4.2.2. Internal Loads

The first simulation is performed for combination ULS1 (without transient and natural events), where the overloading is the internal pressure, during normal conditions, plus 2000 N applied in the vertical direction, for pipelines located 3 m from the ground according to [28]. In order to verify whether the pipeline bridge is designed appropriately, it is necessary to satisfy the following conditions: i the maximum pipeline bending stress, due to pressure, weight and other sustained mechanical loads, is below the ultimate material load limits, given by [29]:

$$\text{S}\_{\text{V}} = \frac{\text{L}}{10\text{Z}} \left[ \text{qL} + 2\left(\text{C} + \text{W}\right) \right] < 35 \text{MPa} \tag{2}$$

where *L* = length between supports (m); *Z* = moment of resistance (cm4); *q* = total distributed loads (N/m); *C* = total concentrated loads (N), in this case *C* " 0; *W* = overload imposed at mid-span;

ii the maximum deflection at mid-span is below 25 mm for the limit value of pipelines with diameters higher than 100 mm [29], calculated as:

$$\delta = \frac{2400\text{L}^3}{\text{EI}} \left[ \frac{\text{C} + \text{W}}{\text{3}} + \frac{\text{qL}}{\text{4}} \right] \tag{3}$$

where *E* = elastic modulus (MPa); *I* = moment of inertia of the pipeline cross section (cm4);

iii the maximum stress due to internal pressure (*Sp max*) is below the acceptable stress of the pipe (*i.e*., *Sh* " 173.8 *MPa* (Table 1)):

$$\mathbf{S\_{p\,max}} \lessapprox \mathbf{S\_h} \tag{4}$$

and the total longitudinal load stresses (*Sl*) (*i.e*., pressure, weight, overloading) is below *Sh*:

$$\rm{S}\_{\rm{l}} \lessapprox \rm{S}\_{\rm{h}} \Leftrightarrow \frac{\rm{PD}}{4e} + \rm{S}\_{\rm{V}} \leqslant \rm{S}\_{\rm{h}} \tag{5}$$

iv the vertical or lateral motion is below to the limit motion given by [29]:

$$\delta\_{\text{max}} = \frac{10^7 \text{qL}^4}{24 \text{EI}} \tag{6}$$

where *δmax* = maximum displacement (mm); *E* = elastic modulus (kgf/cm2).

For combination ULS1, all the conditions are verified. The maximum deflection Equation (3) obtained at mid-span is ~11.5 mm (Figure 4), which is below the allowable limit value.

**Figure 4.** Maximum displacement achieved for permanent loads (ULS1) during normal conditions.
