## 5 Example of Computations  
Numerical calculations of the temporary temperatures and phase share computations were conducted for single-pass SAW overlaying S355J2G3 steel plate. The dimensions of the board used in the padding test and the geometric parameters of the applied padding weld (bead dimensions \(h_{w}\) = 2.6 mm and \(w_{w}\) = 18 mm (\(d_{p}\) = 0)) were used for the calculations. The same properties of the welded plate and electrode material were adopted in the calculations: \(a\) = 8\(\cdot\)10\({}^{\circ}\) \(\mathrm{m}^{2}\)/s, \(C_{p}\) = 670 J/kg K, \(\rho=\rho_{e}\) = 7 800 kg/m\({}^{3}\) and \(L\) = 268 kJ/kg. The power of the heat source was determined by the welding parameters used in the experiment, i.e. \(U\) = 30 V and \(I\) = 400 A. When the efficiency \(\eta\) = 0.99 is taken into account, the assumed power equals 11 880 W. The Gaussian surface distribution was determined by \(\tau_{0}\) = 0.13 s, and volumetric was determined by \(z_{0}\) = 0.008 m.  
The initial electrode temperature was taken at \(T_{e}\) = 100 \({}^{\circ}\)C [45]. Numerical simulations were performed for the cross-section of the plate in half the length of the pad. The calculations were carried out using our own procedures written in the Borland Pascal structural language and made in the Delphi environment. In the area of the padding and the heat affected zone, the distance between the calculation points was 0.5 mm, while in the remaining area 1 mm. The time increment was 0.0002 s in integration functions during temperature calculations. Maximum temperature peaks at individual cross-section points made it possible to determine the maximum temperature isotherms (Figure 6). In turn, isotherms of critical temperatures of phase transformations allowed us to determine the boundaries of individual characteristic areas (Figure 7). The solidus value of 1493 \({}^{\circ}\)C was determined based on the iron-carbon system phase diagram. In the case of determining temperature limits of phase transformations in the solid state, the average heating speed was taken into account at points where the max peak temperature exceeded 720 \({}^{\circ}\)C (\(A_{1}\) temperature value in classical heat treatment processes). Figure 5: Diagram of supercooled austenite transformations. Considering the effect of this average heating speed at the start and at the end temperatures of the austenitic transformation [51; 52] the temperature values \(A_{3}\) = 920 \({}^{\circ}\)C and \(A_{1}\) = 748 \({}^{\circ}\)C were determined. The calculations of individual structural component shares during the phase transformation of heating were made in accordance with the Equations (16)-(18). The constants \(b_{i}\) and \(n_{i}\) were determined depending on the values \(A_{1}\) and \(A_{3}\). In the calculation algorithm of the phase transformation kinetics of cooling, the temperatures of the start and the end of phase transformations are determined using the TTT diagram of S355 steel (Figure 8) developed based on [53], while the volumetric shares of individual components are determined using Equations (19)-(23). Distributions of individual phases in the cross-section after the process is shown in Figures 9-11. The share of bainite in the padded weld varies from 100% (in weld corners) to 69% in its middle part (Figure 9). The volume shares of ferrite and pearlite vary from 28% and 3% in the weld to 70% and 30% in native material, respectively (Figures 10 and 11). Figure 8: Time-temperature-transformation (TTT) diagram of S355 steel. Figure 6: Maximum temperature field (\({}^{\circ}\)C). Figure 7: Cross section with determined heat affected zone. Figure 11: Distribution of ferrite share in padded weld. Figure 10: Distribution of pearlite share in padded weld. Figure 9: Distribution of bainite share in padded weld. Figure 14: Thermal history at point 3: \(T\)—temperature; \(A\), \(B\), \(F\), \(P\)—volume fractions of austenite, bainite, ferrite and pearlite, respectively.