# Slope Deflection Method in Nonlocal Axially Functionally Graded Tapered Beams
## 1 Introduction  
In the present study, static analysis of axially graded nonlocal Euler-Bernoulli beams was performed using the slope deflection method. Firstly, the basic equations of a nonlocal Euler-Bernoulli beam subjected to distributed load are obtained [1; 2; 3; 4]. Then, it is assumed that the modulus of elasticity and the moment of inertia functionally change along the axis (AFGM) in order to provide a general solution to the problem. For the solution to the problem, the transport matrix is formed with the help of [PERSON]'s repeated integral. Stiffness coefficients were obtained by examining different boundary conditions for different loads. Although there are many studies on axially graded beams in previous studies, there is no research in the literature on the solution of unit displacement constants of nanoscale frames. In the above-mentioned studies, the slope deflection method has been solved for beams with constant cross sections and includes a classical structural solution. In the present work, the equations are given in the most general form and the solutions can be easily solved independently of the changes in the cross section and material parameters. Moreover, as this study is examined within the scope of nonlocal elasticity theory for Euler-Bernoulli beams, it brings great innovation to the literature in terms of static analysis of both macro- and nano-scale systems. There are many approaches to solving mechanical problems. Most of these theories are methods used in classical elasticity theory. However, this situation differs in the analysis of nanoscale structures. It has been repeatedly emphasized that the effect of atomic size should be taken into account in the analysis of nanoscale structures. In this context, approaches suggesting enriched deformation theories such as nonlocal elasticity theory, [PERSON]'s higher-order deformation theory, and strain gradient theory have been put forward by early researchers in analyzing such structures [5]. In the literature, there are many methods for solving statically indeterminate systems, such as the force method and the slope deflection method. The slope deflection method is the most widely used among these approaches, especially for systems with a high degree of hyperstaticity. Simplifying the application of the slope deflection method, new reduced equations based on incorporating both the rotations of the joints and the effects of fixed-end moments into one term instead of two terms are presented [6]. The slope deflection method was first introduced by [PERSON] to analyze beam and frame systems in the early 20 th century. This method is basically based on the solution of the continuity equations obtained depending on the displacements of the joints in beam and frame systems. The main advantage of this method is that it can be easily applied to statically indeterminate systems and is suitable for programming [7; 8; 9]. Statically indeterminate space frame systems were analyzed using the slope deflection method [10]. The slope deflection method was used by the researchers to analyze the orthotropic bridge support plates analytically [11]. Static analyses of Timoshenko beam-column connections were investigated using the slope deflection method [12]. With the developing technology, smart materials have replaced the traditional materials used in engineering problems. It is of great importance to determine the properties of advanced technology materials such as functionally graded materials at a good level in order for the structural elements used in almost every field of engineering to have higher mechanical strength. AFGMs are advanced technology materials used in building elements such as wind turbines, helicopter blades, and airframes. Studies have shown that AFGMs subjected to loads in different directions show a higher mechanical strength than metal-ceramic-type FGMs. However, owing to the fact that the equations of motion of AFGMs can take very complex forms, their analytical analysis is quite difficult [13; 14; 15; 16].