## 3 Material and Numerical Models
### Material Models
#### 3.2.1 Model of Concrete in Confining Elements--Elastic-Based Degradation Model  
Principles of the continuum damage-based method were used to describe the development of damage in the discussed model. Effective stress expressed the non-linear behaviour of the material \(o_{c}^{\text{ef}}\). Effective stress should be understood as stress under which deformations of the undamaged body are equal to deformations of the body damaged by true stress, described as equivalent strain \(\varepsilon^{\text{eq}}\), calculated from the following relation [42]:  
\[\varepsilon^{\text{eq}}=\frac{\sigma_{ci}}{E_{ci}} \tag{1}\]  
Effective uniaxial strain is regarded as deformation caused by stress \(\sigma_{ci}\) and modulus of elasticity \(E_{ci}\) in the \(i\)-direction, represented by damage and caused by stress related to stress direction \(\sigma_{ci}\)[40]. A diagram of relationships between equivalent uniaxial stress and strain is shown in Figure 2[41]. Figure 2: Equivalent uniaxial stress-strain relationship for elastic model with material degradation. In a tensile state before cracking (range 1; Figure 2), material exhibited linear-elastic behaviour. Elasticity modulus \(E_{c}\) is the initial modulus of material elasticity, and effective tensile strength \(f_{t}^{\text{ref}}\) was determined on the basis of the criterion of material behaviour under biaxial stress state. The general stress is expressed by the following relationship: \[\sigma_{c}^{\text{ef}}=E_{c}\varepsilon^{\text{eq}},\ 0\leq\sigma_{c}\leq f _{t}^{\text{ref}}.\] (2) In the post-cracking phase (range 2; Figure 2), the material's behaviour was described in two ways:  
* The model of fictitious cracks resulting from the cracking mechanics and the accepted rule for crack width,
* The model of local deformations of a material point. In the model of fictitious cracks, an exponential relationship was applied, as shown in Figure 3, on the basis of [43]. In a compression state (range 3; Figure 2), the stress-strain relationship was adopted from the FIB (Federation internationale du beton) Model Code [44] in the following form:  
\[\sigma_{c}^{\text{ef}}=f_{t}^{\text{ref}}\frac{kx-x^{2}}{1+(k-2)x}, \tag{3}\]  
where \(x=\varepsilon/\varepsilon_{c}\), \(k=E_{o}/E_{c}\), \(\sigma_{c}^{\text{ef}}\) represents normal compressive stress in the material, \(f_{t}^{\text{ref}}\) represents the effective compressive strength of concrete, \(x\) signifies normalised strains, \(\varepsilon\) signifies strains, \(\varepsilon_{c}\) signifies strains corresponding to effective compressive strength \(f_{c}^{\text{ref}}\), \(k\) is the shape coefficient (\(k=1\) representing a linear function, and \(k=2\) representing a parabola), \(E_{o}\) signifies initial modulus of elasticity, and \(E_{c}=f_{t}^{\text{ref}}/\varepsilon_{c}\) is the secant modulus of elasticity. The shape of the function expressing the stress-strain relationship in a compressive state is presented in Figure 4.  
Figure 4: Compressive stress–strain relationship. Figure 3: Exponential relationship determining crack width of the SBeta model. The adopted non-linear stress-strain relationship took into account the material damage developed before maximum stress was reached (as opposed to the material with visible cracks). The linear stress-strain relationships (range 4; Figure 2) referred to the state after achieving maximum compressive stress \(f_{t}^{\text{ref}}\). Displacements after reaching the maximum compressive strength \(f_{c}^{\text{ref}}\) were only observed in the load plane. Displacement was assumed to occur regardless of the specimen size. Such a hypothesis was verified for concrete. The criterion, adjusted according to the results of concrete testing by [PERSON] [45], was applied under biaxial compressive state conditions (Figure 5).