
Kirsanov Mikhail Nikolaevich 
National Research University, “Moscow Power Engineering Institute” (MPEI)
Doctor of Physical and Mathematical Sciences, Professor, Department of Robotics, Mechatronics, Dynamics and Strength of Machines, National Research University, “Moscow Power Engineering Institute” (MPEI), 14 Krasnokazarmennaya str., Moscow, 111250, Russian Federation;
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Analytical solutions have definite advantages over numerical ones. It is quite complicated to find solutions for a whole class of structures different in the number of bars or panels if we speak about girder trusses. The solutions obtained in a symbol form may be analytically investigated depending on all the parameters characterizing the structure. This allows educing the distinctive features of the systems and finding possibilities for optimization in terms of mass, strength and rigidity. At the present time the induction method is the most efficient to obtain analytical solutions for a number of trusses. Using computer mathematics system Maple the author obtained analytical expressions for deflection of a flat statically determinate parallelflanged elastic truss depending on the number of panels at uniform and concentrated load. It is shown that the rake angle of a free support greatly influences the stiffness of a structure. The graphic charts of the dependence of deflection from the number of panels at a fixed span length and a given load show extremum. The author obtains asymptotic characteristics of the deflection and expressions for stresses in the most compressed and tension bars. It is also shown that the solution type for a deflection isn’t changed in case of different loading types and has a form of a polynomial of not more than third degree in correspondence with the number of panels in half of a span.
DOI: 10.22227/19970935.2016.10.3544

Kirsanov Mikhail Nikolaevich 
National Research University Moscow Power Engineering Institute (MPEI)
Doctor of Physical and Mathematical Sciences, Professor, Department of the Theoretical Mechanics and Mechatronics, National Research University Moscow Power Engineering Institute (MPEI), 14 Krasnokazarmennaya str., Moscow, 111250, Russian Federation.
Beamlike spatial twolayer symmetric truss is formed by four plane trusses connected by the long sides, and rests on four corner points. Stresses in truss components are defined in a symbolic manner by the method of joint isolation using the Maple computer algebra system. Matrix of the set of equilibrium equations is formed in a cycle according to the number of bars of the truss. For calculation of deflection the MaxwellMohr formula is used. The solution is framed for the case of various bar sectional areas and is generalized to an arbitrary number of panels by the method of induction. Operators for formation and solution of recurrence equations are involved for determination of general terms of sequences of coefficients. Certain limit performance and asymptotic characteristics of the structure are found. Formulas for stresses in the most compressed and stretched truss components are derived. Model of statically determinate spatial twolayer truss is proposed. Exact analytical expression for deflection of the truss under action of a concentrated force is found. The used algorithm allows to expand the solution to an action of other loads and methods of supporting. Inhomogeneous distribution of material throughout the structure bars is taken into account in the solution. It enables a designer to choose the most optimal combination of design parameters without making numerical calculations in specialized packages. The proposed twolayer trusses may find practical use in roofs of buildings and structures where a natural interior volume creates additional thermal protection, herewith providing an improvement of strength.
DOI: 10.22227/19970935.2017.2.165171

Kirsanov Mikhail N. 
National Research University “Moscow Power Engineering Institute” (MPEI)
Doctor of Physical and Mathematical Sciences, Professor of Department of robotics, mechatronics, dynamics and strength of machines, National Research University “Moscow Power Engineering Institute” (MPEI), 14 Krasnokazarmennaya st., Moscow, 111250, Russian Federation.

Tinkov Dmitriy V. 
National Research University “Moscow Power Engineering Institute” (MPEI)
postgraduate of Department of robotics, mechatronics, dynamics and strength of machines, National Research University “Moscow Power Engineering Institute” (MPEI), 14 Krasnokazarmennaya st., Moscow, 111250, Russian Federation.
Introduction. Analytical solutions for problems of structural mechanics are not only an alternative approach to solving problems of strength, reliability and dynamics of structures, but also the possibility for simple performance evaluations and optimization of structures. Frequency analysis of planar trusses, most often used in construction and engineering, is an important part of the study of structures. Objectives  development of a threeparameter induction algorithm for deriving the analytical dependence of the natural oscillation frequencies of the truss on the number of panels. Materials and methods. A flat, statically definable truss with one additional external link and double braces has been considered. The inertia properties of the truss are modeled by point masses located in the nodes of the lower straight truss belt. Each mass is assumed to have only one vertical degree of freedom. The stiffness of all truss rods is assumed to be the same. The task is to obtain analytical dependences of the oscillation frequencies of the proposed truss model on the number of panels. The derivation of the desired formulas is performed by the method of induction in three stages  according to the numbers of rows and columns of the compliance matrix, calculated using the Maxwell  Mohr formula and the number of panels. To find common members of the obtained sequences of coefficients, an apparatus was used to compile and solve the recurrent equations of the Maple computer mathematics system. The task of determining frequencies has been reduced to the eigenvalue problem of a bisymmetric matrix. Results. For the elements of the compliance matrix, general formulas have been found, according to which the frequency equations are compiled and solved. It is shown that in the frequency spectra of trusses with different numbers of panels there is always one common frequency (middle frequency) located in the middle of the spectrum. An expression is found for the maximum value of the average oscillation frequency as a function of the height of the truss. Conclusions. The proposed truss scheme, despite its external static indeterminacy and the lattice, which does not allow for the calculation of forces by such methods as the method of cutting nodes and the cross section method, allows analytical solutions for the natural frequencies of loads in the nodes. The obtained formulas have a rather simple form, and some general properties, such as frequency coincidences for different numbers of panels and the presence of an analytically calculated maximum of the average frequency function of the truss height, make this solution convenient for practical structural evaluations.
DOI: 10.22227/19970935.2019.3.284292