## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

In this study, the snap-through buckling behaviour of axisymmetric shells, subjected to axisymmetric horizontal peripheral load or displacement for various shell parameters and various boundary conditions, is investigated. Results obtained seem not to have been reported previously. An application of peripheral displacement type of loading is seen in metal-ceramic composite transducers developed by sandwiching a piezoelectric (PZT) ceramic between two metal end caps which serve as mechanical transformers for converting and amplifying the lateral displacement of the ceramic into an axial motion normal to the metal cap. In our numerical search, we have observed that snap-through and snap-back buckling is possible for shallow spherical caps for a very narrow range of the shell parameter used. When a hole is opened around the apex of the cap, buckling is possible for a larger range of the shell parameter. Obtaining the displacement amplification and the blocking or generative force for various material and geometric properties is necessary for the possible application of the findings in transducer design. The numerical results are presented in graphical forms.

To read the full-text of this research,

you can request a copy directly from the authors.

... The goal of this paper is to answer the question whether snap-through from the initial unstrained position to the strained position on the other side of the base plane is possible when the shallow shell is under inplane edge tension. There are several extensions in this paper compared to the study in Akkas and Odeh (2001) . First of all, we employ the von Karman's plate model and use a Galerkin's method to study the deformation of the shallow shell. ...

... The reason we study the case of simply-supported boundary condition is to compare our result with that presented in Akkas and Odeh (2001). In all these cases, both w * and / * are required to be finite at the center r * = 0. Eqs. ...

... In other words, the locus of P 1 s is always on the positive side. These are in contrast with the conclusions reported in Akkas and Odeh (2001) in which the authors claimed that snap-through is possible and the apex is moved to the other side of the base plane even before the so-called snap-through occurs. Although the shell discussed in Akkas and Odeh (2001) is simply supported and the initial shape is spherical, the main features should be the same. ...

In this paper we study the deformation and stability of a shallow shell under uniform edge tension, both theoretically and experimentally. Von Karman’s plate model is adopted to formulate the equations of motion. For a shell with axisymmetrical initial shape, the equilibrium positions can be classified into axisymmetrical and unsymmetrical solutions. While there may exist both stable and unstable axisymmetrical solutions, all the unsymmetrical solutions are unstable. Since the unsymmetrical solutions will not affect the stability of the axisymmetrical solutions, it is concluded that for quasi-static analysis, there is no need to include unsymmetrical assumed modes in the calculation. If the shell is initially in the unstrained configuration, it will only be flattened smoothly when the edge tension is applied. No snap-through buckling is possible in this case. On the other hand, if the shell is initially in the strained position, it will be snapped back to the stable position on the other side of the base plane when the edge tension reaches a critical value. Experiment is conducted on several free brass shells of different initial heights to verify the theoretical predictions. Generally speaking, for the range of initial height H

... Shallow arches have been investigated for their interesting nonlinear behavior due to the effect of curvature [1] and bistability [2]. For microsystem applications, electrostatic actuation has been widely used in micro-electromechanical systems (MEMS) such as micromechanical memory [3], resonant microsensors [4], or other micro-and nano-devices. ...

In this work, the nonlinear dynamics of a microbeam shallow arch actuated through an out-of-plane electrostatic force arrangement is investigated. A reduced order model is developed to analyze the static, free vibration, and nonlinear dynamic response of the microstructure under different direct current and alternating current load conditions. A numerical investigation is conducted by comparing the response of the arch near primary and secondary resonances using a nonparallel plates actuation scheme where the arch itself forms a moving electrode. The results show that the nonparallel excitation can be efficient for primary and secondary resonances excitation. Moreover, unlike the classical parallel plates method, where the structure is vulnerable to the dynamic pull-in instability, this nonparallel excitation arrangement can provide large amplitude motion while protecting the structure from the so-called static and dynamic pull-in instabilities. In addition to primary resonance, secondary resonances are demonstrated at twice and one-half the primary resonance frequency. The ability to actuate primary and/or secondary resonances without reaching the dynamic pull-in instability can serve various applications where large strokes increase their performance, such as for resonator-based sensitive mass sensors.

