1 Study on the calculation method for hydrodynamic pressure of bridge piers in deep water under earthquakes Prof. Bo Song University of Science and Technology Beijing
2 main contents 1 The Background of Research 2 A Simplified Calculation Method Proposed to Study on Interaction of Water and Structure under Earthquake main contents The Effect of Hydrodynamic Pressure on the Slender Structure Based on the Simplified Method The Comparison of the Proposed Simplified Method and the Finite Element Method The Comparison of the Simplified Method and the Shaking Table Experiment 6 Conclusions
3 1. The Background of Research China has a planning of five sea-crossing projects: the Bohai Strait, the Yangtze River Estuary, Hangzhou Bay, Pearl River Estuary and the Qiongzhou strait from north to south. Typical deep water bridges in domestic Total length:639m Type: Cable- stayed Bridge Max water depth :96m Yangtze River Bridge in Yunyang
4 1. The Background of Research The damage of Bridge in 2008 Wenchuan earthquake: The main pier of the Miaoziping bridge is 104 m high and its deepest submerged depth is more than 40 m. Although there is no evidence that the impact of hydrodynamic pressure could directly result in structural damage, the influence of water on structures should nevertheless be considered in order to ensure the safety of structures in water. FIGURE 1 The damages of Mingjiang Bridge in Miaoziping (color figure available online). MiaoZiPing MinJiang Bridge Total length :1440m Type : Cable- stayed Bridge Max water depth :96m
5 曲げモーメント M(tfm) Application: Bending Moment of Girder and Pier> Energy transfer The Application requirements of study Taking the performance design of steel arch bridge for example: the structural design of the pier and abutment, including the main beam are closely related to hydrodynamic effect. 部材番号 n 4000 b Mz 曲率 φ(1/m)
6 1. The Background of Research In this study, based on the ideal potential fluid theory and the Morison equation, a simplified calculation method is put forward. The dynamic response of a typical cylindrical pile was studied by using the simplified method and then the results of the model were compared with those from the finite element method. In order to further validate the simplified method, a comparison of results from simplified method and the experiment was performed.
7 1. The Background of Research Seismic action - water bridge structure Earthquake damage investigation Experimental study Theoretical analysis Field test Laboratory test Numerical method Simplified method Seismic observation Shaking table test Explain test and seismic damage Meets the demand of engineering seismic design method Research Frame
8 2. The Effect of Hydrodynamic Pressure on the Slender Structure Based on the Simplified Method The distribution of hydrodynamic pressure long the height of piers under the earthquake It can be found that hydrodynamic pressure caused by plate boundary type seismic wave acting on the piers is larger than large-scale earthquake occurred at the inland area and the obvious peak value of hydrodynamic pressure appeared on the top of the pier when the plate boundary type occurred. This will make the point of hydrodynamic The distribution of hydrodynamic pressure along the pier height direction pressure getting upper and bigger bending moment at the bottom of the piers.
9 3. The Effect of Hydrodynamic Pressure on the Slender Structure Based on the Simplified Method (a) 20m depth of water (b) 15m depth of water (c) 10m depth of water (d) 5m depth of water T1-II-3 different water depths under seismic action hydrodynamic pressure distribution along the pier height direction (a) 20m depth of water (b) 15m depth of water (c) 10m depth of water (d) 5m depth of water T2-II-1 different water depths under seismic action hydrodynamic pressure distribution along the pier height direction For typical entity rectangular section and cylindrical pile pier structure, studying on the distribution of hydrodynamic pressure acting on the structure. Analyzing the influence factors of dynamic water - structure interaction, hydrodynamic effect on dynamic characteristics of the natural vibration period of piers cannot be ignored when water depth value exceeding the critical depth.
10 2. A Simplified Calculation Method for Hydrodynamic Pressure to Slender Structures FIGURE 2 Hydrodynamic forces on cylindrical pile According to Morison s hydrodynamic theory, the pier in the water will produce the same magnitude of movement as that of the fluid particles. The hydrodynamic force on the pile includes two components: one is when the fluid flows through the piles, the existence of the piles causes the acceleration or deceleration of the fluid, and the fluid exerts a force on the pile known as the inertial force FI ; the other is due to the viscous effect and the swirl in the wake of the fluid which causes a drag force on the pile, denoted FD (as shown in Fig. 2).
