**2. Structural system model**

### **2.1 Mathematical modelling**

In this chapter, a simply supported Rayleigh beam [17, 18] of finite length L with geometric nonlinearities [19, 20], subjected by two kinds of moving loads (wind and train actions) and positioned on a foundation having fractional order viscoelastic physical properties is considered as structural system model and presented in **Figure 1**.

As demonstrated in (Appendix A) and in Refs. [16, 19, 21], the governing equation for small deformation of the beam-foundation system is given by:

$$\rho S \frac{\partial^2 w(\mathbf{x}, t)}{\partial t^2} + EI \frac{\partial^4 w(\mathbf{x}, t)}{\partial \mathbf{x}^4} - \frac{3}{2} EI \frac{\partial^2}{\partial \mathbf{x}^2} \left[ \frac{\partial^2 w(\mathbf{x}, t)}{\partial \mathbf{x}^2} \left( \frac{\partial w(\mathbf{x}, t)}{\partial \mathbf{x}} \right)^2 \right] - \rho I \frac{\partial^4 w(\mathbf{x}, t)}{\partial \mathbf{x}^2 \partial t^2} + \mu \frac{\partial w(\mathbf{x}, t)}{\partial t}$$

$$\left[ -\frac{ES}{2L} \frac{\partial^2 w(\mathbf{x}, t)}{\partial \mathbf{x}^2} \right]\_0^L \left( \frac{\partial w(\mathbf{x}, t)}{\partial \mathbf{x}} \right)^2 \mathbf{dx} + \sum\_{j=1}^{N\_P} (k\_j + c\_j D\_t^{a\_j}) w(\mathbf{x}, t) \delta \left[ \mathbf{x} - \frac{jL}{N\_P + 1} \right] = F\_{ad}(\mathbf{x}, t) + \rho I \frac{\partial^2 w(\mathbf{x}, t)}{\partial \mathbf{x}^2} + \mu \frac{\partial^2 w(\mathbf{x}, t)}{\partial \mathbf{x} \partial \mathbf{x}} \delta \left[ \mathbf{x} - \frac{jL}{N\_P + 1} \right]$$

$$P \sum\_{i=0}^{N-1} c\_i \delta[\mathbf{x} - \mathbf{x}\_i(t - t\_i)].$$

In which *ρS*, *EI*, *ρI*, *μ*, *w x*ð Þ , *t* , are the beam mass per unit length, the flexural rigidity of the beam, the transverse Rayleigh beam coefficient, the damping coefficient and the transverse displacement of the beam at point *x* and time *t*, respec-

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

the foundation-beam interaction force (per unit length of the beam's axis), *EI <sup>∂</sup>*4*w x*ð Þ , *<sup>t</sup>*

the beam essentially due to Euler Law. The nonlinear term is obtained by using the Taylor expansion of the exact formulation of the curvature up to the second order

terms on the right-hand side of Eq. (1) are used to describe the wind an train actions over the beam. In particular, the first term *Fad*ð Þ *x*, *t* is the aerodynamic force given

> *A*1 *U*

where *ρ<sup>a</sup>* is the air mass density, *b* is the beam width, *A <sup>j</sup>* (j = 0, 1, 2) are the aerodynamic coefficients (*A*<sup>0</sup> = 0.0297, *A*<sup>1</sup> = 0.9298, *A*<sup>2</sup> = -0.2400) [24]. *U* is the wind velocity which can be decomposed as *U* ¼ *u* þ *u t*ð Þ, where *u* is a constant (average) part representing the steady component and *u t*ð Þ is a time varying part

According to **Figure 1**, the boundary conditions of the beam are considered

The next section deals with the reduction of the main equation Eq. (1).

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

> *w*ð Þ 0, *t <sup>∂</sup>x*<sup>2</sup> <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup>

þ *A*2 *U*2

� �<sup>2</sup> � � are the linear and the nonlinear term of the rigidity of

*<sup>∂</sup>x*2*∂t*<sup>2</sup> is the rotary inertia force of the beam per unit length, *μ*

*Sketch of a beam-foundation system subjected to wind actions and series of moving forces.*

damping force of the beam per unit length, P*NP*

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

> 1 2

*<sup>ρ</sup>abU*<sup>2</sup> *<sup>A</sup>*<sup>0</sup> <sup>þ</sup>

representing the turbulence. It is assumed in this work that (*u* ≫ *u t*ð Þ).

