The location effects, i.e. the different ground conditions on the bridge supports, also increased the bridge response, so those authors introduced different peak ground accelerations and different spectral gains. The first and second form of the bridge vibration modes are shown in FIG.
A spectral‑representation‑based algorithm
With this coherence function, the loss of the coherence effect is considered in the non-uniform seismic ground motions generated for this study. Φjl are independent sequences of random phase angles generated from a uniform distribution in the interval [0, 2𝜋] ; and Hkjm(𝜔l) is a typical element of the triangular matrix obtained from the Cholesky decomposition of the cross-spectral density function matrix 𝐒k(𝜔) (Eq. 2000) mode, the cost of the digital generation of sample functions, ̈ugk ,j(t) i this case, in a simulated stochastic process, can be drastically reduced using the Fast Fourier Transform (FFT) technique.
Iterative scheme of Deodatis
On the other hand, to generate time history sample functions compatible with a Eurocode prescribed response spectrum for each bridge support, the Deodatis (1996) scheme was implemented in the code developed for this work. To calculate the acceleration response spectra corresponding to the ground acceleration time histories generated ̈ugk,j(t) at step 5 of Table 4 scheme, the following Equations can be used. Generate the stationary time histories agk j,( )t ;j=1,2..n; k=y z, by means of Eq.(14) using the Cholesky decomposition of the cross-spectral density function matrix.
Zerva (2009) states that the convergence criteria for the iterative Deodatis scheme cannot be easily determined, but that the iteration can be stopped when the shape of the target response spectrum is captured by the response spectrum of the simulations. In each new iteration, the power spectral density functions Skj(𝜔) were updated as given in Eq. 22), RSAk,j(𝜔l) is the Eurocode 8 target response spectrum corresponding to bridge girder j, and RSAuk,j̈g(t)(𝜔l) is the calculated response spectrum from the time history ̈ugk,j(t ) generated in the previous iteration (i-1). Out of all 10 iterations performed, the time history corresponding to the iteration in which the differences between the target and calculated response spectra were less than 15% (Eq. 23)) was selected as the most suitable for this study.
Sometimes the time history of iteration 8 met the established criteria for obtaining a suitable time history response spectrum (Eq.
Consistency treatment of earthquake record
In general, only a few iterations, 10 or less than 10 iterations, are required to achieve this goal. If the mentioned goal is not reached within the first few iterations, the simulated time history should be discarded and a new one generated (Deodatis 1996). The iterative scheme of Deodatis (Table 4) to generate the time-history ground motions at each bridge abutment ̈ugk,j(t) has been used in this study, performing 10 iterations.
The authors observed that the last iteration, number 10, was not always the one that met the criterion expressed in Eq. The time-domain high-pass filtering method has been used here as the mentioned record consistency treatment. This method consists in modifying the generated acceleration time histories with a high-pass filter in the form of a critical damping oscillator (Xia et al. 2018).
24), ̈ugk,jo(t) refers to the ground acceleration time histories at the bridge abutments before modification; while ̈ugk,j(t), ̇ugk,j(t), ugk,j(t) are the acceleration, velocity, and displacement time histories at the bridge abutments after treatment.
Seismic ground motion simulations
9 a Generated lateral ground accelerations at bridge supports 1 and 12 (PGA 0.06·g and 0.07·g, respectively) and b lateral ground displacements associated with ground accelerations shown earlier in this figure. 10 a Generated vertical ground accelerations at bridge supports 1 and 12 (simultaneously with lateral ground accelerations shown in Fig. 9) and b vertical ground displacements associated with ground accelerations shown earlier in this figure. The seismic event selected from the 20 samples was used in this study because it was the most unfavorable for road safety.
Figures 9 and 10 plot the time histories of ground acceleration and displacement on several bridge girders corresponding to a sample earthquake event with reference PGAy = 0.06·g. 9b and 10b, the amplitudes of the ground displacements at bridge girder 12 (pillar 11) were greater than at girder 1 (pillar 1). This was due to the fact that the terrain at pier 11 was less stiff than at bridge pier 1, so the corresponding response spectra at these two locations were different.
In these figures, the effect of variable site conditions of this spatially variable seismic ground motion can be observed.
