**4. Application Example**

The proposed motion-based design method under uncertainty conditions was validated herein via the design of three passive damping devices when they are used to control the wind-induced vibrations of the longest cable of a real bridge. For this purpose, the Alamillo bridge (Seville, Spain) was considered (Figure 4). The length of the deck of this bridge is 200 m. Unlike most cable-stayed bridges, the Alamillo bridge has not back-stays. An inclination of its pylon of 32◦ with respect to the vertical axis compensates the lack of the back-stays [55]. A total of 26 stays (13 parallel pairs) with a longitudinal separation of 12 m guarantees an adequate connection between the deck and the pylon.

**Figure 4.** Illustrative scheme of the Alamillo bridge.

Previous research reported that the longest stay cable of this bridge, which has both a low damping and mass ratio, was prone to vibrate due to the wind action. Concretely, it was detected that the main sources of vibration of this cable were the rain–wind interaction phenomenon and the turbulent component of the wind action [56]. Therefore, this stay cable was considered as a benchmark to validate the performance of the proposed design method. For this purpose, three di fferent passive damping devices (viscous, elastomeric and friction dampers) were designed according to the proposed method, and the results obtained were compared with the ones provided by a conventional method adopted by the Standards [6]. Additionally, the uncertainty associated with the variation of the modal properties of the cable due to the modifications of the operational and environmental conditions was taken into account in this design process. The development of this case-study was organized in the following steps: (i) a FE model of the cable was built and its numerical modal properties were obtained via a numerical modal analysis; (ii) a transient analysis was performed to evaluate the vibration serviceability limit state of the structure; (iii) as this limit state was not met, the three passive damping devices were designed according to both methods (the new proposal and the conventional one); and (iv) finally, the results obtained were compared and some conclusions were drawn to close the section.

#### *4.1. FE Model and Numerical Modal Analysis*

The FE model of the cable was built using the software Ansys [57]. The geometrical and mechanical properties of the cable under study were as follows: (i) its length, *L* = 2.92 × 10<sup>2</sup> m; (ii) its outer diameter, *D* = 0.20 m; (iii) the e ffective area of its cross section, *A* = 8.38 × 10−<sup>3</sup> m2; (iv) the e ffective moment of inertia, *I* = 5.58 × 10−<sup>4</sup> m4; (v) its mass per unit length, *m* = 60 kg/m; (vi) an axial force, *H* = 4.13 × 10<sup>6</sup> N; (vii) a Young's modulus, *E* = 1.6 × 10<sup>11</sup> N/ *m*2; and (viii) the angle between the cable and the deck, γ = 26◦. The cable was modelled by a mesh of 100 equal-length beam elements (BEAM188). In order to simulate numerically the sag e ffect, a nonlinear static analysis was previously performed. The objective of this preliminary analysis was to find both the initial tensional state and pre-deformed shape of the cable. The self-weight of the cable and its initial axial force were considered as loads for this preliminary nonlinear static analysis. Subsequently, the results of this analysis were used to update the geometry and tensional state of the cable. Later, the linear perturbation method was considered to perform the modal analysis [57]. Additionally, the stress sti ffening e ffect was taken into account to perform this modal analysis.

As result of this numerical modal analysis, the first six natural frequencies were obtained. Table 1 shows the value of these first six natural frequencies ( *fi* being the natural frequencies of the *ith* vibration mode).


**Table 1.** Numerical natural frequencies of the cable.

#### *4.2. Assessment of the Vibration Serviceability Limit State of the Cable under Uncertainty Conditions*

As it was expected, according to the numerical natural frequencies obtained (Table 1), this cable was prone to vibrate under wind action due to both the turbulent component of the wind (the first two natural frequencies are lower than 1 Hz [58]) and the rain–wind interaction phenomenon (the six natural frequencies are lower than 3 Hz [6]). For this reason, the assessment of the vibration serviceability limit state of this stay cable was performed herein following the recommendations of the Federal Highway Administration (FHWA) guidelines [6].

On the one hand, in order to avoid the wind-induced vibrations associated with the rain–wind interaction phenomenon, it must be checked that the damping ratio of all the vibration modes, whose natural frequencies are lower than 3 Hz, are greater than a recommended value [6,59]. In order to determine this recommended value, the FHWA guidelines [6] establishes that the rain–wind interaction phenomenon can be neglected if the Scruton number, *Sc*, is greater than 10 for all the considered vibration modes. This condition may be expressed as follows:

$$S\_{c,i} = \frac{m\xi\_i}{\rho D^2} > 10\tag{19}$$

where ξ*i* is the damping ratio of the *ith* vibration mode.

