Static and Dynamic Micropolar Linear Elastic Beam Finite Element Formulation, Implementation, and Analysis
Publication: Journal of Engineering Mechanics
Volume 141, Issue 8
Abstract
Starting with static and dynamic micropolar linear plane stress elasticity, and applying Timoshenko beam kinematics with axial stretch, a mixed micropolar small-strain beam finite element (FE) formulation results. The mixed formulation is shown to be convergent upon mesh refinement under static and dynamic loading. The acceleration form of the Newmark family of time integration methods is applied to integrate the coupled hyperbolic linear governing equations. Instantaneous axial and transverse step forces are applied and released to analyze the free longitudinal and transverse vibrations with the mixed formulation FE implementation. The transverse displacement and rotational degrees of freedom (DOF) are coupled, but the axial displacement is decoupled from the rotational DOF because the first area moment of inertia is zero. Applied sinusoidal axial and transverse forces lead to axial and transverse displacement and rotational wave patterns that are a combination of low and high frequency waves. The effect of length scale on elastic couple modulus and spin inertia is demonstrated, which shows a transverse and rotational stiffening through upon increasing , yet a decreasing frequency as also increases with .
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Acknowledgments
Funding for this research was provided by National Science Foundation grant CMMI-0700648, Army Research Office grant W911NF-09-1-0111, the Army Research Laboratory, and Office of Naval Research grant N00014-11-1-0691. This funding is gratefully acknowledged.
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© 2015 American Society of Civil Engineers.
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Received: Sep 11, 2014
Accepted: Nov 25, 2014
Published online: Apr 29, 2015
Published in print: Aug 1, 2015
Discussion open until: Sep 29, 2015
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