Yeni bir çatlaklı çubuk eleman ile eğri eksenli çubuklarda çatlağın konumunun ve derinliğinin tespiti
Crack detection in curved beams with using a new cracked beam element
Yazar: CAN ALTUĞ İPEK
Danışman: PROF. DR. EKREM TÜFEKCİ
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Makine Mühendisliği Ana Bilim Dalı / Katı Cisimlerin Mekaniği Bilim Dalı
Derece: Yüksek Lisans, 2017
Konu: Makine Mühendisliği = Mechanical Engineering
Dizin: Arıza arama = Fault detection ; Eğri kirişler = Curved beams ; Genetik algoritmalar = Genetic algorithms ; Kırılma mekaniği = Fracture mechanics ; Sonlu elemanlar yöntemi = Finite element method

Özet:
Çubuklar, diğer yapı elemanlarına göre, basitliklerinin getirdiği kullanım yaygınlığı 
sebebiyle mühendislik alanında oldukça önemli bir yere sahiptir. Geniş kullanım 
alanları sebebiyle çubuklar, yıllar boyunca araştırmacılar arasında önemli ve popüler 
bir konu olmuş ve olmaya da devam etmektedir. Çubuklar düz veya eğri gibi farklı 
formlara sahip olabilmekle birlikte, bu tez çalışmasında ele alınan eğri eksenli 
çubuklardır.
Bu çalışmada, ekseni üzerindeki herhangi bir konumda ve derinlikte çatlağa sahip 
olan sabit eğrilikli ve sabit kesitli eğri eksenli çubuklar için, yeni bir çatlaklı çubuk 
sonlu eleman formülasyonu geliştirilmiş ve kesin çözümden uyarlanmış sonlu 
eleman yöntemi uygulanmıştır. Ardından bu yeni geliştirilmiş çatlaklı çubuk eleman 
ile kurgulanmış sonlu eleman yöntemi, ters problem için kullanılmış ve doğal frekans 
değerleri bilinen çatlaklı bir çubuk için, çatlağın eksen üzerindeki konumu ve 
derinliği tespit edilmiştir. Ters problemin çözümünde bir optimizasyon yöntemi olan 
genetik algoritma kullanılmıştır
Birinci bölümde, çubuk teorisi hakkında kısaca bilgi verilmiş, çalışmanın amacı ve 
kapsamı belirtilmiştir. Çubuk üzerindeki çatlağa yönelik yapılan çalışmalar 
incelenmiş. Ardından çubuk üzerindeki çatlağın tespiti için kullanılan ters problemin 
çözümü için, literatürde kullanılan yöntemler hakkında bilgi verilmiştir.
İkinci bölümde, değişken eğrilikli ve değişken kesitli eğri eksenli çubukların düzlem 
içi genel denklemleri verilmiştir. Ardından çember eksenli ve sabit kesitli eğri 
eksenli çubuklar için düzlem içi statik denklemlerin başlangıç değerleri problemi 
yoluyla çözümü verilmiştir. Asal matrisin terimleri elde edilmiştir.
Üçüncü bölümde, bir önceki bölümde elde edilen eğri eksenli çubukların statik 
denkleminin analitik çözümü kullanılarak, sonlu eleman formülasyonu uyarlanmıştır. 
Çember eksenli ve sabit kesitli, üzerinde çatlak ve benzeri herhangi bir hata 
bulunmayan bir çubuk elemanının rijitlik ve kütle matrisleri elde edilmiştir.
Dördüncü bölümde, kırılma mekaniği hakkında kısaca bilgi verilmiş, çatlağın çubuk 
üzerindeki etkisinden bahsedilmiş ve çatlağın yarattığı yerel esneklik matrisi elde 
edilmiştir. Ardından çubuk ekseni üzerindeki herhangi bir konumda ve derinlikteki 
çatlağa sahip bir çubuk için, yeni bir çatlaklı çubuk eleman geliştirilmiştir. Bu yeni 
çatlaklı çubuk eleman için rijitlik matrisi elde edilmiştir.
