experimental and numerical study on pounding … · web viewtherefore, performing a hydrodynamic...
TRANSCRIPT
Analysis of hydrodynamic forces acting on submerged decks of coastal bridges under oblique wave action based on potential flow theory
Qinghe Fanga, Rongcan Honga, Anxin Guoa *, Peter K. Stansbyb and Hui Lia
a. Key Lab of Structures Dynamic Behaviour and Control of the Ministry of Education, School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, China
b. School of Mechanical, Aerospace and Civil Engineering, University of Manchester, Manchester M13 9PL, UK
ABSTRACT
The structures of existing coastal bridges without appropriate clearance between the still water level and low chord of the bridge deck, are vulnerable to wave-induced damage due to the strong wave force acting on the bridge deck during a hurricane or typhoon. This paper presents an analytical solution for hydrodynamic wave forces acting on girder-type bridge decks under hurricane-generated oblique water waves based on linear potential theory. First, some necessary assumptions are made, and the boundary value problem is defined. Then, the mathematical formulation and analytical solution of this wave–structure interaction problem are derived using the method of separation of variables. After determining the unknown coefficients of the velocity potentials with the matching eigenfunction expansion method, the wave forces acting on a submerged deck are obtained from the velocity field by applying the Bernoulli theorem. Finally, the analytical solution of the wave force is validated by the data from two different scaled hydrodynamic experiments. Employing the proposed method, the wave forces acting on a bridge deck are investigated considering the effect of the wave propagation direction, wave properties, and structural configuration. The parametrical analysis shows the potential for minimizing the horizontal wave force by the optimization of the structural configuration.
Keywords: coastal bridge, superstructure, hurricane, oblique wave, wave forces, analytical solution.
1 Introduction
Hurricane-generated large waves together with storm surges have significantly damaged, and even destroyed, numerous bridges in the coastal region of the Gulf of Mexico during Hurricane Ivan in 2004 and Hurricane Katrina in 2005. Those bridges had a common characteristic of a small clearance between the still water level and lower chord of the bridge girders. Under the action of large waves, bridge superstructures are vulnerable to the external wave loads if a sufficient vertical connection between the superstructure and substructure is lacking. The damage caused to bridges leads to significant economic losses and delays the
**Corresponding author. Tel.: +86-451-86283190; fax: +86 451 86283190E-mail address: [email protected] (A.X. Guo)
1/23
1
2
3
4567
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
12
rescue and reconstruction of the disaster-prone areas. After the catastrophic disaster of the hurricanes in 2004 and 2005, some researchers
attempted to explore the mechanism of a wave acting on coastal bridges. Owing to the complex configuration of a bridge superstructure, the wave–structure interaction is a complex physical process involving multi-phase flow interaction, air trapping, wave slamming, and overtopping. Therefore, performing a hydrodynamic experiment is a direct and effective approach to investigate this problem. Cuomo et al. (2009) conducted a basin test on a 1:10-scaled bridge model of the US 90 to I-10 ramp bridge over the Mobile bay, which was destroyed in Hurricane Katrina, to measure the wave pressure on its structural members. Marin (2010) performed a hydrodynamic test on a 1:8-scaled bridge model of I-10 bridges of Louisiana and Florida with different configurations to investigate the wave loads acting on the bridge deck. Based on the test results, the American Association of State Highway and Transportation Officials (AASHTO) published a preliminary guideline for the retrofitting of damaged bridges, in which three cases are defined to estimate the wave forces acting on coastal bridges (AASHTO, 2008). Bradner et al. (2011) designed a 1:5-scaled reinforced concrete model of the damaged I-10 bridge over the Escambia bay to test the wave forces acting on the bridge deck under regular and random waves with different clearances. Guo et al. (2015b) also tested a 1:10-scaled bridge model containing a deck and five girders in the wave flume under different monochromatic waves.
In addition to a hydrodynamic experiment, numerical simulation is another effective and practical alternative to investigate the wave action on coastal structures. Huang and Xiao (2009) simulated the wave action process on a bridge deck using the incompressible Reynolds-averaged Navier–Stokes equation and k– equations to investigate the wave loads exerted on bridge structures. With the same model, Xiao et al. (2010) suggested that the most dangerous situation was when the bridge deck was just fully submerged and the maximum uplift wave force acted on the structures. Using commercial software Flow-3D, Jin and Meng (2011) investigated the wave loads acting on a bridge superstructure by considering the effect of green water. In that study, the simulation results were analysed and compared with the AASHTO guideline. A simple method for estimating the wave loads was developed based on the simulation results. In addition, some researchers also conducted numerical simulations to investigate the effect of structural geometry, air trapping, and adjacent bridge lanes on the wave force on bridge superstructures applied by cnoidal waves (Seiffert et al., 2015) and solitary waves (Hayatdavoodi et al., 2014; Xu and Cai, 2015a; Xu and Cai, 2015b; Xu et al., 2015).
