Implementation of Haar wavelet, higher order Haar wavelet, and differential quadrature methods on buckling response of strain gradient nonlocal beam embedded in an elastic medium

  • Subrat Kumar Jena
  • S. ChakravertyEmail author
  • Mohammad Malikan
Original Article


The present investigation is focused on the buckling behavior of strain gradient nonlocal beam embedded in Winkler elastic foundation. The first-order strain gradient model has been combined with the Euler–Bernoulli beam theory to formulate the proposed model using Hamilton’s principle. Three numerically efficient methods, namely Haar wavelet method (HWM), higher order Haar wavelet method (HOHWM), and differential quadrature method (DQM) are employed to analyze the buckling characteristics of the strain gradient nonlocal beam. The impacts of several parameters such as nonlocal parameter, strain gradient parameter, and Winkler modulus parameter on critical buckling loads are studied effectively. The basic ideas of the numerical methods, viz. HWM, HOHWM, and DQM are presented comprehensively. Also, a comparative study has been conducted to explore the effectiveness and applicability of all the three numerical methods in terms of convergence study. Finally, the results, obtained by this investigation, are validated properly with other works published earlier.


Buckling behavior Strain gradient model Nonlocal beam HWM HOHWM DQM 



The first two authors are very much thankful to the Defence Research and Development Organization(DRDO), New Delhi, India (Sanction Code: DG/TM/ERIPR/GIA/17-18/0129/020) for the funding to carry out the present research work.


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© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology RourkelaRourkelaIndia
  2. 2.Department of Mechanics of Materials and Structures, Faculty of Civil and Environmental EngineeringGdansk University of TechnologyGdanskPoland
  3. 3.Department of Mechanical Engineering, Faculty of EngineeringIslamic Azad University, Mashhad BranchMashhadIran

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