Buckling of stepped beams

Aeronautical and Space-Rocket Engineering

Strength and thermal conditions of flying vehicles


Аuthors

Erkov A. P.

Sukhoi Civil Aircraft Company, SCAC, 26, Leninskaya Sloboda str, Moscow, 115280, Russia

e-mail: ap.erkov@yandex.ru

Abstract

The article discusses the problems of stability of two types of beams of variable stiffness: with a stepped change in cross section with two zones and with a step change in section with three zones. Simply supported boundary conditions at two ends are considered, as well as with embedding at one end and with a free second end. Beams of isotropic material and of the laminated composites are discussed.

To study the stability of beams of variable stiffness, the Ritz method was used. Beams with the ratio of the maximum and minimum flexural rigidity in the zones does not exceed 8 are considered, since in practice the ratio greater than 8, as a rule, is not applied. Analytical expressions for determining the critical force are obtained. The calculation results and their verification are given.

The results of analytical calculations were compared with the results obtained by the finite element method (MSC.Nastran / MSC.Patran). Based on a comparative analysis, graphs of the error of analytical solutions (relative to the solution obtained by the finite element method) were constructed. To minimize the error of analytical equations, a correction factor was introduced.

The study showed that the equations applicable for calculating the critical force of isotropic beams are also applicable to composite beams. Correction factors obtained for isotropic beams are also applicable to composite beams.

In addition to assessing the accuracy of analytical equations for the critical force, the influence of local effects in the area of the junction of zones with different flexural rigidity is investigated. In practice, the Bernoulli hypothesis does not work in the junction area of the zones, which has some influence on the magnitude of the critical force.

Results of investigation:

- Analytical equations were obtained for determining the critical force for two types of beams of variable stiffness with two types of boundary conditions;

- The accuracy of analytical equations was investigated. A correction factor was introduced, which allows to obtain a more accurate result for the critical force;

- The technique can be applied to other types of beams of variable stiffness and other boundary conditions not considered in this paper;

- The resulting analytical expressions are easy to automate. For this suit, for example, Microsoft Excel can be used.

Keywords:

Ritz method, critical force, stability of beams, beams of variable cross section, composite beams, stability of beams of variable stiffness, beams of variable stiffness

References

  1. Dinnik A.N. Izvestiya Donskogo politekhnicheskogo instituta, 1913, no. 1, pp. 390-404.

  2. Dinnik A.N. Ustoichivost’ uprugikh sistem (Stability of elastic systems), Moscow – Leningrad, ONTI NKTP SSSR, 1935, 187 p.

  3. Timoshenko S.P. Ustoichivost’ sterzhnei, plastin i obolochek (Stability of rods, plates and shells), Moscow, Nauka, 1971, 808 p.

  4. Volmir A.S. Ustoichivost’ deformiruemykh sistem (Stability of deformable systems), Moscow, Nauka, 1967, 984 p.

  5. Timoshenko S.P. Ustoichivost’uprugikh sistem (Stability of elastic systems), Moscow, Gosudarstvennoe izdatel’stvo tekhniko-teoreticheskoi literatury, 1955, 568 p.

  6. Alfutov N.A. Osnovy rascheta na ustoichivost’ uprugikh sistem (Fundamentals of elastic systems stability calculation), Moscow, Mashinostroenie, 1978, 312 p.

  7. Wang C.M., Wang C.Y., Reddy J.N. Exact Solutions for Buckling of Structural Members. Boca Raton, Florida, CRC Press, 2004, 224 p.

  8. Elishakoff I. Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions, Boca Raton, Florida, CRC Press, 2004, 752 p.

  9. Simitses G.J., Hodges D.H. Fundamentals of Structural Stability, Elsevier Inc., 2006, 480 p.

  10. Bazant Z.P., Cedolin L. Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories. World Scientific Publishing Co. Pte. Ltd., Singapore, 2010, 1011 p.

  11. Zotov A.A. Avtomatizirovannyi raschet na prochnost’ i ustoichivost’ konstruktsii letatel’nykh apparatov (Automated calculation of aircraft structures strength and stability), Moscow, MAI, 1992, 149 p.

  12. Tishkov V. V., Firsanov V. V. About model formulation for longitudinal impact on compound bar with relation to investigation of safety for complex aircraft systems upon emergency conditions. Aerospace MAI Journal, 2004, vol. 11, no. 2, pp. 3-10.

  13. Bondar’ T.A. Vychislitel’nye tekhnologii, 2003, vol. 8, no. 2, pp. 27-35.

  14. Kul’terbaev Kh.P., Karmokov K.A. Vestnik Volgogradskogo gosudarstvennogo arkhitekturno- stroitel’nogo universiteta, 2013, vol. 34, no. 53, pp. 90­–98.

