graduation project - anasayfa
TRANSCRIPT
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GRADUATION PROJECT
JANUARY, 2021
POST-BUCKLING BEHAVIOR OF RECTANGULAR PLATES WITH A HOLE
Thesis Advisor: Prof. Dr. Vedat Ziya DOΔAN
Nihal Γzden YETKΔ°N
Department of Astronautical Engineering
Anabilim DalΔ± : Herhangi MΓΌhendislik, Bilim
ProgramΔ± : Herhangi Program
ISTANBUL TECHNICAL UNIVERSITY FACULTY OF AERONAUTICS AND ASTRONAUTICS
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JANUARY, 2021
ISTANBUL TECHNICAL UNIVERSITY FACULTY OF AERONAUTICS AND ASTRONAUTICS
POST-BUCKLING BEHAVIOR OF RECTANGULAR PLATES WITH A HOLE
GRADUATION PROJECT
Nihal Γzden YETKΔ°N
110150153
Department of Astronautical Engineering
Anabilim DalΔ± : Herhangi MΓΌhendislik, Bilim
ProgramΔ± : Herhangi Program
Thesis Advisor: Prof. Dr. Vedat Ziya DOΔAN
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Thesis Advisor : Prof. Dr. Vedat Ziya DOΔAN ..............................
Δ°stanbul Technical University
Jury Members : Prof. Dr. Halit SΓΌleyman TΓRKMEN .............................
Δ°stanbul Technical University
Asst. Prof. Demet BALKAN ..............................
Δ°stanbul Technical University
Nihal Γzden YETKΔ°N, student of ITU Faculty of Aeronautics and Astronautics
student ID 110150153, successfully defended the graduation entitledβ POST-
BUCKLING BEHAVIOR OF RECTANGULAR PLATES WITH A HOLEβ,
which she prepared after fulfilling the requirements specified in the associated
legislations, before the jury whose signatures are below.
Date of Submission : 30 January 2021
Date of Defense : 08 February 2021
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To my family,
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FOREWORD
This thesis is written as the final assignment to complete Astronautical Engineering at
Istanbul Technical University. The aim is to see buckling and post-buckling behavior of
perforated plates with different configurations. It had been completed with the
knowledge I gained from my classes for four years.
I would like to thank all my teachers throughout my university life for their effort on
relaying information. I would especially like to thank to Prof. Dr. Vedat Ziya DOΔAN
for his guidance and support during this process. Finally, special thanks to my family
who was always by my side and gave limitless support during my education life. I am
grateful for everybody that helped me complete this study successfully.
January 2021 Nihal Γzden Yetkin
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TABLE OF CONTENTS
FOREWORD ............................................................................................................................ v
TABLE OF CONTENTS ........................................................................................................ vii
ABBREVIATIONS .................................................................................................................. ix
SYMBOLS ................................................................................................................................ x
LIST OF TABLES .................................................................................................................. xii
LIST OF FIGURES ...............................................................................................................xiii
SUMMARY ............................................................................................................................ xiv
ΓZET....................................................................................................................................... xv
1. INTRODUCTION ................................................................................................................. 1
1.1 Purpose of Thesis .............................................................................................................. 1
1.2 Literature Review .............................................................................................................. 1
2. BUCKLING .......................................................................................................................... 5
2.1 Introduction ....................................................................................................................... 5
2.2 Buckling of Column .......................................................................................................... 5
2.3 Buckling of Thin Plate ....................................................................................................... 7
2.3.1 Introduction ................................................................................................................ 7
2.3.2 Boundary conditions ................................................................................................... 7
2.3.3 Derivation of elastic buckling equations of plates ........................................................ 9
2.3.4 Energy method ............................................................................................................ 9
2.3.5 Buckling of rectangular plates uniformly compressed in one direction....................... 10
2.3.6 Buckling of SCSC rectangular plates uniformly compressed in simply
supported sides .................................................................................................................. 13
3. POST-BUCKLING ............................................................................................................. 16
3.1 Introduction ..................................................................................................................... 16
3.2 Post-Buckling Approaches ............................................................................................... 16
3.3 Post βBuckling Analysis .................................................................................................. 17
3.3.1 Von Karman large deflection equations with initial imperfections ............................. 17
3.3.2 Empirical approach (Effective width) ........................................................................ 18
3.3.3 Riks method .............................................................................................................. 19
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4. ANALYSIS & RESULTS ................................................................................................... 21
4.1 Material Selection ........................................................................................................... 21
4.2 Finite Element Model ...................................................................................................... 21
4.2.1 Modelling ................................................................................................................. 21
4.2.2 Approximate global size and meshing ....................................................................... 22
4.2.3 Boundary conditions ................................................................................................. 24
4.2.4 Loading in buckling stage ......................................................................................... 25
4.2.5 Solution method in buckling stage ............................................................................ 25
4.2.6 Loading in Post-Buckling Stage ................................................................................ 25
4.2.7 Solution method in post-buckling stage ..................................................................... 26
4.3 Results ............................................................................................................................ 26
4.3.1 Effect of a/b ratio ...................................................................................................... 26
4.3.2 Effect of boundary condition ..................................................................................... 32
4.3.3 Effect of thickness .................................................................................................... 33
4.3.4 Effect of d/b ratio ...................................................................................................... 34
4.3.5 Effect of hole shape .................................................................................................. 36
4.3.6 Effect of plate material.............................................................................................. 38
5. CONCLUSION ................................................................................................................... 39
REFERENCES ....................................................................................................................... 41
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ABBREVIATIONS
AGS : Approximate Global Size
C : Clamped Edge
F : Free Edge
S : Simply Supported Edge
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SYMBOLS
Pcr : Critical load
E : Modulus of elasticity
I : Moment of inertia
L : Unsupported length of the column
K : Column effective length factor
a : Length of plate
b : Width of plate
t : Thickness of plate
Mx, My, Mz : The in-plane moment resultants
Vx, Vy, Vz : The vertical shear forces
W : The out of plane deflection
Nx, Ny, Nxy : The compressive loads
ΞT : The work of external forces
ΞU : The strain energy of bending
u, v, w : Displacement components in x,y and z directions respectively
u0, v0 : Displacement components at the plane of z=0
Οx, Οy, Οz : The in-plane normal stresses
Ξ΅X, Ξ΅Y, Ξ΅Z : The in-plane normal strains
Ξ³xy : The in-plane shear strain
U : The strain energy
β : Total potential energy
E : The Youngβs modulus
G : The shear modulus
Ξ½ : The Poissonβs ratio
D : The flexural rigidity
m : The sinusoidal half-waves
Οcr : The critical value of the compressive stress
K0NM : The stiffness matrix of initial state
KΞNM : The initial stress and load stiffness matrix with respect to applied load
Ξ»i : The eigenvalues
ΚiM : The eigenvectors
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F : The stress variation in the plate
Ar : Coefficient prescribing the magnitude of the deflections
An : Normalized values of the eigenvector
be :Effective width
Οy : Yield stress
PN : The loading pattern
Ξ» : The load magnitude parameter
Ξlin : Initial increment
lperiod : The total arc length scale factor
KiNM : Tangent stiffness
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LIST OF TABLES
Page
Table 2.1 Values of the Factor k.................................................................................. 15
Table 4.1 Change in Critical Buckling Load with Element Size .................................. 23
Table 4.2 Critical Buckling Loads of Plates without Hole ........................................... 27
Table 4.3 Critical Buckling Loads of Plates with Hole ................................................ 28
Table 4.4 Maximum Displacements of Plates with Hole in Post-Buckling Stage ......... 29
Table 4.5 Critical Loads for Plates without Hole and with Different Boundary
Conditions ................................................................................................................... 32
Table 4.6 Critical Loads for Plates with Hole and Different Boundary Conditions ...... 33
Table 4.7 Critical Buckling Loads for Plates with Different Thicknesses..................... 34
Table 4.8 Critical Loads for Plates with Different d/b Ratio ........................................ 35
Table 4.9 Critical Loads for Plates with Different Hole Shapes ................................... 36
Table 4.10 Material Properties and Critical Loads of Steel and Aluminum Plates ........ 38
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LIST OF FIGURES
Page
Figure 1.1 Loadβend displacement path ........................................................................ 3
Figure 2.1 Load-Deformation Behavior for an Axially Loaded Bar ............................... 6
Figure 2.2 Geometry of the Plate ................................................................................... 7
Figure 2.3 Uniaxially compressed plate ....................................................................... 13
Figure 2.4 Uniformly Compressed SCSC Plate ........................................................... 14
Figure 3.1 Unstable Static Response............................................................................ 19
Figure 4.1 Model of the plate without hole .................................................................. 22
Figure 4.2 Model of the plate with hole ....................................................................... 22
Figure 4.3 Meshed Model with Element Size 15 mm................................................... 24
Figure 4.4 Boundary Conditions of the Plate ............................................................... 24
Figure 4.5 Model with Boundary Conditions and Applied Load .................................. 25
Figure 4.6 Critical Load vs. a/b ratio for Plates without Hole ...................................... 27
Figure 4.7 Critical Load vs. a/b ratio for Plates with Hole ........................................... 28
Figure 4.8 Location of the Selected Node for Post-Buckling Analyses ........................ 30
Figure 4.9 Displacement of the Node for a/b=1 and a/b=1.5 ........................................ 30
Figure 4.10 Displacement of the Node for a/b=1.5 and a/b=2 ...................................... 31
Figure 4.11 Displacement of the Node for a/b=2 and a/b=3 ......................................... 31
Figure 4.12 Boundary Conditions of the Simply Supported and Free Plate .................. 32
Figure 4.13 Displacement of the Node for d/b=0.5 and d/b=0.1 ................................... 35
Figure 4.14 Buckled Shape of Plate with Elliptical Hole ............................................. 37
Figure 4.15 Buckled Shape of Plate with Circular Hole ............................................... 37
Figure 4.16 Buckled Shape of Plate with Square Hole ................................................. 37
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POST-BUCKLING BEHAVIOR OF RECTANGULAR PLATES WITH A HOLE
SUMMARY
Steel plates are widely used in engineering industries such as aerospace, automotive,
marine and etc. In aerospace industry it is critical to ensure safety at the aircraftβs critical
areas such as the wings, fuselage, landing gears, tail bottom, rotor blades and the
airframes. At this part, the high resistance to corrosion and high temperature tolerance
are the key parameters of steel to be especially used in landing gear and engine
components. Besides, the high tensile strength and shear modulus makes it more
preferable. However, weight reduction is one of the primary considerations in aerospace
industry which sometimes led to the usage of the perforated plates. Perforated plates are
also useful in aircraft applications for providing ease of access in servicing, inspection
and maintenance. Since perforated plates are highly preferred in some specific
applications and their behavior are different than the unperforated plates, it is important
to investigate their buckling and post-buckling behavior. At the point of critical load
value, the structure suddenly experiences a large deformation and may lose its ability to
carry load. This stage is the buckling stage. After the buckling, applied load may or may
not change, while deformation continues to increase. However, in some cases the plates
continue to carry load after certain amount of deflection which is called the post-
buckling stage. In the analyses, critical buckling load is obtained by eigenvalue buckling
prediction and post-buckling behavior is observed by using Riks method.
