## 5 Calculation Model for Calculating the Remaining Load-Bearing Capacity
### Damage at an Angle  
To determine the value of the bearing capacity of a sample, it is necessary to find the height of compressed zone \(x\) perpendicular to the neutral line, which passes at angle \(\varphi\).  
The equilibrium equation of limiting forces is as follows (13):  
\[N-f_{cd}\cdot A_{c}-\sum_{i=1}^{n}\sigma_{s1-s4}\cdot A_{s1-s4}=0 \tag{13}\]  
where \(A_{c}\) is the area of the compressed zone of concrete. The stresses in the reinforcing bars are found using empirical Formula (11).  
Under oblique, out-of-center compression, the additional condition of parallelism of the planes of action for the moments of external and internal forces dictates that point A (application of external force) (Figure 10), point B (equal of compressive forces in concrete and reinforcement), and point C (equal forces in stretched armature) should lie on one line.  
At inclined damage, there were more unknown quantities than at a direct one.  
We included the following as unknowns:  
N--the bearing capacity of the sample according to the calculation;  
x--the height of the compressed cross-sectional area;  
\(\varphi\)--the angle of inclination of the neutral line;  
\(\beta_{i}\prec\beta\)--values that must be found to describe the position of the coordinates for the center of mass of the compressed zone of concrete.  
Figure 10: Condition of parallelism of force planes.  
We had five unknown quantities, that is, it was necessary to compile five equations that included these quantities.  
The first equation, in the case of direct damage, was the equation of equilibrium relative to the x-axis; the second and third equations were sums of the moments relative to the x- and y-axes. The fourth and fifth equations were the equations of the static moments of the compressed zone of concrete. Their compositions were possible due to the assumption that the stresses were uniform over the area.  
\[N\cdot e-f_{cd}\cdot A_{c}\cdot x_{c}-\sigma_{s1-s2}\cdot A_{s1-s2}\cdot h_{ 01-02}+\sigma_{s3-s4}\cdot A_{s3-s4}\cdot a^{\prime}=0 \tag{14}\]  
\[N\cdot\frac{b}{2}-\sigma_{1}\cdot A_{1}\cdot\left(a^{\prime}+\frac{d}{2} \right)-\sigma_{2}\cdot A_{2}\bigg{(}b-a^{\prime}-\frac{d}{2}\bigg{)}+\sigma_ {3}\cdot A_{3}\cdot a+\sigma_{4}\cdot A_{4}\cdot\left(b-a^{\prime}-\frac{d}{2} \right)-f_{cd}\cdot y_{c}\cdot A_{c}=0 \tag{15}\]  
\[Sx_{1}=A_{1}y_{1}-A_{2}y_{2} \tag{16}\]  
\[Sy_{1}=A_{3}x_{3}-A_{4}x_{4} \tag{17}\]  
The average value of the ratio between the experimental bearing capacity and the one determined by the proposed method is as follows:  
\[\left(\frac{\overline{N}}{N_{\textit{desp}}}\right)=\frac{1}{n}\sum_{i=1}^{n} \left(\frac{N}{N_{\textit{desp}}}\right)=1.26 \tag{18}\]  
The root mean square deviation according to the calculation results is as follows:  
\[\sigma=\sqrt{\frac{1}{n-1}}\sum_{i=1}^{n}\left[\left(\frac{N}{N_{\textit{desp} }}\right)-\left(\frac{\overline{N}}{N_{\textit{desp}}}\right)\right]^{2}=0.375 \tag{19}\]  
The coefficient of variation is as follows:  
\[\
u=\frac{\sigma}{\left(\frac{N}{N_{\textit{desp}}}\right)}=0.297 \tag{20}\]  
Thus, the method of determining the bearing capacities of damaged columns that is described in this section was sufficiently accurate and could be based on verification calculations in cases of detection for similar situations.  
Algorithms for the calculation of damaged, compressed, and obliquely compressed elements were developed that, thanks to which, it was possible to find unknown quantities.