Beşinci bölümde, daha önceki bölümlerde elde edilen çatlaksız ve çatlaklı çubuk 
elemanların rijitlik ve kütle matrisleri kullanılarak modellenmiş çember eksenli ve 
sabit kesitli eğri eksenli çubukların özdeğer problemleri çözdürülmüş ve hem 
çatlaksız hem de çatlaklı eğri eksenli çubuklar için doğal frekans değerleri elde 
edilmiştir. Elde edilen doğal frekanslar, hem literatürdeki çalışmalar ile 
karşılaştırılmış; hem de İstanbul Teknik Üniversitesi Makina Fakültesi'ndeki 
Mukavemet Birimi Laboratuvarı'nda yapılan deneylerden elde edilen sonuçlar ile 
karşılaştırılmıştır.
Altıncı bölümde, ters problem hakkında kısaca bilgi verilmiş, bir optimizasyon 
yöntemi olan genetik algoritma (GA) tanıtılmıştır. Ardından bir önceki bölümde 
verilen çatlaksız ve çatlaklı çubukların doğal frekansları kullanılarak, GA yardımıyla 
çatlağın eksen üzerindeki konumu ve derinliği tespit edilmiştir. 
Yedinci bölümde, çalışmanın kapsamı kısaca ele alınmış, elde edilen sonuçlar 
yorumlanmış  ve yeni önerilerde bulunulmuştur.

Abstract:
Beams, comparing with any other structural elements, have a very important role in 
engineering field considering their common usage due to their simplicity. This makes 
them a very hot topic for researchers from past to present throughout the centuries. 
Beams have a variety of forms according to their geometry such as straight, curved, 
tapered. However, only the curved beams are considered in this study.
A structure has a design aim to fulfil during service. Beams are the main elements to 
build the structure and every one of them is very important for that structure to meet 
the design criteria. However, a notch or crack in the beam, dramatically reduces the 
life span of the beam and this might create a catastrophic failure if it's not detected 
on time. There are some methods to detect the crack in the beam,  however they can 
be very expensive or very time consuming. This study aims to create a faster, easier 
way to detect the crack by using finite element method.
In this study, a finite element formulation is derived from analytically solved static 
problem considering a cracked beam with constant curvature and constant cross-
section. Since the basis of the formulation derived from the exact solution, it has no 
simplification. This means that, shear deformation, axial deformation and rotatory 
inertia is not neglected. Then, a new curved cracked beam element is introduced. 
This new cracked curved beam element contains a crack on an arbitrary location and 
arbitrary depth which can be defined by the user. Using this new cracked curved 
beam element's and also the intact beam element's stiffness and mass matrices, intact 
and cracked curved beams are modelled with the finite element formulation which is 
derived from exact solution. Those modelled intact and cracked curved beams are 
then used to solve the inverse problem in order to find the crack parameters such as 
location and depth. During the inverse problem, genetic algorithm (GA) is 
implemented as an optimization method. Main aim of this study is to detect the 
crack's location and depth with using a new cracked curved beam element.
  
A crack on the beam creates a local flexibility. In the light of this phenomena, a local 
flexibility matrix is constructed and  transfer matrices are used to define this local 
flexibility on the intact curved beam element. By doing so, a new cracked curved 
beam element that contains a crack is then ready to be used to formulate the finite 
element method. Since this new cracked curved beam element contains an inherent 
crack, it can be used to model any number of crack in the beam. In this study, only 
one crack is considered since the main focus is to prove that the new cracked curved 
beam element is sufficient enough to model the curved cracked beam.
In the first chapter,  a general view for curved beams, where they are used and their 
importance is presented. After that, the presence of the crack and its effect on the 
beam is given. A detailed review for the studies in the literature about cracks in the 
beam is stated. These studies contain the effect of the crack to the beam considering 
vibration properties such as natural frequency and mod shapes. It is stated that the 
crack induces a local flexibility and diminishes the natural frequencies of the beam. 
It is seen from these studies that, there are many ways to model the crack, but 
generally the crack is modeled as link or spring. After generic information about the 
crack, the studies in the literature about inverse problem is given. It is declared that, 
the inverse problem needs to be solved in order to detect the cracks in the beam. 
