The Effect of Principal Bending Curvature on the Lateral Buckling of Uniform Slender Beams

1977 ◽  
Vol 44 (2) ◽  
pp. 311-316
Author(s):  
D. A. Peters
Keyword(s):  
Numerical Solutions ◽  
Order Effect ◽  
Buckling Load ◽  
Beam Deflection ◽  
Cantilever Beams ◽  
Pure Bending ◽  
Lateral Buckling ◽  
First Order ◽  
Slender Beams ◽  
Support Conditions

The general lateral buckling equation is developed for a uniform, slender, simply supported beam fixed in torsion and with a load applied at the shear center of the midspan cross section. In this general equation, the effect of principal bending curvature (i.e., beam deflection prior to buckling) is completely accounted for. Therefore, a distinction is made between beams fixed in torsion about the deformed or undeformed elastic axis, and distinct boundary conditions are derived for each case. The equations for each of the two support conditions are then specialized to include only the first-order effect of principal bending curvature and these equations are compared with similar equations for cantilever beams and beams in pure bending. Finally, simplified buckling load formulas are derived and compared with numerical solutions of the general equations for each of the lateral buckling configurations. The comparison shows that the approximate formulas provide good estimates for the buckling load and that the classical buckling load formulas that neglect principal bending curvature are not always conservative for infinitely slender beams.

scholarly journals Experimental and Numerical Buckling Analyses of Laminated Composite Plates under Temperature Effects

2010 ◽  
Vol 19 (4) ◽  
pp. 096369351001900 ◽  
Author(s):  
Emin Ergun
Keyword(s):  
Composite Plate ◽  
Numerical Solutions ◽  
Laminated Composite ◽  
Composite Plates ◽  
Buckling Load ◽  
Fibre Reinforcement ◽  
Stacking Sequence ◽  
Laminated Composite Plates ◽  
Different Temperatures ◽  
Good Agreement

The aim of this study is to investigate, experimentally and numerically, the change of critical buckling load in composite plates with different ply numbers, orientation angles, stacking sequences and boundary conditions as a function of temperature. Buckling specimens have been removed from the composite plate with glass-fibre reinforcement at [0°]i and [45°]i (i= number of ply). First, the mechanical properties of the composite material were determined at different temperatures, and after that, buckling experiments were done for those temperatures. Then, numerical solutions were obtained by modelling the specimens used in the experiment in the Ansys10 finite elements package software. The experimental and numerical results are in very good agreement with each other. It was found that the values of the buckling load at [0°] on the composite plates are higher than those of other angles. Besides, symmetrical and anti-symmetrical conditions were examined to see the effect of the stacking sequence on buckling and only numerical solutions were obtained. It is seen that the buckling load reaches the highest value when it is symmetrical in the cross-ply stacking sequence and it is anti-symmetrical in the angle-ply stacking sequence.

scholarly journals Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Darae Jeong ◽  
Yibao Li ◽  
Chaeyoung Lee ◽  
Junxiang Yang ◽  
Yongho Choi ◽  
...  
Keyword(s):  
Parabolic Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Second Order ◽  
Order Convergence ◽  
Numerical Convergence ◽  
Verification Method ◽  
First Order ◽  
Convergence Test ◽  
Second Order Convergence

In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

On the lateral buckling of uniform slender cantilever beams

1975 ◽  
Vol 11 (12) ◽  
pp. 1269-1280 ◽  
Author(s):  
Dewey H. Hodges ◽  
David A. Peters
Keyword(s):  
Cantilever Beams ◽  
Lateral Buckling

First-Order Frequency Effects in Supersonic Panel Flutter of Finite Cylindrical Shells

1973 ◽  
Vol 40 (2) ◽  
pp. 464-470
Author(s):  
M. Holt ◽  
T. M. Lee
Keyword(s):  
Mach Number ◽  
Elastic Shell ◽  
Order Effect ◽  
Aerodynamic Load ◽  
Panel Flutter ◽  
Thin Cylindrical Shell ◽  
First Order ◽  
Flutter Boundary ◽  
Shell Equation ◽  
Set Up

