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GRADUATION PROJECT

JANUARY, 2021

POST-BUCKLING BEHAVIOR OF RECTANGULAR PLATES WITH A HOLE

Thesis Advisor: Prof. Dr. Vedat Ziya DOĞAN

Nihal Özden YETKİN

Department of Astronautical Engineering

Anabilim Dalı : Herhangi Mühendislik, Bilim

Programı : 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 Özden YETKİN

110150153

Department of Astronautical Engineering

Anabilim Dalı : Herhangi Mühendislik, Bilim

Programı : Herhangi Program

Thesis Advisor: Prof. Dr. Vedat Ziya DOĞAN

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Thesis Advisor : Prof. Dr. Vedat Ziya DOĞAN ..............................

İstanbul Technical University

Jury Members : Prof. Dr. Halit Süleyman TÜRKMEN .............................

İstanbul Technical University

Asst. Prof. Demet BALKAN ..............................

İstanbul Technical University

Nihal Özden YETKİN, student of ITU Faculty of Aeronautics and Astronautics

student ID 110150153, successfully defended the graduation entitled” POST-

BUCKLING BEHAVIOR OF RECTANGULAR PLATES WITH A HOLE”,

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 DOĞAN

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 Özden 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

ÖZET....................................................................................................................................... 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 –Buckling 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

viii

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

ΔT : The work of external forces

ΔU : 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

σx, σy, σz : The in-plane normal stresses

εX, εY, εZ : The in-plane normal strains

γxy : The in-plane shear strain

U : The strain energy

∏ : Total potential energy

E : The Young’s modulus

G : The shear modulus

ν : The Poisson’s ratio

D : The flexural rigidity

m : The sinusoidal half-waves

σcr : The critical value of the compressive stress

K0NM : The stiffness matrix of initial state

KΔNM : The initial stress and load stiffness matrix with respect to applied load

λi : The eigenvalues

ʋiM : 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

σy : Yield stress

PN : The loading pattern

λ : The load magnitude parameter

Δlin : 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 Load–end 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 aircraft’s 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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DELİKLİ DİKTÖRTGEN PLAKALARIN BURKULMA SONRASI DAVRANIŞI

ÖZET

Çelik mühendislik alanı uygulamalarında kullanımı sıklıkla tercih edilen bir malzemedir.

Kullanımı en yaygın olan alanlar arasında havacılık, otomotiv ve denizcilik vardır.

Havacılık endüstrisinde, hava aracının kritik önem taşıyan bölgelerinde güvenliği

sağlamak büyük önem taşımaktadır. Bu bölgeler arasında kanatlar, gövde, iniş takımları

ve iskelet vardır. Bu noktada dikkat edilecek ana unsurlar malzemenin aşınmaya ve

yüksek sıcaklığa toleransının fazla olmasıdır. Dolayısıyla bu niteliklere sahip olan

çeliğin özellikle iniş takımlarında ve motor parçalarında kullanımı oldukça yaygındır.

Ayrıca yüksek çekme direnci ve kesme katsayısı çeliğin daha çok tercih edilmesini

sağlayan unsurlardır. Fakat havacılık endüstrisinde yapının ağırlığını mümkün

olduğunca düşük tutmak üretimin en önemli esaslarından biridir. Bu da düşük ağırlıklı

malzeme tercihiyle veya yapının ağırlığını azaltacak yöntemlerle sağlanmaktadır.

Örneğin plakalarda delik açmak ağırlığı azaltacağı için tercih edilebildiği gibi bazı

alanlarda da servis, denetim ve bakım kolaylığı sağladığı için uygulanmaktadır. Delikli

plakaların bu gibi alanlarda çokça tercih edilmesi ve davranışlarının düz plakalardan

farklılık göstermesi sebebiyle burkulma ve burkulma sonrası davranışlarının ayrıca

incelenmesi gerekmektedir.

Uygulanan yükün kritik noktaya ulaştığı noktada yapıda büyük deformasyonlar oluşur

ve yapı artık yük taşıyamaz hale gelebilir. Ani yapısal değişimin ortaya çıktığı bu

duruma burkulma denir. Burkulmadan sonra uygulanan yükün artıp azalmasına bağlı

olmadan deformasyon artmaya devam edebilir. Fakat bazı durumlarda belli bir

deformasyondan sonra bile plaka yük taşımaya devam edebilir. Bu gibi durumlarda

plakanın burkulma sonrası davranışı incelenmektedir. Yapılan analizlerde kritik yük

hesaplamada eigenvalue burkulma analizi kullanılırken, burkulma sonrası davranışını

incelemede Riks metodu kullanılmıştır.

