# Evaluation of Methods Based on CPTu Testing for Prediction of the Bearing Capacity of CFA Piles
## 2 Materials and Methods
### Statistical Analysis of Pile Bearing Capacity  
Several authors carried out the statistical analysis for evaluation of applicability of various methods for estimation/prediction of pile bearing capacity [50; 51; 61; 62]. Statistical analysis allows to classify given methods based on reliability of pile bearing capacity estimation and this approach should be one of the criteria for evaluation and comparison of methods. The ratio of predicted and measured pile bearing capacity \(Q_{p}/Q_{m}\) is important evaluation criterion. However, visual examination of \(Q_{p}/Q_{m}\) plot can be highly subjective. Therefore, additional tools are required for proper evaluation. Overall, four evaluation criteria were adopted to classify selected methods for prediction of pile bearing capacity based on CPT data:  
* Best fit formula of predicted and measured bearing capacity function \(Q_{p}/Q_{m}\) and coefficient of determination \(R^{2}\);
* Mean \(\mu\) and sample standard deviation \(\sigma\) for \(Q_{p}/Q_{m}\);
* A value of 20% reliability level obtained from log-normal distribution of \(Q_{p}/Q_{m}\);
* A value of 50% and 90% cumulative probability \(P_{50}\) and \(P_{90}\). Predicted bearing capacity based on CPT data \(Q_{p}\) is compared with the interpreted bearing capacity obtained from the static load test of pile \(Q_{m}\). The \(Q_{p}/Q_{m}\) ratio is theoretically in interval from 0 to an unlimited upper value with the optimal value of 1.0. If the ratio \(Q_{p}/Q_{m}<1.0\), the method underestimates the bearing capacity and if the ratio \(Q_{p}/Q_{m}>1.0\), the method overestimates the bearing capacity. Mean \(\mu\) and sample standard deviation \(\sigma\) based on \(Q_{p}/Q_{m}\) ratio are important indicators of reliability and efficiency of method. The ideal accurate method has \(\mu(Q_{p}/Q_{m})=1.0\) and \(\sigma(Q_{p}/Q_{m})=0\), which means that every predicted pile bearing capacity is equal to the measured one. In a real case, \(\mu\) is close to 1.0 and \(\sigma\) is close to 0, respectively. As mentioned above, the \(Q_{p}/Q_{m}\) ratio can be from 0 to an unlimited upper value which provides uneven weight to underestimated and overestimated predictions. If \(Q_{p}\leq 0\), the pile has no bearing capacity in the physical point of view but such a conclusion is unrealistic. In a mathematical point of view, such a case is achieved when \(Q_{m}\) is infinitely large and is also unrealistic. Therefore, [PERSON] and [PERSON] proposed the application of log-normal distribution of \(Q_{p}/Q_{m}\) for evaluation of efficiency of pile bearing capacity prediction method [51]. For log-normal distribution, mean \(\mu\) and sample standard deviation \(\sigma\) are calculated for natural logarithm of \(Q_{p}/Q_{m}\) (Equations (11) and (12)):  
\[\mu_{ln}\bigg{(}\frac{Q_{p}}{Q_{m}}\bigg{)}=\frac{1}{n}\sum_{i=1}^{n}ln\bigg{(} \frac{Q_{p}}{Q_{m}}\bigg{)}, \tag{11}\]  
\[\sigma_{ln}\bigg{(}\frac{Q_{p}}{Q_{m}}\bigg{)}=\sqrt{\frac{1}{n-1}}. \tag{12}\]  
The \(Q_{p}/Q_{m}\) ratio and natural logarithm \(\ln(Q_{p}/Q_{m})\) are determined for each pile. The mean \(\mu_{ln}\), sample standard deviation \(\sigma_{ln}\) and coefficient of variation \(COV\) for \(\ln(Q_{p}/Q_{m})\) are calculated for each method. Log-normal distribution is defined as a distribution with density \(f(x)\) according to Equation (13) and distribution function \(F(x)\) according to Equation (14):  
\[f(x)= \frac{1}{\sqrt{2\pi}\sigma_{ln}}\exp\bigg{[}-\frac{1}{2}\bigg{(}\frac{ \ln(x)-\ \mu_{ln}}{\sigma_{ln}}\bigg{)}^{2}\bigg{]}, \tag{13}\]  
\[F(x)= \frac{1}{\sqrt{2\pi}\sigma_{ln}}\int\limits_{0}^{x}\frac{1}{\mu}\exp \bigg{[}-\frac{1}{2}\bigg{(}\frac{\ln(\mu)-\ \mu_{ln}}{\sigma_{ln}}\bigg{)}^{2}\bigg{]}d\mu. \tag{14}\]  
Log-normal distribution was applied for evaluation of reliability and efficiency of Unique, LCPC, and Eurocode 7--2 prediction methods. According to probability function of log-normal distribution, the probability of a bearing capacity prediction with 20% reliability level was calculated for each method.