An inductive method of calculation of the deflection of the truss regular type

Kirsanov, Mikhail

doi:10.23968/2500-0055-2016-1-3-14-17

Cited by 5 publications

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“…These solutions and their combinations can be used for assessment of accuracy and reliability of the numerical results obtained in specialized packages of structural mechanics (Samofalov, Ziukas, 2015). Compact analytical solutions by induction method on the basis of the program (Kirsanov, 2012) were obtained in (Ilin, Kirsanov, 2016;Kirsanov, 2016a;Kirsanov, 2016b;Ponamareva, 2016;Dong, Kirsanov, 2016;Voropai, Kazmiruk, 2016;Voropai, 2016;Shipaeva, 2016;Bolotina, 2016). The comparative review of some analytical solutions is given in (Tinkov, 2015).…”

Section: Resultsmentioning

confidence: 99%

“…In any case, the obtained equation will be intended for estimate but rather accurate characterization of a structure. The accuracy of the analytic equations obtained by the induction method (Ilin, Kirsanov, 2016;Kirsanov, 2016b;Ponamareva, 2016) particularly manifests itself for structures with a large number of elements where numerical methods (with the solution of a linear equation system as their integral part) have a tendency to accumulate roundoff errors.…”

Section: Subject Tasks and Methodsmentioning

confidence: 99%

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The Mathematical Model of Dome Covering Industrial Facility

Kirsanov

Astahov

2017

AEJ

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A statically determinate flat truss models the industrial facility's arch. To derive dependence of structural deflection on the number of panels in a span, the Maxwell-Mohr equation, computer mathematics system Maple and the induction method are applied. Forces in the rods are determined by the method of cutting nodes. Several cases of loading are considered: uniformly distributed along the top and bottom chords, uniformly distributed along the lateral surface, and by concentrated force. For analytical assessment of the structural strength, equations for forces in the most compressed and tensioned rods and equation for support displacement are derived. Asymptotics of the solution for the number of panels at the fixed span length, and total load is found.

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Section: Resultsmentioning

confidence: 99%

Section: Subject Tasks and Methodsmentioning

confidence: 99%

The Mathematical Model of Dome Covering Industrial Facility

Kirsanov

Astahov

2017

AEJ

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“…Формула для прогиба с применением системы Maple выведена для распорной фермы в труде [2]. Решетчатые фермы различной конфигурации исследованы методом индукции [3][4][5]. Получены простые аналитические решения для арочных ферм [6][7][8].…”

Section: Introductionunclassified

Analytical Solution of the Frequency of the Load Oscillation at an Arbitrary Girder Node in the System Maple

Kirsanov

Tinkov

2019

Stroitel’stvo: nauka i obrazovanie

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Introduction. We study the oscillations of a massive load on a planar statically definable symmetric truss of a regular type with parallel belts. Truss weight is not included. Free vertical oscillations are considered. The stiffness of the truss rods is assumed to be the same, the deformations are elastic. Lattice of the truss is double with descending braces and racks. New in the formulation and solution of the problem is the analytical form of the solution, which makes it possible in practice to easily evaluate the frequency characteristics of the structure depending on an arbitrary number of truss panels and the location of the load. Materials and methods. The operators and methods of the system of computer mathematics Maple are used. To determine the forces in the rods, the knotting method is used. The common terms of the sequence of coefficients of solutions for different numbers of panels are obtained from solving linear homogeneous recurrent equations of various order, obtained by special operators of the Maple system. Dependence on two arbitrary natural parameters is revealed in two stages. First, solutions for fixed load positions are found, then these solutions are summarized into one final formula for frequency. Results. By a series of individual solutions to the problem of load oscillation using the double induction method, it was possible to find common members of all sequences. The solution is polynomial in both natural parameters. Graphs constructed for particular cases, showed the adequacy of the approach. The discontinuous non-monotonic nature of the intermittent change depending on the number of truss panels and some other features of the solution are noted. Conclusions. It is shown that the induction method, previously applicable mainly to statics problems with one parameter (number of truss panels), is fully operational to the problems of the oscillations of system with two natural parameters. It should be noted that significant labor costs and a significant increase in the time symbolic transformations in such tasks

