# Bearing Capacity near Support Areas of Continuous Reinforced Concrete Beams and High Grillages
## 2 Materials and Methods
### Modeling the Complex Stress-Strain State of Research Elements
Empirical expressions for the coefficients \(\mathbf{\kappa}_{c}\) and \(\beta_{w}\) were obtained from the corresponding adequate mathematical models with good information content by replacing the coded values of design factors with natural values:  
\[ \begin{array}{l}\alpha_{s}=1.65-0.94(a/h_{0}-2)-0.13(\frac{s-25 MRa}{10 MRa})-0.15\Big{(}\frac{\varrho_{w}-0.0034}{0.0016}\Big{)}+0.29\times\Big{(}\frac{ \varrho_{w}-0.015}{0.005}\Big{)}+\\ +0.26\Big{(}\frac{\varrho_{Ly}-0.015}{0.005}\Big{)}-0.49(a/h_{0}-2)^{2}+0.14(a/ h_{0}-2)\times\Big{(}\frac{c-25 MRa}{10 MRa}\Big{)}-0.31(a/h_{0}-2)\times\\ \times\Big{(}\frac{\varrho_{Lt}-0.015}{0.005}\Big{)}-0.006\times\Big{(}\frac{s -25 MRa}{10 MRa}\Big{)}\Big{(}\frac{\varrho_{Lt}-0.0034}{0.0016}\Big{)}-0.05 \Big{(}\frac{s-25 MRa}{10 MRa}\Big{)}\times\Big{(}\frac{\varrho_{Lb}-0.015}{0.00 5}\Big{)}+\\ +0.07\times\Big{(}\frac{\varrho_{w}-0.0034}{0.0016}\Big{)}\times\Big{(}\frac{ \varrho_{Lt}-0.015}{0.005}\Big{)},v=7.9\%;\end{array}  \tag{12}\]  
\[ \begin{array}{l}\beta_{w}=0.61-0.17(a/h_{0}-2)-0.17(\frac{s-25 MRa}{10 MRa})-0.03\Big{(}\frac{\varrho_{w}-0.0034}{0.0016}\Big{)}+0.10\times\\ \times\Big{(}\frac{\varrho_{Lt}-0.015}{0.005}\Big{)}+0.15\Big{(}\frac{\varrho_{ Lt}-0.015}{0.005}\Big{)}-0.04\Big{(}\frac{\varrho_{Lt}-0.015}{0.005}\Big{)}^{2}+0.03 (a/h_{0}-2)\times\\ \times\Big{(}\frac{\varrho_{Lt}-0.015 MRa}{0.005 MRa}\Big{)}-0.03(a/h_{0}-2)\Big{(} \frac{\varrho_{Lt}-0.015}{0.005}\Big{)}+0.006\Big{(}\frac{s-25 MRa}{10 MRa}\Big{)} \times\Big{(}\frac{\varrho_{w}-0.0034}{0.0016}\Big{)}-\\ -0.06\Big{(}\frac{\varrho_{w}-0.0034}{0.0016}\Big{)}\times\Big{(}\frac{\varrho _{Lt}-0.015}{0.005}\Big{)}-0.03\times\Big{(}\frac{\varrho_{w}-0.0034}{0.0016} \Big{)}\times\Big{(}\frac{\varrho_{Lt}-0.015}{0.005}\Big{)},v=6.8\%.\end{array}  \tag{13}\]  
For beams of rectangular section, expression (11) takes the form:  
\[F_{v}=F_{c}+F_{sw}=\alpha_{c}f_{ctt}2 bh_{0}+\beta_{w}f_{ywd}\{[c+2(s_{0}/h_{0} )h_{0}]/S_{w}\}A_{sw}, \tag{14}\]  
where \(\mathbf{c}\) is the width of the area (pile caps) of the transmission of the support reaction;  
\(\mathbf{A}_{sw}\) is the cross-sectional area of the transverse rods in one transverse plane;  
\(\mathbf{c}_{0}/\mathbf{h}_{0}\) is the relative length of the horizontal projection of the dangerous inclined crack, which is recommended to be (15) obtained from the mathematical model (8) by replacing the coded values of research factors with their natural values:  
\[ \begin{array}{l}c/h_{0}=1.37+0.56(a/h_{0}-2)-0.06(\frac{s-25 MRa}{10 MRa})-0.0 7\Big{(}\frac{\varrho_{w}-0.0034}{0.0016}\Big{)}-0.21\times\Big{(}\frac{ \varrho_{Lt}-0.015}{0.005}\Big{)}+\\ +0.17\Big{(}\frac{\varrho_{Lt}-0.015}{0.005}\Big{)}-0.05\Big{(}\frac{\varrho_{ Lt}-0.015}{0.005}\Big{)}^{2}-0.05(a/h_{0}-2)\times\Big{(}\frac{c-25 MRa}{10 MRa}\Big{)}-0.07(a/h_{0}-2)\times\\ \times\Big{(}\frac{\varrho_{w}-0.0034}{0.0016}\Big{)}-0.15(a/h_{0}-2)\Big{(} \frac{\varrho_{Lb}-0.015}{0.005}\Big{)}+0.12(a/h_{0}-2)\Big{(}\frac{\varrho_ {Lt}-0.015}{0.005}\Big{)}-0.05\Big{(}\frac{\varrho_{w}-0.0034}{0.0016}\Big{)} \times\\ \times\Big{(}\frac{\varrho_{Lb}-0.015}{0.005}\Big{)}+0.05\Big{(}\frac{\varrho_{ Lb}-0.0034}{0.0016}\Big{)}\Big{(}\frac{\varrho_{Lt}-0.015}{0.005}\Big{)}-0.03 \Big{(}\frac{\varrho_{Lb}-0.015}{0.005}\Big{)}\Big{(}\frac{\varrho_{Lt}-0.015}{0.005}\Big{)},v=6\%.\end{array}  \tag{15}\]  
When determining the normative value of the forcing force \(\text{F}_{\text{b}}\) according to Equation (9) and (12), \(\mathbf{f}_{ctt}\) and \(\mathbf{f}_{yxt}\) should be taken instead of \(\mathbf{f}_{ctd}\) and \(\mathbf{f}_{yxd}\). Comparison of the experimental values and those calculated by the proposed engineering method of the bearing capacity of the experimental continuous beams/gfillages for punching over the middle support showed satisfactory convergence (v \(\leq\) 9%).