Fonksiyonel derecelendirilmiş konsol kirişlerde büyük yer değiştirmelerin analizi
Large deflection analysis for functionally graded cantilever beams
Yazar: AYHAN HACIOĞLU
Danışman: DOÇ. DR. CEMAL BAYKARA
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Makine Mühendisliği Ana Bilim Dalı / Konstrüksiyon Bilim Dalı
Derece: Yüksek Lisans, 2017
Konu: Makine Mühendisliği = Mechanical Engineering
Dizin: Doğrusal olmayan = Nonlinear ; Doğrusal olmayan davranışlar = Nonlinear behaviors ; Kiriş teorisi = Beam theory ; Kirişler = Beams ; Sehim = Deflection

Özet:
Fonksiyonel derecelendirilmiş malzemeler, çeşitli koşullara uyum sağlayabilmeleri için, ihtiyaçlarımız doğrultusunda malzeme bileşenleri değiştirilebilen ve bu özelliği ile mühendislik alanında oldukça geniş bir kullanım alanına sahip olan malzemelerdir. Geçmişi çok eskilere dayanmayan fonksiyonel derecelendirilmiş malzemeler hakkında yapılan çalışmalar günümüzde de yoğun bir şekilde devam etmektedir. Son zamanlarda özellikle malzemenin lineer olmayan davranışlarına yönelik çeşitli çalışmalar yapılmıştır. 
Bu çalışmada homojen lineer olmayan malzemeden imal edilmiş bir konsol kiriş ile Poisson oranı sabit, elastisite modülü kuvvet kuralına göre değişen, fonksiyonel derecelendirilmiş malzemeden imal edilmiş bir konsol kirişin çeşitli sınır koşulları ve yükleme tipleri altında yer değiştirmelerinin analizi sonlu elemanlar metodu ile ANSYS Workbench 15.0 programından faydalanılarak gerçekleştirilmiştir. Fonksiyonel derecelendirilmiş malzemenin sonlu elemanlar modellemesi, malzeme özelliği düşey doğrultuda, kuvvet kuralına göre kademeli olarak süreksiz değişim gösterecek şekilde yapılmıştır. Kuvvet indisi n'nin farklı değerlerine göre oluşturulan çeşitli modellerin analizlerinden elde edilen sonuçlar önceki çalışmalardan analitik ve nümerik yöntemlerle elde edilen sonuçlar ile karşılaştırılmış ve bu sayede model doğrulanmıştır. 
Kullanılan konsol kiriş ince kesitli olduğundan, uygulanan küçük kuvvetlerde dahi büyük yer değiştirmeler oluşur ve bu durum yüksek geometrik lineer olmama durumunu ortaya çıkarır. Farklı yükleme tipleri ile yapılan çalışmada geometrik lineer olmama durumunun yanı sıra malzeme kaynaklı lineer olmama durumu da söz konusudur. Fonksiyonel derecelendirilmiş malzemenin lineer olmayan gerilme – birim şekil değiştirme değerleri Ludwick bağıntısı ve genelleştirilmiş Ludwick bağıntısı ile elde edilmiştir. Bu sayede malzeme kaynaklı lineer olmama durumu iki farklı şekilde incelenmiştir.
Genelleştirilmiş Ludwick tipi homojen ve fonksiyonel derecelendirilmiş malzemeli konsol kiriş kullanılarak gerçekleştirilen analiz sonuçları göstermiştir ki ε_0 parametresindeki artış serbest uçtaki yer değiştirme miktarını da arttırmaktadır. Dikkat edilmesi gereken husus ε_0 için doğru bir değer seçilmez ise elde edilecek sonuçların gerçeklikten hayli uzaklaşacağıdır. 
Fonksiyonel derecelendirilmiş malzemelerde, kuvvet indisinin sıfıra eşit olduğu durumda, homojen kiriş elde edilir ve en düşük çökme bu durumda meydana gelir. Kuvvet indisinin değeri arttıkça konsol kirişte görülen yer değiştirme miktarı da artmaktadır.

Abstract:
Functionally Graded Materials (FGM) have been commonly used in engineering studies because of their adaptability to different situations by changing the material composition in a preferred direction with respect to requirements. Over the past decades in Japan, before FGM' s were suggested, thin layers and laminates were used in the form of plane structures. However, the change of temperatures lead to thermal deformation, and thus, thin layers were separated. Because of this deficiency, engineers attempt to define a new way using different materials or methods in order to avoid the cause of the seperation. Therefore, a new material named as FGM was suggested as a result of several experiments.
FGM's have remarkable advantages over homogenous materials owing to continuously varying material properties. For instance, FGM's composed of ceramic and metal have capability of suffering from high temperature media due to higher termal resistance of the ceramic phase and show stronger mechanical performance of metal phase to guarantee the structural integrity of FGM's. 
