The description of cracking and failure mechanisms of RC structures within finite element analysis has led to two fundamentally different approaches: the discrete and the smeared one.
The smeared model was first introduced by Rashid (1968) and Červenka and Gerstle (1971,1972). It considers the cracked solid as a continuum by reducing stiffness properties and considering cracks smeared over a distinct area, typically finite element or an area corresponding to an integration point of the finite element, Figure 2.1a. The smeared approach permits a description in terms of stress-strain relations passing from an isotropic constitutive law to an orthotropic law after the crack formation. This approach maintains the original mesh and does not impose restrictions on cracks inclination.
Figure 2.1: Example of (a) smeared crack model and (b) discrete crack model.
(b)
σ1 σ2
(a)
On the contrary, discrete crack models represent cracks as a geometrical discontinuity using, for example, interface elements. The first introduction to concrete structures has been made by Saouma and Ingraffea (1981). This method is theoretically more suitable to capture the failure localization. On the other hand, an adaptive re-meshing technique is required to account for phenomena such as progressive failure, Figure 2.1b. Furthermore, the crack is constrained to follow a predefined path, so it is a more suitable method to simulate the behavior of a structure dominated by the presence of one or few cracks. The main disadvantage of the discrete crack concept is the need of more specialized software and it is the main reason why the smeared method has become more widespread.
A further distinction can be done between the smeared rotating crack model (Rots and Blaauwendraad, 1989) and smeared fixed crack model. The first one assumes that, during loading, the crack pattern should change direction and for this reason the shear stresses are not considered (Stevens et al., 1991; Palermo and Vecchio, 2004; He et al., 2008); the second one hypothesizes the starting crack pattern as fixed (Okamura et al., 1991; Sittipunt and Wood, 1995). In this last case, the prediction of shear stresses generated along the cracks become very important, most of all when the structural behaviour is dominated by aggregate interlock phenomena.
More recently, a new strategy based on embedded discontinuities is developed (Belytschko et al. 1988; Dvorkin et al. 1990; Dias-da-Costa et al. 2009). By means elements with embedded discontinuities, the explicit remeshing is obtained by additional degrees of freedom that capture the jump in the displacement field inside the element. The partition-of-unity-based methods exploit the properties of the finite-element shape functions so that the extra degrees of freedom are overlaid to the regular nodes in the domain subjected to the enrichment.
With the advent of the new seismic codes, the interest in more realistic predictions of the non-linear behaviour of RC structures has increased. In fact, one of the main characteristics of RC structures is the highly non-linear response to cyclic loading, in particular seismic one. For this reason, realistic cyclic constitutive models are required to obtain reasonably accurate simulations of RC members. Nevertheless, if numerous are the constitutive models applied for monotonic loading case, as summarized by Bažant (2002) and de Borst (2002), the cyclic ones are less common in literature.
Existing commercial finite-element codes often have limitations in representing cyclic behavior, due to idealizations in material models. For example, to solve convergence problems, the tensile behaviour of concrete is commonly assumed to be secant in the unloading/reloading
phases, Figure 2.2a, even if the experimental evidence demonstrates that irrecoverable tensile strains remain in concrete (Gopalaratnam and Shah, 1985; Yankelevsky and Reinhardt, 1987).
Furthermore, the crack closing process implies that the concrete path does not pass through the origin (Mansour and Hsu, 2005), Figure 2.2b. Indeed, under reversed cyclic loading, concrete may repeatedly experience crack closing and reopening. During this process, due to the aggregate interlock and the bond between steel and concrete, the compressive stress can be still transferred through the crack before it is fully closed. More complexities, as stiffness degradation in concrete and the Bauschinger effect of steel bars, are introduced by cyclic loads.
Figure 2.2: Tensile behaviour for concrete: (a) secant unloading/reloading phases; (b) crack closing process.
For all these reasons, a reliable numerical model able to catch these types of non-linearity and characterized by sufficiently accurate response predictions and simplicity in formulation is necessary. Therefore, a Physical Approach for Reinforced Concrete under Cyclic Loading condition (PARC_CL 2.1) is proposed in this Ph.D. research.
The PARC model, Figure 2.3, was proposed in 2001 for the analysis of the behavior of reinforced-concrete membranes, subjected to plane stresses and monotonically loaded up to failure, (Belletti et al. 2001). The applications of the PARC model demonstrated the reliability of the obtained results; however, even a structure subjected to monotonic loading could experience unloading/reloading cylces due to the redistribution of internal actions (think for example to crack opening and closing). Therefore, it is useful to implement appropriate formulations that consider the possibility of loading and reloading branches. Furthermore, for the design of RC structures in seismic areas, the availability of cyclical constitutive laws becomes essential in order to conduct non-linear dynamic analyses. For all these reasons, the
σ
ε εu
σcracking
εcracking
σcracking
εcracking εu
σclose cracking
(a) (b)
σ
ε
model PARC_CL 1.0 was proposed as an extension to cyclic loads (Belletti et al., 2013a), Figure 2.3. This release was characterised by secant unload/reload path not permitting to consider the real hysteretic behaviour of RC structures, (Belletti et al., 2013b; Damoni et al., 2014). To overcome this limit a new version called PARC_CL 2.1, developed by the author of this research, is proposed. The PARC_CL 2.1 permits to take into account hysteretic cycles and plastic strains in the unloading phase. The model is implemented in a user subroutine UMAT.for for the analyses of RC members by means of ABAQUS code. More specifically, it is possible to assess the static, cyclic and dynamic behavior of slabs, structural walls buildings, and floors by means of multi layered shell or membrane elements, Belletti et al. (2017b) and Belletti et al.
(2018). PARC_CL 2.1 is an improvement of PARC_CL 2.0 (Belletti et al., 2017a), in which some modification in the cyclic concrete formulation and some important contribution like shrinkage, tension stiffening, and buckling effect are added.
Figure 2.3: PARC model evolution.
PARC Model [Belletti et al. 2001]
Unload/reload
PARC_CL 2.1 Model [Belletti et al. 2017a] Unload/reload
Plastic part
F
F u
PARC_CL 1.0 Model [Belletti et al. 2013a]
Unload/reload
F
u
u F