Till date, researches on the wave action on bridge superstructures are mainly focused on the specific case of waves propagating perpendicular to the bridge deck. Studies on the oblique wave action on a bridge superstructure are rarely reported. However, the hindcasting analysis of Hurricane Katrina performed by Smith (2007) and Hu and Chen (2011) indicated that most hurricane waves that induced the damage of coastal bridges attacked the bridges at certain incident angles. Denson (1980) has been the only investigator who conducted a preliminary experiment on oblique wave loads acting on highway bridge decks using a 1:24-
2/23
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
scaled model after Hurricane Camille in 1969. In his study, the wave forces acting on bridge deck models were measured under an oblique monochromatic wave with a period of 3 s. Owing to the scarcity of investigations on this topic, the AASHTO guideline (AASHTO, 2008) suggested researchers to conduct more work on the oblique wave forces acting on coastal bridges.
Hydrodynamic experiment and numerical simulation are both effective approach to investigate wave forces on coastal bridges. However, it is cost- and time-consuming to do wave flume tests. The setup of a robust numerical model needs experienced technique that is not easy for bridge engineers. Considering the cost and efficiency, an analytical method is comparatively more efficient to investigate the wave–structure interaction problem under certain conditions. Using the Green–Naghdi theory, Hayatdavoodi and Ertekin (2015a, b)calculated the wave forces acting on submerged horizontal plate which is similar to the bridge deck. Besides, the potential flow theory is a classic method to analyse wave properties and wave–structure interaction under gravity waves. It has been utilized to analyse oblique waves interacting with structures such as rigid vertical thin barriers (Losada et al., 1992), submerged rectangle breakwaters (Abul-Azm, 1994; Zheng et al., 2007), moored floating membranes (Karmakar and Soares, 2012), and perforated walls (Li et al., 2003; Liu et al., 2016). When a bridge deck is submerged in water owing to a storm surge during a hurricane event, it is possible to analyse the wave–structure interaction. Employing the potential flow method, Guo et al. (2015a) studied the forces acting on submerged bridges generated by normal waves and developed a simple estimation method of wave forces. The hydrodynamic performance of submerged breakwater, has similar geometric characteristics with the bridge deck, under oblique waves were investigated using the potential method (Abul-Azm and Gesraha, 2000; Gesraha, 2004, 2006).
This paper presents a linear potential flow based analytical solution of the wave forces acting on bridge decks under an oblique wave action. The main contribution of this paper can be concluded as: (1) an analytical model based on the potential flow theory is developed for estimating the oblique wave force acting on the submerged bridge decks, which is urgent to be investigated in engineering practice; (2) the effect of the wave propagation direction, wave properties, and structural configuration are investigated by the developed analytical model. This paper is organized as following: First, the boundary value problem is defined based on certain assumptions. Then, the mathematical formula is derived by the method of eigenfunction expansion matching to obtain the analytical solution. Subsequently, the analytical model is validated by the results of a 1:5-scaled wave test and 1:10-scaled oblique wave test. Finally, the proposed method is used to investigate the effects of wave incident direction and bridge geometry to gain a better understanding of the wave action mechanism of bridge superstructures under oblique waves.
2 Mathematical formulation and analytical solution
2.1 Problem definition
3/23
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
In existing coastal bridges, the typical configuration of the superstructure segment is composed of a slab and several inverted T girders. Those girders are always connected with the each other by a diaphragm to enhance the integral rigidity. In this work, a bridge superstructure sustaining the attack of a train of incident oblique waves in a submerged state is simplified as a boundary value problem, as shown in Fig.1. To make the problem manageable, the girders in the prototypes are simplified as rectangular structures in this schematic by ignoring the local geometrical shapes of the girders. Furthermore, the superstructures of the coastal bridges are assumed to be rigidly fixed on the cap beam. The wave is assumed to be an inviscid, incompressible and irrotational fluid, and the superstructure is assumed to be a rigid body without deformation under the wave action. In the proposed method, the viscosity effect is also neglected. Furthermore, it should be noted that the proposed method is also suitable for the specific case when the bridge superstructure is fully submerged in the sea water. The aerial and semi-submerged status of the bridge superstructure cannot be calculated by this method due to the effect of the air.
The coordinate system used for the analysis is also depicted in Fig.1. In this cartesian coordinate system, the x-y plane is defined to be located at the still water level (SWL) with the x-axis coinciding with the transverse direction of the bridge. The coastal bridge is assumed to be infinitely long along the y-direction. The monochromatic wave train is assumed to
obliquely attack the bridge with incident angle ( ).
