# Predicting Tensile Strength for Prestressed Reinforced Concrete-Driven Piles
## 2 Tensile Strength of Reinforced-Concrete-Driven Piles  
The AASHTO [5] Standard and FHWA [6] guidelines prescribed the tensile strength of concrete as:  
\[\sigma_{dr}=\varphi_{da}\Big{(}0.25\sqrt{f_{c}^{\prime}}+f_{pe}\Big{)} \tag{1}\]  
where \(\sigma_{dr}\) is the driving tensile-stress limit or tensile strength (MPa); \(\varphi_{da}\) is the resistance factor during driving (1.0 for concrete piles); \(f^{\prime}c\) is the concrete compression strength  
\begin{table}
\begin{tabular}{l c c c} \hline \hline \begin{tabular}{c} **Standards** \\ **(or Codes)** \\ \end{tabular} & **Correlations in MPa** & \begin{tabular}{c} **Tensile Strength** \\ **Testing Methods** \\ \end{tabular} & \begin{tabular}{c} **Type of Compressive** \\ **Samples** \\ \end{tabular} \\ \hline \hline \begin{tabular}{c} ACI 318-2002 [7] \\ \end{tabular} & \(f_{r}=0.62\sqrt{f_{cl}^{\prime}}\) & Modulus of Rupture & Cylinder \\ \hline \begin{tabular}{c} BS 8007 [8] \\ \end{tabular} & \(f_{t}=0.12f_{c}\)\({}^{0.27}\) & Split Cylinder & Cube \\ \hline \begin{tabular}{c} CEN (2002) [9] \\ \end{tabular} & \(f_{r}=0.342f_{c}^{2/3}\) & Modulus of Rupture & Cylinder \\ \hline \begin{tabular}{c} JCI 2016 [10] \\ \end{tabular} & \(f_{t}=0.13f_{c}^{0.85}\) & Split Cylinder & Cylinder \\ \hline \begin{tabular}{c} JSCE 2002 [11] \\ \end{tabular} & \(f_{t}=0.44f_{c}^{0.5}\) & Split Cylinder & Cylinder \\ \hline \begin{tabular}{c} AJJ 2008 [12] \\ \end{tabular} & \(f_{t}=0.18f_{c}^{0.75}\) & Split Cylinder & Cylinder \\ \hline \begin{tabular}{c} CAN/CSA A23.3-04 (2007) [13] \\ \end{tabular} & \(f_{r}=0.6f_{c}^{0.5}\) & Modulus of Rupture & Cylinder \\ \hline \begin{tabular}{c} NZS-3101 [14] \\ \end{tabular} & \(f_{r}=0.6f_{c}^{0.5}\) & Modulus of Rupture & Cylinder \\ \hline \begin{tabular}{c} IS 456-2000 [15] \\ \end{tabular} & \(f_{r}=0.626f_{c}^{0.5}\) & Modulus of Rupture & Cylinder \\ \hline \
\begin{tabular}{c} CEB-FIB 2010 [16] \\ \end{tabular} &
\begin{tabular}{c} \(f_{r}=0.3f_{c}^{0.5}\) \\ (For concrete grades \(\leq\) C50) \\ \end{tabular} & Axial tension & Cylinder \\ \hline \hline \end{tabular}
\end{table}
Table 1: Tensile Strength and Compressive Strength Empirical Correlations in Standards.  
at 28 days, unless otherwise specified (MPa); \(f_{pe}\) is the effective prestressing stress in concrete (MPa).  
Figure 1 presents the modulus of rupture data from the 1330 tests from research over the past 80 years [22; 23; 24; 25]. These data can be used to evaluate Equation (1). The observations from the data are: (1) when the compressive strength of concrete is greater than 50 MPa, the correlation of the compressive and tensile strengths of concrete, according to ACI 363R-10 [24], which is  
\[f_{r}=0.623\sqrt{f_{c}^{\prime}} \tag{2}\]  
corresponds to the lower boundary of the data points, and (2) when the compressive strength of concrete is less than 50 MPa, the data lower bound is the \(f_{r}\) curve, which is  
\[f_{r}=0.415\sqrt{f_{c}^{\prime}} \tag{3}\]  
As in Equation (1), AASHTO [5] recommends using the tensile strength of concrete at \(0.25\sqrt{f_{c}^{\prime}}\). Compared with the observations from Figure 1, it can be seen that the AASHTO [5] approach is significantly conservative, and especially when the compressive strength is greater than 50 MPa. It is proposed that the equation for the tensile strength should be adjusted to:  
\[\sigma_{dr}=\varphi_{da}\Big{(}k\times\sqrt{f_{c}^{\prime}}+f_{pe}\Big{)} \tag{4}\]  
where \(k=0.415\) for \(f_{c}^{\prime}<50\) MPa and \(k=0.623\) for \(f_{c}^{\prime}\geq 50\) MPa. Equation (4) predicts the tensile strength of the piles more accurately. It should be noted that the compressive strength itself is a parameter that already includes a factor of safety; therefore, the tensile strength, when determined through the compressive strength, does not require an additional factor of safety. Therefore, Equation (4) also provides the maximum allowed induced tensile stress during pile driving.  
Figure 1: Relationship between Modulus of Rupture and Compressive Strength of Concrete (Data from 1330 tests).