## 5 Numerical Results
### Reinforced Concrete Beam
#### 5.1.2 Analysis of the Load-Carrying Capacity and Displacement  
Figure 9 presents the numerical results for the C1 beam using the dynamic relaxation method with the arc-length parameter (DRM + AL). These results were compared with experimental, theoretical, and numerical finite element method results that were taken from the literature. The diagram illustrates the transversal displacements in the mid-span of the C1 beam in node 15 as the load function. The solid line represents the results using the DRM + AL with limit the load value \(P_{0}^{\text{DRM+AL}}\,=\,72.48\,\text{kN}\) and displacement value \(\omega_{0}^{\text{DRM+AL}}\,=\,4.0\,\text{cm}\). These results were compared to [PERSON]'s experimental results (double thin solid line), and the numerical results of [PERSON] [35] (dense dashed line), and [PERSON] [36] (dotted line). The numerical results presented in Reference [35] by [PERSON] were made by combining the procedures of the finite elements method with the arc-length method using ANSYS. The basis of the numerical results that are presented in Equation [36] by [PERSON] was developed using the calibration of the finite element model in ANSYS. The sparse, dashed line is the theoretical limit load-carrying capacity that is determined according to principles of the plasticity theory for bar elements.  
\[P_{0}=\frac{M_{0}}{l_{1}} \tag{35}\]  
where \(M_{0}\) is the limit bending moment for the reinforced concrete cross-section, and \(l_{1}\) is the distance from the support axis to the application point of force \(P\). The theoretical limit load-carrying capacity of the beam calculated in this way is \(P_{0}\,=\,74.32\,\text{kN}\).  
These numerical results agree with [PERSON]'s experimental results. [PERSON] experimentally determined the limit displacement value \(\omega_{0}^{B}\,=\,9.27\,\text{cm}\). This displacement was achieved by the DRM + AL for load value \(P_{0}^{\text{DRM+DL}}\,=\,75.48\,\text{kN}\), which agrees with the experimental results by more than 98%.  
The arc-length method allowed for observing the limit points for the displacement load function. There is a local softening area in Figure 9, which characterizes a decrease in the load that is related to an increase in the displacement. This phenomenon has been observed in experiments when the first  
Figure 9: The numerical results using the proposed method for node 15 of the C1 beam compared with experimental and theoretical results from [PERSON] [27], [PERSON] [28], [PERSON] [35], and [PERSON] [36].  
crack appears and when the cracking zone increases in size. After the reinforcing steel achieved a full plastic state, subsequent softening areas were also observed.  
The observed nature of the changes in transverse displacements and rotations over time indicates that the proposed numerical method is stable for the pseudo-time step \(\Delta t=4.74\times 10^{-5}\), which was determined using Equation (29).  
The stress state in subsequent effort steps of the C1 beam is presented in Figure 10. The load steps are marked by numbers \(\left|1\right|-\left|6\right|\) and correspond with the load steps that are indicated on the graph in Figure 9. The development of the element's effort is observed from the elastic states \(\left|1\right|\) through the appearance of the first crack \(\left|2\right|\) to the plasticizing of the reinforcing steel \(\left|4\right|\) and the propagation of concrete softening processes \(\left|5\right|\) and \(\left|6\right|\), in particular, areas of the structural element. The tracking of the stress state allows for a detailed description of the degradation in the critical cross-section leading to the destruction of the structural element.  
Figure 10: Distribution of normal stresses depending on the load of C1 beam.