Jimenez et al 2018a - Revision history

Cinmemj at 11:31, 29 October 2018

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Cinmemj at 14:16, 28 September 2018

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Cinmemj at 14:10, 28 September 2018

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Cinmemj at 14:03, 28 September 2018

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Cinmemj at 13:58, 28 September 2018

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 Se incluyen tres ejemplos de aplicación de la metodología presentada, los cuales pretenden servir de validación y demostrar el potencial de cálculo de la misma. En los dos primeros casos se analizan dos vigas isoestáticas que permiten la comparativa con los resultados que se obtienen del ánalisis mediante métodos analíticos y, en el tercer caso se presentan los resultados de un Benchmark en el cual se ha trabajado en los últimos meses en que se estudia el comportamiento de un modelo a escala de un edificio de contención de una central nuclear.
-=Abstract=
+==Abstract==
 The main objective of this monograph is to present a novel methodology for the analysis of post-tensioned concrete structures, based on the potential offered by the use of the Serial-Parallel Rule of Mixtures when modelling composite materials. The advantages of this methodology are studied in comparison to the available approaches, i.e. the formulation proposed by the standards used in the design of structures and the mechanism used in numerical simulation based on the finite element method (FEM).

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-=Resumen=
+==Resumen==
-El objeto de la presente tesina es el de presentar una metodología novedosa para el cálculo de estructuras de hormigón posteso, basada en las posibilidades que brinda el uso de la teoría Serie Paralelo para la modelización de materiales compuestos. Se exponen, así mismo, las ventajas de esta metodología frente a los mecanismos actuales de cálculo que aparecen en los estándares de diseño de estructuras y aquellos otros basados en la simulación numérica a través del método de elementos finitos (MEF).
+El objeto de la presente monografía es el de presentar una metodología novedosa para el cálculo de estructuras de hormigón posteso, basada en las posibilidades que brinda el uso de la teoría Serie Paralelo para la modelización de materiales compuestos. Se exponen, así mismo, las ventajas de esta metodología frente a los mecanismos actuales de cálculo que aparecen en los estándares de diseño de estructuras y aquellos otros basados en la simulación numérica a través del método de elementos finitos (MEF).
 La teoría Serie Paralelo permite la modelización individualizada de los materiales componentes, actuando como una gestora de modelos constitutivos con el objetivo de simular el comportamiento del material compuesto en cuestión. Se modela el hormigón pretensado como un material compuesto de fibras largas en que la orientación de la fibra la marca el tendón de acero.
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 Se incluyen tres ejemplos de aplicación de la metodología presentada, los cuales pretenden servir de validación y demostrar el potencial de cálculo de la misma. En los dos primeros casos se analizan dos vigas isoestáticas que permiten la comparativa con los resultados que se obtienen del ánalisis mediante métodos analíticos y, en el tercer caso se presentan los resultados de un Benchmark en el cual se ha trabajado en los últimos meses en que se estudia el comportamiento de un modelo a escala de un edificio de contención de una central nuclear.
 =Abstract=

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 When defining the serial and parallel directions that control the composite behaviour, it is important to know which composite material is being modelled. Depending on their topology, composite materials can be classified as <span id='citeF-37'></span>[[#cite-37|[37]]]:
 * Materials with composite matrix. This is the case of ''cermet'' (ceramic and metal) and concrete.
 * Materials with composite matrix and short fibres. This is the case of fibre-reinforced concrete and some aeronautical materials.

