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ÖgeMechanical behavior of the bi-directional beams(Lisansüstü Eğitim Enstitüsü, 2022) Çelik, Murat ; Artan, Reha ; 693155 ; Yapı MühendisliğiNanotechnology, which is one of the most developed areas of today's technology, has a widespread usage area because it can be easily integrated into many areas of engineering. The design and production of very small-scale functional materials in fields such as aerospace and aviation industry, medicine, energy and civil engineering, and their effective use in related fields make significant contributions to engineering science. The fact that these very small-scale materials used in many areas of technology have spread to almost every field of engineering has revealed the need to determine their internal structure and mechanical properties. Materials exhibit some mechanical behavior such as bending, torsion and buckling under loads. It is an important issue for the stability of the structure to be able to detect such mechanical effects in the most accurate way. In this sense, the dimensions of nanoscale structures, which are comparable to the distance between atoms that make up that material, have shown that the classical elasticity theory, which is currently widely used, is not sufficient in determining the mechanical behavior of such structures. Since the size effect is an important factor in nanoscale structures, defining a more general continuum model that includes such parameters increases the accuracy of the solution of the related problems. In this thesis, the bending and buckling behavior of beams is investigated within the framework of the strain gradient theory. In determining the mechanical behavior of nanostructures, the non-local elasticity theory, the couple stress theory and the strain gradient theory, which are proposed by respectable scientists such as Eringen and Mindlin, are widely used. The calculation load of the problem may increase depending on the loading, boundary conditions and geometric properties of the analyzed structure. Thus, the Initial Values Method is used to solve the problem in the thesis. The transport matrix approach is used for the static bending analysis of the beam within the scope of Initial Values Method. In the first part of the study, the bending behavior of the Euler-Bernoulli nano beam whose material properties change in two directions (bi-directional) is investigated. Therefore, it is assumed that the modulus of elasticity is variable along the axis and thickness of the beam. In the literature, the modulus of elasticity for a functionally graded material (FGM) is expressed in terms of arbitrary functions and the bending behavior is investigated for the first time within the scope of the gradient elasticity theory. In this context, the basic equations and boundary conditions (simply supported and fixed at both ends) are obtained with the help of Hamilton's principle for the beam under uniformly distributed load. While 4 end conditions can be written depending on the boundary conditions in the classical elasticity approach, 6 end conditions are obtained by using the gradient elasticity theory. While each loading case is required for the solution of the 6th order differential equation in the classical solution, the vertical displacements for two different types of loading cases are calculated by solving 3 linear equation systems based on 3 unknowns using the initial values method. As a result of the study, it was observed that the size effect decreased depending on the increase of nano beam length. In other words, the difference between local and non-local theory becomes important in small scales. However, it was observed that there is a lower vertical displacement in the bi-directional Euler-Bernoulli nano beam due to the increase in the inhomogeneous material constant (β). The accuracy and importance of the study for both boundary conditions are demonstrated with the help of graphics. In the second part of the study, the buckling behavior of the Euler-Bernoulli nano beam (FGM) whose material properties change in two directions (bi-directional) is investigated. Basic equations and boundary conditions are derived with the help of Hamilton's principle. Since the transport matrix cannot be calculated analytically for buckling analysis, the approximate transport matrix (Matricant) is used in the solution. Critical buckling loads are calculated for the classical theory of elasticity and the gradient elasticity theory depending on the number of intervals. In the results of study; It is observed that the first two terms of the transport matrix reflect the exact result if the number of intervals is chosen large enough. Remarkably, it is concluded that the first and second type of boundary conditions in the buckling calculation may depend on the type of material used. The accuracy of such a proposition can only be demonstrated more clearly by experimental studies. In addition, it is observed that the buckling resistance of the beam increases depending on the increase in the material characteristic length (γ). The accuracy and scientific contribution of the study is expressed with the help of the diagrams. The greatest contribution of the subject investigated herein to science is that it can guide the analysis and design of nanostructures used in many areas of technology. The importance of parameters such as the size effect and inhomogeneous material coefficient in the analysis of very small-scale structures has once again been demonstrated with strong propositions and results. Thus, it is supported by previous studies that theories such as the non-local elasticity theory and the gradient elasticity theory give more realistic results compare to the classical theory. With the design of micro and nano electro-mechanical systems (MEMS and NEMS), which are frequently encountered in the electronic device industry, as functional graded materials, it has become more important to determine the mechanical properties of such small-scale structures. In this sense, it is aimed that the relevant thesis study and the international publications published by us will make significant contributions to the literature.