Convergence of the Rayleigh–Ritz Method for buckling analysis of arbitrarily configured I-section beam–columns

Abstract

Design of the slender members requires calculation of buckling loads in addition to stress and deflection demand/capacity ratios. The Rayleigh–Ritz Method, which allows one to present approximate closed-form solutions for certain cases, is one of the simplest methods for this purpose. This study evaluates the buckling analysis of the I-section prismatic beam–columns with the Rayleigh–Ritz Method in detail. First, algebraic, trigonometric, and exponential trial functions for various restraint configurations are derived carefully in finite series form. Then, an iterative procedure to calculate buckling loads and modes is described. Finally, a software is developed with Mathematica and the sensitivity of the results and performance to trial function type and the number of terms is investigated over 1000 computer-generated numerical examples, which include doubly and singly symmetric sections, simply supported and cantilever members, intermediate torsional and lateral restraints, transversal concentrated and distributed loads acting above/below the shear center, and axial loads.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

References

  1. 1.

    ANSI/AISC 360-16: Specification for Structural Steel Buildings. American Institute of Steel Construction (2016)

  2. 2.

    Eurocode 3: Design of Steel Structures (EN 1993). European Committee for Standardization (CEN) (2005)

  3. 3.

    Chen, F., Qiao, P.: Buckling of delaminated bi-layer beam--columns. Int. J. Solids Struct. 48, 2485–2495 (2011). https://doi.org/10.1016/j.ijsolstr.2011.04.020

    Article  Google Scholar 

  4. 4.

    Douville, M.-A., Le Grognec, P.: Exact analytical solutions for the local and global buckling of sandwich beam--columns under various loadings. Int. J. Solids Struct. 50, 2597–2609 (2013). https://doi.org/10.1016/j.ijsolstr.2013.04.013

    Article  Google Scholar 

  5. 5.

    Sad Saoud, K., Le Grognec, P.: An enriched 1D finite element for the buckling analysis of sandwich beam--columns. Comput. Mech. 57, 887–900 (2016). https://doi.org/10.1007/s00466-016-1267-1

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Law, K.H., Gardner, L.: Global instability of elliptical hollow section beam--columns under compression and biaxial bending. Int. J. Steel Struct. 13, 745–759 (2013). https://doi.org/10.1007/s13296-013-4015-9

    Article  Google Scholar 

  7. 7.

    Saoula, A., Meftah, S.A., Mohri, F., Daya, E.M.: Lateral buckling of box beam elements under combined axial and bending loads. J. Constr. Steel Res. 116, 141–155 (2016). https://doi.org/10.1016/j.jcsr.2015.09.009

    Article  Google Scholar 

  8. 8.

    Bedair, O.: Economical design procedures for built-up box sections subject to compression and bi-axial bending. Structures. 1, 51–59 (2015). https://doi.org/10.1016/j.istruc.2014.09.001

    Article  Google Scholar 

  9. 9.

    Giżejowski, M.A., Szczerba, R., Gajewski, M.D., Stachura, Z.: Buckling resistance assessment of steel I-section beam--columns not susceptible to LT-buckling. Arch. Civ. Mech. Eng. 17, 205–221 (2017). https://doi.org/10.1016/j.acme.2016.09.003

    Article  Google Scholar 

  10. 10.

    Tankova, T., Marques, L., Andrade, A., Simões da Silva, L.: A consistent methodology for the out-of-plane buckling resistance of prismatic steel beam--columns. J. Constr. Steel Res. 128, 839–852 (2017). https://doi.org/10.1016/j.jcsr.2016.10.009

    Article  Google Scholar 

  11. 11.

    Ioannidis, G.I., Avraam, T.P.: Lateral-torsional buckling of simply supported beams under uniform bending and axial tensile force. Arch. Appl. Mech. 82, 1393–1402 (2012). https://doi.org/10.1007/s00419-012-0680-y

    Article  MATH  Google Scholar 

  12. 12.