*<sup>w</sup>*ð Þ¼ 0, *<sup>t</sup> w L*ð Þ¼ , *<sup>t</sup>* 0; *<sup>∂</sup>*<sup>2</sup>

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

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

*<sup>∂</sup>t*<sup>2</sup> represents the inertia term of the beam per unit length,

*<sup>j</sup>*¼<sup>1</sup> *<sup>k</sup> <sup>j</sup>* <sup>þ</sup> *cjD<sup>α</sup> <sup>j</sup>*

*t* � �*w x*ð Þ , *<sup>t</sup> <sup>δ</sup> <sup>x</sup>* � *jL*

d*x* is the inplane tension of the beam [19, 20]. The

� �<sup>2</sup> " #, (2)

*w L*ð Þ , *t*

*<sup>∂</sup>x*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:* (3)

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

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

*NP*þ1 h i is

*∂x*<sup>4</sup>

tively. In Eq. (1), *ρS <sup>∂</sup>*2*w x*ð Þ , *<sup>t</sup>*

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

2*L*

*<sup>∂</sup>*2*w x*ð Þ , *<sup>t</sup> ∂x*<sup>2</sup> Ð *L* 0

after some derivations by [24–26]:

*Fad*ð Þ¼ *x*, *t*

*ρI <sup>∂</sup>*4*w x*ð Þ , *<sup>t</sup>*

**Figure 1.**

and <sup>3</sup> <sup>2</sup> *EI <sup>∂</sup>*<sup>2</sup> *∂x*<sup>2</sup>

as [27]

**15**

[19, 22, 23]. *ES*

*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 1.** *Sketch of a beam-foundation system subjected to wind actions and series of moving forces.*

In which *ρS*, *EI*, *ρI*, *μ*, *w x*ð Þ , *t* , are the beam mass per unit length, the flexural rigidity of the beam, the transverse Rayleigh beam coefficient, the damping coefficient and the transverse displacement of the beam at point *x* and time *t*, respectively. In Eq. (1), *ρS <sup>∂</sup>*2*w x*ð Þ , *<sup>t</sup> <sup>∂</sup>t*<sup>2</sup> represents the inertia term of the beam per unit length, *ρI <sup>∂</sup>*4*w x*ð Þ , *<sup>t</sup> <sup>∂</sup>x*2*∂t*<sup>2</sup> is the rotary inertia force of the beam per unit length, *μ <sup>∂</sup>w x*ð Þ , *<sup>t</sup> <sup>∂</sup><sup>t</sup>* is the damping force of the beam per unit length, P*NP <sup>j</sup>*¼<sup>1</sup> *<sup>k</sup> <sup>j</sup>* <sup>þ</sup> *cjD<sup>α</sup> <sup>j</sup> t* � �*w x*ð Þ , *<sup>t</sup> <sup>δ</sup> <sup>x</sup>* � *jL NP*þ1 h i is the foundation-beam interaction force (per unit length of the beam's axis), *EI <sup>∂</sup>*4*w x*ð Þ , *<sup>t</sup> ∂x*<sup>4</sup> and <sup>3</sup> <sup>2</sup> *EI <sup>∂</sup>*<sup>2</sup> *∂x*<sup>2</sup> *<sup>∂</sup>*2*w x*ð Þ , *<sup>t</sup> ∂x*<sup>2</sup> *<sup>∂</sup>w x*ð Þ , *<sup>t</sup> ∂x* � �<sup>2</sup> � � are the linear and the nonlinear term of the rigidity of the beam essentially due to Euler Law. The nonlinear term is obtained by using the Taylor expansion of the exact formulation of the curvature up to the second order [19, 22, 23]. *ES* 2*L <sup>∂</sup>*2*w x*ð Þ , *<sup>t</sup> ∂x*<sup>2</sup> Ð *L* 0 *<sup>∂</sup>w x*ð Þ , *<sup>t</sup> ∂x* � �<sup>2</sup> d*x* is the inplane tension of the beam [19, 20]. The terms on the right-hand side of Eq. (1) are used to describe the wind an train actions over the beam. In particular, the first term *Fad*ð Þ *x*, *t* is the aerodynamic force given after some derivations by [24–26]:

$$F\_{ad}(\mathbf{x},t) = \frac{1}{2}\rho\_a bU^2 \left[ A\_0 + \frac{A\_1}{U} \frac{\partial w(\mathbf{x},t)}{\partial t} + \frac{A\_2}{U^2} \left(\frac{\partial w(\mathbf{x},t)}{\partial t}\right)^2 \right],\tag{2}$$

where *ρ<sup>a</sup>* is the air mass density, *b* is the beam width, *A <sup>j</sup>* (j = 0, 1, 2) are the aerodynamic coefficients (*A*<sup>0</sup> = 0.0297, *A*<sup>1</sup> = 0.9298, *A*<sup>2</sup> = -0.2400) [24]. *U* is the wind velocity which can be decomposed as *U* ¼ *u* þ *u t*ð Þ, where *u* is a constant (average) part representing the steady component and *u t*ð Þ is a time varying part representing the turbulence. It is assumed in this work that (*u* ≫ *u t*ð Þ).

