Snap-through Phenomenon Research Articles

Non-linear random response of laminated composite shallow shells using finite element modal method

This paper investigates the large-amplitude multi-mode random response of thin shallow shells with rectangular planform at elevated temperatures using a finite element non-linear modal formulation. A thin laminated composite shallow shell element and the system equations of motion are developed. The system equations in structural node degrees-of-freedom (DOF) are transformed into modal co-ordinates, and the non-linear stiffness matrices are transformed into non-linear modal stiffness matrices. The number of modal equations is much smaller than the number of equations in structural node DOF. A numerical integration is employed to determine the random response. Thermal buckling deflections are obtained to explain the intermittent snap-through phenomenon. The natural frequencies of the infinitesimal vibration about the thermally buckled equilibrium positions (BEPs) are studied, and it is found that there is great difference between the frequencies about the primary (positive) and the secondary (negative) BEPs. All three types of motion: (i) linear random vibration about the primary BEP, (ii) intermittent snap-through between the two BEPs, and (iii) non-linear large-amplitude random vibration over the two BEPs, can be predicted. Copyright © 2006 John Wiley & Sons, Ltd.

POST-BUCKLING OF AN ELASTIC COLUMN WITH VARIOUS ROTATIONAL END RESTRAINTS

In this technical note, the post-buckling behavior of a simply supported elastic column with various rotational end conditions of the supports is investigated. The compressive force is applied at the tip of the column. The characteristic equation for solving the critical loads is obtained from the boundary value problem of linear systems. In the post-buckling state, a set of nonlinear differential equations with boundary conditions is established and numerically solved by the shooting method. The interesting features associated with this problem such as the limit load point, snap-through phenomenon and the secondary bifurcation point will be highlighted herein.

Threefold-symmetric Bricard linkages for deployable structures

A closed-loop overconstrained spatial mechanism composed of six hinge-jointed bars, which has three planes of symmetry in any position, is called a threefold-symmetric Bricard linkage. In this paper a kinematic analysis of these linkages is presented. It is pointed out that for particular parameter values, kinematic bifurcation of the linkages can occur. Features of the kinematic bifurcation are discussed in detail. The applicability of threefold-symmetric Bricard linkages and of their alternative forms to deployable structures is investigated. In addition, by using the theory of kinematic bifurcation, a snap-through phenomenon appearing in a deployable hexagonal ring is explained.

A concept for the generation of out-of-plane distortion from tailored FRP laminates

The phenomenon of thermally induced distortions in unsymmetrical laminates is well understood, and it may be shown that a square, unsymmetrical 0.90 laminate will tend to form two stable geometries with a snap-through phenomenon between them. This paper discusses laminates in which at each point the lay-up is symmetrical across the laminate mid-plane, but which still exhibit multiple stable geometries. The number of stable geometries can be controlled by the details of the lay-up from the minimum of two to, in principle, an unlimited number. In addition it will be shown how a similar process can be used to generate multiple stable stress states and geometries in unidirectional laminates. This paper represents a very preliminary experimental investigation of the design space available for such composite laminates. Possible applications of composites with multiple stable geometries are noted.

Geometric design of arbitrarily curved bi-stable deployable arches with discrete joint size

The deployable structures presented in this work are bi-stable in the sense of being self-standing and stress-free when fully closed or fully deployed, but exhibit incompatibilities between the member lengths at intermediate geometric configurations during the deployment process, which lead to the occurrence of second-order strains and stresses resulting in a snap-through phenomenon that “locks” the structures in their deployed configuration. Until now the geometric shapes that were possible in the deployed configuration were only flat or curved with constant curvature. This limitation is addressed in the present paper by proposing a geometric design methodology for deployable arches of arbitrary curvature, accounting also for the discrete joint size, and applying it successfully for the geometric design of a semi-elliptical arch. The arch is then modeled with finite elements, and a geometrically non-linear analysis is performed in order to verify the deployability feature. Further verification is provided by the construction of a small-scale physical model. A preliminary structural design indicates the overall feasibility of the arch for short to medium spans and light loads.

Development and application of a laterally driven electromagnetic microactuator

A laterally driven electromagnetic microactuator (LaDEM) is introduced, and a micro-optical switch is designed and fabricated as an application. LaDEM offers parallel movement of the microactuator to the silicon substrate surface (in-plane mode). Polysilicon-on-insulator wafers and a reactive ion etching process were used to fabricate high-aspect-ratio vertical microstructures, which allowed the equipping of vertical micromirrors. A fabricated single leaf spring had a width of 1.2 μm, thickness of 16 μm, and length of 920 μm. The resistance of the fabricated leaf spring for the optical switch was 5 Ω. The deflection of the leaf spring started to profoundly increase at about 400 mA, and it showed snap-through phenomenon over that current value. Owing to the snap-through phenomenon, a large deflection of 60 μm was detected at 566 mA.

