Buckling of Multi-Span Frames with Newmark Method

Ashraf Badir

doi:10.56748/ejse.26835

Authors

Ashraf Badir Florida Gulf Coast University

DOI:

https://doi.org/10.56748/ejse.26835

Keywords:

Newmark method, buckling loads, stability, effective length, sway frames, non-prismatic columns

Abstract

The Newmark’s numerical method of computing deflections, moments and buckling loads of isolated columns is extended for the analysis of elastic buckling loads and buckling modes of prismatic and non-prismatic single story, multi-span frames with combinations of hinged and fixed columns. Step-by-step description of the developed procedure is presented, using statics equilibrium, slope deflection equations and boundary conditions. The elastic line of the buckling mode is determined as a major part of the solution, and the numerical procedure is used to calculate the buckling loads and the columns’ effective length factor for multi-span frames. The most favorable variation of cross-section of tapered frame columns is calculated, giving the maximum possible elastic critical load of the frame for constant columns’ volume.

Downloads

Download data is not yet available.

References

ACI 318 (2025), Building code requirements for structural concrete and commentary, American Concrete Institute; Farmington Hills, MI, USA.

AISC (2023), Steel Construction Manual 16th ed., American Institute of Steel Construction, USA.

Badir, A. (2011), “Elastic buckling loads of hinged frames by the Newmark method,” International Journal of Applied Sciences and Technology, 1(3), 14-21.

Badir, A. (2020), “Elastic buckling loads of fixed frames by the Newmark method,” Electronic Journal of Structural Engineering, 20(1), 1-5.

Bazeos, N. and Karabalis D. “Efficient Computation of buckling loads for plane steel frames with tapered members,” Engineering Structures 28, 771-775.

Behjati-Avval, M. and Kalatjari, V. (2015) “Determination of the effective length factor for non-prismatic columns in two-span gable frames,” Current World Environment, 10(special issue 1), 421-428,

Belabed, Z., Tounsi, A., Bousahla, A.A. and Yaylacı, M. (2024) “Accurate free and forced vibration behavior prediction of functionally graded sandwich beams with variable cross-section: A finite element assessment,” Mechanics Based Design of Structures and Machines, 52(11), 9144–9177.

Belabed, Z., Bousahla, A.A. and Tounsi, A. (2024), “Vibrational and elastic stability responses of functionally graded carbon nanotube reinforced nanocomposite beams via a new Quasi-3D finite element model,” Computers and Concrete, 34(5), 625–648.

Belabed, Z., Tounsi, A., Bousahla, A.A., Khedher, K.M., Salem and M.A. (2024), “Mechanical behavior analysis of FG-CNTRC porous beams resting on Winkler and Pasternak elastic foundations: A finite element approach,” Computers and Concrete, 34(4), 447–476.

Belabed, Z., Tounsi, A., Bousahla, A.A., Bourada, M. and Al-Osta, M.A. (2024) “Free vibration analysis of Bi-Directional Functionally Graded Beams using a simple and efficient finite element model,” Structural Engineering and Mechanics, 90(3), 233–252.

Belabed, Z., Tounsi, A., Al-Osta, M.A. and Minh, H.L. (2024) “On the elastic stability and free vibration responses of functionally graded porous beams resting on Winkler-Pasternak foundations via finite element computation,” Geomechanics and Engineering, 36(2), 183–204.

Bentrar, H., Chorfi, S.M., Belalia, S.A., Tounsi, A., Ghazwani, M.H. and Alnujaie, A. (2023), “Effect of porosity distribution on free vibration of functionally graded sandwich plate using the P-version of the finite element method,” Structural Engineering and Mechanics, 88(6), 551–567

Bradford, M. A. and Abdoli Yazdi, N. (1999) “ A Newmark-based method for the stability of columns,” Computers & Structures, 71, 689-700.

BSI (2005) Design of steel structures general rules for buildings, London 1993-1

Duan L and Chen WF (1988) “Effective length factor for columns in braced frames,” Journal of Structural Engineering, ASCE 114(10), 2357–2370.

Duan L and Chen WF (1989) Effective length factor for columns in unbraced frames. Journal of Structural Engineering: ASCE 115(l): 149–165.

Duan, L. and Chen, W.F. (1999) “Effective Length Factors of Compression Members” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999

Euler, L (1778), “Determinatio onerum, quae columnae gestare valent,” Acta Academiae Scientiarum Petropolitanae 1(a), 121-145 (in Latin).

Godoy, L.A. and Elishakoff, I. (2020), “The experimental contribution of Petrus Van Musschenbroek to the discovery of a buckling formula in the early 18th century”, International Journal of Structural Stability and Dynamics, 20(5), 1-28.

