Scientific and applied communication

Mathematical models of optimization problems at shakedown

Mathematical models for the optimization problems of metal structures at shakedown definitely contains strength (ULS – ultimate limit state) and stiffness (SLS – serviceability limit state) constraints. Residual displacements determined by plastic deformations appear in stiffness constraints. The residual displacements developing during shakedown process of ideally elastic-plastic structures under variable repeated load can vary non-monotonically. Therefore, it is important to determine variation bounds of residual displacements (particularly in the cases when only variation bounds of loading rather than a particular history of a load are known). Thus, numerical methods, extremum energy principles, the assumption of a small displacement and mathematical programming theory have been used in the paper to develop mathematical models for the analysis problem of residual displacements. Using the proposed scanning method, it is acceptable to define the revised bounds of residual displacements without analysing the detailed history of loading. However, scanning technique is approximate method comparing by comprehensive analysis of elastic-plastic structure, when to exact values of displacement are obtained. The application of the proposed scanning method is illustrated by a benchmark of a plane multi-supported beam.

The article analyses one of possible optimization methodologies for ideal elastic–plastic structures at shakedown and its application for shallow spherical shells having prescribed geometry and affected by a variable repeated load (VRL) – a system of external forces that may vary independently of each other. The paper accepts that only time-independent upper and lower bounds of variations in external forces are given. The pronounced effect of external forces, i.e. in this context, possible histories of variations in forces, is not examined (the unloading phenomenon of cross-sections is ignored in the course of plastic deformation). The discussed concept of the structure at shakedown refers to the Melan theorem related to statically allowable admissible internal forces. Thus, for the discretization of the spherical shell, with the help of an assumption about small displacements, the equilibrium finite element method based on internal force approximation is applied. The limit axial force of the cross-section is supposed to be constant within the bounds of the finite element, and only the optimal distribution of limit internal forces among elements, according to the selected criterion, is in need of search. The article presents a discrete mathematical model for determining the optimal allocation problem with strength and stiffness requirements. The model conforms to the limit axial force of the shallow spherical shell of the variable repeated load and takes into account ultimate and serviceability limit states of EC3 with corresponding reliability levels. Structural optimization methods refer to extreme energy principles of mechanics and are illustrated with the numerical examples of spherical shell optimization.

A shakedown-frame plastic moments minimization and load-optimization nonlinear mathematical model with strength, stiffness, and stability constraints is investigated. A methodology and algorithms for stability evaluation have been developed according to various standards (Eurocode 3 (EC3) and the Dutch NEN 6771) by integrating the MatrixFrame commercial software for the building industry and the nonlinear mathematical programming software developed by the authors. For other investigators, this work makes it possible to integrate the solutions of nonlinear programming problems (plastic state variables – residual forces and displacements) into their structural design software. Numerical examples of optimization of frame structures are presented.

Using the concept of a variable repeated load and shakedown theory, a unified technique is proposed for formulating mathematical models for the optimization of frame- and truss-like structures under different loads. Strength, stiffness and stability (for trusses only) constraints are included in non-linear mathematical models of structure volume minimization and load optimization problems. Even though the load is prescribed within certain limits, the mathematical models allow the variational bounds of the displacement (the stiffness of the structure depends on them) to be evaluated in the deformed state. Numerical example concerning calculation of frame structure is presented. The results are valid for small displacements.