In this project, rectangular perforated thin plates are used which are simply supported on
all sides. Plates are subjected to axial compression from the right short edge in x-
direction in all cases. The critical buckling load formula was derived in a form which is
applicable to various boundary conditions. The post-buckling behavior of plates cannot
be fully observed by analytical solutions which leads the researchers to the numerical
solutions. Among the post-buckling methods, Riks method is mostly used to predict
unstable, geometrically nonlinear behavior of structures. In the analyses, variation in the
aspect ratio, thickness, hole diameter and hole shape are investigated. Aluminum is also
investigated to compare with steel plates. Different boundary conditions are defined to
make comparison with simply supported plates and finally, flat plates are also
investigated to observe and determine the effect of a hole on the structure. It is clearly
seen that making a hole on the plate decreases the critical buckling load and as the result
of decreased buckling load, variation in displacement decreases in post-buckling stages.
Detailed evaluation had been made in the results section of this thesis.
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DELΔ°KLΔ° DΔ°KTΓRTGEN PLAKALARIN BURKULMA SONRASI DAVRANIΕI
ΓZET
Γelik mΓΌhendislik alanΔ± uygulamalarΔ±nda kullanΔ±mΔ± sΔ±klΔ±kla tercih edilen bir malzemedir.
KullanΔ±mΔ± en yaygΔ±n olan alanlar arasΔ±nda havacΔ±lΔ±k, otomotiv ve denizcilik vardΔ±r.
HavacΔ±lΔ±k endΓΌstrisinde, hava aracΔ±nΔ±n kritik ΓΆnem taΕΔ±yan bΓΆlgelerinde gΓΌvenliΔi
saΔlamak bΓΌyΓΌk ΓΆnem taΕΔ±maktadΔ±r. Bu bΓΆlgeler arasΔ±nda kanatlar, gΓΆvde, iniΕ takΔ±mlarΔ±
ve iskelet vardΔ±r. Bu noktada dikkat edilecek ana unsurlar malzemenin aΕΔ±nmaya ve
yΓΌksek sΔ±caklΔ±Δa toleransΔ±nΔ±n fazla olmasΔ±dΔ±r. DolayΔ±sΔ±yla bu niteliklere sahip olan
Γ§eliΔin ΓΆzellikle iniΕ takΔ±mlarΔ±nda ve motor parΓ§alarΔ±nda kullanΔ±mΔ± oldukΓ§a yaygΔ±ndΔ±r.
AyrΔ±ca yΓΌksek Γ§ekme direnci ve kesme katsayΔ±sΔ± Γ§eliΔin daha Γ§ok tercih edilmesini
saΔlayan unsurlardΔ±r. Fakat havacΔ±lΔ±k endΓΌstrisinde yapΔ±nΔ±n aΔΔ±rlΔ±ΔΔ±nΔ± mΓΌmkΓΌn
olduΔunca dΓΌΕΓΌk tutmak ΓΌretimin en ΓΆnemli esaslarΔ±ndan biridir. Bu da dΓΌΕΓΌk aΔΔ±rlΔ±klΔ±
malzeme tercihiyle veya yapΔ±nΔ±n aΔΔ±rlΔ±ΔΔ±nΔ± azaltacak yΓΆntemlerle saΔlanmaktadΔ±r.
ΓrneΔin plakalarda delik aΓ§mak aΔΔ±rlΔ±ΔΔ± azaltacaΔΔ± iΓ§in tercih edilebildiΔi gibi bazΔ±
alanlarda da servis, denetim ve bakΔ±m kolaylΔ±ΔΔ± saΔladΔ±ΔΔ± iΓ§in uygulanmaktadΔ±r. Delikli
plakalarΔ±n bu gibi alanlarda Γ§okΓ§a tercih edilmesi ve davranΔ±ΕlarΔ±nΔ±n dΓΌz plakalardan
farklΔ±lΔ±k gΓΆstermesi sebebiyle burkulma ve burkulma sonrasΔ± davranΔ±ΕlarΔ±nΔ±n ayrΔ±ca
incelenmesi gerekmektedir.
Uygulanan yΓΌkΓΌn kritik noktaya ulaΕtΔ±ΔΔ± noktada yapΔ±da bΓΌyΓΌk deformasyonlar oluΕur
ve yapΔ± artΔ±k yΓΌk taΕΔ±yamaz hale gelebilir. Ani yapΔ±sal deΔiΕimin ortaya Γ§Δ±ktΔ±ΔΔ± bu
duruma burkulma denir. Burkulmadan sonra uygulanan yΓΌkΓΌn artΔ±p azalmasΔ±na baΔlΔ±
olmadan deformasyon artmaya devam edebilir. Fakat bazΔ± durumlarda belli bir
deformasyondan sonra bile plaka yΓΌk taΕΔ±maya devam edebilir. Bu gibi durumlarda
plakanΔ±n burkulma sonrasΔ± davranΔ±ΕΔ± incelenmektedir. YapΔ±lan analizlerde kritik yΓΌk
hesaplamada eigenvalue burkulma analizi kullanΔ±lΔ±rken, burkulma sonrasΔ± davranΔ±ΕΔ±nΔ±
incelemede Riks metodu kullanΔ±lmΔ±ΕtΔ±r.
Bu projede delikli dikdΓΆrtgen Γ§elik plakalar ΓΌzerinde Γ§alΔ±ΕΔ±lmΔ±ΕtΔ±r ve plakalar bΓΌtΓΌn
kenarlarΔ±ndan basit mesnetlenmiΕtir. TΓΌm durumlarda plakalara saΔ kΔ±sa kenarΔ±ndan tek
doΔrultulu basΔ±nΓ§ uygulanmΔ±ΕtΔ±r. Kritik yΓΌkΓΌn hesaplanmasΔ± iΓ§in formΓΌl Γ§Δ±karΔ±mΔ±
yapΔ±lmΔ±ΕtΔ±r ve bu formΓΌl farklΔ± sΔ±nΔ±r ΕartlarΔ±nΔ±n uygulandΔ±ΔΔ± durumlarda da
kullanΔ±labilecek halde sunulmuΕtur. Burkulma sonrasΔ± davranΔ±ΕΔ± incelemede net sonuΓ§
veren analitik bir çâzΓΌm yΓΆntemi bulunmamaktadΔ±r, bu durum araΕtΔ±rmalarda daha Γ§ok
nΓΌmerik çâzΓΌmlere yΓΆnelimi arttΔ±rmΔ±ΕtΔ±r. Riks metodu yapΔ±larΔ±n lineer olmayan ve
deΔiΕken davranΔ±ΕlarΔ±nΔ±n analizinde kullanΔ±lΔ±r ve burkulma sonrasΔ± metotlarΔ± arasΔ±nda en
Γ§ok tercih edilendir. Riks metodunda hem uygulanan yΓΌk hem de yer deΔiΕtirme
bilinmeyendir ve ikisi eΕzamanlΔ± olarak çâzΓΌlΓΌr. ABAQUS çâzΓΌm uzayΔ±nda maksimum
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yer deΔiΕtirmeyi gΓΆsterir ve çâzΓΌm eΔrisi uygulanan yΓΌk ile yer deΔiΕtirmenin
dengelendiΔi noktalar serisinden oluΕur.
