Abstract

A statically determinate planar truss has rectilinear belts and a triangular lattice consisting of double braces. The four links make it seemingly statically indeterminate. The derivation of the formula for the dependence of the deflection on its dimensions and the number of panels is given. Forces in rods are determined in symbolic form by cutting out nodes from the solution of a system of linear equations in the system of computer mathematics Maple. To determine the deflection, the Maxwell - Mohr’s formula is used. Rods (except all rigid support) are assumed to be elastic with the same rigidity. The generalization of individual solutions to an arbitrary number of panels is done by induction. Operators of the Maple system from the calculation data yield linear homogeneous recurrence equations for the coefficients of the desired formula. The solutions of these equations give the general terms of the obtained sequences. Formulas for three types of loads are obtained and compared (the uniform loading of the nodes of the lower and upper belts and the concentrated force in the middle of the span). Curves of the dependence of deflection on the number of panels have weakly expressed minima. The dependencies of the forces in the most compressed and stretched rods on the number of panels are derived. Also given are asymptotic estimates for solutions in accordance with the number of panels in fixed spans of the construction and at a given total load.

Mikhail Nikolaevich Kirsanov

National Research University “MPEI”

Author for correspondence.
Email: c216@ya.ru
14 Krasnokazarmennaya St., Moscow, 111250, Russian Federation

Professor, National Research University “Moscow Power Engineering Institute”, Professor of M.V. Lomonosov Moscow State University. Author of ten monographs and textbooks on mathematics and mechanics, member of the Russian National Committee on Theoretical and Applied Mechanics. Research interests: structural mechanics, analytical solutions, Maple, differential equations, discrete mathematics, artificial intelligence methods, rheology

References

1. Kirsanov, M.N. (2016). Analytical calculation of truss girder with the «Butterfly» lattice. Structural Mechanics and Analysis of Constructions, No 4, 2–5. (In Russ.)
2. Kirsanov, M.N., Razananairina, Р.C. (2017). The formula for deflection of truss with cases of kinematic variability. Postulat, No 9.
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4. Bolotina, T.D. (2016). The deflection of the flat arch truss with a triangular lattice depending on the number of panels. Bulletin of Scientific Conferences, No 4–3 (8), 7–8.
5. Tinkov, D.V. (2015). Comparative analysis of analytical solutions to the problem of truss structure deflection. Magazine of Civil Engineering, No 5 (57), 66–73. (In Russ.)
6. Kirsanov, M.N. (2016). Static analysis and mounting diagram of flat truss. Vestnik Gosudarstvennogo universiteta morskogo i rechnogo flota imeni admirala S.O. Makarova [Bulletin of State University of Marine and River Fleet named after Admiral S.O. Makarov], 5 (39), 61–68. (In Russ.)
7. Kirsanov, M. (2016). An inductive method of calculation of the deflection of the truss regular type. Architecture and Engineering, 1 (3), 14–17.
8. Kirsanov, M.N. (2016). Analysis of the buckling of spatial truss with cross lattice. Magazine of Civil Engineering, (4), 52–58. doi: 10.5862/MCE.64.5. (In Russ.)
9. Kirsanov, M.N. (2012). Maple i Maplet. Resheniya zadach mekhaniki [Maple and Maplet. Solving the problems of mechanics], St. Petersburg: “Lan'” Publ., 512. (In Russ.)

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