## 5 Calculation Model for Calculating the Remaining Load-Bearing Capacity  
The theoretical bases and preconditions for the development of a calculation model for the calculation of the remaining load-bearing capacity of compressed damaged reinforced concrete elements of different flexibilities were as follows. Basic prerequisites for the calculation:  
1. For calculating, the average cross-section of the element was considered as the one in which there were the greatest stresses;
2. The hypothesis of flat sections was accepted;
3. Models with heights of 1.0 m were considered short, and flexibility was not taken into account. In samples with lengths of 1.75 m and 2.5 m, the effect of deflection was proposed to be taken into account by multiplying the moment by the coefficient \(\eta\), which was found using a method that took into account the change in stiffness in the damaged area;
4. The work of naked reinforcing bars was taken into account by introducing coefficients that took into account its flexibility;
5. The stresses in the compressed zones of concrete were evenly distributed and taken as equal to \(f_{cd}\):
6. The forces in the stretched zone were completely perceived by the reinforcement, and the work of the stretched concrete was not taken into account in the perception of external forces;
7. The tensile stress in the reinforcement was not more than the calculated tensile strength \(f_{t}\), and the compressive strength was no more than \(f_{yd}\). Stresses in the reinforcement were determined based on the position of the neutral line and the height of the compressed zone of concrete;
8. The force planes of the outer and inner pairs of forces coincided or were parallel. The approach of taking into account the deflection separately in each of the planes was chosen because, in the columns of real building structures, the calculated lengths of the columns in the plane and from the plane, as a rule, have different values. Figure 8 shows the calculated scheme of a reinforced concrete column with damage in the middle that does not lead to the rotation of the neutral line, supposing that the stiffness of the column of the undamaged part \(D_{1}\) damaged \(D_{2}\), while \(D_{1}>D_{2}\). This problem was solved using the well-known theory for the stability of the classical resistance of materials. Figure 8: Determination of critical force: (**a**) calculation scheme and (**b**) boundary conditions. The differential equations of the curved axis for each of the parts of the column are presented in Formula (1):  
\[\left\{ \begin{array}{l}\frac{\partial^{2}f_{1}(x_{1})}{\partial x_{1}^{2}}+k_{1} ^{2}\cdot f_{1}(x_{1})=0;\\ \frac{\partial^{2}f_{2}(x_{2})}{\partial x_{2}^{2}}+k_{2}^{2}\cdot f_{2}(x_{2}) =0\end{array} \right. \tag{1}\]  
In Formula (1), \(k_{1}^{2}=\frac{P}{D_{1}};k_{2}^{2}=\frac{P}{D_{2}};D_{1},D_{2}\) is the stiffness of the undamaged and damaged parts of the columns; P is the compressive force; \(f_{1}(x_{1})\), \(f_{2}(x_{2})\) is the deflection function for the corresponding section; and \(x_{1}\), \(x_{2}\) are the corresponding coordinates of the points. The solution to Formula (1) is usually sought as follows in Equation (2):  
\[\left\{ \begin{array}{l}f_{1}(x_{1})=A\cdot\sin(k_{1}\cdot x_{1})+B\cdot\cos (k_{1}\cdot x_{1});\\ f_{2}(x_{2})=C\cdot\sin(k_{2}x_{2})+D\cdot\cos(k_{2}\cdot x_{2})\end{array} \right. \tag{2}\]  
The boundary conditions for the rod (Figure 8) are given in Equation (3):  
\[f_{1}(0)=0;f_{1}(l_{1})=f_{2}(0);\frac{\partial f_{1}(l_{1})}{\partial x_{1}} =\frac{\partial f_{2}(0)}{\partial x_{2}};\frac{\partial f_{2}(l_{2})}{ \partial x_{2}}=0 \tag{3}\]  
After simple transformations, a homogeneous linear formula, Equation (4), is obtained:  
\[\left\{ \begin{array}{l}B=0\\ A\cdot k_{1}\cdot\cos(k_{1}l_{1})=C\cdot k_{2};\\ A\cdot\sin(k_{1}l_{1})=D;\\ C\cdot\cos(k_{2}l_{2})-D\cdot\sin(k_{2}l_{2})=0\end{array} \right.