... Bistability of a structural element, enabling transition between two coexisting stable configurations, is an attractive feature, which can be utilized for various applications in the realm of micro-and nanoelectromechanical systems (MEMS/NEMS) ( Hu and Burgueo (2015) . Examples range from energy harvesters ( Arrieta, Hagedorn, Erturk, & Inman, 2010;Harne & Wang, 2013;Zhu & Zu, 2013 ), to transducers ( Akkas & Odeh, 2001 ), sensors ( Harne & Wang, 2014 ), switches ( Intaraprasonk & Fan, 2011 ), non-volatile memories ( Charlot, Sun, Yamashita, Fujita, & Toshiyoshi, 2008 ) and micro-pumps ( Liu, 2010;Machauf, Nemirovsky, & Dinnar, 2005;Nisar, Afzulpurkar, Mahaisavariya, & Tuantranont, 2008;Pan, Ng, Liu, Lam, & Jiang, 2001;Tavakol & Holmes, 2016;Wagner, Quenzer, Hoerschelmann, Lisec, & Juerss, 1996 ). These structures are often actuated using electrostatic forces, allowing design of low power consumption, short response time devices, conveniently compatible with the current fabrication processes and driving electronic circuity. ...

The criterion defining the geometric parameters guaranteeing bistable behavior of electrostatically actuated curved axisymmetric circular plate is established. The usage of Berger’s approximation for von-Kármán nonlinear plates, combined with single degree of freedom (DOF) reduced order (RO) modeling, allowed derivation of a simple semi-analytical bistability criterion, obtained in the form of an implicit algebraic equation in terms of critical deflection and plate geometric parameters. The criterion is verified by direct numerical solutions, combined with the arc-length method. Case studies are presented, illustrating the implementation of the suggested criterion as a useful tool for the early design stage for MEMS/NEMS devices.

... Traditionally, the buckling phenomenon was considered as a failure, which should be avoided. However, progress in the design of devices where snap-through behavior could be beneficial ( Hu and Burgueño, 2015 ) stimulated renewed interest in the mechanics of bistable beams ( Zhu and Zu, 2013;Pi and Bradford, 2013 ) and shells ( Akkas and Odeh, 2001;Sabzikar Boroujerdy and M.R. Eslami, 2014 ). In the realm of micro-and nanoelectrome- et al. (2004) , Nayfeh et al. (20 05) and Vogl and Nayfeh (20 03) , the pull-in and natural frequencies of circular and rectangular initially flat microplates were analyzed. ...

Micro- and nanolectromechanical systems (MEMS/NEMS) incorporating two-dimensional structural elements such as plates and shells attracted significant interest in recent years. These structures demonstrate rich electromechanical behavior and could be advantageous in applications. In this work, we explore implementation of two models describing axisymmetric behavior of initially curved circular micro plates, subjected to a distributed nonlinear electrostatic force. While both models are based on the Kirchoff hypothesis and on the nonlinear Föppl von Kármán (FvK) strain-displacements relations, the second model employs the Berger approximation, which significantly simplifies the formulation and describes the plate by a single governing equation. In both cases, the solution is based on the Galerkin decomposition with buckling modes of an initially flat plate used as the base functions. To track the unstable branches of the equilibrium curve, continuation methods in conjunction with the Riks algorithm are implemented. The validation of the models is conducted for two loading cases, namely “mechanical” deflection-independent load, and electrostatic displacement-dependent load. Results of a finite elements (FE) analysis, as well as of a finite differences (FD) solution of the differential equations, were used as a reference. We estimate the accuracy of the RO models and provide recommendations concerning the number of degrees of freedom (DOF) required to reach a desired accuracy. We show that a simple RO model based on Berger plate theory, can be conveniently used for analysis of electrostatically actuated plates with low initial curvature and small thickness to electrostatic gap ratio.

