# A Novel Approach Proposal for Estimation of Ultimate Being Capacity Based on Pile Loading Test Data
## 2 Materials and Methods
### Method
#### 2.2.5 Description of The Proposed Method  
According to the results of the examination and analysis made in this study, a mathematical model-based method that can evaluate the failure load based on the data of 8 pile loading tests has been proposed. The proposed method has been turned into a mathematical model as a result of the process of the load settlement curve. The pile loading test evaluation methods based on the mathematical model in the literature give values far away from the failure load determined by theoretical methods and the failure load determined by graphical methods even in piles up to failure. The main reason for this situation is thought to be due to the excessive iteration of the methods. When the pile loading test curves of the piles which reached failure are examined, it is seen that these curves consist of approximately three separate sections and are shown in Figure 1.  
By iterating based on the mathematical model, it has been determined that the distribution group to be selected to estimate the pile failure load and the ordinate-apse correlation of the distribution should be determined precisely. 1. The load-settlement relationship needs to be examined at the logarithmic level. 2. While extrapolating based on the mathematical model, it has been determined that using the distribution within the linear part before a failure will give a conservative failure load, while using the distribution within the linear section after failure will give an excessive failure load. For this reason, it has been seen that extrapolating over the distribution in the parabola during the load will give the ideal result. Figure 1: Typical pile loading test curves of the piles that reached failure. Application of the proposed method:  
The application of the proposed method is expressed below:  
At each load-settlement intersection, the settlement value \(\left(log_{\mathcal{Q}_{l}}^{\Delta_{i}}\right)\) is found at the algorithm load base. \(\Delta_{i}\): Settlement value in each loading stage (mm),  
Q\({}_{i}\): Load value in each loading stage (tons). The The average of all values \(\left(log_{\mathcal{Q}_{l}}^{\Delta_{i}}\right)\) is calculated: \(\frac{\sum_{i=1}^{n}log_{\mathcal{Q}_{l}}^{\Delta_{i}}}{n}\)  
n: Number of loading stages  
For each load stage, \(K=\frac{\sum_{i=1}^{n}log_{\mathcal{Q}_{l}}^{\Delta_{i}}}{n}\)  
\(\Delta_{i}\): value is calculated (this value is represented by \"K\" symbol) and K-Q distribution is drawn. K: It is the correspondence of the value of \(\left(\frac{\sum_{i=1}^{n}log_{\mathcal{Q}_{l}}^{\Delta_{i}}}{n}\right)\) (Unitless). In the resulting distribution, the distribution formed by the points corresponding to the parabola during the failure, shown in Figure 2, is combined on an ideal line, and the point where this line intersects the apse gives the ultimate bearing capacity (failure load). If it is desired to work more precisely, the regression line is drawn by performing regression analysis, and the equation of the regression line is found as:  
\[y=a\cdot x+b \tag{11}\]  
a: Incline of the line (tons\({}^{-1}\)),  
b: The point where the line intersects the ordinate (y) axis (Unitless)  
Failure load is found as:  
\[Q_{ult}=-b/a \tag{12}\]  
where:  
a: Incline of the line(tons\({}^{-1}\)),  
b: The point where the line intersects the ordinate (y) axis (Unitless). As can be seen in Figure 2, the distribution of the pile loading test of the piles that reached failure has two breaking points, A and B, and three different trendlines. Since there is no linear part after failure in piles that do not reach failure, the distribution formed by the parabola part during failure will be clearly seen with this method. Thanks to the related equation, the \"linear region before failure\", \"linear region after failure,\" and \"linear region after failure\" intersection corners can be seen more clearly in the load-settlement graph. Figure 2: Typical graph of the proposed method for piles up to failure.