Thus, this requirement is equivalent to guaranteeing a minimum damping ratio for each considered vibration mode. The minimum required damping ratio may be determined as follows:

$$
\kappa\_i^{\varepsilon} > \frac{10\rho D^2}{m} \tag{20}
$$

As expected, due to the results of previous experimental tests, the damping ratio associated with the first six vibration modes of this cable did not meet this condition [56]. Hence, it was necessary to increase the value of these damping ratios. A passive damping device can be designed and installed on the cable for this purpose.

On the other hand, in order to analyze the e ffect of the turbulent component of wind action on the dynamic behavior of the cable, a transient analysis was performed. As a result of this transient analysis, the dynamic response of the cable under wind action can be obtained and the vibration serviceability limit state of the cable can be assessed. According to the FHWA guidelines [6], this limit state is met if the maximum displacement of the cable is lower than an allowable displacement which is defined in terms of the user tolerance. Table 2 shows the allowable displacement of the cable in terms of the design level required [6]. In this study, a recommended design level was established for the vibration serviceability limit state.

**Table 2.** User tolerance limits for the di fferent design levels [6].


1 *D* is the outer diameter of the cable.

Additionally, as the dynamic response of the stay cable was sensitive to the variation of its modal properties associated with the change of the operational and environmental conditions during its overall life cycle, a reliability analysis about the compliance of the vibration serviceability limit state was performed. For this purpose, it was assumed that the axial force of the cable is a random variable normally distributed. According to the results provided by Stromquist-LeVoir et al., it could be also assumed that this random variable has a range of variation of ± 10% [60]. A sample of stay cables with di fferent values of the axial force was generated. The vibration serviceability limit state was assessed on this sample. For this purpose, the vibration serviceability limit state must be reformulated in order to take into account the uncertainty conditions. According to this, this limit state is met if a reliability index, β*s*(θ), is greater than an allowable reliability index, β*<sup>t</sup>*.

In order to compute the reliability index, β*s*(θ), the maximum displacement of the stay cable (obtained from the di fferent transient analyses performed on the sample of stay cables), which constitutes the demand of the wind action, *Da*(θ), and the allowable displacement of the stay cable (established by the FHWA guidelines [6]), which constitutes the capacity of the structure, *Cs*, were determined. Additionally, as the wind action is defined according to a return period of 50 years, the corresponding value of the allowable reliability index is β*t* = 1.35, according to the European guidelines [34].

As a numerical method in order to both determine the sample and compute the reliability index, β*s*(θ), the Monte Carlo method was considered herein. A convergence analysis was performed to

determine the size of the sample [61]. As a result of this convergence analysis, the size of the sample was established at 100.

Finally, in order to evaluate the demand of the wind action, *Da*(θ), the wind forces must be determined. For this purpose, simulations of the wind velocities were generated. The simulation of these wind velocities was addressed employing the wave superposition spectral-based method [8]. Both the von Karma spectra and a coherence function, as they are defined by the European guidelines [54], were employed herein. The following design parameters were considered for the wind simulation [54]: (i) basic wind velocity, *vb*,<sup>0</sup> = 26 m/s; (ii) a directional factor, *cdir* = 1; (iii) a season factor, *csea* = 1; (iv) a orography factor, *coro* = 1; (v) an terrain type III category (which involves a terrain factor, *kr* = 0.216; a roughness length, *zo* = 0.3 m; and a minimum height, *zmin* = 5 m); (vi) a duration of each simulation of 300 s; and (vii) a time step of 5 × 10−<sup>3</sup> s [62]. In this study, the wind velocities were generated at ten different heights of the cable (resulting from dividing the cable into ten equal-length segments), as Figure 5 depicts. This mesh density was considered for all the simulations conducted in the paper, in order to ensure that all the obtained results were consistent. Although preliminary analyses performed by the authors concluded that the meshing in Figure 5 was adequate for our aims (illustrating the performance of the proposed motion-based approach), the reader should be aware of the fact that the numerical simulation of the structural response under wind excitation depends on such mesh density, so that further analyses are recommended. A graphical user interface [63] was developed in the commercial software Matlab [64] to evaluate the wind action following the above guidelines.

**Figure 5.** Representation of the ten different heights where the wind action is applied.

The application of Equations (11) and (12) allows the wind-induced forces in terms of the wind velocities to be computed. For this purpose, the following values for the characteristic parameters were adopted: (i) a density of the air, ρ = 1.23 kg/m3; (ii) a drag coefficient, *CD* = 1.2 [2]; and (iii) a lift coefficient, *CL* = 0.3 [6].

Finally, a transient analysis (time history simulation) was performed for each element of the sample. The nonlinear geometrical behavior of the stay cables was considered for this analysis. A Newmark-beta method (an unconditionally stable method with parameters β*m* = 1/4 and γ*m* = 1/2) was considered to solve the transient analysis. Hence, the reliability index, β*s*(θ), was computed from the results of this set of transient analysis. Subsequently, the vibration serviceability limit state of the stay cable under uncertainty conditions was assessed. Thus, the reliability index, β*s*(θ), was lower than the allowable reliability index, β*<sup>t</sup>*, so this limit state was not met.

