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CivilComp Proceedings
ISSN 17593433 CCP: 79
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 162
Optimal Shakedown Design of Bar Systems: Strength, Stiffness and Stability Constraints J. Atkociunas and D. Merkeviciute
Department of Structural Mechanics, Vilnius Gediminas Technical University, Lithuania J. Atkociunas, D. Merkeviciute, "Optimal Shakedown Design of Bar Systems: Strength, Stiffness and Stability Constraints", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 162, 2004. doi:10.4203/ccp.79.162
Keywords: optimal shakedown design, elasticplastic bar systems, energy principle, mathematical programming.
Summary
Elastic perfectly plastic trusses are considered. The geometry of the trusses, bar
material properties and variable repeated load are known. The quasistatic load is defined by
the timeindependent 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 cyclicplastic collapse. Usually, optimal design of elasticplastic
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 stressstrain 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 stressstrain 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 stressstrain 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 nonmonotonically (that is the result of unloading phenomenon of the crosssections [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 nonlinear problem of truss minimum volume, stated on the basis of extremum energy principles, is not traditional mathematical programming problem:
Complementary slackness conditions of mathematical programming do not allow evaluation of unloading phenomenon of bar crosssections. That is why this problem is solved stepby step (the Rozen project gradient method is applied). The objective of this paper is application of mathematical programming and KuhnTucker optimality conditions in optimization problems of truss at shakedown. References
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