... The snap-through buckling phenomenon in thin shells structures is one of buckling modes which possesses the characteristics of large displacements [2][3][4]. Due to the complexity of nonlinear analysis of shells, the solutions are usually solved by using the finite element method [5][6][7][8], Generalized differential quadrature method (GDQ method) [9][10][11][12][13][14][15][16][17], Ritz' method [18][19][20][21][22][23][24], Galerkin method [25,26]. As indicated by Shen [1], the perturbation method differs from the Galerkin method, Ritz method and other numerical methods, it is a numerical method of solving mathematical problems and can gives explicit analytical expressions of all the variables. ...

This paper pays attention to predicting the nonlinear bending behaviors of functionally graded materials (FGM) infinite cylindrical shallow shells with a two-parameter elastic foundation by using a two-step perturbation method. The shells are subjected to uniform temperature rise and temperature dependency of the constituents is also taken into account. Two ends of the shells are assumed to be clamped or pinned and in-plane boundary conditions are immovable. The governing equations are derived based on physical neutral surface concept and high order shear deformation theory. The explicit expressions between the transverse load and the deflection are obtained by perturbation method. In numerical examples, some comparisons are shown to verify the correctness of the present research and solution method. It can be concluded that FGM cylindrical shallow shells subjected to uniform bending loadings will bring about snap-through buckling and jump changes, and the foundation can enhance the stability of the shells.

... Traditionally, the buckling phenomenon was considered as a failure which should be avoided. However, progress in the design of smart devices where snap-through behavior could be beneficial [3] stimulated renewed interest in the mechanics of bistable beams [4,5] and shells [6,7]. In the realm of microand nanoelectromechanical systems (MEMS/NEMS), applications of these elements include switches [8], sensors [9], nonvolatile memories [10] and micro-pumps [11][12][13][14], just to name a few. ...

The axisymmetric snap-through of an initially curved circular micro plate, subjected to a transversal distributed electrostatic force is studied. The analysis is based on a reduced order (RO) model resulting from the Galerkin decomposition, with buckling modes of a flat plate used as the base functions. In order to check the validity of the RO model, the corresponding problem for a displacement-independent ("mechanical") load is solved, and a comparison between the RO model and those obtained using finite elements (FE) analysis is carried out. It is shown, that the two are in good agreement, indicating that the RO model can be used for a plate undergoing electrostatic loading. However, the study shows that at least three degrees of freedom (DOF) are required for an accurate prediction of the equilibrium path and bistability. The coupled electromechanical analysis shows that due to the nonlinearity of the electrostatic load, the snap-through occurs at a lower displacement than in the case of the "mechanical" load. Moreover, the study concludes that actuation of plates of realistic dimensions can be achieved by reasonably low voltages.

... The buckling, snap-though buckling and post-buckling behavior of spherical shells have been studied with a greater interest in Refs. [14][15][16][17][18][19][20][21][22][23][24] under various loading combinations. These papers report analytical, numerical, and few cases of experimental work. ...

Previously, buckling behavior of several conical and spherical shells have been studied with great rigor. In this paper, snap through buckling behaviour for metallic dished shells under uniform external pressure is investigated. These shells are geometrically complex since it consists of a shallow conical frustum with a flat closed top. Such shells find many engineering applications, for instance as actuator elements in control components in cryogenic engines. Currently, no clear guidelines exist for design performance evaluation of such peculiar shells. This paper aims to establish a valid FE methodology for buckling and post buckling analysis of such shells using ABAQUS in tandem with experiments. A parametric study is carried out to understand the effect of geometrical parameters and imperfection sensitivity of these shells to snap-through buckling. Moreover, experiments were carried out using 3-D Digital Image Correlation(3D-DIC) for measuring whole-field deflection and strains. Numerical analysis was carried out, using generalized Eigen value analysis and non-linear analysis using modified-Riks technique with various material models, to correlate with the experimental observations. Non-linear elasto-plastic analysis with perfectly elastic-plastic material model agrees well with the experimental observations. Comparison of results from the numerical study indicates that material plasticity has a major effect on critical buckling pressure.

... There is a large amount of literature available regarding the stability of dome structures. For instance, [3][4][5][6] are concerned with continuum domes and shells, [7][8] single-layer space frames, [9][10] double-layer grids [11][12] the stability of other types of space structures. ...

