Torsional Buckling of Columns by the Lyapunov Method

1985 ◽  
Vol 29 (03) ◽  
pp. 189-193
Author(s):  
T. A. I. Akeju
Keyword(s):  
Metric Space ◽  
Lyapunov Functional ◽  
Lyapunov Method ◽  
Direct Method ◽  
Stability Theorem ◽  
Torsional Buckling ◽  
Buckling Loads ◽  
Relative Ease ◽  
Flexural Buckling ◽  
Cubic Equation

The paper presents an application of Lyapunov's direct method to torsional and torsional-flexural buckling of columns. A metric space and a Lyapunov functional are proposed for each of the problems. Making use of Zubov's stability theorem and appropriate eigenvalue inequalities, the functionals yield the expressions for the buckling loads for simple and fixed supports. Of particular interest is the relative ease with which the expressions are derived, especially in the torsional-flexural buckling case, where it has not been necessary to seek the roots of the traditional cubic equation which governs the critical loads of the member.

The Computation of Flexural–Torsional Buckling Loads

1952 ◽  
Vol 19 (2) ◽  
pp. 214-219
Author(s):  
H. F. Michielsen
Keyword(s):  
Buckling Load ◽  
Trial And Error ◽  
Torsional Buckling ◽  
Buckling Loads ◽  
Cubic Equation ◽  
Flexural Torsional Buckling ◽  
Ordinary Means

Abstract Solution by ordinary means of the cubic equation for the torsional buckling load, first derived by Kappus, often leads to insufficiently accurate results. A method is presented which avoids the loss of accuracy involved. The procedure is straightforward and no graphical means are involved. However, one formula is to be solved by trial and error.

Flexural-Torsional Buckling of Pultruded Fiber-Reinforced Polymer Angle Beams under Eccentric Loading

2020 ◽  
Vol 982 ◽  
pp. 201-206
Author(s):  
Jaksada Thumrongvut ◽  
Natthawat Pakwan ◽  
Samaporn Krathumklang
Keyword(s):  
Buckling Load ◽  
Fiber Reinforced Polymer ◽  
Width Ratio ◽  
Fiber Reinforced ◽  
Torsional Buckling ◽  
Eccentric Load ◽  
Buckling Loads ◽  
Reinforced Polymer ◽  
Cross Sectional Dimension ◽  
Flexural Torsional Buckling

In this paper, the experimental study on the pultruded fiber-reinforced polymer (pultruded FRP) angle beams subjected to transversely eccentric load are presented. A summary of critical buckling load and buckling behavior for full-scale flexure tests with various span-to-width ratios (L/b) and eccentricities are investigated, and typical failure mode are identified. Three-point flexure tests of 50 pultruded FRP angle beams are performed. The E-glass fibre/polyester resin angle specimens are tested to examine the effect of span-to-width ratio of the beams on the buckling responses and critical buckling loads. The angle specimens have the cross-sectional dimension of 76x6.4 mm with span-to-width ratios, ranging from 20 to 40. Also, four different eccentricities are investigated, ranging from 0 to ±2e. Eccentric loads are applied below the horizontal flange in increments until beam buckling occurred. Based upon the results of this study, it is found that the load and mid-span vertical deflection relationships of the angle beams are linear up to the failure. In contrast, the load and mid-span lateral deflection relationships are geometrically nonlinear. The general mode of failure is the flexural-torsional buckling. The eccentrically loaded specimens are failed at critical buckling loads lower than their concentric counterparts. Also, the quantity of eccentricity increases as buckling load decreases. In addition, it is noticed that span-to-width ratio increases, the buckling load is decreased. The eccentric location proved to have considerable influence over the buckling load of the pultruded FRP angle beams.

Cross-sectional properties of cold-formed angles

1984 ◽  
Vol 11 (3) ◽  
pp. 649-655 ◽  
Author(s):  
Murty K. S. Madugula ◽  
Sujit K. Ray
Keyword(s):  
Key Words ◽  
Cross Sectional Area ◽  
Sectional Area ◽  
Cross Sectional ◽  
Buckling Loads ◽  
Moments Of Inertia ◽  
Flexural Buckling ◽  
Failure Loads ◽  
Shear Centre

Cross-sectional properties of both equal and unequal leg cold-formed angle sections are presented. Besides cross-sectional area, location of centroid, moments of inertia, and torsional constant, the properties listed include the location of shear centre and the magnitude of warping constant. These two latter properties are required for determining failure loads of angles subjected to torsional–flexural buckling. Also listed are two important parameters, β1, and β2, that are required for the calculation of theoretical buckling loads of eccentrically loaded columns. Key words: buckling, cold-formed angles, columns, cross-sectional properties, shear centre, stability, torsional–flexural buckling, warping constant.

