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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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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 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
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
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
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
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
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
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
Background PLANAR SYSTEM • established by assembling the planar responses of individual cells NODES INDIVIDUAL UNIT CELLS
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
Background REPRESENTATIVE ACTIONS • each cell utilising a reduced number of freedoms –four nodes located at the T-centroids
Background PLANAR SYSTEM • system is solved globally using a standard discrete solution • realistic unit-based planar stress distribution is obtained x y xy
Background GEOMETRIC STIFFNESS MATRIX • according to RSA:
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
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 ...
Buckling analysis strategy • SHIFTING LOCAL REGION • calculate KG from planar response • determine KE from out-of-plane analysis • eigenvalue analysis + iteration
Buckling analysis strategy • SHIFTING LOCAL REGION • calculate KG from planar response • determine KE from out-of-plane analysis • eigenvalue analysis + iteration
Buckling analysis strategy • SHIFTING LOCAL REGION • calculate KG from planar response • determine KE from out-of-plane analysis • eigenvalue analysis + iteration
Application examples 1. WEB-POST BUCKLING • symmetric cellular beams • parent I-section = 1016305222UB • depth, Dp = 1603mm • diameter, Do = 1280mm • spacing, S = 1472mm • web thickness, tw = 16mm
Application examples 1. WEB-POST BUCKLING • horizontal normal stress, x FEA:ADAPTIC PROPOSED EFG/RSA
Application examples 1. WEB-POST BUCKLING • vertical normal stress, y FEA:ADAPTIC PROPOSED EFG/RSA
Application examples 1. WEB-POST BUCKLING • shear stress, xy FEA:ADAPTIC PROPOSED EFG/RSA
Application examples 1. WEB-POST BUCKLING c = 33.621 c = 33.173
Application examples 1. WEB-POST BUCKLING FEA:ADAPTIC PROPOSED EFG/RSA
Application examples 2. TEE BUCKLING • symmetric cellular beams • parent I-section = 1016305222UB • depth, Dp = 1603mm • diameter, Do = 840mm • spacing, S = 1472mm • web thickness, tw = 16mm
Application examples 2. TEE BUCKLING c = 80.100 c = 79.695
Application examples 2. TEE BUCKLING FEA:ADAPTIC PROPOSED EFG/RSA
Application examples 3. BUCKLING AROUND THE OPENINGS • symmetric cellular beams • parent I-section = 1016305222UB • depth, Dp = 1603mm • diameter, Do = 1280mm • spacing, S = 2944mm • web thickness, tw = 16mm
Application examples 3. BUCKLING AROUND THE OPENINGS c = 68.598 c = 67.122
Application examples 3. BUCKLING AROUND THE OPENINGS FEA:ADAPTIC PROPOSED EFG/RSA
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
Thank you A.R. Zainal Abidin and B.A. IzzuddinDepartment of Civil and Environmental Engineering
Appendix ITERATIVE RANK 2 REDUCED EIGENVALUE PROBLEM • determine the 2 probing modes: an initial assumed mode (UA) and its complementary mode (UB)
Appendix ITERATIVE RANK 2 REDUCED EIGENVALUE PROBLEM • the 2 modes are then used to formulate a rank 2 eigenvalue problem ...