## 2 Materials and Methods
### FE Model of the Masonry Vault  
The principal objective of the FE simulations was the comparison of the dynamic performance of the unstrengthened and strengthened masonry vault (Figure 8). Hence, two variants of the structure's FE model were analyzed. In the first variant it was assumed that the vaults are unstrengthened. In the second variant, the strengthening was applied to the top as well as bottom surfaces of the vault. They were upgraded with strengthening system realized by application of composite materials, i.e., carbon net + mortar = carbon fiber reinforced cementitious matrix (C-FRCM). The masonry elements of the vault were realized with solid elements [27] for both variants of the structure. For the second variant, the C-FRCM strengthening system was modeled as shell elements. The 'Tie' option was used as a connection between the masonry vault and C-FRCM [27]. For the purpose of this study, i.e., analysis of the plastic behavior of the vault under a series of seismic shocks, the ABAQUS/Standard software program [27] was introduced and a finite element (FE) model of the vault was composed. The 3D numerical model was prepared using the quadratic tetrahedral elements of type C3D10 (for the masonry structural parts) and quadratic triangular elements of type STRI65 (for C-FRCM strengthening system) available in the ABAQUS element library. The complete number of nodes and elements equaled 109,964 and 62,260, correspondingly. For the investigated variants of the FE model of the masonry vault, the non-linear plastic material model was implemented, since the evaluation of seismic-induced damages was the main purpose of the numerical simulations. Hence, the concrete damage plasticity model (CDP), also known as the Barcelona Model (BM) was used for masonry parts of the vault [25]. Figure 7: Destructive laboratory tests (DT) of the materials used in the analyzed vault part: (**a**) masonry brick compressive strength; (**b**) carbon net tensile strength; (**c**) example diagram of compressive strength; (**d**) example diagram of strength—slip. The CDP model, applied in the ABAQUS/Standard software program, makes it possible to present all occurrences typical for the material that occurs during cyclic loading [19]. The CDP model is generally dedicated to concrete material structures and it describes the isotropic features of the concrete. This model uses conceptions of the combination of non-associated multi-hardening plasticity and scalar damaged elasticity to represent the inelastic behavior of concrete and to describe the irreversible damage that occurs during the fracturing process. Further material factors for tension and compression are specified in the mathematical formulation of the CDP model that allows simulating accurately the behavior of the material in a complex state of stress. The yield surface is provided by hardening variables expressing equivalent plastic strains: by two hardening variables representing equivalent plastic strains: \(\tilde{e}_{t}^{pl}\) and \(\tilde{e}_{c}^{pl}\), linked to failure mechanisms under tension and compression loading, respectively. To control damage evolution of the material stiffness degradation scalar value d, tensile damage variable \(d_{t}\), compressive damage variable \(d_{c}\) are applied (SDEG, DAMAGET and DAMAGEC, respectively). Equivalent plastic strain \(\tilde{e}^{pl}\) (marked in the ABAQUS/Standard as PEEQ--for compression, PEEQT--for tension) describes total plastic strain level in elements. This parameter allows to determine increases of plastic strain for every step of analysis and to follow the process of strain accumulation. Tensile and compressive stiffness degradation parameters depend on the level of equivalent plastic strain \(\tilde{e}^{pl}\):  
\[d_{c}=d_{c}(\tilde{e}_{c}^{pl}) \tag{6}\]  
\[d_{t}=d_{t}(\tilde{e}_{t}^{pl}) \tag{7}\]  
The behavior of masonry material structural parts is diverse compared to concrete material elements (the main feature of masonry is anisotropy).