Melan shakedown theorem interpretation for optimization of framed structures

Liudas Liepa

Doctoral dissertation

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The dissertation investigates the elastic-plastic beam-type frame structures, which are subjected to external variable repeated loads (VRL). Only the bounds of loads variation are known, which allows realization of any loading program (history). Material model of the structures is based according to the Prandtl’s diagram, small deformations assumption is considered. Mathematical models of optimization problems, in this thesis, are created according to the extremum energy principles of mechanics and mathematical programming theory.

The introduction reveals the investigated problem, importance of the thesis, the object of research, describes the purpose and tasks of the thesis, research meth-odology, scientific novelty, the practical significance of results and defended statements. The introduction ends with a list of author’s publication on the subject of defended dissertation.

Chapter 1 provides overview of the literature. Shakedown – Melan theorem and non-shakedown – Koiter theorems, of the structures subjected to VRL, are described in historic context. The duality of these theorems is discussed. General mathematical model of the optimization problem for the structures which experi-ences cyclic plastic collapse is presented.

In Chapter 2 optimization methods of elastic perfectly plastic structures sub-jected to VRL are improved by means of Melan‘s shakedown theorem. The main focus here is to improve the methods of determination of residual displacements variation bounds. Created methodology for estimation of residual displacement is presented in this thesis, based on the interpretation of Melan theorem and does not require formation of new influence matrices at each new scanning stage. This can-not be ignored by analyzing particular history of loading, because calculation scheme of the structure changes due to plastic deformations.

Chapter 3 explains methodology of steel and reinforced concrete structures optimization at shakedown conditions through numerical experiments. Volume optimization of steel structure is explained in the first part of Chapter 3. In the second part of Chapter 3 a modified mathematical model of optimization at shake-down problem of reinforced concrete structures is presented and numericaly tested. Yielding conditions in the mathematical model is derived according to the strength locus of reinforced concrete cross-section considering Eurocode 2 re-quirements. Design of required reinforcement is integrated into the optimization algorithm.

In Chapter 4 optimization solution quality validation methodology is pre-sented. Numerical results are validated according to the Rosen‘s gradient projec-tion method. Optimization solution validation methodology can be adapted and for continuum structures (plates, shells).

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145×205 mm
126 p.
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