An improved calculation of the supersonic panel flutter characteristics of a thin cylindrical shell of finite length is presented. The aerodynamic load is determined with account taken of first-order terms in vibration frequency, and when this is introduced into the elastic shell equation an integro differential equation results. An equivalent eigenvalue problem is set up by applying Galerkin’s method to this equation. The flutter boundary, for given Mach number and circumferential mode n, corresponds to the shell thickness ratio at which the real part of any one of the eigenvalues first becomes non-negative. It is found that the most severe flutter condition, for given Mach number, occurs for a circumferential mode n = 7. The present calculations exclude second-order frequency terms in the elastic part of the flutter equation, even though they may have a first-order effect. A subsequent calculation referred to here shows that these terms indeed have no significant influence on the first-order analysis.

Application of Homotopy Perturbation Method for Harmonically Forced Duffing Systems

2011 ◽  
Vol 110-116 ◽  
pp. 2277-2283 ◽  
Author(s):  
Xiang Meng Zhang ◽  
Ben Li Wang ◽  
Xian Ren Kong ◽  
A Yang Xiao
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Primary Resonance ◽  
Effective Technique ◽  
Duffing System ◽  
First Order ◽  
Homotopy Perturbation ◽  
Analytical Approximations ◽  
Case Based

In this paper, He’s homotopy perturbation method (HPM) is applied to solve harmonically forced Duffing systems. Non-resonance of an undamped Duffing system and the primary resonance of a damped Duffing system are studied. In the former case, the first-order analytical approximations to the system’s natural frequency and periodic solution are derived by HPM, which agree well with the numerical solutions. In the latter case, based on HPM, the first-order approximate solution and the frequency-amplitude curves of the system are acquired. The results reveal that HPM is an effective technique to the forced Duffing systems.

Parametric Deflection Approximations for End-Loaded, Large-Deflection Beams in Compliant Mechanisms

1995 ◽  
Vol 117 (1) ◽  
pp. 156-165 ◽  
Author(s):  
L. L. Howell ◽  
A. Midha
Keyword(s):  
Closed Form ◽  
Large Deflection ◽  
Numerical Solutions ◽  
Compliant Mechanisms ◽  
Elliptic Integral ◽  
Cantilever Beams ◽  
Simple Method ◽  
Body Angle ◽  
Analysis And Design ◽  
Integral Solutions

Geometric nonlinearities often complicate the analysis of systems containing large-deflection members. The time and resources required to develop closed-form or numerical solutions have inspired the development of a simple method of approximating the deflection path of end-loaded, large-deflection cantilever beams. The path coordinates are parameterized in a single parameter called the pseudo-rigid-body angle. The approximations are accurate to within 0.5 percent of the closed-form elliptic integral solutions. A physical model is associated with the method, and may be used to simplify complex problems. The method proves to be particularly useful in the analysis and design of compliant mechanisms.

Buckling and Post Buckling Behavior of a Transversely Stiffened Ship Hull Model

1964 ◽  
Vol 8 (04) ◽  
pp. 7-21
Author(s):  
H.G. Schultz
Keyword(s):  
Reasonable Agreement ◽  
Ship Hull ◽  
Buckling Load ◽  
Experimental Results ◽  
Postbuckling Behavior ◽  
Pure Bending ◽  
Box Girder ◽  
Buckling Behavior ◽  
Theoretical Investigations ◽  
Post Buckling

In the paper presented the behavior of a transversely formed box-girder model subjected to pure bending is discussed, where the deck plating of the model is loaded above the buckling load. The experimental results obtained are in reasonable agreement with theoretical investigations and show the influence of fabrication initiated plate deflections on the buckling and postbuckling behavior of the deck plating clearly. A method is suggested for determining the buckling load of plates having large initial deformations.

scholarly journals Experimental Investigations of Impact Damage Influence on Behavior of Thin-Walled Composite Beam Subjected to Pure Bending

