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Exact transcendental stiffness matrices of general beam-columns embedded in elastic mediums

Sondipon Adhikari

Computers & Structures, Volume: 255, Start page: 106617

Swansea University Author: Sondipon Adhikari

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DOI (Published version): 10.1016/j.compstruc.2021.106617

Abstract

Stiffness matrices of beams embedded in an elastic medium and subjected to axial forces are considered. Both the bending and the axial deformations have been incorporated. Two approaches for deriving the element stiffness matrix analytically have been proposed. The first approach is based on the dir...

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Published in: Computers & Structures
ISSN: 0045-7949
Published: Elsevier BV 2021
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URI: https://cronfa.swan.ac.uk/Record/cronfa57434
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spelling 2021-08-19T16:24:39.4945588 v2 57434 2021-07-22 Exact transcendental stiffness matrices of general beam-columns embedded in elastic mediums 4ea84d67c4e414f5ccbd7593a40f04d3 Sondipon Adhikari Sondipon Adhikari true false 2021-07-22 FGSEN Stiffness matrices of beams embedded in an elastic medium and subjected to axial forces are considered. Both the bending and the axial deformations have been incorporated. Two approaches for deriving the element stiffness matrix analytically have been proposed. The first approach is based on the direct force–displacement relationship, whereas the second approach exploits shape functions within the finite element framework. The displacement function within the beam is obtained from the solution of the governing differential equation with suitable boundary conditions. Both approaches result in identical expressions when the exact transcendental displacement functions are used. Exact closed-form expressions of the elements of the stiffness matrix have been derived for the bending and axial deformation. Depending on the nature of the axial force and stiffness of the elastic medium, seven different cases are proposed for the bending stiffness matrix. A unified approach to the non-dimensional representation of the stiffness matrix elements and system parameters that are consistent across all the cases has been developed. Through Taylor-series expansions of the stiffness matrix coefficients, it is shown that the classical stiffness matrices appear as an approximation when only the first few terms of the series are retained. Numerical results shown in the paper explicitly quantify the error in using the classical stiffness compared to the exact stiffness matrix derived in the paper. The expressions derived here gives the most comprehensive and consistent description of the stiffness coefficients, which can be directly used in the context of finite element analysis over a wide range of parameter values. Journal Article Computers & Structures 255 106617 Elsevier BV 0045-7949 Beam-column, Elastic foundation, Bending deformation, Stiffness matrix, Exact solutions, Transcendental shape function 15 10 2021 2021-10-15 10.1016/j.compstruc.2021.106617 COLLEGE NANME Science and Engineering - Faculty COLLEGE CODE FGSEN Swansea University 2021-08-19T16:24:39.4945588 2021-07-22T08:47:27.2527630 Faculty of Science and Engineering School of Engineering and Applied Sciences - Uncategorised Sondipon Adhikari 1 57434__20444__50ce79c571204534be8faea0ccc6a14a.pdf 57434.pdf 2021-07-23T10:35:33.1194728 Output 1160091 application/pdf Accepted Manuscript true 2022-07-17T00:00:00.0000000 ©2021 All rights reserved. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution Non-Commercial No Derivatives License (CC-BY-NC-ND) true eng http://creativecommons.org/licenses/by-nc-nd/4.0/
title Exact transcendental stiffness matrices of general beam-columns embedded in elastic mediums
spellingShingle Exact transcendental stiffness matrices of general beam-columns embedded in elastic mediums
Sondipon Adhikari
title_short Exact transcendental stiffness matrices of general beam-columns embedded in elastic mediums
title_full Exact transcendental stiffness matrices of general beam-columns embedded in elastic mediums
title_fullStr Exact transcendental stiffness matrices of general beam-columns embedded in elastic mediums
title_full_unstemmed Exact transcendental stiffness matrices of general beam-columns embedded in elastic mediums
title_sort Exact transcendental stiffness matrices of general beam-columns embedded in elastic mediums
author_id_str_mv 4ea84d67c4e414f5ccbd7593a40f04d3
author_id_fullname_str_mv 4ea84d67c4e414f5ccbd7593a40f04d3_***_Sondipon Adhikari
author Sondipon Adhikari
author2 Sondipon Adhikari
format Journal article
container_title Computers & Structures
container_volume 255
container_start_page 106617
publishDate 2021
institution Swansea University
issn 0045-7949
doi_str_mv 10.1016/j.compstruc.2021.106617
publisher Elsevier BV
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Engineering and Applied Sciences - Uncategorised{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Engineering and Applied Sciences - Uncategorised
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description Stiffness matrices of beams embedded in an elastic medium and subjected to axial forces are considered. Both the bending and the axial deformations have been incorporated. Two approaches for deriving the element stiffness matrix analytically have been proposed. The first approach is based on the direct force–displacement relationship, whereas the second approach exploits shape functions within the finite element framework. The displacement function within the beam is obtained from the solution of the governing differential equation with suitable boundary conditions. Both approaches result in identical expressions when the exact transcendental displacement functions are used. Exact closed-form expressions of the elements of the stiffness matrix have been derived for the bending and axial deformation. Depending on the nature of the axial force and stiffness of the elastic medium, seven different cases are proposed for the bending stiffness matrix. A unified approach to the non-dimensional representation of the stiffness matrix elements and system parameters that are consistent across all the cases has been developed. Through Taylor-series expansions of the stiffness matrix coefficients, it is shown that the classical stiffness matrices appear as an approximation when only the first few terms of the series are retained. Numerical results shown in the paper explicitly quantify the error in using the classical stiffness compared to the exact stiffness matrix derived in the paper. The expressions derived here gives the most comprehensive and consistent description of the stiffness coefficients, which can be directly used in the context of finite element analysis over a wide range of parameter values.
published_date 2021-10-15T04:13:10Z
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