According to **Figure 1**, the boundary conditions of the beam are considered as [27]

$$w(\mathbf{0},t) = w(L,t) = 0; \quad \frac{\partial^2 w(\mathbf{0},t)}{\partial \mathbf{x}^2} = \frac{\partial^2 w(L,t)}{\partial \mathbf{x}^2} = \mathbf{0}.\tag{3}$$

The next section deals with the reduction of the main equation Eq. (1).

moving train; however, the interaction between the wind and the train dynamics has been altogether neglected. In this limit the suspension bridge response is dominated by wind force. The coupled dynamic analysis of vehicle and cable-stayed bridge system under turbulent wind has also been recently conducted by Xu and Guo [2] under the other limit of low wind speed. In the same view, the both effects

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

of turbulent wind and moving loads on the brigde response are investigated

gated the pros, and the cons, of the elastic bearings.

(analytical [13–15] and numerical [13–16]) are used.

**2. Structural system model**

**2.1 Mathematical modelling**

*<sup>∂</sup>t*<sup>2</sup> <sup>þ</sup> *EI <sup>∂</sup>*<sup>4</sup>*w x*ð Þ , *<sup>t</sup>*

ð *L*

0

*εiδ x* � *xi*ð Þ *t* � *ti* ½ �*:*

*<sup>∂</sup>x*<sup>4</sup> � <sup>3</sup> 2 *EI <sup>∂</sup>*<sup>2</sup> *∂x*<sup>2</sup>

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

*ρS ∂*2 *w x*ð Þ , *t*

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

*P* X *N*�1

**14**

*i*¼0

numerically by Chen and Wu [3]. Another interesting results related to the problem of the dynamics of bridges subjected to the combined dynamic loads of vehicles and wind are presented in Refs. [4–6]. To summarize: from the standpoint of bridge engineering the disturbances, either due to wind (in low or high speed limits), passage of heavy loads (single massive trains or disordered aggregates of smaller freight carriers, result in a complex interaction with the bridge vibrations. However, the elastic properties of the bridges are enhanced by the insertion of bearings (the part ranging between the bridge deck and the piers) as a possible protection against severe earthquakes. For if one wants the bearings to protect the bridge, they should isolate the structure from ground vibrations and/or transfer the load to the foundation [7]. Noticed that the bearings can be constituted by some elastic or viscoelastic material. In the literature, the dynamics analysis of bridges with elastic bearings to moving loads has received limited attention. nevertheless, some authors like Yang *et al.* [8], Zhu and Law [9], Naguleswaran [10] and Abu Hilal and Zibdeh [11] have adressed a very interesting resultats about this subjet. There are investi-

The bearings can also be constituted by some viscoelastic materials (such as elastomer)[12]. Therefore, The viscoelastic property of the materials may be modelled by using the constitutive equation of Kelvin-Voigt type, which contain fractional derivatives of real order. In this Chapter we aim to investigate first the pros, and the cons, of the viscoelastic bearings and second the turbulence effect of the wind actions on the response of beam. To accomplish our goal some methods

In this chapter, a simply supported Rayleigh beam [17, 18] of finite length L with geometric nonlinearities [19, 20], subjected by two kinds of moving loads (wind and train actions) and positioned on a foundation having fractional order viscoelastic physical properties is considered as structural system model and presented in **Figure 1**. As demonstrated in (Appendix A) and in Refs. [16, 19, 21], the governing equation for small deformation of the beam-foundation system is given by:

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

� �*w x*ð Þ , *<sup>t</sup> <sup>δ</sup> <sup>x</sup>* � *jL*

� *ρI*

*NP* þ 1 � �

*<sup>∂</sup>*<sup>4</sup>*w x*ð Þ , *<sup>t</sup> <sup>∂</sup>x*<sup>2</sup>*∂t*<sup>2</sup> <sup>þ</sup> *<sup>μ</sup>* *<sup>∂</sup>w x*ð Þ , *<sup>t</sup> ∂t*

(1)

¼ *Fad*ð Þþ *x*, *t*

� �<sup>2</sup> " #

*∂*2 *w x*ð Þ , *t ∂x*<sup>2</sup>

*<sup>k</sup> <sup>j</sup>* <sup>þ</sup> *cjD<sup>α</sup> <sup>j</sup> t*

<sup>d</sup>*<sup>x</sup>* <sup>þ</sup><sup>X</sup> *NP*

*j*¼1
