A note on the integrability of a remarkable static Euler-Bernoulli beam equation

A. Fatima; Ashfaque H. Bokhari; F. M. Mahomed; F. D. Zaman

doi:10.1007/s10665-012-9583-8

A note on the integrability of a remarkable static Euler-Bernoulli beam equation

A. Fatima
, Ashfaque H. Bokhari
, F. M. Mahomed
, F. D. Zaman

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

It has recently been shown that the fourth-order static Euler-Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, in the maximal case has three symmetries. This corresponds to the negative fractional power law y ^-5/3, and the equation has the nonsolvable algebra sl(2, R). We obtain new two- and three-parameter families of exact solutions when the equation has this symmetry algebra. This is studied via the symmetry classification of the three-parameter family of second-order ordinary differential equations that arises from the relationship among the Noether integrals. In addition, we present a complete symmetry classification of the second-order family of equations. Hence the admittance of sl(2, R) remarkably allows for a three-parameter family of exact solutions for the static beam equation with load a fractional power law y ^-5/3.

Original language	English
Pages (from-to)	101-108
Number of pages	8
Journal	Journal of Engineering Mathematics
Volume	82
Issue number	1
DOIs	https://doi.org/10.1007/s10665-012-9583-8
State	Published - Oct 2013

Bibliographical note

Funding Information:
Acknowledgments We are grateful to the Department of Mathematics and Statistics at the King Fahd University of Petroleum and Minerals, Saudi Arabia, for providing enabling facilities. FM also thanks the Department for its hospitality during the time this work was completed. AF acknowledges an enabling Ph.D. scholarship from the University of the Witwatersrand.

Keywords

Exact solution
Fourth-order ordinary differential equation
Static beam equation
Symmetry classification

ASJC Scopus subject areas

General Mathematics
General Engineering

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10.1007/s10665-012-9583-8

Cite this

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title = "A note on the integrability of a remarkable static Euler-Bernoulli beam equation",

abstract = "It has recently been shown that the fourth-order static Euler-Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, in the maximal case has three symmetries. This corresponds to the negative fractional power law y -5/3, and the equation has the nonsolvable algebra sl(2, R). We obtain new two- and three-parameter families of exact solutions when the equation has this symmetry algebra. This is studied via the symmetry classification of the three-parameter family of second-order ordinary differential equations that arises from the relationship among the Noether integrals. In addition, we present a complete symmetry classification of the second-order family of equations. Hence the admittance of sl(2, R) remarkably allows for a three-parameter family of exact solutions for the static beam equation with load a fractional power law y -5/3.",

keywords = "Exact solution, Fourth-order ordinary differential equation, Static beam equation, Symmetry classification",

author = "A. Fatima and Bokhari, \{Ashfaque H.\} and Mahomed, \{F. M.\} and Zaman, \{F. D.\}",

note = "Funding Information: Acknowledgments We are grateful to the Department of Mathematics and Statistics at the King Fahd University of Petroleum and Minerals, Saudi Arabia, for providing enabling facilities. FM also thanks the Department for its hospitality during the time this work was completed. AF acknowledges an enabling Ph.D. scholarship from the University of the Witwatersrand.",

year = "2013",

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doi = "10.1007/s10665-012-9583-8",

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AU - Fatima, A.

AU - Bokhari, Ashfaque H.

AU - Mahomed, F. M.

AU - Zaman, F. D.

N1 - Funding Information: Acknowledgments We are grateful to the Department of Mathematics and Statistics at the King Fahd University of Petroleum and Minerals, Saudi Arabia, for providing enabling facilities. FM also thanks the Department for its hospitality during the time this work was completed. AF acknowledges an enabling Ph.D. scholarship from the University of the Witwatersrand.

PY - 2013/10

Y1 - 2013/10

N2 - It has recently been shown that the fourth-order static Euler-Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, in the maximal case has three symmetries. This corresponds to the negative fractional power law y -5/3, and the equation has the nonsolvable algebra sl(2, R). We obtain new two- and three-parameter families of exact solutions when the equation has this symmetry algebra. This is studied via the symmetry classification of the three-parameter family of second-order ordinary differential equations that arises from the relationship among the Noether integrals. In addition, we present a complete symmetry classification of the second-order family of equations. Hence the admittance of sl(2, R) remarkably allows for a three-parameter family of exact solutions for the static beam equation with load a fractional power law y -5/3.

AB - It has recently been shown that the fourth-order static Euler-Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, in the maximal case has three symmetries. This corresponds to the negative fractional power law y -5/3, and the equation has the nonsolvable algebra sl(2, R). We obtain new two- and three-parameter families of exact solutions when the equation has this symmetry algebra. This is studied via the symmetry classification of the three-parameter family of second-order ordinary differential equations that arises from the relationship among the Noether integrals. In addition, we present a complete symmetry classification of the second-order family of equations. Hence the admittance of sl(2, R) remarkably allows for a three-parameter family of exact solutions for the static beam equation with load a fractional power law y -5/3.

KW - Exact solution

KW - Fourth-order ordinary differential equation

KW - Static beam equation

KW - Symmetry classification

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DO - 10.1007/s10665-012-9583-8

M3 - Article

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SN - 0022-0833

VL - 82

SP - 101

EP - 108

JO - Journal of Engineering Mathematics

JF - Journal of Engineering Mathematics

IS - 1

ER -

A note on the integrability of a remarkable static Euler-Bernoulli beam equation

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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