## Abstract

The paper addresses a mathematical model describing the dynamic response of an elongated bridge supported by elastic pillars. The elastic system is considered as a multi-structure involving subdomains of different limit dimensions connected via junction regions. Analytical formulae have been derived to estimate eigenfrequencies in the low frequency range. The analytical findings for Bloch–Floquet waves in an infinite periodic structure are compared with the finite element numerical computations for an actual bridge structure of finite length. The asymptotic estimates obtained here have also been used as a design tool in problems of asymptotic optimization.

## 1. Introduction

The present paper introduces an asymptotic model of a periodic elastic multi-structure, consisting of thin elements of different scales. The junction conditions between different elements of the structure are derived and discussed in the text. Furthermore, the combined model is considered in the dynamic regime, and applications are offered in the design of a bridge in a practical engineering project.

In practical engineering systems, the elastic bodies are finite, and even long structures such as bridges have a finite number of spans separating the supporting pillars. It is possible however to use the results of the spectral analysis for Bloch waves in a relatively simple low-dimensional periodic system to characterize eigenfrequencies of vibration of finite size elongated bodies. We also note that for the accurate study of the response of the structural system, the full-scale finite-element computation is needed, and for an elastic structure such as a long bridge, it is an industrial scale task. On the other hand, description of the Bloch waves in a low-dimensional system can be a straightforward analysis, and many estimates can be obtained in an explicit analytical form.

Bloch–Floquet waves are considered as a classical object in problems of optics and acoustics, as described in detail in the classical texts of Brillouin (1946), Kittel (1953) and Kunin (1975). Dispersion of elastic waves in continuous systems with prestress was discussed in Parnell (2007), Gei *et al.* (2009) and Gei (2010). Dispersion and filtering properties of periodic elastic systems were studied in lattices (Maradudin & Weiss 1958; Maradudin *et al.* 1958; Movchan & Slepyan 2007), discrete media with internal microstructure (Martinsson & Movchan 2003; Colquitt *et al.* 2011) and structured media (Bigoni & Movchan 2002; Brun *et al.* 2010), whereas analytical models were built for monodimensional periodic structure by Mead (1973) and Heckl (2002) at a different level of generality; an extended review of different beam models can be found in Mead (1996). In particular, such structures support propagation of waves within a certain frequency range (pass bands) and, on the contrary, there are intervals of frequencies (stop bands) where waves become evanescent.

We offer a mathematical model of a practical problem, linked to the design and construction of the S'Adde Bridge in Sardinia, Italy. The bridge is shown in figure 1*a*, and the general geometry and dimensions are depicted in figure 1*b*.

The S'Adde Bridge is located on the SS 129 road in Macomer (NU), Italy. The viaduct was inaugurated on 13 June 2007 under the supervision of the Construction and Project Manager Dr G. F. Giaccu. The structure is a prestressed continuous hollow box-girder bridge and it consists of two-side spans of 45 m and a central span of 90 m, with a total length of 180 m. The 24.36 m tall piers have been properly designed to allow for longitudinal displacements, along the *x*-axis (see figure 3), during the launching when the deck cables are prestressed. Additional details of the geometry of the supporting pillars and of the deck are given in the §3.

The novelty of the approach advocated in the present paper is in the asymptotic analysis of the Bloch–Floquet waves applied to a multi-scale elastic system, referred to as a *multi-structure*. The problem studied here cannot be covered in the framework of a one-dimensional model involving, for example, a quasi-periodic Green's function: the model incorporates the data on the main upper deck of the bridge as well as the supporting pillars and the interaction between them, via the junction conditions on the displacement components and their derivatives. Despite the complexity of the overall system, the asymptotic model is elegant and enables one to proceed analytically to the dispersion equation, band gaps for Bloch–Floquet waves and explicit analytical estimates of the eigenfrequencies of standing waves within such an elastic structure.

The analytical and numerical approaches are then extended to the longer bridge structures, with the examples including six-span and 12-span systems. These are finite structures and their eigenvalues are computed via an finite element method (FEM) algorithm. It is shown that prediction of the spectral analysis for Bloch waves in an infinite periodic system leads to an accurate estimate of intervals characterizing location of eigenvalues of a finite elongated system. Some typical eigenmodes for a six-span bridge are shown in figure 2. These include the cases of longitudinal motion, as well flexural vibrations in the vertical and horizontal directions, which are orthogonal to the axis of the bridge.

