**5. Conclusions**

In this Chapter we have revised some aspects of the response of viscoelastic foundation of bridges to a simplified model of moving loads and wind random perturbations. The results have been compared to the numerical solution for the modal equation, obtained with the deterministic and stochastic version of Newton-Leipnik algorithm. The analysis has begun modeling the steady-state vibration of the beam suspensions made of a fractional-order viscoelastic material. The resulting mathematical model consists of a component for the beam, and the Kelvin-Voigt foundation type containing fractional derivative of real order, as well as a stochastic term to account for wind pressure. We have highlighted the simplifications. Perhaps the most significant, in the very model formulation, has been the assumption that the load passage consists of concentrated masses, spatially periodic and moving at constant speed. This simplification is crucial to reduce the full partial differential equation to a single smode model — Wind load is modeled as the aerodynamic force related to the wind that blows orthogonally to the beam axis with random velocity. The whole system has then been modeled with a partial differential equation that can be reduced to a one-dimensional modal equation. The beam response under moving and/or stochastic wind loads has been estimated analytically assuming that

*Stationary probability distribution function versus the amplitude a and the phase ϕ for (a) θ*<sup>0</sup> ¼ 0*:*09 *and*

*Stationary probability distribution function as function of the amplitude for different values of the parametric wind turbulence parameter. Analytical curves (-), numerical curve (o). All The parameters are given in [21].*

*Vibrations of an Elastic Beam Subjected by Two Kinds of Moving Loads and Positioned on…*

*DOI: http://dx.doi.org/10.5772/intechopen.96878*

**Figure 5.**

**Figure 6.**

**25**

*(b) θ*<sup>0</sup> ¼ 0*:*2*. All The parameters are given in [21].*

*Vibrations of an Elastic Beam Subjected by Two Kinds of Moving Loads and Positioned on… DOI: http://dx.doi.org/10.5772/intechopen.96878*

**Figure 5.**

decreases and progressively shifts toward larger amplitude values, while the average center position stays in the same position. Thus, the additive (*θ*<sup>0</sup> ) and parametric (*θ*<sup>1</sup> ) wind turbulence decreases the chance for the beam to quickly reach the amplitude resonance. It is also demonstrated that the PDF has only one maximum

*Stationary probability distribution function as function of the amplitude for different values of the additive wind turbulence parameter. Analytical curves (-), numerical curve (o). All The parameters are given in [21].*

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

We have plotted curves **Figure 6(a)** and **(b)** that presents the stationary probability distribution function *Ps*ð Þ *a*, *ϕ* versus the amplitude *a* and the phase *ϕ*. This graph just confirm the results obtained in **Figures 4** and **5** and the highest value of

**Figure 7** presents the times histories of the maximum vibration of the beam. The case where the beam is subjected to moving loads (*a*), to wind actions (*b*) and to the

In this Chapter we have revised some aspects of the response of viscoelastic foundation of bridges to a simplified model of moving loads and wind random perturbations. The results have been compared to the numerical solution for the modal equation, obtained with the deterministic and stochastic version of Newton-Leipnik algorithm. The analysis has begun modeling the steady-state vibration of the beam suspensions made of a fractional-order viscoelastic material. The resulting mathematical model consists of a component for the beam, and the Kelvin-Voigt foundation type containing fractional derivative of real order, as well as a stochastic

situated in the vicinity of *am* ¼ 0*:*2.

the PDF is more visible.

both wind and loads (*c*).

