Kabuk yapıların sonlu elemanlar yöntemi kullanılarak yapısal olarak incelenmesi
Structural analysis of shell structures by using the finite element method
Yazar: HASAN KIYIK
Danışman: PROF. DR. METİN ORHAN KAYA
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Uçak ve Uzay Mühendisliği Ana Bilim Dalı / Uçak Mühendisligi Bilim Dalı
Derece: Yüksek Lisans, 2020
Konu: Uçak Mühendisliği = Aircraft Engineering

Özet:
Bu yüksek lisans tezinin amacı katlanmış plakalar yardımıyla kabuk yapıların plaka teorileri ile sonlu elemanlar yöntemi uygulanarak serbest titreşim değerlerinin hesaplanıp elde edilen sonuçların literatürde elde edilen veriler ve NASTRAN paket programı ile kıyaslanarak kabuk yapıların en iyi hangi yöntemle modellenip çözüleceği konusunda bize yol göstermesidir.
Serbest titreşim değerlerinin yapılan analizler ile literatürde elde edilen değerler ve NASTRAN paket programı kadar iyi yakınsadığı görülmüştür. Dolayısıyla kabuk yapıların katlanmış plakalar yardımıyla modellenip plaka teorileri ile değerlendirilebildiği gösterilmiştir. Kabuk yapıları modellenirken farklı plaka teorileri ile analiz edilmiş olup serbest titreşim değerlerinin iyi sonuçlar verdiği belirlenmiştir. Yapılan çalışmada çok sayıda farklı plaka teorileri ve uygulama yöntemleri ile Mathematica programı yardımıyla bilgisayar kodu yazılmıştır.
Matematiksel modelleme gerçek dünyada karşılaşılan bir sorunun matematiksel olarak gerçeğe en yakın şekilde modellenip bu sorunu çözmek için matematiğin kullanılması sürecidir. Dolayısıyla doğru bir sonuç alınabilmesi için problemin gerçek dünyadaki özelliklerinin iyi bir şekilde belirlenip modelin sınırlamalarının ve özelliklerinin modele doğru bir şekilde yansıtılması gerekmektedir.
Başlangıçta düz olan plaka enine yüklere karşı koymak için kesme kuvvetleri, eğilme ve burulma momentleri üretir. Yükler genelde her iki yönde de taşındığı ve izotropik plakalardaki burulma rijitliği oldukça önemli olduğundan, bir plaka karşılaştırılabilir uzunluk ve kalınlıktaki bir kirişe göre daha serttir. Plaka yapıları hafif olması, yüksek yük taşıma kapasitesi ve ekonomik olması nedeniyle tercih edilir bir yapı konumundadır.
Plaka yapıları kullanılarak elde edilen katlamalı yapılar ve kabuklar üç boyutlu olarak düşünülüp ince ve kalın yapılar oluşturularak yapısal analizleri yapılmıştır.
Bu tezde Kirchhoff ve Reissner-Mindlin plaka teorileri kullanılarak elde edilen yapıların sonlu elemanlar yöntemi ile serbest titreşim analizleri yapılmış olup ince kabuk yapıları için Kirchhoff plaka yöntemi ve Reissner-Mindlin plaka yöntemi, kalın kabuk yapıları için ise Reissner-Mindlin plaka yöntemi kullanılmıştır.
Problemlerde sınır şartları belirlenirken literatürde yapılan çalışmalar göz önünde bulundurularak basit mesnetlenmiş ve her iki ucundan sabitlenmiş yapılar kullanılmıştır. Serbest titreşim frekans değerlerinin yanı sıra mod şekilleri de göz önünde bulundurularak yapıların davranışları incelenmiştir.
Yapılan çalışmalar ve elde edilen sonuçlar neticesinde gerçeğe yakın sonuçlar elde edebilmek için en uygun hangi teorilerin, modelleme yöntemlerinin, durumların ve sınır şartlarının kullanılması konusunda bilgiler elde edilmiştir.

Abstract:
The purpose of this master's thesis is to evaluate shell structures, which are accepted as folded plates and solve the shell structures by calculating the free vibration values by applying the plate theory with the help of the finite element method and comparing the datas with obtained in the literature and the NASTRAN package program. The result of this study will show us which method is the best method for solving the shell structures.
