## Methodology and Results
### Vibrations Induced in the Side Branches
Due to the high mean stresses due to internal pressure, it is not possible to easily compare the measured amplitudes of the strain from vibrations with their safe limit against fatigue crack. It is necessary to separate the individual oscillation cycles and take into account the mean value of the  
Figure 2: Vibration of the pipe and branch: (**a**) in-phase in the same direction (without the relative deflection) (**b**) out p-base in the same direction (inducing deflection of branch), and (**c**) in a different direction (inducing deflection of the branch). cycle [13; 14]. The real-time monitoring of dangerous vibrations can be accomplished by processing the measured time process of strains in time batches, e.g., in 30 or 60 s.  
For reliable information about the operational capability of the oscillating pipeline, it is, however, necessary to solve two more problems:  
* to determine cyclic properties of the critical volume of the material (often the weld joint) and
* the condition for assessing the criticality of the vibrations. The critical volume of the weld joint material in terms of oscillation is the heat affected area or the area with defects in the weld metal. In view of the validity of the cyclic properties of the material, it is appropriate to perform cyclic tests of the welded samples, as shown in Figure 5. The cyclic tests of specimens taken along the pipeline circumference in a place of the weld joint include all the effects of the microstructure of all individual weld joint zones and its geometry, as well as possible defects. Figure 4: (**a**) The arrangement of strain sensors along the perimeter of the pipeline, and (**b**) the application of the strain sensors (gauges) on the real pipe. Figure 3: Frequency composition of signals measured by accelerometers and strain gauges on the vibrating pipe branch. The condition of the vibration permissibility of the T-branch of the welded pipeline is then given by the relation  
\[\varepsilon_{aqq}\leq\frac{0.8\sigma_{c}^{5.10^{6}}}{E}\, \tag{3}\]  
where \(\sigma_{c}^{5.10^{6}}\) is a fatigue limit for 5 million cycles to failure (for the selected probability of survival, i.e., 95%), and \(\varepsilon_{aqq}\) is an amplitude of equivalent strain in the critical cross-section, taking into account the mean value of the cycle ([PERSON] [16] or [PERSON]'s [17] method). Such a cycle is mainly induced due to stresses from the internal pressure and possible combination of normal (axial plus bending loading) and shear stresses (torsional loading). \(E\) is the Young modulus of the material. The amplitudes of strain (stress) lower than value of the cyclic anelasticity limit induced by oscillation may be considered as safe, as was presented in [18]. These amplitudes of strains below this cyclic anelasticity limit are thus neglected. Thus, the monitoring system takes into account only the amplitudes of strains above this limit. The example of putting such a pipeline monitoring system in practice for the gas pipeline courtyard compressor station is presented by its main online information screen in Figure 6. This screen shows the information about critical cross-section points of the pipeline system at the compressor courtyard. KM1-KM9 are those critical monitoring places where there are installed 5 strain gauges, according to the methodology displayed in \[v_{n}=\frac{f_{ne}d}{S_{t}} \tag{1}\]0 These strain sensors continuously sense the deformation state of each monitoring cross-section KM1-KM9. The evaluated and visualized magnitudes are:  
* safety in terms of the pressure integrity of the pipe,
* fatigue damage accumulation state [19], and
* vibration permissibility. The long-term operation of the monitoring system places high demands on the stability of the sensors and the associated electronics, which can lead to the undesired valuating offset effect of the measured signals.