# Practical Considerations in the Design of Passive Free Piles in Sliding Soil
## 3 Materials and Method
The volume weight is \(\gamma\) = 25 kN/m\({}^{3}\) and the design tensile strength, \(\mathrm{f_{t,d}}\), is obtained from the ratio of the cross-section areas of both steel and concrete:  
\[f_{t,d}=\frac{A_{S,l}}{A_{C}}f_{y,d}=0.02\times 391=7.82\ \mathrm{MPa} \tag{2}\]  
The design compressive strength of the pile and its longitudinal modulus of elasticity are assumed coincident with those of a class C25/30 concrete, i.e., \(\mathrm{f_{c,d}}\) = 14.16 MPa, E = 31,500 MPa and Poisson's ratio \(\gamma\) = 0.2. The introduction of the aforementioned simplified constitutive model for the pile, although not able to capture shear failure, allows us to predict with reasonable approximation firstly the displacement of the pile head when the yield moment, \(\mathrm{M_{y,d}}\), is reached and subsequently the displacement of the pile head when the ultimate moment, \(\mathrm{M_{ult,d}}\), is attained in the most stressed cross section of the pile. Therefore, the shear failure mode is detected by comparing the induced shear stress with the pile shear strength (Equation (1)). The mobilised shear strength along the discontinuity surfaces of the model is simulated by using interface elements arranged along the sliding surface and the pile lateral surface. With these elements, it is possible to reach large displacements along the model discontinuities, as well as to take into account the possible loss of contact between the pile and the ground. The strength of these elements follows the Mohr-Coulomb failure criterion in the case of non-associated flow (\(\psi\) = 0). For these elements, it is necessary to define:  
* The angle of residual strength, \(\varphi_{r}\), along the slip surface;
* Soil-pile adhesion a\({}^{\prime}\);
* Soil-pile angle of shear strength \(\delta^{\prime}\);
* The normal stiffness modulus, \(\mathrm{K_{n}}\);
* The shear stiffness modulus \(\mathrm{K_{t}}\). All parameters and cases considered for the numerical models are summarised in Table 2.  
In the vicinity of the pile head, these displacements amount to just over 50 cm and coincide with the displacement of the upper unstable formation. Along the shaft, all the three models show the achievement of the design failure moment in simple bending according to the kinematic mechanisms of type BY for Model 1, of type B2 for Model 2 and of type B1 for Model 3. The results of the numerical models show that during pile-soil interaction, the concrete first reaches the yield curvature, \(\chi_{\text{y,d}}\), and then the ultimate one, \(\chi_{\text{ult,d}}\). As reported in Figure 6, the moment-curvature relationships obtained for the most stressed pile sections of the Models 1, 2 and 3 of Figure 5 are compared with that of the concrete pile cross-section subjected to simple bending. Although the first moment-curvature relationships derive from a simplified constitutive model for concrete, they allow us to outline two representative reference stages of the pile, i.e., the elastic limit and the failure. The complete pile-soil interactions for the aforementioned models of Figure 5 are synthetised in Figures 7-9, in terms of normal stress, shear stress, bending moment and horizontal displacement of the pile, both when the yield moment is reached (Model 1 Figure 7a, Model 2 Figure 8a and Model 3 Figure 9a) and when the ultimate moment is achieved in the most stressed cross-section of the pile (Model 1 Figure 7b, Model 2 Figure 8b and Model 3 Figure 9b). From these distributions, much information of practical interest can be deduced. Firstly, it can be observed that in correspondence to the slip surface the axial load and the shear load are maximum, whereas the bending moment is low, almost close to zero. Therefore, in the stability analyses of translational slopes reinforced with free-head flexible single-piles, it is sufficient to introduce the former two forces for a global stability check.