Wave and wave-current actions on a bridge tower: An experimental study

Abstract

With a scale of 1:100, an experimental model was set up to investigate the dynamic responses of a bridge tower subjected to ocean waves and wave-currents. The bridge tower was designed for a sea-crossing bridge. Based on a pile-group foundation, it was designed to be a typical gate-type structure. Wave-induced base shear forces on the foundation and motion responses of the tower were measured and analyzed. The experimental results show that when a wave period is close to the natural period of the structure, an obvious resonance will be induced on the structure. For different wave action angles, the longitudinal incident waves induced the largest longitudinal base shear forces on the foundation and the greatest dynamic motions on the upper tower of the structure. Because of the pile-group effectiveness, the incident directions of the waves and the wave periods affect the acting forces on the foundation of the structure. For wave-current actions, forward currents increase the forward wave forces on the foundation and decrease the backward forces, but do not significantly affect the motion responses of the upper tower. The experimental results can be used as the verification data for numerical calculations. With the structural forms of the pile-group and the gate-type tower being typical, the results given in this study can be used as a reference for similar engineering designs.

Keywords

base shear force bridge tower dynamic response experiment wave wave-current

Introduction

With the engineering technique consistently improving, the construction of sea bridges, including coastal bridges, bay bridges, and strait-crossing bridges, is developing rapidly. Compared with the environment of an inland bridge, marine environments are more threatening and complex to a sea-crossing bridge. Deep water, strong waves, and currents are serious to the stability and safety of the structure during its construction and service periods. When a sea-crossing bridge, such as a cable-stayed bridges or a suspension bridge, is designed to have a long span to satisfy the water depth condition and the navigation requirement, the structure becomes to be a flexible structure under ocean wave actions. When a bridge tower is freestanding during the bridge construction period, its natural period possibly may be within the period range of ocean waves; the wave forces on the foundation and the wave-induced movements on the upper tower are likely to cause the local damage, strength failure, or fatigue fracture on the structure. Therefore, understanding the hydrodynamic characteristics and dynamic responses of a freestanding bridge tower subjected to waves and wave-currents is important to engineering designs and constructions.

It is known that pile-group and caisson are two typical forms for bridge foundations, and A-type and gate-type are two common structural forms for bridge towers. In a previous work, the load forces and the motion responses of an A-type bridge tower based on a caisson foundation due to waves and wave-currents have been experimentally studied by Wei et al. (2017). Even so, it is necessary to further study the dynamic responses of a gate-type bridge tower based on a pile-group foundation due to waves and wave-currents.

The top of a pile cap or a caisson usually designed to be higher than the water surface, and then, ocean waves can generate a horizontal force and an overturning moment on the structure. As wave forces are important to the strength designs of bridge foundations, the estimation of wave and wave-current forces on bridge foundations has been an important research project. The Morison equation (Morison et al., 1950) is an essential method for computing the wave forces on a pile-group foundation. With the pile-group effectiveness considered, the wave forces on a pile-group are obtained by superposing the Morison forces on each single pile in the group. A bridge caisson foundation is commonly designed to be a vertical cylinder, with its cross section being circular, rectangular, or quasi-elliptical. As a caisson foundation is generally large compared with a common wave length, wave diffraction effect induced by the structure is obvious and must be considered in the wave force calculations. Based on linear wave diffraction, an earliest analytical solution that developed by MacCamy and Fuchs (1954) has always been used to calculate the wave forces on a large circular cylinder. Based on this solution, also with the assumptions of linear theory, wave diffraction by an array of bottom-mounted circular cylinders has been analyzed (Maniar and Newman, 1997). Furthermore, wave diffractions on multiple, in-line vertical circular cylinders due to solitary waves have been investigated (Neill et al., 2018). The wave force solution on a uniform vertical with cosine-type radial perturbations has been developed (Mansour et al., 2002). An analytical solution of the wave forces on a uniform vertical cylinder with an arbitrary cross section has been presented in recent years (Disibüyük et al., 2017; Liu et al., 2016). A formulation of the nonlinear loads exerted on floating bodies by steep irregular surface waves has been presented (Sclavounos, 2012). The formulation can be applied for ships and offshore platforms and the extreme wave loads and responses of offshore wind turbines. These solutions will help in estimating the wave forces on large uniform caisson foundations. For foundations with irregular submersed geometric profile, such as a pile-group foundation with a pile cap or a gravity-type foundation, the wave diffractions on the structures are complex, and then, the wave force estimations are generally solved by numerical calculations or model experiments. Numerical methods like the boundary element method (Liu et al., 2015; Newman, 1985; Newman and Lee, 2002; Teng and Eatock Taylor, 1995) and the finite element method (Kim et al., 2006; Ma et al., 2001) have been well applied in calculating the wave forces on underwater structures. However, few model experiments have been done on wave forces on bridge foundations. The study of this article will be an experimental supplement on this research.

