# Experimental Study on Mechanical Properties of Graded EPS-Steel Fiber-Reinforced Concrete
## 3 Analysis of Experimental Phenomena and Results
### Conversion Formula and Accuracy of Compression Ratio  
There are few existing formulas for calculating the bending strength of graded EPS-fiber-reinforced concrete. Based on the bending strength calculation formula for ordinary concrete, combined with the formula for calculating the bending strength of recycled aggregate graded concrete proposed by [PERSON] [32], the (\(f_{f}=0.75\sqrt{f_{cu}}\)) formula was used to predict the bending strength of graded EPS-steel fiber concrete in this study. It was found that the predicted values showed significant deviation from the experimental values. The reason for this discrepancy lies in the differences in aggregates and grading, which cause substantial variations in the concrete skeleton structure, leading to a reduced accuracy of the formula. As a result, the prediction formula is no longer effective for estimating the bending strength of graded EPS-steel fiber concrete.  
Considering that analyzing graded EPS or steel fibers in isolation cannot accurately predict the impact of their coupling on the bending-to-compression ratio of concrete, a coefficient equation (\(f(\alpha,\beta)\)) was introduced. Using the experimental data from this study,  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline
**EPS** & **Steel Fiber** & \multicolumn{5}{c}{**Stress Peak Value (MPa)**} & \multicolumn{5}{c}{**Growth Rate (\%)**} \\ \cline{3-13}
**(\%)** & **(\%)** & **C30A** & **C30a** & **C40B** & **C40b** & **C50C** & **C50c** & **A** & **a** & **B** & **b** & **C** & **c** \\ \hline \multirow{3}{*}{15} & 1.5 & 6.18  
the average values of the bending-to-compression ratios corresponding to the gravel group and fine aggregate group were taken as shown in Figure 9. The variation range of the class bending-to-compression ratio [33] (\(f_{f}/\sqrt{f_{cu}}\)) values is within 20%, and merging the data to calculate the average value has minimal impact on the deviation of the results. Through regression analysis, the formula coefficients for the bending-to-compression ratio under the coupled effect of graded EPS and steel fibers were adjusted.  
\[f_{f}=\gamma(\alpha,\beta)\sqrt{f_{cu}} \tag{1}\]  
\[\gamma(\alpha,\beta)=A+A_{1}\alpha+A_{2}\beta+A_{3}\alpha^{2}+A_{4}\beta^{2}+A _{5}\alpha\beta\text{--Correction coefficient formula} \tag{2}\]  
The value of \(\alpha\) ranges from 1.5 to 2.25, and the value of \(\beta\) ranges from 15 to 35.  
In the equation, \(\alpha\) represents the steel fiber volume fraction, \(\beta\) represents the graded EPS volume fraction, and \(A,\ A_{1},\ A_{2},\ A_{3},\ A_{4},\ A_{5}\) is the variable control parameter. The formula for the concrete-like bending-to-compression ratio under the coupling effect of graded EPS and steel fibers is derived through fitting. The resulting formula coefficients are as shown in Equation (2) \(\gamma(\alpha,\beta)=2.56-1.4\alpha+0.0045\beta+0.4\alpha^{2}+7.68\beta^{2}-0.012\alpha\beta\). The adjusted RRR value is 0.989, The numerical accuracy of the test values and the prediction model in this paper is shown in Figure 10, indicating a good fit.  
After deriving the specific control parameters, Equation (2) is substituted back into Equation (1) to obtain the modified formula for the flexural strength of graded EPS-steel fiber concrete, which is then validated using the compressive strength test values. By substituting \(\frac{\partial\gamma(\alpha,\beta)}{\partial\alpha}=0\) and \(\frac{\partial\gamma(\alpha,\beta)}{\partial\beta}=0\) from Equation (2), the maximum bending-to-compression ratio under the combined effect of graded EPS and steel fibers can be calculated, allowing for the prediction of the optimal volume fraction.  
Figure 9: Curve fitting of flexural compression ratio under the coupling effect of graded EPS and steel fibers.