4.2.3. Transient Loads

For combination ULS3, the effects of pressure, weight, other sustained loads and occasional loads including earthquakes shall meet the requirements of Equation (7):

$$S\_I \ll S\_h \Leftrightarrow \frac{PD}{4\varepsilon} + S\_{\upsilon} + S\_{\upsilon} \leqslant kS\_h \tag{7}$$

where k = 1.2 for occasional loads acting less than 1% of the operating period; So = stress due to occasional loads, such as thrusts from pressure, flow transients and earthquakes.

The other assumption is related to the critical pressure (*i.e*., burst pressure) [29]:

$$p\_{\mathbb{C}} = \frac{2E}{1 - \lambda^2} \left(\frac{t}{D}\right)^3 \tag{8}$$

where *E* the elastic modulus of steel, *λ* the poisson modulus, *D* and *t* the outer diameter and the thickness of the pipe, respectively. By Equation (8) with the specified parameters of the pipeline (Table 1), the critical pressure is 24 MPa, which is higher than the maximum pressure of the transient event. This means that the characteristics of the pipeline (*i.e*., diameter and material) guarantee extreme conditions between 10 MPa and 24 MPa.

Considering combination ULS3, the water-hammer event is defined as a concentrated force applied at the upstream point of the pipeline. To simulate the transient event, a path was set to the concentrated force (*i.e*., *F* = 3456.4 kN) obtained through the maximum pressure registered. This generates a moving load starting along the pipeline (Figure 5).

**Figure 5.** Combination ULS3: water-hammer defined as a moving load starting in 1. Maximum displacements achieved under the transient event (in cm).

Calculating the design assumptions described in Section 4.2.2, it is verified that this combination fails in the ULS3 conditions presented in Table 4.


**Table 4.** Results obtained for the two selected load conditions.

The vibration of the suspended pipeline under the action of a transient phenomenon described by a moving load gives results similar to the ones presented *in situ*. Moreover, the numerical results indicate that the maximum deflection at mid-span is 702.40 mm, which is 1.19 times the theoretical value (588.44 mm) and 0.94 less than the real value (750 mm). The difference between FEM and *in situ* measurements is not significant, confirming the dynamic behaviour of the structure under an impulsive load can be simulated using this model.

Figure 6 illustrates the maximum moving effect in each point of the structure under the moving load (*i.e*., a water-hammer event).

**Figure 6.** Maximum displacement in each point of the pipeline during a transient event.

From the deformed shape of the structure, some inconveniences can be observed, starting from the 1st to 20th m of length of the analysed pipe branch. As an overview, the deformed shape of this suspended pipeline has a point where the anchor supports, exceed their damping limit (*i.e*., maximum expansion), permitting some elements to vibrate with a self-period. This influences the global behaviour of the entire structure leading to its fall, as it can be seen from *in situ* pictures (see Figure 1).