    Orun, A.E., Guler, M.A.: Effect of hole reinforcement on the buckling behaviour of thin-walled beams subjected to combined loading. Thin-Walled Struct. 118, 12–22 (2017). https://doi.org/10.1016/j.tws.2017.04.034

    Article  Google Scholar 

  13. 13.

    Unterweger, H., Taras, A., Feher, Z.: Lateral-torsional buckling behaviour of I-section beam--columns with one-sided rotation and warping restraint. Steel Constr. 9, 24–32 (2016). https://doi.org/10.1002/stco.201610009

    Article  Google Scholar 

  14. 14.

    Mohri, F., Damil, N., Potier-Ferry, M.: Buckling and lateral buckling interaction in thin-walled beam--column elements with mono-symmetric cross sections. Appl. Math. Model. 37, 3526–3540 (2013). https://doi.org/10.1016/j.apm.2012.07.053

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Lei, J., Li, L.: Combined web distortional and lateral-torsional buckling of partially restrained I-section beams. Int. J. Mech. Sci. 131–132, 107–112 (2017). https://doi.org/10.1016/j.ijmecsci.2017.06.057

    Article  Google Scholar 

  16. 16.

    Magnucka-Blandzi, E.: Critical state of a thin-walled beam under combined load. Appl. Math. Model. 33, 3093–3098 (2009). https://doi.org/10.1016/j.apm.2008.10.014

    Article  MATH  Google Scholar 

  17. 17.

    Mohri, F., Bouzerira, C., Potier-Ferry, M.: Lateral buckling of thin-walled beam--column elements under combined axial and bending loads. Thin-Walled Struct. 46, 290–302 (2008). https://doi.org/10.1016/j.tws.2007.07.017

    Article  Google Scholar 

  18. 18.

    Aristizabal-Ochoa, J.D.: Matrix method for stability and second-order analysis of Timoshenko beam--column structures with semi-rigid connections. Eng. Struct. 34, 289–302 (2012). https://doi.org/10.1016/j.engstruct.2011.09.010

    Article  Google Scholar 

  19. 19.

    Monsalve-Cano, J.F., Aristizábal-Ochoa, J.D.: Stability and free vibration analyses of orthotropic 3d beam--columns with singly symmetric section including shear effects. Eng. Struct. 113, 315–327 (2016). https://doi.org/10.1016/j.engstruct.2016.01.036

    Article  Google Scholar 

  20. 20.

    Zhu, Y., Hu, Y., Cheng, C.: Analysis of nonlinear stability and post-buckling for Euler-type beam--column structure. Appl. Math. Mech. 32, 719–728 (2011). https://doi.org/10.1007/s10483-011-1451-x

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Sophianopoulos, D.S., Papachristou, K.S.: In-plane stability of uniform steel beam--columns on a Pasternak foundation with zero end-shortening. Arch. Appl. Mech. 82, 1653–1662 (2012). https://doi.org/10.1007/s00419-012-0664-y

    Article  MATH  Google Scholar 

  22. 22.

    Yuan, Z., Wang, X.: Buckling and post-buckling analysis of extensible beam--columns by using the differential quadrature method. Comput. Math. with Appl. 62, 4499–4513 (2011). https://doi.org/10.1016/j.camwa.2011.10.029

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Kumar, M., Yadav, N.: Buckling analysis of a beam--column using multilayer perceptron neural network technique. J. Franklin Inst. 350, 3188–3204 (2013). https://doi.org/10.1016/j.jfranklin.2013.07.016

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Kucukler, M., Gardner, L., Macorini, L.: Flexural-torsional buckling assessment of steel beam--columns through a stiffness reduction method. Eng. Struct. 101, 662–676 (2015). https://doi.org/10.1016/j.engstruct.2015.07.041

    Article  Google Scholar 

  25. 25.

    White, D.W., Jeong, W.Y., Toğay, O.: Comprehensive stability design of planar steel members and framing systems via inelastic buckling analysis. Int. J. Steel Struct. 16, 1029–1042 (2016). https://doi.org/10.1007/s13296-016-0070-3

    Article  Google Scholar 

  26. 26.

