## 2 Proposed Finite Element Model
### Validation of Finite Element Model
#### 2.1.4 Lateral Loads and Numerical Simulation Procedure  
Lateral earth pressures on large diameter pipe are important in order to model the behaviour of the pipe-soil system. However, the study of this lateral loads during pipe installation is limited. [PERSON] et al. [10] developed a method to model construction stages across layers, using an algorithm to activate and deactivate soil layers. Figure 7 shows the last three layers considered in the study in contact with the pipe, when only the pipe and a group of soil layers are active. The shared nodes allow the model parts to capture and track modified geometry, according to the deformed shape of neighbouring parts.  
For the case of lateral pressure induced by compaction, a horizontal load is applied to each constructive layer of the compacted soil [10]. The calculated stresses are applied to each soil layer using the equivalent temperature loading (Equation (5)) [10]:  
\[\alpha\Delta T=\sigma s\left(-\frac{A_{s}}{L_{s}K_{pipe}}-\frac{A_{s}}{L_{s}K _{wall}}-\frac{1}{E_{soil}}\right) \tag{5}\]  
where:  
* the cross-sectional area of the soil layer is \(A_{s}\);
* the length of each layer is \(L_{s}\);
* \(K_{pipe}\) values are obtained as the product of the pipe thickness and its elasticity modulus;
* \(K_{wall}\), is obtained multiplying the trench wall's elasticity modulus by the trench wall's length;
* \(E_{soil}\) is the elasticity modulus of the lateral fill;
* \(\alpha\) is a virtual thermal coefficient;  
Figure 7: Shared nodes between active and inactive parts: (**a**) Black dots are shared nodes between soil and pipe, and grey dots are shared nodes between soil layers. (**b**) Analysis in finite element software (Deflections U, U2 in meters).  
* \(\sigma\)s is the lateral soil stress. It is calculated from the value of the undrained cohesion Cu of the soil. Cu is measured in units of pressure. According to [PERSON] and [PERSON] [30] the lateral soil pressure reaches up to 80% of Cu for soil plasticity indexes of 48%.  
The undrained cohesion values, Cu, have been defined according to the sensitivity analysis developed in Section 2.2.1.  
Finally, the lateral stresses are introduced into the model through the temperature variation \(\Delta T\) obtained for each layer.  
In the finite element model developed, the pipe does not reach its elastic limit and the soil works in its elastoplastic state. The soil failure criterion proposed by [PERSON] is used. The Mohr-Coulomb criterion relates the maximum and minimum triaxial stresses to the friction angle \(\phi\) and the cohesion \(c\) of the material (Equation (6)). The Mohr-Coulomb criterion defines the yield function, \(F(\sigma_{n},\tau)\), as a measure of how close the normal and shear stress are to the failure surface. The graphical representation is a circumference perpendicular to the hydrostatic axis and corresponds to half of the first invariant of the stress tensor. When the friction angle of the soil is not considered, that is, \(\phi\) = 0, the expression is simplified to the Tresca criterion.  
\[F(\sigma_{n},\tau)=|\tau|-(c+\sigma_{n}\tan\phi) \tag{6}\]  
This mathematical model has been chosen considering that its parameters are easily obtained with a conventional soil study and the accuracy of the results is sufficient for trend analysis.