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

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.

Evaluating the Applicability of Bernoulli’s Hypothesis in Beam Analysis

“…The widespread use of modern software systems for structural analysis in construction necessitates their verification to determine the reasonable level of detail with regard to the analyzed analytical model and the required accuracy of calculations (ANSYS, 2009;Simulia, 2012: SOFiSTiK AG, 2014. Thus, it is required to obtain accurate solutions for typical problems related to the analysis of new building structure types in order to use analytical solutions (Karpov et al, 2021) for verification of various software systems. S. P. Timoshenko (Timoshenko, 1945;Timoshenko and Woinowsky-Krieger, 1963) is rightfully considered the author of the refined theory Volume 7 Issue 3 (2022) considering transverse shear in the analysis of beams, plates, and shells.…”

“…By using Bernoulli's hypothesis, Karpov et al (2021) presented a method to find an accurate solution to the beam bending equation for a beam of uniform cross-section height, subjected to different types of transverse load (distributed along the entire length of the beam, distributed along a part of the beam, concentrated force, or a moment of a couple of forces), with different types of beam end restraint. An analytical solution for a beam can also be obtained by using the hypothesis taking into account transverse shears.…”

“…The method of solving Eq. ( 1) described by Karpov et al (2021) can also be applied to solve system (2).…”

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“…The widespread use of modern software systems for structural analysis in construction necessitates their verification to determine the reasonable level of detail with regard to the analyzed analytical model and the required accuracy of calculations (ANSYS, 2009;Simulia, 2012: SOFiSTiK AG, 2014. Thus, it is required to obtain accurate solutions for typical problems related to the analysis of new building structure types in order to use analytical solutions (Karpov et al, 2021) for verification of various software systems. S. P. Timoshenko (Timoshenko, 1945;Timoshenko and Woinowsky-Krieger, 1963) is rightfully considered the author of the refined theory considering transverse shear in the analysis of beams, plates, and shells.…”

“…By using Bernoulli's hypothesis, Karpov et al (2021) presented a method to find an accurate solution to the beam bending equation for a beam of uniform cross-section height, subjected to different types of transverse load (distributed along the entire length of the beam, distributed along a part of the beam, concentrated force, or a moment of a couple of forces), with different types of beam end restraint. An analytical solution for a beam can also be obtained by using the hypothesis taking into account transverse shears.…”