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On the energy theorems of elastic pin-jointed frames.

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Title: On the energy theorems of elastic pin-jointed frames.
Authors: Sands, C. M.1 c.sands@abdn.ac.uk, Chandler, H. W.1
Source: International Journal of Mechanical Engineering Education. Jul2010, Vol. 38 Issue 3, p260-276. 17p. 2 Black and White Photographs, 1 Illustration, 2 Diagrams, 5 Graphs.
Subjects: Virtual work, Frames (Vector analysis), Mathematical inequalities, Thermal expansion, Potential energy surfaces, Elasticity
Abstract: This paper shows, using only elementary mathematics, that, for a pin-jointed structure with non-linear elastic members, the classical minimum principles can be derived from one fundamental inequality. This is constructed from: (1) Young's inequality, which is applied to the force-extension law, which has been expanded to include terms for thermal expansion or lack of fit; (2) the principle of virtual work, which is first demonstrated using a compact version of an existing proof. The method clearly shows why the theorems of minimum total potential energy and minimum complementary energy are extremum principles and not just stationary ones. The energy theorems of Castigliano and Engesser are also derived from the same inequality. The case of Hookean behaviour is particularly simple, as Young's inequality reduces to the sum of the squares of the error in satisfying the force-extension law. The stiffness method for linear elastic pin-jointed frames is established without the need to use rotation matrices to generate the set of equations for the full frame. Matlab code for the analysis of a three-dimensional pin-jointed frame is included in an appendix. [ABSTRACT FROM AUTHOR]
Copyright of International Journal of Mechanical Engineering Education is the property of Sage Publications Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22International+Journal+of+Mechanical+Engineering+Education%22">International Journal of Mechanical Engineering Education</searchLink>. Jul2010, Vol. 38 Issue 3, p260-276. 17p. 2 Black and White Photographs, 1 Illustration, 2 Diagrams, 5 Graphs.
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  Data: <searchLink fieldCode="DE" term="%22Virtual+work%22">Virtual work</searchLink><br /><searchLink fieldCode="DE" term="%22Frames+%28Vector+analysis%29%22">Frames (Vector analysis)</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+inequalities%22">Mathematical inequalities</searchLink><br /><searchLink fieldCode="DE" term="%22Thermal+expansion%22">Thermal expansion</searchLink><br /><searchLink fieldCode="DE" term="%22Potential+energy+surfaces%22">Potential energy surfaces</searchLink><br /><searchLink fieldCode="DE" term="%22Elasticity%22">Elasticity</searchLink>
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  Data: This paper shows, using only elementary mathematics, that, for a pin-jointed structure with non-linear elastic members, the classical minimum principles can be derived from one fundamental inequality. This is constructed from: (1) Young's inequality, which is applied to the force-extension law, which has been expanded to include terms for thermal expansion or lack of fit; (2) the principle of virtual work, which is first demonstrated using a compact version of an existing proof. The method clearly shows why the theorems of minimum total potential energy and minimum complementary energy are extremum principles and not just stationary ones. The energy theorems of Castigliano and Engesser are also derived from the same inequality. The case of Hookean behaviour is particularly simple, as Young's inequality reduces to the sum of the squares of the error in satisfying the force-extension law. The stiffness method for linear elastic pin-jointed frames is established without the need to use rotation matrices to generate the set of equations for the full frame. Matlab code for the analysis of a three-dimensional pin-jointed frame is included in an appendix. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of International Journal of Mechanical Engineering Education is the property of Sage Publications Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Text: English
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      – SubjectFull: Mathematical inequalities
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      – SubjectFull: Thermal expansion
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      – SubjectFull: Potential energy surfaces
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              Text: Jul2010
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