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This study carries out a parametrical analysis of the seismic response to asynchronous earthquake ground motion of a long multispan rc bridge, the Fener bridge, located on a high seismicity area in the north-east of Italy. A parametrical analysis has been performed investigating the influence of the seismic input correlation level on the structural response: a series of nonlinear time history analyses have been executed, in which the variation of the frequency content in the accelerograms at the pier bases has been described by considering the power spectral density function (PSD) and the coherency function (CF). In order to include the effects due to the main nonlinear behaviours of the bridge components, a 3D finite element model has been developed, in which the pounding of decks at cap-beams, the friction of beams at bearings, and the hysteretic behaviour of piers have been accounted for. The sensitivity analysis has shown that the asynchronism of ground motion greatly influences pounding forces and deck-pier differential displacements, and these effects have to be accurately taken into account for the design and the vulnerability assessment of long multispan simply supported bridges.

Earthquake ground motion is usually assumed as a spatially uniform dynamic input in seismic analysis; this assumption is correct for structures standing on a reasonably restricted area, in which the soil characteristics are presumed to be homogeneous and the seismic wave propagating velocity can be neglected, but becomes inadequate for spatial structures standing on large sites such as extended foundations or dams, and long-estending structures such as bridges, viaducts, tunnels, and pipelines. In these cases the spatial variability of ground motion should be considered to avoid gross evaluation errors or at least underestimation of the dynamic response, since the phenomenon affects the response considerably and, hence, the level of protection of these structures (Lupoi et al. [

For this type of bridges are required quite complex numerical models to represent with acceptable approximation the global structural response taking into account the inchoerency of the seismic excitation at the supports, the impact phenomena between neighbouring structural segments, and the nonlinear behaviour of the substructural components (piers and decks).

In the present study, the acceleration and displacement time histories at the several prescribed locations on the ground surface corresponding to the bridge supports, are generated using the spectral representation method [

As regards evaluation of the pounding effects, it has to be said that the interest of researchers is quite recent; the problem was first investigated by [

From the aforementioned studies interesting conclusions can be drown for an improved modelling of the pounding effect:

pounding between adjacent segments can be described with fair accuracy through an impact element characterised by stiffness and damping (which accounts for energy dissipation);

it has been noted in [

in the finite element model the stiffness

it is important to define opportunely the time step used for the integration in the time-domain to avoid that colliding adjacent segments of neighbouring decks may behave like rigid bodies, since they influence the dynamic response with their axial deformation.

In the present study state-of-the-art models have been used to simulate the asynchronous ground motion as a multisupport seismic excitation and describe the pounding effects, as described in Section

This study carries out a parametrical analysis of the seismic response to asynchronous earthquake ground motion of a long multispan rc bridge, the Fener bridge (see Figures

Fener bridge: (a) lateral view of the bridge on the Piave river.

Typical transverse section of the superstructure.

Pier elevation and longitudinal section.

Cross sections of (a) typical column with reinforcement and (b) typical longitudinal precast concrete beam.

It represents an important overcrossing of the Piave river for the region road network. It was built in the mid nineteen seventies, and it consists of 24 regular spans having the same length of 24.75 m, except for the lateral spans near abutments, which are shorter (in particular at one end there are two spans with reduced lengths of 18 m and 17.5 m, respectively, whilst at the other end only the last span has a slightly shorter lenght of 23.75 m). The overall structure is about 579 m long.