Because of the interactions of waves and currents, wave-current forces on a bridge foundation will be more complex compared with pure wave forces. A simple superposition method of wave-current force calculation has been presented. The method suggests that wave force and current force are calculated separately and then added together, that is, wave force is calculated by the Morison equation or the boundary element method while current force is calculated by the drag force formula (Pan et al., 2013). Moreover, because the interaction of wave-currents with the structure is very complex, especially for the foundations with arbitrary pile caps on piles, model experiments are necessarily conducted to find out the wave-current forces. As the foundations of the East Sea Bridge of China were designed to be pile-group with circular pile cap, two model experiments have been carried out to study the wave and wave-current forces on those foundations for engineering application (Lan et al., 2005; Liu et al., 2007). However, as the structural forms of bridge foundations are various, such experimental studies are still rare.

Although much research has been devoted to the load forces and damages on bridge deck subjected to ocean waves in previous works (Bradner et al., 2011; Hayatdavoodi and Ertekin, 2016; Seiffert et al., 2014; Xu and Cai, 2015), some attentions have been paid to the dynamic responses of a bridge tower or a whole bridge due to ocean loads in recent years. Because the stiffness of a caisson foundation is large, the vibrations on the foundation induced by waves are tiny. Nonetheless, because the bridge tower based on the foundation is generally designed to be high and consists of relatively slender columns, the tower will be liable to shake obviously under a weak vibration on the foundation, especially when the tower is freestanding during its construction period. With wind always being a significant load on a bridge, two model experiments have been carried out to study the wave influences on the dynamic responses of bridge towers due to wind, that is, the coupled actions of wind and wave loads on freestanding bridge towers (Bai et al., 2016; Guo et al., 2016). The relevant investigations of a bridge tower under the wave and wave-current loads are still lacking in previous studies. In addition, by a model experiment driven by flume wave and wave-current loads, this study is to reveal the hydrodynamic responses of a classical gate-type bridge tower based on a common pile-group foundation.

Experimental design and setup

Description of the bridge tower and model design

The bridge tower served as the prototype structure for the experiment is a component of a planned sea-crossing bridge of the Qiongzhou Strait of China. The foundation of the tower is designed to be a pile-group foundation. The tower is designed to be a double-column, gate-type structure, as shown in Figure 1. The height of the tower is 203.95 m from the top of the pile cap to the tower top. The pile-group is composed of 16 piles. Each pile is 3 m in diameter and 23 m in length from the seabed to the bottom of the pile cap. The cross sections and the main design sizes of the foundation, tower columns, and crossbeams are shown in Figure 1(c).

Figure 1.

Structural form and primary design sizes of the bridge tower: (a) longitudinal view, prime cross sections are labeled with A-F; (b) transverse view; and (c) design dimensions of the cross sections labeled in (a).

The experiment model was marked with a geometric scale of 1:100 with consideration of the experimental contents and the conditions of the experimental facilities. Geometric similarity, Froude similarity, and elastic-gravity similarity (Chi and Lin, 2004) criteria were used to design the model. Geometric similarity is to ensure the length, width, and height of the structure model and the water depth, wave length, and wave height of the flume wave to have a same geometric scale. As the gravity and inertia forces of fluid are the main determinants of the wave and wave-current forces on a bridge structure. The flow field of flume wave was designed to satisfy the Froude similarity, that is

\frac{λ_{g} λ_{l}}{λ_{v}^{2}} = 1, \frac{λ_{l}}{λ_{t} λ_{v}} = 1

(1)

where $λ_{l}$ , $λ_{g}$ , $λ_{v}$ , and $λ_{t}$ are geometric scale, gravity scale, velocity scale, and time scale, respectively. The structure model of the tor was designed to satisfy the elastic-gravity similarity. The similarity relation is

λ_{E} = λ_{l} λ_{ρ}, λ_{t} = λ_{l}^{1 / 2}

(2)

where $λ_{E}$ and $λ_{ρ}$ are elasticity modulus scale and density scale, respectively.

