特征值屈曲分析：Column Buckling

__________________________________________________________________________ Copyright © 2008 D. G. Taggart, University of Rhode Island. All rights reserved. Disclaimer. 1 ABAQUS Tutorial – Column Buckling Consider a 5 m column with a 10 cm circular cross-section (R=.05m) loaded in axial compression. The column is pinned at its ends. Determine the critical buckling modes and corresponding mode shapes Theoretical Solution The theoretical Euler buckling loads are given by For a steel column (E = 200 GPa) with I = 4.909e-6 m 4 , the critical buckling loads and mode shapes are given by Table 1. Theoretical Buckling Loads n Pcr 1 3.876e5 2 1.550e6 3 3.488e6 4 6.202e6 5 9.690e6 6 1.395e7 Finite Element solution (ABAQUS) Start => Programs => ABAQUS 6.7-1 => ABAQUS CAE Select 'Create Model Database' File => Save As => create directory for files Module: Sketch Sketch => Create Add=> Point => enter coordinates (0,0), (0,5) => select 'red X' Add => Line => Connected Line => select point at (0,0) with mouse, then (0,5) , right click => Cancel Procedure => Done 2 22 L IEn P cr π = First three mode shapes __________________________________________________________________________ Copyright © 2008 D. G. Taggart, University of Rhode Island. All rights reserved. Disclaimer. 2 Module: Part Part => Create => select 2D Planar, Deformable, Wire, Approx size 10 => Continue Add => Sketch => select 'Sketch-1' => Done => Done Module: Property Material => Create => Name: Material-1, Mechanical, Elasticity, Elastic => set Young's modulus = 200e9, Poisson's ratio = 0.3 => OK Profile => Create => Circular => r=.05 => OK Section => Create => Name: Section-1, Beam, Beam => Continue => Profile Name: Profile-1 => Linear Properties => E=200e9, G=77e9 => OK => OK Assign Section => select all elements by dragging mouse => Done => Section-1 => Done Assign Beam Section Orientation => select full model => Done => n1 direction = 0.0,0.0,-1.0 => Done Module: Assembly Instance => Create => Part-1 => OK Module: Step Step => Create => Name: Step-1, Linear Perturbation, Buckle => Continue => Number of Eigenvalues requested: 6 => OK Module: Load Load => Create => Name: Step-1, Step: Step 1, Mechanical, Concentrated Force => Continue => select point at (0,5) => Done => set CF 1 =0, CF 2 = -1 => OK BC => Create => Name: BC-1, Step: Step-1, Mechanical, Displacement/Rotation => Continue => select point at (0,0) => Done => U1=U2=0 BC => Create => Name: BC-1, Step: Step-1, Mechanical, Displacement/Rotation => Continue => select point at (0,5) => Done => U2=0 Module: Mesh Seed => Edge by Size => select full model by dragging mouse => Done => Element Size=.25 => press Enter => Done Mesh => Element Type => select full model by dragging mouse => Done => Element Library: Standard, Geometric Order: Linear, Family: Beam, Cubic interpolation (B23)=> OK => Done Mesh => Instance => OK to mesh the part Instance: Yes => Done Module: Job Job => Create => Name: Job-1, Model: Model-1 => Continue => Job Type: Full analysis, Run Mode: Background, Submit Time: Immediately => OK Job => Manager => Submit => Job-1 Results __________________________________________________________________________ Copyright © 2008 D. G. Taggart, University of Rhode Island. All rights reserved. Disclaimer. 3 Module: Visualization Result => Step.Frame => view Eigenvalues (Buckling Loads) - see Table 2 below Plot => Deformed Shape View => Graphics Options => Background Color => White Ctrl-C to copy viewport to clipboard => Open MS Word Document => Ctrl-V to paste image Plot=> Contours => Result => Field Output => select S, Max. Principal => Section Points => Category: 'beam general' => select section points at +/- 2.5 to view stress contours. Ctrl-C to copy viewport to clipboard => Open MS Word Document => Ctrl-V to paste image Report => Field Output => Setup => Number of Significant Digits => 6 Report => Field Output => Variable => Position: Unique Nodal => select U: Spatial Displacements, UR3: Rotational Displacements, S: Max. Principal => Apply Cut and paste tabulated results from 'abaqus.rpt' file to MS Word document. Table 2. Buckling Loads (FEA) __________________________________________________________________________ Copyright © 2008 D. G. Taggart, University of Rhode Island. All rights reserved. Disclaimer. 4 Buckled Mode Shapes: