Method of variation approximations in the theory of nonlinear deformation of irregular spatial systems

Kobelev, Evgeny

doi:10.23968/1999-5571-2018-15-6-30-36

Cited by 1 publication

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“…Currently, almost all structural calculations are performed with the use of computer technologies, which makes it possible to automate the process and ensure a quite high accuracy of the calculations (Akimov and Mozgaleva, 2014;Karpov, 2006Karpov, , 2010Karpov, , 2011Kobelev, 2018).…”

Section: Introductionmentioning

confidence: 99%

“…When a concentrated load is described by delta functions, it means that the load is applied to the point, and that is impossible in real practice. In the course of structural calculations for structures having various irregularities or exposed to local loads, the mathematical apparatus of the theory of generalized functions is widely used (Alyukov, 2011(Alyukov, , 2012Belostochny, 1999;Ilin et al, 2005;Kobelev, 2018;Kobelev andLukashevich, 2020a, 2020b;Mikhailov, 1980;Mikhailov et al, 1990). To specify correctly the location of stiffeners reinforcing a shell, Karpov (Ilin et al, 2005) introduced special unit column functions equal to the difference between two unit functions and showed how to describe local loads applied to a small area, by using those functions.…”

Section: Introductionmentioning

confidence: 99%

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Application of Analytical Solutions for Bending Beams in the Method of Movement

Karpov

Kobelev

Panin

2021

AEJ

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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.

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Section: Introductionmentioning

confidence: 99%