Delikli plakalarΔ±n kenar uzunluklarΔ± oranΔ±, kalΔ±nlΔ±ΔΔ±, delik Γ§apΔ± ve delik Εekli
deΔiΕtirilerek Γ§eΕitli analizler yapΔ±lmΔ±ΕtΔ±r. FarklΔ± bir materyal tercihi yaparak Γ§elik
plakayla kΔ±yaslanabilmesi iΓ§in alΓΌminyum tercih edilmiΕtir. AyrΔ±ca basit mesnetlenmiΕ
plakalarΔ±n farklΔ± sΔ±nΔ±r ΕartlarΔ± uygulanmΔ±Ε plakalarla karΕΔ±laΕtΔ±rΔ±lmasΔ± yapΔ±lmΔ±ΕtΔ±r.
Bunlarla beraber, ana konu olan delikli plakalar hakkΔ±nda en doΔru sonuca varabilmek
adına düz plakalarda da analizler yapılmıŠve plakada delik açmanın etkileri
gΓΆzlemlenmiΕtir.
Γncelikle plakanΔ±n kenar uzunluklarΔ± oranΔ±nΔ±n kritik yΓΌke etkisi incelenmiΕtir. Uzun
kenar sabit tutulmuΕ, kΔ±sa kenar giderek küçültΓΌlmΓΌΕtΓΌr. Analizler hem delikli hem
deliksiz plakalar ΓΌzerinde yapΔ±lmΔ±ΕtΔ±r. SonuΓ§ olarak plaka merkezinde bir deliΔin
varlΔ±ΔΔ±nΔ±n, kritik yΓΌkΓΌn deliksiz plakalara gΓΆre daha dΓΌΕΓΌk olmasΔ±na sebep olduΔu
gΓΆzlemlenmiΕtir. Fakat hem delikli hem de deliksiz plakalarda kΔ±sa kenar küçüldΓΌkΓ§e
kritik yΓΌkΓΌn arttΔ±ΔΔ± gΓΆzlemlenmiΕtir. Bu durumda plakanΔ±n bΓΌtΓΌn kenarlarΔ±ndan basit
mesnetlenmiΕ olmasΔ± etkili olmuΕtur. PlakanΔ±n kΔ±sa kenar uzunluΔu küçüldΓΌkΓ§e
mesnetlenmiΕ uzun kenarlar birbirine yaklaΕmakta ve plakanΔ±n daha sabit durmasΔ±nΔ±
saΔlamaktadΔ±r. Bu da plakanΔ±n burkulma yΓΌkΓΌnΓΌn artmasΔ±nΔ± ifade etmektedir. ArdΔ±ndan
elde edilen kritik yΓΌk verileri Riks metodunda kullanΔ±larak burkulma sonrasΔ± analizi
yapΔ±lmΔ±Ε ve plakalarΔ±n maksimum yer deΔiΕtirme miktarlarΔ± bulunmuΕtur. Analiz
sonucunda maksimum yer deΔiΕtirmenin x, y ve z eksenlerindeki bileΕenleri incelenmiΕ
ve z eksenindeki yer deΔiΕtirmenin toplam yer deΔiΕtirmenin %99βunu oluΕturduΔu
sonucuna varΔ±lmΔ±ΕtΔ±r. Bu nedenle burkulma sonrasΔ± davranΔ±ΕΔ±n incelenmesinde plakanΔ±n
z eksenindeki hareketi temel alΔ±nmΔ±ΕtΔ±r. Kenar uzunluklarΔ± oranΔ±nΔ±n etkisini
gâzlemleyebilmek amacıyla bütün plakalarda delik yanındaki bir nokta seçilmiŠve o
noktadaki yer deΔiΕtirme izlenmiΕtir. SonuΓ§ olarak, kΔ±sa kenar küçüldΓΌkΓ§e noktanΔ±n yer
deΔiΕtirme grafiΔinde dalgalanmanΔ±n arttΔ±ΔΔ± gΓΆzlemlenmiΕtir. Buradan da kritik yΓΌkΓΌn
artΔ±ΕΔ±na baΔlΔ± olarak plakanΔ±n pozitif ve negatif z eksenlerindeki yer deΔiΕtirme
sapmalarΔ±nΔ±n arttΔ±ΔΔ± sonucuna varΔ±lmΔ±ΕtΔ±r.
SΔ±nΔ±r ΕartlarΔ±nΔ±n kritik yΓΌke olan etkisini araΕtΔ±rmak amacΔ±yla kΔ±sa kenarlarΔ±ndan basit
mesnetlenmiΕ, uzun kenarlarΔ±ndan serbest bΔ±rakΔ±lmΔ±Ε ve saΔ kΔ±sa kenarΔ±ndan yΓΌk
uygulanmΔ±Ε plakalar incelenmiΕtir. Bir ΓΆnceki analizdeki gibi kenar uzunluklarΔ± oranΔ±
deΔiΕtirilerek her birinde sΔ±nΔ±r ΕartlarΔ±nΔ±n kritik yΓΌke etkisi karΕΔ±laΕtΔ±rΔ±lmΔ±ΕtΔ±r. SonuΓ§lara
bakΔ±ldΔ±ΔΔ±nda uzun kenarlarΔ±ndan serbest bΔ±rakΔ±lan plakalarΔ±n kritik yΓΌkΓΌnΓΌn oldukΓ§a
dΓΌΕΓΌk olduΔu gΓΆrΓΌlmΓΌΕtΓΌr. TΓΌm kenarlarΔ±ndan basit mesnetlenmiΕ plakalarΔ±n aksine,
uzun kenarlarΔ± serbest bΔ±rakΔ±lan plakalarda kΔ±sa kenar uzunluΔu küçüldΓΌkΓ§e kritik yΓΌkΓΌn
azaldΔ±ΔΔ± gΓΆzlemlenmiΕtir. Bunun sebebinin kΔ±sa kenar kΔ±saldΔ±kΓ§a plakanΔ±n kolon gibi
davranmasΔ± ve direnme gΓΌcΓΌnΓΌn azalmasΔ± olduΔu sonucuna varΔ±lmΔ±ΕtΔ±r.
Plaka kalΔ±nlΔ±ΔΔ±nΔ±n davranΔ±Εa etkisini araΕtΔ±rmak amacΔ±yla kenar uzunluklarΔ± ve delik Γ§apΔ±
sabit tutulan plakalarda kalΔ±nlΔ±k 2 ile 10 mm arasΔ±nda deΔiΕtirilmiΕ ve analizler
yapΔ±lmΔ±ΕtΔ±r. Plaka kalΔ±nlΔ±ΔΔ± arttΔ±kΓ§a kritik yΓΌkΓΌn de arttΔ±ΔΔ± gΓΆrΓΌlmΓΌΕtΓΌr. KalΔ±nlΔ±k ile kritik
yΓΌk arasΔ±nda oransal bir iliΕki tespit edilmiΕtir. Plaka kalΔ±nlΔ±ΔΔ± 2 katΔ±na Γ§Δ±ktΔ±ΔΔ± durumda
kritik yΓΌkte de 2βnin kΓΌpΓΌ oranΔ±nda artΔ±Ε gΓΆrΓΌlmΓΌΕtΓΌr.
xvii
ArdΔ±ndan delik Γ§apΔ±nΔ±n etkisi incelenmiΕ ve delik Γ§apΔ±nΔ±n kΔ±sa kenara oranΔ± deΔiΕtirilerek
analiz yapΔ±lmΔ±ΕtΔ±r. Oran arttΔ±kΓ§a kritik yΓΌkΓΌn azaldΔ±ΔΔ± gΓΆrΓΌlmΓΌΕtΓΌr. Kritik yΓΌkΓΌn
azalmasΔ±yla burkulma sonrasΔ± davranΔ±Εta yer deΔiΕtirmenin dalgalanmasΔ±nΔ±n azalmasΔ±
beklenmektedir. SaΔlama yapmak amacΔ±yla oranΔ±n en küçük ve en bΓΌyΓΌk olduΔu
plakalarΔ±n burkulma sonrasΔ± davranΔ±ΕΔ±nΔ±n grafiΔi Γ§izdirilerek beklenen sonucun doΔru
olduΔu anlaΕΔ±lmΔ±ΕtΔ±r.
Delik Εeklinin plaka ΓΌzerindeki etkisini anlamak iΓ§in 3 farklΔ± Εekilde delik
oluΕturulmuΕtur. Yuvarlak, elips ve kare delikler oluΕturulup her durumda deliΔin alanΔ±
sabit tutulmuΕtur. Kritik yΓΌkΓΌn en dΓΌΕΓΌk olduΔu plaka deliΔin elips Εeklinde olduΔu
plaka, en yΓΌksek olduΔu durum kare delikli plakada olmuΕtur.
