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