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1 Department of Civil and Environmental Engineering Sungkyunkwan University 비대칭 박벽보의 개선된 해석이론 및 방법 An Improved Theory and Analysis Procedures of Nonsymmetric Thin-walled Beams

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Contents Introduction - Historical review - Scope of this research Linearized Principle of Virtual Work Displacement Field of Nonsymmetric Thin-walled beam Derivation of Exact Dynamics Stiffness Matrix - 14 displacement parameters - 14 displacement parameters - Force-displacement relations - Force-displacement relations - Exact dynamic stiffness matrix - Exact dynamic stiffness matrix Straight Beam Element Numerical Examples Conclusions2

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Introduction Historical review (1) Argyris, J.H. Argyris, J.H. Dunne, P.C. Scharpf, D.W. Scharpf, D.W. Kim, M.Y. Chang, S.P. Chang, S.P. Kim, S.B. “On large displacement-small strain analysis of structures with rotational degrees of freedom” Saleeb, A.F. Chang, T.Y.P. Chang, T.Y.P. Gendy, A.S. “Spatial stability and free vibration of shear flexible thin- walled elastic beams. I: Analytical approach II: Numerical approach” II: Numerical approach” 1994 “Effective modelling of spatial buckling of beam assemblages accounting for warping constraints and assemblages accounting for warping constraints and ratation-dependency of moments” ratation-dependency of moments” Kim, M.Y. Chang, S.P. Chang, S.P. Kim, S.B. “Spatial stability analysis of thin-walled space frame”

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Historical review (2) Friberg, P.O. Banejee, J.R. Kim, S.B. Kim, M.Y. “New numerical scheme based on the quadratic eigenproblem of thin-walled beam with open cross section” “Generalized formulation which is believed to improve Friberg’s method” “Improved formulation for spatial stability and free vibration of thin-walled tapered beams and space frame” Leung, A.Y.T Zeng, S.P. “Coupled bending-torsional dynamic stiffness considering for Timoshenko beam” 1998

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5 In order to demonstrate the accuracy of this study, the natural frequencies and buckling loads are evaluated and compared with analytic solutions and F.E solutions Most of previous finite element formulations of thin-walled beam introduce approximate displacement fields by using the shape function In this research, a improved formulation for free vibration and spatial stability of thin-walled beam is developed For the general case of loading and boundary conditions, it is very difficult to obtain closed form solutions for natural frequencies and buckling loads of thin-walled beam A clearly consistent numerical procedure which generates an exact dynamic stiffness matrix of thin-walled beam is presented. Scope of this research

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Linearized Principle of Virtual Work 6 Equilibrium Equation for General Continuum where Strain Linearized Equilirium Equation where v

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Displacement Field of Nonsymmetric Thin-walled beam 7 Notation for displacement parameters and stress resultants (a) Displacement parameters (a) Displacement parameters (b) Stress resultants (b) Stress resultants

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8 Displacement Field Warping function: First-order terms of displacement parameter Second-order terms of displacement parameter

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9 Cross-section constants

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10 Stress Resultants

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11 Potential energy of the thin-walled beam Where

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13 Governing equations of thin-walled beam

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14 Force-displacement relationship

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Derivation of Exact Dynamic Stiffness Matrix 14 displacement parameters where,15

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16and, Differential equation of the first order with constant coefficient

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17where, In matrix form

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General solution Complex eigen analysis by using IMSL subroutine DGVCRG eigenvalues 14 14 eigenvectors Eigen problem of nonsymmetric matrix

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20where, In matrix form is obtained from Nodal displacement vector

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Compute complex inverse matrix by using IMSL subroutine DLINCG Compute complex inverse matrix by using IMSL subroutine DLINCG21 Displacement state vector

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22where, In matrix form

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23 Exact dynamic stiffness matrix where, Nodal force vector where,

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Straight Beam Element Straight Beam Element24 Shape functions : Linear Hermitian polynomial : Cubic Hermitian polynomial Equilibrium equation

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Numerical Examples Numerical Examples25 1. Free vibration of simply supported thin-walled beam Sectional property

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Analysis results Analysis results26 Tab.1 Natural frequencies of simply supported beam

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27 2. Free vibration of thin-walled cantilever and fixed beam Cantilever beam Fixed beam Nonsymmetric channel section

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28 Tab.2 Natural frequencies of cantilever beam

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29 Tab.3 Natural frequencies of fixed beam

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30 3. Buckling of thin-walled cantilever beam under axial load Tab.4 Flexural-torsional buckling loads for cantilever beam [N]

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Conclusions Conclusions31 A consistent numerical procedure which generates an exact dynamic stiffness matrix of nonsymmetric thin-walled beam is presented Numerical results by the present method are in a good agreement with those by thin-walled beam elements and ABAQUS’s shell elements. Present procedure is general and provides a systematic tool for the numerical evaluation of exact solution of ordinary differential equation A improved formulation for spatial stability and free vibration of nonsymmetric thin-walled beam is developed

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