Fig.1. Typical schematic of the wave–bridge system. (a) Top view and (b) Cross section.
As shown in Fig.1(b), the entire fluid domain of the bridge superstructure is divided into
subdomains. Based on the assumption of an inviscid, incompressible, and irrotational fluid, velocity potential can be written as
11\* MERGEFORMAT ()
where denotes the real part of the complex expression, , is the wave number in the y-axis direction, is the angular frequency of the incident wave, t is the time, and k is the wavenumber obtained from the following dispersion equation:
4/23
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
22\* MERGEFORMAT ()where g is the gravity acceleration.
Within the entire fluid domain, the spatial velocity potentials should satisfy the three-dimensional Laplace equation, i.e.
33\* MERGEFORMAT ()Owing to the periodic variation of the incident and scattered potentials in the y-axis
direction, the governing equation can be expressed by the Helmholtz equation after substituting Eq.1 into Eq.3.
44\* MERGEFORMAT ()Furthermore, the velocity potentials of the different subdomains should satisfy the
corresponding linearized free surface, seabed, structural surface and far field boundary conditions, respectively.
for or , 55\*MERGEFORMAT ()
for 66\* MERGEFORMAT ()
for 77\*MERGEFORMAT ()
for 88\*MERGEFORMAT ()
99\* MERGEFORMAT ()
2.2 Analytical solution
The governing equation in Eq.4 can be solved by using the separation variable method
under the boundary conditions. Velocity potential of subdomain (j=1, 2, …, J) is determined by the following scheme:
1010\* MERGEFORMAT ()
5/23
143
144
145
146
147
148
149
150
151
152
153
154
155156
157
158
159
160
161
162
163
164
165
166
167
168
1111\* MERGEFORMAT ()
1212\*MERGEFORMAT ()
where is the wave amplitude, ( j=1, 2, …, J) and ( j=2, 3, …, J-1) are unknown
coefficients to be determined, and are the eigenvalues of the open and covered
subdomains, respectively, and is the eigenfunction of the corresponding subdomain. The eigenvalues and eigenfunction in the open subdomains can be expressed as follows:
1313\* MERGEFORMAT ()
1414\*MERGEFORMAT ()
For the covered subdomain of with an upper water elevation of , the corresponding eigenvalue and eigenfunction are given by
1515\*MERGEFORMAT ()
1616\*MERGEFORMAT ()
where is the covered water depth and is the positive real root of the equation,
1717\* MERGEFORMAT ()
Furthermore, eigenfunctions ( )jnZ z are orthonormal to each other in the corresponding
subdomain, i.e.
1818\*
6/23
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
MERGEFORMAT ()
It is easily found that the eigenvalues and eigenfunction of subdomain
have the same expressions and orthogonality with ( 3,5, )j j by replacing under-keel
water depth with in Eqs. 15 and 16 as well as substituting inundation depth with in Eq. 18.
At the interfaces of two adjacent subdomains, velocity potential and horizontal velocity of the fluid satisfy the following continuity conditions:
1919\* MERGEFORMAT ()
2020\* MERGEFORMAT ()According to the continuity conditions, two sets of linear equations of the unknown
coefficients can be obtained at each interface by substituting the velocity potential into the continuity equations defined in Eqs. 19 and 20. It should be noted that the continuity along the y-axis direction is neglected owing to the assumption of infinite length for the bridge. Based on the continuity conditions at the interface, the unknown coefficient of adjacent subdomains
can be determined. For brevity, only the matching processes at the interface of and
are presented in this study. For the other subdomains, the unknown coefficients can also be easily determined with the same procedure.
By substituting Eqs. 10 and 11 into Eq. 19, the continuity of the velocity potential at
interface between subdomains and can be expressed as
. 2121\*MERGEFORMAT ()
Multiplying eigenfunction of subdomain 2 on both sides of Eq. 21, and
integrating the equation from to , a set of linear equations can be obtained by
truncating at
2222\* MERGEFORMAT()
where , and are the N-dimensional vectors of the unknown coefficients in the
velocity potential; {G} is an N-dimensional vector; and , , and are dimensional matrices as presented in the appendix.
In addition, the continuity condition of the horizontal velocity at the interface changes to the following form after substituting Eqs .10 and 11 into Eq.20.
7/23
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
2323\*MERGEFORMAT ()
Considering the orthogonality of the eigenfunctions, the above equation can also be
expressed in a matrix form by multiplying vertical eigenfunction on both sides of the
equation, integrating from to , and truncating at .
2424\* MERGEFORMAT()
where is also an N-dimensional vector, is the N N dimensional identity matrix, and
and are N N dimensional matrices presented in the appendix:
A similar matching procedure can be conducted at interface by substituting Eq. 11 with j = 2 and 3 into Eqs. 19 and 20, respectively, as given below.