Mechanical behavior and stability of the internal membrane of the InCor ventricular assist device.

This paper describes and analyzes the mechanical behavior of the internal membrane of the InCor VAD (Heart Institute [InCor], University of São Paulo, Brazil), applying the knowledge and tools of structural engineering analysis. This membrane plays an important role in the operation of the ventricular assist device (VAD) because it separates the blood chamber from the pneumatic one, transmitting the pneumatic load to the blood, thus making the desired blood flow possible. The loading repeats itself every time the VAD beats. Therefore the performance, reliability, and durability of the membrane are critical for the performance of the VAD. The mathematical model is based on the large deflection theory of thin shells and on the finite element method. The snap-through instability phenomenon, which is responsible for transmission of the pneumatic load to the blood, was observed in the membrane both when modeled mathematically and experimentally. Principal stresses and strain distributions were obtained with this model at certain load levels along the pre- and postbuckling paths.

Snap-through behaviour of cables in flexible structures

A study of the static and dynamic behaviour of flexible structures that exhibit the snap-through phenomenon is presented in this paper. A single cable example is presented to show that dynamic analysis can be an effective method of solving static snap-through problems. This paper presents an in-depth study of the behaviour of a guyed mast structure that has received attention in a number of previous elastic investigations [Peterson C. Abgespannte Maste und Schornsteine, Statik und Dynamik. Berlin: Wilhelm Ernst and Sohn Verlag, 1970; Argyris JH, Dunne PC, Angelopoulos T. Comp Meth Appl Mech Engng 1973;2:203–50; El-Katt MTH. Non-linear dynamic analysis of cable structures. PhD Thesis, UMIST, 1987]. The Williams toggle is an example of a rigid frame snap-through behaviour [Williams FW. Quart J Mech Appl Mech 1964;17:456–69]. Meek and Xue [Meek JL, Xue Q. Comput Meth Appl Mech Engng 1996;136:347–61] present a similar approach to the authors using dynamic analysis to trace the snap-through behaviour of rigid framed structures without having to follow the drooping load path. The very flexible guy snap-through response has not been previously reported.

Non-linear response of geometrically imperfect sandwich curved panels under thermomechanical loading

This paper deals with the non-linear response of sandwich curved panels exposed to thermomechanical loadings. The mechanical loads consist of compressive/tensile edge loads, and a lateral pressure while the temperature field is assumed to exhibit a linear variation through the thickness of the panel. Towards obtaining the equations governing the postbuckling response, the Extended Galerkin’s Method is used. The numerical illustrations concern doubly curved, circular cylindrical and as a special case, flat panels, all the edges being simply supported. Moveable and immoveable tangential boundary conditions in the directions normal to the edges are considered and their implications upon the thermomechanical load-carrying capacity are emphasized. Effects of the radii of curvature and of initial geometric imperfections on the load-carrying capacity of sandwich panels are also considered and their influence upon the intensity of the snap-through buckling are discussed. It is shown that in special cases involving the thermomechanical loading and initial geometric imperfection, the snap-through phenomenon can occur also in the case of flat sandwich panels.

Optimum Design Of Deployable Structures Using Genetic Algorithms

The prefabricated deployable space frames investigated in this work are smart structures in the sense that they have the ability to adapt to changing needs by using the, usually destructive, snap-through phenomenon as a form of prestressing. Their design is very challenging due to the need to balance the conflicting requirements of desired flexibility during deployment and desired stiffness under service loads. This paper describes a first effort to escape from a heuristically driven, manual design by employing optimization via a genetic algorithm.

An improved analytical model for the prediction of the nonlinear behavior of flat and curved deployable space frames

The deployable space frames investigated in this paper consist of straight bars linked together in the factory as a compact bundle, which can then be unfolded into large-span, load-bearing structural forms. During the deployment process incompatibilities between the member lengths lead to the occurrence of strains and stresses resulting in a snap-through phenomenon that ‘locks’ the structures in their deployed configuration. The structural response during deployment is, therefore, characterized by geometric nonlinearities and hence accurate simulation of the deployment process requires sophisticated finite element modeling. In the present work, three versions of a simple analytical model are proposed that predict the intensity of the snap-through phenomenon based on geometric compatibility considerations. Deployable units for flat roofs or spherical domes are addressed. The objective of this approximation is to minimize the number of required finite element analyses during preliminary design. The proposed analytical model is validated through comparison to numerical results obtained by detailed finite element simulation.