    Ranjbaran, A., Ranjbaran, M.: State-based buckling analysis of beam-like structures. Arch. Appl. Mech. 87, 1555–1565 (2017). https://doi.org/10.1007/s00419-017-1273-6

    Article  MATH  Google Scholar 

  27. 27.

    Ritz, W.: Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. J. für die reine und Angew. Math. (Crelle’s Journal). 1909, 1–61 (1909). https://doi.org/10.1515/crll.1909.135.1

    Article  MATH  Google Scholar 

  28. 28.

    Leissa, A.W.: The historical bases of the Rayleigh and Ritz methods. J. Sound Vib. 287, 961–978 (2005). https://doi.org/10.1016/j.jsv.2004.12.021

    Article  Google Scholar 

  29. 29.

    Desai, Y.M., Eldho, T.I., Shah, A.H.: Finite Element Method with Applications in Engineering. Pearson Education, London (2011)

    Google Scholar 

  30. 30.

    Ozbasaran, H.: Optimal design of I-section beam--columns with stress, non-linear deflection and stability constraints. Eng. Struct. 171, 385–394 (2018). https://doi.org/10.1016/j.engstruct.2018.05.110

    Article  Google Scholar 

  31. 31.

    Ozbasaran, H., Aydin, R., Dogan, M.: An alternative design procedure for lateral-torsional buckling of cantilever I-beams. Thin-Walled Struct. 90, 235–242 (2015). https://doi.org/10.1016/j.tws.2015.01.021

    Article  Google Scholar 

  32. 32.

    Yilmaz, T., Kirac, N., Kilic, T.: Lateral-Torsional Buckling of European Wide Flange I-Section Beams. In: Proceedings of the 2nd World Congress on Civil, Structural and Environmental Engineering (CSEE’17), Barcelona (2017)

  33. 33.

    Yilmaz, T., Kirac, N.: Analytical and parametric investigations on lateral torsional buckling of European IPE and IPN beams. Int. J. Steel Struct. 17, 695–709 (2017). https://doi.org/10.1007/s13296-017-6024-6

    Article  Google Scholar 

  34. 34.

    Dou, C., Guo, Y.-L., Pi, Y.-L., Zhao, S.-Y., Bradford, M.A.: Effects of shape functions on flexural-torsional buckling of fixed circular arches. Eng. Struct. 59, 238–247 (2014). https://doi.org/10.1016/j.engstruct.2013.10.028

    Article  Google Scholar 

  35. 35.

    Liu, A., Lu, H., Fu, J., Pi, Y.: Lateral-torsional buckling of circular steel arches under arbitrary radial concentrated load. J. Struct. Eng. 143, 04017129 (2017). https://doi.org/10.1061/(ASCE)ST.1943-541X.0001858

    Article  Google Scholar 

  36. 36.

    Liu, A., Lu, H., Fu, J., Pi, Y.-L.: Lateral-torsional buckling of fixed circular arches having a thin-walled section under a central concentrated load. Thin-Walled Struct. 118, 46–55 (2017). https://doi.org/10.1016/j.tws.2017.05.002

    Article  Google Scholar 

  37. 37.

    Dou, C., Jiang, Z.-Q., Pi, Y.-L., Gao, W.: Elastic buckling of steel arches with discrete lateral braces. Eng. Struct. 156, 12–20 (2018). https://doi.org/10.1016/j.engstruct.2017.11.028

    Article  Google Scholar 

  38. 38.

    Ozbasaran, H., Yilmaz, T.: Shape optimization of tapered I-beams with lateral-torsional buckling, deflection and stress constraints. J. Constr. Steel Res. 143, 119–130 (2018). https://doi.org/10.1016/j.jcsr.2017.12.022

    Article  Google Scholar 

  39. 39.