The structure of the paper can be described as follows. Section 2*a* incorporates the analytical description of the low-dimensional model for Bloch waves in the periodically supported/constrained flexural beam. The dispersion equation and its analysis are included in §2*b*. We present the comparative analysis for pass bands for Bloch waves versus eigenfrequencies of finite-size elongated bridge structures. An example of a structural optimization based on the analytical solution is given. Furthermore, the Concluding remarks section includes some additional details on the actual systems and the related comparative analysis.

## 2. Three-dimensional bridge as a multi-structure

We shall construct an asymptotic approximation to the solutions of a class of spectral problems, which describes the dynamic response of the elongated bridge structure.

We use the notion of a multi-structure introduced by Ciarlet (1990) and Kozolov *et al.* (1999) as the set involving subdomains of different limit dimensions connected through junction regions. In the case of the bridge, the upper deck can be thought of an elongated plate or a beam, and the supporting pillars are thin elastic rods, with the characteristic dimension of their cross section being a small parameter *ε*. The multi-structure in question is shown in figure 3.

The aim of the asymptotic approximation is to obtain the analytical estimate of low-range eigenfrequencies of the bridge structure. We identify two regimes.

— For the first eigenfrequency, the bridge deck is approximated as a rigid solid, whereas the supporting pillars are treated as thin flexural elastic beams. In the case where the inertia of the beams is neglected, the asymptotic approximation of the first eigenfrequency can be derived as follows. The time-harmonic transverse displacement

*h*(*z*,*t*)=*H*(*z*)e^{iωt}, , of each pillar is a cubic function of the space variable*z*, namely 2.1where*H*(*z*) satisfies the boundary conditions*H*(0)=0,*H*′(0)=0,*H*(*l*_{p})=*F*_{M},*H*′(*l*_{p})=0 (*H*′(*z*)=d*H*(*z*)/*dz*), with*F*_{M}the displacement amplitude of the deck, assumed to be rigid, and*l*_{p}the length of the pillar. The deck has total mass*M*_{T}and it is characterized by the displacement*F*(*x*,*t*)=*F*_{M}e^{iωt}satisfying the equation of motion 2.2where the subscripts*tt*and*zzz*stand for the second- and third-order partial derivatives with respect to*t*and*z*, respectively. In equation (2.2), is the amplitude of the total shear force transmitted by the*N*_{p}pillars of bending stiffness*E*_{p}*J*_{p}each, where the inertia*J*_{p}has the order of magnitude*O*(*ε*^{4}). Then, the first eigenfrequency*ω*_{1}can be deduced from equation (2.2), which is equal to 2.3in the time-harmonic regime, yielding 2.4It is clear that the order of magnitude of the first frequency is*O*(*ε*^{2}).— To obtain the estimates for the higher frequencies, we need to take into account the deformation of the bridge deck. Hence the bridge deck can be approximated as an elastic flexural beam interacting with thinner (of smaller cross section) supporting pillars of a given height

*l*_{p}. Instead of considering a full-scale structure, we restrict ourselves to the analysis of Bloch waves in an infinite periodic structure, so that the computation can be reduced to the elementary cell. Although analytical models for the elastodynamic behaviour of elongated beams on periodic elastic supports have already been proposed by Heckl (2002), our analysis runs over the more general configuration of*multi-structure*, describing the full interaction between low-dimensional models of different parts of the bridge, and the connection of Bloch–Floquet waves with the vibration modes of the finite system. In this case, we employ a quasi-periodic Green's function as a model solution. The dispersion equation is derived accordingly as described in the following text for the corresponding low-dimensional model.

### (a) Low-dimensional model

For sufficiently low frequencies *ω* of vibrations, the upper deck is treated as a one-dimensional massive elastic beam resting on concentrated elastic supports disposed periodically with span length *d* approximating the pillars contribution and we assume that the leading approximation of the elastic displacement field within the deck is
2.5where **e**_{x}, **e**_{y}, **e**_{z} stand for the basis Cartesian vectors in the direction of the principal *xyz*-axes, and *t* is the time.

For the equivalent elastic beam, we consider three possible decoupled vibration modes—the vertical and the horizontal bending modes and the longitudinal axial mode. The equation of motion for the vertical bending mode is
2.6where the subscripts stand for the partial derivatives with respect to the relevant independent variable, *EJ*_{y}(*x*) is the vertical bending stiffness, *ρ* is the mass density, *A*(*x*) is the cross-sectional area and *q*(*x*,*t*) is the vertical distributed load.

