Keywords: optimal shakedown design, elastic-plastic bar systems, energy principle, mathematical programming.

Elastic perfectly plastic trusses are considered. The geometry of the trusses, bar material properties and variable repeated load are known. The quasi-static load is defined by the time-independent lower and upper variation bounds. The detailed loading history is not considered, as the loading fits into the aforementioned variation bounds. Adapted to variable repeated load the structure satisfies strength conditions and it is safe with respect to cyclic-plastic collapse. Usually, optimal design of elastic-plastic structures is obtained by neglecting stiffness constraints and does not satisfy the serviceability requirements of the structure [1,2,3,4]. Therefore, not only strength, but also stiffness and stability constraints should be included in mathematical models of optimal design of structures at shakedown.

In this paper, the optimum design problem is described as follows: when geometry of truss, also lower and upper bounds of loading are prescribed, the truss of minimum volume is to be found. The stress-strain state of optimized truss should satisfy all requirements of strength, stiffness and stability. Usually, residual forces, strains and displacements are used for characterization of structure stress-strain state in shakedown theory. The mathematical model for residual force determination are based on the principle of minimum complimentary deformation energy. Residual strains and displacements can be found due the principle of minimum total potential energy. Thus, in both cases mathematical programming problems are solved [5].

The stress-strain state of structures at shakedown depends on the loading history (that makes difficulties for direct application of mathematical programming duality theory). Analysis of residual displacements for trusses that have undergone plastic deformation is a very important constituent of mathematical models of optimization problems: the stiffness of truss is ensured by restricting the nodal displacements [6]. Determination of truss residual displacements is a difficult problem of dissipative systems mechanics [7,8] because during a shakedown process residual displacements are varying non-monotonically (that is the result of unloading phenomenon of the cross-sections [6]). It becomes more difficult, when load is characterized only by lower and upper load variation bounds. In that case it is possible to find only variation bounds of residual displacements. In this paper a new method is proposed for determination of residual displacement variation bounds (the linear mathematical programming problem is solved).

In this research stability constraints are related with recommendations of EUROCODE 3, where admissible forces of the bars in compression are obtained by reduction of their material yield limit [9]. Thus, stability requirements are introduced into optimized structure yield conditions by changing the admissible limit force of the compressible bars.

The non-linear problem of truss minimum volume, stated on the basis of extremum energy principles, is not traditional mathematical programming problem:

1. during the solution process stiffness of truss elements is changing;
2. it is necessary to determine lower and upper bounds of residual displacements.

Complementary slackness conditions of mathematical programming do not allow evaluation of unloading phenomenon of bar cross-sections. That is why this problem is solved step-by step (the Rozen project gradient method is applied). The objective of this paper is application of mathematical programming and Kuhn-Tucker optimality conditions in optimization problems of truss at shakedown.

1

Dorosz, S., Konig, J. A., Sawczuk, A., Z. Kowal, W. Seidel, "Deflections of elastic-plastic hyperstatic beams under cyclic loading", Archives of Mechanics, 33(5), 611-624, 1981.

2

Kaneko, L., Maier, G., "Optimum design of plastic structures under displacement's constraints", Computer Methods in Applied Mechanics and Engineering, 27(3), 369-392, 1981. doi:10.1016/0045-7825(81)90139-0

3

Kaliszky, S., Logo, J., "Optimal plastic limit and shakedown design of bar structures with constraints on plastic deformation", Engineering structures, 19(1), 19-27, 1997. doi:10.1016/S0141-0296(96)00066-1

4

Atkociunas, J., "Mathematical models of optimization problems at shakedown", Mechanics Research Communications, 26(3), 319-326, 1999. doi:10.1016/S0093-6413(99)00030-0

5

Cyras, A., "Extremum principles and optimization problems for linearly strain hardening elastoplastic structures", Applied mechanics, 22(4), 89-96, 1986.

6

Merkeviciute, D., Atkociunas, J., "Incremental method for unloading phenomenon fixation at shakedown", Journal of Civil Engineering and Management, IX (3), 178-191, 2003.

7

Weichert, D., Maier, G. (Editors), "Inelastic behavior of structures under variable repeated loads", Springer, New York, Vienna. 2002.

8

Lange-Hasen, P., "Comparative study of upper bound methods for the calculation of residual deformation after shakedown", Technical University of Denmark, Dept. of Struct. Engineering and Materials, Lyngby, 1998.

9

Kaliszky, S., Logo, J., "Plastic behaviour and stability constraints in shakedown analysis and optimal design of trusses", Structural and Multidisciplinary Optimization, 24, 118-124, 2002. doi:10.1007/s00158-002-0222-2

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