YapΔ±lan tΓΌm analizlerde plaka malzemesi Γ§elik olarak belirlenmiΕtir. Son olarak Γ§eliΔin
baΕka bir malzemeyle karΕΔ±laΕtΔ±rmasΔ±nΔ± yapmak amacΔ±yla alΓΌminyum plakayla analiz
yapΔ±lmΔ±Ε ve sonuΓ§lar karΕΔ±laΕtΔ±rΔ±lmΔ±ΕtΔ±r. AlΓΌminyumun gerilme katsayΔ±sΔ± Γ§eliΔinkinin
ΓΌΓ§te biri kadardΔ±r ve gerilme katsayΔ±sΔ± burkulma yΓΌkΓΌyle doΔrudan orantΔ±lΔ± olduΔu iΓ§in
burkulma yΓΌkΓΌ de Γ§eliΔinkinin ΓΌΓ§te biri olarak bulunmuΕtur. Bu sebeple alΓΌminyum
plakalarΔ±n burkulma sonrasΔ±nda yΓΌk altΔ±nda yer deΔiΕtirme dalgalanmasΔ±nΔ±n daha az
olmasΔ± beklenmektedir.
xviii
1
1. INTRODUCTION
Perforated metal (steel) plates are widely used in aerospace industry due to their
structural and applicable advantages. Perforated plates are useful for providing ease of
access in servicing, inspection and maintenance. Besides, they are highly preferred in
aircraft applications for their weight reducing sufficiency. It is necessary to investigate
the perforated plateβs behavior since they are also applicable in many other engineering
industries. Buckling and post-buckling behavior of plates with hole are different from
plates without hole and there are many parameters that need to be considered in the
analyses.
1.1 Purpose of Thesis
In this study, post-buckling behavior of steel plates with holes will be analyzed and
discussed. The plates are rectangular and the hole is placed at center. The analyses will
be conducted for different boundary conditions and different sizes and shapes of holes in
order to see their effects on the structural response in buckling and post-buckling. The
aim is to find out the changes in post-buckling behavior of plates with and without hole.
Numerical results will be obtained by using a finite element program, ABAQUS.
1.2 Literature Review
Structures that are used in engineering industries need to be analyzed particularly for the
investigation of their buckling and post-buckling behavior. The load carrying capacity of
a body is the most essential point to consider in design processes. Under axial
compression or shear loadings, a sudden change in the structural component can occur.
The point of failure in the shape of the structure is called buckling and the critical
buckling load is the maximum load supported by the body just before it is buckled. Euler
2
is the first to derive the critical buckling load formula for long slender columns [1].
Buckling of plates have been investigated since the early years of the 19th century.
Yamaguchi derived the buckling stress coefficient which depends on the ratio a/b and
sinusoidal half wave m [2]. Love determined the stresses and deformations in thin plates
subjected to forces and moments using the assumptions proposed by Kirchhoff [3].
Timoshenko used energy approach and developed solution for buckling load of plates
with different boundary conditions and loading cases [4]. Bryan formulated the critical
buckling stress equation using the energy method for a plate that is simply supported and
under uniaxial compression [5]. Buckling of a variable thickness rectangular plate had
been studied by Whittrick and Ellen [6]. Also, to determine the critical buckling stress of
variable thickness, Chehil and Dua used a perturbation technique on simply supported
rectangular plates [7]. Xiang and Wang developed buckling and vibration solution for
perforated plates having two opposite sides simply supported while the other two sides
can be free, simply supported and clamped [8]. Shanmugam and Narayanan [9] and
Azizian and Roberts [10] explored the buckling of square plates with square and circular
holes under biaxial and shear loadings. Paik examined perforated plates under combined
biaxial and shear loading and developed empirical formulas for the critical buckling load
[11].
For a column, its critical buckling load should be considered as its collapse load since it
cannot withstand any compressive loading afterwards. However for a thin plate, collapse
does not occur when the elastic buckling is reached and it can withstand additional
compressive load. Thus, a plate can resist post-buckling loadings and its behavior
depends on the change in its structural parameters. In engineering applications plates
with a hole are widely used in thin-walled structures to provide access for services,
inspection and maintenance, for example in webs of plates, aero plane fuselages, box
girders and ship grillages. Despite the advantages, the presence of holes in structural
members causes the stress distribution within the member to change and generally a
reduction occurs in the critical buckling capacity of the plate. From the structural design
point of view investigation of post buckling behavior of these plates are as critical as the
investigation of local buckling of such plates. Thus, research into the post-buckling
3
behavior of thin plates was carried out especially in the aircraft industry since the early
times. Schuman and Back performed series of compression tests on plates of various
materials and having different width [12]. Shanmugam et al. [13] and Shanmugam and
Dhanalakshmi [14] used finite element method to investigate the load carrying capacity
of perforated plates with different boundary conditions. They come up with an effective
formula to determine the load carrying capacity for perforated plates with different plate
slenderness, hole size, boundary conditions and the nature of loading based on the post-
buckling behavior. The nonlinear mathematical theory was established by Cheng and
Fan for perforated thin plates using the von Karmanβs assumptions [15]. Rhodes made
research on the elastic and plastic post-buckling behavior of plates and in the study, non-
linear differential equations set up by Von Karman are used in the examination of thin
plates [16]. In this study, the effect of load on the displacement of plate after buckling is
demonstrated in Figure 1.1.
Figure 1.1 Loadβend displacement path [16]
Point A in figure is the buckling point. After buckling, axial stiffness drops right away in
the post-buckling stage for a perfect plate. Since the stress increase as the load increase,
the axial stiffness continues to decrease and results into raising in the end displacement.
Besides, Botman and Besselling found out that by using elastic analysis effective width
gave sensible predictions of failure when applied to plates with non-linear behavior [17].
4
Bakker et al. examined square plates with initial imperfections and discussed analytical
and semi-analytical formulas to define the post-buckling behavior [18]. Experimental
investigation carried out by Yu and Davis on the buckling and post-buckling behavior of
rectangular plates containing centrally located holes [19]. Horne and Narayanan
developed theoretical post-buckling analysis for stiffened plates that are axially
compressed [20]. Kumar and Singh investigated the effect of boundary conditions on the
buckling and post-buckling behavior of the axially compressed quasi-isotropic laminates
with shaped and different sized cutouts (circular, square, diamond, ellipticalβvertical,
and ellipticalβhorizontal) by using finite element method [21]. For numerical post-
buckling analysis the Riks method is widely used. In the Riks method, the load
magnitude and displacements are unknown and it solves simultaneously for both. For
unstable problems the load displacement response can show high nonlinear behavior.
Riks method is an algorithm that gives effective solution in such cases. Novoselac et al.
performed numerical analysis and used eigenvalue buckling prediction and Riks method
for investigation of linear and nonlinear buckling and post buckling behavior of a bar
with imperfections [22].
5
2. BUCKLING
2.1 Introduction
The failure of a mechanical component can be divided into two major categories:
material failure and structural instability which is usually called buckling. Buckling of a
member occurs when compressive or shear loadings reach a critical level which causes a
deformation, a sudden change on the structure. It is necessary to investigate and analyze
the buckling characteristics of the structures so that they can safely support their
intended loadings. Generally two types of buckling exist. They are bifurcation-type and
deflection-amplification type. The deflection-amplification type is the buckling in
empirical design and on the other hand, the bifurcation-type is the theoretical approach.
In this chapter, buckling of columns and thin plates are briefly explained.
2.2 Buckling of Column
Columns are long slender members subjected to axial compressive loading. The
maximum axial loading that supported safely by the column is called the critical load
(Pcr), also the Euler load. Leonhard Euler is the first to investigate the behavior of
columns and derive the solution of bifurcation-type critical buckling load formula for
long slender (ideal) columns. Beyond this critical load, the column will buckle and
deflect laterally.
πππ =π2πΈπΌ
(πΎπΏ)2 (2.1)
Where,
Pcr = critical axial load on the column
E = modulus of elasticity for the material
6
I = moment of inertia for the columnβs cross-sectional area
L = unsupported length of the column
K, column effective length factor, whose value depends on the conditions of end support
of the column, as follows.
For both ends pinned (hinged, free to rotate), K=1.
For both ends fixed, K=0.5
For one end fixed and the other end pinned, K = β2/2 = 0.7071
For one end fixed and the other end free to move laterally, K=2
KL is called the effective length and represents the length of the equivalent Euler
column
The structural instability of the component occurs when the applied loads reach the
Euler buckling load which may also be called as the bifurcation buckling load. The
stability conditions depending on the applied loads on such component are represented
in Figure 2.1. The bifurcation point in figure shows the critical buckling load.
Figure 2.1 Load-Deformation Behavior for an Axially Loaded Bar [23]
7
2.3 Buckling of Thin Plate
2.3.1 Introduction
Thin plate is a geometrical structure which has a thickness, t, much smaller than the
dimensions of its two edges, a and b. Similar to long columns, thin plates are tend to
buckle out of their plane in the presence of compressive loads. The buckled shape
depends on the loading and support conditions along the edges. Buckling behavior of
plates has a great importance and should be considered since plates are widely used in
many engineering applications, particularly in aeronautical engineering.