Global stability of an innovative dome comprising of double-layer space frame sections together with curved flexural members has been studied. The dome had both an outer flat and an inner spherical double-layer grid space frames and formed the roof over a joint ballroom and meeting space of 3,100 square metres. The relatively wide spans between the outer and the inner space structures were covered using thirty six curved flexural pipe members laid on a synclastic surface. These members were employed for architectural purposes to provide a clear glazed area for the roof. Introducing relatively large span flexural members joining the two lattice space frames together complicated the overall structural response of the dome. It also made the dome susceptible to different premature instability modes. The non-linear FE program ABAQUS [1] was utilised to study the global stability of the dome and to investigate fully both its pre and post failure behaviour.
Different member configurations were considered for the dome and their effects on the behaviour; stability, snap through buckling and the collapse load of the structure were investigated. In the dome responses, several failure modes such as overall torsional buckling, in-plane ring buckling and symmetrical and non-symmetrical vertical snap through were identified.
It was noted that the presence of restraints placed between the flexural members could prevent the occurrence of premature torsional and vertical snap through buckling in the dome. The global behaviour of the dome appeared to be very sensitive to both the type and configuration of the restraints applied along the flexural members. The dome was also found to be remarkably sensitive to non-symmetric loads.

The study brings forth extended criteria for electrostatically actuated micro-plates, which take into account the presence of prestress. Following previous studies that analysed the veracity of a reduced order (RO) model, based on Berger’s plate approximation, we present here an analysis of a single degree-of-freedom (DOF) model. The resulting simplified model has enabled a semi-analytical derivation of criteria, for which the plate must adhere to in order to gain bistability in the presence of electrostatic load. The study entails a precursor analysis of a “mechanically” loaded plate, so as to inspect the reliability of Berger’s RO (BRO) model, whilst laying the analytical-numerical foundation for the subsequent electrostatic analysis. The end result is a set of criteria in the space of the plate elevation, thickness-to-gap ratio, and membrane load, splitting it into mono- and bistable domains. Numerical fits, obtained for various cases are also presented. In the process of the study, an upper bound for the condition was also disclosed, representing a necessary condition for bistability. The resulting criteria were compared to numerically obtained ones, extracted from the validated Föppl-von-Kármán’s RO (vKRO) model, as well as from direct finite differences (FD) based solutions (vKFD). The result has shown a remarkable qualitative resemblance, especially in low enough thickness-to-gap ratios, providing a much needed tool in the design of such structures.

The behavior of spherical caps under radial dynamic edge loading in theform of step loading of infinite duration in the time domain has beeninvestigated. The aim here is to present a mathematical model of shallowshells of revolution which may undergo snap-through buckling as a resultof the radial displacement of the circular boundary. The snap-throughbuckling under such a loading condition is contrary to intuition and itseems not to have been previously observed. In this search it isobserved that snap-through buckling is also possible under peripheraldynamic loading conditions. The amount of snapping is remarkable whenthe cap has an opening around the apex. A technological application ofthe peripheral type of loading is seen in metal-ceramic compositetransducers achieved by installing a piezoelectric ceramic disk betweentwo metal end caps. The radial motion of the ceramic is converted into aflex-tensional motion in the spherical caps. As a result, a largedisplacement is obtained in the perpendicular direction, which mayresult in snap-through buckling. For the numerical solution of theproblem a computer program, using a linearized finite elementincremental-iterative approach based on updated Lagrangian formulationis developed, and the whole process is accomplished using the Newmarkmethod as the time integration scheme.

A bibliographical review of the finite element methods (FEMs) applied for the linear and nonlinear, static and dynamic analyses of basic structural elements from the theoretical as well as practical points of view is given. The bibliography at the end of the paper contains 1,726 references to papers, conference proceedings and theses/dissertations dealing with the analysis of beams, columns, rods, bars, cables, discs, blades, shafts, membranes, plates and shells that were published in 1996-1999.

