A.R. Zainal Abidin and B.A. Izzuddin

Department of Civil and Environmental Engineering

Brief Introduction Cellular Beams – BehaviourCurrent Method of Assessments

Background of Proposed ModelPlanar Response – Geometric StiffnessOut-of-plane Analysis – Material Stiffness

Buckling Analysis ApproachIterative Rank 2 Reduced Eigenvalue

ProblemShifting Local Region

Application Examples

IntroductionCELLULAR BEAMSsteel I-section beams with regular openings

of circular shape throughout the webadvantages:

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

IntroductionBEHAVIOURpresence 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

FAILURE MODESdevelopment of local buckling, typically most

critical in web-post and compressive regions around the openings

WEB-POST TEE BUCKLING BUCKLING BUCKLING NEAR HOLES

Introduction

IntroductionCURRENT ASSESSMENT METHODS1.Finite Element Analysis (FEA) – continues to

be computationally demanding2.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

IntroductionTHE MAIN OBJECTIVElooking 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

IntroductionWHY EFG METHOD?1.can be easily applied to irregular domains2.potential efficiency in separating planar and out-

of-plane responses unlike FEM3.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

Background

BackgroundPLANAR SYSTEMestablished by assembling the planar

responses of individual cells

INDIVIDUAL UNIT CELLS

NODES

BackgroundUNIT CELL ANALYSISdiscritised using the EFG

method – via the moving least squares (MLS) technique

rigid body movement is preventedby means of simple supports atthe web-post

BackgroundREPRESENTATIVE ACTIONS each cell utilising

a reduced number of freedoms –four nodes located at the T-centroids

PLANAR SYSTEMsystem is solved globally using a standard

discrete solution realistic unit-based planar stress

distribution is obtained x y

xy

Background

BackgroundGEOMETRIC STIFFNESS MATRIXaccording to RSA:

;

T

T T

x xyT

xy y

d

d

d

G θ θ θ

xy θ θ θ xy

xy xy

K B diag k B

T T diag k T T

T T

, 3

, 2

,

;

1 0

; 1 0 ;12 1

2 0 0 1 / 2

T

j xx

j yy

j xy

d

N vEt

N and vv

N v

E E E

E

K B D B

B D

BackgroundOUT-OF-PLANE RESPONSEis obtained using the EFG method with

Kirchhoff’s theory for thin plates

planar displacements assumed to be reasonably small – KE is determined with reference to the undeformed geometry

Buckling analysis strategyaims for efficiency and accuracydiscrete 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 region2.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

Application examples1. WEB-POST BUCKLING

symmetric cellular beams parent I-section = 1016305222UB depth, Dp = 1603mm

diameter, Do = 1280mm spacing, S = 1472mm web thickness, tw = 16mm

Application examples1. WEB-POST BUCKLING horizontal normal stress, x

FEA:ADAPTIC

PROPOSED EFG/RSA

Application examples1. WEB-POST BUCKLING vertical normal stress, y

FEA:ADAPTIC

PROPOSED EFG/RSA

Application examples1. WEB-POST BUCKLING shear stress, xy

FEA:ADAPTIC

PROPOSED EFG/RSA

Application examples1. WEB-POST BUCKLING

c = 33.621

c = 33.173

Application examples1. WEB-POST BUCKLING

FEA:ADAPTIC PROPOSED EFG/RSA

Application examples2. TEE BUCKLING

symmetric cellular beams parent I-section = 1016305222UB depth, Dp = 1603mm

diameter, Do = 840mm spacing, S = 1472mm web thickness, tw = 16mm

Application examples2. TEE BUCKLING

c = 80.100

c = 79.695

Application examples2. TEE BUCKLING

FEA:ADAPTIC PROPOSED EFG/RSA

Application examples3. 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 examples3. BUCKLING AROUND THE OPENINGS

c = 68.598

c = 67.122

Application examples3. BUCKLING AROUND THE OPENINGS

FEA:ADAPTIC PROPOSED EFG/RSA

Conclusion1. 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

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

AppendixITERATIVE RANK 2 REDUCED EIGENVALUE PROBLEMdetermine the 2 probing modes: an initial assumed mode (UA) and its complementary mode (UB)

1 ;

; ;T

Twhere and

B E B

A E AB A A G A A

A G A

U K P

U K UP P K U

U K U

AppendixITERATIVE RANK 2 REDUCED EIGENVALUE PROBLEMthe 2 modes are then used to formulate a rank 2 eigenvalue problem

...

, ;

;

;

;

c

T

T

eig with being the lowest positive value in

where

in which

E G

E m E m

G m G m

m A B

λ k k λ

k U K U

k U K U

U U U