An Incremental-iterative Method of Static nonlinear Analysis to Trace the Equilibrium Path of Shallow Space Dome Structures

Document Type : Original Research

Authors
J. Mashhadi
A. Hajizadeh
S. Rostami
Department of Civil Engineering, Technical and Vocational University (TVU), Tehran, Iran
10.22034/23.3.27
Abstract
Nowadays, non-linear analysis of structures has become an attractive matter and appears necessary for most structural engineering applications, whereas tendency for more accurate structural analysis has increased by many engineers. In general, nonlinear analysis is divided into two main parts: geometrical and material nonlinearity. In some cases, both nonlinear types are used simultaneously in the analysis process. In this paper, a new Incremental-iterative method for the analysis of truss structures, including both geometric and material nonlinearity behavior, is proposed. Nonlinear equilibrium equations are solved using an Incremental-iterative method based on the displacement control process. The basis of geometric nonlinear process is based on rotational formulation and material nonlinearity is based on tracing the stress-strain relationship for post-buckling behavior of the truss members. In this proposed method, it is assumed that the magnitude of the displacement increment vector to the size of the total displacement vector at the beginning of each load increment, is a fixed value. Based on this idea and thinking, the corresponding equations are written and a new formulation has been developed based on the displacement control scheme, as, by employing a specified displacement, the corresponding load will be obtained. Using fixed incremental displacement algorithm, this paper, proposed a novel method that has a possibility of passing the limit points in the case of highly nonlinear behavior state. The proposed method, is able to pass the limit points including snap-through and snap-back. Some examples are provided for the proposed method and by solving them, the efficiency of the proposed algorithm is examined. A limit point refers to the turning point for the equilibrium path of a structure, which can be further considered as the transition point from stable to unstable equilibrium states or vice versa. In analysis of such structures, the simple incremental iterative methods unable to pass this limit points. The simple incremental iterative methods couldn’t trace the equilibrium path after the limit points. For resolving such disadvantages, advanced analysis methods have been developed numerical examples demonstrate the feasibility and accuracy of the proposed algorithm, to be highly suitable in predicting nonlinear response of structures with multiple limit points and snap-back points, and trace the equilibrium path accurately. The results show that the method developed in this paper, traces the equilibrium curve as well as the modified arc-length method with a small difference. In spite of the fact that the procedure herein explained is only implemented for truss structures, it is possible to generate the proposed method for other structures. This paper is organized as follows. In the first section, we will have a brief review of nonlinear analysis of trusses, including both geometric and material nonlinearity. A literature review is done and a number of available methods in this field is describe. In the second section we explained the basic concepts in nonlinear analysis. The third section geometric and material nonlinear behavior of trusses are reviewed. Section four is allocated to implementation of proposed method for nonlinear analysis. In section five, the validity of the proposed method is illustrated with some examples.

Keywords

Subjects

[1] P. Wriggers, "Non‐linear finite element analysis of solids and structures, volume 1: Essentials, MA Crisfield, John Wiley. ISBN: 0‐471‐92956‐5," ed: Wiley Online Library, 1994.
[2] J. Oden, "Finite element applications in nonlinear structural analysis," in Proceedings of the ASCE Symposium on Application of Finite Element Methods in Civil Engineering, 1969: Vanderbilt University, pp. 419-456.
[3] M. J. Turner, H. Martin, and R. Weikel, "Further development and applications of the stiffness method," Matrix methods of structural analysis, vol. 1, pp. 203-266, 1964.
[4] J. H. Argyris, Recent Advances in Matrix Methods of Structural Analysis/progress in Aeronautical Sciences, Vol. 4. Pergamon Press, 1964.
[5] G. A. Wempner, "Discrete approximations related to nonlinear theories of solids," International Journal of Solids and Structures, vol. 7, no. 11, pp. 1581-1599, 1971.
[6] E. Riks, "An incremental approach to the solution of snapping and buckling problems," International journal of solids and structures, vol. 15, no. 7, pp. 529-551, 1979.
[7] M. A. Crisfield, "A fast incremental/iterative solution procedure that handles “snap-through”," in Computational methods in nonlinear structural and solid mechanics: Elsevier, 1981, pp. 55-62.
[8] E. Ramm, "Strategies for tracing the nonlinear response near limit points," in Nonlinear finite element analysis in structural mechanics: Springer, 1981, pp. 63-89.
[9] K.-J. Bathe and E. N. Dvorkin, "On the automatic solution of nonlinear finite element equations," Computers & Structures, vol. 17, no. 5-6, pp. 871-879, 1983.
[10] Y.-B. Yang, Linear and nonlinear analysis of space frames with nonuniform torsion using interactive computer graphics. Cornell University, 1984.
[11] S. L. Chan, "Geometric and material non‐linear analysis of beam‐columns and frames using the minimum residual displacement method," International Journal for Numerical Methods in Engineering, vol. 26, no. 12, pp. 2657-2669, 1988.
[12] H. Saffari, S. Shojaee, S. Rostami, and M. Malekinejad, "Application of cubic spline on large deformation analysis of structures," International Journal of Steel Structures, vol. 14, no. 1, pp. 165-172, 2014.
[13] A. Kassimali and E. Bidhendi, "Stability of trusses under dynamic loads," Computers & structures, vol. 29, no. 3, pp. 381-392, 1988.
[14] C. D. Hill, G. E. Blandford, and S. T. Wang, "Post-buckling analysis of steel space trusses," Journal of Structural Engineering, vol. 115, no. 4, pp. 900-919, 1989.
[15] S. S. Tezcan and P. Krishna, "Discussion of “Numerical solution of nonlinear structures”," Journal of the Structural Division, vol. 94, no. 6, pp. 1613-1627, 1968.
[16] M. Papadrakakis, "Inelastic post-buckling analysis of trusses," Journal of Structural Engineering, vol. 109, no. 9, pp. 2129-2147, 1983.
[17] R. De Borst, M. A. Crisfield, J. J. Remmers, and C. V. Verhoosel, Nonlinear finite element analysis of solids and structures. John Wiley & Sons, 2012.
[18] M. Crisfield, "Advanced Topics, Volume 2, Non-Linear Finite Element Analysis of Solids and Structures," 1997.
[19] G. Ramesh and C. Krishnamoorthy, "Inelastic post‐buckling analysis of truss structures by dynamic relaxation method," International journal for numerical methods in engineering, vol. 37, no. 21, pp. 3633-3657, 1994.
[20] H.-T. Thai and S.-E. Kim, "Large deflection inelastic analysis of space trusses using generalized displacement control method," Journal of Constructional Steel Research, vol. 65, no. 10-11, pp. 1987-1994, 2009.
[21] G. A. Hrinda, "Geometrically nonlinear static analysis of 3D trusses using the arc-length method," Computational Methods and Experimental Measurements XIII, Prague, Czech Republic, pp. 243-252, 2007.
Modares Civil Engineering journal
Volume 23, Issue 3
July and August 2023
Pages 27-40