    Schillinger, D., Papadopoulos, V., Bischoff, M., Papadrakakis, M.: Buckling analysis of imperfect I-section beam--columns with stochastic shell finite elements. Comput. Mech. 46, 495–510 (2010). https://doi.org/10.1007/s00466-010-0488-y

    Article  MATH  Google Scholar 

  40. 40.

    Ozbasaran, H.: Finite differences approach for calculating elastic lateral torsional buckling moment of cantilever I sections. Anadolu Univ. J. Sci. Technol. A Appl. Sci. Eng. 14, 143–152 (2013). https://doi.org/10.18038/BTD-A.89119

    Article  Google Scholar 

  41. 41.

    Ozbasaran, H.: A parametric study on lateral torsional buckling of European IPN and IPE cantilevers. Int. J. Civil Environ. Struct. Constr. Archit. Eng. 8, 783–788 (2014)

    Google Scholar 

  42. 42.

    Monterrubio, L.E., Ilanko, S.: Proof of convergence for a set of admissible functions for the Rayleigh–Ritz analysis of beams and plates and shells of rectangular planform. Comput. Struct. 147, 236–243 (2015). https://doi.org/10.1016/j.compstruc.2014.09.008

    Article  Google Scholar 

  43. 43.

    Moreno-García, P., dos Santos, J.V.A., Lopes, H.: A Review and Study on Ritz method admissible functions with emphasis on buckling and free vibration of isotropic and anisotropic beams and plates. Arch. Comput. Methods Eng. 25, 785–815 (2018). https://doi.org/10.1007/s11831-017-9214-7

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability. Dover Publications, Mineola (1961)

    Google Scholar 

  45. 45.

    Chajes, A.: Principles of Structural Stability Theory. Prentice-Hall, Upper Saddle River (1974)

    Google Scholar 

  46. 46.

    Yoo, C.H., Lee, S.C.: Stability of Structures: Principles and Applications. Butterworth-Heinemann, Oxford (2011)

    Google Scholar 

  47. 47.

    Vetyukov, Y.: Nonlinear Mechanics of Thin-Walled Structures. Springer, Vienna (2014)

    Google Scholar 

  48. 48.

    Chen, W.-F., Lui, E.M.: Structural Stability: Theory and implementation. PTR Prentice Hall, Upper Saddle River (1993)

    Google Scholar 

  49. 49.

    Trahair, N.S.: Flexural-Torsional Buckling of Structures. CRC, Boca Raton (1993)

    Google Scholar 

  50. 50.

    Galambos, T.V.: Guide to stability design criteria for metal structures. Wiley, London (1998)

    Google Scholar 

  51. 51.

    Chen, W.-F., Atsuta, T.: Theory of Beam--Columns, Volume 1: In-plane Behavior and Design. J. Ross Publishing, Plantation (2008)

    Google Scholar 

  52. 52.

    Chen, W.-F., Atsuta, T.: Theory of Beam--Columns, Volume 2: Space Behavior and Design. J. Ross Publishing, Plantation (2008)

    Google Scholar 

  53. 53.

    Galambos, T.V., Surovek, A.E.: Structural Stability of Steel: Concepts and Applications for Structural Engineers. Wiley, London (2008)

    Google Scholar 

  54. 54.

    Ziemian, R.D.: Guide to Stability Design Criteria for Metal Structures. Wiley, London (2010)

    Google Scholar 

  55. 55.

    Bažant, Z.P., Cedolin, L.: Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories. World Scientific, Singapore (2010)

    Google Scholar 

  56. 56.

    Pi, Y.L., Trahair, N.S.: Prebuckling deflections and lateral buckling. I: theory. J. Struct. Eng. 118, 2949–2966 (1992). https://doi.org/10.1061/(ASCE)0733-9445(1992)118:11(2949)

    Article  Google Scholar 

  57. 57.

    Pi, Y.L., Trahair, N.S.: Prebuckling deflections and lateral buckling. II: applications. J. Struct. Eng. 118, 2967–2985 (1992). https://doi.org/10.1061/(ASCE)0733-9445(1992)118:11(2967)

    Article  Google Scholar 

  58. 58.

