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[1] Crandall, S.H., Engineering Analysis, McGraw-Hill: New York, 1956.
[2] Hetényi, M., Beams on Elastic Foundations: Theory with Applications in the Fields of Civil and Mechanical Engineering, The University of Michigan Press: Ann Arbor, Michigan, 1946.
[3] Connor, J.J., Analysis of Structural Members Systems, Ronald Press: New York, 1976.
[4] Sokolnikoff, I.S., Mathematical Theory of Elasticity, 2nd edn, McGraw-Hill: New York, 1956.
[5] Timoshenko, S.P. & Woinowsky-Krieger, S., Theory of Plates and Shells, 2nd edn, McGraw-Hill: New York, 1959.
[6] González, E., Solución a Problemas de Elasticidad mediante Series de Potencia, Civil Engineering Thesis, Universidad de Chile: Santiago, 2003.
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Open Access
Research article

Solution of Solid Mechanic Equilibrium Problems by Power Series

E. González,
M. Sarrazin
Universidad de Chile, School of Engineering, Chile
International Journal of Computational Methods and Experimental Measurements
|
Volume 3, Issue 1, 2015
|
Pages 33-48
Received: N/A,
Revised: N/A,
Accepted: N/A,
Available online: N/A
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Abstract:

The paper presents the application of power series to the numerical solution of equilibrium problems in elasticity. Complete bases of power series that satisfy the differential equations are developed, first for unidimensional problems like the equilibrium of a beam on elastic foundation, second for the harmonic differential equation in two dimensions, with application to the Saint-Venant’s torsion problem and, finally, for the biharmonic equation, which can be applied to plane elasticity problems as well as to the plate-bending problem. In the case of unidimensional problems the solution is exact, because the number of boundary conditions is equal to the number of parameters involved in the series expansion, whereas in the two-dimensional problems the solution satisfies exactly the differential equation but only approximately the boundary conditions. The approximation of the solution will depend on the number of points selected at the boundary. The method presented here can also be used for developing high-order finite elements of any number of nodes and boundary shapes using complete polynomial expansions that satisfy the differential equation. Selected practical applications are shown.

Keywords: Beam on elastic foundation, Biharmonic equation, Boundary conditions, Equilibrium problems, Finite elements, Harmonic equation, Plane elasticity, Plate-bending problem, Power series, Saint-Venant’s torsion problem

1. Introduction

2. Power Series Solution of a Differential Equation

3. Power Series Solution for $d^vw(x)/dx^v + rw(x) = 0$

4. Power Series Solution for the Harmonic Equation $\left(\nabla^2 \phi(x, y)=0\right)$

5. Power Series Solution for the Biharmonic Equation $\left(\nabla^4 \phi(x, y)=0\right)$

6. Conclusions

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Acknowledgments

The authors thank the Department of Civil Engineering of the University of Chile, where this research was carried out.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References
[1] Crandall, S.H., Engineering Analysis, McGraw-Hill: New York, 1956.
[2] Hetényi, M., Beams on Elastic Foundations: Theory with Applications in the Fields of Civil and Mechanical Engineering, The University of Michigan Press: Ann Arbor, Michigan, 1946.
[3] Connor, J.J., Analysis of Structural Members Systems, Ronald Press: New York, 1976.
[4] Sokolnikoff, I.S., Mathematical Theory of Elasticity, 2nd edn, McGraw-Hill: New York, 1956.
[5] Timoshenko, S.P. & Woinowsky-Krieger, S., Theory of Plates and Shells, 2nd edn, McGraw-Hill: New York, 1959.
[6] González, E., Solución a Problemas de Elasticidad mediante Series de Potencia, Civil Engineering Thesis, Universidad de Chile: Santiago, 2003.
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González, E. & Sarrazin, M. (2015). Solution of Solid Mechanic Equilibrium Problems by Power Series. Int. J. Comput. Methods Exp. Meas., 3(1), 33-48. https://doi.org/10.2495/CMEM-V3-N1-33-48
E. González and M. Sarrazin, "Solution of Solid Mechanic Equilibrium Problems by Power Series," Int. J. Comput. Methods Exp. Meas., vol. 3, no. 1, pp. 33-48, 2015. https://doi.org/10.2495/CMEM-V3-N1-33-48
@research-article{González2015SolutionOS,
title={Solution of Solid Mechanic Equilibrium Problems by Power Series},
author={E. GonzáLez and M. Sarrazin},
journal={International Journal of Computational Methods and Experimental Measurements},
year={2015},
page={33-48},
doi={https://doi.org/10.2495/CMEM-V3-N1-33-48}
}
E. GonzáLez, et al. "Solution of Solid Mechanic Equilibrium Problems by Power Series." International Journal of Computational Methods and Experimental Measurements, v 3, pp 33-48. doi: https://doi.org/10.2495/CMEM-V3-N1-33-48
E. GonzáLez and M. Sarrazin. "Solution of Solid Mechanic Equilibrium Problems by Power Series." International Journal of Computational Methods and Experimental Measurements, 3, (2015): 33-48. doi: https://doi.org/10.2495/CMEM-V3-N1-33-48
GONZÁLEZ E， SARRAZIN M. Solution of Solid Mechanic Equilibrium Problems by Power Series[J]. International Journal of Computational Methods and Experimental Measurements， 2015， 3(1)： 33-48. https://doi.org/10.2495/CMEM-V3-N1-33-48