# Calculation of the deflection of an arched truss with suspended elements depending on the number of panels

## Abstract

The aim of the work - to propose a scheme and analytical calculation of a statically definable planar truss with a suspended lower belt. Methods. The formula for the dependence of the deflection of the truss under the action of a uniform load on the lower belt on its size and the number of panels is derived in the computer mathematics system Maple. The forces in the rods are found from the solution of the general system of equilibrium equations of all nodes in symbolic form. The deflection is calculated using the Maxwell - Mohr's formula. Generalization of a number of formulas for deflection obtained by increasing the number of panels sequentially to an arbitrary number is performed by double induction using two independent parameters. In this case, special operators of the Maple system are used, allowing for a sequence of coefficients in the desired formula to create and solve recurrent equations that satisfy the elements of the sequences. Results. The obtained solutions have a polynomial form for the number of panels. Curves of deflection dependence on the number of panels are constructed and analyzed. Asymptotic properties of solutions are found in the case of a fixed span length of the structure and a given total load. The proposed scheme is a statically determinate structure with two independent parameters of regularity allows for the finding of a fairly simple analytical solution. The resulting formula is most effective in calculating systems with a large number of elements, where numerical methods tend to accumulate rounding errors.

## Keywords

### Mikhail N. Kirsanov

National Research University “Moscow Power Engineering Institute”

Author for correspondence.
Email: c216@ya.ru

Doctor of Physical and Mathematical Sciences, Professor, Department of Robotics, Mechatronics, Dynamics and Strength of Machines

14 Krasnokazarmennaya St, Moscow, 111250, Russian Federation

## References

1. Mathieson C., Roy K., Clifton G., Ahmadi A., Lim J.B.P. Failure mechanism and bearing capacity of cold-formed steel trusses with HRC connectors. Engineering Structures. 2019;201:109741.
2. Villegas L., Moran R., Garcia J.J. Combined culm-slat Guadua bamboo trusses. Engineering Structures. 2019; 184:495–504.
3. Dong L. Mechanical responses of snap-fit Ti-6Al-4V warren-truss lattice structures. International Journal of Mechanical Sciences. 2020;173:105460.
4. Tinkov D.V., Safonov A.A. Design Optimization of Truss Bridge Structures of Composite Materials. Journal of Machinery Manufacture and Reliability. 2017;46(1):46–52.
5. Bolotina T.D. The deflection of the flat arch truss with a triangular lattice depending on the number of panels. Bulletin of Scientific Conferences. 2016;(8)4–3:7–8.
6. Timofeeva Т.А. Formulas for calculating the deflection of a flat lattice frame with an arbitrary number of panels. Structural mechanics and structures. 2019;4(23):26–33. (In Russ.)
7. Boyko A.Y., Tkachuk G.N. Derivation of the formulas for the deflection of a flat hinged-rod frame in the of symbol mathematics Maple system. Structural mechanics and structures. 2019;4(23):15–25. (In Russ.)
8. Belyankin N.A., Boyko A.Y. Formula for deflection of a girder with an arbitrary number of panels under the uniform load. Structural mechanics and structures. 2019;1(20): 21–29. (In Russ.)
9. Tkachuk G.N. The formula for the dependence of the deflection of an asymmetrically loaded flat truss with reinforced braces on the number of panels. Structural mechanics and structures. 2019;2(21):32–39. (In Russ.)
10. Tinkov D.V. Comparative analysis of analytical solutions to the problem of deflection of truss structures. Magazine of civil Engineering. 2015;5(57):66–73. (In Russ.)
11. Osadchenko N.V. Analytical solutions of problems of deflection of flat trusses of arch type. Structural mechanics and structures. 2018;1(16):12–33. (In Russ.)
12. Kompaneets A.K.The calculation of the displacement of the movable support of flat arched diagonal truss with a load at Midspan. Youth and science. 2017;4–2:108. (In Russ.)
13. Savinyh A. Analysis of deflection of the arch truss loaded at the upper belt. Construction and Architecture. 2017; 5;3(6):12–17. (In Russ.)
14. Tinkov D.V. Calculation of the deflection of a flat arched truss with a cross-shaped grid. Postulat. 2017;12(26):74. (In Russ.)
15. Kirsanov M.N. Analysis of the buckling of spatial truss with cross lattice. Magazine of Civil Engineering. 2016; 4(64):52–58. (In Russ.)
16. Domanov J.V. The analytical dependence of the deflection of the spatial console of the triangular profile on the number of panels. Science Almanac. 2016;6–2(19):214–217. (In Russ.)
17. Larichev S.A. Inductive analysis of the effect of a building lift on the stiffness of a spatial beam truss. Trends in Applied Mechanics and Mechatronics. 2015;1:4–8. (In Russ.)
18. Kirsanov M.N. Planar trusses. Schemes and Formulas. Cambridge Scholars Publishing; 2019.