  15. Gabbasov R.F., Filatov V.V. Chislennyi metod rascheta sostavnykh sterzhnei i plastin s absolyutno zhestkimi poperechnymi svyazyami (Numerical method for composite rods and plates with absolutely rigid transverse links computing), Moscow, ASV, 2014, 200 p.

  16. Bandurin N.G., Kalashnikov S.Yu. Stroitel’stvo i rekonstruktsiya, 2015, vol. 2, no. 58, pp. 4-11.

  17. Gorbachev V.I., Moskalenko O.B. Mekhanika tverdogo tela, 2011, no. 4, pp. 181-192.

  18. Krutii    Yu.S.      Stroitel’naya    mekhanika   i    raschet sooruzhenii, 2010, no. 6, pp. 22-29.

  19. Krutii    Yu.S.      Stroitel’naya    mekhanika   i    raschet sooruzhenii, 2011, no. 2, pp. 27-33.

  20. Krutii Yu.S. Vestnik Odesskoi Gosudarstvennoi Akademii Stroitel’stva i Arkhitektury, 2014, no. 56. URL: http://mx.ogasa.org.ua/handle/123456789/1464

  21. Senitskii Yu.E., Ishutin A.S. Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta, 2015, vol. 19, no. 2, pp. 341-357.

  22. Tsarenko S.N. Izvestiya vysshikh uchebnykh zavedenii. Stroitel’stvo, 2016, no. 1, pp. 5-13.

  23. Vlasova E.V. Aktual’nye problemy zheleznodorozhnogo transporta. Sbornik statei. Voronezh, filial Rostovskogo gosudarstvennogo universiteta putei soobshcheniya, 2018, pp. 127-130.

  24. Elishakoff I., Rollot O. New closed-form solutions for buckling of a variable stiffness column by Matematica®. Journal of Sound and Vibration, 1999, vol. 224, no. 1, pp. 172-182. DOI: 10.1006/ jsvi.1998.2143

  25. Liao S. Series solution of large deformation of a beam with arbitrary variable cross section under an axial load, ANZIAM Journal, 2009, vol. 51, pp. 10-33. DOI: 10.1017/S1446181109000339

  26. Coskun S.B. Advances in Computational Stability Analysis, InTech, Rijeka, 2012, 140 p.

  27. Taha M., Essam M. Stability behavior and free vibration of tapered columns with elastic end restraints using the DQM method. Ain Shams Engineering Journal, 2013, vol. 4, no. 3, pp. 515-521. DOI: 10.1016/ j.asej.2012.10.005

  28. Afsharfard A., Farshidianfar A. Finding the buckling load of non-uniform columns using the iteration perturbation method. Theoretical and Applied Mechanics Letters, 2014, vol. 4, no. 4. DOI: 10.1063/2.1404111

  29. Elishakoff I., Eisenberger M., Delmas A. Buckling and Vibration of Functionally Graded Material Columns Sharing. Structures, 2016, vol. 5, pp. 170-174. DOI: 10.1016/j.istruc.2015.11.002

  30. Shvartsman B., Majak J. Numerical method for stability analysis of functionally graded beams on elastic foundation. Applied Mathematical Modelling, 2016, vol. 40, no. 5-6, pp. 3713-3719. DOI: 10.1016/ j.apm.2015.09.060

  31. Ioakimidis N.I. The energy method in problems of buckling of bars with quantifier elimination. Structures, 2018, vol. 13, pp. 47-65. DOI: 10.1016/ j.istruc.2017.08.002

  32. Golfam B., Nazarimofrad E., Zahrai S.M. Bending, second-order and buckling analysis of non prismatic beam-columns by differential quadrature method. Applied MethematicalModelling, 2018, vol. 63, pp. 362–­373. DOI: 10.1016/j.apm.2018.06.054

  33. Uribe-Henao A.F., Zapata-Medina D.G., Arboleda- Monsalve L.G., Aristizabal-Ochoa J.D. Static and Dynamic Stability of a Multi-stepped Timoshenko Column Including Self-weight. Structures, 2018, vol. 15, pp. 28-42. DOI: 10.1016/j.istruc.2018.05.004

  34. Kulinich I.I., Litvinov V.V., Blyagoz A.M. Novye tekhnologii, 2012, no. 4, pp. 75-81.

  35. Sapountzakis E.J., Tsiatas G.C. Elastic flexural buckling analysis of composite beams of variable cross-section by BEM. Engineering Structures, 2007, vol. 29, no. 5, pp. 675-681. DOI: 10.1016/j.engstruct.2006.06.010

  36. Lellep J., Sakkov E. Buckling of stepped composite columns. Mechanics of Composites Materials, 2006, vol. 42, no. 1, pp. 63-72. DOI: 10.1007/s11029-006-0017-4

mai.ru — informational site of MAI

Copyright © 1994-2024 by MAI