2525\* MERGEFORMAT ()
2626\
* MERGEFORMAT ()The following group of linear equations can also be obtained by applying the orthogonality of
the eigenfunctions of subdomains and , respectively:
2727\*MERGEFORMAT ()
2828\*MERGEFORMAT ()
where , , , , , , , and are all N N dimensional matrices as defined in the appendix:
A similar operation can be performed at other interfaces following the same matching
procedure at interfaces and . Finally, the unknown coefficients of the velocity
potential in the entire fluid domain can be calculated by solving all the groups of linear equations obtained from all the interfaces between the adjacent subdomains.
2.3 Reflection, transmission coefficients, and hydrodynamic forces
The reflection and transmission coefficients of the bridge superstructure can be
8/23
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
calculated from the coefficients of the velocity potential in the onshore and offshore domains, respectively, as follows:
and 2929\*MERGEFORMAT ()
such that
3030\* MERGEFORMAT ()to satisfy the conservation of the wave energy in the entire fluid domain.
The wave forces acting on the bridge superstructure can be calculated by integrating the pressure over the wet surface. Applying the linearized Bernoulli equation, the wave forces are expressed as
3131\*MERGEFORMAT ()
3232\* MERGEFORMAT ()
3333\* MERGEFORMAT ()
where and are the horizontal and vertical wave forces acting on the bridge
superstructure, respectively, is the overturning moment about the ending support exerted by the wave, and is the density of the wave water.
3 Validation
3.1 Normal wave test of a 1:5-scaled bridge model
Test data from two hydrodynamic experiments performed for bridge decks were used to validate the proposed model. The first experiment was conducted by Bradner et al. (2011) for investigating the mechanism of wave action on coastal bridges after the catastrophic accidents in 2004 and 2005. Fig.2 shows the cross-section of the bridge, at a 1:5 geometrical scale, employed in the wave flume test. For more details on the experiment refer to Bradner et al.
(2011). In the experiment, all the test trials were conducted at wave angle = , which implies that the waves impact the deck model perpendicularly. The test results from the submerged case with a still water depth of 2.17 m and wave period of 2.0–4.5 s were used to validate the present analytical model. To be consistent with the assumptions of the present
9/23
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
analytical model, all I-shape girders of the test model are simplified as rectangular cross section. Similar simplification is commonly implemented in other research work about wave forces acting on coastal bridges (Cai et al., 2018; Hayatdavoodi et al., 2014; Huang and Xiao, 2009; Jin and Meng, 2011; Wei and Dalrymple, 2016). With this simplification, the projected vertical or horizontal area of whole structure are not changed, while the local boundaries and the flow field of the chamber surrounded by the girders and deck would slightly influenced. However, according to the research results published in the open literature, the effects of such simplification would not significant. Detailed analysis on the effects caused from the simplification will be investigated in the future.
Fig.2. Cross-section of the I-10 bridge (a) 1:5-scaled model, (b) simplified cross-section used in present model.
The convergence of present analytical model was first examined under the evanescent modes in the velocity potentials affecting the accuracy and convergence of the solution. The process of obtaining the analytical solution can be implemented by truncating the infinite series in the velocity potentials defined in Eqs. (8–10) at n = N. The parameters used were b = 0.97 m, w = 0.09 m, s = 0.28 m, d1 = 0.05 m, d2 = 0.28 m, d = 2.17 m, girder number of 6 (as shown in Fig.2), and wave angle = 0o that was actually replaced by 0.1o during the implementation because of the divergence of the analytical model at= 0o.
Fig.3 shows the dimensionless wave force results of the present solution with different truncation orders of the evanescent modes. It is obvious that when truncating number N exceeds 8, 3, and 3 for the horizontal and vertical wave forces and overturning moment, respectively, the contribution of the higher evanescent modes to the wave forces is small. Therefore, truncating number N is selected as 10 in the following analysis considering the balance between the accuracy and computational efficiency.
10/23
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
Fig.3. Effect of the truncation order on the wave forces (a) Horizontal wave force, (b) Vertical wave force, (c) Overturning moment.
