Buckling of Shallow Arches
TL;DR: In this article, the exact solution to the nonlinear problem of the buckling of a shallow, circular arch subjected to a uniform pressure or a concentrated load is given, and an explicit bifurcated equilibrium path is presented.
Abstract: The exact solution to the nonlinear problem of the buckling of a shallow, circular arch subjected to a uniform pressure or a concentrated load is given. In addition an explicit bifurcated equilibrium path is presented. The buckling loads, according to several criteria, are found and their range of validity is investigated in detail. It is shown that even if the smallest buckling load is associated with an asymmetric bifurcation this alone is not a sufficient condition for the asymmetric buckling criterion to govern the problem.
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TL;DR: In this paper, an incremental approach to the solution of buckling and snapping problems is explored, where the authors use the length of the equilibrium path as a control parameter, together with the second order iteration method of Newton.
1,821 citations
TL;DR: In this paper, finite element incremental formulations for non-linear static and dynamic analysis are reviewed and derived starting from continuum mechanics principles, and a consistent summary, comparison, and evaluation of the formulations which have been implemented in the search for the most effective procedure.
Abstract: SUMMARY Starting from continuum mechanics principles, finite element incremental formulations for non-linear static and dynamic analysis are reviewed and derived. The aim in this paper is a consistent summary, comparison, and evaluation of the formulations which have been implemented in the search for the most effective procedure. The general formulations include large displacements, large strains and material non-linearities. For specific static and dynamic analyses in this paper, elastic, hyperelastic (rubber-like) and hypoelastic elastic-plastic materials are considered. The numerical solution of the continuum mechanics equations is achieved using isoparametric finite element discretization. The specific matrices which need be calculated in the formulations are presented and discussed. To demonstrate the applicability and the important differences in the formulations, the solution of static and dynamic problems involving large displacements and large strains are presented.
789 citations
TL;DR: In this paper, a reduced basis technique and a computational algorithm are presented for predicting the nonlinear static response of structures, where a total Lagrangian formulation is used and the structure is discretized by using displacement finite element models.
Abstract: A reduced basis technique and a computational' algorithm are presented for predicting the nonlinear static response of structures. A total Lagrangian formulation is used and the structure is discretized by using displacement finite element models. The nodal displacement vector is expressed as a linear combination of a small number of basis vectors and a Rayleigh-Ritz technique is used to approximate the finite element equations by a reduced system of nonlinear equations. The Rayleigh-Ritz approximation functions (basis vectors) are chosen to be those commonly used in the static perturbation technique namely, a nonlinear solution and a number of its path derivatives. A procedure is outlined for automatically selecting the load (or displacement) step size and monitoring the solution accuracy. The high accuracy and effectiveness of the proposed approach is demonstrated by means of numerical examples.
414 citations
TL;DR: A stabilized conforming (SC) nodal integration, which meets the integration constraint in the Galerkin mesh‐free approximation, is generalized for non‐linear problems by introducing a Lagrangian strain smoothing to the deformation gradient, and by performing a nodal Integration in the undeformed configuration.
Abstract: A Stabilized Conforming (SC) nodal integration, which meets the integration constraint in the Galerkin meshfree approximation, is generalized for nonlinear problems. In SC nodal integration, a strain smoothing stabilization is introduced and applied to the deformation gradient using a Lagrangian meshfree discretization. The assumed strain method expressed in terms of deformation gradient is employed to formulate an efficient meshfree discrete equation for nonlinear problems. A significant gain in computational efficiency is achieved, as well as enhanced accuracy, in comparison to the solution of meshfree method using Gauss integration method. The performance of the proposed method appears to be quite robust when dealing with irregular discretization.
272 citations
Cites methods from "Buckling of Shallow Arches"
...The SC nodal integration and Gauss integration solutions are compared with the analytical solution [20] using three model re nements and two normalized support sizes R=1:0 and 2....
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TL;DR: In this article, the results of theoretical and experimental investigation of an initially curved clamped-clamped microbeam actuated by a distributed electrostatic force were presented, and the influence of various parameters on the stability was investigated.
Abstract: The results of theoretical and experimental investigation of an initially curved clamped–clamped microbeam actuated by a distributed electrostatic force are presented. Reduced-order Galerkin and consistently constructed lumped models of the shallow Euler–Bernoulli arch were built and verified by numerical analysis, and the influence of various parameters on the stability was investigated. Due to the unique combination of generic mechanical and electrostatic nonlinearities, the voltage–deflection characteristic of the device may have two maxima implying the existence of sequential snap-through buckling and pull-in instability and of bistability of the beam. The first critical voltage can be higher or lower than the second one, while the stable deflections are significantly larger than in a straight beam. The minimal initial elevation required for the appearance of the snap-through in the electrostatically actuated beam is smaller than in the case of uniform deflection-independent loading; a closed-form approximation of this elevation was evaluated. The devices were fabricated from silicon on insulator (SOI) wafer using deep reactive ion etching and in-plane responses were characterized by means of optical and scanning electron microscopy. Model results obtained for the actual dimensions of the device were in good agreement with the experimental data.
271 citations