Bu projede delikli dikdörtgen çelik plakalar üzerinde çalışılmıştır ve plakalar bütün

kenarlarından basit mesnetlenmiştir. Tüm durumlarda plakalara sağ kısa kenarından tek

doğrultulu basınç uygulanmıştır. Kritik yükün hesaplanması için formül çıkarımı

yapılmıştır ve bu formül farklı sınır şartlarının uygulandığı durumlarda da

kullanılabilecek halde sunulmuştur. Burkulma sonrası davranışı incelemede net sonuç

veren analitik bir çözüm yöntemi bulunmamaktadır, bu durum araştırmalarda daha çok

nümerik çözümlere yönelimi arttırmıştır. Riks metodu yapıların lineer olmayan ve

değişken davranışlarının analizinde kullanılır ve burkulma sonrası metotları arasında en

çok tercih edilendir. Riks metodunda hem uygulanan yük hem de yer değiştirme

bilinmeyendir ve ikisi eşzamanlı olarak çözülür. ABAQUS çözüm uzayında maksimum

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yer değiştirmeyi gösterir ve çözüm eğrisi uygulanan yük ile yer değiştirmenin

dengelendiği noktalar serisinden oluşur.

Delikli plakaların kenar uzunlukları oranı, kalınlığı, delik çapı ve delik şekli

değiştirilerek çeşitli analizler yapılmıştır. Farklı bir materyal tercihi yaparak çelik

plakayla kıyaslanabilmesi için alüminyum tercih edilmiştir. Ayrıca basit mesnetlenmiş

plakaların farklı sınır şartları uygulanmış plakalarla karşılaştırılması yapılmıştır.

Bunlarla beraber, ana konu olan delikli plakalar hakkında en doğru sonuca varabilmek

adına düz plakalarda da analizler yapılmış ve plakada delik açmanın etkileri

gözlemlenmiştir.

Öncelikle plakanın kenar uzunlukları oranının kritik yüke etkisi incelenmiştir. Uzun

kenar sabit tutulmuş, kısa kenar giderek küçültülmüştür. Analizler hem delikli hem

deliksiz plakalar üzerinde yapılmıştır. Sonuç olarak plaka merkezinde bir deliğin

varlığının, kritik yükün deliksiz plakalara göre daha düşük olmasına sebep olduğu

gözlemlenmiştir. Fakat hem delikli hem de deliksiz plakalarda kısa kenar küçüldükçe

kritik yükün arttığı gözlemlenmiştir. Bu durumda plakanın bütün kenarlarından basit

mesnetlenmiş olması etkili olmuştur. Plakanın kısa kenar uzunluğu küçüldükçe

mesnetlenmiş uzun kenarlar birbirine yaklaşmakta ve plakanın daha sabit durmasını

sağlamaktadır. Bu da plakanın burkulma yükünün artmasını ifade etmektedir. Ardından

elde edilen kritik yük verileri Riks metodunda kullanılarak burkulma sonrası analizi

yapılmış ve plakaların maksimum yer değiştirme miktarları bulunmuştur. Analiz

sonucunda maksimum yer değiştirmenin x, y ve z eksenlerindeki bileşenleri incelenmiş

ve z eksenindeki yer değiştirmenin toplam yer değiştirmenin %99’unu oluşturduğu

sonucuna varılmıştır. Bu nedenle burkulma sonrası davranışın incelenmesinde plakanın

z eksenindeki hareketi temel alınmıştır. Kenar uzunlukları oranının etkisini

gözlemleyebilmek amacıyla bütün plakalarda delik yanındaki bir nokta seçilmiş ve o

noktadaki yer değiştirme izlenmiştir. Sonuç olarak, kısa kenar küçüldükçe noktanın yer

değiştirme grafiğinde dalgalanmanın arttığı gözlemlenmiştir. Buradan da kritik yükün

artışına bağlı olarak plakanın pozitif ve negatif z eksenlerindeki yer değiştirme

sapmalarının arttığı sonucuna varılmıştır.