The deck lodging two lanes has an overall width of 9 metres; the deck structure is made up by four I-shaped precast beams with a constant height of 1.4 m and by a 16 cm high rc slab. The transverse distribution of traffic loads is obtained through 3 orthogonal rc girders positioned in the middle and at both ends of each span. Piers have a portal-shaped structure with circular rc columns, whose height varies gradually along the plan from a minimum of 5 to 8 m roughly, since the deck slope in the longitudinal direction is about 2%, while the extrados levels of plinths at the base remain constant. Piers raise on deep foundations as illustrated in Figure

The pier section is shown in Figure

The materials used for piers can be classified as follows:

concrete: grade C25/30;

reinforcing steel: smooth bars, characteristic yield stress

According to the national seismic zonation map, Fener bridge is located in an area characterised by PGA = 0.25 g, on a soil of medium stiffness (type B soil, according to the national zonation map [

In the numerical model of the bridge elements with linear and nonlinear behaviour have been adopted in order to represent effectively the global structural response: main beams, cap-beams, and transverse girders have been modelled with linear beam elements, the rc slab has been modelled with plate elements, whilst a nonlinear behaviour has been adopted for columns to simulate their hysteretic behaviour (using the Takeda-model [

Models for (a) hysteretic behaviour of piers (Takeda model) and (b) gap element between adjacent structural segments.

For the pier-element it has been necessary to assign in input a nonlinear force-displacement law, which has been obtained through a push-over analysis both in the longitudinal direction, where the column has a cantilever deflection, and in the transverse direction where the piers behaviour is that of a portal frame. A lumped plasticity element has been employed for modelling the piers; the derived force-displacement curves are plotted below (see Figures

Girders sit on cap-beams without any bearing devices therefore restraint of superstructure segments from longitudinal displacement is given only by friction; the force-displacement law for frictional bearings is assumed as an idealized rigid-plastic behaviour, with a friction coefficient taken as

Connection between girder and cap-beam (frictional behaviour).

Piers behaviour in the longitudinal direction: (a) deflection shape; (b) force-displacement curve.

(a) Piers behaviour in the transverse direction: deflection shape; (b) force-displacement curve.

Pier-deck pounding has been modelled through nonlinear gap elements which react only under compression, after the initial gap closure corresponding to the joint width (2 cm).

The gap element stiffness

As to external restraints, they have been considered fixed both in translation and rotation, because foundations are plinths on piles and in a first approximation the soil-structure interaction can be neglected. The superstructure segments not considered in the model (which represent a boundary condition to it) have been substituted with gap elements as illustrated in [

The FE model of the bridge and the related nonlinear dynamic analyses have been performed using

Three-dimensional FE model of the seven central spans.

In the present study, the acceleration and displacement time histories at several prescribed locations on the ground surface corresponding to the bridge supports are generated using the spectral representation method. A uniform soil type is considered. As mentioned before, in order to generate the stochastic field, three basic components are required: (i) power spectral density function, (ii) coherency function, and (iii) shape function.

Different analytical models for PSD are advanced by some authors; in this study the expressions given in EC8 [

Assuming that the seismic wave field can be completely described by a single plane vawe, its spatial variation can be quantified by means of the coherency function, which expresses the dependence in the frequency domain between the PSD of time histories ground motions occuring at two different stations

In general

There are several models available in literature for the coherency function; in the present study the formulation given in [

Coherency function modulus obtained for different correlation levels corresponding to a set of 4 values of parameters

The shape function of the oscillatory process is defined in a general exponential form as suggested in [

In this study the formulation proposed in [

Ray of the 8 stations implemented in the FE model of the structure.

Different patterns of coherency have been selected in the parametric study, in order to represent the intermediate levels between the full correlation and the total uncorrelation of the time histories: 16 combinations of parameters

Combinations of

Vawe-passage effect | Geometrical incoherence | |||

300 | 600 | 900 | 1200 | |

300 | x | x | X | x |

600 | x | x | X | x |

900 | x | x | X | x |

1200 | x | x | X | x |

A superimposition of the displacement time-histories generated in the simulation is reported as an example in Figure

Displacement time histories for stations 1 to 8: (a) highly correlated time-histories (set 1/5

In order to determine the nonlinear response of the structure to a large set of earthquake ground excitations, it is necessary to use an efficient and not much time-consuming time-integration algorithm; in the present study the mode superimposition procedure based on load-dependent Ritz vectors [

The values calculated for

Integration time step adopted

Elastic modulus | 24821 |

Density ^{3} | 2500 |

Deck span lenght | 24.75 |

Impact duration | 0.016 |

Integration time step | 0.01 |

0.625 |

It should be noticed that in fact a superstructure segment does not hit the neighbouring deck directly, due to the presence of the cap beam, but this element has been assumed as transmitting the impact rigidly and not influencing the wave propagation.

