## 1 Introduction
[PERSON] [3] defined Equation (1), which approximates the horizontal deflection of a steel pipe under the assumption shown in Figure 2.  
\[\frac{\Delta X}{D_{e}}\%_{0}=\frac{K_{\alpha}[D_{L}(W_{r}+P_{L})]r^{3}}{EI+0. 061E^{\prime}r^{3}} \tag{1}\]  
Figure 1: Example of pipeline installation crossing different accesses and a haul road. In accordance with Equation (1), the deflection is directly proportional to the following:  
* The pipe bedding effect constant \(K_{\alpha}\), which depends on the pipe bedding angle \(\alpha\);
* The deflection factor for installation time \(D_{L}\);
* The dead load \(W_{r}\left(\frac{kN}{m}\right)\), which is commonly estimated using the [PERSON]'s equation or its derivatives;
* The live load \(P_{L}\left(\frac{kN}{m}\right)\), which can be estimated using the [PERSON] or Timoshenko equations or their derivatives;
* The cube of the mean radius of the pipe r.  
On the other hand, it is inversely proportional to the sum of the pipe stiffness defined by the following:  
* The product of the elasticity modulus, \(E\), and the inertia moment of the pipe, \(I\);
* The lateral soil stiffness of the pipe, with value of \(0.061E^{\prime}r^{3}\), where the value \(0.061\) corresponds to the value of \(\beta\) for an angle of \(50^{\circ}\) (a typical value in this type of installations, but which may vary depending on the installation). \(E^{\prime}\) is the modulus of soil reaction obtained from AWWA M11 standard tables [1], which has different recommended values and depends on soil properties, degree of compaction and depth [4]. In some cases, the recommended values of the hybrid module \(E^{\prime}\) are limiting for certain soil types. For the soil type considered in this article, the most appropriate alternative is the one proposed by [PERSON] et al. [5], restricted soil modulus \(M_{s}\) (kPa). The restricted modulus is an inherent characteristic of the material and is determined as the gradient of the secant line on the stress-strain curve derived from a soil compression test. This test, which is carried out inside cylindrical containers to avoid lateral expansion of the soil, measures the capacity of the soil to withstand axial compression loads. Alternatively, the restricted modulus \(M_{s}\) can be calculated, based on the [PERSON]'s modulus \(E_{s}\) and the [PERSON]'s ratio \(v_{s}\) of the soil, using Equation (2):  
\[M_{s}=\frac{E_{s}(1-v_{s})}{(1+v_{s})(1-2v_{s})} \tag{2}\]  
To determine the value of \(M_{s}\), the load applied to the pipe is taken into account. Using previous research, it is established that E\({}^{\prime}\) is in the range of \(0.7M_{s}\) to \(1.5M_{s}\), although AWWA M11 [1] suggests considering \(E^{\prime}=M_{s}\). On the other hand, the surface  
Figure 2: [PERSON]’s stress distribution hypothesis [3]. loads \(W_{r}\) produced by the weight of the vehicle and transmitted to the trench through its wheels [6], represent a force on the crown of the pipe, \(P_{L}\), whose value can be estimated using the Timoshenko expression (Equation (3)). \[P_{L}=\frac{I_{f}W_{r}}{a_{c}}\Bigg{(}1-\frac{H^{3}}{\left(r_{t}^{2}+H^{2} \right)^{1.5}}\Bigg{)}D_{e} \tag{3}\]  
Its impact depends on the following:  
* The backfill height over pipe H;
* The external diameter of the pipe \(D_{e}\);
* The radius of contact \(r_{t}\), equivalent to the width of the tire;
* The equivalent contact area of the wheel track on the trench \(a_{c}\);
* The impact factor \(I_{f}\), which for the present study was considered to be 2. The AWWA M11 standard [1] also suggests verifying the resistance of the pipe taking into account the buckling limit \(q_{a}\) (kPa) which is calculated by Equation (4). This limit, \(q_{a}\), depends on the following:  
* The pipe stiffness \(EI\);
* Its outside diameter \(D_{e}\);
* Modulus of soil reaction \(E^{\prime}\);
* Backfill height over pipe H;
* The water buoyancy factor \(R_{w}\), which is considered 1.0 for dry soils or those far from humid areas.