Contents Introduction Introduction - - Historical review Historical review - - Scope of this research Scope of this research Linearized Principle of Virtual Work Linearized Principle of Virtual Work Displacement Field of Nonsymmetric Thin- Displacement Field of Nonsymmetric Thin- walled beam walled beam Derivation of Exact Dynamics Stiffness M Derivation of Exact Dynamics Stiffness M atrix atrix - 14 displacement parameters - 14 displacement parameters - Force-displacement relations - Force-displacement relations - Exact dynamic stiffness matrix - Exact dynamic stiffness matrix Straight Beam Element Straight Beam Element Numerical Examples Numerical Examples Conclusions Conclusions 2
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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.
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Kim, M.Y.Kim, M.Y. Chang, S.P.Chang, S.P.Kim, S.B.Kim, S.B.
““On large displacement-small strain analysis of structures On large displacement-small strain analysis of structures with rotational degrees of freedom”with rotational degrees of freedom”
““Spatial stability and free vibration of shear flexible thin-wSpatial stability and free vibration of shear flexible thin-walled elastic beams. I: Analytical approachalled elastic beams. I: Analytical approach
II: Numerical approach”II: Numerical approach”
19941994
““Effective modelling of spatial buckling of beamEffective modelling of spatial buckling of beam assemblages accounting for warping constraints andassemblages accounting for warping constraints and
ratation-dependency of moments”ratation-dependency of moments”
Kim, M.Y.Kim, M.Y. Chang, S.P.Chang, S.P.Kim, S.B.Kim, S.B.
““Spatial stability analysis of thin-walled space frame”Spatial stability analysis of thin-walled space frame”
Historical review (2)Historical review (2)
Friberg, P.O.Friberg, P.O.
Banejee, J.R.Banejee, J.R.
Kim, S.B.Kim, S.B.Kim, M.Y. Kim, M.Y.
““New numerical scheme based on the quadratic eigenproblNew numerical scheme based on the quadratic eigenproblem of thin-walled beam with open cross section”em of thin-walled beam with open cross section”
““Generalized formulation which is believed to improve FriGeneralized formulation which is believed to improve Friberg’s method”berg’s method”
““Improved formulation for spatial stability and free Improved formulation for spatial stability and free vibration of thin-walled tapered beams and space frame”vibration of thin-walled tapered beams and space frame”
In order to demonstrate the accuracy of this study, the natural frequencies In order to demonstrate the accuracy of this study, the natural frequencies and buckling loads are evaluated and compared with analytic solutions and buckling loads are evaluated and compared with analytic solutions and F.E solutionsand F.E solutions
Most of previous finite element formulations of thin-walled beam Most of previous finite element formulations of thin-walled beam introduce approximate displacement fields by using the shape functionintroduce approximate displacement fields by using the shape function
In this research, a improved formulation for free vibration and spatial In this research, a improved formulation for free vibration and spatial stability of thin-walled beam is developedstability of thin-walled beam is developed
For the general case of loading and boundary conditions, it is very difficult For the general case of loading and boundary conditions, it is very difficult to obtain closed form solutions for natural frequencies and buckling loads to obtain closed form solutions for natural frequencies and buckling loads of thin-walled beamof thin-walled beam
A clearly consistent numerical procedure which generates an exact dynamic A clearly consistent numerical procedure which generates an exact dynamic stiffness matrix of thin-walled beam is presented.stiffness matrix of thin-walled beam is presented.
Scope of this researchScope of this research
Linearized Principle of Virtual WorkLinearized Principle of Virtual Work
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Equilibrium Equation for General ContinuumEquilibrium Equation for General Continuum
Tab.2 Natural frequencies of cantilever beamTab.2 Natural frequencies of cantilever beam 2( / sec)rad
mode Present study 5 beam element 10 beam element ABAQUS
1 0.027 0.027 0.027 0.028
2 0.336 0.337 0.336 0.331
3 0.707 0.709 0.707 0.696
4 1.074 1.076 1.075 1.074
5 4.859 4.880 4.860 4.766
6 7.186 7.232 7.189 7.083
7 18.22 18.45 18.24 17.95
8 20.15 20.26 20.16 19.36
9 24.39 24.73 24.42 23.58
10 47.34 48.77 47.54 46.52
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Tab.3 Natural frequencies of fixed beamTab.3 Natural frequencies of fixed beam 2( / sec)rad
mode Present study 5 beam element 10 beam element ABAQUS
1 0.910 0.912 0.910 0.914
2 3.115 3.129 3.116 3.046
3 5.803 5.850 5.806 5.780
4 17.66 17.82 17.68 17.24
5 18.60 19.03 18.63 18.31
6 20.61 20.66 20.61 19.20
7 44.83 46.77 45.02 43.99
8 59.01 60.61 59.12 56.81
9 91.21 115.0 92.07 87.77
10 150.1 152.9 150.6 127.6
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3. Buckling of thin-walled cantilever beam under axial load3. Buckling of thin-walled cantilever beam under axial load
mode Present study 8 Beam elements ABAQUS
1 13.800 13.800 14.001
2 112.55 112.56 113.10
3 191.84 191.84 190.08
4 258.54 258.77 256.67
5 414.76 416.05 408.53
6 526.71 526.73 509.74
7 571.33 571.33 546.49
Tab.4 Flexural-torsional buckling loads for cantilever beam [N]Tab.4 Flexural-torsional buckling loads for cantilever beam [N]
ConclusionsConclusions
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A consistent numerical procedure which generates an exact dynamic sA consistent numerical procedure which generates an exact dynamic stiffness matrix of nonsymmetric thin-walled beam is presentedtiffness matrix of nonsymmetric thin-walled beam is presented
Numerical results by the present method are in a good agreement withNumerical results by the present method are in a good agreement withthose by thin-walled beam elements and ABAQUS’s shell elements.those by thin-walled beam elements and ABAQUS’s shell elements.
Present procedure is general and provides a systematic tool for the Present procedure is general and provides a systematic tool for the numerical evaluation of exact solution of ordinary differential equationnumerical evaluation of exact solution of ordinary differential equation
A improved formulation for spatial stability and free vibration of noA improved formulation for spatial stability and free vibration of nonsymmetric thin-walled beam is developednsymmetric thin-walled beam is developed