A modified arc-length method was proposed to solve the problem that the traditional Newton-Raphson method fails in nonlinear finite element analysis on structures with buckling in the structure or softening in the material. The unbalanced load vector in the nonlinear equations is decomposed into two orthogonal vectors. A new constraint equation was derived, and solved to obtain the current load step factor. Complex roots are avoided by modifying the arc length. Two examples of nonlinear analyses on arch structures with geometric and material nonlinearity were presented, respectively. The snap-through in the post-bulking period was also revealed in the examples. The results demonstrate that solutions can be obtained with the proposed method when buckling in the structure or softening in the material occurs.

In this study, the effect of compressibility on the nonlinear buckling of the simply supported polyurethane spherical shells subjected to an apical concentrated load is presented. The problem is strongly nonlinear both physically and geometrically. Since the closed form solution of the corresponding algebraical and differential equations is not possible, numerical methods are used unavoidably. The governing equations of the problem are converted to algebraical difference equations via the finite difference method and the obtained algebraical equations are solved numerically by using the Newton— Raphson method. Several numerical experiments corresponding to the various values of a thickness parameter, a depth parameter and a material constant related with the compressibility of polyurethane are performed and the force— apex deflection curves are drawn. Concluding remarks pertaining to the effect of the variation of the material constant related with the compressibility on the buckling loads and buckling deflections of the simply supported polyurethane spherical shells, subjected to an apical load, with various thicknesses and depths are presented.

Geometrically nonlinear analysis of initially imperfect shallow spherical shells under uniformly distributed axisymmetrical load is investigated in this computational study. The thickness of the shell is considered to be uniform and the material is assumed to be isotropic. The numerical treatment of the nonlinear fundamental shallow spherical shell equations is carried out by the finite difference method and the Newton—Raphson method. The influence of the parameters (Poisson's ratio, parameters of thickness, depth and initial imperfection) on the critical load, the value and the location of the maximum displacement components and the maximum stress resultants is examined by various numerical experiments each of which is made for two distinct types of supports along the edge: clamped and simply supported. Despite the dominant role of the nonlinearity, some general statements are obtained and the influence of the parameters is highlighted.

The asymmetric bifurcation problem for a shallow spherical cap is examined. The applied pressure can act either external or internal to the cap and both cases are treated here. Assuming a non-linear axisymmetric basic state, the linearised bifurcation equations for the pressurised shell are investigated in the limit when the thickness of the cap is much less than the maximum rise of the shell mid-surface. Within this regime the wrinkling patterns in both cases are confined to a narrow zone near the edge of the shell, making it possible to solve asymptotically the corresponding equations and derive analytical predictions for both the critical pressure and the corresponding number of wrinkles. Some comparisons with direct numerical simulations are included as well.

The axisymmetric snap-through of an initially curved circular micro plate, subjected to a distributed electrostatic force is studied. The analysis is based on a reduced order (RO) model resulting from the Galerkin decomposition, with buckling modes of a flat plate used as the base functions. The results of the RO model are compared with the results available in the literature for initially flat plate and with numerical results obtained for both cases of displacement-independent "mechanical" load and displacement-dependent electrostatic load. The study indicates that a model with at least three degrees of freedom (DOF) is required for an accurate prediction of the equilibrium path, under both types of loading. The analysis shows that due to the nonlinearity of the electrostatic load, the snap-through occurs at a lower displacement than under the "mechanical" load. The presented results also indicate that micro plates of realistic dimensions can be actuated by reasonably low voltages, suggesting the feasibility of the usage of such elements for various applications.

Introduces a new method for fabricating capacitive micromachined ultrasonic transducers (CMUTs) that uses a wafer bonding technique. The transducer membrane and cavity are defined on an SOI (silicon-on-insulator) wafer and on a prime wafer, respectively. Then, using silicon direct bonding in a vacuum environment, the two wafers are bonded together to form a transducer. This new technique, capable of fabricating large CMUTs, offers advantages over the traditionally micromachined CMUTs. First, forming a vacuum-sealed cavity is relatively easy since the wafer bonding is performed in a vacuum chamber. Second, this process enables better control over the gap height, making it possible to fabricate very small gaps (less than 0.1 μm). Third, since the membrane is made of single crystal silicon, it is possible to predict and control the mechanical properties of the membrane to within 5%. Finally, the number of process steps involved in making a CMUT has been reduced from 22 to 15, shortening the device turn-around time. All of these advantages provide repeatable fabrication of CMUTs featuring predictable center frequency, bandwidth, and collapse voltage.

