## 1 Introduction  
In recent years, calculation of pile bearing capacity data in-situ test has been broadly applied by geotechnical engineers and building foundations, because these data are more accurate and reliable than small-scale laboratory tests. In fact, bored-piles have been considered empirically more as an art-work than science [1], and are formed using appropriate machines (capacity-type) to fill the holes with applicable concrete and reinforcement. Their usual sizes are 400 mm to 3,000 mm diameter, with a capacity that reaches up 45,000 kN based on the pile size and geological profile close to the pile, so an excellent pile capacity has reduced the pile cap size and pile numbers in the group [2, 3]. Therefore, bored-pile designing in most countries has relied on the results of the conventional standard penetration test (SPT). According to the literature, the procedure of bored-pile design has consisted of stages such as: (1) the calculation of the end bearing capacity (\(f_{b}\)) in bored-pile; (2) calculation of the shaft bearing capacity in bored-pile (\(f_{s}\)), note that, the sum of both values is the ultimate bearing capacity of an individual pile; and (3) pile working load which has been presumed from the ultimate bearing capacity by using a safety factor that permits the piles' interaction within the group [4; 5]. Regarding the empirical approach of \(f_{s}\) in relation to \(K_{s}\times\) SPT, \(f_{b}\) has been related to \(K_{b}\times\) SPT widely applied in pile designing. To evaluate \(K_{s}\) (i.e., skin friction ratio, \(f_{s}/SPT\)) and \(K_{b}\) (i.e., end bearing ratio, \(f_{b}/SPT\)), the value including local soil condition has required vibrating wire strain gauges (VWSG) and a mechanical tell-tale rod installed within the pile to measure the axial stress-strain relation and movement in different levels down to the pile's toe and shaft. Bored-pile of length (L) and diameter (D\({}_{s}\)) supporting a vertical head load (\(P_{h}\)) by the mobilized shaft and base resistance (\(P_{s}-P_{b}\)) is illustrated in Figure 1a. The vertical displacement of the pile's base and head are defined as \(\Delta_{b}-\Delta_{h}\), followed by the pile compressing (shortening) as \(e_{p}\) ignoring the pile weight as it is in Equation (1) [6]:  
\[P_{h}=P_{s}+P_{b} \tag{1}\]  
According to the cross-sectional area of concrete \(A\), the stress along with the pile \(\sigma_{h}\) is as Equation (2) [6]:  
\[\sigma_{h}=\frac{P_{h}}{A} \tag{2}\]  
\[\Delta_{h}=e_{p}+\Delta_{b} \tag{3}\]  
The length \(\delta_{z}\) has been located at depth \(z\) below the pile head level as shown in Figure 1a-c. Shaft resistance \(\delta P_{s}\) mobilized along the segment \(\delta_{z}\) equals the axial force changing in axial force \(\delta P_{z}\). The unit shaft resistance \(f_{s}(z)\) mobilized in \(\delta_{z}\) has been related to \(P(z)\) as shown in Equation (4):  
\[\frac{dP(z)}{dz}=-\pi D_{s}f_{s}(z) \tag{4}\]  
The negative sign shows that \(P_{z}\) decreased as \(z\) increased (Figure 1c). [PERSON] et al. [7] mention that if the pile's elastic feature has been denoted (\(E_{p}\)), ignoring all vertical soil moving and pile displacement \(w(z)\) (e.g., pile movement at depth \(z\) below the pile head level) at depth \(z\) have been offered as Equation (5):  
\[\frac{dw(z)}{dz}=\frac{-4P(z)}{\pi D_{s}{}^{2}E_{p}} \tag{5}\]  
Equation (5) is different with \(z\) and \(dP(z)/dz\), replaced by using Equations (1)-(4) as Equation (6):  
\[\frac{d^{2}w(z)}{dz^{2}}=\frac{4}{D_{s}E_{p}}f_{s}(z) \tag{6}\]  
All the mentioned equations according to force equilibrium and displacement correspondents are valid irrespective of pile type and soil grouping [7]. Figure 1: Displacement and force of loaded bored pile–soil; (**a**) schematic representation, (**b**) forces on a pile element in “equilibrium., (**c**) force details in element length of \(\delta_{x}\) length of \(\delta_{y}\).