Fig.4 compares the wave forces obtained from the present analytical solution and hydrodynamic tests. In Fig.4, the experimental results of the wave forces are provided in a statistical form, where the circles represent the mean value and short lines depict the standard deviation of the maximum horizontal or vertical force. The buoyancy is excluded from the test results for the convenience of comparison. It should be noted that several wave trials were conducted at different wave heights for a single wave period. Consequently, there is a suit of test data for each kb. It can be observed from Fig.4 that the normalized horizontal wave force
significantly increases with the increase in wavenumber when . After reaching the
peak value at , the normalized horizontal wave force is slightly decreased with further
increase in . Comparatively, the vertical force exhibits a decreasing trend with the increase
in , which contains an obvious reduction when wavenumber is smaller than 1.0 and a
slow decline when wavenumber . From the comparison with the experiment results, it can be observed that the theoretical solution can reasonably predict the maximum wave forces acting on costal bridges in some extent, especially at the cases with smaller kb. When kb becomes larger, some errors exists between the test results and the analytical solution. The difference maybe induced by the experimental error and the nonlinear effects such as wave impact, breaking and aeration during the wave acting on the offshore side of the bridge superstructure, especially for the horizontal wave force.
Fig.4. Comparison of the results of the present model and 1:5-scaled model test. (a) Horizontal force, (b) Vertical force.
11/23
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
3.2 Oblique wave test of 1:10-scaled bridge model
The authors conducted a 1:10-scaled model test of a coastal bridge in a wave flume, as shown in Fig.5(a), to validate the proposed analytical model. The system contained three spans and the forces acting on the middle one was measured. The superstructure consisted of five scaled-AASHTO type III girders and a deck, and the substructure included two piers and a single bent cap. A guard rail with the height of 0.15 m, which is higher than the height of the maximum wave crest of 0.125 m, was specifically designed to stop the wave water from overtopping the deck, ensuring the experimental setup was consistent with the analytical model. The overtopping of the water wave would reduce the vertical wave force. The analytical solution neglecting the overtopping of the water wave is a conservative result of the wave force estimation from the engineering point of view. The wave forces were measured by load cells installed between the girders and bent caps, as shown in Fig.5(b). The forces acting on the guard rail were transferred to the neighbouring spans. The test plan covered four different wave periods, five different wave heights, and seven wave angles changing from 0o
to 90o with an increment of 15o. During the tests, the incident regular waves were well calibrated in the empty wave flume before installing the specimen.
Fig.5. Experiment setup (a) Photo of the wave flume test; (b) Cross-section of the structure.
Fig.6 displays the comparison between the experimental data and calculated results of the proposed model. The scatters represent the dimensionless wave forces measured under different waves, in which each hollow dot stands for the wave force in a single cycle under the given wave period and wave angle. The calculated results are plotted as solid lines in the figure. Fig.6 (a) and (b) show the comparison between the experimental and calculated horizontal and vertical forces for wave angle = 0 o, respectively. It can be seen from Fig.6 (a) that the normalized horizontal wave forces increases quickly then decreases slowly about
the wave number kb after peaking at about . The vertical wave force shows a continuous declining with the wave number kb. It can be summarised that the proposed model gives an acceptable estimation to the horizontal force but a relative conservative prediction for vertical wave at kb of approximately 0.4. This phenomenon can be attributed to the size of the experimental model relative to the wave length. For the wave number kb about 0.4, the wave length is about 5.5 m that means the B/ = 0.182 smaller than 0.2 for submerged bridge
12/23
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
deck. The wake effects and drag forces cannot be considered in the present model which leads the difference between calculated and measured results.
Results for wave angle = 15o are shown in Fig.6 (c) and (d). It can be observed that the horizontal wave force shares similar tendency about the wave number kb, but the value of the peak is slightly decrease and moves forward to large kb comparing with = 0o. The horizontal wave force for kb = 2.8 exhibits some difference with the calculated result. This can be attributed to the shading effect of neighbouring spans in the test which cannot be nicely considered in present analytical model. Same with = 0o, the vertical wave force peaks at the minimum wave number kb then goes down with the increase of the wave number. The calculated vertical force is still conservative compared with the test data for kb about 0.4.
With the further increase of wave angle =30o to 90o, the calculated horizontal and vertical wave forces shares similar tendency versus the wave number with case = 0 and 15o, as shown in Fig.6 (e)-(n). The difference between the calculated and measured vertical force is reduced. It can be explained that the submerged bridge deck becomes a large subject relative to the wave length with the change of wave angle. It is also noted that there is a difference between the analytical solution and test results in some trails due to so many effects, such as model difference, viscous effect, nonlinear effect, test setting and linear theory assumption. This difference would be further investigated in the future research.
13/23
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
14/23
381
382
383
384
Fig.6. Comparison of the wave forces obtained from the test results and present analytical solution for a 1:10-scaled bridge model at different wave angles.
4 Results and discussion
The wave forces acting on the bridge superstructure are affected by several parameters including the properties of the incident wave and structural configuration. In this section, based on the proposed analytical method, a parametric analysis of the bridge desk shown in Fig. 2 is presented to investigate the sensitivity of the wave and geometrical parameters on this wave–structure system.
