Keywords: optimal shakedown design, frames, stability, energy principles, mathematical programming.

The development of state-of-the-art design methods for the shakedown based design of structures currently lacks the requirement for stability evaluation in the plastic state. The solution of this concern is a real design problem for frames required by the various design codes. In this paper the structures optimal design methodology applied to the shakedown frames is based on newly created mathematical models and their close relationship with the design software MatrixFrame. This paper continues the work of [1].

In this paper the main mathematical models are created with strength, stiffness and stability constraints for the general nonlinear optimization problem of frame parameters (minimal volume, partially) and load distribution optimization. The total displacements (residual and elastic) are included in the stiffness constraints. The external actions on the frame are characterized by their upper and lower variation bounds that are independent of time. The volume minimization problem of frame elements investigated in this work is based on the solution of a similar limit moments problem. By using the limit moments obtained not only include the plastic resistance moments but also the cross-sections (and the volume in the same time). The paper is oriented towards to linear yield conditions and "I" shaped cross-sections. While solving the load optimization problem load upper and lower bounds satisfying shakedown, stiffness and stability constraints are determined.

The software created by authors is based on Rosen's project gradient method [2] and is applied to the solution of nonlinear problems. The problem solution part that is related to stability is transferred to the design software with the design codes (EC3, NEN 6771) implemented. Solution procedures become iterative: structural or load constraints of ordinary iteration of the main optimization problem are calculated using the design software MatrixFrame. In the proposed methodology initial data for the design software MatrixFrame becomes the residual forces and residual displacements obtained from the solution of the optimization problem i.e. the influence of plastic deformations is evaluated. Convergence with the desirable precision of the main optimization problem objective function is a criterion of the optimal solution. It is worth noting, that the proposed methodology allows the load combinations, occurring in the engineering practise to be realised as separate cases of variable repeated load.

The numerical examples concerning optimization of frame structures are presented. The results are valid for small displacements.

1

J. Atkociunas, D. Merkeviciute, A. Venskus, "Optimal shakedown design of bar systems: Strength, stiffness and stability constraints", Computers & Structures, In Press, Corrected Proof. doi:10.1016/j.compstruc.2008.01.0082

M.S. Bazaraa, H.D. Sherali, C.M. Shetty, "Nonlinear programming: theory and algorithms", New York: Brijbasi Art Press Ltd., John Wiley & Sons, Inc., 652, 2004.

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