Being an elastic dynamic structure, the model consists of the skeleton, clump weights, and polyethylene shells, as shown in Figure 2(a). The skeleton was welded by 16 circular iron bars that served as the pile group, a steel plate that served as the pile cap, two long circular iron bars that served as the tower columns, and two rectangular iron bars that served as the crossbeams. The cross-sectional dimensions of the iron bars were determined by the flexural stiffness of the cross sections of the corresponding components on the prototype structure. The skeleton is designed to simulate the overall flexural stiffness of the bridge tower and is the main stress component of the model. To satisfy the gravity similarity, iron clump weights were fixed on the tower columns of the model. Under wave actions, the surface of the underwater part of the prototype tower would be simulated by the model. Therefore, some polyethylene shells were pasted on the underwater part of the model to satisfy its geometric similarity.

Figure 2.

Test model: (a) photograph of the model in the wave flume and (b) placement of the sensors.

Test equipment and measuring instruments

The experiment was carried out at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China. The wave flume used for the experiment possesses the capability of simulating waves and currents. As shown in Figure 3, it is 60 m in length, 4 m in width, and 2.5 m in height. Flume waves are generated by a piston-type wavemaker at one side of the flume and absorbed by an absorber at the other side of the flume. With a current inlet and outlet, uniform currents can be generated by a circulatory pump system. Joint wave and current then can be simulated by cooperative works of the wavemaker and the pump system. The test model was located at the middle zone of the flume at a distance of 30 m from the wavemaker. The incident wave and current were recorded by a capacitance-type wave gauge and an acoustic Doppler velocimetry (ADV) placed at upstream of the model. Before the model tests were carried out, the flume waves and currents were carefully calibrated to ensure their target values.

Figure 3.

Schematic of the wave flume (unit: m).

The layout of the measuring instruments for recording the dynamic responses of the model are shown in Figure 2. Three two-component accelerometers were installed at three elevations on the model to measure the acceleration responses in the x and y directions. Laser displacement sensors were installed on the traversing carriage above the flume to record the displacements of the tower top. The model was freestanding with its foundation fixed on an assembled force balance. Then, the wave-induced base shear forces in the x and y directions would be measured by this force balance.

Load conditions

Load cases for the experiment were designed according to the sea state. A flume water depth of 28.5 cm, which is corresponding to the designed water depth of 28.5 m shown in Figure 1, was considered during the whole experiment. The hydrological data at the bridge site shows that the wave heights with a cumulative probability of 1%, H_1%, for 25-year and 100-year return periods are 3.0 and 4.7 m, respectively, and the corresponding average wave periods, $\bar{T}$ , are 7.0 and 9.6 s, respectively. In addition, with the resonant period of the structure focused on, the wave period of 7.8 s was also considered for the experiment. According to the geometric scale of 1:100 and the Froude similarity, six regular waves were determined for the experiment. Including the wave heights, H, wave periods, T, and water depth, d, Table 1 shows the model and prototype parameters of these six load cases.

Table 1.

Regular wave conditions.