Sınır şartlarının kritik yüke olan etkisini araştırmak amacıyla kısa kenarlarından basit

mesnetlenmiş, uzun kenarlarından serbest bırakılmış ve sağ kısa kenarından yük

uygulanmış plakalar incelenmiştir. Bir önceki analizdeki gibi kenar uzunlukları oranı

değiştirilerek her birinde sınır şartlarının kritik yüke etkisi karşılaştırılmıştır. Sonuçlara

bakıldığında uzun kenarlarından serbest bırakılan plakaların kritik yükünün oldukça

düşük olduğu görülmüştür. Tüm kenarlarından basit mesnetlenmiş plakaların aksine,

uzun kenarları serbest bırakılan plakalarda kısa kenar uzunluğu küçüldükçe kritik yükün

azaldığı gözlemlenmiştir. Bunun sebebinin kısa kenar kısaldıkça plakanın kolon gibi

davranması ve direnme gücünün azalması olduğu sonucuna varılmıştır.

Plaka kalınlığının davranışa etkisini araştırmak amacıyla kenar uzunlukları ve delik çapı

sabit tutulan plakalarda kalınlık 2 ile 10 mm arasında değiştirilmiş ve analizler

yapılmıştır. Plaka kalınlığı arttıkça kritik yükün de arttığı görülmüştür. Kalınlık ile kritik

yük arasında oransal bir ilişki tespit edilmiştir. Plaka kalınlığı 2 katına çıktığı durumda

kritik yükte de 2’nin küpü oranında artış görülmüştür.

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Ardından delik çapının etkisi incelenmiş ve delik çapının kısa kenara oranı değiştirilerek

analiz yapılmıştır. Oran arttıkça kritik yükün azaldığı görülmüştür. Kritik yükün

azalmasıyla burkulma sonrası davranışta yer değiştirmenin dalgalanmasının azalması

beklenmektedir. Sağlama yapmak amacıyla oranın en küçük ve en büyük olduğu

plakaların burkulma sonrası davranışının grafiği çizdirilerek beklenen sonucun doğru

olduğu anlaşılmıştır.

Delik şeklinin plaka üzerindeki etkisini anlamak için 3 farklı şekilde delik

oluşturulmuştur. Yuvarlak, elips ve kare delikler oluşturulup her durumda deliğin alanı

sabit tutulmuştur. Kritik yükün en düşük olduğu plaka deliğin elips şeklinde olduğu

plaka, en yüksek olduğu durum kare delikli plakada olmuştur.

Yapılan tüm analizlerde plaka malzemesi çelik olarak belirlenmiştir. Son olarak çeliğin

başka bir malzemeyle karşılaştırmasını yapmak amacıyla alüminyum plakayla analiz

yapılmış ve sonuçlar karşılaştırılmıştır. Alüminyumun gerilme katsayısı çeliğinkinin

üçte biri kadardır ve gerilme katsayısı burkulma yüküyle doğrudan orantılı olduğu için

burkulma yükü de çeliğinkinin üçte biri olarak bulunmuştur. Bu sebeple alüminyum

plakaların burkulma sonrasında yük altında yer değiştirme dalgalanmasının daha az

olması 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 plate’s 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 Karman’s 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 Load–end 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, elliptical–vertical,

and elliptical–horizontal) 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.

𝑃𝑐𝑟 =𝜋2𝐸𝐼

(𝐾𝐿)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 column’s 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 = √2/2 = 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 𝜕2𝑤

𝜕𝑥2 + 𝜈𝜕2𝑤

𝜕𝑦2 = 0

(2.4)

My(x,y)=0 𝜈𝜕2𝑤

𝜕𝑥2 +𝜕2𝑤

𝜕𝑦2 = 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; 𝜕𝑤(𝑥,𝑦)

𝜕𝑥= 0

(2.5)

For y=0 and y=b ; w(x,y)=0; 𝜕𝑤(𝑥,𝑦)

𝜕𝑦= 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];

𝜕4𝑤

𝜕𝑥4+ 2

𝜕4𝑤

𝜕𝑥2𝜕𝑦2+

𝜕4𝑤

𝜕𝑦4=

1

𝐷(𝑁𝑥

𝜕2𝑤

𝜕𝑥2+ 𝑁𝑦

𝜕2𝑤

𝜕𝑦2+ 2𝑁𝑥𝑦

𝜕2𝑤

𝜕𝑥𝜕𝑦) (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.