Julian, O.G. and Lawrence, L.S. (1959), Notes on J and L nomographs for determination of effective lengths (unpublished report).

Katiyar, V., Gupta, A. and Tounsi, A. (2022) “Microstructural/geometric imperfection sensitivity on the vibration response of geometrically discontinuous bi-directional functionally graded plates (2D-FGPs) with partial supports by using FEM,” Steel and Composite Structures, 45(5), 621–640.

Kishi N., Chen W.F. and Goto, Y. (1997) “Effective length factor of columns in semi-rigid and unbraced frames,” Journal of Structural Engineering: ASCE 123(3): 313–320.

Krynicki, E.J. and Mazurkiewicz, Z.E. (1964), “Frames of solid bars of varying cross sections,” J. Struct. Div., ASCE, 90(ST4), 145-174.

Lakhdar, Z., Chorfi, S.M., Belalia, S.A., Khedher, K.M., Alluqmani, A.E., Tounsi, A. and Yaylacı, M. (2024) “Free vibration and bending analysis of porous bi-directional FGM sandwich shell using a TSDT p-version finite element method,”. Acta Mechanica, 235, 3657–3686.

Lee, B. K. and Lee, J. K (2021) “Buckling of Tapered Heavy Columns with Constant Volume,” Mathematics, 9(6), 657.

Li, Q., Zou, A. and Zhang, H. (2016), “A simplified method for stability analysis of multi-story frames considering vertical interactions between stories.” Advances in Structural Engineering, 19(4), 599-610.

Musschenbroek, P. (1729) Physicæ experimentales, et geometricæ, de magnete, tuborum capillarium vitreorumque speculorum attractione, magnitudine térrea, cohaerentia corporum firmorum. Dissertationes, ut et Ephemerides meteorologicae Ultrajectinae, Lugdun Batavonum, apud Samuelem Luchtmans, (in Latin).

Naidoo, D. and Li, K. (2019), “Optimization of no sway plane rigid frames against buckling,” Journal of Civil Engineering Research, 9(1), 1-15.

Newmark, N.M. (1943), “Numerical procedure for computing deflections, moments, and buckling loads,” Trans. ASCE. 108,1161-1943.

Nikolić, A. and Šalinić, S. (2017), “Buckling Analysis of non-prismatic columns: A rigid multibody approach,” Engineering Structures, 143, 511-521.

Oppe, M., Muller, C., and Iles, C. (2005) NCCI: buckling lengths of columns; rigorous approach. London: Access Steel.

Pomares JC, Pereiro-Barceló J, González A, Aguilar R. Safety Issues in Buckling of Steel Structures by Improving Accuracy of Historical Methods. Int J Environ Res Public Health. 2021 Nov 22;18(22):12253. doi: 10.3390/ijerph182212253. PMID: 34832006; PMCID: PMC8618944.

Rayhan, S. B., Rahman, M. M., Sultana, J., Szávai, S., & Varga, G. (2025), “Finite Element and Machine Learning-Based Prediction of Buckling Strength in Additively Manufactured Lattice Stiffened Panels,” Metals, 15(1), 81.

Rezaiee-Panjad, M., Shahabian, F. and Bambaeechee, M. (2016), “Stability of non-prismatic frames with flexible connections and elastic supports,” KSCE Journal of Civil Engineering, 20(2), 832-846.

Schilling, Charles G. (1983). "Buckling of One-Story Frames," Engineering Journal, American Institute of Steel Construction, Vol. 20, pp. 49-57.

Tati, A. (2021), “Finite Element Analysis of thermal and mechanical buckling behavior of functionally graded plates,” Archive of Applied Mechanics, vol. 91, pp. 4571–4587

Tian, W., Hao, J. and Zhong, W (2021a) “Buckling of stepped columns considering the interaction effect among columns”, Journal of construction steel research, 177

Tian, W., Sun, J, and Hao, J. (2021b) “Effective length factors of three- and two-segment stepped columns”, Journal of construction steel research, 181

Timoshenko, S.P. and Gere, J.M. (1964) Theory Of Elastic Stability, (2nd ed.) McGraw-Hill, New York, NY, USA.

Tounsi, A., Belabed, Z., Bounouara, F., Balubaid, M., Mahmoud, S.R., Bousahla, A.A., and Tounsi, A. (2024), “A finite element approach for forced dynamical responses of porous FG nanocomposite beams resting on viscoelastic foundations,”

International Journal of Structural Stability and Dynamics, Online Ready.

Weber, A., Orr, J.J., Shepherd, P. and Crothers, K. (2015) “The effective length of columns in multi-storey frames”, Engineering Structures, 102, 132-143.