# Slope Deflection Method in Nonlocal Axially Functionally Graded Tapered Beams
## 7 Conclusions  
In this paper, the slope deflection method for AFG beams with a variable cross section was presented in the frame of nonlocal elasticity. First, the transfer matrix was presented for AFG beams with a variable cross section. The stiffness coefficients and fixed-end moments were calculated systematically using MIV. Fixed-end moments of a beam subjected to distributed loads include the nonlocal parameter (\(\gamma\)) (see Section 4). It was observed that the nonlocal parameter was not included in the stiffness coefficients. That is, stiffness coefficients are independent of the nonlocality parameter (see Section 5). The difference between the moment values obtained by local and nonlocal theories grows significantly when the \(\gamma/L\) ratio increases (see Figures 5a,b and 8a,b). A similar situation also occurred in the rotation of joints in the frames (see Figure 12b). Therefore, the fixed-end moments calculated by nonlocal elasticity should be used in the static analysis of the nanostructures. All of the obtained results were validated by comparison to the reference study.  
Figure 15: The variation of the rotation at the joint \(D\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**End Moments** & **Reference Study (\(\gamma\)/\(L\) = 0) (kN.m) [64]** & **Present (\(\gamma\)/\(L\) = 0) (kN.m)** \\ \hline \(M_{AD}\) & \(-25.9\) & \(-25.9\) \\ \(M_{DA}\) & \(3.32\) & \(3.32\) \\ \(M_{DC}\) & \(3.32\) & \(3.32\) \\ \(M_{CD}\) & \(-6.32\) & \(-6.32\) \\ \(M_{BC}\) & \(9.43\) & \(9.48\) \\ \(M_{CB}\) & \(-6.32\) & \(-6.37\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: End moments for a frame system subjected to horizontal triangular load [64].  
With the help of the present work, the unit displacement constants of the axially graded beams in different loading conditions were calculated using the given formulas. The obtained results were validated by comparison to the reference study. As statically indeterminate systems may contain many unknowns, solving such systems statically increases the computational load. This study offers an innovative approach in terms of easier analysis of systems such as frames of different dimensions that we encounter in engineering structures. The fact that the \(\gamma/L\) ratio is zero in the study indicates the abandonment of the nanoscale and the transition to the classical state. Another important aspect of this study is that the relevant solutions are in a general form. That is, the solutions do not depend on the dimensions of the problem.  
The authors agree that this study will be an important resource for structural engineering problems, wind turbine design, relatively small-sized electronic devices such as NEMS and MEMS, and the design of biomedical devices. We also believe that a more advanced composite material model can be obtained by designing the beam as a bi-directional structure, in addition to the structure of axially graded beams in the current study.  
Numerical work and writing, E.D.; theoretical work, review and editing, M.C.; theoretical and numerical work and review, R.A. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The necessary datas are avaliable in references section.  
This research is supported by the Alexander von Humboldt Foundation.  
The authors declare no conflict of interest.