**5. Conclusions**

**24**

**Figure 4.**

*Stationary probability distribution function as function of the amplitude for different values of the parametric wind turbulence parameter. Analytical curves (-), numerical curve (o). All The parameters are given in [21].*

**Figure 6.**

*Stationary probability distribution function versus the amplitude a and the phase ϕ for (a) θ*<sup>0</sup> ¼ 0*:*09 *and (b) θ*<sup>0</sup> ¼ 0*:*2*. All The parameters are given in [21].*

term to account for wind pressure. We have highlighted the simplifications. Perhaps the most significant, in the very model formulation, has been the assumption that the load passage consists of concentrated masses, spatially periodic and moving at constant speed. This simplification is crucial to reduce the full partial differential equation to a single smode model — Wind load is modeled as the aerodynamic force related to the wind that blows orthogonally to the beam axis with random velocity. The whole system has then been modeled with a partial differential equation that can be reduced to a one-dimensional modal equation. The beam response under moving and/or stochastic wind loads has been estimated analytically assuming that

albeit in practice it is more realistic to set a maximum speed at which the bridge should be crossed. In other words, to keep resonance at bay, it is necessary to set a speed limit below the resonance insurgence. Analogously, one could think to limit the vehicles number, not to have a minimum number of vehicles across the bridge. We conclude the parameters analysis, namely the stiffness and the viscoelastic properties of the foundations, noticing that such parameters can be optimized with an appropriated tuning, see e.g., **Figure 3**, or the analogous indications that stems

*Vibrations of an Elastic Beam Subjected by Two Kinds of Moving Loads and Positioned on…*

By way of conclusion, let us summarize that the special properties of the viscoelastic foundations and of the time dependent perturbations, vehicles and wind, interact. As a result also the construction and management parameters are not to be considered independent procedures, for they are deeply interwoven if safe trans-

Part of this work was completed during a research visit of Prof. Nana Nbendjo at the University of Kassel in Germany. He is grateful to the Alexander von Humboldt

To deal with the modelling, let us consider the dynamic equilibrium of a beam element of length *dx*; *w* ¼ *w x*ð Þ , *t* and *θ* ¼ *θ*ð Þ *x*, *t* be the transversal displacement and the angle of rotation of the beam element respectively. We denote the internal bending moment by *M*, the internal shear force by *V*, the inplane tension due to the inplane strain, issue of the assumed negligible longitudinal displacement of the beam by *T*, the foundation-beam interaction force (per unit length of the beam's axis) by *QF*ð Þ *x*, *t* and the external distributed loading by *Fad*ð Þ *x*, *t* and *f x*ð Þ , *t* .

Setting the vertical forces on the element equal to the mass times acceleration

*w x*ð Þ , *t*

*<sup>∂</sup>*<sup>4</sup>*w x*ð Þ , *<sup>t</sup> <sup>∂</sup>t*<sup>2</sup>*∂x*<sup>2</sup> � *<sup>T</sup> <sup>∂</sup>*<sup>2</sup>

*<sup>∂</sup>t*<sup>2</sup> (41)

(42)

(43)

*w x*ð Þ , *t ∂x*<sup>2</sup>

(44)

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *QF*ð Þ� *<sup>x</sup>*, *<sup>t</sup> f x*ð Þ� , *<sup>t</sup> Fad*ð Þþ *<sup>x</sup>*, *<sup>t</sup> <sup>ρ</sup><sup>S</sup> <sup>∂</sup>*<sup>2</sup>

*∂*2 *θ*ð Þ *x*, *t*

*<sup>∂</sup><sup>x</sup>* , Eq. (42) becomes:

*<sup>∂</sup>*<sup>3</sup>*w x*ð Þ , *<sup>t</sup>*

*∂*2 *w x*ð Þ , *t <sup>∂</sup>t*<sup>2</sup> � *<sup>ρ</sup><sup>I</sup>*

*<sup>∂</sup>t*<sup>2</sup> � *<sup>T</sup> <sup>∂</sup>w x*ð Þ , *<sup>t</sup>*

*<sup>∂</sup>t*<sup>2</sup>*∂<sup>x</sup>* � *<sup>T</sup> <sup>∂</sup>w x*ð Þ , *<sup>t</sup>*

*∂x*

*∂x*

from the results of **Figures 4** and **5** for the wind features *θ*<sup>0</sup> and *θ*1.

Foundation for financial support within the Georg Forster Fellowship.

portation is to be guaranteed.

