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A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering

Buckling Analysis of Cellular Beams using the Element-Free Galerkin Method with the Rotational Spring Analogy. A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering. Contents. Brief Introduction Cellular Beams – Behaviour Current Method of Assessments

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A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering

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  1. Buckling Analysis of Cellular Beams using the Element-Free Galerkin Method with the Rotational Spring Analogy A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering

  2. Contents Brief Introduction Cellular Beams – Behaviour Current Method of Assessments Background of Proposed Model Planar Response – Geometric Stiffness Out-of-plane Analysis – Material Stiffness Buckling Analysis Approach Iterative Rank 2 Reduced Eigenvalue Problem Shifting Local Region Application Examples

  3. Introduction CELLULAR BEAMS • steel I-section beams with regular openings of circular shape throughout the web • advantages: • Better in-plane flexural resistance – enabling long clear spans • Significant building height reduction by integrating M&E services with the floor depth – reduced cost • Aesthetical value – large space without screening effects

  4. Introduction BEHAVIOUR • presence of web holes causes a highstress concentration in the narrow parts of the beams horizontal normal stress, x vertical normal stress, y shear stress, xy

  5. Introduction FAILURE MODES • development of local buckling, typically most critical in web-post and compressive regions around the openings WEB-POST TEE BUCKLING BUCKLING BUCKLING NEAR HOLES

  6. Introduction CURRENT ASSESSMENT METHODS • Finite Element Analysis (FEA) – continues to be computationally demanding • Simplified models • Lawson-2006 – a strut model to explain web-post buckling phenomena. • Ward-1990– semi-empirical models for web-post & tee buckling assessments. – calibrated against detailed FEA models – limited to specific geometries including layout and range of dimensions

  7. Introduction THE MAIN OBJECTIVE • looking for more efficient buckling analysis of cellular beams, with emphasis on elastic local buckling effects • extend the use of Element-Free Galerkin (EFG) method developed by Belytschkocombined with Rotational Spring Analogy (RSA) proposed by Izzuddin

  8. Introduction WHY EFG METHOD? • can be easily applied to irregular domains • potential efficiency in separating planar and out-of-plane responses unlike FEM • compared to MLPG, it ensures external equilibrium at sub-domain level between internal loading and boundary actions • facilitates the application of the RSA; - for example, the same fixed integration points can be used unlike MLPG

  9. Background

  10. Background PLANAR SYSTEM • established by assembling the planar responses of individual cells NODES INDIVIDUAL UNIT CELLS

  11. Background UNIT CELL ANALYSIS • discritised using the EFGmethod – via the moving least squares (MLS) technique • rigid body movement is preventedby means of simple supports atthe web-post

  12. Background REPRESENTATIVE ACTIONS • each cell utilising a reduced number of freedoms –four nodes located at the T-centroids

  13. Background PLANAR SYSTEM • system is solved globally using a standard discrete solution • realistic unit-based planar stress distribution is obtained x y xy

  14. Background GEOMETRIC STIFFNESS MATRIX • according to RSA:

  15. Background OUT-OF-PLANE RESPONSE • is obtained using the EFG method with Kirchhoff’s theory for thin plates • planar displacements assumed to be reasonablysmall – KE is determined with reference to the undeformed geometry

  16. Buckling analysis strategy • aims for efficiency and accuracy • discrete buckling assessment performed within a local region that consists of at most 3 unit cells • the lowest buckling load factor is determined by: • shifting the local region • using an iterative rank 2 reduced eigenvalue problem ...

  17. Buckling analysis strategy • SHIFTING LOCAL REGION • calculate KG from planar response • determine KE from out-of-plane analysis • eigenvalue analysis + iteration

  18. Buckling analysis strategy • SHIFTING LOCAL REGION • calculate KG from planar response • determine KE from out-of-plane analysis • eigenvalue analysis + iteration

  19. Buckling analysis strategy • SHIFTING LOCAL REGION • calculate KG from planar response • determine KE from out-of-plane analysis • eigenvalue analysis + iteration

  20. Application examples 1. WEB-POST BUCKLING • symmetric cellular beams • parent I-section = 1016305222UB • depth, Dp = 1603mm • diameter, Do = 1280mm • spacing, S = 1472mm • web thickness, tw = 16mm

  21. Application examples 1. WEB-POST BUCKLING • horizontal normal stress, x FEA:ADAPTIC PROPOSED EFG/RSA

  22. Application examples 1. WEB-POST BUCKLING • vertical normal stress, y FEA:ADAPTIC PROPOSED EFG/RSA

  23. Application examples 1. WEB-POST BUCKLING • shear stress, xy FEA:ADAPTIC PROPOSED EFG/RSA

  24. Application examples 1. WEB-POST BUCKLING c = 33.621 c = 33.173

  25. Application examples 1. WEB-POST BUCKLING FEA:ADAPTIC PROPOSED EFG/RSA

  26. Application examples 2. TEE BUCKLING • symmetric cellular beams • parent I-section = 1016305222UB • depth, Dp = 1603mm • diameter, Do = 840mm • spacing, S = 1472mm • web thickness, tw = 16mm

  27. Application examples 2. TEE BUCKLING c = 80.100 c = 79.695

  28. Application examples 2. TEE BUCKLING FEA:ADAPTIC PROPOSED EFG/RSA

  29. Application examples 3. BUCKLING AROUND THE OPENINGS • symmetric cellular beams • parent I-section = 1016305222UB • depth, Dp = 1603mm • diameter, Do = 1280mm • spacing, S = 2944mm • web thickness, tw = 16mm

  30. Application examples 3. BUCKLING AROUND THE OPENINGS c = 68.598 c = 67.122

  31. Application examples 3. BUCKLING AROUND THE OPENINGS FEA:ADAPTIC PROPOSED EFG/RSA

  32. Conclusion • effective method for local buckling analysis of cellular beams, combining EFG with RSA • shifting local region approach provides significant computational benefit • ability to predict accurately different forms of local buckling • not only applicable to regular cellular beams but also to other irregular forms

  33. Thank you A.R. Zainal Abidin and B.A. IzzuddinDepartment of Civil and Environmental Engineering

  34. Appendix ITERATIVE RANK 2 REDUCED EIGENVALUE PROBLEM • determine the 2 probing modes: an initial assumed mode (UA) and its complementary mode (UB)

  35. Appendix ITERATIVE RANK 2 REDUCED EIGENVALUE PROBLEM • the 2 modes are then used to formulate a rank 2 eigenvalue problem ...

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