ΔT1 is the work of external forces and ΔU is the strain energy of bending, we find the

critical values of forces from the equation;

Δ𝑇1 = ΔU (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

2∬ [𝑁𝑥 (

𝜕𝑤

𝜕𝑥)

2

+ 𝑁𝑦 (𝜕𝑤

𝜕𝑦)

2

+ 2𝑁𝑥𝑦

𝜕𝑤

𝜕𝑥

𝜕𝑤

𝜕𝑦] 𝑑𝑥𝑑𝑦

+ 𝐷

2∬ (

𝜕2𝑤

𝜕𝑥2+

𝜕2𝑤

𝜕𝑦2)

2

− 2(1 − ʋ) [𝜕2𝑤

𝜕𝑥2

𝜕2𝑤

𝜕𝑦2− (

𝜕2𝑤

𝜕𝑥𝜕𝑦)

2

] 𝑑𝑥𝑑𝑦 = 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;

𝑡 ≪ 𝑎, 𝑏 (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;

𝑢 = 𝑢0 − 𝑧𝜕𝑤

𝜕𝑥

𝑣 = 𝑣0 − 𝑧𝜕𝑤

𝜕𝑦 (2.10)

𝑤 = 𝑤0

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;

𝜎𝑧 = 0

휀𝑧 = 휀𝑥𝑧 = 휀𝑦𝑧 = 0 (2.11)

11

The non-zero linear strains associated with the displacement field are;

휀𝑥𝑥 =𝜕

𝜕𝑥= −𝑧

𝜕2𝑤

𝜕𝑥2

휀𝑦𝑦 =𝜕

𝜕𝑦= −𝑧

𝜕2𝑤

𝜕𝑦2 (2.12)

𝛾𝑥𝑦 =𝜕

𝜕𝑦+

𝜕

𝜕𝑥= −2𝑧

𝜕2𝑤

𝜕𝑥𝜕𝑦

The virtual strain energy U of the Kirchhoff plate theory is given by;

𝛿𝑈 = ∫ [∫ (𝜎𝑥𝑥𝛿휀𝑥𝑥 + 𝜎𝑦𝑦𝛿휀𝑦𝑦 + 𝜎𝑥𝑦𝛿𝛾𝑥𝑦)𝑑𝑧ℎ/2

−ℎ/2

] 𝑑𝑥𝑑𝑦Ω0

= − ∫ (𝑀𝑥𝑥

𝜕2𝛿𝑤

𝜕𝑥2+ 𝑀𝑦𝑦

𝜕2𝛿𝑤

𝜕𝑦2+ 2𝑀𝑥𝑦

𝜕2𝛿𝑤

𝜕𝑥𝜕𝑦) 𝑑𝑥𝑑𝑦

Ω0

(2.13)

where Ω0 denotes the domain occupied by the mid-plane of the plate, (σxx, σyy) the

normal stresses, σxy 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

γyz= γxz=0 in the Kirchhoff plate theory. The relationship between the moments and

stresses are given by [2];

𝑀𝑥𝑥 = ∫ 𝜎𝑥𝑥𝑧𝑑𝑧ℎ/2

−ℎ/2

𝑀𝑦𝑦 = ∫ 𝜎𝑦𝑦𝑧𝑑𝑧ℎ/2

−ℎ/2

(2.14)

𝑀𝑥𝑦 = ∫ 𝜎𝑥𝑦𝑧𝑑𝑧ℎ/2

−ℎ/2

The work W done by the uniaxial load Nx, due to displacement w only, equals;

𝑊 = −1

2∫ 𝑁𝑥 (

𝜕𝑤

𝜕𝑥)

2

Ω0

𝑑𝑥𝑑𝑦 (2.15)

12

The virtual work δW due to the uniaxial load Nx is given by

𝛿𝑊 = ∫ 𝑁𝑥

𝜕𝑤

𝜕𝑥

𝜕𝛿𝑤

𝜕𝑥Ω0

𝑑𝑥𝑑𝑦 (2.16)

The principle of virtual displacements requires that first variation of total potential

energy ∏ is equal to zero.

