff stress and Lagrangian strain tensors. Degenerate isoparametric elements, permitting relaxation of the Kirchhoff–Love hypothesis, are used in a finite element formulation specialized for the analysis of shells of revolution subjected to axisymmetric loading. The linearized incremental equations of motion are solved using direct integration procedures, with added accuracy obtained from application of equilibrium correction at each step. The effectiveness of the numerical techniques is illustrated by the dynamic response analyses carried out on a shallow spherical cap subjected to uniform external step pressure loadings.

Buckling of Shallow Spherical Caps and Truncated Hemispheres

B of shallow spherical shells and truncated hemispheres subjected to nearly axisymmetric loads is studied using a digital computer program for the geometrically nonlinear analysis of arbitrarily loaded shells. Small asymmetric perturbations in the load are used to disclose the minimum bifurcation buckling loads and to illuminate the imperfection-sensitivity of the shells. The pressure-loaded spherical cap was selected because previous results had indicated that it exhibits all of the traits desired: its axisymmetric behavior is snap-buckling, bifurcation loads exist below the axisymmetric snap-buckling loads, and it is sensitive to imperfections. The truncated hemisphere under axial tension was selected because the experimental buckling loads are approximately one-half of the theoretical bifurcation loads. Imperfection-sensitivity was suspected to cause the discrepancy.

Buckling of Reinforced Concrete Shells

When the designers of reinforced concrete shell structures consider buckling, they have to apply data derived from elastic analyses or elastic model tests to a nonelastic material. The research described was designed to reduce the uncertainties involved by investigating the effect of tension cracking and nonlinearity on the buckling of concrete shells. The information desired was obtained experimentally by comparing the buckling of reinforced mortar models to geometrically similar models fabricated from plastics. The program included tests on 4 long-span single-barrel cylindrical roof shells, on 5 shallow spherical caps, and on 11 cylindrical panels. The results from the cylindrical shells and spherical caps are summarized, while the cylindrical panel tests are described in some detail. The complete curve for interaction of axial and circumferential buckling was established for the cylindrical panels. Conclusions are made about the use of appropriate models in buckling studies, and on the effects of tension cracking and nonlinearity in concrete shells.

A New Method for Computing Axisymmetric Buckling of Spherical Caps

A modification of Newton’s method is applied to the solution of the nonlinear differential equations for clamped, shallow spherical caps under uniform pressure. The linear form of Newton’s method or quasi-linearization breaks down at limit points of the differential equations. A simplified “quadratic form” is derived in the paper and shown to be satisfactory for continuing the solution past the limit point and into the postbuckling region. Results for the buckling pressures defined by the limit points agree with published results for perfect caps. New results are presented for imperfect caps that check experiment.

Buckling of thin spherical shells

This paper deals with the buckling of a shallow spherical cap subjected to uniform edge moment and a clamped deep spherical shell under uniform pressure. The first problem is formulated in integral equations which are solved by an iterative procedure. The buckling moments are determined for a wide range of the shell geometrical parameter. The second problem is based on the concept that the highly deformed region around the apex is treated as a shallow spherical cap elastically supported by the rest of the shell. The stability of a thin sphere is treated as a special case. The results obtained in both problems are compared with existing solutions.

Buckling and postbuckling behavior of spherical caps under axisymmetric load

Clamped shallow spherical cap elastic buckling and initial postbuckling behavior under uniformly distributed load over circular region centered at apex