An Analytical Study on the Buckling Behavior of Cracked Column Based on the Physical Property of the Dirac delta Function

Subject Areas : Analysis of Structure and Earthquake

Hadi  Dehghan Manshadi ¹ (Department of Civil Engineering, Yazd Branch, Islamic Azad University, Yazd, Iran)
Roham  Reyhani ² (M.S. Student, Department of Civil Engineering, Yazd Branch, Islamic Azad University, Yazd, Iran)
Mohammad Ali  Dashti Rahmat Abadi ³ (Assistant Professor and Scientific Secretary of Yazd Islamic Azad University)

Received: 2023-07-04 Accepted : 2023-10-27 Published : 2024-02-28

Keywords: Cracked column, Buckling, Dirac’s delta function, crack opening and closing, torsion spring ,

Abstract :

The Dirac delta function provides a simple and effective tool for representing point loads and singularities in structural problems, often leading to closed-form solutions. In this study, buckling of simple double-ended columns with one and two cracks has been analyzed analytically. Although in recent years this issue has attracted the attention of many researchers, the methods presented to solve the problem usually have a significant computational load. Therefore, in this study, a new approach has been used to solve the problem using the property of the Dirac's delta function. This approach simplifies the problem-solving process and significantly reduces the computational cost. Based on this, the crack is modeled with a bilateral behaviour via Dirac delta function. This model takes into account the crack closure effect on buckling behaviour of column by introducing a suitable switching criterion, which allows each crack to be open or closed depending on the sign of the axial strain at the crack centre. The proposed method was used to finding the buckling load, determining the effects of crack stiffness for both one and two-crack scenarios, and accounting for the effect(s) of crack opening and closing on the buckling load. For validation purposes, the finite element software SAP2000 was utilized.

References:

[1] Krauberger N, Bratina S, Saje M, Schnabl S, Planinc I. Inelastic buckling load of a locally weakened reinforced concrete column. Engineering Structures. 2012 Jan 1;34:278-88.
https://doi.org/10.1016/j.engstruct.2011.09.006

[2] Vadillo G, Loya JA, Fernández-Sáez J. First order solutions for the buckling loads of weakened Timoshenko columns. Computers & Mathematics with Applications. 2012 Oct 1;64(8):2395-407.
https://doi.org/10.1016/j.camwa.2012.05.009

[3] Nikpour K. Buckling of cracked composite columns. International Journal of Solids and Structures. 1990 Jan 1;26(12):1371-86.
https://doi.org/10.1016/0020-7683(90)90084-9

[4] Dehghani Mahmoodabadi MA, Dehghan Manshadi S.H., Ranjbaran A, Esfandiari M.J., Dehghan Manshadi S.M. Analysis of localization in the buckling of a two-span column with elastic end connections. European Journal of Environmental and Civil Engineering. 2018 Jul 3;22(7):811-35.
https://doi.org/10.1080/19648189.2016.1219879

[5] Kisa M. Vibration and stability of multi-cracked beams under compressive axial loading. International Journal of the Physical Sciences. 2011 Jun 4;6(11):2681-96.
https://doi.org/10.5897/IJPS11.493

[6] Yazdchi K, Gowhari Anaraki AR. Carrying capacity of edge-cracked columns under concentric vertical loads. Acta mechanica. 2008 Jun;198(1-2):1-9.
https://doi.org/10.1007/s00707-007-0523-z

[7] Toygar ME, Kıral Z, Sayman O, Arman Y, Özen M. Effect of interface crack on lateral buckling behavior and free vibration response of a sandwich composite beam. Journal of composite materials. 2013 Jul;47(15):1843-51.
https://doi.org/10.1177/0021998312451611

[8] Okamura H, Liu HW, Chu CS, Liebowitz H. A cracked column under compression. Engineering Fracture Mechanics. 1969 Apr 1;1(3):547-64.
https://doi.org/10.1016/0013-7944(69)90011-3

[9] Zhou L, Huang Y. Crack effect on the elastic buckling behavior of axially and eccentrically loaded columns. Structural engineering and mechanics: An international journal. 2006;22(2):169-84.
https://doi.org/10.12989/sem.2006.22.2.169

[10] Challamel N, Lanos C, Casandjian C. Localization in the buckling or in the vibration of a two-span weakened column. Engineering Structures. 2006 Apr 1;28(5):776-82.
https://doi.org/10.1016/j.engstruct.2005.10.005

[11] Fu C, Yang X. Bending of a Viscoelastic Timoshenko Cracked Beam Based on Equivalent Viscoelastic Spring Models. Advances in Civil Engineering. 2021 Oct 21;2021:1-6.
https://doi.org/10.1155/2021/8663213

[12] Soliman ES. Investigation of modal and damage parameters of isotropic cantilever beam under double-sided crack. Journal of Failure Analysis and Prevention. 2020 Feb;20:120-36.
https://doi.org/10.1007/s11668-020-00806-z

]13[ دهقان منشادی، هادی؛ علوی نسب، عماد؛ امیری، حمیدرضا. تاثیر ترک روی رفتار کمانشی ستون ضعیف شده. نشریه مهندسی سازه و ساخت;1400، 8 (شماره ویژه 1): 493-506.
doi: 10.22065/jsce.2020.208226.1998

[14] Zill DG. Advanced engineering mathematics. Jones & Bartlett Learning; 2020 Dec 1.