### APPLICATION OF ANALYTICAL SOLUTIONS FOR BENDING BEAMS IN THE METHOD OF MOVEMENT

Vladimir Karpov, Evgeny Kobelev, Aleksandr Panin

#### Abstract

Introduction: Usually, to analyze statically indeterminate rod systems, the classical displacement method and preprepared tables for two types of rods of the main system are used. A mathematically correct representation of local loads with the use of generalized functions makes it possible to find an accurate solution of the differential equation for the equilibrium of a beam exposed to an arbitrary transverse load. Purpose of the study: We aimed to obtain analytical expressions for functions of deflection, rotation angles, transverse forces, and bending moments depending on four types of local loads for beams with different boundary conditions, so as to apply accurate solutions in the displacement method. Methods: We propose an analytical form of the displacement method to analyze rod structural models. For beams exposed to different types of transverse load (uniformly distributed force, concentrated force, or a couple of forces), accurate analytical solutions were obtained for functions of deflection, bending moments, and transverse forces at different types of beam ends’ restraint. This is possible due to the fact that concentrated load and load in the form of the moment of force can be specified by using unit column functions. By transforming Mohr’s integrals, using integration by parts, we show that the system of canonical equations of the displacement method was obtained based on the Lagrange principle. Results: Based on the analysis of a statically indeterminate frame, the effectiveness of the proposed analytical method is shown as compared with the classical displacement method.

#### Keywords

Rod systems, displacement method, beam bending equation, Mohr’s integral, mathematical model, work of internal forces, work of external forces, Lagrange principle.

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#### References

Akimov, P. A. and Mozgaleva, M. L. (2014). Multi-level discrete and discrete-continuous for the local analysis of building structures. Moscow: Moscow State University of Civil Engineering, 632 p.

Alyukov, S. V. (2011). Approximation of step functions in problems of mathematical modeling. Mathematical Models and Computer Simulations, Vol. 3, Issue 5, 661. DOI: 10.1134/S2070048211050036.

Alyukov, S. V. (2012). Approximations of piecewise linear and generalized functions. Germany: LAP LAMBERT Academic Publishing, 120 p.

Babanov, V. V. (2011). Structural mechanics. In 2 volumes. Vol. 2. Moscow: Academia Publishing Center, 288 p.

Belostochny, G. N. (1999). Analytical methods for the determination of closed path integrals of singular differential equations for the thermoelastic behavior of geometrically irregular shells. Reports of Military Sciences Academy. No. 1. Volga Regional Office. Saratov, pp. 14–26.

Ignatyev, V. A. (1979). Analysis of regular statically indeterminate rod systems. Saratov: Publishing House of Saratov University, 296 p.

Ilin, V. P., Karpov, V. V. and Maslennikov, A. M. (2005). Numerical methods for solving structural mechanics problems. 2nd edition. Moscow: ASV Publishing House, 426 p.

Karpov, V. V. (2006). Mathematical modeling, model analysis algorithms, simulation experiment in the theory of shells. Saint Petersburg: Saint Petersburg State University of Architecture and Civil Engineering, 330 p.

Karpov, V. V. (2010). Strength and stability of stiffened shells of revolution. In 2 parts. Part 1. Models and algorithms for the analysis of strength and stability of stiffened shells of revolution. Moscow: Fizmatlit, 288 p.

Karpov, V. V. (2011). Strength and stability of stiffened shells of revolution. In 2 parts. Part 2. Simulation experiment under static mechanical action. Moscow: Fizmatlit, 248 p.

Kobelev, E. A. (2018). Method of variation approximations in the theory of nonlinear deformation of irregular spatial systems. Bulletin of Civil Engineers, No, 6 (71), pp. 30–36. DOI: 10.23968/1999–5571–2018–15–6–30–36.

Kobelev, E. A. and Lukashevich, N. K. (2020a). Solving the problem of unilateral contact of the slab with the strengthening beams by the method of variational approximations. IOP Conference Series: Materials Science and Engineering, Vol. 953, 012089. DOI: 10.1088/1757–899X/953/1/012089.

Kobelev, E. A. and Lukashevich, N. K. (2020b). Solving the contact problem when strengthening the slab with a beam using discontinuous functions. IOP Conference Series: Materials Science and Engineering, Vol. 962, 022046. DOI: 10.1088/1757-899X/962/2/022046.

Korn, G. and Korn, T. (1974). Mathematical handbook (for scientists and engineers). Moscow: Nauka, 831 p.

Korneyev, S. A. (2011). Technical rod theory. Use of generalized functions to solve problems related to the strength of materials. Omsk: Omsk State Technical University, 82 p.

Leontyev, N. N., Sobolev, D. N. and Amosov, A. A. (1996). Fundamentals of structural mechanics of rod systems. Moscow: ASV Publishing House, 541 p.

Maslennikov, A. M. (1987). Structural analysis with numerical methods. Leningrad: Publishing House of Leningrad State University, 224 p.

Maslennikov, A. M., Kobelev, E. A. and Maslennikov, N. A. (2020). Fundamentals of structural mechanics of rod systems: study guide. Saint Petersburg: Petropolis Publishing House, 342 p.

Mikhailov, B. K. (1980). Plates and shells with discontinuous parameters. Leningrad: Publishing House of Leningrad State University, 196 p.

Mikhailov, B. K., Kobelev, E. A. and Gayanov, F. F. (1990). Structural analysis with the use of generalized functions. Leningrad: Leningrad Civil Engineering and Construction Institute, 99 p.

Smirnov, V. I. (1967). A course of higher mathematics. Vol. 2. Moscow: Nauka, 655 p.

Zolotov, A. B., Akimov, P. A., Sidorov, V. N. and Mozgaleva, M. L. (2008). Mathematical methods in structural mechanics (with fundamentals of the generalized function theory). Moscow: ASV Publishing House, 336 p.

DOI: https://doi.org/10.23968/2500-0055-2021-6-4-42-53

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