L IST OF S YMBOLS
Chapter 3 P RELIMINARY N UMERICAL I NVESTIGATION
3.2. N UMERICAL M ODELLING
3.2.2. Abaqus FE Code
Pushover analyses are also carried out using Abaqus (2010) assuming a Concrete Damage Plasticity (CDP) material model for masonry, which is available within the software and within many other FE codes. Although a CDP approach is conceived for isotropic fragile materials like concrete, it has been widely shown that its basic constitutive law can be also adapted to masonry (Acito et al., 2014; Barbieri et al., 2013; Bayraktar et al., 2010;
Brandonisio et al., 2013; Clementi et al., 2018). It is worth noting, indeed, that experimental results reported by Page, (1981) on regular masonry wallets and successive numerical models (Milani et al. 2006) show that such material exhibits a moderate orthotropy ratio (around 1.2) under biaxial stress states in the compression-compression region. Obviously, such feature cannot be considered when an isotropic model, like the present one, is utilized.
However, the utilization of isotropic models is commonly accepted in the literature (like concrete smeared crack approach available in both Ansys and Adina) after an adaptation of the parameters to fit an average behavior between vertical and horizontal compression. A suitable model should also consider the ratio between the ultimate compression strength in biaxial stress states and in uniaxial conditions. Such ratio, which exhibits some similarities between concrete and masonry, is reasonably set equal to 1.16. The CDP model allows analyzing materials with different strength in tension and compression, assuming distinct damage parameters. Compressive crushing is also described by means of the introduction of plastic deformation with a parabolic softening law.
In tension, see Figure 3.1a, the stress-strain response follows a linear-elastic relationship, until the peak stress σt0 is reached. Then, micro-cracks start to propagate in the material, a phenomenon which is macroscopically represented by softening in the stress-strain relationship. Under axial compression, the response is linear up to the value of the yield stress σc0 (Figure 3.1b). After the yield stress, the response is typically characterized by hardening, which anticipates compression crushing, represented by a softening branch beyond the peak stress σcu. Damage variables in tension and compression are defined by means of the following standard relationships given in Eq. 3.1a and Eq. 3.1b:
(
1)
0(
pl)
t d Ec c c
σ = − ε ε− (3.1a)
(
1)
0(
pl)
t d Et t t
σ = − ε ε− (3.1b)
where σt and σc are the uniaxial tensile and compressive stress, respectively, E0 is the initial elastic modulus, εt and εc are the total strain in tension and compression, respectively; εtpl andεcpl are the equivalent plastic strain in tension and compression, respectively. The terms dt and dc are the tensile and compressive damage variables.
(a) (b)
(c) (d)
Figure 3.1. Abaqus material non-linear behavior in uniaxial (a) tension, and (b) compression, Stress-strain curve of masonry prism in (c) tension, and (d) compression.
In the present study, damage is assumed active in tension only, since the tensile strength of the masonry material is very low, especially in comparison with the compressive one. When strain reaches a critical value, the material elastic modulus degrades in the unloading phase to E < E0. In particular, within the simulations, a reduction equal to 5% of the Young’s modulus with respect to the initial value is assumed for a plastic deformation equal to 0.003 (Valente and Milani, 2016).
The strength domain is a standard Drucker Prager DP surface modified with a so- called Kc parameter (Figure 3.2) representing the ratio between the distance from the
0 0.02 0.04 0.06 0.08 0.1 0.12
0 0.05 0.1 0.15
σ,Axial Tensile Stress (MPa)
ε, Axial Tensile Strain (%)
0 0.5 1 1.5 2 2.5 3 3.5
0 0.1 0.2 0.3 0.4
σ,Axial Comp Stress (MPa)
ε, Axial Compressive Strain (%) εplt
σto
Eo
(1-dt)Eo
σt
εelt
εt
(1-dc) E0 E0
σc0 σcu σc
εplc εelc
hydrostatic axis of the maximum compression and tension, respectively. As per user’s guide it is kept equal to 0.667 in all computations (López-Patiño et al.,2017; Bertolesi et al.,2016). The tension corner is regularized with a correction parameter referring to eccentricity, see Figure 3.3. The user guidelines suggest a default value of 0.1 for eccentricity. A value of 10o was adopted for the dilatation angle for the inelastic deformation in the nonlinear range, which is in agreement with the data suggested by Pluijim and Van Der (1993). The ratio between the bi-axial (fbo) and uniaxial (fco) compression strength has been kept equal to 1.16 as suggested by Page (1981) for concrete (masonry behavior found to be similar). The values of the various inelastic parameters adopted for the analyses are defined in Table 3.1. Details on building geometry and mechanical properties of the materials are given in the next section.
(a) (b)
Figure 3.2. Abaqus: (a) modified Drucker-Prager strength domain, and (b) yield surface in the deviatoric plane corresponding to different values of Kc (-σʹ1, -σʹ2, -σʹ3 are principal stresses in three axisand – S1, – S2, – S3 are yield/flow surfaces in the deviatoric plane).
Figure 3.3. Smooth Druker-Prager failure criterion adopted in the simulations, p-q plane.
Damage Plasticity Model Mohr-Coulomb Model -σʹ1
-σʹ2
-σʹ3
KC=2/3
-S2 -S1
-S3
KC=1 CM
CM
TM
CM = Compressive Meridian, TM = Tensile Meridian
Material Point Δεα
Δεα
q
pa p βo
q = Mises equivalent stress p = Hydrostatic pressure stress pa = Evolution parameter Δεα= Equivalent creep strain rate βo = Material angle of friction
Table 3.1. Concrete damage plasticity properties
Dilatation Angle Eccentricity fbo/fco Kc Viscosity Parameter
10 0.1 1.16 0.667 0.0001
Whilst the utilization of the CDP model is probably more in agreement with the actual behavior of masonry, its utilization requires experienced users and huge computational time, especially when 3D FE models with many elements are used. Steel bands are modelled in Abaqus exactly in the same way as done within Strand 7.