# Practical Considerations in the Design of Passive Free Piles in Sliding Soil
## 3 Materials and Method  
The ground hypothesised in the simulations is a medium-dense clay with a volume weight of \(\gamma\) = 20 kN/m\({}^{3}\), unaffected by the presence of the groundwater. The simple failure criterion of Mohr-Coulomb has been introduced in the analyses for soil, by considering effective parameters of cohesion \(c^{\prime}\), friction angle \(\varphi^{\prime}\), normal stress \(\sigma^{\prime}\), and non-associated flow rule (\(\psi\) = 0). Before failure, the soil has a linear elastic behaviour characterised by a longitudinal stiffness modulus E\({}_{s}\) and Poisson's ratio \(\
u\). In some cases, the stiffness modulus of the upper soil is differentiated from the lower one in order to highlight the influence of the soil stiffness on the displacement levels. For the same reason, the shear strength parameters of both the sliding and the stable soils are differentiated in some cases. The pile, 1200 mm in diameter, is made of concrete C25/30, with characteristic cylindrical strength \(f_{\mathrm{ck}}\) = 25 MPa, characteristic cubic strength R\({}_{\mathrm{ck}}\) = 30 MPa and design compression strength \(f_{\mathrm{c,d}}\) = 14.16 MPa. A maximum longitudinal steel reinforcement, B450C, is arranged inside the pile cross section, made up of 28 \(\O\)\(\O\) 32 mm, with a design strength of \(f_{\mathrm{y,d}}\) = 391.30 MPa, a concrete cover of 50 mm in thickness and a gap distance of 90 mm between reinforcements. Under these conditions, the design's simple-bending moments at yield and at failure are, respectively, M\({}_{\mathrm{y,d}}\) = 2800 kN \(\times\) m and M\({}_{\mathrm{ult,d}}\) = 3840 kN \(\times\) m (Figure 4). In addition to longitudinal reinforcement, a transverse spiral stirrup, B450C, 16 mm in diameter and with a pitch of 180 mm, is considered. The shear strength has been evaluated with different approaches, such as that suggested by [24] for reinforced rectangular cross-sections subjected to shear-compression or  
Figure 4: Design moment–curvature relationship of a concrete circular cross-section, 1200 mm in diameter, reinforced with 28 \(\O\) 32 mm and subjected to simple bending. Identification of both the design yield moment M\({}_{\mathrm{y,d}}\) and the design failure moment M\({}_{\mathrm{ult,d}}\). Figure 3: Numerical model constraints in the various calculation phases: (**a**) initial application of gravity, (**b**) application of displacements U\({}_{\mathrm{xx}}\) to the upper layer, (**c**) plan view. shear-tensile and with the simplified expression proposed by [25], specifically calibrated for a reinforced circular cross-section. The results of the different calculation criteria appear quite homogeneous; the most conservative method comes from the approach put forward by [25] according to which the design shear strength is evaluated through the expression:  
\[V_{ult,d}=\left[0.232d^{2}(100\rho_{l}f_{ck})^{1/s}\right](1+238\rho_{w}) \tag{1}\]  
where:  
\(d\) = pile diameter (mm);  
\(\rho_{l}=\mathrm{A_{sl}}/\mathrm{A_{c}}\);  
\(A_{sl}\) = total cross-section area of the longitudinal reinforcement;  
\(A_{c}\) = pile cross-section area;  
\(f_{ck}\) = characteristic cylindrical strength of the concrete (MPa);  
\(\rho_{w}=A_{sw}/(s\times d)\);  
\(A_{sw}\) = twice the cross-section area of the \(\mathcal{O}\) 16 mm spiral stirrup (mm\({}^{2}\));  
\(s\) = pitch of the spiral stirrup (mm). Under these assumptions, the design shear strength is \(\mathrm{V_{ult,d}}\cong\) 1770 kN and represents, more or less, the maximum pile shear strength achievable for the pile under investigation, which is considerably reinforced. In order to discretise the reinforced concrete pile with brick elements, a simplified approach is introduced by considering an equivalent elastic-perfectly plastic material which can reach failure under compression or tension.