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In engineering practice structures are often subject to variable or cycling loading. While the actual history of the loading is unknown, the loads are usually bounded inside a load domain. The behaviour of a structure subject to variable loads can be classified by one of the following cases: 1. If the loads remain at sufficiently low levels, structure’s response is purely elastic. 2. However if the applied loading is sufficiently high, the structure exceeds its load-carrying capacity and eventually it reaches the point of plastic collapse. 3. If the plastic strain increments does not change sign with each load cycle, after a number of cycles, the total strains and displacements grow large enough to ultimately cast the structure unserviceable due to excessive plastic deformation accumulation. This steady state response is called incremental plasticity or ratcheting. 4. On the other hand if the plastic strain increments do alternate sign with each load cycle, they tend to cancel each oth ...
In engineering practice structures are often subject to variable or cycling loading. While the actual history of the loading is unknown, the loads are usually bounded inside a load domain. The behaviour of a structure subject to variable loads can be classified by one of the following cases: 1. If the loads remain at sufficiently low levels, structure’s response is purely elastic. 2. However if the applied loading is sufficiently high, the structure exceeds its load-carrying capacity and eventually it reaches the point of plastic collapse. 3. If the plastic strain increments does not change sign with each load cycle, after a number of cycles, the total strains and displacements grow large enough to ultimately cast the structure unserviceable due to excessive plastic deformation accumulation. This steady state response is called incremental plasticity or ratcheting. 4. On the other hand if the plastic strain increments do alternate sign with each load cycle, they tend to cancel each other and while the total structure deformation remains small, in the points of intense stress local material failures develop, after a sufficient number of cycles. This failure mode is called alternating plasticity or low-cycle fatigue. 5. After some time plastic deformations cease to develop further and the accumulated dissipated energy remains bounded. Response of the structure is purely elastic after the initial transient response, where plastic deformation appears during the initial cycles. This phenomenon is called adaptation or elastic shakedown. The essence of shakedown analysis is to determine how much we can expand or contract a load domain of variable loads so to be certain that the elastoplastic structure subject to these loads will shakedown. Applications of shakedown analysis include, but are not limited to, pipes and pipe junctions, storage tanks and silos, pressure vessels, nuclear reactors, road pavements and train rails. The goals of this thesis are summarized to the following: - To study the computational realization of shakedown analysis of metal structures under three dimensional stress state. - To make use of the classic elastoplastic material model and the commonly used in metal structures, Tresca yield criterion. - To use - if possible - proved reference mathematical programming software and algorithms. - To develop computer programs capable of analyzing structures under three dimensional stress state utilizing the finite elements method and produce the necessary input data for feeding the mathematical programming software. - To perform numerical applications on model structures of practical value from the engineering point of view and to evaluate the performance of the suggested formulation. The body of the dissertation comprises seven chapters. In the first introductory chapter a short mention of the adaptation phenomenon and its applications along with a review of state of the art is written. Also the motives and aims of this thesis are outlined. In the second chapter some basic concepts and relations of elastoplasticity, that will be used later on, are described along with the elastic perfectly plastic material model and the Tresca yield criterion. The third chapter introduces elements of the finite elements methods (FEM) and in particularly the formulation based on the displacement method. Special mention of the eight-node 3D isoparametric solid element (brick element) is made, as this is the type of element that will be used in the numerical examples. The fourth chapter deals with the shakedown phenomenon and the definition of shakedown analysis. The two fundamental approaches of Melan (static theorem) and Koiter (kinematic theorem) are described. The time elimination from the shakedown problem and its connection with the finite element method are also given. In the fifth and most important chapter, the formulation of the shakedown problems under the three-dimensional Tresca criterion into a mathematical programming problem under linear and semidefinite constraints, is presented. To accomplish this the Tresca criterion is expressed at first as the difference of maximum and minimum stress tensor eigenvalues. Then the previous expression of the criterion is written as a system of semipositivness constraints. A suitable transformation of the stress and strain tensors is applied for the elimination of the spherical part from the Tresca criterion. Finnaly the descritized shakedown problems are formulated as semidefinite programming problems. Some of the used definitions on semidefinite programming along with relevant algorithms and their software implementations are given for the shake of completeness. 82B $
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