a.r. zainal abidin and b.a. izzuddin department of civil and environmental engineering
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/1.jpg)
A.R. Zainal Abidin and B.A. Izzuddin
Department of Civil and Environmental Engineering
![Page 2: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/2.jpg)
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
![Page 3: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/3.jpg)
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
![Page 4: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/4.jpg)
![Page 5: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/5.jpg)
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
![Page 6: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/6.jpg)
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
![Page 7: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/7.jpg)
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
![Page 8: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/8.jpg)
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
![Page 9: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/9.jpg)
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
![Page 10: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/10.jpg)
Background
![Page 11: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/11.jpg)
BackgroundPLANAR SYSTEMestablished by assembling the planar
responses of individual cells
INDIVIDUAL UNIT CELLS
NODES
![Page 12: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/12.jpg)
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
![Page 13: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/13.jpg)
BackgroundREPRESENTATIVE ACTIONS each cell utilising
a reduced number of freedoms –four nodes located at the T-centroids
![Page 14: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/14.jpg)
PLANAR SYSTEMsystem is solved globally using a standard
discrete solution realistic unit-based planar stress
distribution is obtained x y
xy
Background
![Page 15: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/15.jpg)
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
![Page 16: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/16.jpg)
, 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
![Page 17: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/17.jpg)
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 ...
![Page 18: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/18.jpg)
Buckling analysis strategy
SHIFTING LOCAL REGION- calculate KG from planar response- determine KE from out-of-plane analysis- eigenvalue analysis + iteration
![Page 19: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/19.jpg)
Buckling analysis strategy
SHIFTING LOCAL REGION- calculate KG from planar response- determine KE from out-of-plane analysis- eigenvalue analysis + iteration
![Page 20: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/20.jpg)
Buckling analysis strategy
![Page 21: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/21.jpg)
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
![Page 22: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/22.jpg)
Application examples1. WEB-POST BUCKLING horizontal normal stress, x
FEA:ADAPTIC
PROPOSED EFG/RSA
![Page 23: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/23.jpg)
Application examples1. WEB-POST BUCKLING vertical normal stress, y
FEA:ADAPTIC
PROPOSED EFG/RSA
![Page 24: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/24.jpg)
Application examples1. WEB-POST BUCKLING shear stress, xy
FEA:ADAPTIC
PROPOSED EFG/RSA
![Page 25: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/25.jpg)
Application examples1. WEB-POST BUCKLING
c = 33.621
c = 33.173
![Page 26: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/26.jpg)
Application examples1. WEB-POST BUCKLING
FEA:ADAPTIC PROPOSED EFG/RSA
![Page 27: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/27.jpg)
Application examples2. TEE BUCKLING
symmetric cellular beams parent I-section = 1016305222UB depth, Dp = 1603mm
diameter, Do = 840mm spacing, S = 1472mm web thickness, tw = 16mm
![Page 28: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/28.jpg)
Application examples2. TEE BUCKLING
c = 80.100
c = 79.695
![Page 29: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/29.jpg)
Application examples2. TEE BUCKLING
FEA:ADAPTIC PROPOSED EFG/RSA
![Page 30: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/30.jpg)
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
![Page 31: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/31.jpg)
Application examples3. BUCKLING AROUND THE OPENINGS
c = 68.598
c = 67.122
![Page 32: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/32.jpg)
Application examples3. BUCKLING AROUND THE OPENINGS
FEA:ADAPTIC PROPOSED EFG/RSA
![Page 33: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/33.jpg)
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
![Page 34: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/34.jpg)
A.R. Zainal Abidin and B.A. IzzuddinDepartment of Civil and Environmental Engineering
![Page 35: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/35.jpg)
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
![Page 36: A.R. Zainal Abidin and B.A. Izzuddin Department of Civil and Environmental Engineering](https://reader038.vdocuments.us/reader038/viewer/2022110212/568142ff550346895daf5512/html5/thumbnails/36.jpg)
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