Eğri eksenli çubuklarda çatlak modellemesi
Crack modelling in curved rods
Yazar: UĞURCAN EROĞLU
Danışman: PROF. DR. EKREM TÜFEKCİ
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Makine Mühendisliği Ana Bilim Dalı / Makine Mühendisliği Bilim Dalı
Derece: Doktora, 2019
Konu: Makine Mühendisliği = Mechanical Engineering
Dizin: Global optimizasyon yöntemi = Global optimization method ; Uzay çubuklar = Space bars ; Yapısal hasar = Structural damage

Özet:
Bu tezde çatlak benzeri hasar bulunan eğri eksenli çubukların statik ve dinamik problemleri ele alınmıştır. Bir boyutlu sürekli ortam olarak modellenen çubukta eksenel deformasyon, kayma deformasyonu ve dönme eylemsizliği etkileri hesaba dahil edilmiştir. Çatlağın, yapının genel davranışına etkisi lineer elastik kırılma mekaniğinin temel tanımları ve Castigliano Teoremi yardımıyla ortaya konulan yay modeli ile hesaplanmıştır. Çatlak varlığı neticesinde oluşacak tüm esneklik bileşenleri hesaplara dahil edilmiştir. Buna ek olarak, literatürde daha önce göz önüne alınmayan çatlağın kesit üzerindeki konumu da esneklik matrisinin köşegen üzerinde olmayan elemanlarının işaretleri yardımıyla hesaba katılmıştır. 

İnce çubuk kabulü ile incelenen statik ve dinamik problemlerde, çatlağın kesit üzerindeki konumunun eksenel kuvvetin eğilme momenti ile karşılaştırılabilir olduğu problemlerde oldukça etkili olduğu görülmüştür. Açık çatlak kabulü ile yapılan hesaplamalarda özellikle iki ucunun hareketi engellenmiş sığ çubukların simetrik modlarına ait frekansların, çatlağın kesit üzerindeki konumuna oldukça hassas olduğu görülmüştür. Statik problemlerde ise, azalan narinlik ile çatlağın kesit üzerindeki konumunun daha etkili olduğu saptanmıştır. Ayrıca eş frekans eğrileri kullanılarak yapılan basit analizde, literatürde yapıldığı gibi çatlağın kesit üzerindeki konumunun hesaplara dahil edilmemesi halinde çatlağın eksen üzerindeki konumunun yanlış öngörüleceği sonucuna varılmıştır. 

Diferansiyel Evrim Algoritması'ndan yararlanılarak çatlak tespiti problemi çözülmüştür. Literatürden farklı olarak çatlağın kesit üzerindeki konumu da tespit edilmiştir. Literatürde yer alan deneysel sonuçlar ve tez kapsamında yapılan deneylerin sonuçları kullanılarak çatlağın konumunu ve derinliğini tarif eden değişkenler pek çok farklı senaryo için başarılı bir şekilde tahmin edilmiştir.

Eğri eksenli çubuğun kalın olması halinde lineer olmayan gerilme dağılımı neticesinde gerilme şiddeti çarpanlarının hem eğriliğe hem de çatlağın kesit üzerindeki konumuna bağlı olacağı açıktır. Buradan hareketle, ilk defa bu tezde, kalın eğri eksenli çubuklar için gerilme şiddeti faktörleri elde edilmiştir. Lineer elastik kırılma mekaniğinin öngördüğü çatlak ucu civarındaki gerilme dağılımı ve kesitlerin dengesi kullanılarak oldukça basit bir yaklaşımla elde edilen bu yeni gerilme şiddeti çarpanlarının gerçekten de hem eğriliğe hem de çatlağın kesit üzerindeki konumuna bağlı olduğu görülmüştür. Eğilme momentinin ile eksenel kuvvetin açılma moduna ve kesme kuvetinin kayma moduna etkilerini tarif eden bu fonksiyonların, çatlağın kesit üzerindeki konumuna eğrilikten daha çok bağımlı oldukları gösterilmiştir. Bu fonksiyonlar ve kalın çubuk modeli kullanılarak çatlak bulunan eğri eksenli çubukların dinamik analizi yapılmış ve deneysel çalışmalar ile oldukça yakın sonuçlar elde edilmiştir.

Abstract:
Structural damage has many causes, e.g., cyclic loading, collisions, corrosive environment, imperfect technological operations, material defects. Whatever the reason might be, the physical-mathematical model of damage and the investigation of its effects on the structural 
mechanical behaviour is of major importance, since it may lead to early damage identification and structural maintenance. The aim of this study is to examine thoroughly the mechanical behavior of curved rods, which are fundamental structural elements being used extensively in many engineering applications, with crack-like defects and their identification by means of dynamic behavior. 

First chapter is devoted to the historical background of solid mechanics, focusing on the development of one-dimensional beam models and initiation of fracture mechanics. Starting from the initiation of mechanics, major works of the important contributors, such as Galilei, Young, Coulomb, Navier, Possion and Cauchy, are mentioned. In addition, studies on examination of beam-type structures with cracks by using diferent one-dimensional structural models are reviewed. Briefly, survey of modern literature justifies using one-dimensional structural models along with spring anology for crack-type damages as far as the global behavior of the structure is of interest. Moreover, it is evident that using relatively new optimization techniques for the crack identification problem instead of utilizing iso-frequency curves as it is done in the early works on crack identification provides sufficient accuracy.