A sensitivity analysis of the structure dynamic behaviour due to different spatially varying ground motion sets has been carried out, evaluating the influence of the seismic input correlation on the structural response, in terms of the following:

differential displacement between piers and deck segments;

pounding forces between cap-beams and decks;

effects on piers: shear forces at the bases and maximum displacements at the tops.

The response analysis focuses on the central span of the FE model, in order to provide results unaffected by the boundary coditions; as previously said, for each prefixed level of ground motion correlation (16 in total, each one determined by a couple of the parameters

Differential displacements between piers and decks are represented in Figure

Pier-deck differential displacement varying with seismic input correlation level (represented by parameters

The limited amplitude of differential displacements prevents pull-off-and-drop collapse of deck segments and can be explained considering that joint gaps at span ends are small (2 cm) and do not allow the development of high inertia forces at the deck level; consequently the displacements cannot be considerably amplified. These results are in accordance with the observations reported in [

Impact forces between cap beams and decks are highly influenced by the correlation level of ground motions at the structural supports: as Figure _{I}, obtained in the extreme case of weakly correlated time histories (

(a) Pounding forces varying with correlation level of input ground motions; (b) Total number of registered impacts (mean value).

Registered impacts follow a similar tendency (see Figure

As regards the effects on piers in the longitudinal direction, they are represented in Figure

Longitudinal direction: (a) shear forces at the pier base: maximum values obtained for each correlation level of input ground motions; (b) maxima values of displacement at pier top.

These effects can be explained considering that when ground excitations are weakly correlated or uncorrelated, the movement of deck segments can be in opposite direction due to out-of-phase vibrations, and this fact determines collisions that reduce displacements at the top of the pier (and consequently the shear and bending moment at the base induced by the deformation of the pier itself). When the ground excitation is highly correlated, the responses of the bridge spans are in phase, the inertial forces at the pier tops are maximised, and in consequence displacements at the top and shear forces increase.

Regarding the response in the transverse direction, it should be noted that structural behaviour is not clearly affected by seismic input correlation (see Figure

Transverse direction: maximum shear forces.

A parametrical analysis has been performed with the aim of investigating the influence of the seismic input correlation level on the structural response of a long multispan girder bridge. A series of nonlinear time history analyses have been performed, in which the main nonlinear behaviours of the bridge components, have been included: (i) the pounding of decks at cap-beams, (ii) the friction of beams at bearings, and (iii) the hysteretic behaviour of piers. The following conclusions can be drawn:

differential displacements between decks and pier-tops are affected by input correlation level but remain within a limited range (under the threshold of 5 cm) with the maximum value obtained for the extreme case of maximum coherency loss. The fact that they are relatively small prevents decks from unseating and can be explained by the limited width of the bridge joints;

asynchronous ground motion influences greatly the pounding forces between decks and pier-tops, which can assume values 3 times larger than those calculated by an analysis with uniform input (represented by the case with the highest correlation level between the time-histories). The amplified pounding effects might determine considerable damage to the local area of bridge decks;

as regards the effects on piers, it can be observed that in the longitudinal direction there is a general trend for displacements and shear forces, which increase with higher correlation levels of input ground motions. In the transverse direction the seismic response is not clearly influenced by the correlation level of ground excitations.

The results highlight that the spatial variation properties of the earthquake ground motion can significantly change the structural response especially in terms of pounding forces and deck unseating, and consequently these effects have to be taken into account for the design or the vulnerability assessment of long multispan simply supported bridges.