δ∏ = δU − δW = 0 (2.17)

δ∏ = − ∫ (𝑀𝑥𝑥

𝜕2𝛿𝑤

𝜕𝑥2+ 𝑀𝑦𝑦

𝜕2𝛿𝑤

𝜕𝑦2+2𝑀𝑥𝑦

𝜕2𝛿𝑤

𝜕𝑥𝜕𝑦+ 𝑁𝑥

𝜕𝑤

𝜕𝑥

𝜕𝛿𝑤

𝜕𝑥) 𝑑𝑥𝑑𝑦 = 0

Ω0

(2.18)

Assuming the material of the plate to be isotropic and obeys Hooke’s law, then the

stress-strain relations are given by;

∫ (𝑀𝑥𝑥

𝜕2𝛿𝑤

𝜕𝑥2+ 𝑀𝑦𝑦

𝜕2𝛿𝑤

𝜕𝑦2+2𝑀𝑥𝑦

𝜕2𝛿𝑤

𝜕𝑥𝜕𝑦+ 𝑁𝑥

𝜕𝑤

𝜕𝑥

𝜕𝛿𝑤

𝜕𝑥) 𝑑𝑥𝑑𝑦 = 0

Ω0

(2.19)

𝜎𝑥𝑥 =𝐸

1 − 𝑣2(휀𝑥𝑥 + 𝑣휀𝑦𝑦)

𝜎𝑦𝑦 =𝐸

1 − 𝑣2(휀𝑦𝑦 + 𝑣휀𝑥𝑥) (2.20)

𝜎𝑥𝑦 = 𝐺𝛾𝑥𝑦 =𝐸

2(1 + 𝑣)𝛾𝑥𝑦

where E denote the Young’s modulus, G the shear modulus, and ν the Poisson’s ratio.

By substituting Equations (2.20) into Equation (2.14) and carrying out the integration

over the plate thickness:

𝑀𝑥𝑥 = ∫ 𝜎𝑥𝑥𝑧𝑑𝑧 =𝐸

1 − 𝑣2∫ (휀𝑥𝑥 + 𝑣휀𝑦𝑦)𝑧𝑑𝑧 = −𝐷 (

𝜕2𝑤

𝜕𝑥2+ 𝑣

𝜕2𝑤

𝜕𝑦2)

ℎ/2

−ℎ/2

ℎ/2

−ℎ/2

𝑀𝑦𝑦 = ∫ 𝜎𝑦𝑦𝑧𝑑𝑧 =𝐸

1 − 𝑣2∫ (휀𝑦𝑦 + 𝑣휀𝑥𝑥)𝑧𝑑𝑧 = −𝐷 (

𝜕2𝑤

𝜕𝑦2+ 𝑣

𝜕2𝑤

𝜕𝑥2)

ℎ/2

−ℎ/2

ℎ/2

−ℎ/2

(2.21)

𝑀𝑥𝑦 = ∫ 𝜎𝑥𝑦𝑧𝑑𝑧 = 𝐺 ∫ 𝛾𝑥𝑦𝑧𝑑𝑧 = −(1 − 𝑣)𝐷𝜕2𝑤

𝜕𝑥𝜕𝑦

ℎ/2

−ℎ/2

ℎ/2

−ℎ/2

13

where D is the flexural rigidity;

𝐷 =𝐸ℎ3

12(1 − 𝑣2) (2.22)

the governing equation for buckling of plate subjected to a uniaxial load is obtained:

𝐷 (𝜕4𝑤

𝜕𝑥4+ 2

𝜕4𝑤

𝜕𝑥2𝜕𝑦2+

𝜕4𝑤

𝜕𝑦4) + 𝑁𝑥

𝜕2𝑤

𝜕𝑥2= 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];

𝜕4𝑤

𝜕𝑥4+ 2

𝜕4𝑤

𝜕𝑥2𝜕𝑦2+

𝜕4𝑤

𝜕𝑦4= −

𝑁𝑥

𝐷

𝜕2𝑤

𝜕𝑥2 (2.24)

Assuming that under the action of compressive forces the plate buckles in m sinusoidal

half-waves;