4.1 Effects of the wave angle
The effect of wave direction on the wave forces acting on the bridge superstructure in a submerged state is depicted in Fig.7. It can be observed that the wave forces reach maximum
values when the waves perpendicularly attack the bridge segment at for each wave period. The horizontal wave forces at all the wave directions exhibit the same trend with the
increase in . As shown in the figure, the horizontal force increases rapidly to reach its peak
value for wave incident angle , but escalates gradually with increasing kb when the
wave angle is larger than . After reaching the peak value, the horizontal wave forces
decrease and finally converge to the same value when , for all wave angles. Similarly,
the vertical wave force also achieves a maximum value at the perpendicular case when . In the vertical direction, the wave force exhibits a tendency to decrease continuously with the
increase in kb , for all wave angles. With the wave angle changing from to , the
vertical wave force reduces drastically, and finally converges to 0 with increasing kb . The calculated results suggest that the bridge sustains a larger vertical force under longer period oblique waves, whereas a maximum horizontal wave force is experienced at different kb values ranging from 1.2 to 5.0 with the wave angle varying changing from 0o to 85o.
The overturning moment exhibits a trend similar to the vertical wave force, versus wave
number kb , as shown in Fig.7(c). It can be concluded that the overturning moment is primarily induced by the vertical wave force because the arm of the vertical force is much
15/23
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
larger than that of the horizontal wave force. In the following analysis, the overturning moment has not been discussed specifically owing to its close similarity with the vertical wave force.
Fig.7. Effect of the wave incident direction on dimensionless wave forces: (a) Horizontal (b) Vertical (c) Overturning moment
The relationships between the wave incident direction and wave forces at each specific kb can be observed more clearly in Fig.8. It can be seen that the horizontal wave force decreases with the increase in the wave incident angle for a given kb and converges to 0 when the wave angle reaches approximately 90o. This is owing to the fact that the wavenumber (
) depicted on the x-axis reaches 0 at the wave angle of 90o. The vertical wave force at a different wave number shows a distinct trend with the variation in the wave direction. For the condition kb < 0.4, first the vertical wave force remains stable, particularly for when the dimensionless wave number is small (kb = 0.1). Following the slow variation stage, the dimensionless vertical wave force drastically reduces to approximately 0 with further increase
in . Moreover, for kb > 0.6, the dimensionless vertical wave force shows a continuous
decrease, becoming approximately 0, as the incident angle changes from o0 to
o90 . The reason for the convergence of the dimensionless vertical wave forces to zero is the decrease in the integral length, expressed in Eq.32, to 0 when the wave propagates parallelly with the bridge spans.
16/23
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
Fig.8. Dimensionless wave forces along wave directions at different kb values: (a) Horizontal (b) Vertical
4.2 Effects of the structural configuration
This sub-section discusses the effect of the structural configuration on the wave forces. It should be noted that some parameters of the structural configuration, which are irrationally applied in engineering practices, are considered here only to demonstrate clearly the relationships between the influencing factors and wave forces. It can be concluded from the former analysis that the bridge sustains maximum forces when the waves propagate normally, and so, only wave direction = 0 o is considered in the parametrical analysis of the structural configuration.
Fig.9 displays the effect of submergence depth d2 (as shown in Fig.1) on the dimensionless wave forces for the wave–structure interaction system shown in Fig.2, with changing d2, invariable d1/d2= 0.18, and wave direction = 0 o. It can be noticed from Fig.9 that bridges with a low d2/d experience a smaller horizontal wave force for small kb, but a large wave force after the horizontal wave force reaches its peak with the increase in kb. Dimensionless submergence depth d2/d only slightly affects the vertical wave force when wave number kb < 0.5. Furthermore, the bridge with a lower d2/d endures a larger vertical wave force when wave number kb > 0.5. It is recognized that an increase in d2/d implies a relatively large geometrical size of the wet surface sustaining the wave pressure in the horizontal direction. A stronger reflection will result in a larger horizontal wave force and smaller vertical wave force.
The effect of the relative thickness of the bridge deck is shown in Fig.10. In this analysis, deck thickness d1/d2 (as shown in Fig.1) varies from 0.0 to 1.0 and invariable dimensionless submergence depth d2/d is set at 0.13. The other parameters are the same as that in Fig.2. Relative thickness d1/d2 = 0 implies that the deck thickness is zero that is implemented by setting d1 as a very small value in the calculation process. The bridge superstructure converts into a box when d1/d2 = 1.0. From Fig.10, it can be found that the horizontal wave forces at different deck thickness d1/d2 exhibit the same trend versus wavenumber kb. The deck thickness affects the peak value of the dimensionless horizontal wave force significantly. The peak value of the horizontal wave force increases with the increase in deck thickness d1/d2 and
17/23
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
the occurrence of oscillations within the range of [0.0, 0.7]. It is also found that the vertical wave forces are only slightly affected by d1/d2. Because relative deck thickness d1/d2 does not affect the vertical wave force, it shows a possibility to optimize the deck thickness to reduce the horizontal wave force acting on the bridge superstructure.
