Keywords: optimization, optimal design, elasto-plastic structures, impact, seismic loading, explosion, plastic deformation, residual displacements.

When in the design of structures extreme loadings such as short time high intensity dynamic pressure (explosion), impact or earthquake have to be taken into consideration then, except for special cases, the plastic reserves of the material can be utilized, but the development of excessive plastic deformations and residual displacements and the collapse have to be prevented. Following this design concept in this paper three appropriate methods are presented for the determination of the optimal layout of material of elasto-plastic structures (beams, frames, trusses and plates) subjected to extreme loading. The investigation is extended also to the case when in the optimal design in addition to one of the extreme loads the normal (working) loads can be separately or simultaneously taken into consideration. In case of dynamic pressure the widely used approximation is used that, because during explosion the pressure is very high and its duration is short, the pressure loading can be replaced by impulsive loading represented by an initial velocity fields [1,2,3]. Since during the motion the displacements are generally large the elastic deformations are neglected and the development of moderately large displacements are taken into account. To control the plastic behaviour of the structure an upperbound theorem is used for the determination of the expected residual displacements. Because the high velocity of the of the motion the viscosity of the material is also taken in the design into consideration. The mathematical formulation is based on the finite element method and on the concept of porous material where the material distribution of the material is described by the unknown material densities of the elements which are design variables. The proposed method is based on the requirements, that: during the response the entire kinetic energy is dissipated by plastic work, the residual displacements do not exceed a prescribed limit and the total mass of the structure is minimum.

Next consider protective structures, containments, shelters etc., struck by a mass traveling with an initial velocity [2,3]. In the investigation the elastic deformations and local damage of the structure are neglected and it is assumed that during the response the mass and the structure are moving together. Then making use of the principle of conservation of momentum and assuming a yield mechanism the initial velocity field of the structure can be determined. In this manner the problem is reduced to that of the dynamic pressure described above. The formulation of the two problems is described by two nonlinear mathematical programming problems coupled by the design variables. It can be solved by iteration and the SQP method was successfully applied for the subproblem solutions.

Considering the problem of seismic optimal design the appropriate method to be presented is confined to elasto-plastic frame structures and based on the idea of the pushover analysis is proposed in [4,5]. In the design procedure first the modal analysis of the elastic and the limit analysis of the elasto-plastic structure have to be conducted. Then making use of the elastic response spectra given in [4,5] the investigation can be reduced to the analysis of an equivalent single-degree-of-freedom system, from which the approximate value of the (target) displacement of the original elasto-plastic structure can be determined. This should not exceed the value determined by the damage limitation requirements given in [5]. Finally the general case is considered when in the optimal design the states of the structure under normal (working) loads and under one of the extreme loads can be separately or simultaneously taken into consideration [2]. It is assumed that the normal loads have to be carried in elastic stage with limited displacements while in case of the extreme loads, as it was described above the structure undertakes plastic deformations. The application of the above methods is illustrated by examples.

Acknowledgement

The present study was supported by The Hungarian National Scientific and Research Foundation (OTKA) (grant T 042993) and by Janos Bolyai Scholarship.

1

N. Jones, "Structural Impact", Cambridge University Press, 1997.

2

S. Kaliszky, J. Lógó, "Layout and Shape Optimization of Elastoplastic Disks with Bounds on Deformation and Displacement", J. Mech. Struct. and Mach., 30(2), 179-191, 2002. doi:10.1081/SME-120003014

3

S. Kaliszky, J. Lógó, "Layout Optimization of Rigid-Plastic Structures under High Intensity Short Time Dynamic Pressure", Mechanics Based Design of Structures and Machines.31(2), 131-149, 2003. doi:10.1081/SME-120020288

4

S. Kaliszky, J. Lógó, "A Unified Model for the Elasto-plastic Optimal Design of Structures with Complience Constraints", Proc. of 5th World Congress of Structural and Multidisciplinary Optimization, May 19-23, 2003, Lido di Jesolo, Venice, Italy, 181-182.

5

Eurocode 8, Structures in Seismic Regions, Commission of the European Communities, 2002.

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