The equation of motion for the horizontal bending mode is
2.7where *EJ*_{z}(*x*) is the transverse bending stiffness and *p*(*x*,*t*) is the transverse distributed load.

Finally, the equation of motion for the longitudinal mode is
2.8where *r*(*x*,*t*) is the longitudinal distributed load.

### (b) Dispersion and structural optimization

In order to derive the dispersion relation for the bridge structure, we consider time-harmonic vibrations of radian frequency *ω* and we look for a displacement solution in the form
2.9By assuming, for simplicity equivalent uniform bending stiffness and cross-sectional area of the deck, we compute the quasi-periodic Green's function for the vertical and transverse bending modes involving displacement components *W*(*x*) and *V* (*x*), respectively. We start from Green's function for an infinite beam: the equation of motion for Green's displacement at the point *x* owing to a concentrated force of unit amplitude applied in *x*_{0} and vibrating harmonically with radian frequency *ω* is
2.10where *δ*(*x*−*x*_{0}) is the Dirac-delta function and . Fourier transform of equation (2.10) gives
2.11where is the Fourier transform of *g*^{Tot} that has the solution
2.12

The inverse transform of is then 2.13and 2.14

#### (i) Quasi-periodic Green's function

The quasi-periodic Green's function is defined by the relation
2.15for −*d*/2<*x*,*x*_{0}<*d*/2, where the forces are applied at the points of the periodic array, with the phase shift determined by the Bloch parameter *k*.

In the following, we show that *G*^{Tot}(*x*,*x*_{0};*ω*) is quasi-periodic; that is, it satisfies the Bloch–Floquet condition
2.16Let us start considering
2.17and, introducing *p*=*n*−m, we obtain
2.18

Let us now introduce in the definition (2.15) the infinite body Green's function (2.13)
2.19that can be recast in the form
2.20where
2.21For the purpose of applications, we are now interested to evaluate the configuration in which *x*=*x*_{0}=0; in this case, the quasi-periodic Green's function takes the form
2.22where
2.23and
2.24have been computed in the sense of distribution, giving the final form
2.25

#### (ii) Dispersion diagrams

First, we consider the vertical bending mode involving vertical displacement *W*(*x*) in the deck and longitudinal displacement in the pillar; the displacement at the contact point between the deck and the pillar is *W*(0)=*C*_{z}*G*_{0}(*ω*,*k*), where *C*_{z} identifies the amplitude of the corresponding concentrated force and in Green's function (2.25) the bending stiffness of the deck is . Equating the vertical displacement at the top of the pillar *C*_{z}/*γ*_{z} with the deck displacement *W*(0), we obtain the dispersion relation
2.26which describes the nonlinear relation between the frequency *ω* and the wavenumber *k*. The equivalent vertical stiffness of the pillar is equal to *γ*_{z}=20.79×10^{3} MPa m; the analytical estimate of the stiffness (*EA*_{p})/*l*_{p}=21.114×10^{3} MPa m; (*A*_{p}=15 m^{2}, *l*_{p}=24.36 m) is slightly above the FEM numerical value.

Analogously, we consider the transverse bending mode involving horizontal displacement *V* (*x*)=*C*_{y}*G*_{0}(*ω*,*k*) of the deck and transverse displacement of the pillar; the bending stiffness of the deck is now and the dispersion relation takes the form
2.27where *γ*_{y}=1.50×10^{3} MPa m. This numerical value was computed from an FEM model for a real structure where the supporting pillar is split into two interconnected blades, as shown in figures 1 and 8. Furthermore, this stiffness is used in the simplified asymptotic model for a homogeneous supporting pillar (figure 3).

In figure 4*a*, the dispersion relations (2.26) and (2.27) have been represented. The deck present a low-frequency stop-band; that is, there is a cut-off frequency below which the waves are evanescent. The first low-frequency pass band for the two types of flexural vibrations are in the ranges *f*=(1.4 Hz,2.207 Hz) and *f*=(1.717 Hz,2.223 Hz) for the vertical and transverse vibrations, respectively. We note that in the first pass band, the two modes of vibration have a comparable range of frequency with partial overlapping and that there are two frequency intervals 0 Hz<*f*<1.4 Hz and 2.223 Hz<*f*<3.354 Hz of complete band gap.