Fig.9. Effect of submergence depth d2 on the dimensionless wave forces: (a) Horizontal (b) Vertical
Fig.10. Effect of the relative thickness of the deck on the dimensionless wave forces: (a) Horizontal (b) Vertical
As shown in Fig.11, the effect of the girder width and space on the dimensionless wave forces is investigated by setting girder width w/b (Fig.1) within the range [0.005, 0.333] for the same wave–structure interaction system shown in Fig.2. This indicates that the girder is very thin when width w/b equals to 0.005. Moreover, w/b = 0.333 represents the case when there is no space between adjacent girders. The following can be observed from Fig.11: (1) Relative girder width w/b only affects the horizontal wave force markedly when wave number kb > 0.5, (2) A bridge superstructure with a wider girder will experience a larger horizontal wave force, (3) The vertical forces are hardly effected by girder width w/b.
The dimensionless wave force acting on the bridge superstructure as a function of the number of girders is plotted in Fig.12. The parameters of the wave–structure system used here are consistent with Fig.2, except the number of girders. Fig.12 (a) indicates that the horizontal wave force acting on the superstructure increases with the increase in the number of girders.
18/23
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
The peak of the horizontal wave force, which is attained at wave number kb = approximately 1.2, increases by approximately 10% with the number of girders added but before the total reaches five. When the superstructure consists more than six girders, the horizontal wave force is less affected by the addition of the girders. In addition, the vertical wave force is negligibly affected by the number of girders.
Fig.11. Effect of the girder width on the dimensionless wave forces: (a) Horizontal (b) Vertical
Fig.12. Effect of the number of girders on the dimensionless wave forces: (a) Horizontal (b) Vertical
5 Conclusions
Coastal bridges are vulnerable to the large waves produced during hurricane storms, particularly existing ones without sufficient clearance. In this study, an analytical solution is proposed based on the potential flow theory and matching eigenfunction method with the objective of developing an efficient method to predict the wave forces acting on a submerged bridge. The effectiveness of present analytical model is validated by the tested results of two hydrodynamic experiments conducted on bridge models. The main conclusions are summarized as follows.
(1) For submerged bridge superstructure, the horizontal wave force increases quickly then decreases slowly about the wave number kb after peaking at different value of kb for
19/23
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
different wave angle. The vertical wave force shows a continuous declining with the wave number kb for all wave angle.
(2) The wave forces acting on the superstructure of the submerged bridge experiences maximum horizontal and vertical wave forces when the waves propagate normally. Wave with long period is very dangerous for bridges without sufficient vertical connection even at the oblique incidences.
(3) The geometry of the bridge structure affects the horizontal wave force more significantly than the vertical wave force. It shows the promise that the horizontal wave force can be minimized by using a thin deck, and a small number of narrow girders without obvious influence on vertical wave force.
(4) It should be noted that the analytical solution of this study is based on the potential theory in the linear frame work. The proposed method is limited when the strong nonlinear of the water wave is included in the analysis for some specific cases.
Acknowledgement
The authors highly appreciate the financial support provided by the National Natural Science Foundation of China (51725801)341Equation Section (Next)
Appendix
, (A.1)
, (A.2)
, (A.3)
, (A.4)
(A.5)
, (A.6)
, (A.7)
, (A.8)
, (A.9)
, (A.10)
, (A.11)
, (A.12)
20/23
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
, (A.13)
, (A.14)
, (A.15)
References
AASHTO, 2008. Guide specifications for bridges vulnerable to coastal storms. American Association of State Highway and Transportation Officials, Washington, DC.
Abul-Azm, A., 1994. Diffraction through wide submerged breakwaters under oblique waves. Ocean Engineering 21 (7), 683-706.
Abul-Azm, A.G., Gesraha, M.R., 2000. Approximation to the hydrodynamics of floating pontoons under oblique waves. Ocean Engineering 27 (4), 365-384.
Bradner, C., Schumacher, T., Cox, D., Higgins, C., 2011. Large scale laboratory observations of wave forces on a highway bridge superstructure. Oregon State University, Portland, OR.
Cai, Y., Agrawal, A., Qu, K., Tang, H.S., 2018. Numerical Investigation of Connection Forces of a Coastal Bridge Deck Impacted by Solitary Waves. Journal of Bridge Engineering 23 (1), 04017108.
Cuomo, G., Shimosako, K., Takahashi, S., 2009. Wave-in-deck loads on coastal bridges and the role of air. Coastal Engineering 56 (8), 793-809.
