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

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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

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Introduction CELLULAR BEAMS steel I-section beams with regular openings of circular shape throughout the web advantages: 1. Better in-plane flexural resistance – enabling long clear spans 2. Significant building height reduction by integrating M&E services with the floor depth – reduced cost 3. Aesthetical value – large space without screening effects

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Introduction BEHAVIOUR presence of web holes causes a high stress concentration in the narrow parts of the beams horizontal normal stress, x vertical normal stress, y shear stress, xy

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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 Introduction

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CURRENT ASSESSMENT METHODS 1. Finite Element Analysis (FEA) – continues to be computationally demanding 2. 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

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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 Belytschko combined with Rotational Spring Analogy (RSA) proposed by Izzuddin

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Introduction WHY EFG METHOD? 1. can be easily applied to irregular domains 2. potential efficiency in separating planar and out-of-plane responses unlike FEM 3. compared to MLPG, it ensures external equilibrium at sub-domain level between internal loading and boundary actions 4. facilitates the application of the RSA; - for example, the same fixed integration points can be used unlike MLPG

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Background

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PLANAR SYSTEM established by assembling the planar responses of individual cells INDIVIDUAL UNIT CELLS NODES

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Background UNIT CELL ANALYSIS discritised using the EFG method – via the moving least squares (MLS) technique rigid body movement is prevented by means of simple supports at the web-post

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

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

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GEOMETRIC STIFFNESS MATRIX according to RSA:

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Background OUT-OF-PLANE RESPONSE is obtained using the EFG method with Kirchhoff’s theory for thin plates planar displacements assumed to be reasonably small – K E is determined with reference to the undeformed geometry

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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: 1. shifting the local region 2. using an iterative rank 2 reduced eigenvalue problem......

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

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

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

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Application examples 1. WEB-POST BUCKLING symmetric cellular beams parent I-section = 1016 305 222UB depth, D p = 1603mm diameter, D o = 1280mm spacing, S = 1472mm web thickness, t w = 16mm

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

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

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

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Application examples 1. WEB-POST BUCKLING c = c =

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

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Application examples 2. TEE BUCKLING symmetric cellular beams parent I-section = 1016 305 222UB depth, D p = 1603mm diameter, D o = 840mm spacing, S = 1472mm web thickness, t w = 16mm

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Application examples 2. TEE BUCKLING c = c =

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

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Application examples 3. BUCKLING AROUND THE OPENINGS symmetric cellular beams parent I-section = 1016 305 222UB depth, D p = 1603mm diameter, D o = 1280mm spacing, S = 2944mm web thickness, t w = 16mm

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Application examples 3. BUCKLING AROUND THE OPENINGS c = c =

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

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Conclusion 1. effective method for local buckling analysis of cellular beams, combining EFG with RSA 2. shifting local region approach provides significant computational benefit 3. ability to predict accurately different forms of local buckling 4. not only applicable to regular cellular beams but also to other irregular forms

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

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Appendix ITERATIVE RANK 2 REDUCED EIGENVALUE PROBLEM determine the 2 probing modes: an initial assumed mode (U A ) and its complementary mode (U B )

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

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