2.3.2 Boundary conditions
The geometry of the plate having a long edge, a, short edge, b, and thickness, t, is
represented in Figure 2.2 and placed in the coordinate system accordingly.
Figure 2.2 Geometry of the Plate
8
Free Edge (F)
Such an edge is free of moment and vertical shear force. That is;
For x=0 and x=a ; Mx(x,y)=0; Vy(x,y)=0
(2.2)
For y=0 and y=b ; My(x,y)=0; Vx(x,y)=0
Simply Supported Edge (S)
The plate that will be analyzed is simply supported on all edges. The out of plane
deflection and the bending moment are both zero along the simply supported edges.
Hence;
For x=0 and x=a ; w(x,y)=0; Mx(x,y)=0
(2.3)
For y=0 and y=b ; w(x,y)=0; My(x,y)=0
Bending moments expressed as;
Mx(x,y)=0 π2π€
ππ₯2 + ππ2π€
ππ¦2 = 0
(2.4)
My(x,y)=0 ππ2π€
ππ₯2 +π2π€
ππ¦2 = 0
Clamped Edge (C)
In the clamped edge, both the deflection and slope vanishes. That is;
For x=0 and x=a ; w(x,y)=0; ππ€(π₯,π¦)
ππ₯= 0
(2.5)
For y=0 and y=b ; w(x,y)=0; ππ€(π₯,π¦)
ππ¦= 0
9
2.3.3 Derivation of elastic buckling equations of plates
It is assumed that the forces applied in the middle plane, cause the plate to buckle
slightly. In the investigation of the structural instability of the structure, the magnitudes
of the forces that keep the plate in that slightly buckled shape are calculated. The
equation for buckled plate is represented as [24];
π4π€
ππ₯4+ 2
π4π€
ππ₯2ππ¦2+
π4π€
ππ¦4=
1
π·(ππ₯
π2π€
ππ₯2+ ππ¦
π2π€
ππ¦2+ 2ππ₯π¦
π2π€
ππ₯ππ¦) (2.6)
Applying uniform forces means that the forces Nx, Ny and Nxy are constant throughout
the plate. For definite values of these forces, the desired critical value can be determined
by the use of the given boundary conditions.
2.3.4 Energy method
Another approach that can be used in investigation of buckling of plates is the energy
method. It is mostly used when an accurate result could not be obtained by using
Equation (2.6) and only when a close result is needed for the critical load value. On this
approach, formulations are reduced to prevent complexity.
ΞT1 is the work of external forces and ΞU is the strain energy of bending, we find the
critical values of forces from the equation;
Ξπ1 = ΞU (2.7)
By using the above equation and assuming the edges of the plate are prevented from
movement in the xy-plane during buckling, it is obtained that;
1
2β¬ [ππ₯ (
ππ€
ππ₯)
2
+ ππ¦ (ππ€
ππ¦)
2
+ 2ππ₯π¦
ππ€
ππ₯
ππ€
ππ¦] ππ₯ππ¦
+ π·
2β¬ (
π2π€
ππ₯2+
π2π€
ππ¦2)
2
β 2(1 β Κ) [π2π€
ππ₯2
π2π€
ππ¦2β (
π2π€
ππ₯ππ¦)
2
] ππ₯ππ¦ = 0 (2.8)
The first integral in this equation represents the change in strain energy due to stretching
of the middle plane of the plate during buckling, and the second represents the energy of
bending of the plate and the critical value is obtained by equating these two integrals.
10
2.3.5 Buckling of rectangular plates uniformly compressed in one direction
A rectangular thin plate having length a, width b, and thickness t, is subjected to uniaxial
compressive loads Nx. The thickness of the plate is much smaller than the edges;
π‘ βͺ π, π (2.9)
The simplest plate theory is proposed by Kirchhoff and the assumptions for the
Kirchhoff plate theory are [25]:
deflections are small (i.e. less than the thickness of the plate),
the middle plane of the plate does not stretch during bending, and remains a
neutral surface,
plane sections rotate during bending to remain normal to the neutral surface, and
do not distort, so that stresses and strains are proportional to their distance from
the neutral surface,
the loads are entirely resisted by bending moments induced in the elements of the
plate and the effect of shearing forces is neglected
The following displacement field could be expressed based on the assumptions;
π’ = π’0 β π§ππ€
ππ₯
π£ = π£0 β π§ππ€
ππ¦ (2.10)
π€ = π€0
where u, v and w are displacement components in the directions of x, y, and z axes,
respectively. u0 and v0 are displacement components associated with the plane of z=0.
These assumptions yield to the following stress and strain components;
ππ§ = 0
νπ§ = νπ₯π§ = νπ¦π§ = 0 (2.11)
11
The non-zero linear strains associated with the displacement field are;
νπ₯π₯ =π
ππ₯= βπ§
π2π€
ππ₯2
νπ¦π¦ =π
ππ¦= βπ§
π2π€
ππ¦2 (2.12)
πΎπ₯π¦ =π
ππ¦+
π
ππ₯= β2π§
π2π€
ππ₯ππ¦
The virtual strain energy U of the Kirchhoff plate theory is given by;
πΏπ = β« [β« (ππ₯π₯πΏνπ₯π₯ + ππ¦π¦πΏνπ¦π¦ + ππ₯π¦πΏπΎπ₯π¦)ππ§β/2
ββ/2
] ππ₯ππ¦Ξ©0
= β β« (ππ₯π₯
π2πΏπ€
ππ₯2+ ππ¦π¦
π2πΏπ€
ππ¦2+ 2ππ₯π¦
π2πΏπ€
ππ₯ππ¦) ππ₯ππ¦
Ξ©0
(2.13)
where Ξ©0 denotes the domain occupied by the mid-plane of the plate, (Οxx, Οyy) the
normal stresses, Οxy the shear stress, and (Mxx, Myy, Mxy) the moments per unit length.
Note that the virtual strain energy associated with the transverse shear strains is zero as
Ξ³yz= Ξ³xz=0 in the Kirchhoff plate theory. The relationship between the moments and
stresses are given by [2];
ππ₯π₯ = β« ππ₯π₯π§ππ§β/2
ββ/2
ππ¦π¦ = β« ππ¦π¦π§ππ§β/2
ββ/2
(2.14)
ππ₯π¦ = β« ππ₯π¦π§ππ§β/2
ββ/2
The work W done by the uniaxial load Nx, due to displacement w only, equals;
π = β1
2β« ππ₯ (
ππ€
ππ₯)
2
Ξ©0
ππ₯ππ¦ (2.15)
12
The virtual work Ξ΄W due to the uniaxial load Nx is given by
πΏπ = β« ππ₯
ππ€
ππ₯
ππΏπ€
ππ₯Ξ©0
ππ₯ππ¦ (2.16)
The principle of virtual displacements requires that first variation of total potential
energy β is equal to zero.
Ξ΄β = Ξ΄U β Ξ΄W = 0 (2.17)
Ξ΄β = β β« (ππ₯π₯
π2πΏπ€
ππ₯2+ ππ¦π¦
π2πΏπ€
ππ¦2+2ππ₯π¦
π2πΏπ€
ππ₯ππ¦+ ππ₯
ππ€
ππ₯
ππΏπ€
ππ₯) ππ₯ππ¦ = 0
Ξ©0
(2.18)
Assuming the material of the plate to be isotropic and obeys Hookeβs law, then the
stress-strain relations are given by;
β« (ππ₯π₯
π2πΏπ€
ππ₯2+ ππ¦π¦
π2πΏπ€
ππ¦2+2ππ₯π¦
π2πΏπ€
ππ₯ππ¦+ ππ₯
ππ€
ππ₯
ππΏπ€
ππ₯) ππ₯ππ¦ = 0
Ξ©0
(2.19)
ππ₯π₯ =πΈ
1 β π£2(νπ₯π₯ + π£νπ¦π¦)
ππ¦π¦ =πΈ
1 β π£2(νπ¦π¦ + π£νπ₯π₯) (2.20)
ππ₯π¦ = πΊπΎπ₯π¦ =πΈ
2(1 + π£)πΎπ₯π¦
where E denote the Youngβs modulus, G the shear modulus, and Ξ½ the Poissonβs ratio.
By substituting Equations (2.20) into Equation (2.14) and carrying out the integration
over the plate thickness:
ππ₯π₯ = β« ππ₯π₯π§ππ§ =πΈ
1 β π£2β« (νπ₯π₯ + π£νπ¦π¦)π§ππ§ = βπ· (
π2π€
ππ₯2+ π£
π2π€
ππ¦2)
β/2
ββ/2
β/2
ββ/2
ππ¦π¦ = β« ππ¦π¦π§ππ§ =πΈ
1 β π£2β« (νπ¦π¦ + π£νπ₯π₯)π§ππ§ = βπ· (
π2π€
ππ¦2+ π£
π2π€
ππ₯2)
β/2
ββ/2
β/2
ββ/2
(2.21)
ππ₯π¦ = β« ππ₯π¦π§ππ§ = πΊ β« πΎπ₯π¦π§ππ§ = β(1 β π£)π·π2π€
ππ₯ππ¦
β/2
ββ/2
β/2
ββ/2
13
where D is the flexural rigidity;
π· =πΈβ3
12(1 β π£2) (2.22)
the governing equation for buckling of plate subjected to a uniaxial load is obtained:
π· (π4π€
ππ₯4+ 2
π4π€
ππ₯2ππ¦2+
π4π€
ππ¦4) + ππ₯
π2π€
ππ₯2= 0 (2.23)
In, Figure 2.3 uniaxially compressed plate is shown.
