A numerical analysis of large deflections of beams

Wang, T.M.; Lee, S.L.; Zienkiewicz, O.C.

doi:10.1016/0020-7403(61)90005-4

Cited by 56 publications

(25 citation statements)

References 12 publications

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“…By substituting Eqs. (1) and (2) into Eq. (3), the moment curvature relationship can be expressed as Case…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…Many models were studied in order to investigate the stability of the beam models under variety of load cases. Focusing on the simple beam model, there are a number of research works dealing with the postbuckling of the beams under several load cases such as transverse point load [1], axial point load [2][3][4], follower point load [5], uniform load [6], follower uniform load [7], and concentrated moment [8,9]. Majority of previous studies focuses on the postbuckling behaviors of beams under only two specific load cases: point and uniform loads.…”

Section: Introductionmentioning

confidence: 99%

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Postbuckling of elastic beam subjected to a concentrated moment within the span length of beam

Phungpaingam

Chucheepsakul

2007

Acta Mech Sin

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This paper aims to present the exact closed form solutions and postbuckling behavior of the beam under a concentrated moment within the span length of beam. Two approaches are used in this paper. The nonlinear governing differential equations based on elastica theory are derived and solved analytically for the exact closed form solutions in terms of elliptic integral of the first and second kinds. The results are presented in graphical diagram of equilibrium paths, equilibrium configurations and critical loads. For validation of the results from the first approach, the shooting method is employed to solve a set of nonlinear differential equations with boundary conditions. The set of nonlinear governing differential equations are integrated by using Runge-Kutta method fifth order with adaptive step size scheme. The error norms of the end conditions are minimized within prescribed tolerance (10 −5 ). The results from both approaches are in good agreement. From the results, it is found that the stability of this type of beam exhibits both stable and unstable configurations. The limit load point existed. The roller support can move through the hinged support in some cases of β and leads to the more complex of the configuration shapes of the beam.

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“…By substituting Eqs. (1) and (2) into Eq. (3), the moment curvature relationship can be expressed as Case…”

Section: Formulation Of the Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Postbuckling of elastic beam subjected to a concentrated moment within the span length of beam

Phungpaingam

Chucheepsakul

2007

Acta Mech Sin

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“…Goto et al [55] used elliptic integrals to derive a solution for plane elastica with axial and shear deformations. Zienkiewicz [57] to investigate a uniform simply supported beam subjected (1) to a nonsymmetrical concentrated load and (2) to a uniformly distributed load over a portion of its span. Denman and R. Schmidt [56] solved the problem of large deflection of thin elastica rods subjected to concentrated loads by using a Chebyshev approximation method.…”

Section: Basic Theories and Principles Of Nonlinear Beam Deformationsmentioning

confidence: 99%

Nonlinear Structural Engineering

2006

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“…Lippmann et al (1961) used an analog computer to integrate the elastica equation for end angles less than 60°. Wang et al (1961) used a finite difference scheme to solve the same problem and compared the numerical results with experiment. Christensen (1962) used a Ritz method to study the large deflection of a simply supported beam under combined axial and lateral loads.…”

Section: Introductionmentioning

confidence: 99%

Slip-through of a heavy elastica on point supports

Chen

2010

International Journal of Solids and Structures

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a b s t r a c tIn this paper, we study the deformation and stability of a heavy elastica resting symmetrically on two frictionless point supports on the same horizontal level. The static analysis finds multiple equilibria when the distance of the two point supports is smaller than a certain value. In order to determine whether these equilibria are stable, a dynamic analysis is conducted to calculate their natural frequencies. In order to take into account the sliding between the elastica and the smooth point supports during vibration, a dynamic analysis based on an Eulerian description is adopted. It is found that stable equilibrium can exist only when the half support span a is between two limits a ðsÞ min and a ðsÞ max . This range depends on a dimensionless ratio between the weight density and the flexural rigidity of the elastica. When an a between a ðsÞ min and a ðsÞ max decreases and approaches a ðsÞ min , the elastica will slip away from the side. On the other hand, when an a between a ðsÞ min and a ðsÞ max increases and approaches a ðsÞ max , the elastica will slip through between the two supports.

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A numerical analysis of large deflections of beams

Cited by 56 publications

References 12 publications

Postbuckling of elastic beam subjected to a concentrated moment within the span length of beam

Postbuckling of elastic beam subjected to a concentrated moment within the span length of beam

Nonlinear Structural Engineering

Slip-through of a heavy elastica on point supports

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