    Ozbasaran, H.: An approximate lateral-torsional buckling mode function for Cantilever I-Beams. Int. J. Civil Environ. Struct. Constr. Archit. Eng. 9, 1401–1404 (2015)

    Google Scholar 

  59. 59.

    Burden, R.L., Faires, J.D.: Numerical Analysis. Cengage Learning, Boston (2010)

    Google Scholar 

  60. 60.

    Wolfram Research Inc.: Mathematica (2016)

Download references

Acknowledgements

This study is supported by the Eskisehir Osmangazi University’s Scientific Research Projects Department (Project ID: 2015-806).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hakan Ozbasaran.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A: Derivation of the exponential trial function for restraint configuration 2

The boundary conditions for torsional rotation of the cantilever restraint configuration 2 are \(\phi \left( 0 \right) =0\) and \(\phi _{,z} \left( 0 \right) =0\) (see Table 1). Equations A.1 and A.2 can be obtained by applying these boundary conditions to the exponential series (Eq. 2.6).

$$\begin{aligned}&\phi \left( 0 \right) =\mathop \sum \limits _{i=1}^n k_i i^{\frac{0}{L}}=k_1 +k_2 +\cdots +k_n =\mathop \sum \limits _{i=1}^n k_i =0 \end{aligned}$$
(A.1)
$$\begin{aligned}&\phi _{,z} \left( 0 \right) =\mathop \sum \limits _{i=2}^n \frac{\ln i}{L}k_i i^{\frac{0}{L}}=\frac{\ln 2}{L}k_2 +\frac{\ln 3}{L}k_3 +\cdots +\frac{\ln n}{L}k_n =\mathop \sum \limits _{i=2}^n \frac{\ln i}{L}k_i =0 \end{aligned}$$
(A.2)

The coefficients of the last terms (\(k_n \)) can be solved from Eqs. A.1 and A.2 as given in Eqs. A.3 and A.4, respectively.

$$\begin{aligned}&k_n =-k_1 -k_2 -\cdots -k_{n-1} =-\mathop \sum \limits _{i=1}^{n-1} k_i \end{aligned}$$
(A.3)
$$\begin{aligned}&k_n =-\frac{1}{\ln n}\left[ {\ln 2\,k_2 +\ln 3\,k_3 +\cdots +\ln \left( {n-1} \right) k_{n-1} } \right] =-\frac{1}{\ln n}\mathop \sum \limits _{i=2}^{n-1} \ln ik_i \end{aligned}$$
(A.4)

Since the solutions for the \(k_n \) coefficients must be equal, it can be shown that the coefficient \(k_{n-1} \) can be obtained in terms of the other coefficients from \(k_1 \) to \(k_{n-2} \). Equation A.5 can be written by separating the terms with the coefficient \(k_{n-1} \) in Eqs. A.3 and A.4.

$$\begin{aligned}&-\mathop \sum \limits _{i=1}^{n-2} k_i -k_{n-1} =-\frac{1}{\ln n}\mathop \sum \limits _{i=2}^{n-2} \ln i\,k_i -\frac{1}{\ln n}\ln \left( {n-1} \right) k_{n-1}\nonumber \\&\quad \rightarrow \frac{1}{\ln n}\mathop \sum \limits _{i=2}^{n-2} \ln i\,k_i -\mathop \sum \limits _{i=1}^{n-2} k_i =\left( {1-\frac{\ln \left( {n-1} \right) }{\ln n}} \right) k_{n-1} \end{aligned}$$
(A.5)

Arranging the series and simplifying Eq. A.5 by using the properties of the logarithms leads to the solution for \(k_{n-1} \) (Eq. A.6).

