Eğri eksenli enine çatlaklı çubukların düzlem dışı analizi
Out of plane analysis of curved beams with transverse crack
Yazar: BARIŞ AYBER
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 ; Mühendislik Bilimleri = Engineering Sciences
Dizin: Eksenel çatlak = Axial cracks ; Çember kiriş = Circular arches

Özet:
Eğri eksenli çubuklar uzun yıllar boyunca araştırmacıların ilgisini çeken ve üzerine sayısız araştırma yapılmış en basit ve en yaygın yapı elemanlarıdır. Günümüzde makina, inşaat, havacılık gibi alanlarda kullanım örnekleri bulunmaktadır. Bu nedenle önemi hala yerini korumakta ve en doğru çözüm ile analiz yöntemleri için üzerine yapılan araştırmalar devam etmektedir. 
Bu çalışmada düzlemsel eğri eksenli ve enine çatlaklı çubukların düzlem dışı statik ve dinamik davranışları sonlu eleman yöntemiyle farklı yükleme ve mesnetleme koşullarında incelenmiş ve sonlu eleman paket program sonuçları ile karşılaştırılmıştır.
Kullanılan denklemler de basitleştirici bir kabul yapılmaksızın kayma deformasyonu ve eksenel uzama etkileri dahil edilmiş, esneklik matrisini oluşturan elemanlardan biri olan burulma momenti etkisi hesaba katılmıştır. Bu denklemler kullanılarak bir sonlu eleman yöntemi uygulanmış, narinlik derecesi, eksen açıklığı, çatlağın derinliği ve çatlağın kesit üzerindeki konumu gibi değişkenlerin etkileri analiz edilmiş ve sonuçlar değerlendirilmiştir.
Birinci bölümde genel olarak çubuk teorisi hakkında bilgiler verilmiş, çalışmanın amacı ve kapsamı paylaşılmıştır.
İkinci bölümde literatürde eğri eksenli çubuklar üzerine yapılan araştırmalar incelenmiş ve bu çalışmalarda statik ve dinamik problemlerin çözümleri için kullanılan yöntemler kısaca anlatılmıştır. Genelde Ritz, Galerkin ve sonlu elemanlar gibi yaklaşık yöntemlerin kullanıldığı çözümlemelerde kayma deformasyonu, eksenel uzama ve dönme eylemsizliği etkilerinin ihmal edildiği görülmüştür. Statik ve dinamik çalışmaların ardından çatlak etkisi üzerine yapılan araştırmalar paylaşılmış ve eğri eksenli çubuklarda çatlak etkisinin düzlem dışı analizi üzerine yeterli çalışma bulunamadığı görülmüştür.
Üçüncü bölümde kısaca çubuk statiğinin düzlem dışı genel denklemleri verilmiş ve kullanılan sonlu eleman yöntemi paylaşılmıştır.
Dördüncü bölümde kırılma mekaniğine girilmiş, çatlak tipleri ve etkileri paylaşılmıştır. Gerilme şiddeti faktörleri, enerji salınım oranı ve esneklik matrisi denklemleri ve ilişkileri incelenmiş, bu çalışma kapsamında ele alınmış olan dairesel kesitli çubuğun yüzeysel enine çatlak durumu modellenmiştir. Bu modelleme ile esneklik matrisi hesaplanarak mevcut eleman için çatlağın etkisini de içeren rijitlik matrisine geçiş yapılmış ve sonlu eleman formülasyonuna eklenmiştir.
Beşinci bölümde çatlağın etkisi de dahil edilerek oluşturulmuş olan sonlu eleman formülasyonu kullanılarak düzlemsel eğri eksenli enine çatlaklı çubukların çeşitli mesnetleme durumları için dairesel profilde örnekler çözülmüştür. Bu çözümlemelerde narinlik derecesi, kirişin eksen açıklığı çatlağın konumu ve derinliği gibi parametrelerin etkisi incelenmiş ve sonuçlar grafikler-çizelgeler yardımıyla  paylaşılmıştır.
Altıncı bölümde çalışmanın kapsamı kısaca ele alınmış ve elde edilen sonuçlar tartışılmıştır.