It was observed that the free vibration values converged as well as the values obtained in the literature and the NASTRAN package program. Therefore, it has been shown that shell structures can be modeled with the help of folded plates and evaluated with plate theories. While modeling shell structures, it was analyzed with different plate theories and it was determined that free vibration values gave good results. In the study, computer code was written with the help of Mathematica program with many different plate theories and application methods. Also, shell models are modeled in a cad program similar to the Mathematical model.
Finite element method; the approximate solution of a boundary value problem is to calculate the region by dividing the region into a certain number of sub-regions, assuming that the unknown function is approximately defined within each element. At common joint points of the functions of the said elements by collecting the values they receive, a solution for the whole region is obtained.
Mesh generation can be defined as the process of dividing a physical definition range into smaller definition intervals (elements). The aim is to facilitate the solution of a differential equation. Therefore, the approximation accuracy of the results to be obtained in the finite element method depends on the type of element and the number of elements.
Finite element method is a method that produces approximate solutions. The solution can be repeated by increasing the number of elements, changing the element type, changing the mesh production method. Thus, the difference between the results can be observed.
Apart from the finite element method, there are other approximate methods. These include finite difference method, weighted residual method, Rayleigh-Ritz method, Galerkin method. The most obvious difference between the finite element method and others is that the approximate solution is searched in very small sub-regions. In the finite element method, instead of using a function that will meet the boundary conditions for the entire solution region, the functions are defined on elements with simple geometries and boundary conditions are not considered on the basis of elements.
Mathematical modeling is the process of modeling a problem encountered in the real world in the most realistic way and using mathematics to solve this problem. Therefore, in order to get a correct result, the real world properties of the problem should be determined well and the limitations and properties of the model should be reflected on the model correctly.
The original flat plate produces shear forces, bending and twisting moments to withstand transverse loads. Since the loads are generally carried in both directions and the twisting rigidity of the isotropic plates is very important, a plate is stiffer than a beam of comparable length and thickness. Plate structures are a preferred structure due to their light weight, high load carrying capacity and economical structure.
A shell can be thought of as extending a plate to a non-planar surface. The non-planar state creates axial (membrane) forces in addition to the bending and shear forces, thus providing higher structural strength.
Shell structures are widely used in the aviation, automobile and ship industries, as they are very lightweight and easy to manufacture.
Unlike beams or plates, which are normally one or two-dimensional structures, shell structures can freely vibrate in three directions. This situation, complicated the movements of the shell structures at resonance frequencies. For this reason, besides the frequency behavior, the definition of modal with the amplitudes of the cylindrical shell structures is always of great importance. Areas such as engineering design, acoustic and sound radiation largely depend on the amplitude ratios of the cylindrical shell structures.
The folding structures and shells obtained by using plate structures are considered in three dimensions, and thin and thick structures are formed and their structural analyzes are made.
Although analytical methods are very important, the use of numerical models to solve mathematical models of complex shell structures has become an important component in the design process. Finite element method is used as the basic numerical method for the analysis of shells.
In this thesis, the free vibration analysis of the structures obtained by using Kirchhoff and Reissner-Mindlin plate theories was done, using the Kirchhoff plate method and Reissner-Mindlin plate method for thin shell structures, and the Reissner-Mindlin plate method for thick shell structures.
Shear lock occurs in Mindlin plate elements as the plate thickness decreases (closer to the thin plate limit). Because as the thickness decreases, the sliding effects are dominant besides the bending effects and the plate behaves more rigid than it actually is. As a result, bending energy terms are calculated as zero. In order to solve this problem, some stabilization techniques have been applied on the theory. This technique is called reduced integrations.
While determining the boundary conditions in the problems, structures that are simply supported and fixed at both ends are used by taking into consideration the studies in the literature.
The behaviors of the structures are examined by taking into account the mode shapes as well as the free vibration frequency values.
As a result of the studies carried out and the results obtained, information was obtained on which theories, modeling methods, situations and boundary conditions are most suitable for achieving realistic results.
As expected, the Kirchhoff theory gave better results than Reissner-Mindlin theory in thin cylindrical tube structures formed with thin plates. In addition, as it is known that Reissner-Mindlin theory converges better as a result of the previous studies for the thick cylindrical tube structure, analysis has been made for the thick cylindrical tube structure and the results have been obtained.