- STRUCTURAL MECHANICS
- Exact Analytical Solutions To Non-Stationary Problems For Rods Based On S. P. Timoshenko Theory
- UDC 624.04:534.11 DOI: 10.33622/0869-7019.2020.07.16-25
Alexander L. ZONENBERG, e-mail: firstname.lastname@example.org
Moscow State University of Civil Engineering (National Research University), Yaroslavskoe shosse, 26, Moscow 129337, Russian Federation
Abstract. Based on the theory of S.P. Timoshenko, exact analytical solutions of non-stationary problems for rods in general form are given. Such solutions are necessary for the study of transient wave processes of deformation in rods under transverse short-term and fast-changing effects. The work uses operational calculus based on the integral Laplace- Carson transformation, methods of structure dynamics, Mathcad program (drawing of graphs). Exact formulas are given in general form, including integrals from Bessel functions, to determine efforts in semi-infinite rods of various types, as well as in an infinite rod when solving an auxiliary problem. For each task, a conclusion is made on the existence of discontinuities. The situation where force impacts are pulsed is highlighted. As an illustration, graphs of changes in time of efforts in rods calculated according to the given formulas under effects of special type are given. The results of the work can be used in the practice of calculating buildings and structures for the effect of impact and explosive loads, with a refined assessment of the parameters of oscillations excited by industrial enterprises, vehicles, etc.
Key words: Transient wave processes, S.P. Timoshenko theory, rods, traveling waves, operational ratio, Bessel functions.
1. Kolsky H. Volny napryazheniya v tverdyh telah [Stress waves in solids]. Moscow, Izd-vo inostrannoy literatury Publ.,1955. 192 p. (In Russian).
2. Zonenberg A. L. New operational ratios and their application to non-stationary tasks for rods based on S. P. Timoshenko theory. Stroitel'naya mekhanika inzhenernyh konstrukcij i sooruzhenij, 2020, vol.16, no. 1, pp. 62-75. (In Russian).
3. Timoshenko S. P. Kurs teorii uprugosti [Course in the theory of elasticity]. Kiev, Naukova dumka Publ., 1972. 508 p. (In Russian).
4. Slepyan L. I., Yakovlev Yu. S. Integral'nye preobrazovaniya v nestacionarnyh zadachah mekhaniki [Integral transformations in non-stationary problems of mechanics]. Leningrad, Sudostroenie Publ., 1980. 344 p. (In Russian).
5. Grigolyuk E. I., Selezov I. T. Nonclassical theories of vibrations of bars, plates and shells. Itogi nauki i tekhniki. Seriya: Mekhanika tverdyh deformiruemyh tel [Advances in sciences and engineering. Mechanics of deforming solids]. Moscow, VINITI Publ., 1973. 272 p. (In Russian).
6. Selezov I. T. On the development of the Timoshenko theory of transversal oscillations of elastic rods. Journal of Machinery Manufacture and Reliability, 2016, no. 1, pp. 16-23. (In Russian).
7. Su Yu-Chi, Ma Chien-Ching. Theoretical analysis of transient waves in a simply-supported Timoshenko beam by ray and normal mode methods. International Journal of Solids and Structures, 2001, vol. 48, no. 3-4, pp. 535-552.
8. Su Yu-Chi, Ma Chien-Ching. Transient wave analysis of a cantilever Timoshenko beam subjected to impact loading by Laplace transform and normal mode methods. International Journal of Solids and Structures, 2012, vol. 49, no. 9, pp. 1158-1176.
9. Wang X. Q., So R. M. C. Timoshenko beam theory: A perspective based on the wave-mechanics approach. Wave Motion, 2015, vol. 57, pp. 64-87.
10. Abramyan A. K., Indeitsev D. A., Postnov V. A. Running and standing waves of Timoshenko beam. Izvestiya RAN. Mehanika tverdogo tela, 2018, no. 2, pp. 101-109. (In Russian).
11. Leonard R. W., Budiansky B. On traveling waves in beams. NACA Repts, 1954, no. 1173, pp. 389-415.
12. Dengler M. A. Transversale Wellen in Stäben und Platten unter stoßförmiger Belastung. Österr. Ingenieur-Archiv, 1956, vol. 10, no. 1, pp. 39-66.
13. Flügge W., Zajac E. E. Bending impact waves in beams. Ingenieur-Archiv,1959, vol. 28, iss. 1, pp. 59-70.
14. Lurie A. I. Operacionnoe ischislenie i ego prilozheniya k zadacham mekhaniki [Operational calculus and its application to the problems in mechanics]. Moscow, Leningrad, Gostekhizdat Publ., 1950. 432 p. (In Russian).
15. Watson G. N. Teoriya besselevyh funkcij [À treatise on the theory of Bessel functions]. Part 1. Moscow, Izd-vo inostrannoi literatury, 1949. 799 p. (In Russian).
16. Ditkin V. A., Prudnikov A. P. Spravochnik po operacionnomu ischisleniyu [Handbook of operational calculations]. Moscow, Vysshaya shkola Publ., 1965. 468 p. (In Russian).
17. Ditkin V. A., Prudnikov A. P. Operacionnoe ischislenie [Operational calculus]. Moscow, Vysshaya shkola Publ., 1966. 408 p. (In Russian).
18. Doetsch G. Rukovodstvo k prakticheskomu primeneniyu preobrazovaniya Laplasa [Guide to the applications of laplace transforms]. Moscow, Nauka Publ., 1965. 288 p. (In Russian).
19. Efros A. M., Danilevsky A. M. Operacionnoe ischislenie i konturnye integraly [Operational calculus and contour integrals]. Kharkiv, Gosudarstvennoe nauchno-tekhnicheskoe izdatelstvo Publ., 1937. 383 p. (In Russian).
- For citation: Zonenberg A. L. Exact Analytical Solutions to Non-Stationary Problems for Rods Based on S. P. Timoshenko Theory. Promyshlennoe i grazhdanskoe stroitel'stvo [Industrial and Civil Engineering], 2020, no. 7, pp. 16-25. (In Russian). DOI: 10.33622/0869-7019.2020.07.16-25.