Since the early days of commercial nuclear power generation, bursting discs have been used as safety devices for the protection of nuclear power plants against excessive pressure or vacuum. Asme code requirements vary according to the quality assurance classification of the plant which they protect. For Class 3 they may be used as the sole pressure relief device; in Class 2 their use is restricted to secondary pressure relief for main steam or liquid service; while for Class 1 they may be used only for secondary overpressure relief. This approach must now be considered excessively conservative in view of the significant improvements which have been achieved in the design and operation of modern bursting discs.

The sixth editions of these seminal books deliver the most up to date and comprehensive reference yet on the finite element method for all engineers and mathematicians. Renowned for their scope, range and authority, the new editions have been significantly developed in terms of both contents and scope. Each book is now complete in its own right and provides self-contained reference; used together they provide a formidable resource covering the theory and the application of the universally used FEM. Written by the leading professors in their fields, the three books cover the basis of the method, its application to solid mechanics and to fluid dynamics.* This is THE classic finite element method set, by two the subject's leading authors * FEM is a constantly developing subject, and any professional or student of engineering involved in understanding the computational modelling of physical systems will inevitably use the techniques in these books * Fully up-to-date; ideal for teaching and reference

An updated Lagrangian formulation of a quadratic degenerated isoparametric shell element is presented for geometrically nonlinear elasto-plastic shell problems. A finite rotation effect is included in the formulation by adopting a co-rotational scheme. The load stiffness matrix has been derived for the treatment of a pressure load. For elasto-plastic behavior, the layered element model is used. The Newton-Raphson iteration method is employed to solve incremental nonlinear equations. For tracking of post-buckling behavior, the work control method is taken into account. Verification of the present technique is obtained by analyzing the available reference problems. Good correlations between the computed results and referenced data can be drawn.

Buckling of arches is studied using a corotational finite element model in conjunction with a modified Riks-Wempner technique. The corotational formulation allows for separation of rigid body displacements from deformation displacements of the element. The program can equally be applied to rigid or prestressed arches, as it takes into account the sequential fabrication of stressed (i.e. prebuckled) arches. Looping paths tracing nonlinear response of arches have been successfully obtained for several cases, including rigid and prestressed elastica arches with both symmetric and asymmetric modes of buckling. Comparisons are made with the results for prestressed arches using shooting method, and with the benchmark results for various rigid arches using discrete element method. When compared to a discrete element technique, the present method appears to be more cost-effective, as a finite element mesh with 60% fewer elements can result in virtually the same degree of accuracy.

An overview is presented of qualitative and quantitative studies of the equilibrium of geometrically nonlinear bodies subjected to conservative multiparametrical loading. The effects of multiparametrical loading paths are studied by means of a dynamic approach in which the equilibrium states are the steady states. An incremental numerical solution is also described for examining the static effects on the nonlinear structures. The loading paths of conservative multiparametrical load configurations strongly influence incremental algorithms based on finite-element and finite-difference techniques. Only stable states contribute to the formation of the final states, and equilibrium states do not depend on changes in the times of the loadings. External viscous friction is thought to have no influence on the final state; the incremental approach is found to yield good results in the solutions of static problems.

The loss of stability by so-called bifurcation in tension of geometrically nonlinear plates and spherical caps loaded by axisymmetrical cross forces is examined. These phenomena are manifested by a transition from axisymmetrical forms to the nonsymmetrical ones with formation of wrinkles. They may occur on plates, and also on caps when the loading acts on their inner (concave) side. In these cases, the meridians are stretched at every point. It is necessary to distinguish between these phenomena and bifurcation in compression. The latter is impossible for plates and shallow caps and may occur only on relatively deep caps when the loading acts outside (convex side). Experimental observations of the stability loss by bifurcation in tension are described. Qualitative investigations of the writer indicate, in particular, that the considered phenomena cannot occur when the edge supports are fixed.