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“…Формулы для расчета плоских статически определимых ферм получены в работах [1][2][3][4][5], простран-ственных -в работах [6][7][8][9].…”

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“…The formulae for calculation of flat statically determinate trusses were derived in papers [1][2][3][4][5], spatialin [6][7][8][9].…”

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Exact Solution of the Problem of Deflection of a Truss With an Arbitrary Number of Panels in the Maple System

Kirsanov¹

2017

SNO

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Аннотация. Получены выражения для прогиба статически определимой плоской балочной фермы под действием равномерно распределенной по нижнему поясу нагрузки в зависимости от числа панелей. Решет-ка фермы с нисходящими раскосами симметрична от-носительно середины. Для определения усилий в стержнях используется метод вырезания узлов. С по-мощью пакета компьютерной математики Maple решена система уравнений равновесия в символьной форме. При вычислении прогиба по формуле Максвелла-Мора учитываются разные жесткости стержней поясов и решетки. Для обобщения серии решений для ферм с различным числом панелей на случай произвольного числа панелей применен метод индукции и операторы для составления и решения рекуррентных уравнений системы Maple. В результате найдены некоторые пре-дельные свойства решения.Ключевые слова: ферма, точное решение, про-гиб, индукция, Maple DOI: 10.22227/2305-5502.2017.1.1 M.N. Kirsanov MPEI EXACT SOLUTION OF THE PROBLEM OF DEFLECTION OF A TRUSS WITH AN ARBITRARY NUMBER OF PANELS IN THE MAPLE SYSTEMAbstract. This article presents the formulae derived for statically determinate flat truss under load evenly distributed in the lower boom depending on the number of panels. The truss lattice with lowering diagonals is symmetric with respect to the middle. In order to calculate the forces in the rods, method of joints is used. Simultaneous equations in symbol presentation were solved using the Maple computer mathematics software package. Different stiffness coefficients for boom and lattice rods are considered when calculating the deflection using Maxwell-Mohr formula. For the purpose of synthesis of a number of solutions for the trusses with different numbers of panels in case of arbitrary number of panels, the induction method and functions for creation and solution of recurrent equations in Maple system were used. As a result, some limit properties of solution were discovered.Key words: truss, exact solution, deflection, induction, Maple DOI: 10.22227/2305-5502.2017.1.1Аналитические решения задач строи-тельной механики имеют не только теорети-ческое, но и практическое значение как для оценки проектируемых, так и для анализа экс-плуатационных характеристик. Формулы для расчета плоских статически определимых ферм получены в работах [1-5], простран-ственных -в работах [6][7][8][9].В рассматриваемой симметричной кон-струкции (рис. 1) число панелей в половине пролета четное n = 2k. Ферма статически определима, содержит s = 4n + 5 шарнирных узла и m = 8n + 4 стержня,включая три стерж-ня, моделирующие опоры. Нагрузка приложе-на к узлам нижнего пояса.Analytical solutions of the problems of the construction engineering possess not only theoretical, but practical value as well, both for assessment of design parameters, and for analysis of operational parameters. The formulae for calculation of flat statically determinate trusses were derived in papers [1-5], spatialin [6-9].The symmetrical structure being considered ( fig. 1) features even number of panels in one half of the span n = 2k. The truss is statically ...

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An inductive method of calculation of the deflection of the truss regular type

Cited by 5 publications

References 7 publications

The Mathematical Model of Dome Covering Industrial Facility

The Mathematical Model of Dome Covering Industrial Facility

Analytical Solution of the Frequency of the Load Oscillation at an Arbitrary Girder Node in the System Maple

Exact Solution of the Problem of Deflection of a Truss With an Arbitrary Number of Panels in the Maple System

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