Denson, K.H., 1980. Wave forces on causeway-type coastal bridges: Effects of angle of wave incidence and cross-section shape. Mississippi State University, Starkville, MS.
Gesraha, M.R., 2004. An eigenfunction expansion solution for extremely flexible floating pontoons in oblique waves. Applied Ocean Research 26 (5), 171-182.
Gesraha, M.R., 2006. Analysis of shaped floating breakwater in oblique waves: I. Impervious rigid wave boards. Applied Ocean Research 28 (5), 327-338.
Guo, A., Fang, Q., Li, H., 2015a. Analytical solution of hurricane wave forces acting on submerged bridge decks. Ocean Engineering 108 (16), 519-528.
Guo, A.X., Fang, Q.H., Bai, X.D., Li, H., 2015b. Hydrodynamic Experiment of the Wave Force Acting on the Superstructures of Coastal Bridges. Journal of Bridge Engineering 20 (12), 04015012.
Hayatdavoodi, M., Ertekin, R.C., 2015a. Nonlinear Wave Loads on a Submerged Deck by the Green–Naghdi Equations. Journal of Offshore Mechanics and Arctic Engineering 137 (1), 011102.
Hayatdavoodi, M., Ertekin, R.C., 2015b. Wave forces on a submerged horizontal plate - Part II: Solitary and cnoidal waves. Journal of Fluids and Structures 54, 580-596.
Hayatdavoodi, M., Seiffert, B., Ertekin, R.C., 2014. Experiments and computations of
21/23
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
solitary-wave forces on a coastal-bridge deck. Part II: Deck with girders. Coastal Engineering 88 (6), 210-228.
Hu, K., Chen, Q., 2011. Directional spectra of hurricane-generated waves in the Gulf of Mexico. Geophysical Research Letters 38 (19), L19608.
Huang, W., Xiao, H., 2009. Numerical modeling of dynamic wave force acting on Escambia Bay Bridge deck during Hurricane Ivan. Journal of Waterway, Port, Coastal, and Ocean Engineering 135 (4), 164-175.
Jin, J., Meng, B., 2011. Computation of wave loads on the superstructures of coastal highway bridges. Ocean Engineering 38 (11), 2185-2200.
Karmakar, D., Soares, C.G., 2012. Oblique scattering of gravity waves by moored floating membrane with changes in bottom topography. Ocean Engineering 54, 87-100.
Li, Y., Dong, G., Liu, H., Sun, D., 2003. The reflection of oblique incident waves by breakwaters with double-layered perforated wall. Coastal Engineering 50 (1), 47-60.
Liu, Y., Li, Y.-c., Teng, B., 2016. Interaction between oblique waves and perforated caisson breakwaters with perforated partition walls. European Journal of Mechanics - B/Fluids 56, 143-155.
Losada, I.J., Losada, M.A., Roldan, A.J., 1992. Propagation of oblique incident waves past rigid vertical thin barriers. Applied Ocean Research 14 (3), 191-199.
Marin, J.M., 2010. Wave loading on bridge superstructures, Civil and Coastal Engineering Department. University of Florida, Gainesville, Florida.
Seiffert, B.R., Ertekin, R.C., Robertson, I.N., 2015. Wave loads on a coastal bridge deck and the role of entrapped air. Applied Ocean Research 53, 91-106.
Smith, J.M., 2007. Modeling Nearshore Waves for Hurricane Katrina. Army Engineer Research and Development Center, Vicksburg, MS, p. 15.
Wei, Z., Dalrymple, R.A., 2016. Numerical study on mitigating tsunami force on bridges by an SPH model. Journal of Ocean Engineering and Marine Energy 2 (3), 365-380.
Xiao, H., Huang, W., Chen, Q., 2010. Effects of submersion depth on wave uplift force acting on Biloxi Bay Bridge decks during Hurricane Katrina. Computers & Fluids 39 (8), 1390-1400.
Xu, G., Cai, C., 2015a. Numerical simulations of lateral restraining stiffness effect on bridge deck–wave interaction under solitary waves. Engineering Structures 101, 337-351.
Xu, G., Cai, C.S., 2015b. Wave Forces on Biloxi Bay Bridge Decks with Inclinations under Solitary Waves. Journal of Performance of Constructed Facilities 29 (6), 04014150.
Xu, G., Cai, C.S., Han, Y., 2015. Investigating the Characteristics of the Solitary Wave Induced Forces on Coastal Twin Bridge Decks. Journal of Performance of Constructed Facilities.
Zheng, Y., Liu, P., Shen, Y., Wu, B., Sheng, S., 2007. On the radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas. Ocean
22/23
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
Engineering 34 (3), 436-450.
23/23
595
596