Meshless Local Buckling Analysis of Steel Beams with

Meshless Local Buckling Analysis of Steel Beams with Web Openings A. R. Zainal Abidin, B. A. Izzuddin Department of Civil and Environmental Engineering

Introduction Steel Beams with Web Openings • effective way for fulfilling the needs of integrating M&E services within the floor depth – benefiting from lower fabrication cost, enhanced structural performance and improved artistic quality • some modifications are necessary – causing different regularity and even irregularity • exposed to local effects e. g. web-post buckling, tee buckling, Vierendeel bending due to high force concentrations from secondary effects – potentially leading to a markedly different structural response and buckling resistance.

Scope & objectives of the study • concerned with the development of sophisticated and yet efficient methods for assessing the elastic buckling of beams with regular and irregular web openings – particularly on local buckling effects in the web region • a novel method is proposed – based on the Element Free Galerkin (EFG) method for the numerical discretisation – together with a simplified buckling assessment approach from the Rotational Spring Analogy (RSA)

Plate buckling by the EFG/RSA OUT-OF-PLANE BUCKLING - for simply supported perforated beams under UDL GEOMETRIC STIFFNESS - determined from the planar stress field (according to RSA) MATERIAL STIFFNESS - directly obtained using the EFG method

Plate buckling by the EFG/RSA EFG method • moving least squares (MLS) approximation with C 1 continuity is adopted to construct the displacement field function • essential boundary conditions are imposed via a penalty factor – preserves the size and positive-definiteness of the stiffness matrix compared to Lagrange multipliers approach RSA method • offers simplified buckling assessment – involves only first-order kinematics as familiar in linear structural problems • requires pre-determined stresses from planar analysis Combination of both methods – facilitates in further simplifications – e. g. application of a ‘local region’ and a rank 2 reduced eigenvalue problem

Planar response • by dividing the beams into unit cells – resembles a super-element with a reduced number of freedoms • subjected to EFG discretisation and considered under representative actions – a standard discrete assembly procedure is employed • rigid body movement is prevented by means of simple supports at the web-post • the approach is of most benefit in problems where the irregularity in the openings profile is not extensive over the beam length

Planar response • the obtained stresses are continuous only within the local unit cell domain – though the associated inaccuracy is negligible for typical beams • continuous field of in-plane stresses can still be achieved by applying the MLS approximation at the overall beam level – requires much more computational effort

Out-of-plane response Material stiffness • employing the EFG method – in view of Kirchhoff’s thin plate theory • the sum of two contributions – the bending stiffness matrix and the penalty stiffness matrix Geometric stiffness • established from the planar stress field in accordance with the RSA – preserves the same discretisation Contribution of flange stiffness • assumption – the flanges offer full restraint to the out-of-plane displacement of the web along the interconnecting edges • considered only in terms of their material and geometric rotational stiffness

Simplified buckling analysis Local region approach • consisting of at least three unit cells – successively shifted along the beam span – reduces the size of the stiffness matrices significantly Rank 2 reduced eigenvalue problem • an assumed mode and its complementary mode are combined – enables the original MDOF problem to be solved effectively via an iterative approach – accelerates the convergence of iterations to the lowest buckling mode

Application example 1 • symmetric castellated beam with regular hexagonal web openings • has a geometry – length L = 30 m, depth Dp = 1. 603 m, spacing S = 1. 472 m, size of the regular hexagonal openings h 1 = 0. 84 m, b 1 = 1. 1554 x h 1 and b 2 = 0. 578 x h 1 • employs two circular holes near the left support – considering the same relative width of the existing holes, do = b 1 • uniformly distributed loading, UDL = 1. 0 k. N/m on top of the beam

Web-post buckling FEA-ADAPTIC Proposed EFG/RSA model

Application example 2 • typical cellular beam originating from 1016 x 305 x 222 UB • length L = 30 m, depth Dp = 1. 603 m, spacing S = 1. 472 m and hole diameter do = 0. 84 m • allowing several holes with a 0. 1 m vertical offset – together with one infilled hole near the right end of the beam support • transverse UDL of 1. 0 k. N/m – applied along the top edge considering a simply supported condition.

Tee buckling Proposed EFG/RSA model FEA-ADAPTIC

Application example 3 • consequence of elongated openings is studied for a beam with regular rectangular holes along its web • originated from 1016 x 305 x 222 UB – length L = 30 m, depth Dp = 1. 603 m, spacing S = 1. 472 m and hole diameter do = 0. 84 m • modified to incorporate three elongated openings –located at the middle of the beam, near the right support and between these two positions • subjected to 1. 0 k. N/m UDL under conditions of simple support.

Combined tee and web-post buckling FEA-ADAPTIC Proposed EFG/RSA model

Conclusion • • • a new approach by combining the EFG method with the RSA – proved efficient for local buckling assessment of steel beams with web openings of various shapes and sizes enables accurate and efficient buckling predictions – different forms of local buckling are shown to be accurately predicted, even for beams with irregular openings offers an interactive assessment approach – via the shifting local region strategy and assumed modes – leading to a much reduced problem size significant advantages – due to the separation of the planar from the out-of-plane response and discretisation benefits of meshless methods – avoid the need for excessively small element meshes most computational benefits for beams with regular repeated cells – also applicable to more complicated beam profiles – including irregular holes, tapered beams and curved beams

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