Figure 2.3 Uniaxially compressed plate
Equation (2.23) can be used for plates having different boundary conditions by
modifying the equation according to the boundary condition requirements. In the
following section, a rectangular plate which is clamped at two opposite sides and simply
supported along the other two sides is represented and the buckling equation of the plate
is obtained by using Equation (2.23).
2.3.6 Buckling of SCSC rectangular plates uniformly compressed in simply
supported sides
A rectangular plate clamped at two opposite sides, simply supported along the other two
sides, and uniformly compressed in the direction of the simply supported sides is
represented in Figure 2.4.
14
Figure 2.4 Uniformly Compressed SCSC Plate
By using the method of integration and Eq. (2.23), which is for the case of uniform
compression along the x axis, and with Nx considered positive for compression [4];
π4π€
ππ₯4+ 2
π4π€
ππ₯2ππ¦2+
π4π€
ππ¦4= β
ππ₯
π·
π2π€
ππ₯2 (2.24)
Assuming that under the action of compressive forces the plate buckles in m sinusoidal
half-waves;
π€ = π(π¦) sinπππ₯
π (2.25)
Expression (2.25) satisfies the boundary conditions along the simply supported sides
x = 0 and x = a of the plate, since at;
x=0 and x=a w=0 π2π€
ππ₯2 + ππ2π€
ππ¦2 = 0 (2.26)
The critical value of the compressive stress is given by the equation;
Οππ = ππ2π·
π2π‘ (2.27)
in which k is a numerical factor depending on the ratio a/b of the sides of the plate.
15
Several values of this factor are given in Table 2.1.
Table 2.1 Values of the Factor k [4]
a/b 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0
k 13.38 8.73 6.74 5.84 5.45 5.34 5.18 4.85 4.52 4.41
It can be seen that the effect of clamping the edges on the magnitude of Οcr decreases as
the ratio a/b increases.
16
3 POST-BUCKLING
3.1 Introduction
As it is mentioned in the previous chapter, buckling is the sudden failure of a structure in
the presence of compressive stress. When small loads applied to a slender structure, it
deforms with hardly noticeable change in the geometry and load carrying capacity. At
the point of critical load value, the structure suddenly experiences a large deformation
and may lose its ability to carry load. This stage is the buckling stage. After the buckling
applied load may or may not change, while deformation continues to increase. However,
in some cases the plates continue to carry load after certain amount of deflection which
is called the post-buckling stage as the follow-up of buckling.
Buckling and post-buckling characteristics are of great importance in the analysis and
design processes. Analyzing and designing a thin plate without unseasonable failure
makes such configurations desirable from structural design stand point.
3.2 Post-Buckling Approaches
Eigen value buckling prediction is used to determine critical buckling loads and
imperfection sensitivity of structures. It is a linear perturbation procedure and performed
by solving the following equation [26];
(πΎ0ππ + πππΎβ
ππ)π£ππ = 0 (3.1)
where, K0NM is the stiffness matrix of initial state of the model, KΞ
NM is the initial stress
and load stiffness matrix with respect to applied reference load, Ξ»i is the eigenvalues,
17
ΚiM is the eigenvectors or buckling mode shapes, M and N are the degree of freedom of
the FE model and i is the number of buckling mode.
In order to determine the eigenvalues, a reference load and required boundary conditions
should be applied to finite element model first. Eventually, when eigenvalues are
obtained from the analysis, they are multiplied with the reference applied load to the
model, so that the critical buckling load of the structure is obtained by;
πππππ = πππ (3.2)
An eigenvalue analysis is not a post-buckling approach. However, it is a required step
for further analysis. Therefore, non-linear static analysis is required to investigate post-
buckling response of structure.
3.3 Post βBuckling Analysis
3.3.1 Von Karman large deflection equations with initial imperfections
An analysis of post-buckling behavior was carried out, assuming that after the
buckling, the out of plane deflections remain the same form but only the deflection
magnitude changes. This assumption gives reasonable results for plates when the applied
load is less than about twice the buckling load. Since detailed formulation is lengthy and
complex an outline of the post-buckling analysis will be given. There are two
simultaneous non-linear differential equations that set up by von Karman and then
reformed by Marguerre. These equations give exact solution only for the simplest
loading and support conditions [27].
The Equilibrium Equation is;
π4π€
ππ₯4+ 2
π4π€
ππ₯2ππ¦2+
π4π€
ππ¦4=
π
π·+
π‘
π·[π2πΉ
ππ¦2
π2(π€ + π€0)
ππ₯2β 2
π2πΉ
ππ₯ππ¦2
π2(π€ + π€0)
ππ₯ππ¦+
π2πΉ
ππ₯2
π2(π€ + π€0)
ππ¦2] (3.3)
18
The Compatibility Equation is;
π4πΉ
ππ₯4+ 2
π4πΉ
ππ₯2ππ¦2+
π4πΉ
ππ¦4= πΈ [(
π2π€
ππ₯ππ¦)
2
βπ2π€
ππ₯2
π2π€
ππ¦2β (
π2π€0
ππ₯ππ¦)
2
+π2π€0
ππ₯2
π2π€0
ππ¦2] (3.4)
Where F is the stress variation in the plate, w is the out-of-plane deflection at any point
on the plate, w0 is the initial deflection (or imperfection) system in the plate, q specifies
the lateral load on the plate, D is the plate flexural rigidity factor. From the buckling
solution, the deflected form can be written as;
π€ = π΄π sinπππ₯
πβ π΄
π cosπππ¦
π
π
π=1
(3.5)
Where Ar is a coefficient prescribing the magnitude of the deflections, An are
normalized values of the eigenvector obtained in the buckling analysis, a is the length of
plate in x direction, b is the width of plate in y direction, k is the number of half-waves
in loaded direction, n is the number of half-waves across loaded direction and x, y, z are
cartesian coordinates of plate.
3.3.2 Empirical approach (Effective width)
According to the empirical research that made to increase the accuracy of the analysis on
the post-buckling behavior of thin plates, it was observed that beyond a certain width the
ultimate load that could be carried was not effected from the actual width. That brings
out the expression of effective width which was developed by von Karman. This
expression states that for a plate of actual width βbβ, an effective width βbeβ can be used
in the evaluation of the load-carrying capacity.
Von Karmanβs effective width expression in terms of the critical stress and yield stress
is;
ππ
π= β
ππΆπ
ππ (3.6)
19
where,
ππΆπ =ππ2πΈπ‘2
12(1 β π£2)π2 (3.7)
Which later modified by Winter [28];
ππ
π= β
ππΆπ
ππ[1 β 0.22β(
ππΆπ
ππ)] (3.8)
Effective width method is used to calculate the strength of the plate. This concept
replaces the non-uniform stress distribution by an equivalent uniform stress distribution
which is equal to the stress at the edges over a reduced width of the plate.
3.3.3 Riks method
The Riks method is used to predict stable and unstable post-buckling behavior of the
structure and the approach of geometrically nonlinear failure requires incremental
solution. In the Riks method, the load magnitude is used as an additional unknown and it
solves for loads and displacements simultaneously. During periods of response in
unstable problems, the load and/or the displacement may decrease as the solution
progresses and form a type of behavior as shown in Figure 3.1.
Figure 3.1 Unstable Static Response [22];
20
PN is the loading pattern which expresses one or more loads (N is the degree of freedom)
and Ξ» is the load magnitude parameter. Thus, the actual load state at any time is defined
as Ξ»PN and uM is the displacements at that time (M is the degree of freedom). In
ABAQUS the maximum value of all displacement variables processes in the solution
space and the solution path is defined to be the continuous set of equilibrium points
described by the vector (uM;Ξ»). In scaled load-displacement space, an initial increment
Ξlin is defined by the user. Another parameter lperiod, the total arc length scale factor, may
also be specified, if unspecified default value of 1 is assumed by software. By using
these two parameters, the initial load proportionality factor Ξ»in is calculated by the
following formula [29];
βπππ =βπππ
πππππππ (3.9)
Assuming a solution has been formed at ith time (π’ππ , Ξ»π), then tangent stiffness Ki
NM is
formed. The finite element formula of the structure:
πΎππππ’π
π = πππ (3.10)
Through the iterations of Riks method, Ξ», the load magnitude parameter is calculated
automatically. By defining Ξlmin and Ξlmax, the incrementation can be limited according
to the analysis requirements otherwise it will continue until ABAQUS completes the
step.
21
4 ANALYSIS & RESULTS
In this chapter, post-buckling behavior of perforated plates is investigated by evaluating
the results obtained from the analysis. Since it is not possible to obtain exact solutions
by analytical methods, the numerical post-buckling calculations are performed by using
ABAQUS. As one of the most effective methods to perform post-buckling analysis, Riks
Method is used to examine thin perforated plates with various properties.