Load cases	Model			Prototype
Load cases	H (cm)	T (s)	d (cm)	H (m)	T (s)	d (m)
W1	3.0	0.70	28.5	3.0	7.0	28.5
W2	4.7	0.70	28.5	4.7	7.0	28.5
W3	3.0	0.78	28.5	3.0	7.8	28.5
W4	4.7	0.78	28.5	4.7	7.8	28.5
W5	3.0	0.96	28.5	3.0	9.6	28.5
W6	4.7	0.96	28.5	4.7	9.6	28.5

Two random waves, RW1 and RW2, were also considered in the experiment. JONSWAP spectrum is used as the frequency spectrum of the random waves. Table 2 gives the significant wave heights, H_s, and the spectral peak periods, T_p, of these two load cases. RW1 in the table was designed to simulate the once-in-a-century random wave condition, and RW2 was designed to simulate a more extreme condition for testing the bearing capacity and the responses of the tower structure. In addition, to study the effect of forward currents on the dynamic responses of the structure, two corresponding joint wave and current actions, RW1+C1 and RW2+C2, were taken into account. With considerations of the hydrological condition and the capacity of the water pump system of the flume. The velocities, V, of the horizontal averaged currents were designed to be 25 and 35 cm/s for C1 and C2, respectively, as shown in Table 2. In order to conduct comparative analyses on the results, the coexistences of waves and currents were ensured to be the forced wave-current couplings for the test, that is, the target spectrums tested by the wave gauge of the wave-currents were ensured to approximate to the spectrums of the corresponding pure waves. Free coupling and forced coupling are two wave-current interaction modes. For this experiment, the free coupling is too random and is not convenient for the comparisons of the test results due to wave and wave-current actions. So, the forced couplings of wave-current actions were carried out for the study.

Table 2.

Random wave and wave-current conditions.

Load cases	Model				Prototype
Load cases	H_s (cm)	T_p (s)	d_p (cm)	V (cm/s)	H_p (m)	T_p (s)	d_p (m)	V (m/s)
RW1	2.9	0.81	28.5	-	2.9	8.1	28.5	-
RW2	6.1	1.11	28.5	-	6.1	11.1	28.5	-
RW1 + C1	2.9	0.81	28.5	25	2.9	8.1	28.5	2.5
RW2 + C2	6.1	1.11	28.5	35	6.1	11.1	28.5	3.5

Using the wave gauge located in the flume at a distance of 6 m from the test model, surface elevations of all the regular waves, random waves, and wave-currents would be recorded during the experiment. Representative time histories of the surface elevations are shown in Figure 4. The test model was installed at the middle zone of the wave flume at a distance of 25 m from the wave absorber, as shown in Figure 3. The transverse width of the foundation of the test model is 0.52 m, and the width of the flume is 4 m. The length and width of the flume are large enough to avoid the wave reflection effects induced by the flume. The stabilities of the incoming waves were satisfactory within the test time.

Figure 4.

Surface elevation records of RW1 and RW1 + C1 in the experiment.

By rotating the test model with the force balance, variable wave incident angles were simulated in the experiment. A body-fitted coordinate was labeled on the model structure in Figure 2(a). Its x-direction is the longitudinal direction of the structure and the y-direction is the transverse direction of the structure. The angles between the wave incident direction and the x-direction of the model would be adjusted during the tests. With 0° corresponding to a longitudinal incident wave and 90° corresponding to a transverse incident wave, including 0°, 22.5°, 45°, 67.5°, and 90°, waves act on the model structure in five different acting directions were simulated in the experiment.

Experimental results and discussion

Dynamic characteristics of the tower model

Before testing the dynamic responses of the model under the flume wave and wave-current actions, free-decay tests was conducted with and without water to find out the dynamic characteristics of the structure. Natural frequencies and damping ratios of the model structure were obtained from the acceleration and displacement records of the tests. Figure 5 shows a free-decay displacement record of tower top.

Figure 5.

Displacement record of the free-decay test (x-direction, dry condition).

Flexural deformations along the x and y directions are the main mode shapes of the structure, as shown in Figure 6. They are corresponding to the first two natural frequencies of the structure. Because the bending rigidity of the pile-group foundation is very large than the bending rigidity of the upper tower, the upper tower has obvious bending deformations while the pile-group foundation has almost no deformation for the main modes. Because the tower was designed to be a gate-type structure, flexural stiffness of the structure along the y-direction is greater than that along the x-direction. Accordingly, the natural frequency of the structure along the y-direction is greater than that along the x-direction. The test natural frequencies of the model structure and the design frequencies of the prototype structure that calculated by the finite element method are presented in Table 3. Because the flexural stiffness of the upper tower is weaker than the stiffness of the pile-group foundation, the swings of the upper tower were obvious, while the movements of the foundation were tiny in free-oscillation tests. Because the water level is lower than the top of the pile cap, the water does not have an obvious effect on the free oscillation of the upper tower. Therefore, Table 3 shows that the test frequencies with and without water are identical with each other. Meanwhile, by comparison, it is found that the test frequencies are close to the design frequencies for both the x and y directions. This result verifies the rationality of the model design.

Figure 6.

Main mode shapes of the structure: (a) flexural mode along the x-direction and (b) flexural mode along the y-direction.

Table 3.

Natural frequencies of the structure.

Mode shape	Design frequency (Hz)		Test frequency (Hz)
Mode shape	Prototype	Model	Without water	With water
x-direction	0.1241	1.241	1.277	1.268
y-direction	0.2670	2.670	2.830	2.808

Dynamic responses driven by regular waves at 0°

Under the regular wave actions, base shear forces of the foundation and the dynamic responses, including the displacements and accelerations, of the upper tower were measured. The regular wave periods that T = 0.70, 0.78, and 0.96 s are, respectively, lesser than, approximately equal to, and greater than the first-order self-oscillation period of the structure. For different wave periods, Figure 7 shows that the wave-induced base shear forces on the foundation do not have an obvious resonance effect. Because the bottom of the foundation is fixed, horizontal movements of the structure were restricted under the wave actions. The horizontal wave forces did not motivate a dynamic effect on the base shear forces, so the base shear forces directly reflect the wave forces on the foundation without resonance effect. Nevertheless, because the upper tower is a slender structure and its flexural stiffness is weak compared with that of the foundation, a slight flexural vibration of the foundation induced by a wave moment may cause a strong swing on the upper tower, as shown in Figure 8. When the wave period is close to the natural period of the structure, an obvious resonance was excited by the wave load.

Figure 7.

Time histories of the longitudinal base shear forces due to 0° incident waves (H = 3 cm): (a) T = 0.70 s, (b) T = 0.78 s, and (c) T = 0.96 s.

Figure 8.

Time histories of the longitudinal top displacements for different wave periods (0°, H = 3 cm).

Effects of the wave action angle

Amplitudes of the base shear forces and the top displacements were extracted to find out the response properties of the structure for different wave directions. For six load cases listed in Table 1, W1–W6, Figure 9 shows the variations of the longitudinal base shear forces, F_x, and the transverse base shear forces, F_y, along with the wave directions, 0°, 22.5°, 45°, 67.5°, and 90°. With the flume water depth being 28.5 cm, the water level is higher than the bottom of the pile cap; so, together with the pile-group, the pile cap was impacted by the waves during the tests, increasing the complexity of the wave forces on the foundation.

Figure 9.

Base shear forces versus the wave action angles.

For both F_x and F_y, by comparing data of W1 with data of W2, data of W3 with data of W4, and data of W5 with data of W6, Figure 9 shows that, when the wave periods are same, it is reasonable that the shear forces induced by waves with H = 4.7 cm are larger than that induced by waves with H = 3.0 cm. The wave height is higher, the wave force is greater.

It is shown in Figure 1 that the pile-group is composed of two paratactic groups of that each consists of 8 piles and that the pile cap is round-ended with its transverse width of 52.1 m being larger than its longitudinal width of 21.8 m. The longitudinal blocking area of the foundation is larger than its transverse blocking area. In this, for a same wave, such as for W6, the longitudinal base shear forces, F_x, of 12.93 N by the wave at 0° is greater than the transverse base shear force, F_y, of 6.65 N by the wave at 90°. Meanwhile, when the waves act at 0°, the longitudinal base shear forces, F_x, appear to be largest for all load cases. With increase in the action angles, the projected area of the longitudinal blocking area of the foundation on wave propagation direction decreases and then F_x decreases.

It is also found in Figure 9(b) that the largest F_y does not appear at 90° for all load cases. For W5 and W6, the largest F_y appears at 90°, but for W1–W4, the largest F_y appears at 45°. Being different from the wave force on a solitary pile, the wave force on a pile-group would be affected by the pile-group effectiveness. Because there are phase differences between the piles in the group, the largest values of the wave forces on each pile do not appear at the same time. Therefore, the total wave force on a pile-group is not a simple multiplication of wave force on a solitary pile with the pile number in the group. However, based on the pile layout of the group, because of the diffraction effect, the front piles would generate a protection effect to the back piles, resulting that the wave forces on the front piles are larger than that on the back piles. Both the phase difference effect and the protection effect will decrease the total wave forces on the pile-group. By computing the wave length, L, by the wave periods and the water depth, it is known that the wave length of W1–W6 is approximate to be 75, 75, 91, 91, 127, and 127 m, respectively. With the transverse width of the foundation found to be D_y = 52.1 m, and the longitudinal width of the foundation found to be D_x = 21.8 m, D_y/L for W1–W6 is 0.69, 0.69, 0.57, 0.57, 0.41, and 0.41, respectively, and D_x/L for W1–W6 is 0.29, 0.29, 0.24, 0.24, 0.17, and 0.17, respectively. It is found that when the waves of W1–W4 acted on the foundation at 90°, D/L ≥ 0.57 (D is the feature width of the structure along wave propagation direction. When waves act on the foundation at 90°, D = D_y). In this condition, the incident waves are defined to be short waves for this study. When the foundation was acted by a short wave, the phase difference effect and the protection effect on the pile-group are more evident. For 90° acting direction, because the long axis of the foundation is along the wave direction, compared with the cases for other acting directions, the phase differences between the piles and the protection of the front-side piles to the backside pile are more evident. Therefore, for W1–W4, the base shear forces, F_y, at 90° are lesser than the that at the oblique angles of 45° and 67.5°. But for the long waves of W5–W6, as the pile-group effectiveness is relatively weaker, the forces, F_y, reasonably increase with the rise of the angle and reach its greatest values at 90°. In addition, when the waves act on the pile cap, the rectangular, round-ended structure has similar phase differences for its front part and backside part.

On the whole, the experimental results show that wave direction and structure form of such pile-group foundation will significantly determine the load forces on the structure. For reducing the wave forces on a sea-crossing bridge, the wave height, wave period, and wave direction at the bridge site should be carefully considered, and the bridge foundation should be carefully designed. The experimental results can provide some references for the structural design of pile-group foundation.

It is known from above that the transverse flexural stiffness of the tower is greater than its longitudinal flexural stiffness, so its transverse natural vibration period is away from the wave periods of the given wave loads. Therefore, under a wave action, it was observed that the longitudinal flexural movement of the tower was obvious, while the transverse movement of the tower was very tiny and negligible. So, the study focuses on the longitudinal flexural movement of the upper tower. Figure 10 shows the longitudinal top displacements of the tower for different wave action directions. At 0°, the movement of the structure is largest and decreases with the rise of the wave acting angles. Because the dynamic behaviors of the tower are mainly induced by the longitudinal wave forces on the foundation, the variation trend of the displacements is corresponding to the variation trend of the longitudinal base shear forces for different wave action angles.

Figure 10.

Longitudinal top displacements of the model versus the wave action directions.

Effects of the current

It is found from the experiment that the base shear forces driven by a joint wave and current action are different from that driven by a pure wave. Figure 11 shows that, when the model was acted by the random wave, RW1, the mean values of the irregular base shear forces on the foundation are near zero. But for RW1+C1, affected by a forward current, the mean value of the shear forces is 3.2 N, which is greater than zero. However, the mean square error of the forces by RW1+C1 is 3.60 N, which is close to the mean square error of 3.38 N by RW1, reflecting that the existence of the current does not significantly change the fluctuation intensity of the induced forces. As a result, it can be known from Figure 11 that, affected by the forward current, the forward wave forces on the foundation were increased and greater than that induced by the pure wave, and the backward wave forces were decreased and lesser than that induced by the pure wave. This phenomenon is similar to a natural wind, which is seen as the superposition of a mean wind and a fluctuating wind. Joint action of a wave and a forward current can be seen to include a mean component and a fluctuating component. The mean component induces a positive mean force on the foundation, and the fluctuating component induces a fluctuating force around the mean force. On the whole, affected by a forward current, wave-current forces are greater than the pure wave forces. Current effect should be considered for the strength design of such pile-group foundation.

Figure 11.

Time histories of the longitudinal base shear forces driven by the random wave, RW1, and the wave-current, RW1 + C1, at 0°.

The corresponding displacement vibrations of the tower top induced by RW1 and RW1+C1 are shown in Figure 12. By comparison, it is found that, being different from the base shear forces, the mean values of the displacements both for RW1 and RW1+C1 are close to each other and near zero. Meanwhile, the mean square errors of the displacements by RW1 and RW1+C1 are 0.360 and 0.378 mm, respectively, meaning that the vibration intensity of the upper tower induced by the wave-current are close to that by the pure wave. It can be known that when a static force acts on the foundation, because of the huge stiffness of the foundation, the induced static displacement on the upper tower is very tiny and negligible. The flexural swings of the upper tower are dynamic behaviors and are mainly caused by the fluctuating forces acting on the foundation. Under a wave-current, the mean component of the wave-current force does not induce a significant displacement on the tower, and the fluctuating component of the wave-current force induces a main flexural swing on the tower. Therefore, as the fluctuating component of the acting force driven by RW1+C1 is similar to the fluctuation force driven by RW1, the vibration responses of the tower induced by them are similar. As shown in Figure 13, it is also found that the power spectral densities (PSD) of the displacement driven by RW1+C1 are similar to that by RW1, with their peak frequencies being near the natural frequency of the structure.

Figure 12.

Time histories of the top displacements driven by RW1 and RW1 + C1 at 0°.

Figure 13.

Power spectral densities of the displacements driven by RW1 and RW1 + C1 at 0°.

Similar characteristics were observed for RW2 and RW2+C2 and for the other action angles; because of the length of this article, not all measurements are presented.

Conclusion

This article presents an experiment result of the dynamic responses of a typical bridge tower due to ocean waves and wave-currents. The main conclusions are as follows.

From the test data, it is found that the wave-induced base shear forces directly reflect the wave acting forces on the foundation without obvious dynamic effects. Nonetheless, because of the weak longitudinal flexural stiffness of the tower, the acting waves motivate swing movements on the upper tower. Especially, for a wave that the wave period is close to the natural period of the structure, an obvious resonance was observed. In engineering practice, for structural safety, the design frequencies of such bridge tower should be kept off the wave frequencies.

The wave-induced longitudinal and transverse base shear forces were systematically analyzed. For different wave acting directions, the longitudinal incident waves induced the largest longitudinal wave forces on the foundation. With a longitudinal incident wave gradually changed to a transverse incident wave, the longitudinal wave force decreases. For the long waves, the largest transverse base shear forces appear in the transverse direction. Nonetheless, because of the significant pile-group effectiveness, the transverse wave forces induced by the oblique incident waves around 45° are larger than that by the transverse waves. This conclusion reveals that, when a pile-group is used as the foundation of a sea bridge, in the design of the arrangement of the piles, the incident directions and the periods of ocean waves should be taken into consideration.

In the experiment, as the upper tower designed to be a gate-type structure, the main dynamic behaviors of the tower induced by the waves were longitudinal swings. The longitudinal incident waves induced the largest wave forces on the foundation and then induced largest dynamic movements on the tower. This reminds that the veers of such plane bridge towers should be under the consideration of the common wave directions at bridge site.

By comparing the dynamic responses driven by combined wave and current actions with that driven by the corresponding pure waves, it is shown that, affected by a forward current, the forward wave forces on the foundation are increased and the backward wave forces are decreased. Nonetheless, the motion responses of the upper tower are not significantly affected by the currents. With a bridge tower being commonly rigid on its foundation and being relatively flexible on its upper structure, this experimental phenomenon is reasonable for wave-current actions.

When a pile-group or caisson foundation is applied for a sea-crossing bridge, for reducing the wave effect on the structure, considering the wave height and usual wave direction at the bridge sites, the direction of the bridge and the structural forms of the foundation and the bridge tower should be carefully designed. The experimental results can provide some significant references for the structure form selection and size design of similar engineering.

Footnotes

Acknowledgements

The authors acknowledge the State Key Laboratory of Coastal and Offshore Engineering of Dalian University of Technology.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Technology Project of the Ministry of Transport of China (Grant Number 2011318494150).

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