#### (iii) Evaluation of the pillar inertial effects

In §2*b*(ii), dispersive relations (2.26) and (2.27) have been obtained considering non-massive pillars and evaluating only their elastic effects in terms of equivalent stiffnesses *γ*_{z} and *γ*_{y}. If we consider the time-harmonic behaviour of the pillars, the spatial distribution of the vertical displacement of the pillar has the form
2.28where *F*_{1} and *F*_{2} can be computed from the boundary conditions *W*_{p}(0)=0 and *γ*_{z}/*l*_{p}*W*′_{p}(*l*_{p})=−*C*_{z} (*C*_{z} is an arbitrary constant), yielding
2.29

Then, the dispersion relation is obtained from the boundary condition *W*_{p}(*l*_{p})=*C*_{z}*G*_{0}(*ω*,*k*), and can be written in the form
2.30which is different from equation (2.26) by virtue of the *inertial factor*
2.31which tends to 1 as .

Analogously, the spatial distribution of the transverse displacement *V* _{p}(*x*) of the pillar has the form
2.32where *δ*=12*ρA*_{p}/*l*_{p}. The constants *G*_{1}, *G*_{2}, *G*_{3} and *G*_{4}, obtained from the boundary conditions *V* _{p}(0)=0, *V* ′_{p}(0)=0, *V* ′_{p}(*l*_{p})=0 and −(*l*^{3}_{p}*γ*_{y}/12)*V* ′′′_{p}(*l*_{p})=*C*_{y} (*C*_{y} is an arbitrary constant), are
2.33

Then, the dispersion relation is obtained from the boundary condition *V* _{p}(*l*_{p})=*C*_{y}*G*_{0}(*ω*,*k*), and can be written in the form
2.34which is different from equation (2.27) by virtue of the *inertial factor*
2.35that tends to 1 as .

The inertial factors *η*_{z}(*ω*) and *η*_{y}(*ω*) are plotted in figure 4*b* as a function of the frequency *f* and it is shown that for the structure under consideration the pillar inertial effects can be neglected.

#### (iv) Pass bands for Bloch waves versus eigenfrequencies of finite systems

It is illustrated in the present section that the dispersion properties of the infinite periodic system can be used to determine the viable range of eigenfrequencies for different classes of eigenmodes within a structure of the finite bridge. The dispersion curves shown in figure 4 can be projected to the vertical axis (the frequency axis) to define the ‘pass band’ intervals. Then for a structure, which incorporates a finite number of cells, the corresponding eigenfrequencies can be found in the predicted intervals. This simple approach appears to be efficient even when the number of spans of the bridge structure is small.

For the infinite periodic system, we use the quasi-periodic Green's function (2.25) and the related dispersion relations (2.26) and (2.27). For a finite size bridge, the eigenfrequencies are obtained from an FEM numerical analysis, with different bridge structures being taken into account.

*Low-frequency flexural modes of the upper deck of the bridge.* We first consider the low-frequency range corresponding to the two lowest dispersion curves, representing the vertical and horizontal flexural Bloch–Floquet waves, in figure 4. The comparison is made with the bridge, including just two support pillars. Figure 5 includes the rescaled dispersion curves, the pass band intervals for flexural waves and the lowest flexural frequencies of the finite bridge with two pillars. It is noted that the finite-element computation leading to the eigenfrequencies of the finite three-dimensional bridge is a computational task of an industrial scale. The proposed algorithm enables one to estimate the required frequencies analytically. Indeed, as shown in figure 5, the location of the lowest frequencies for flexural vibrations is fully consistent with the prediction of the analytical model of Bloch–Floquet waves. The corresponding flexural eigenmodes are shown in figure 6.

*Low-frequency flexural modes of the supporting pillars.* The lowest frequency, corresponding to the longitudinal horizontal motion of the bridge, is common for all finite structures supported by thin elastic pillars and can be easily estimated by considering the simple oscillator sketched in figure 6. The estimate is derived by using an asymptotic approximation of the upper deck of the bridge by the rigid solid, which is connected to the flexural elastic pillars. In the following formula, *M* stands for the mass of the bridge deck for the elementary cell of the structure, whereas *γ*_{x} is the lowest flexural stiffness of the pillar, which also depends on its length and the geometry of the cross section. The numerical values (*M*=2.165×10^{6} *kg*, *γ*_{x}=53 MPa m) have been computed for the case of the real structure depicted in figure 1:
2.36The simple oscillator model for the computation of the first eigenfrequency is shown on the bottom right of figure 6. The analytical approximation above is very accurate, with the relative difference not exceeding 4 per cent when compared with the values obtained from the finite-element computations for different number of spans in the finite bridge structure. We note that the formula (2.36) is similar to equation (2.4), with the difference that the stiffness *γ*_{x} was evaluated numerically from the FEM model for the actual two-blade pillar.