11 2. A Simplified Calculation Method for Hydrodynamic Pressure to Slender Structures The inertial force is therefore equivalent to the product of the quantity of water and the corresponding acceleration. On the other hand, considering the fluid around the pile, the inertial force at a unit length (Δd) of pile at depth z is defined as Then 2 D fi ( CM 1) A u x ( CM 1) u x z 4 F f d ( C 1) V ( u x) I I M where C M is mass coefficient and also could be called inertial coefficient, ρ is the density of water, and ΔV is the volume of the underwater pier structure. z (1) (2)
12 2. A Simplified Calculation Method for Hydrodynamic Pressure to Slender Structures For the fixed rigid pile the drag force is related to the relative velocity of the fluid particles and the piles. The drag force is proportional to the square of velocity of the fluid particles and the projected area of the object in the direction perpendicular to the moving direction of the fluid. As the movement of the fluid is a reciprocating motion, in order to keep the direction, the drag force at a unit height (Δd) of pile at depth z is defined as Then (3) (4) where A p is the projected area of the unit length of the cylinder in the vertical direction and C D is the drag force coefficient considering the shape of the pile and the frictional drag force of the column wall.
13 2. A Simplified Calculation Method for Hydrodynamic Pressure to Slender Structures Then the hydrodynamic pressure on the surface of the circular cylinder could be defined as (5) As the influence of the structure on the water is ignored, the movement of the structure does not cause a movement of water, so the speed and acceleration of the water particles are zero. The hydrodynamic force on the unit length of the cylinder along the X-axis direction can be expressed as (6)
14 2. A Simplified Calculation Method for Hydrodynamic Pressure to Slender Structures A dynamic equation of the entire pier system under seismic conditions could be expressed as (7) The water equivalent added mass on i could be defined as (8) FIGURE 3 Dynamic response model for the proposed simplified method Therefore, a dynamic equation of the entire pier system under seismic conditions can be expressed as (9) (10)
15 3. The Effect of Hydrodynamic Pressure on the Slender Structure Based on the Simplified Method FIGURE 4 Model of cylindrical pile in deep water. For a water-structure interaction simulation analysis, a pile from the high-pile foundation of the main tower of the Nanjing Yangtze River Bridge was selected as a prototype. The bridge is located in a Class II site with eight degrees seismic fortification intensity. The pile is 50 m high, its diameter D is 3 m, the superstructure mass on the pile is m=2000 t. In order to make the prototype equivalent to a cylindrical pile, the equivalent consolidation depth method was used to simplify the dynamic effect of soil on the pile. The bottom of the pile is fixed as shown in Fig. 4.
16 3. The Effect of Hydrodynamic Pressure on the Slender Structure Based on the Simplified Method Acceleration (Gal) Acceleration (Gal) Acceleration (Gal) Acceleration (Gal) Acceleration (Gal) T1-II Time (s) Time (s) T2-II FIGURE 5 Acceleration-time curves of seismic waves Time (s) T1-II Time (s) T2-II Time (s) T2-II-1 Selected interplate earthquake (Type 1) and intraplate earthquake (Type 2) in Class II sites as farfault and near-fault seismic waves, included four seismic waves (T1- II-1, T1-II-3, T2-II-1, T2-II-3). Taking the project s information into account, the proportion method was used to modify the waves and the peak accelerations of 4 seismic waves were adjusted to 0.2 g when the seismic waves were put along the horizontal direction. The four adjusted waves are shown in Fig. 5.
17 3. The Effect of Hydrodynamic Pressure on the Slender Structure Based on the Simplified Method In order to study the effect of hydrodynamic pressure on the internal forces of structures in water subjected to earthquakes, use R to express the influence of the hydrodynamic water on the maximum value of structural dynamic responses:
18 3. The Effect of Hydrodynamic Pressure on the Slender Structure Based on the Simplified Method Pier Height (m) Pier Height (m) Pier Height (m) Pier Height (m) The cylindrical pile model was studied by using the proposed simplified method. The value of the hydrodynamic inertial force coefficient C M is chosen as 2.0 in accordance with Code of Hydrology for Sea Harbour. According to the model in Fig. 4, study on the dynamic behavior of the model was made under different seismic waves and from that internal forces of the model were attained. Figures 6 and 7 show how the hydrodynamic pressure influences the shear forces and flexural moments of the pile respectively m depth 30m depth 50m depth m depth 30m depth 50m depth m depth 30m depth 50m depth m depth 30m depth 50m depth R shear (%) R shear (%) R shear (%) R shear (%) (a) T1-II-1 (b) T1-II-3 (c) T2-II-1 (d) T2-II-3 FIGURE 6 Influence of hydrodynamic pressure on the shear forces along the cylindrical pile (color figure available online).