A theoretical study of buckling of clamped shallow spherical shells under uniform external pressure is presented. For sufficiently large deflection, deformations of such shells are not proportional to the applied pressure. The shell deforms axisymmetrically under sufficiently low pressure. The problem of axisymmetrical snapping has been solved by different numerical methods and the results agree with each other. The buck ling pressures obtained in such a manner are too high as compared with experimental results ob tained in References. Initial imperfections of the shell and unsymmetrical buckling are presumed to be the sources of this discrepancy between axisymmetrical buckling theory and experiment.

In this study procedures for overcoming limit and bifurcation points in large-scale structural analysis problems are described and evaluated. The methods are based on Newton's method for the outer iterations, while for the linearized problem in each iteration the preconditioned truncated Lanczos method is employed. Special care is placed upon line search routines for accelerating the convergence properties and enhancing the stability of the outer method. The proposed methodology retains all characteristics of an iterative method by avoiding the factorization of the current stiffness matrix. The necessary eigenvalue information is retained in the tridiagonal matrix of the Lanczos approach.

The non-linear, axisymmetrical behavior of truncated shallow spherical shells under transverse loading is studied. Load-deflection relations are obtained through iteration and numerical integration. Shells subjected to uniform pressure and combined uniform pressure and concentrated ring loading have been investigated.

The paper presents a classification of mathematical commonly encountered in connection with solution of non−linear finite element problems. The principal methods for numerical solution of the non−linear equations are surveyed and discussed. Special emphasis is placed upon the description of an automatic load incrementation procedure with equilibrium iterations. It is shown how this algorithm can be adapted for solving problems involving instabilities, snap−through and snap−back. A simple scalar quantity denoted the current stiffness parameter is suggested; this parameter is used to characterize the overall behaviour of non−linear problems. It can also be used as a steering parameter in the solution process. The use of the present technique is illustrated by several examples.

The paper describes a study of incremental-iterative solution techniques for geometrically non-linear analyses. The solution methods documented are based on a modified Newton-Raphson approach, meaning that the tangent stiffness matrix is computed at the commencement of each load step but is then held constant throughout the equilibrium iterations. A consistent mathematical notation is employed in the description of the iterative and load incrementation strategies, enabling the simple inclusion of several solution options in a computer program. The iterative strategies investigated are iteration at constant load, iteration at constant displacement, iteration at constant ‘arc-length’, iteration at constant external work, iteration at minimum unbalanced displacement norm, iteration at minimum unbalanced force norm and iteration at constant ‘weighted response’. The load incrementation schemes investigated include strategies based on the number of iterations required to achieve convergence in the previous load step, strategies based on the ‘current stiffness parameter’ and a strategy based on a parabolic approximation to the load-deflection response. Criteria for detecting when the applied external load increment should reverse sign are described.A challenging example of a circular arch exhibiting snap-through (load limit point) behaviour and snap-back (displacement limit point) behaviour is solved using several different iterative and load incrementation strategies. The performance of the solution schemes is evaluated and conclusions are drawn.

A solution strategy for the analysis of nonlinear structures is described. The strategy is a simple extension of existing Newton-type procedures, and can easily be incorporated into existing computer programs.
Earlier work which contributed to the development of the strategy is reviewed and the theory of the procedure is presented. Six examples, covering several different types of structural behaviour are described. These examples suggest that the strategy is remarkably stable and efficient.

The superior performance of the consistent shell element in the small deflection range has encouraged the authors to extend the formulation to large displacement static and dynamic analyses. The nonlinear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear stiffness matrix and the unbalanced load vector for the consistent shell element is presented in this study. Meanwhile, a simplified method for coding the nonlinear formulation is provided by relating the components for the nonlinear B-matrices to those of the linear B-matrix. The consistent mass matrix for the shell element is also derived and then incorporated with the stiffness matrix to perform large displacement dynamic and free vibration analyses of shell structures. Newmark's method is used for time integration and the Newton-Raphson method is employed for iterating within each increment until equilibrium is achieved. Numerical testing of the nonlinear model through static and dynamic analyses of different plate and shell problems indicates excellent performance of the consistent shell element in the nonlinear range.

