GENERAL ANALYTICAL SOLUTION OF AN ELASTIC BEAM UNDER VARYING LOADS WITH VALIDATION

Authors:

Hafeezullah Channa,Muhammad Mujtaba Shaikh,Kamran Malik,

DOI NO:

https://doi.org/10.26782/jmcms.2022.11.00004

Keywords:

Elastic beam,General analytical solution,Deflection,Slope,

Abstract

In this paper, we take into account the system of differential equations with boundary conditions of a fixed elastic beam model (EBM). Instead of finding a solution of EBM for a particularly specified load, which is the usual practice, we derive the general analytical solution of the model using techniques of integrations. The proposed general analytical solutions are not load-specific but can be used for any load without having to integrate successively again and again. We have considered load in a general polynomial form and obtained a general analytical solution for the deflection and slope parameters of EBM. Direct solutions have been determined under two types of loads: uniformly distributed load and linearly varying load. The formulation derived has been validated on the known cases of uniformly distributed load as appears frequently in the literature.

Refference:

I. Babak Mansoori, Ashkan Torabi, Arash Totonch (2020). ‘Numerical investigation of the reinforced concrete beams using cfrp rebar, steel sheets and gfrp’. J. Mech. Cont.& math. Sci., vol.-15, no.-3, pp 195-204.
II. Cheng XL, Han W, Huang HC (1997). Finite element methods for Timoshenko beam, circular arch and Reissner-Mindlin plate problems. J. Comput. Appl. Math.,79(2): 215- 234.
III. Li L (1990). ‘Discretization of the Timoshenko beam problem by the p and h/p versions of the finite element method’. Numer. Math., 57(1): 413-420.
IV. Malik, Kamran, Shaikh, Abdul Wasim and Shaikh, Muhammad Mujtaba. (2021). “An efficient finite difference scheme for the numerical solution of Timoshenko beam model. Journal of mechanics of continua and mathematical sciences”, 16 (5): 76-88..
V. Malik, Kamran, Shaikh, Muhammad Mujtaba and Shaikh, Abdul Wasim. (2021).“On exact analytical solutions of the Timoshenko beam model under uniform and variable loads. Journal of mechanics of continua and mathematical sciences”, 16 (5): 66-75.
VI. Timoshenko SP (1921). On the correction for shear of the differential equation for transverse Vibrations of prismatic bars, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 41(245): 744-746.
VII. Timoshenko, S. (1953). History of strength of materials. New York: McGraw-Hill Charles V. Jones, “The Unified Theory of Electrical Machines”, London, 1967.

View Download