Abstract:
Curved beam elements are one of the common used structural elements in different field of engineering such as mechanical, aerospace and civil engineering. As being critical part of these areas, the behaviour of the design is important for static and dynamic characteristics and the potential effects to the system. Due to that fact, there are a lot of studies in literature which is related to curved beam elements and still engineers and scientists are interested in those to enlight more facts about their characteristics and potential effects.
This study focus on out of plane analysis of curved beams with transverse crack by a finite element formulation which is based on an exact solution equations in literature. First of all, curved beams, which is dealing with in this study, has symmetric axes on cross section with respect to circular arches section. This gives a chance to evaluate the analysis in two type: in-plane and out of plane. Equations are shared for a beam element, stiffness matrix is obtained. Mass matrix is derived with expressions of kinetic energy including rotatory inertias. Supporting examples are shared with a literature and an engineering analysis program – ABAQUS result comparison done before crack location and depth effects to natural frequencies.
There are a lot of study aims curved beam elements, characteristics and the potential effects in literature. It is obtained that despite there are several works concerning the analysis of curved beams, most of them consider a curved beam without a crack or having an analysis on curved beam with a crack for in-plane results. In addition to this, there are simplified approaches to have abbreviated equations in lots of the studies. Axial deformation, shear deformation and rotatory inertia are neglected due to this fact. Simplified solutions are easy to manage however results have to be accurate as well. This is why it is important to have a finite element formulation from exact solution to have accurate results.
In first chapter, a general view of curved beams and the aim of the study is given with importance. Concept and the differences to literature studies are presented.
In second chapter, studies in literature are given including the summaries. As this study contains two important subjects-curved beams and crack effects, literature studies are explained under 3 title; 2 are related to curved beams static and dynamic analysis and third one is the effect of crack existence.
In third chapter, curved beam equations are given briefly for out-of plane format and the proposed finite element formulation approach is shared. Stiffness matrix of an element and mass matrix are presented.

In fourth chapter, introduction of fracture mechanics are summarized to have a general view of crack existence and the effects. Also limited criterias and two basic approach are explained. Afterwards, stress intensity factors, energy release rate and compliance matrix equations are presented with all relations to each other. These crack existence is modelled on an circular section curved beam. At the end of the chapter, the finite element formulation which is presented in third chapter, is updated with the effect of compliance matrix in order to include the effects of crack.
In fifth chapter, this proposed finite element formulation is tested for a circular section curved beam problem which was analized on one of the previous study in literature. Results are compared and presented to demonstrate the accuracy level for clamped-clamped condition. After a certain confidence, same analysis done for same curved beam for clamped-free condition including crack effect. Afterwards, a curved beam analysis done and same results taken from an engineering program – ABAQUS to have the comparison with CAE and results are evaluated with error percentages. Moreover, new analysis performed for different curved beams to evaluate the effects of the crack against curvature radius, axial span, crack depth ratio and crack location. In first section, a curved beam is analysed for clamped-free boundary condition and analysis results are shared by graphics for first to fifth natural frequency changes based on the location change of the crack. Results are shared in frequency ratio to have dimensionless comparison. Secondly, same analysis is repeated for clamped-clamped condition and evaluations done. In next analysis, crack depth ratio range defined and curved beam analysed for defined crack depth ratio to monitor the changes in frequency ratio from first to fifth natural frequency with graphs. To evaluate the vibrational behavioural change of curvature beam with an existence crack, different location of cracks are analysed and results shared.
In sixth chapter, study importance and aim presented again and the analysis of curved beams in fifth chapter shared. The effects of curvature radius, axial span, crack depth ratio and crack locations are discussed based on the analysis results in graphs. Recommendations for next steps are shared and the opportunities defined.
As a conclusion, curved beams which is used as a basic structural elements in many of engineering field such as mechanical engineering, aerospace engineering and civil engineering, are basis of lots of the studies in literature and still keep the importance in engineering science. Additionally, crack existence is important due to static and dynamic effects to the system. Sometimes, this existence is critical due to strength risks and sometimes, this existence cause the dynamic behaviour changes and a potential root cause of a failure mode. In such cases, it is important to prevent a failure due to crack existence and growth if there is any strength risks obtained or it is important to evaluate the vibrational effects to system if dynamic behaviour changes become critical. For instance, crack effect to a curved beam structure for any of the engineering fields which has the lower tolerance targets for natural frequencies become significantly important due to potential effects to dynamic responses in the system and cause a deviation from the target. With this deviation, a failure mode can be obtained due to dynamic behaviour changes and frequency conflict to other structural parts in the system. This is a main risk for an engineering system and its functionality. Due to that fact, it is important to evaluate the behaviour of curved beam elements with crack existence and share the results to increase the know-how in order to understand the potential effects of each parameter. For in-plane analysis, there are studies which focus on defining the location of the crack based on measured natural frequencies of curved beam. The results in this study also can be used to define the crack location for an analysed curved beam with known frequencies. Additionally, these results and the outcome can be supported with real testing results to increase the confidence level. In next studies, these can be part of the main focus.