# Slope Deflection Method in Nonlocal Axially Functionally Graded Tapered Beams
## 1 Introduction
The thermal buckling and vibration behavior of a sandwich plate with a viscoelastic core have been studied by taking into account the effect of temperature [37]. Researchers studied the thermoelastic behavior of FGM beams by obtaining polynomials of the exact results of different equations expressed with interpolation functions [38]. Wave propagation behavior of FGM beams under high-frequency loading was analyzed using a finite element approach [39]. The ambient-temperature-dependent static behavior of beams in an FGM structure was investigated with the help of a high-order beam theory [40]. A new model was presented for calculating the buckling load of columns made of FGMs [41]. The mechanical behavior of Euler-Bernoulli beams was presented in which material properties vary in either the horizontal or vertical direction [42]. Functionally graded materials are a type of composite material whose material properties are designed to show continuity from one surface to another. As for laminated structures, stress discontinuities occur between two surfaces. These discontinuities are eliminated in FGMs. Owing to these properties, FGMs are widely used today in many areas such as the design of medical devices, as well as in the fields of construction and ship and aircraft engineering. Beams made of FGMs are classified into two basic groups according to their design status and their grading according to the thickness and length of the element, with their mechanical properties varying in different directions (axially or longitudinal) [43; 44]. These types of beams are called axially graded beams (AFGs). There are a limited number of studies in the literature on the analytical approach of AFGs [45; 46]. There are many methods in the literature to determine the natural frequency of non-uniform axial FGM beams. The bending and buckling behaviors of bi-directional FGM beams are investigated within the scope of strain gradient theory [47; 48]. These are the complementary functions approach [49], finite element method [42; 43; 44; 45; 46; 47; 48; 49; 50], power series approach [51] used to define mode shapes, Chebyshev polynomials [52], differential quadrature method [53], differential form including initial values [54], and Fredholm integral equations [55]. Classical elasticity theory is not sufficient to analyze the behavior of nanoscale structures in the most accurate way. Therefore, it has emerged that an enriched continuum model is needed to describe the mechanical behavior of such structures. Although the molecular dynamics approach (MD) is a method used to analyze such structures, it contains a high level of computational load, making it difficult for such simulations to express a general solution. The nonlocal elasticity theory approach states that the stress field at a point is affected not only by the strain at that point, but also by the strain of all adjacent points. In this context, nonlocal elasticity theory has an important place in the analysis of size-dependent mechanical behavior [56; 57]. In recent years, numerous studies are presented to display the importance of the size effect in small-scale structures [58; 59; 60]. The slope deflection method is a method based on expressing the displacement componentsat the nodes as unknowns. There are many publications in the literature related to this method. However, there is no publication describing the method in nonlocal theory for structures composed of AFG elements. Stiffness coefficients and end moments are analyzed within the scope of nonlocal elasticity theory using the method of initial values (MIV).