In general, NLFEAs allow to consider realistic material modelling in order to obtain accurate structural behaviour. However, the results obtained from NLFEAs strongly depend on the modelling strategies. For this reason, attending blind predictions and round robin competitions is a fundamental step of NLFEA effectiveness check. The SMART 2013 international benchmark experience is an example of blind prediction in which the University of Parma took part.
Participants to the benchmark were invited to provide the response assessment of scaled RC nuclear structure facility, Figure 1.1a, tested under strong motion input (Richard et al., 2015). The aim of the research was to evaluate the predictive capacity of the modelling in terms of global and local response, in order to try to partially fill the gap between the need of non-linear calculation for large structures and the modelling technique. Figure 1.1 shows the different modelling approaches adopted by participant teams, that represent the main modelling strategies used in engineering practice.
Figure 1.1b-c is an example of frame approach using beam elements. The beam elements approach is particularly indicated for modelling frame structures (Yazgan and Dazio, 2011). By means of this approach the structure is modelled as an assembly of interconnected frame elements in which the non-linear flexural behaviour can be assessed using two different models:
the lumped plasticity models allow to concentrate the non-linearity in some limited part of the
element. The lumped plasticity model is simpler and computationally lighter but, on the other hand, it requires a lot of experience by the operator to establish the non-linear behaviour and the characteristic length of the non-linear parts. On the contrary, the distributed plasticity models (also known as fibre models) allow capturing the spread of non-linearity through the whole member. The cross section of the element is discretized into smaller subregions referred to as fibres. A uniaxial cyclic stress-strain model is assigned to each fibre depending on the material it represents. The main advantages are related to the accuracy in the global response prediction and to the computational efficiency for the non-linear dynamic response analysis of structures with several hundred members (Spacone et al., 1996a; Spacone et al., 1996b).
However, it is difficult to capture the shear failure (associated with the development of inclined crack) or phenomena associated with torsional effects.
Figure 1.1: SMART2013 International Benchmark (Richard et al., 2018): (a) pictures of the RC specimen and modelling approaches adopted by participants: (b) lumped mass based models, (c) beam element based model,
(d) plate and shell element model, (e) solid element model, (f) shell and solid elements.
Another modelling approach that can be adopted is the solid elements one, Figure 1.1e. In this case, the structural member is subdivided into three dimensional (3-D) finite elements characterized at each node by translational degrees of freedom. This approach can describe the geometry of the structure, especially the connection of adjacent members (e.g. walls and slabs),
more accurately than beam or shell element models. Moreover, due to the fact that these elements are defined in the 3-D space, they can capture different failure modes, related to flexural and shear behaviour both in-plane and out-of-plane. However, prediction of the non-linear response of the structure, modelled using brick elements, is high time and memory demanding; convergence issues are usually affecting the NLFEA solutions and the response is strongly dependent on material models which requires a 3D formulation. Solid elements have the tendency to produce large systems of equations and for this reason are usually applied when other approaches are unsuitable or would produce inaccurate analysis results.
Figure 1.1d-f is an example of modelling using 2D elements. In particular Figure 1.1f shows the modelling strategies adopted by the team from the University of Parma using shell elements (of which the author of this Ph.D. thesis takes part), (Belletti et al., 2017). Shell elements can be considered as a simplification, with respect to solid element, based on two main hypotheses. The first one is that a plane section remains a plane before and after deformation.
The second consideration is that stresses, acting normal to the mid-surface of the shell elements, are negligible (Maekawa et al., 2003)
Within the same modeling approach is also possible to assign the non-linear behaviour of materials in different ways. For example, during the CASH benchmark (organized by OEDC-NEA - Nuclear Energy Agency), to which the University of Parma participated, most of the participants have been using shell elements to assess the capacity of full scale RC walls extracted from a nuclear power plants (NPP) building subject to various seismic loading intensities. The non-linearity at the integration points of shell elements has been assigning by means of different strategies: plasticity models, damage mechanics models, and non-linear elastic models.
Stress-based plasticity models are useful for modelling concrete subjected to triaxial stress states (Grassl et al., 2002; Cervenka and Papanikolaou, 2008). In these models, the elastic part of the strain is separated from the plastic one in order to realistically represent the observed deformations in confined compression. However, plasticity models are not able to describe the reduction of the unloading stiffness, experimentally observed.
Conversely, strain based damage mechanics models are able to consider the gradual reduction of the elastic stiffness (Mazars and Pijaudier-Cabot, 1989). On the other hand, isotropic damage mechanics models are often unable to describe irreversible deformations
observed in experiments and are mainly limited to tensile and low confined compression stress states.
For this reason, combinations of isotropic damage and plasticity are widely used to model both tensile and compressive failure (Grassl and Jirasek, 2006). These kinds of constitutive laws allow a physical description of crack initiation and development but are less useful in case of severe loadings.
A different approach is based on non-linear elastic models (discrete crack models or smeared crack models, described in Chapter 2). The smeared crack approach is the procedure most commonly employed. The basic assumption is that the first crack is developed perpendicular to the principal tensile direction. After cracking, the material can be described with an orthotropic model. In this context, this Ph.D. thesis is placed. A smeared crack model for the prevision of RC elements subjected to cyclic and dynamic loading is proposed.