19 3. The Effect of Hydrodynamic Pressure on the Slender Structure Based on the Simplified Method Pier Height (m) Pier Height (m) Pier Height (m) Pier Height (m) Figures 7 show how the hydrodynamic pressure influences the shear forces and flexural moments of the pile respectively m depth 30m depth 50m depth m depth 30m depth 50m depth m depth 30m depth 50m depth m depth 30m depth 50m depth R moment (%) R moment (%) R moment (%) R moment (%) (a) T1-II-1 (b) T1-II-3 (c) T2-II-1 (d) T2-II-3 FIGURE 7 Influence of hydrodynamic pressure on the flexural moments along the cylindrical pile (color figure available online).
20 3. The Effect of Hydrodynamic Pressure on the Slender Structure Based on the Simplified Method Depth (m) Depth (m) Study on the influence of hydrodynamic force on the shear force and the flexural moment at the bottom of the cylindrical pile was made, as shown in Fig T1-II-1 T1-II-3 T2-II-1 T2-II T1-II-1 T1-II-3 T2-II-1 T2-II R shear (%) R moment (%) FIGURE 8 Influence of hydrodynamic pressure on the shear force and the flexural moment at the bottom of the cylindrical pile (color figure available online).
21 3. The Effect of Hydrodynamic Pressure on the Slender Structure Based on the Simplified Method In Fig. 8, subjected to two types of seismic waves, the bottom shear force and the bottom flexural moment of the cylindrical pile are impacted by different water depths. R shear and R moment appear to increase with increasing water depth, R shear and R moment under Type 1 seismic waves are larger than them under Type 2 seismic waves. The maximum R shear reaches 77%, which is at 50 m depth and under T2-II-1 seismic wave, and the maximum R moment in 40 m depth is only 16.4% when subjected to T2-II-3 seismic wave.
22 3. The Effect of Hydrodynamic Pressure on the Slender Structure Based on the Simplified Method The influence of boundary effect on the hydrodynamic pressure Based on the study of boundary effect on the hydrodynamic pressure of bridge pier, carrying out the regulation study of the variation of the hydrodynamic pressure based on the different depths of water and water models (respectively 10 times, 20 times, 30 times of the short side and infinite boundary waters), establishing the optimal method of the boundary selecting, lay a foundation for the performance design of the long span bridges. Using the El Centrowave(NS) as loadwaveform, duration of 40s, adjusting the peak acceleration to 400gal according to the site condition. Rectangular pier-water model El Centro - wave curve
23 3. The Effect of Hydrodynamic Pressure on the Slender Structure Based on the Simplified Method The influence of boundary effect on the hydrodynamic pressure By the increasing of the water area boundary, the value of the hydrodynamic pressure is decreasing. When the water model is 30 times of the side length of one pier, the maximum of the hydrodynamic pressure is 13159Pa. Comparing with infinite boundary (the maximum value is 13137Pa), the error is 1.3%. Based on this, it can be considered the water model, 30 times of the side length of one pier, basically meets requirements.
24 3. The Effect of Hydrodynamic Pressure on the Slender Structure Based on the Simplified Method X Z Y The comparison of the dynamic characteristics of the cylindrical pile received from the simplified method and the finite element method. The assumption of ideal potential fluid was adopted, so the fluid satisfied the properties such as no heat transfer, viscidity, irrotationality, and imcompressibility, and the fluid boundary also satisfied the small deformation conditions. FIGURE 9 Structural model for the finite element method (color figure available online). The structure model and water model were established, respectively, and were in the coupled state through setting contacts between them. The body of water within a 50 m range from the pile was considered. The calculation model is shown in Fig. 9.