Type of earthquakes generated
- Type I earthquakes
- Type II earthquakes
- Type III and IV earthquakes
- Earthquake outside the bridge
The terrain effects that are taken into account in this type of seismic movements are those indicated in Table 2. The terrain at piers j = 9 to j = 12 (piers 8–11) is loose gravel (type D soil according to Ec-8), and the terrain at the rest of the piers consists of stiff clay (type C soil according to Ec-8). In this type of earthquake, no effects of the spatial variation of seismic motion along the bridge are taken into account.
These seismic motions were obtained by copying the generated ground motions at girder j = 1 of type I earthquakes. In all the above described types of earthquakes, seismic ground motions occur at each bridge support (pillars and abutments), and seismic ground motions also occur under the track outside the bridge on which the train travels before entering the bridge and after leaving the bridge. . The aforementioned type of earthquake was considered outside the bridge, as it is a reference calculation to assess the effect of the bridge on the train's response to an earthquake.
Seismic forces on the bridge
In the above equations, ld is the length of the first and last finite element of the deck (Fig. 11). Izd, Iyd are section moments of inertia of the first and last deck finite element. Ed and Ecp are respectively the longitudinal elastic modulus of the deck and piers considered in the model.
For the systems of train equations (Eq. 4)), in the present study, nonlinear functions are located in the interaction force vectors acting on the train and on the bridge ( 𝐅vb and 𝐅bv). To obtain the numerical solution (dynamic response in the time domain) of a dynamic system from its equations of motion requires the use of a finite difference method to integrate said equations over time. Since they are nonlinear equations, Wilson's method was combined with the Newton-Raphson method.
To solve the connection between the two systems (the train and the bridge), an iterative procedure was used by which, at each time step of the analysis, and at each iteration of the Newton-Raphson procedure to solve the bridge system. of equations, another Newton-Raphson procedure was used to solve the system of train equations.
Running safety analysis performed
Influence of earthquake intensity and train speed on the bridge and train responses
The bridge's importance factor 𝛾I and the coefficient S for the terrain type C (firm clay) are taken into account in Eq. We can consider the seismic event of 0.03 g PGA and the passage of the train over the bridge as independent events. So the probability of the frequent 0.03 g PGA earthquake occurring during the life of the bridge while a train is on the bridge in Granada is also Pr6 = 2.21%.
The formulation used in the calculations of the mentioned probabilities is shown in appendix 2 (table 6). An earthquake that brings the safety of the Thalys train running at 300 km/h over high-speed tracks has a PGA of 0.35 g. If we consider that in the area of action (15 km long) for both earthquakes (one of 0.03 PGA and the other of 0.35 PGA) are two bridges that it studied, the probability of the train passing over a bridge at 300 km/h that is at risk of derailment because a frequent earthquake occurs, is now Pr′6 = 4.42%.
We would like to emphasize that the train passage frequency and the duration of the train journey over the bridge are essential parameters when calculating the probability of occurrence of this new proposed seismic serviceability limit state.
Influence of wave passage effect, loss of coherence, site effect and bridge lateral stiffness
- Bridge response
- Vehicle response
- Traffic safety indices
However, this amplification of the deck response, due to a less firm ground at pier 9, was not observed in the deck lateral accelerations (Fig. 17a). However, type III considered seismic wave passage, while type II did not. 16a and 17a, the values of the type III earthquake (diamond line) were greater than those corresponding to the type II earthquake (star line).
In the case of passenger cars (Fig. 20a), it was observed that the spatial variation of the seismic action did not affect the vehicle's response. On the other hand, the speed of the train had a major influence on the maximum accelerations of passenger cars. 20b, that a resonance phenomenon occurred in the behavior of the locomotive at a speed of 280 km/h.
The mechanical properties of the ICE3 train can be found in the documents of Antolín (2013) and Olmos (2016).
Conclusions
It has been verified that the highest values of the studied road safety indices were reached when passenger cars entered the bridge. 21 we can see that the maximum values of the road safety indices (Prud'Homme index and derailment factor) were very similar for all tested earthquake types. However, the ICE3-type train can run safely over the viaduct at any of the speeds tested.
These road safety values, obtained with the seismic event number 4, correspond to the highest value of all the 20 events in the case of IPH,. 22 a Maximum Prud'Homme index for a train wheelset and b maximum wheelset derailment factor in simulations of a Thalys train running at 240 km/h during each of the 20 generated earthquake events of PGA 0.8 g. Simulation of Thalys train, traveling at 240 km/h during several of the generated earthquake events of PGA 0.8 g.
In this appendix, the wording is used in the calculation of the probabilities referred to in art.
