Numerically Stable Solutions to the State Equations for Structural Analyses
Publication: Journal of Engineering Mechanics
Volume 146, Issue 3
Abstract
The state space method has been widely used to analyze the static and dynamic characteristics of homogeneous, laminated, functionally graded, or even intelligent structures. However, the solution of the state equation using the traditional transfer matrix generally encounters the problem of numerical instability. This work, therefore, derives the general solution to the state equation by making use of similarity transformation to convert the system matrix into a matrix in Jordan canonical form (including the diagonal matrix as a special case), so as to avoid the previously stated problem. A special form of the exponential function is also introduced according to the characteristics of the eigenvalues of the system matrix. Furthermore, the undetermined coefficients in the general solution—rather than the original state variables—are considered as the primary unknowns. Consequently, a new solution with numerical robustness to the state equation is obtained. Finally, numerical examples for the free vibration analyses of beams and plates as well as interfacial shear stress analysis of fiber-reinforced polymer (FRP)-strengthened concrete beams are presented to verify that the proposed procedure can circumvent numerical instability completely.
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Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. 51478422) and the Natural Science Foundation of Zhejiang province, China (No. LY18A020006).
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Published In
Journal of Engineering Mechanics
Volume 146 • Issue 3 • March 2020
Copyright
©2019 American Society of Civil Engineers.
History
Received: Jan 2, 2019
Accepted: May 3, 2019
Published online: Dec 21, 2019
Published in print: Mar 1, 2020
Discussion open until: May 21, 2020
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