4.1 Material Selection
Steel plates are widely used in engineering industries such as aerospace, automotive,
marine and etc. In aerospace applications, aerospace components should have damage
tolerance under both static and dynamic load. The high tensile strength, shear modulus,
resistance to corrosion and high temperature tolerance are the key parameters of steel to
be especially used in landing gear and engine components. The properties of Steel are;
E=200 GPa
Ξ½ = 0.3
4.2 Finite Element Model
4.2.1 Modelling
The rectangular plate having a long edge (a) of 750 mm, a short edge (b) of 375 mm and
thickness (t) of 3 mm is created on ABAQUS in order to compare the behavior of the
perforated plates with unperforated plates. The plate is shown in Figure 4.1.
22
Figure 4.1 Model of the plate without hole
At the center of the plate, a hole with radius of 37.5 mm had been extruded as shown in
Figure 4.2, resulting in a d/b ratio of 0.2.
Figure 4.2 Model of the plate with hole
In the analysis, the effect of various a/b ratio, thickness and d/b ratio are investigated.
4.2.2 Approximate global size and meshing
In order to perform finite element analysis, the plates need to be meshed. Before
meshing the structure, element size must be determined. Since the analyses give more
accurate results as the approximate global size (AGS) decreases, it is important to make
23
a sensible selection. For this reason, analysis performed starting from an AGS of 50 mm
and decreased until the critical value converges. The dimensions of the plate are 750 mm
x 375 mm and the plate is unperforated. Critical load values depending on the element
size are represented in Table 4.1.
Table 4.1 Change in Critical Buckling Load with Element Size
AGS
(mm)
Number of
Elements
Critical
Load
(N/mm)
Change
(%)
50 120 101.25
40 171 100.67 0.58
30 325 99.519 1.143
25 450 99.253 0.267
20 722 98.944 0.311
15 1250 98.715 0.229
10 2850 98.510 0.205
As it is seen from the table the change in the critical load value does not show a major
change. In order to achieve more accurate results the element size is chosen to be 15 mm
in the analyses. The meshed structure with element size of 15 mm is shown in
Figure 4.3.
24
Figure 4.3 Meshed Model with Element Size 15 mm
4.2.3 Boundary conditions
The finite element model is simply supported on all edges, therefore the out of plane
deflection and the bending moment are both zero along the edges. Figure 4.4 represents
the simply supported plate, where the degrees of freedom UX, UY, UZ are node
displacements and URX, URY, URZ are rotational displacements.
Figure 4.4 Boundary Conditions of the Plate
25
4.2.4 Loading in buckling stage
In the analyses, uniaxial compression force is applied to the rectangular body. Therefore,
thin plate is subjected to βShell Edge Loadβ in the short edge at the right. The magnitude
of the load is 1 N/mm in the buckling stage and the model is shown in Figure 4.5.
Figure 4.5 Model with Boundary Conditions and Applied Load
4.2.5 Solution method in buckling stage
Eigenvalue buckling prediction is used to determine critical buckling loads. The critical
buckling load of the structure is obtained by multiplying the applied reference load with
the eigenvalue found by the analysis.
πππππ = πππ
Since the reference load is 1 N/mm, critical buckling load is equal to the eigenvalue
achieved as a result of the buckling analysis in ABAQUS.
4.2.6 Loading in Post-Buckling Stage
In the post-buckling analysis, all of the structural properties are same as the buckling
stage and only the magnitude of load changes. The magnitude of Shell Edge Load is
needed to be updated with the critical load obtained in the previous buckling analysis.
26
4.2.7 Solution method in post-buckling stage
Further in the analysis, the linear buckling step is needed to change with nonlinear Static
Riks analysis. It is asked to set incrementation and in order to determine that, tests had
been done on plates. Starting with 100, number of increment is increased and maximum
displacement at the nodes are observed. By 280 incrementation, it was seen that
maximum displacement was converging. Thus, it is determined to be suitable to set the
incrementation as 300. As the result of the Riks analyses node displacements are
obtained.
4.3 Results
Finite element analyses are completed in ABAQUS and the results are represented in
this chapter. In the analyses plates with different boundary conditions, aspect ratio (a/b),
materials, thickness, hole size and hole shape are investigated. Also, analyses on plates
without hole are performed to compare with plates with hole.
4.3.1 Effect of a/b ratio
In the analyses, plates with length (a) of 750 mm and thickness (t) of 3 mm are used.
The plates are simply supported from all edges. The height (b) of the plate varies and
each a/b ratio analysis is performed for both perforated and unperforated plates. The
radius (r) is 37.5 mm for the perforated plates. Critical buckling loads of plates with and
without hole for varying a/b ratio are represented in Table 4.2 and Table 4.3 and plotted
as in Figure 4.6 an Figure 4.7.
27
Table 4.2 Critical Buckling Loads of Plates without Hole
Plate without hole
a/b Critical Load
(N/mm)
1 26.675
1.5 54.708
2 98.715
3 219.45
5 615.28
7.5 1402.9
Figure 4.6 Critical Load vs. a/b ratio for Plates without Hole
0
200
400
600
800
1000
1200
1400
1600
0 1 2 3 4 5 6 7 8
Cri
tica
l Lo
ad (
N/m
m)
a/b ratio
Effect of a/b ratio on the critical load
28
Table 4.3 Critical Buckling Loads of Plates with Hole
Plate with hole
a/b Critical Load
(N/mm)
1 25.644
1.5 51.269
2 89.640
3 195.10
5 485.48
7.5 1080
Figure 4.7 Critical Load vs. a/b ratio for Plates with Hole
As is seen in the tables and figures, critical buckling load increases as a/b ratio increase
for fixed length (a). Reducing the length of the short edges causes the long (side) edges
0
200
400
600
800
1000
1200
0 1 2 3 4 5 6 7 8
Cri
tica
l Lo
ad (
N/m
m)
a/b ratio
Effect of a/b ratio on the critical load
29
come closer to each other and since these long edges are simply supported the plate
becomes stiffer and more stable, which yield higher critical buckling loads.
Comparing the perforated and unperforated plates, the critical buckling loads are higher
in the unperforated cases which means making a hole at the center of a rectangular
decreases the endurance of a thin plate.
After performing buckling analysis, Riks method is used to investigate the post-buckling
behavior of the perforated plates. In the Riks method, the initial load applied to the body,
is the critical load value obtained as a result of the eigenvalue buckling prediction. The
post-buckling analyses had been performed on the perforated plates with a/b ratio 1 to 3.
At the end of the Riks analyses, it is seen that the displacement in z-direction comprises
around %99 of the total displacement. Table 4.4 represents the maximum displacements
and its z-component.
Table 4.4 Maximum Displacements of Plates with Hole in Post-Buckling Stage
Plate with hole
a/b
Critical
Load
(N/mm)
Umax
(mm)
(U3)max
(mm)
1 25.644 8.872 8.868
1.5 51.269 7.548 7.528
2 89.640 8. 370 8.257
3 195.10 7.603 7.533
Therefore, in the results of post-buckling analyses the z-component of the displacement
is used.
In order to be able to make a comparison between plates with different a/b ratio, a node
near the hole had been selected as reference point at all plates. The exact location of the
selected node is shown in Figure 4.8.
30
Figure 4.8 Location of the Selected Node for Post-Buckling Analyses
The displacement of the node with incrementation is plotted on ABAQUS and then
transferred to MS Excel. The plots for plates with different aspect ratio had been
superimposed to better observe the fluctuation. Length (a) is 750 mm for all plates and
(b) varies. Figure 4.9, Figure 4.10 and Figure 4.11 show the change in post-buckling
behavior of plates.
Figure 4.9 Displacement of the Node for a/b=1 and a/b=1.5
-3
-2
-1
0
1
2
3
4
5
6
7
1 10 19
28 37 46
55
64
73
82
91
10
0
10
9
11
8
127
13
6
14
5
15
4
16
3
17
2
18
1
19
0
19
9
20
8
217
22
6
23
5
24
4
25
3
26
2
27
1
28
0
28
9
29
8Dis
pla
cem
ent (
mm
)
Number of increment
Effect of a/b on the displacement of the node
a/b=1 a/b=1.5
31
Figure 4.10 Displacement of the Node for a/b=1.5 and a/b=2
Figure 4.11 Displacement of the Node for a/b=2 and a/b=3
As it is seen from the figures, the variation in displacement in +z and βz direction
increases as the aspect ratio increases. This is related to the increase in critical load and
decrease in axial stiffness. The critical load is highest at a/b=3 and this results in to more
variation in displacement at the node near the hole.
