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

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.

Automated summary

This study focuses on the buckling behavior of strain gradient nonlocal beams on Winkler elastic foundations. By combining the first-order strain gradient model with the Euler-Bernoulli beam theory using Hamilton's principle, the study employs three numerical methods (HWM, HOHWM, and DQM) to analyze buckling characteristics and study the impact of various parameters on critical buckling loads. A comprehensive presentation of the numerical methods and a comparative study on their convergence demonstrate the effectiveness and applicability of the methods in analyzing the buckling behavior. The findings of this investigation are properly validated with previous works.