Yeni bir çatlaklı çubuk eleman ile eğri eksenli çubuklarda çatlağın konumunun ve derinliğinin tespiti

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 kullanılmıştırBirinci 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. 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. 75