Table 4 Analytical models presenting relationships σ-ε during the compression of concrete reinforced with steel fibers.

Ezeldin and Balaguru¹¹

Equation (6)

\({f}_{c}={f}_{c}{\prime}+\mathrm{11,232}{V}_{f}\lambda ; \beta =\mathrm{1,093}+0.2429{V}_{f}{\lambda }^{-0.926};\)

\({\varepsilon }_{0}={\varepsilon }_{0}{\prime}+1427\cdot 1{0}^{-6}{V}_{f}\lambda\)  

Someh and Saeki^37,38

Equation (6)

\(\beta =\mathrm{1,032}[{f}_{c}\left(1+{V}_{f}\lambda \right){]}^{0.113}; {\varepsilon }_{0}=1.84\cdot 1{0}^{-3}{f}_{c}^{0.147}\)  

Nataraja et al.²²

Equation (6)

\({f}_{c}={f}_{c}{\prime}+6.9133{V}_{f}\lambda ; \beta =0.5811+0.8155{V}_{f}{\lambda }^{-0.7406};\)

\({\varepsilon }_{0}={\varepsilon }_{0}{\prime}+1.92\cdot 1{0}^{-3}{V}_{f}\lambda\)  

Mansur et al.⁵

Equation (6) dla \(0\le \frac{\varepsilon }{{\varepsilon }_{0}}\le 1;\)

\(\sigma ={f}_{c }\frac{{k}_{1}\beta \left(\frac{\varepsilon }{{\varepsilon }_{0}}\right)}{{k}_{1}\beta -1+{\left(\frac{\varepsilon }{{\varepsilon }_{0}}\right)}^{{k}_{2}\beta }} dla 1<\frac{\varepsilon }{{\varepsilon }_{0}};\)

\(\beta =\frac{1}{\left[1-\left(\frac{{f}_{c}}{{\varepsilon }_{0}E}\right)\right]}; E=\left(10300-4000{V}_{f}\right){f}_{c}^{0.33} ;\)

\({\varepsilon }_{0}=(5\bullet 1{0}^{-4}+7.2\bullet 1{0}^{-4}{V}_{f}\lambda ){f}_{c}^{0.35}\)

Baros et al.¹⁰

\(\sigma ={f}_{c }\frac{\frac{\varepsilon }{{\varepsilon }_{0}}}{\left(1-p-q\right)+q\left(\frac{\varepsilon }{{\varepsilon }_{0}}\right)+p{\left(\frac{\varepsilon }{{\varepsilon }_{0}}\right)}^{\frac{1-q}{p}}};\)

\(q=1-p-\frac{{E}_{1}}{E}; p=1-0.919exp\left(-0.394{W}_{f}\right); {E}_{1}=\frac{{f}_{c }}{{\varepsilon }_{0}};\)

\(E=21500{\left(\frac{{f}_{c}}{10}\right)}^{0.33}; {W}_{f}=\frac{{w}_{f}}{\rho }; {\varepsilon }_{0}=2.2\cdot 1{0}^{-3}+2\cdot 1{0}^{-4}{W}_{f};\)  

\({W}_{f}=\frac{{w}_{f}}{\rho }\); ρ—gęstość kompozytu [kg/m³]; w_f—zawartość włókien stalowych [kg/m³]

Ou et al.²¹

Equation (6)

\({f}_{c}={f}_{c}{\prime}+2.35{V}_{f}\lambda ; \beta =0.75({V}_{f}\lambda {)}^{2}-2{V}_{f}\lambda +3.05;\)

\({\varepsilon }_{0}={\varepsilon }_{0}{\prime}+7\cdot 1{0}^{-4}{V}_{f}\lambda\)  

Lee et al.⁸

\(\sigma ={f}_{c }\frac{A\left(\frac{\varepsilon }{{\varepsilon }_{0}}\right)}{A-1+{\left(\frac{\varepsilon }{{\varepsilon }_{0}}\right)}^{B}}; A=B=\left(\frac{1}{1+\frac{{f}_{c}}{E{\varepsilon }_{0}}}\right) gdy\frac{\varepsilon }{{\varepsilon }_{0}}\le 1;\)

\(A=1+0.723({V}_{f}\lambda {)}^{-0.957} gdy \frac{\varepsilon }{{\varepsilon }_{0}}>1;\)

\(B=\left(\frac{{f}_{c}}{50}\right)\mathrm{0,064}\left(1+0.882{\left({V}_{f}\lambda \right)}^{-0.882}\right)\ge A gdy\frac{\varepsilon }{{\varepsilon }_{0}}>1;\)

\(E=\left(-367{V}_{f}\lambda +5520\right){f}_{c}^{0.41};\)

\({\varepsilon }_{0}=(3\cdot 1{0}^{-4}{V}_{f}\lambda +1.8\cdot 1{0}^{-3}){f}_{c}^{0.12}\)