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 The SP RoM can be included in the category of Micro-Mechanic Methods, specifically it can be catalogued as a Mean Field Method (MFM). It is the result of a century of evolution of the original idea but the bases have the essence of these methodologies. Micro-Mechanic Models study the strain and stress fields at a micro-scale level in order to build the constitutive laws and MFMs add to this conception the fact that:
 *  Mean stresses and strains at each component of the composite material are representative of the constituent behaviour.
 * Stresses and strains at the composite material are related with the stresses and strains at each constituent.
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 The SP RoM formulation is ''strain driven'' <span id='citeF-8'></span><span id='citeF-9'></span>[[#cite-8|[8,9]]], this means that the problem is controlled by the strain field, which is the independent driving variable. Therefore, the current state at a point <math display="inline"> x_i </math> in any of the component materials that constitute a two-phased composite (fibre and matrix) is completely defined just with:
 *  The strain field at the studied point, i.e. <math display="inline"> {^f}\boldsymbol{\varepsilon }\left(x_1\right)</math> and <math display="inline"> {^m}\boldsymbol{\varepsilon }\left(x_2\right)</math>, where <math display="inline"> x_1 \subset \Omega _f  </math>, <math display="inline"> x_2 \subset \Omega _m  </math> and <math display="inline"> \Omega = \Omega _f \cup \Omega _m  </math>.
 * A finite set of internal variables denoted by the vector <math display="inline"> {^f}\boldsymbol{\beta } </math> for the fibre and <math display="inline"> {^m}\boldsymbol{\beta } </math> for the matrix. These internal variables are correlated with the constitutive model used when modelling the component materials.
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 This algorithm is used to obtain the unknowns at all the composite elements of the FE mesh used in the simulation. This provides the local solution of the problem, but the global behaviour is still unknown. The global solution for the structure is obtained through the Equation [[#eq-3.21|3.21]], where the global stiffness matrix is made up of all the composite tangent constitutive tensors <math display="inline"> \mathbb{C} </math> from each element of the FE mesh. Two approaches can be followed in order to obtain these tensors:
 * Calculation using numerical perturbations <span id='citeF-37'></span>[[#cite-37|[37]]]. At this stage, the stress and strain fields of the composite material are known, therefore the composite <math display="inline"> \mathbb{C} </math> tensor can be obtained ''activating small strains'' and analysing the results obtained, i.e. <math display="inline"> \mathbb{C} = \boldsymbol{\sigma } : \boldsymbol{\varepsilon }^{-1} </math>.
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 |}
 * Evolution law of the damage variable, <math display="inline"> d </math>. The general equation that determines the evolution of this variable is:

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 This method has quadratic convergence which is really helpful for the calculation but it also has some drawbacks that must be considered:
 * Jacobian operator cost. This approach computes the Jacobian matrix at each iteration which is usually costly. There are alternatives that can be used instead (''modified Newton-Raphson'') but then the speed of convergence is lost. Thus, it is important to know which situation is better.
 * Antisymmetric Jacobian operators. There are scenarios where the Jacobian matrix is not symmetric and so its inversion is difficult.

← Older revision	Revision as of 11:09, 11 October 2018
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 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>N_i\left(\xi _j, \eta _j, \zeta _j\right)=\left\{1 \quad \mathrm{if} \quad i=j \atop     0 \quad \mathrm{if} \quad  i \neq j    \right.  </math>
+| style="text-align: center;" | <math>N_i\left(\xi _j, \eta _j, \zeta _j\right)=\left\{\begin{array}{lll} 1 & \mathrm{if} & i=j \\ 0 & \mathrm{if} &  i \neq j\end{array}    \right.  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.9)

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 * Generate an appropriate model. The first thing to be done is a correct modelization of the problem. This includes:
 * Build a geometrical model that represents the studied structure, paying attention to the possibility of including simplifications that could lighten the simulation, e.g. symmetries, plane stress case, plane strain case <span id='citeF-1'></span>[[#cite-1|[1]]], etc.
 * Consider how the real constraints and the applied forces should be included in the model and so define the boundary conditions.

← Older revision		Revision as of 13:58, 28 September 2018
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	Now, only one question remains: How is the internal variable of damage <math display="inline"> d </math> controlled? Of course <math display="inline"> d </math> is not constant. It is defined by two properties:		Now, only one question remains: How is the internal variable of damage <math display="inline"> d </math> controlled? Of course <math display="inline"> d </math> is not constant. It is defined by two properties:

−	-
	* Damage threshold criterion. This makes the distinction between an elastic behaviour of the material and another in which the degradation process of the material's properties takes place. This criterion is specific to the material that is being modelled but can be defined in general terms as:		* Damage threshold criterion. This makes the distinction between an elastic behaviour of the material and another in which the degradation process of the material's properties takes place. This criterion is specific to the material that is being modelled but can be defined in general terms as: