Scaling law study for earthquake induced pier–water interaction experiments

Abstract

Froude number similarity has been used in all the physical experiments of earthquake induced hydrodynamic forces acting on deep water bridge piers without validation. This study focuses on the validation of the viability of Froude number similarity experimentally and numerically. Results show that Froude number similarity is viable, although the gravity wave force does not dominate the hydrodynamic force on bridge piers under earthquakes. Further, Euler number similarity is proven to be viable and Froude number similarity is proven to be the special case of Euler number similarity when $\lambda_{f} = \lambda_{L}^{ - 1/2}$($\lambda_{f}$ and $\lambda_{L}$ indicate frequency scale and length scale respectively) in Euler number similitude. Numerical studies of the application demonstrate both Froude number similarity and Euler number similarity have almost the same precision when they are used to scale up the in-line force of model pier to prototype pier under ordinary earthquakes. However, Euler number similarity is recommended because of the advantages of providing a chance to choose frequency scale, which may be better supported by equipment and allow a Reynolds number closer to the one observed in the prototype.

Appendices

Appendix 1

A real deep water bridge, Miaoziping Bridge, standing in Zipingpu reservoir, not far from Chengdu city, Sichuan province, China, was seriously damaged by the 2008 Sichuan Earthquake (Mw 7.9). The width of the reservoir where the bridge crosses it was about 700 m and the submerged depth of the main pier was about 50 m when the 2008 Sichuan Earthquake occurred. Numerical simulation of the flow velocity is conducted to show the importance of the waves generated by river banks. The river banks are assumed to be vertical and the river bed is assumed to be flat, as shown in Fig. 18. Both the river banks and river bed are set as moving walls stimulated by the earthquake motion detected at the nearest earthquake monitoring station in 2008. Three points P1 (350, 0, − 2), P2 (350, 0, − 20), P3 (350, 0, − 49.8) are defined to measure the velocity time histories at the center of the reservoir.

Fig. 18

The 2D model of reservoir (unit: m)

Figure 19 shows the comparison of time history curves between the three monitoring points and the river bank (or river bed). Results show that the curves of the three monitoring points coincide with each other and are extremely close to zero, demonstrating that the three points almost keep still during the earthquake. Because bridge piers are always tens or hundreds of meters away from river banks, a small amount of time is required for the waves generated by river banks to propagate to the locations of the piers. In other words, the main shock of an earthquake may end before the waves arrive at the positions of piers. In addition, the waves decay during the propagation. The horizontal movement of the flat river bottom can strongly influence the extremely near zone to boundary layer but cannot significantly influence the flow field some distance above the river bed. Therefore, the disturbance to the flow filed around the pier induced by river banks and river bed are very small and could be ignored.

Fig. 19

The velocity time histories comparison between the river bank (river bed) and the three monitoring points in the reservoir. Velocity curves at these three points almost coincide with each other and are extremely close to zero during the main shock of the earthquake

In practice, the piers are fixed at the bottom of the river bed, standing in quiescent water before earthquakes, and burst into oscillating simultaneously with the river bed when earthquakes happen. Ignoring the influences of river banks and river bed, the flow filed around a real bridge pier is similar as that stimulated by a cylinder oscillating in initially quiescent water. Therefore, the first scheme which uses an amphibious shake table in a still water flume [13] is better than the second scheme [10, 15, 18], which fixes a water tank on an ordinary shake table. Because in the second scheme, the dimensions of water tank are much smaller than they should be (values scaled down from the dimensions of the reservoir by the length ratio), which results in the water surrounding the model pier oscillating simultaneously with the pier.

Appendix 2

The four cases with different mesh sizes listed in Table 7 are compared to show the influence of the mesh size. The average peak values of the in-line force based on 15 cycle stable oscillations are also listed in Table 7. The average peak value of the “Middle” case is closest to the experimental value, namely 1.152 N, and the “Middle” case consumed less time. Therefore, the mesh sizes in the “Middle” case are employed in this study.

Table 7 The mesh size influence on calculation accuracy and efficiency, $d_{volu}^{max}$ is the maximum grid size of the computational domains, $d_{surf}^{max}$ is the maximum surface element size of the cylinder surface, $d_{prism}$ is the innermost prism layer height, $N$ is the number of the prism layer, and $F_{x}^{peak}$ is the average peak value of the in-line force