# Verification of Actual Prestressing in Existing Pre-Tensioned Members
## 4 Numerical Analysis  
For the verification of experimental measurements, a nonlinear numerical analysis using the ATENA 3D software (version ATENA 5.7.0n, Cervenka Consulting, Prague, Czech Republic) was performed [40; 41]. The pre-tensioned concrete sleeper was modelled to simulate stress relief after the application of saw-cuts. The numerical 3D model is shown in Figure 15. Individual parts of the prestressed concrete sleeper (macro-elements) were assigned the real dimensions, but some minor simplifications were applied in regard to preserving the global behaviour of the specimen. Consequently, stress relief was observed only in the middle area where the saw-cuts were performed. A cross-section of the investigated middle area of the prestressed concrete sleeper was modelled with real geometry. The line supports were modelled at a distance of 100 mm from the ends of the specimen, and as a result, the effective length of the prestressed concrete sleeper with the static scheme of the simply supported beam was 2400 mm. Thus, the actual position of supports in the experimental program was respected. The prestressing wires were modelled as prestressing reinforcement bars (1D elements) using the bond model \"perfect connection\". Material parameters were adopted in compliance with material properties testing, which was part of the experiment. Nevertheless, detailed material properties that were not determined experimentally were defined according to the recommendations listed in [40]. Prestressed concrete sleepers were modelled using material \"3D Nonlinear Cementitious 2\". The finite element (FE) mesh has a crucial influence on the applicability of the results. Generally, it is recommended to use at least four to six elements per thickness of macro-element. All used elements were tetrahedral element CCIso Tetra, see Figure 15. Global element size was set to 0.065 m, whereas the decisive central macro-element important for saw-cut testing analysis was refined into tetrahedral elements with a size of 0.015 m.  
Figure 14: MBN as a function of stress (Specimen 4; Wire 1). Figure 15: 3D numerical model (**a**); FE mesh (**b**); prestressing reinforcement wires (**c**). The contacts between macro-elements were attributed mesh refinement with a magnitude of 0.50. Based on previously mentioned parameters, an automatic mesh generator implemented in the ATENA 3D software generated meshes and data of the FE model. The ATENA FE library includes 3D solid tetrahedral element CCIso Tetra, which the detailed mathematical description containing interpolation functions is given in [40]. The geometry of the used tetrahedral element CCIso Tetra is presented in Figure 16.  
The Fracture-Plastic \"3D Nonlinear Cementitious 2\" Model combines constitutive models for tensile--fracturing, and compressive--plastic behaviour. The fracture model of the numerical simulation is established on the classical orthotropic smeared crack formulation and crack band model. The analysis employs the Rankine failure criterion, exponential softening, and it can be used as a rotated or fixed crack model. The Menetrey-Willam failure surface is the basis for the hardening and softening plasticity model. Eventually, the model operates a return mapping algorithm for the integration of constitutive equations. The development of an algorithm for the combination of the two models is considered. The combined algorithm is based on a recursive substitution, and it allows for the two models to be developed and formulated separately. The material model formulation is established on the strain decomposition into elastic, plastic, and fracturing components. More information about the Fracture-Plastic Constitutive Model can be found in [40]. The relation between uniaxial stress and strain for concrete material can be seen in Figure 17a. In performed numerical analysis, the prestressing reinforcement was modelled as discrete in the form of reinforcement bars (truss elements). The assumption of elastic-perfectly plastic (bilinear law) was applied.