In order to improve the dynamic behavior of this stay cables, different passive damping devices were installed at this stay cable. These passive damping devices were designed according to the proposed method. This design problem is described in next section.

#### *4.3. Motion-Based Design of Passive Damping Devices under Uncertainty Conditions*

Three different passive damping devices were considered for this study: (i) viscous damper; (ii) elastomeric damper; and (iii) friction damper. The FE method was employed to simulate the behavior of these damping devices. The software Ansys [57] was employed for this purpose. Figure 2 depicts the mechanical models, which simulate the behavior of each damper. For each passive damper, the following model was considered: (i) the viscous damper was modelled by a 1D element (COMBIN14) whose characteristic parameter was the damping coefficient, *cd*,*<sup>v</sup>* [sN/m]; the elastomeric damper was also modelled by a 1D element (COMBIN14) whose characteristic parameters were the damping coefficient, *cd*,*<sup>e</sup>* [sN/m], and the stiffness coefficient, *kd*,*e*. [N/m]; and (iii) the frictioin damper was modelled by a 1D element (COMBIN40) whose characteristic parameters were the damping coefficient, *cd*, *f* [sN/m], the stiffness coefficient, *kd*, *f* [N/m], and the friction force, *ff* [N].

Consequently, the different dampers were implemented in the numerical model and designed according to the motion-based design method under uncertainty conditions. The three dampers were installed at a length of *xc* = 0.03*L* according to the recommendations of Ref. [2]. The damper-cable interaction model is shown in Figure 2.

A search domain, [<sup>θ</sup>min, <sup>θ</sup>max], for the characteristic parameters of the dampers was included in the optimization problem to ensure the physical meaning of the solutions obtained. The search domain was defined as follows: (i) the lower bound of the search domain, θmin, was defined as θmin = *cmin*, *kmin*, *ffmin* (where *cmin* is the minimum value of the damping coefficient; *kmin* is the minimum value of the stiffness coefficient, and *ffmin* is the minimum value of the friction force); and (ii) the upper bound of the search domain, θmax, was defined as θmax = *cmax*, *kmax*, *ffmax* (where *cmax* is the maximum value of the damping coefficient; *kmax* is the maximum value of the stiffness coefficient, and *ffmax*is the maximum value of the friction force).

The lower, *cmin*, and upper, *cmax*, bounds of the damping coefficient were determined considering both the requirement of the Scruton number [6] and the optimum damping coefficient of the Pacheco's universal curve [25]. According to this, the following bounds were established: (i) *cmin* = 4.8 × 10<sup>4</sup> sN/m; and (ii) *cmax* = 1.64 × 10<sup>5</sup> sN/m. This search range guarantees that any solution of this design problem avoids the occurrence of the rain–wind interaction phenomenon.

The search domain of the stiffness coefficient and the friction force were based on the results of previous research [2]. According to these results, the following bounds were established: (i) for the stiffness coefficient, *kmin* = 5 × 10<sup>4</sup> N/m and *kmax* = 5 × 10<sup>5</sup> N/m; and (ii) for the friction force, *f*<sup>f</sup>*min*= 1 × 10<sup>4</sup> N and *f*<sup>f</sup>*max*= 4 × 10<sup>4</sup> N.

In order to avoid falling into a local minimum, a global computational algorithm was considered for this optimization problem. Among the different computational algorithms, genetic algorithms were considered herein [65] for its simplicity and grea<sup>t</sup> efficiency to solve structural optimization problems.

Genetic algorithms are nature-inspired computational algorithms based on Darwin's natural selection theory. According to this, each possible value of the characteristic parameters of the damper is identified as a chromosome. Subsequently, each set of characteristic parameters is grouped into an individual (parameter vector). Later, the value of this parameter vector is improved via an iterative process where the value of the objective function is optimized. The optimization process can be summarized in the following steps: (i) an initial random population of parameter vectors is generated; (ii) the objective function is evaluated for all the individuals; (iii) a new population is created using three mechanisms (selection, crossover, and mutation); (iv) the objective function is evaluated for the individuals of the new population; (v) the steps (iii) and (iv) are repeated until some convergence criterion is met. The following parameters were considered for the considered genetic algorithms: (i) an initial population of 5 individuals; (ii) a crossover fraction of 0.4; (iii) a mutation fraction of 0.9; and (iv) a total number of iterations equal to 6.

As result of the optimization process, a Pareto front was obtained. Subsequently, a decisionmaking problem should be solved, the selection of the best solution among the different elements of the Pareto front. In order to address this problem, an additional condition was included. Among the

di fferent elements of the Pareto front, the element of the Pareto front with a lower value of the damping coe fficient was selected as best solution. The commercial software Ansys [57] and Matlab [64] were used to solve this design problem. The results of the optimization problem are summarized in the next sub-sections.