Figure 6 gives the comparative data for a wider range of frequencies and includes the data for pass bands for Bloch waves in the periodic systems, as well as the values of eigenfrequencies for finite bridge structures, which incorporate two, six and 12 pillars. It is observed that the eigenfrequencies of the finite bridge are localized within well-defined frequency intervals corresponding to the pass bands of the periodic structure and that the number of eigenfrequencies within each interval increases with the increase of the number of spans, i.e. with the number of unit cells composing the finite structure.

It is emphasized that the finite-element analysis of finite bridge structures and the corresponding eigenmodes in figures 6 and 2 show that the lowest frequencies correspond to the modes that have been described analytically. These include the vertical flexural modes of the upper deck linked to longitudinal deformations of the pillars, and horizontal flexural modes of the upper deck linked to lateral flexural deformations of the pillars. Torsional vibrations occur for higher frequencies (*f*>8 Hz) where the beam model of the deck is not valid any more and at least a plate model should be considered.

The quasi-periodic Green's function is used in the analytical form. Although the corresponding geometry has been strongly simplified, the analytical prediction of low frequencies of the finite system appears to be accurate and hence such a model is suitable as a design tool for elongated bridge structures.

#### (v) Structural optimization

In figure 7, the dispersive behaviour for different level of equivalent stiffness transmitted by the pillars to the deck is shown. In figure 7*a*, dispersion curves are given for the stiffness values MPa m for the vertical flexural modes and MPa m for the transverse one.

The dispersion curves lay between a lower and an upper bound; the lower bound corresponds to the case of a beam in a free space with the simple dispersion relation 2.37The upper bound corresponds to a periodically simply supported beam that is obtained for and it is characterized by the dispersion relation 2.38

A problem of technological interest is the design of a structure that maximize the filtering of elastic waves; this can be achieved by minimizing the frequency interval amplitudes corresponding to the bands of propagating waves; if we focus the attention on the first band, the optimal stiffness is given by the simple relation 2.39

The dashed curves in figure 7*a* show the dispersion curves obtained for the optimal stiffnesses in equation (2.39).

In figure 7*b*, the frequency interval of the first propagating band is shown as a function of the pillar equivalent stiffness for the vertical and transverse flexural modes. Some of the stiffnesses considered in figure 7*a* are detailed with dashed curves. Such a figure is an ‘example’ of an application of the analytical method that can be used in order to optimize the filtering behaviour of a structure in a wide frequency range.

## 3. Concluding remarks

The numerical computations show a remarkable agreement with predicted analytical estimates obtained from a low-dimensional asymptotic model.

For the sake of completeness, we present figure 8 and table 1, which include the geometry of the pillars and the overall bridge structure. The bridge is a prestressed structure where pre-tensioned cables are incorporated in the upper deck. The numerical analysis was performed with the Finite Element Software Strand7. The discretized model includes three-dimensional brick elements; each span consists of 12 518 brick elements for a total of 58 326 degrees of freedom. The material is the reinforced concrete with Young's modulus *E*=34 290 MPa, Poisson's ratio *ν*=0.2 and density *ρ*=2500 kg m^{−3}. Linear analysis is justified by the presence of prestressing that leads to results in agreement with the experimental measurements.

For the case of high-order frequencies associated with the deformation of the bridge deck, we have used the tools of spectral analysis of Bloch waves in a periodic system. This is shown to be the right approach to evaluate the frequency intervals for a finite elongated bridge.

Finally, the asymptotic formulae obtained here enable one to develop asymptotic optimization algorithms to control the dynamic response of the bridge structure within the low-frequency range.

## Acknowledgements

Part of this research was performed while A.B.M. was at the University of Cagliari under the Visiting Professor Programme 2010 financed by the Regione Autonoma Sardegna. The financial support of the Research Centre in Mathematics and Modelling of the University of Liverpool and Italian Ministry of University and Research under PRIN 2008 ‘Complex materials and structural models in advanced problems of engineering’ (M.B.) are gratefully acknowledged. We thank the referees for their time and for their useful constructive comments.

- Received July 8, 2011.
- Accepted September 7, 2011.

- This journal is © 2011 The Royal Society