Nowadays, FGM's have been commonly used in thermal, optical, structural and electronic materials. Thanks to the development of manufacturing processes, FGM's could be produced into numerous forms like beam, plate and shell. The material property of the FGM can be specifically produced to meet the demands in various engineering applications to utilize the advantage of the properties of individual material. It is achieved due to the sequential change of the material composition of the FGM in a preferred direction. In a simple FGM, two different material ingredients change in a discontinuous way or gradually from one to the other. On the upper and lower surfaces of FGM, the mechanical properties such as Young's modulus and Poisson's ratio of materials are different, but preassigned. The Young's modulus and Poisson's ratio of the beams change continuously only in the z-direction. The Young's modulus of FGM beams in corresponding direction vary with power-law, exponential or sigmoid functions. In this thesis, Young's moduli of FGM beams change with power-law function.
There are two different modeling techniques based on finite different algorithms are used in order to analyse FGM structures. In the present thesis, multi-layered structure method used in order to model beam geometries. In this method, it is assumed that the materials of each individual layers are isotropic, homogenous, incompressible and non-linear. In addition to this, layers are assumed to be bounded rigitly. Therefore, the number and thickness of layers effect the results of analysis. 
The aim of this thesis is to deal with analyzing a beam made of non-linear FGM. In this way, a nonlinear functionally graded beam subjected to concentrated and combined loads was examined. In this thesis, the modeling and analysis of FGM beam is carried out using ANSYS software. ANSYS provide a number of elements to choose from the modeling of gradient materials. The associated element type for FGM beam was SOLID186 which is one of the most suitable element type for layered structures due to it's capability of large deflection. It is a solid element with twenty node and three degree of freedom. It can be associated SOLID186 Layered Structural Solid with a shell section. The layered structure specifications such as material, layer thickness, orientation are decided by shell section commands.
The large deflection of thin flexible beams play a significant role in engineering applications. Therefore, it has received much attention during the past two decades. Thin beams displays large deflections and slopes under loading, which results an increase in geometrical nonlinearity when strains do not change.  In order to solve this problem, beams including large deflections must be formulated with respect to non- linear bending theory. Although there are several solutions to the problem of large deflections of nonlinear materials, these are limited in some ways due to the relations characterizing nonlinear properties of materials.
The nonlinear stress-strain properties of material can be described by Ludwick's constitutive law. The relationship between stress and strain in this law is determined by a purely empirical curve fitting technique. However, there is a major shortcoming of the law when the relation between stress and strain is described. The stress gradient goes to infinity for material nonlinearity constant less than one and goes to zero for the constant greater than one while the strain value shrinks to zero. To deal with this problem, Jung and Kang (2005) suggests generalized Ludwick's law mathematically describing stress and strain relation with adding a new parameter. Thus, in this thesis, it is  specifically focused on generalized Ludwick type material's behavior in order to overcome the shortcoming of Ludwick type materials by utilizing three - parametric law and here, for FGM beams are taken into consideration whose Poisson's ratio is constant and Young's modulus varies with power-law through the thickness. Based on the finite element modeling, numerous analyses were performed by utilizing ANSYS Workbench. The FGM beam was modelled by discontinuously changing material ingredients. 
In this thesis, it was modeled generalized Ludwick type non-linear FGM cantilever beams in order to show large deflections at the free ends under different types of loads such as point loads, linearly distributed loads and combined loads. It was used FGM beams having homogenous and isotropic layers. The Poisson's ratio of beams in each layer is constant, and modulus of elasticity changes with power-law through the vertical direction. In this way, we examine the effect of the parameter describing the material variation profile through the thickness. Then, it was correlated with two different parameters related to generalized Ludwick's law and power-law, respectively. It was also established a link between the additional parameter of generalized Ludwick's law and material nonlinearity constant. Several analyses were performed in order to demonstrate these relations. 
The small and large deflection theories were compared for a linear elastic FGM beams and nonlinear FGM beams in this thesis. The results show that an FGM beam with smaller gradient index can bear larger loadings for Young's modulus of the bottom surface that is less than the top surface of beam. Furthermore, when material nonlinearity constant has value less than one, the beam behaves stiffer than a linear FGM beam. When we compare the large deflections with small ones under small loadings, we notice that both large and small deformation theories shows quite accurate results. Consequently, in terms of Young's modulus at the bottom and top surfaces and the direction of loadings in engineering applications, it probably gives an optimum design of a suitable gradient index such that FGM cantilever beams are capable of bearing large loading. 
The effects of Young's modulus varying with power-law and material non-linearity parameter on the deflection were analyzed. The results of these analysis show that the different gradient indexes and material constant of generalized Ludwick's law have great influences on the bending strength of the beam. Therefore, a FGM beam can be produced stronger than a homogenous beam. Furthermore, the bending stress distribution shows significant differences between FGM and homogenous beam. The obtained results were compared with the analytical results of previous studies considered different materials, load and compositions in order to characterize the robustness of the model and the advantages issued by the properties.