scholarly journals Stability with Respect to a Part of Variables under Constant Perturbations of the Partial Equilibrium Position of Differential Equation Nonlinear Systems

2018 ◽  
Vol 28 (3) ◽  
pp. 344-351
Author(s):  
Pavel P. Lipasov ◽  
Vladimir N. Shchennikov
Keyword(s):  
Nonlinear Systems ◽  
Equilibrium Position ◽  
Lyapunov Method ◽  
Stability Theorem ◽  
Partial Equilibrium ◽  
Dynamic Processes ◽  
Stability Theorems ◽  
Research Objects ◽  
Controlled Motion ◽  
The Stability

Introduction. It is impossible to take into account all the forces acting in the process of mathematical modeling of dynamic processes. In order that mathematical models the most accurately describe the dynamic processes, they must include the terms that correspond the constant perturbations. These problems arise in applied tasks. In this paper we consider the case when the system allows for the partial equilibrium position. The aim of this work is to prove the stability theorem for the partial equilibrium position at constant perturbations, which are small at every instant. Materials and Methods. The research objects are nonlinear systems of differential equations that allow for a partial equilibrium position. Using the second Lyapunov method, there are proved the stability theorems for the constant perturbations of the partial equilibrium position, which are small at every instant. Results. Together with the introduction of stability for a part of the variables, it has become necessary to introduce stability for the part of phase variables under constant perturbations. The first stability theorem of the part of phase variables under constant perturbations was obtained by A. S. Oziraner. In this work, we prove a theorem of the stability of the constant perturbations of the partial equilibrium position, small at every instant. It should be noted that there is no stability theorems of constant perturbations for the partial equilibrium position. Thus, the theorem proved in this work is of a pioneer nature. Conclusions. The theorem 3 proved in the work is the development of the mathematical theory of stability. The results of this work are applicable in the mechanics of controlled motion, nonlinear system.

Stable Control of Multi-link Manipulator Using Collision Phenomena

1991 ◽  
Vol 3 (6) ◽  
pp. 482-490
Author(s):  
Yasumasa Shoji ◽  
◽  
Makoto Inaba ◽  
Toshio Fukuda ◽  
Hidemi Hosokai ◽  
...  
Keyword(s):  
Lyapunov Method ◽  
Direct Method ◽  
Type Model ◽  
Lyapunov Direct Method ◽  
Loss Parameter ◽  
Flexible Wall ◽  
Stabilization Effect ◽  
The Stability ◽  
The Impact ◽  
Stable Control

In this paper, a methodology using the Lyapunov direct method is proposed to analyze the stability of a multi-link manipulator system, which is positioned on a flexible wall, with collision phenomenon. The stability and response of the system are examined by parameter studies of numerical simulation. Because industrial demands for rapid motion of robotics have been increasing in order to achieve higher efficiency, collision has become a problem because every task involves contact when a manipulator interacts with an object. However, few research has been initiated to overcome this problem. In this paper, we employ a Hertz-type model which includes an energy loss parameter to express the impact force between the manipulator and the wall. Using this model, we have verified the stabilization effect of collision by the Lyapunov method. The effect has been confirmed by simulation. As a result, stable positioning of the manipulator on a flexible wall is assured, and the use of collision is sometimes effective to control the manipulator to performs tasks with rapid contact.

Derivatives of Buckling Loads and Vibration Frequencies With Respect to Stiffness and Initial Strain Parameters

1990 ◽  
Vol 57 (1) ◽  
pp. 18-24 ◽  
Author(s):  
Raphael T. Haftka ◽  
Gerald A. Cohen ◽  
Zenon Mro´z
Keyword(s):  
Virtual Work ◽  
Direct Method ◽  
General Purpose ◽  
Initial Strain ◽  
Nonlinear Structures ◽  
Vibration Frequencies ◽  
Finite Element Program ◽  
Buckling Loads ◽  
Element Program ◽  
Derivatives Of

A uniform variational approach to sensitivity analysis of vibration frequencies and bifurcation loads of nonlinear structures is developed. Two methods of calculating the sensitivities of bifurcation buckling loads and vibration frequencies of nonlinear structures, with respect to stiffness and initial strain parameters, are presented. A direct method requires calculation of derivatives of the prebuckling state with respect to these parameters. An adjoint method bypasses the need for these derivatives by using instead the strain field associated with the second-order post-buckling state. An operator notation is used and the derivation is based on the principle of virtual work. The derivative computations are easily implemented in structural analysis programs. This is demonstrated by examples using a general purpose, finite element program and a shell-of-revolution program.