$$\begin{aligned} k_{n-1} =\frac{1}{\ln \left( {\frac{n}{n-1}} \right) }\left( {\mathop \sum \limits _{i=2}^{n-2} \ln i\,k_i -\ln n\mathop \sum \limits _{i=2}^{n-2} k_i -\ln n\,k_1 } \right) =-\frac{1}{\ln \left( {\frac{n-1}{n}} \right) }\mathop \sum \limits _{i=1}^{n-2} \ln \left( {\frac{i}{n}} \right) k_i \end{aligned}$$
(A.6)
Fig. 13
figure13

Buckling analysis of a doubly symmetric cantilever beam with a torsional restraint at \(z=0.4L\) using trigonometric series

Fig. 14
figure14

Buckling analysis of a singly symmetric cantilever column with a lateral restraint at \(z=0.8L\) using exponential series

Then, the coefficient \(k_n \) can be written (Eq. A.7) by substituting \(k_{n-1} \) (Eq. A.6) into Eq. A.4.

$$\begin{aligned}&k_n =-\frac{1}{\ln n}\mathop \sum \limits _{i=2}^{n-2} \ln i\,k_i -\frac{\ln \left( {n-1} \right) }{\ln n}k_{n-1}\nonumber \\&\quad \rightarrow k_n =-\frac{1}{\ln n}\left[ {\mathop \sum \limits _{i=2}^{n-2} \ln i\,k_i +\frac{\ln \left( {n-1} \right) }{\ln \left( {\frac{n}{n-1}} \right) }\mathop \sum \limits _{i=1}^{n-2} \ln \left( {\frac{i}{n}} \right) k_i } \right] \nonumber \\&\quad \rightarrow k_n =-\frac{1}{\ln \left( {\frac{n}{n-1}} \right) }\mathop \sum \limits _{i=1}^{n-2} \ln \left( {\frac{i}{n-1}} \right) k_i \end{aligned}$$
(A.7)

The last (\(k_n \)) and the one before the last (\(k_{n-1} \)) coefficients of the exponential series are obtained in terms of the other coefficients (\(k_1 ,k_2 \ldots ,k_{n-2} \)) to satisfy the boundary conditions. Substituting these new expressions for \(k_n \) and \(k_{n-1} \) into the exponential trial functions template (Eq. 2.6) gives Eq. A.8, which is a proper trial function for configuration 2.

$$\begin{aligned} \phi \left( z \right) =\mathop \sum \limits _{i=1}^{n-2} k_i i^{\frac{z}{L}}-\left( {n-1} \right) ^{\frac{z}{L}}\left[ {\frac{1}{\ln \left( {\frac{n-1}{n}} \right) }\mathop \sum \limits _{i=1}^{n-2} \ln \left( {\frac{i}{n}} \right) k_i } \right] -n^{\frac{z}{L}}\left[ {\frac{1}{\ln \left( {\frac{n}{n-1}} \right) }\mathop \sum \limits _{i=1}^{n-2} \ln \left( {\frac{i}{n-1}} \right) k_i } \right] \end{aligned}$$
(A.8)

Thus, it is shown that the last two coefficients (\(k_n \) and \(k_{n-1} \)) of the exponential series for the considered restraint configuration are dependent on the rest (\(k_1 ,k_2 \ldots ,k_{n-2} \)), no matter how many terms are considered. In other words, if it is desired to approximate the buckling eigenvalue and mode with n different coefficients, \(n+2\) terms must be considered. As the final touch, replacing n with \(n+2\) in Eq. A.8 gives the trial function presented in Table 4 for the configuration 2.

Appendix B: Selected cases

See Figs. 1314, and 15.

Fig. 15
figure15

Buckling analysis of a singly symmetric cantilever beam–column with torsional and lateral restraints at \(z=0.5L\) using power series

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ozbasaran, H. Convergence of the Rayleigh–Ritz Method for buckling analysis of arbitrarily configured I-section beam–columns. Arch Appl Mech 89, 2397–2414 (2019). https://doi.org/10.1007/s00419-019-01508-1

Download citation

Keywords

  • Buckling
  • I-section
  • Beam–column
  • Rayleigh–Ritz
  • Trial function
  • Intermediate restraint