Analytical Predictions Of The Snap-throughCharacteristics Of Deploy Able Structures

The deployable structures investigated in this paper are prefabricated space frames consisting of straight bars linked together in the factory as a compact bundle, which can then be unfolded into large-span, load-bearing structural forms. During the deployment process incompatibilities between the member lengths lead to the occurrence of strains and stresses resulting in a snap-through phenomenon that locks the structures in their deployed configuration. The structural response during deployment is, therefore, characterized by geometric nonlinearities and, hence, accurate simulation of the deployment process requires sophisticated finite element modeling. In the present work, a simple analytical model is proposed that uses geometric considerations to predict the intensity of the snap-through phenomenon. The underlying reasoning is that snap-through must be related to the maximum shortening of the bars due to geometric compatibility requirements. The objective of this approximation is to minimize the number of finite element analyses during preliminary design. The proposed analytical model is validated through comparison to numerical results obtained by detailed finite element simulation. 1 Description of the Problem As mentioned above, this study deals with deployable structures that are prefabricated space frames. They consist of straight bars that are linked together in the factory as a compact bundle, and can then be unfolded into large-span, load bearing structural forms by applying forces on selected nodes. Because of this feature they offer significant advantages in comparison to conventional, Transactions on the Built Environment vol 21, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509

Stochastic bifurcation in non-linear structural systems near 1 : 1 internal resonance

This paper examines the problem of stochastic bifurcation and response statistics of a clamped-clamped beam in the neighborhood of 1 : 1 internal resonance. The beam is initially exposed to an axial static load and subjected to a lateral wide band random excitation. The static load exceeds the Euler buckling load and thus results in the internal resonance condition 1 : 1 between the first two modes. The Fokker-Planck equation is used to generate a general first-order differential equation in the joint moments of response coordinates. Gaussian and non-Gaussian closure schemes are used to close the infinite coupled moment equations. The closed equations are then solved for response statistics and bifurcation boundaries in terms of system and excitation parameters. The theoretical results are compared with those determined by Monte Carlo simulation. The numerical simulation is also used to estimate the marginal probability density of response coordinates. It is found that the numerical simulation results are in good agreement with those obtained by the Gaussian closure, particularly in predicting stochastic bifurcation of the second mode. Under relatively higher excitation levels the Monte Carlo simulation exhibits the occurrence of snap-through phenomenon in the first mode. Both Gaussian and non-Gaussian closures fail to predict the snap-through.

Nonlinear and multiple snapping responses of cylindrical panels comparing displacement control and riks method

A finite element method, incorporating the modified Riks technique and Newton-Raphson method, is used to trace a highly nonlinear response such as snap-through, single and multiple snap-back phenomenon of cylindrical shells. A through the thickness parabolic transverse shear strain is considered. The multiple snapping responses of an isotropic panel are investigated, and their results are compared with those not considering the transverse shear deformation. Also comparison differences are noted with respect to a ‘displacement control’ strategy.

Thermal Buckling of Shallow Bimetallic Two-Hinged Arches

The snap-through phenomenon of shallow bimetallic two-hinged arches due to temperature rise is analyzed. Two types of arch geometry are included. The effect of the position of supporting hinges on the critical temperature rise for snap-through is discussed. Comparisons are made between the present more accurate solution of the critical temperature rise and the approximate solution obtained by Timoshenko. Striking discrepancies are found between these two solutions.

A Nonlinear Model Study of the Thermal Buckling of Thin Elastic Shells

Analysis of the simple model proposed yields equilibrium curves in agreement with those obtained by other investigators for axially compressed, thin-walled, circular cylindrical shells. A rigorous calculation of the stability of the equilibrium of the model indicates that the snap-through phenomenon can be entirely absent when an imperfect shell is heated in a perfectly rigid testing machine. When the testing machine is sufficiently elastic, snap through will occur. It may take place at a temperature smaller than, equal-to, or greater than T = εcr/α, where εcr is the theoretical critical strain of the perfect shell and α is the coefficient of expansion of the material of the shell, but the maximum stress produced in the imperfect shell will always be less than εcrE, where E is Young’s modulus of the material.