Effects of Changing Double Angles Configuration Used as Braces in Seismic Resisting Systems

2018 ◽  
Vol 763 ◽  
pp. 279-286
Author(s):  
Carlos Bermudez ◽  
Oscar Gutierrez
Keyword(s):  
Compressive Strength ◽  
Slenderness Ratio ◽  
Limit State ◽  
Experimental Results ◽  
Limit States ◽  
Torsional Buckling ◽  
Critical Limit ◽  
Flexural Buckling ◽  
Flexural Torsional Buckling ◽  
Buckling Length

Seismic resisting systems consisting of double angles are used in many parts of the world. Generally, these double angles are arranged in the shape of a T, with a very small distance between them. However, sometimes these angles are distanced and faced in order to improve their mechanical characteristics about the axis of symmetry. In the past, their design was made in the same way as the double angles arranged in a T shape, that is, considering the limit states of flexural buckling and buckling by flexural-torsional, but ignoring the properties of the connectors and their effect on the modified slenderness ratio, as well as the fact that in this case the warping constant is not negligible. These parameters are taken into account in this research in order to study the effects of increasing the distance between the connectors and their possible use as braces in seismic resisting systems. The theoretical results were compared with the experimental results of fifty-seven specimens tested in the laboratory of structures of the Universidad Nacional de Colombia – Sede Manizales. The models were classified according to the main angles, the connectors, the total lengths, and the width of separation. All of them were subjected to axial compressive stress, with free rotation at both ends. Three identical specimens of each model were constructed. The flexural buckling length about x-axis was limited to two meters in all specimens tested whereas the flexural bucking length about y-axis and flexural-torsional buckling length were not limited, i.e. these lengths are equivalent to the total length of each specimen tested. This in order that the critical limit state was to be the flexural-torsional buckling as a function of the torsional buckling term in Z, except in the models of class 2 in which this induced condition was not reached. This was proposed to better evaluate the torsional buckling term in Z. The experimental results show that the nominal compressive strength for the flexural-torsional buckling limit state, when it is governed by torsion, is undervalued. A new methodology is proposed for the calculation of the nominal compressive strength for the flexural-torsional buckling limit state, when it is governed by torsion.

Torsional Buckling of the Hubble Space Telescope Solar Arrays

1998 ◽  
Vol 13 (2) ◽  
pp. 65-74
Author(s):  
Earl A. Thornton ◽  
David L Eby ◽  
Peter W. Chung
Keyword(s):  
Cross Section ◽  
Hubble Space Telescope ◽  
Solar Array ◽  
Torsional Buckling ◽  
Buckling Loads ◽  
Space Telescope ◽  
Linear Elastic ◽  
Solar Arrays ◽  
Buckling Behavior ◽  
Open Cross Section

The buckling behavior of a flexible rolled-up solar array used on the Hubble Space Telescope (HST) is investigated analytically and experimentally. Analytically, the solar array is modeled assuming the booms are linear, elastic beams of open cross-section, and the solar blanket is represented as an inextensible membrane. The analyses determine critical buckling loads for flexure and torsion. The behavior of the solar array is also investigated by experiments conducted using a model solar array with (1) closed cross-section booms, and (2) tubular booms like those used on the HST. The analyses and experiments show that the Hubble's solar arrays were deployed with a preload that caused them to buckle in torsion. Based on the study, an hypothesis is suggested for the failure of the solar array's booms that was discovered by the astronauts in 1993.

Characterizing Natural-Fracture Permeability From Mud-Loss Data

SPE Journal ◽  
2010 ◽  
Vol 16 (01) ◽  
pp. 111-114 ◽  
Author(s):  
Jinsong Huang ◽  
D. V. Griffiths ◽  
Sau-Wai Wong
Keyword(s):  
Direct Method ◽  
Transient State ◽  
Natural Fracture ◽  
Curve Matching ◽  
Fracture Permeability ◽  
Type Curves ◽  
Cubic Equation ◽  
Matching Technique ◽  
Mud Loss ◽  
Insight Into

Summary Liétard et al. (1999, 2002) have provided important insight into the mechanism and prediction of transient-state radial mud invasion in the near-wellbore region. They provided type curves describing mud-loss volume vs. time that allow the hydraulic width of natural fractures to be estimated through a curve-matching technique. This paper describes a simpler and more direct method for estimating the hydraulic width by the solution of a cubic equation, with input parameters given by the well radius rw, the overpressure ratio Δp/τy, and the maximum mud loss volume (Vm)max.

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