𝑤 = 𝑓(𝑦) sin𝑚𝜋𝑥

𝑎 (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 𝜕2𝑤

𝜕𝑥2 + 𝜈𝜕2𝑤

𝜕𝑦2 = 0 (2.26)

The critical value of the compressive stress is given by the equation;

σ𝑐𝑟 = 𝑘𝜋2𝐷

𝑏2𝑡 (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 σcr 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];

(𝐾0𝑁𝑀 + 𝜆𝑖𝐾∆

𝑁𝑀)𝑣𝑖𝑀 = 0 (3.1)

where, K0NM is the stiffness matrix of initial state of the model, KΔ

NM is the initial stress

and load stiffness matrix with respect to applied reference load, λi is the eigenvalues,

17

ʋiM 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;

𝜆𝑃𝑟𝑒𝑓 = 𝑃𝑐𝑟 (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 –Buckling 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;

𝜕4𝑤

𝜕𝑥4+ 2

𝜕4𝑤

𝜕𝑥2𝜕𝑦2+

𝜕4𝑤

𝜕𝑦4=

𝑞

𝐷+

𝑡

𝐷[𝜕2𝐹

𝜕𝑦2

𝜕2(𝑤 + 𝑤0)

𝜕𝑥2− 2

𝜕2𝐹

𝜕𝑥𝜕𝑦2

𝜕2(𝑤 + 𝑤0)

𝜕𝑥𝜕𝑦+

𝜕2𝐹

𝜕𝑥2

𝜕2(𝑤 + 𝑤0)

𝜕𝑦2] (3.3)

18

The Compatibility Equation is;

𝜕4𝐹

𝜕𝑥4+ 2

𝜕4𝐹

𝜕𝑥2𝜕𝑦2+

𝜕4𝐹

𝜕𝑦4= 𝐸 [(

𝜕2𝑤

𝜕𝑥𝜕𝑦)

2

−𝜕2𝑤

𝜕𝑥2

𝜕2𝑤

𝜕𝑦2− (

𝜕2𝑤0

𝜕𝑥𝜕𝑦)

2

+𝜕2𝑤0

𝜕𝑥2

𝜕2𝑤0

𝜕𝑦2] (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;

𝑤 = 𝐴𝑟 sin𝑘𝜋𝑥

𝑎∑ 𝐴

𝑛 cos𝑛𝜋𝑦

𝑏

𝑁

𝑛=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 “b”, an effective width “be” can be used

in the evaluation of the load-carrying capacity.

Von Karman’s effective width expression in terms of the critical stress and yield stress

is;

𝑏𝑒

𝑏= √

𝜎𝐶𝑅

𝜎𝑌 (3.6)

19

where,

𝜎𝐶𝑅 =𝑘𝜋2𝐸𝑡2

12(1 − 𝑣2)𝑏2 (3.7)

Which later modified by Winter [28];

𝑏𝑒

𝑏= √

𝜎𝐶𝑅

𝜎𝑌[1 − 0.22√(

𝜎𝐶𝑅

𝜎𝑌)] (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 λ is the load magnitude parameter. Thus, the actual load state at any time is defined

as λPN 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;λ). In scaled load-displacement space, an initial increment

Δlin 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 λin is calculated by the

following formula [29];

∆𝜆𝑖𝑛 =∆𝑙𝑖𝑛

𝑙𝑝𝑒𝑟𝑖𝑜𝑑 (3.9)

Assuming a solution has been formed at ith time (𝑢𝑖𝑀 , λ𝑖), then tangent stiffness Ki

NM is

formed. The finite element formula of the structure:

𝐾𝑖𝑁𝑀𝑢𝑖

𝑀 = 𝑃𝑖𝑁 (3.10)

Through the iterations of Riks method, λ, the load magnitude parameter is calculated

automatically. By defining Δlmin and Δlmax, 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

ν = 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 ‘Shell Edge Load’ 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.

𝜆𝑃𝑟𝑒𝑓 = 𝑃𝑐𝑟

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 –z 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 plate’s 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 Young’s Modulus of 200

GPa and Poisson’s 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 Young’s Modulus is directly proportional to

critical buckling load. It is also confirmed with the analyses since critical load triples as

the Young’s 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 Poisson’s ratio, 0.3 and 0.33 respectively, it was

possible to confirm the proportionality between the critical load and the Young’s

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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