Materials ◽  
2019 ◽  
Vol 12 (7) ◽  
pp. 1127 ◽  
Author(s):  
Tomasz Kubiak ◽  
Lukasz Borkowski ◽  
Nina Wiacek
Keyword(s):  
Impact Damage ◽  
Full Range ◽  
Image Correlation ◽  
Bending Test ◽  
Beam Deflection ◽  
Experimental Investigations ◽  
Pure Bending ◽  
Channel Section ◽  
Thin Walled ◽  
The Impact

The paper deals with buckling, postbuckling, and failure of pre-damaged channel section beam subjected to pure bending. The channel section beams made of eight-layered GFRP laminate with different symmetrical layups have been considered. The specimens with initially pre-damaged web or flange were investigated to access the influence of impact damage on work of thin-walled structure in the full range of load till failure. The bending tests of initially pre-damage beams have been performed on a universal tensile machine with especially designed grips. The digital image correlation system allowing to follow the beam deflection have been employed. The experimentally obtained results are presented in graphs presenting load-deflection or load vs. angle of rotation relations and in photos presenting impact damages areas before and after bending test. The results show that the impact pre-damages have no significant influence on the work of channel section beams.

scholarly journals On incorporating warping effects due to transverse shear and torsion into the theories of straight elastic bars

2020 ◽  
Author(s):  
T. Lewiński ◽  
S. Czarnecki
Keyword(s):  
Cross Sections ◽  
Transverse Shear ◽  
Explicit Construction ◽  
Effective Characteristics ◽  
Warping Function ◽  
Torsional Buckling ◽  
Lateral Buckling ◽  
First Order ◽  
Thin Walled ◽  
Scientific Tool

Abstract By endowing El Fatmi’s theories of bars with first-order warping functions due to torsion and shear, a family of theories of bars, of various applicability ranges, is effectively constructed. The theories thus formed concern bars of arbitrary cross-sections; they are reformulations of the mentioned theories by El Fatmi and theories by Kim and Kim, Librescu and Song, Vlasov and Timoshenko. The Vlasov-like theory thus developed is capable of describing the torsional buckling and lateral buckling phenomena of bars of both solid and thin-walled cross-sections, which reflects the non-trivial correspondence, noted by Wagner and Gruttmann, between the torsional St.Venant’s warping function and the contour-wise defined warping functions proposed by Vlasov. Moreover, the present paper delivers an explicit construction of the constitutive equations of Timoshenko’s theory; the equations linking transverse forces with the measures of transverse shear turn out to be coupled for all bars of asymmetric cross-sections. The modeling is hierarchical: the warping functions are numerically constructed by solving the three underlying 2D scalar elliptic problems, providing the effective characteristics for the 1D models of bars. The 2D and 1D problems are indissolubly bonded, thus forming a unified scientific tool, deeply rooted in the hitherto existing knowledge on elasticity of elastic straight bars.

Improved Approach to the Streamline Curvature Method in Turbomachinery

1987 ◽  
Vol 109 (3) ◽  
pp. 213-217 ◽  
Author(s):  
S. Abdallah ◽  
R. E. Henderson
Keyword(s):  
Flow Field ◽  
Velocity Gradient ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Second Order ◽  
Curvilinear Coordinates ◽  
First Order ◽  
Streamline Curvature ◽  
Stators And Rotors ◽  
Streamline Curvature Method

Quasi three dimensional blade-to-blade solutions for stators and rotors of turbomachines are obtained using the Streamline Curvature Method (SLCM). The first-order velocity gradient equation of the SLCM, traditionally solved for the velocity field, is reformulated as a second-order elliptic differential equation and employed in tracing the streamtubes throughout the flow field. The equation of continuity is then used to calculate the velocity. The present method has the following advantages. First, it preserves the ellipticity of the flow field in the solution of the second-order velocity gradient equation. Second, it eliminates the need for curve fitting and strong smoothing under-relaxation in the classical SLCM. Third, the prediction of the stagnation streamlines is a straightforward matter which does not complicate the present procedure. Finally, body-fitted curvilinear coordinates (streamlines and orthogonals or quasi-orthogonals) are naturally generated in the method. Numerical solutions are obtained for inviscid incompressible flow in rotating and non-rotating passages and the results are compared with experimental data.

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