Second chapter deals with the construction of one-dimensional continua reduced from three-dimensional elasticity. Equations of motion of a slender body are derived in finite form; then, the governing equations for linear static and dynamic problems of planar curved rods are obtained by a perturbation approach. The effects of axial extension, shearing deformation and rotatory inertia are taken into account. The conditions for the existence of an exact solution, and a novel approximate solution technique using Peano's Series and Volterra's Multiplicative Integral are presented. Treatment of concentrated loads by suitable continuity and balance equations are given.

In the third chapter, foundations of spring anology for the crack modelling are re-visited. Stress field around the vicinity of the crack is obtained by eigenfunction expansion initially performed by Williams. All possible compliances caused by the crack are derived based on the concepts of linear elastic fracture mechanics, and Castigliano's Theorem. As a difference from the studies existing in the literature, effect of crack location on the cross-section is considered by the sign of off-diagonal terms in compliance matrix of crack which represent the coupling between axial and bending deformations due to existence of a crack. The jump conditions and balance equations for two sub-rods divided by the crack are presented. 

Fourth chapter focuses on numerical results of many example problems. Static problem of a parabolic arch with line load and parabolic arch with a point load are examined. It has been demonstrated that the effect of crack location on the section becomes more pronounced as the arch gets thicker. Also, very good agreement with the experimental results provided in the literature is observerd. As for dynamic problems, circular beams of different opening angles and boundary conditions, and parabolic arches of different shallownesses are examined. In addition, a perturbation approach to the modelling of crack is also presented to look for the possibility of obtaining rather simple closed-form solutions for static problems. Indeed, very simple and accurate closed-form solutions are obtained for a ring with a crack. Overall, even in slender curved rods for which the Navier's assumption of stress distribution is reasonable, the effects of crack position turn out to be non-negligible, especially when the axial force is dominant. For dynamic problems, it has been shown that wrong estimation of or ignoring the crack position on the section may lead to erronous identification of the crack along the axis. 

In the light of numerical evidences given in the fourth chapter, the fifth chapter, deals with crack identification problem by using Differential Evolution Algorithm. As the first step, the Differential Evolution Algorithm is introduced by explaining each step explicitly. Then, two different objective function are presented, one of which does not require the calculation of natural frequencies and the other uses the frequency differences. Advantages and drawbacks of those objective functions are discussed. Using the results of numerical experiments, the validation of in-house optimization code is performed. Once exactly same crack variables as the inputs are obtained, results of actual experimental studies existing in the literature are used as inputs. Identification of crack position along the rod axis, its severity and position on the cross-section are performed for circular and parabolic beams with pinned ends with a very good accuracy. Even if for the considered structure geometries the crack position on the section was not found to be very effective in the previous chapter, it is also identified with a very good accuracy. Moreover, an experimental modal analysis on fixed-free circular beams with a single crack are performed. The structural model of cracked curved rod is found to perform quite accurately. The inverse problem is also tackled by using the experimental results as inputs for the optimization procedure. A very accurate estimation of crack variables in case of fixed-free boundary condition is achieved.

As there are no complete set of stress intensity factors from which the local compliances due to crack are calculated, the sixth chapter is devoted to the introduction of a simple and effective approach for the calculation of stress intensity factors for thick curved rods. Closed-form expressions for stress intensity factors are derived by ensuring the balance of cross-sections. These new results are validated by re-examination of dynamic behavior of curved rods with cracks of different severities. Both direct and inverse problem by using these new expressions for stress intensity factors, which account for the ratio of height of cross section to the curvature and the position of the crack on the section, and very promising numerical results are obtained. Furthermore, stress intensity factor for C-shaped specimen existing in the literature is calculated by the new approach of this study and very good agreement is observed. 

The main novelties introduced herein are: an approximate solution strategy for linear differential equation systems of variable coefficients, introduction of a non-material parameter to take into account the crack position on the cross-section of the beam and stress intensity factors for thick curved beams including the effect of initial curvature and crack position on the section through the non-symmetric stress distribution in case of no crack. The approximate solution strategy is very effective and accurate. Mechanics of beams with varying cross-section, made of axially functionally graded materials and many different mechanical problems of beam-type structures may well be examined with it. The effect of crack position on the section plays a very important role especially for arches where the axial deformations are comparable with bending deformations. Its importance is not only pronounced for the direct problem but also for the inverse problem of crack identification. It has been proved numerically that ignoring the effects of crack position on the section may lead to erronous estimation of crack position along the rod axis. As the stress intensity factors provided in the literature for straight beams may well be used with a sufficient accuracy for slender curved rods, the effect of initial curvature definetely plays an important role as the rod gets thicker. To improve the performance of spring anology for thick curved beams, the complete set of stress intensity factors derived herein turns out to be a very effective tool.