## 3 Fundamental Equations
### Differential Discretization of the Bar Element  
The equilibrium Equation (9), geometric relationships (10), material models for reinforcing steel (1) and concrete (8), and the cross-sectional model defined by Equations (11) and (12) form the problem within the technical theory of bar structures. The basic system of equations is presented in differential form on the basis of the proposed discretization of the computational model. To discretize the model, we divided the central axis of the bar into nodes with coordinates \((x,z)_{i}\), for \(i=1,\,2,\ldots,\,i-1,\,i,\,i+1,\,\ldots,\,I\), which is shown in Figure 6.  
Figure 5: Model of the reinforced concrete cross-section. In the proposed model the element was divided into inner and boundary nodes. The internal nodes contain the main nodes (odd), which correspond to the division points, and the intermediate nodes (even), which correspond to the line segments that join the division points. The main nodes are treated as cross-sections in which the curvatures and the bending moments are accurately calculated. Then, the longitudinal and transverse forces are determined using the average longitudinal strains on the central axis and the angles of the non-dilatational strains determined in adjacent segments. For the intermediate nodes, the longitudinal strains of the central axis, the angles of the non-dilatational strains, and the longitudinal and transverse forces are accurately calculated. Then, the bending moments are determined using the average curvatures of the adjacent main nodes. After including differential discretization of the bar element, the equilibrium Equation (9) can be rewritten for the nodes of the spatial division in the form below. \[\left\{ \begin{array}{c}-N_{i1}\cos\theta_{i1}+N_{i0}\cos\theta_{i0}+Q_{i1} \sin\theta_{i1}-Q_{i0}\sin\theta_{i0}+P_{xi}-m(s_{i})\ddot{u}_{i}-C_{i}\dot{u}_ {i}=0\\ -N_{i1}\sin\theta_{i1}+N_{i0}\sin\theta_{i0}-Q_{i1}\cos\theta_{i1}+Q_{i0}\cos \theta_{i1}-P_{zi}+m(s_{i})\ddot{w}_{i}+C_{i}\dot{w}_{i}=0\\ M_{i+2}-M_{i}-Q_{i1}\Delta s_{i1}-I_{i1}\ddot{\Phi}_{i1}-C_{f1}\dot{\Phi}_{i1}= 0\end{array} \right. \tag{13}\]  
where \(i1=i+1\) and \(i0=i-1\) represent the segments of the internal spatial division, \(N_{i1}\) and \(Q_{i1}\) are the longitudinal and transverse forces in segment division \(i1=i+1\), \(M_{i}\) is the bending moment on main node \(i\) and \(P_{xi}\), \(P_{zi}\) are the components of the load in node \(i\). The lumped mass of main node \(i\) is below. \[m(s_{i})=\mu(s_{i})\Delta\overline{s}_{i} \tag{14}\]  
where \(\Delta\overline{s}_{i}=\frac{\Delta s_{i1}+\Delta s_{i0}}{2}\)--average length of the segment division, \(\mu(s_{i})=\rho_{cs}A_{cs}\) is the unitary mass of the reinforced concrete element with cross-section \(A_{cs}\), and \(\rho_{cs}\) is the density of the reinforced concrete. The unitary mass inertia moment of the reinforced concrete segment \(j(s_{i1})=\rho_{cs}l_{cs}^{0}\) is characterized by the central (main) moment of the rotary inertia \(J_{cs}^{o}\). The damping factor for displacements in the main node \(i\) determines the relationship. \[C_{i}=c(s_{i})\Delta\overline{s}_{i} \tag{15}\]  
Figure 6: Discretization of the bar’s structural element. The damping factor for the rotations of the segment division \(i=2,\;I-1,\;2\) is below. \[C_{fi}=c_{f}(s_{i})\Delta s_{i} \tag{16}\]  
in which \(c(s_{i})=\alpha\mu(s_{i})\) is the unitary damping factor for the displacement of the reinforced concrete element characterized by a frequency and vibration period of \(\alpha=2\omega_{o}=\frac{4\pi}{I}\) and \(c_{f}(s_{i1})=\alpha_{f}j(s_{i1})\) is a unitary damping factor for the rotation of the reinforced concrete element that is characterized by a frequency and rotational vibration period of \(\alpha_{f}=2\omega_{o}=\frac{4\pi}{I_{f}}\). Let  
\[\left\{ \begin{array}{c}\alpha_{kr}=2\omega_{o,kr}=\frac{4\pi}{I_{kr}}\\ \alpha_{f,kr}=2\omega_{o,fkr}=\frac{4\pi}{I_{f,kr}}\end{array} \right.