-8
-6
-4
-2
0
2
4
1 10
19
28
37
46
55
64
73
82
91
10
0
10
9
11
8
12
7
13
6
14
5
15
4
16
3
17
2
18
1
19
0
19
9
20
8
21
7
22
6
23
5
24
4
25
3
26
2
27
1
28
0
289
298
Dis
pla
cem
ent
(mm
)
Number of increment
Effect of a/b on the displacement of the node
a/b=1.5 a/b=2
-8
-6
-4
-2
0
2
4
6
8
1 10
19
28
37
46 55 64
73
82
91
10
0
10
9
11
8
12
7
13
6
14
5
15
4
16
3
17
2
18
1
19
0
19
9
20
8
21
7
22
6
23
5
24
4
25
3
26
2
27
1
28
0
28
9
29
8
Dis
pla
cem
ent
(mm
)
Number of increment
Effect of a/b on the displacement of the node
a/b=2 a/b=3
32
4.3.2 Effect of boundary condition
The analyses presented in the previous part for investigating the effect of different a/b
ratio, are also performed for plates having different boundary conditions. In this case,
plates are simply supported from the short edges and free at the long edges as shown in
Figure 4.12.
Figure 4.12 Boundary Conditions of the Simply Supported and Free Plate
Critical buckling load values are represented in Table 4.5 for plates without hole and
having different boundary conditions.
Table 4.5 Critical Loads for Plates without Hole and with Different Boundary
Conditions
Plate without hole
a/b SS
Critical Load
(N/mm)
SF
Critical Load
(N/mm)
1 25.644 8.3017
1.5 51.269 8.1552
2 89.640 8.0691
3 195.10 7.9864
33
Critical buckling load values are represented in Table 4.6 for plates with hole and having
different boundary conditions.
Table 4.6 Critical Loads for Plates with Hole and Different Boundary Conditions
Plate with hole
a/b SS
Critical Load
(N/mm)
SF
Critical Load
(N/mm)
1 25.644 8.0443
1.5 51.269 7.7943
2 89.640 7.6048
3 195.10 7.3043
It can be seen from the tables that when the long sides are free, major decrease is
observed in the critical load values. This can be related to that the plates are behaving
more like columns when they are not supported from the two sides. Especially, as the
short edge decreases the plate approaches more to a column and lose its resistance.
However, when the plate is simply supported at all sides, the effect of the boundary
condition increases as the short edge decreases. The supports obstruct the buckling and
the critical load increases.
It can be concluded that since the critical buckling loads are very low when the long
edges are not supported, the plateβs behavior in the post-buckling stage will be stable
and not much variation in displacement will be observed.
4.3.3 Effect of thickness
Analyses performed on plates with different thicknesses (t) and critical buckling loads
are obtained. The dimensions of the plates are 750 mm x 375 mm and the hole radius is
18.75 mm. The results are represented in Table 4.7.
34
Table 4.7 Critical Buckling Loads for Plates with Different Thicknesses
Thickness
(mm)
Critical Load
(N/mm)
2 28.561
3 96.314
4 228.06
6 767.64
8 1814.0
10 3531.1
As seen in Table 4.7 the critical buckling load increases with increasing plate thickness
and the plate becomes stiffer. It can also be deduced that as the thickness is multiplied
by 2, the critical load multiplies by the cube of 2.
In the previous sections, it was concluded that the increase in critical buckling load
makes the variation in displacement more frequent. Meaning that the 10 mm plate will
endure more motion in the post-buckling stage compared to the thinner plates.
4.3.4 Effect of d/b ratio
Critical buckling loads are obtained for plates with different diameter to short edge ratio
(d/b). The dimensions of the plates are 750 mm x 375 mm and the thickness is 3 mm.
Table 4.8 shows the results of the buckling loads for plates with different d/b ratio.
35
Table 4.8 Critical Loads for Plates with Different d/b Ratio
d/b Critical Load
(N/mm)
0.1 96.314
0.2 89.638
0.3 85.529
0.4 77.127
0.5 73.865
It is seen from the table that increasing the diameter of the hole at the center of a thin
plate, reduces the critical buckling load. Since the critical load decreases as the d/b ratio
increase, variation is expected to be less in displacement of the plates with higher d/b
ratio. Post-buckling analyses are performed to crosscheck the behavior of the plates. As
in the previous analyses, the same node near the hole is selected for the displacement
plots. Figure 4.13 shows the displacements in plates with d/b ratio of 0.1 and 0.5.
Figure 4.13 Displacement of the Node for d/b=0.5 and d/b=0.1
-10
-5
0
5
10
15
20
1
10
19
28
37
46
55
64
73
82
91
100 10
9
118
127
136
145
154
163
172
181
190
199
208
217
226
235
244 25
3
262
271
280
289
298D
isp
lace
men
t (m
m)
Number of increment
Effect of d/b on the displacement of the node
d/b=0.5 d/b=0.1
36
It is verified from the figure that variation in displacement increases as d/b ratio
decreases since the critical load is found to be higher.
4.3.5 Effect of hole shape
Critical buckling loads are obtained for plates with dimensions 750 mm x 375 mm and 3
mm thickness. The aim was to observe the effect of hole shape on the buckling load and
post-buckling behavior. Circular, elliptical and square cutouts with equal cross-sectional
areas had been placed at the center of the plates. For the circular cutout the diameter is
determined to be 112.5 mm which corresponds to d/b ratio of 0.3 and area of 9940.2
mm2. The square hole with an edge length of 99.701 mm and the elliptical hole with
radii r1=40 mm and r2=79.102 mm are generated. In Table 4.9 buckling loads of plates
with different hole shapes are represented.
Table 4.9 Critical Loads for Plates with Different Hole Shapes
Hole
shape
Critical
Load
(N/mm)
Ellipse 81.819
Circle 85.529
Square 86.364
The critical buckling load is highest for the plate with square shaped hole and lowest for
the elliptical. It can be concluded that in post-buckling stage, the plate with square
shaped hole endures more variation in displacement in z-axis.
Figure 4.14, Figure 4.15 and Figure 4.16 show the plates with different hole shapes after
buckling analyses performed.
37
Figure 4.14 Buckled Shape of Plate with Elliptical Hole
Figure 4.15 Buckled Shape of Plate with Circular Hole
Figure 4.16 Buckled Shape of Plate with Square Hole
38
4.3.6 Effect of plate material
In all analyses, the material of the plates were steel which has Youngβs Modulus of 200
GPa and Poissonβs ratio of 0.3. In order to observe the effect of material, analysis
performed on an Aluminum plate with dimensions 750 mm x 375 mm, thickness of 3
mm and d/b ratio of 0.1. Table 4.10 shows the material properties and critical buckling
loads of steel and aluminum plates.
Table 4.10 Material Properties and Critical Loads of Steel and Aluminum Plates
Material
Young's
Modulus
(GPa)
Poisson's
Ratio
Critical
Load
(N/mm)
Steel 200 0.3 96.314
Aluminum 70 0.33 32.619
It is known from the buckling formula that Youngβs Modulus is directly proportional to
critical buckling load. It is also confirmed with the analyses since critical load triples as
the Youngβs Modulus triples. Thus, the post-buckling behavior prediction can be made
that steel plates will undergo more variation in displacement compared to the aluminum
plates.
39
5. CONCLUSION
This study presents the investigation of buckling and post-buckling behavior of
perforated rectangular plates. The methods and formulations developed for buckling and
post-buckling analysis of plates had been examined and the ones that are effective for
this study are determined. The behavior of thin rectangular perforated plates subjected to
uniform compressive force is studied using finite element analysis. For the numerical
analysis, eigenvalue buckling prediction and non-linear static Riks method is used and
the investigations are performed on ABAQUS. The effects of plate-support conditions,
aspect ratio, thickness, hole size, hole shape and material on the buckling and post-
buckling strength of the perforated plates was studied. The acquired results are
introduced briefly;
1. It is seen in analysis that for a simply supported plate the critical buckling load
increases as the a/b ratio increase independent from the existence of a central
hole.
In the post-buckling analysis, the maximum displacement in z-direction
comprises %99 of the total displacement thus it is concluded that considering the
displacement in z-axis is sufficient for comparison. As the result of post-buckling
analysis of plates with different aspect ratio, the variation in displacement is
observed to be more for plates of higher ratio. This is related to increased critical
buckling load and decreased axial stiffness.
2. The effect of plate support conditions on the behavior of plate is investigated.
Considerable amount of difference in critical buckling load had been observed
between plates that are simply supported at all edges and that are free at the
unloaded long edges.
40
3. In the analysis of thickness variation, increase in critical load is observed as the
plate get thicker. As a result, in the post-buckling stage plates with higher
thicknesses had withstand more fluctuation in displacement compared to thinner
plates.
4. Effect of hole size had been investigated by deriving the ratio d/b that is the
relation between the diameter of the hole (d) and short edge of the plate (b).
Decrease in critical buckling load had been noticed as the ratio increase.
5. The change in critical load is investigated on plates with different hole shapes. In
the analysis circular, elliptical and square holes with same cross-sectional areas
are placed at the center of the plates. It is seen that the plate with elliptical hole
would be the first to collapse and the one with square hole at last.
6. The last topic of comparison was the material of the plate. Since steel and
aluminum have almost same Poissonβs ratio, 0.3 and 0.33 respectively, it was
possible to confirm the proportionality between the critical load and the Youngβs
Modulus. Since the critical load of aluminum plate is less than the steel plate, it
shows fewer variation in displacement at post-buckling stage.
41
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