*DOI: http://dx.doi.org/10.5772/intechopen.96878*

*∂V*

For small rotation *<sup>θ</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup>* <sup>≈</sup>*<sup>∂</sup>w x*ð Þ , *<sup>t</sup>*

While summing moments produces:

*∂M*

*∂M*

Combining Eq. (41) and Eq. (43) then yields:

*<sup>∂</sup>x*<sup>2</sup> <sup>¼</sup> *QF*ð Þ� *<sup>x</sup>*, *<sup>t</sup> f x*ð Þ� , *<sup>t</sup> Fad*ð Þþ *<sup>x</sup>*, *<sup>t</sup> <sup>ρ</sup><sup>S</sup>*

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>V</sup>* � *<sup>ρ</sup><sup>I</sup>*

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>V</sup>* � *<sup>ρ</sup><sup>I</sup>*

**Acknowledgements**

**Appendix A**

gives:

*∂*2 *M*

**27**

#### **Figure 7.**

*Vibration amplitude of the beam χ(τ) as function of the time τ. All The parameters are given in [21].*

the first mode contains the essential information, and using the stochastic averaging method. The analysis has therefore some limitations, namely the limited values of parameters that have been explored, and to have retained the first mode only in the Galerkin's method. Also, the vehicles' train has been (over?) simplified as a simple periodic drive. With this limitations in mind, let us summarize the main findings. To start with, in this framework it is possible to investigate how the main parameters of the moving loads and of the bearings affect the beam response, and especially how the driving frequency, the loads number, the stiffness coefficient, fractional-order of the viscosity term and the number of bearings affect the dynamic behaviour of the beam. The resonance phenomenon and the stability in the beam system strongly depends on the stiffness and fractional-order of the derivative term of the viscous properties of the bearings. There are a number of quantitative results that are worth mentioning. Firstly, as the number of moving loads increases, the resonant amplitude of the beam increases as well. Secondly, it has been established that as the number of bearings increases, the resonant amplitude decreases and, more importantly, shifts toward larger frequency values. Thirdly, the system response becomes more stable as the order of the derivative increases, for the multivalued solution only appears for the smallest order and quickly disappears as the order increases. All the above results are a consequence of the analysis of the oscillations. However, the stochastic analysis has allowed to estimate the probabilistic distribution as a consequence of the wind random effects. The beam response, and more specifically the stationary probability density function of its amplitude, can also be retrieved. This type of analysis indicates that as the additive wind turbulence parameter increases, the peak value of the probability density function decreases and progressively shifts toward larger amplitude values, while the average center position stays in the same position. Thus, the additive and parametric wind turbulence decreases the chance for the beam to quickly reach the amplitude resonance.

Numerical simulations have confirmed these predictions. This behavior is depicted in **Figures 2**–**5**, which have practical implications, on which we would like to comment. To make an example, the beam system frequency *Ω* displays the bridge response as the vehicles speed changes, see Eq. (7). In principle it could be possible to avoid large oscillations by controlling the speed of the freight vehicles,

*Vibrations of an Elastic Beam Subjected by Two Kinds of Moving Loads and Positioned on… DOI: http://dx.doi.org/10.5772/intechopen.96878*

albeit in practice it is more realistic to set a maximum speed at which the bridge should be crossed. In other words, to keep resonance at bay, it is necessary to set a speed limit below the resonance insurgence. Analogously, one could think to limit the vehicles number, not to have a minimum number of vehicles across the bridge. We conclude the parameters analysis, namely the stiffness and the viscoelastic properties of the foundations, noticing that such parameters can be optimized with an appropriated tuning, see e.g., **Figure 3**, or the analogous indications that stems from the results of **Figures 4** and **5** for the wind features *θ*<sup>0</sup> and *θ*1.

By way of conclusion, let us summarize that the special properties of the viscoelastic foundations and of the time dependent perturbations, vehicles and wind, interact. As a result also the construction and management parameters are not to be considered independent procedures, for they are deeply interwoven if safe transportation is to be guaranteed.