Main idea of the inverse problem is the find the crack parameters such as location 
and depth of the crack. The inverse problem is approached from many different ways 
by researchers. Conventional hard computing methods are very time consuming 
considering an inverse problem. It is observed that the overall consensus if to use an 
optimization method to solve the inverse problem. That's why, it is mainly solved by 
the optimization methods such as genetic algorithm, neural networks, fuzzy logic. 
Papers which uses those different optimization methods for the inverse problem to 
detect the crack are documented. In this study, an optimization method called genetic 
algorithm is used in order to perform inverse problem.
In the second chapter,  in-plane governing equations of a curved beam are presented 
for the static problem. After that, initial value boundary problem is used to solve 
those in-plane equations. The resultant equations are given. These exact solutions are 
valid for any curved beam, including those curved beams which have variable 
curvature and variable cross-section. The solution also does not depend on the 
loading and boundary condition. Then, considering a curved beam with a circular 
curvature and constant cross section, analytical expressions of the fundamental 
matrix is derived.
In the third chapter, finite element method is derived from the exact solution. Two-
node six degree of freedom beam element is presented for the flawless curved beam. 
Using the fundamental matrix which was obtained in the previous chapter, the 
stiffness matrix for the intact beam element is constructed. Also the generic kinetic 
energy expression is used to obtain the mass matrix of the beam element. 
Considering those matrices, the classic finite element assembly is sufficient to model 
the flawless curved beam.
In the fourth chapter, a brief review about the fracture mechanics is given. Stress 
intensity factors and the fracture modes are presented. Using the energy method 
approach, the local flexibility induced by the crack is found. A local flexibility 
matrix is constructed. With the help of the transfer matrices, a new cracked curved 
beam element which has a crack on an arbitrary location and depth is obtained. Using 
this new beam element, a curved beam with a crack can be easily modelled.
In the fifth chapter, previously found stiffness and mass matrices of intact and 
cracked beam element are used to assemble the beam and then eigenvalue problem is 
solved to obtain the natural frequencies. It needs to be stated that the mass matrix for 
the intact and the cracked beam is the same. After that, studies in the literature which 
calculates the natural frequencies analytically and experimentally are presented. A 
comparison is made between the results of this study and the studies in the literature. 
It is stated that the results of the studies in the literature and this thesis study are well 
met.
In the sixth chapter, the importance of the detection of the crack is given. The inverse 
problem is defined. It is stated that the any hard computing method is inadequate 
when the problem is very complex and hard to solve. It would require a lot of time to 
solve those problems with hard computing methods. It is described that, the inverse 
problem in this thesis study is the detecting the crack's location and depth on a 
curved beam with a circular curvature and constant cross-section. In order to detect 
the crack, previously found natural frequencies are used. Those frequencies are the 
product of a beam that is modeled with the new cracked beam element. Methods to 
solve the inverse problem is then defined. Although there are many ways to solve the 
inverse problem, an optimization method which is called genetic algorithm is chosen. 
General review of the genetic algorithm is presented. The cost function of the 
optimization problem is given. Relative error is used as the cost function. The 
difference between intact and cracked beam regarding experimentally determined 
natural frequencies is presented. After that, the difference between intact and cracked 
beam, which are constructed with the method of finite element, considering natural 
frequencies is found. These two sets of differences are then used on the cost function. 
By doing so, it is wanted to diminish the errors of the experiment. Also the weighting 
factor is determined. This factor is used to reduce the effect of the measurement 
noise. Moreover, using the natural frequencies of an intact and the cracked beams as 
an input, inverse problem is solved to detect the crack's location and depth. At first, 
arbitrary chosen crack parameters are tried to be found. Secondly, the natural 
frequencies, which is presented in the study in the literature, is used to see the 
efficiency of the new cracked beam element. Thirdly, an experiment in the laboratory 
of Istanbul Technical University is made and the natural frequencies from that 
experiment is used to solve the inverse problem to find the crack's location and 
depth. It is stated that the finite element formulation with a new cracked beam 
element which is presented in this thesis study is giving coherent results with an 
acceptable absolute error.
In the seventh chapter,  the scope of the study is given with results and discussions. 
Also some suggestions are made for the further studies.