Numerical characterization of the structural behaviour of the Basilica of Pilar in Zaragoza (Spain). Part 2: Constructive process effects
Introduction
The consideration of the effects of the constructive process is mandatory in the design of several types of structures like tunnels or excavation works, dam construction, long span bridges or high buildings [1], [2]. Nevertheless, its inclusion in the analysis of historical structures is unusual, being difficult to find papers on the matter [3], [4], [5]. As it is verified later in the case of the Basilica of Pilar, the influence of considering the constructive process in the obtained results can be very important.
The historical constructive process has been incorporated in the study in two ways, using in both linear material models. In the first place, after modifying the initial 1927 model, with additional elements for the repairing works and new structures such as the river towers, we obtained the actual state model, which was again recalculated with the same loads [6]. The results of this approach are presented in part I of this article. In the second place we developed an evolutionary model considering several hypotheses about the historical construction process for both of them, the 1927 model and the actual state model, in order to verify if the numerical results are affected by the constructive process.
It is evident that in a massive structure, as the Basilica of Pilar, the effects of the geometric non-linearity, considering the deformed geometry in the structural analysis, may not be significant. However the simulation of the constructive process is based on a sequential analysis by stages or time steps, and the activation or deactivation of mesh elements in each analysis stage, starting from an initial mesh with all the elements involved in the analysis previously defined.
In each stage, the activated elements are incorporated into the analysis from that stage with a null initial stress–strain state, simulating new constructed parts originally deactivated. Once activated, their contributions to the global rigidity and loads are taken into account, along with the contributions of the rest of elements in an “alive” state. Deactivated or “killed” elements have a null elemental stiffness matrix, simulating the elimination of parts of construction as in excavation process, or structural elements still not constructed. Therefore, it is necessary to update the global stiffness matrix and load vector, in each stage, to consider the variation of contributions from the alive or killed elements. To accomplish this approach with the finite element program Cosmos/m [7] employed, we must include the geometrical non-linearity in order to force to the program to update these matrix, and each stage is considered as an independent analyses starting from the previous one through a restart. That allows the user to indicate the activated or deactivated elements between each stage. In other programs, like Sap2000 [8], it is possible to consider the non-linear staged construction effect with or without non-linear geometrical effects.
This type of analysis is clearly non-linear. The obtained results depend on the constructive sequence and the history of loads, and could be combined with other non-linearities coming from materials or geometrical, in a static or a dynamic context.
To avoid numerical instabilities, deactivated elements have usually a non-null value in the diagonal positions of their elementary stiffness matrix, but sufficiently reduced so that it does not affect the results of the active structure. In all the models developed in this paper we work with a residual stiffness value of 1 × 10−6 kN/m for deactivated elements. Similarly, the application of the loads must be made carefully, never applying loads on deactivated elements.
Usually finite elements programs do not allow us to modify the mesh in the different stages of an analysis. For that reason the structure is defined completely in the initial stage, indicating the elements that are deactivated. When an element is deactivated, its rigidity and its loads are not added to the analysis, and therefore their nodes do not undergo movements except in the case of having common nodes with active elements. It is important that the displacements of the active elements do not distort excessively the deactivated elements in contact to avoid erroneous results and numerical problems when these elements are activated. If the displacements undergone by the active elements are important, it is necessary to correct the position of the deactivate elements before their activation, for example, by means of rigid solid global movements placing these new elements in an appropriate position with respect to the already active ones.
Section snippets
Finite element formulation considering constructive process
Each constructive stage can be divided in a series of sequential steps with the purpose of improve the convergence. Assuming a geometrical non-linear analysis [9], [10], the solution in each step is obtained in an iterative way by means of the Newton–Raphson method with force control.
The problem in step t and iteration i could be expressed aswhere tF is the external load vector, Δai is the increment of displacements, and are respectively the stiffness tangent
Numerical results for global models
Since in order to apply the constructive process in the global models, a geometric non-linear analysis has been used, we present in the first place the results obtained considering solely the geometrical non-linearity, and later the results adding the staged construction. In all cases linear material model were used [11], [12], and the loads considered are gravitational (self-weight and cover loads) since they are the most important. Results obtained from uniform or varying thermal loads are
Conclusions
The results obtained with geometrical non-linearity analysis are very similar to those obtained considering the linear theory, due to the small movements that take place in this massive and rigid structure. Nevertheless, this non-linear analysis presents convergence problems generated by local instabilities that force to eliminate some elements of the model.
The consideration of simplified non-linear constructive processes and its application with a staged geometrical non-linear analysis in the
Acknowledgements
Special thanks to A. Sánchez, I. Valcarce, J. Cascales and P. Loscos, former research assistants of the University of Coruña, for their contribution to this study.
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