This paper presents a p-version geometrically nonlinear (GNL) formulation based on total Lagrangian approach for a three-node axisymmetric curved shell element. The approximation functions and the nodal variables for the element are derived directly from the Lagrange family of interpolation functions of order pξ and pη. This is accomplished by first establishing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ and η directions for the three- and one-node equivalent configurations that correspond to pξ + 1 and pη+ 1 equally spaced nodes in the ξ and η directions and then taking their products. The resulting element approximation functions and the nodal variables are hierarchical and the element approximation ensures C⁰ continuity. The element geometry is described by the coordinates of the nodes located on the middle surface of the element and the nodal vectors describing top and bottom surfaces of the element.

Based on K. Marguerre's shallow shell theory, a family of higher-order finite elements each consisting of 17–25 nodes and with separate in-plane and bending displacement variables has been developed for the geometrically nonlinear analysis of shallow shells subjected to lateral loads. A step-iteration Newton-Raphson scheme has been adopted in solving the final system of recurrent nonlinear equations. Several numerical examples, including a spherical cap and a square shallow shell with surface in double sine curves, are presented to demonstrate the versatility and convenience of the use of higher-order elements in modelling shallow shells and also the sufficient accuracy of the predictions made by the present formulation in the context of geometrically nonlinear analysis.

This paper is concerned with the numerical solution of systems of equations of discrete variables, which represent the nonlinear behaviour of elastic systems under conservative loading conditions. In particular, an incremental approach to the solution of buckling and snapping problems is explored.The topics that are covered can be summarized as follows:—The computation of nonlinear equilibrium paths with continuation through limit points and bifurcation points.—The determination of critical equilibrium states.Characteristic to the procedures employed is the use of the length of the equilibrium path as control parameter. This feature, together with the second order iteration method of Newton, offers a reliable basis for the procedures described. Actual computations, carried out on a finite element model of a shallow circular arch, illustrate the effectiveness of the methods proposed.

Riks [1] has recently proposed a new solution procedure for overcoming limit points. To this end, he adds, to the standard equilibrium equations, a constraint equation fixing the length of the incremental load step in load/deflection space. The applied load level becomes an additional variable.The present paper describes a means of modifying Rik's approach so that it is suitable for use with the finite element method. The procedure is applied in conjunction with the modified Newton-Raphson method in both its original and accelerated forms. The resulting techniques not only allow limit points to be passed, but also, improve the convergence characteristics of the unconstrained iterative procedures. Illustrative examples include the large deflection analysis of shallow elastic shells and the collapse analysis of a stiffened steel diaphragm from a box-girder bridge.

A digital computer program for the geometrically nonlinear static and dynamic response of arbitrarily loaded shells of revolution is described. The governing partial differential equations are based upon Sanders' nonlinear thin shell theory for the conditions of small strains and moderately small rotations. The governing equations are reduced to uncoupled sets of four linear, second order, partial differential equations in the meridional and time coordinates by expanding the dependent variables in a Fourier sine or cosine series in the circumferential coordinate and treating the nonlinear modal coupling terms as pseudo loads. The derivatives with respect to the meridional coordinate are approximated by central finite differences, and the displacement accelerations are approximated by the implicit Houbolt backward difference scheme with a constant time interval. At every load step or time step each set of difference equations is repeatedly solved, using an elimination method, until all solutions have converged. Results from the program are compared to previously published data for several problems, and the versatility, efficiency, and limitations of the program are candidly evaluated.

Buckling and postbuckling of spherical caps

- Akkas N Bauld
- Nr

Akkas N, Bauld NR. Buckling and postbuckling of spherical caps. ASCE J Engng Mech Div 1971;97:727±39.