The exact element stiffness matrices of stochastically parametered beams

Adhikari, S.; Mukherjee, Shuvajit

doi:10.1016/j.probengmech.2022.103317

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The exact element stiffness matrices of stochastically parametered beams

S. Adhikari

, Shuvajit Mukherjee

Probabilistic Engineering Mechanics, Volume: 69, Start page: 103317

Swansea University Author: Shuvajit Mukherjee

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DOI (Published version): 10.1016/j.probengmech.2022.103317

Abstract

Stiffness matrices of beams with stochastic distributed parameters modelled by random fields are considered. In stochastic finite element analysis, deterministic shape functions are traditionally employed to derive stiffness matrices using the variational principle. Such matrices are not exact becau...

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Published in:	Probabilistic Engineering Mechanics
ISSN:	0266-8920
Published:	Elsevier BV 2022
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URI:	https://cronfa.swan.ac.uk/Record/cronfa60307
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spelling	2022-07-15T13:53:28.1724449 v2 60307 2022-06-23 The exact element stiffness matrices of stochastically parametered beams 0d6adf4c1873dddc78ba26dba8b1c04f Shuvajit Mukherjee Shuvajit Mukherjee true false 2022-06-23 AERO Stiffness matrices of beams with stochastic distributed parameters modelled by random fields are considered. In stochastic finite element analysis, deterministic shape functions are traditionally employed to derive stiffness matrices using the variational principle. Such matrices are not exact because the deterministic shape functions are not derived from the exact solution of the governing stochastic partial differential equation with the relevant boundary conditions. This paper proposes an analytical method based on Castigliano’s approach for a beam element with general spatially varying parameters. This gives the exact and a simple closed-form expression of the stiffness matrix in terms of certain integrals of the spatially varying function. The expressions are valid for any integrable random fields. It is shown that the exact element stiffness matrix of a stochastically parametered beam can be expressed by three basic random variables. Analytical expressions of the random variables and their associated coefficient matrices are derived for two cases: when the bending rigidity is a random field and when the bending flexibility is a random field. It is theoretically proved that the conventional stochastic element stiffness matrix is a first-order perturbation approximation to the exact expression. A sampling method to obtain the basic random variables using the Karhunen–Loève expansion is proposed. Results from the exact stiffness matrices are compared with the approximate conventional stiffness matrix. Gaussian and uniform random fields with different correlation lengths are used to illustrate the numerical results. The exact closed-form analytical expression of the element stiffness matrix derived here can be used for benchmarking future numerical methods. Journal Article Probabilistic Engineering Mechanics 69 103317 Elsevier BV 0266-8920 Random field; Euler–Bernoulli beams; Stiffness matrix; Finite element method; Stochastic mechanics 1 7 2022 2022-07-01 10.1016/j.probengmech.2022.103317 COLLEGE NANME Aerospace Engineering COLLEGE CODE AERO Swansea University 2022-07-15T13:53:28.1724449 2022-06-23T08:36:00.9111493 Faculty of Science and Engineering School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Aerospace Engineering S. Adhikari 0000-0003-4181-3457 1 Shuvajit Mukherjee 2 60307__24613__f1a246ba6de947cc9f3e263f1f8c5a81.pdf 60307.pdf 2022-07-15T13:51:57.8797890 Output 1141897 application/pdf Version of Record true © 2022 The Author(s). This is an open access article under the CC BY license true eng http://creativecommons.org/licenses/by/4.0/
title	The exact element stiffness matrices of stochastically parametered beams
spellingShingle	The exact element stiffness matrices of stochastically parametered beams Shuvajit Mukherjee
title_short	The exact element stiffness matrices of stochastically parametered beams
title_full	The exact element stiffness matrices of stochastically parametered beams
title_fullStr	The exact element stiffness matrices of stochastically parametered beams
title_full_unstemmed	The exact element stiffness matrices of stochastically parametered beams
title_sort	The exact element stiffness matrices of stochastically parametered beams
author_id_str_mv	0d6adf4c1873dddc78ba26dba8b1c04f
author_id_fullname_str_mv	0d6adf4c1873dddc78ba26dba8b1c04f_***_Shuvajit Mukherjee
author	Shuvajit Mukherjee
author2	S. Adhikari Shuvajit Mukherjee
format	Journal article
container_title	Probabilistic Engineering Mechanics
container_volume	69
container_start_page	103317
publishDate	2022
institution	Swansea University
issn	0266-8920
doi_str_mv	10.1016/j.probengmech.2022.103317
publisher	Elsevier BV
college_str	Faculty of Science and Engineering
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hierarchy_top_title	Faculty of Science and Engineering
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department_str	School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Aerospace Engineering{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Aerospace, Civil, Electrical, General and Mechanical Engineering - Aerospace Engineering
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description	Stiffness matrices of beams with stochastic distributed parameters modelled by random fields are considered. In stochastic finite element analysis, deterministic shape functions are traditionally employed to derive stiffness matrices using the variational principle. Such matrices are not exact because the deterministic shape functions are not derived from the exact solution of the governing stochastic partial differential equation with the relevant boundary conditions. This paper proposes an analytical method based on Castigliano’s approach for a beam element with general spatially varying parameters. This gives the exact and a simple closed-form expression of the stiffness matrix in terms of certain integrals of the spatially varying function. The expressions are valid for any integrable random fields. It is shown that the exact element stiffness matrix of a stochastically parametered beam can be expressed by three basic random variables. Analytical expressions of the random variables and their associated coefficient matrices are derived for two cases: when the bending rigidity is a random field and when the bending flexibility is a random field. It is theoretically proved that the conventional stochastic element stiffness matrix is a first-order perturbation approximation to the exact expression. A sampling method to obtain the basic random variables using the Karhunen–Loève expansion is proposed. Results from the exact stiffness matrices are compared with the approximate conventional stiffness matrix. Gaussian and uniform random fields with different correlation lengths are used to illustrate the numerical results. The exact closed-form analytical expression of the element stiffness matrix derived here can be used for benchmarking future numerical methods.
published_date	2022-07-01T04:18:19Z
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score	11.012678

The exact element stiffness matrices of stochastically parametered beams

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