Slope Deflection Method in Nonlocal Axially Functionally Graded Tapered Beams

Erol Demirkan, Murat Çelik*, Reha Artan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In this study, the slope deflection method was presented for structures made of small-scaled axially functionally graded beams with a variable cross section within the scope of nonlocal elasticity theory. The small-scale effect between individual atoms cannot be neglected when the structures are small in size. Therefore, the theory of nonlocal elasticity is used throughout. The stiffness coefficients and fixed-end moments are calculated using the method of initial values. With this method, the solution of the differential equation system is reduced to the solution of the linear equation system. The given transfer matrix is unique and the problem can be easily solved for any end condition and loading. In this problem, double integrals occur in terms of the transfer matrix. However, this form is not suitable for numerical calculations. With the help of Cauchy’s repeated integration formula, the transfer matrix is given in terms of single integrals. The analytical or numerical calculation of single integrals is easier than the numerical or analytical calculation of double integrals. It is demonstrated that the nonlocal effect plays an important role in the fixed-end moments of small-scaled beams.

Original languageEnglish
Article number4814
JournalApplied Sciences (Switzerland)
Issue number8
Publication statusPublished - Apr 2023

Bibliographical note

Publisher Copyright:
© 2023 by the authors.


This research is supported by the Alexander von Humboldt Foundation.

FundersFunder number
Alexander von Humboldt-Stiftung


    • axially functionally graded materials
    • method of initial values
    • nonlocal elasticity
    • size effect
    • slope deflection method
    • transfer matrix


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