Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T14:18:56.116Z Has data issue: false hasContentIssue false
  • English
  • Français

Static Analysis of the Free-Free Trusses by Using a Self-Regularization Approach

Published online by Cambridge University Press:  29 March 2017

J. T. Chen ,
Y. L. Chang ,
S. Y. Leu  and
J. W. Lee
J. T. Chen*
Affiliation:
Department of Harbor and River EngineeringDepartment of Mechanical and Mechatronic EngineeringNational Taiwan Ocean UniversityKeelung, Taiwan
Y. L. Chang
Affiliation:
Department of Harbor and River EngineeringNational Taiwan Ocean UniversityKeelung, Taiwan
S. Y. Leu
Affiliation:
Department of Aviation Mechanical EngineeringChina University of Science and TechnologyHsinchu, Taiwan
J. W. Lee
Affiliation:
Department of Harbor and River EngineeringNational Taiwan Ocean UniversityKeelung, Taiwan
*
*Corresponding author (jtchen@mail.ntou.edu.tw)
Get access

Abstract

Following the success of static analysis of free-free 2-D plane trusses by using a self-regularization approach uniquely, we further extend the technique to deal with 3-D problems of space trusses. The inherent singular stiffness of a free-free structure is expanded to a bordered matrix by adding r singular vectors corresponding to zero singular values, where r is the nullity of the singular stiffness matrix. Besides, r constraints are accompanied to result in a nonsingular matrix. Only the pure particular solution with nontrivial strain is then obtained but without the homogeneous solution of no deformation. To link with the Fredholm alternative theorem, the slack variables with zero values indicate the infinite solutions while those with nonzero values imply the case of no solutions. A simple space truss is used to demonstrate the validity of the proposed model. An alternative way of reasonable support system to result in a nonsingular stiffness matrix is also addressed. In addition, the finite-element commercial code ABAQUS is also implemented to check the results.

Keywords

Self-regularization approach3D free-free structureBordered matrixStiffness matrixSpace truss
Type
Research Article
Information
Journal of Mechanics , Volume 34 , Issue 4 , August 2018 , pp. 505 - 518
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Felippa, C. A., Park, K. C. and Justino Filho, M. R., “The Construction of Free-Free Flexibility Matrices as Generalized Stiffness Inverses,” Computers and Structures, 68, pp. 411–418 (1998).Google Scholar
2. Blazquez, A., Mantic, V., Paris, F. and Canas, J., “On the Removal of Rigid Body Motions in the Solution of Elastostatic Problems by Direct BEM,” International Journal for Numerical Methods in Engineering, 36, pp. 4021–4038 (1996).Google Scholar
3. Vodicka, R., Mantic, V. and Paris, F., “Note on the Removal of Rigid Body Motions in the Solution of Elastostatic Traction Boundary Value Problems by SGBEM,” Engineering Analysis with Boundary Elements, 30, pp. 790–798 (2006).Google Scholar
4. Vodicka, R., Mantic, V. and Paris, F., “On the Removal of the Non-Uniqueness in the Solution of Elastostatic Problems by Symmetric Galerkin BEM,” International Journal for Numerical Methods in Engineering, 66, pp. 1884–1912 (2006).Google Scholar
5. Lutz, E., Ye, W. and Mukherjee, S. “Elimination of Rigid Body Modes from Discretized Boundary Integral Equations,” International Journal of Solids and Structures, 35, pp. 4427–4436 (1998).Google Scholar
6. Chen, J. T., Chou, K. S. and Hsieh, C. C., “Derivation of Stiffness and Flexibility for Rods and Beams by Using Dual Integral Equation,” Engineering Analysis with Boundary Elements, 32, pp. 108–121 (2008).Google Scholar
7. Felippa, C. A. and Park, K. C., “The Construction of Free-Free Flexibility Matrices for Multilevel Structural Analysis,” Computer Methods in Applied Mechanics and Engineering, 191, pp. 2139–2168 (2002).Google Scholar
8. Chen, J. T., Huang, W. S., Lee, J. W. and Tu, Y. C., “A Self-Regularized Approach for Deriving the Free-Free Stiffness and Flexibility Matrices,” Computers and Structures, 145, pp. 12–22 (2014).Google Scholar
9. Chen, J. T., Han, H., Kuo, S. R. and Kao, S. K., “Regularization Methods for Ill-Conditioned System of the Integral Equation of the First Kind with the Logarithmic Kernel,” Inverse Problem in Science and Engineering, 22, pp. 1176–1195 (2014).Google Scholar
10. Ben-Israel, A. and Greville, T. N. E., Generalized Inverses: Theory and Applications, Springer, New York (2003).Google Scholar
11. Felippa, C. A., “A Direct Flexibility Method,” Computer Methods in Applied Mechanics and Engineering, 149, pp. 319–337 (1997).Google Scholar
12. Lin, T. W., Shiau, H. T. and Huang, J. T., “Decomposition of Singular Large Sparse Matrix by Adding Dummy Links and Dummy Degrees,” Journal of the Chinese Institute of Engineers, 15, pp. 723–727 (1992).Google Scholar
13. Sharifi, P. and Popov, E. P., “Nonlinear Buckling Analysis of Sandwich Arches,” Journal of the Engineering Mechanics Division, 97, pp. 1397–1412 (1971).Google Scholar
14. Stewart, G. W., “Modifying Pivot Elements in Gaussian Elimination,” Mathematics of Computation, 28, pp. 537–542 (1974).Google Scholar
15. Hartwig, R. E., “Singular Value Decomposition and the Moore–Penrose Inverse of Bordered Matrices,” SIAM Journal on Applied Mathematics, 31, pp. 31–41 (1976).Google Scholar
16. Chang, S. C. and Lin, T. W., “Constraint Relation Implementation for Finite Element Analysis from an Element Basis,” Advances in Engineering Software (1978), 10, pp. 191–194 (1988).Google Scholar
17. Davis, H. T. and Thomson, K. T., Linear Algebra and Linear Operators in Engineering: with Applications in Mathematica, Academic press, New York (2000).Google Scholar