25 4. The Comparison of the Proposed Simplified Method and the Finite Element Method The natural vibration period of the model was analyzed by using the simplified method and the finite element method; the results are shown in Table 1. TABLE 1 Comparison of the natural vibration periods obtained from the simplified method and the finite element method Depth (m) The simplified method First order The finite element method The natural vibration period (s) Deviation rate (%) The simplified method Third order The finite element method Deviati on rate (%)
26 4. The Comparison of the Proposed Simplified Method and the Finite Element Method Table 2 shows the peak displacement and peak acceleration at the top of the cylindrical pile obtained from the two methods. TABLE 2 Comparison of the dynamic responses at the top of the cylindrical pile obtained from the simplified method and the finite element method Seismic Wave Peak displacement at the top of the pile (m) The finite The simplified element method method R dis (%) Peak acceleration at the top of the pile (m/s 2 ) The finite The simplified element method method R acc (%) T1-II T1-II T2-II T2-II
27 Displacement (m) Displacement (m) Displacement (m) Displacement (m) 4. The Comparison of the Proposed Simplified Method and the Finite Element Method Figures 10 display that the displacement-time curves attained from the simplified method and the finite element method coincide with each other, and so do the acceleration-time curves The Finite Element Method The Simplified Method The Finite Element Method The Simplified Method (a) T1-II (b) T1-II Time (s) The Finite Element Method The Simplified Method Time (s) (c) T2-II Time (s) FIGURE 10 Comparison of displacement-time curves of the point at the top of the cylindrical pile obtained from the simplified method and the finite element method (color figure available online) The Finite Element Method The Simplified Method Time (s) (d) T2-II-3
28 Acceleration (m/s 2 ) Acceleration (m/s 2 ) Acceleration (m/s 2 ) Acceleration (m/s 2 ) 4. The Comparison of the Proposed Simplified Method and the Finite Element Method Figures 11 display that the displacement-time curves attained from the simplified method and the finite element method coincide with each other, and so do the acceleration-time curves Time (s) The Finite Element Method 2 The Simplified Method The Finite Element Method The Simplified Method Time (s) (a) T1-II-1 (c) T2-II FIGURE 11 Comparison of acceleration-time curves of the point at the top of the cylindrical pile obtained from the simplified method and the finite element method (color figure available online) The Finite Element Method The Simplified Method Time (s) The Finite Element Method The Simplified Method Time (s) (b) T1-II-3 (d) T2-II-3
29 5. The Comparison of the Simplified Method and the Shaking Table Experiment FIGURE 12 Layout of the south pier of the Nanjing Third Yangtze River Bridge (unit: m). The South Tower of the Nanjing Third Yangtze River Bridge selected for the experiment has 84 m 29 m elevated pile caps [The Construction Headquarters of Nanjing Third Yangtze River Bridge, 2005].The layout of the south pier of the Nanjing Third Yangtze River Bridge is shown in Fig. 12. There are 30 piles under the elevated pile caps and there are 15 piles whose diameter is 3 m under each round section. The maximum water depth of the elevated pile caps is 45 m.
30 5. The Comparison of the Simplified Method and the Shaking Table Experiment As there are limitations to the testing conditions, only half of the prototype was used in the model. The scale model and the prototype are similar in size and the scales of all physical quantities are shown in Table 3. TABLE 3 Similarity coefficient of pier model Physical Quantity Similarity Coefficient Physical Quantity Similarity Coefficient Length Sl 1.50 Inertial Force S F SES l Modulus of Elasticity S 6.87 Hydrodynamic S P S l 0.02 Equivalent Density Acceleration Stress Strain SE 6.87 S 1 S SE Flexural moment Frequency SES Time St Sl SE S Displacement Su Sl 0.02 S S SE S M 1 S f S S SE l l 33.5
31 5. The Comparison of the Simplified Method and the Shaking Table Experiment FIGURE 13 Arrangement diagram of observation points (unit: mm) (color figure available online). The elevated pile caps are made of reinforced concrete and the piles are made of steel pipes. Wave absorbing devices were installed on the sides of the water tank which are perpendicular to the direction of the input waves in order to eliminate the rebound wave. The arrangement of observation points is shown in Fig.13.
32 5. The Comparison of the Simplified Method and the Shaking Table Experiment The experimental device in air and in water is shown in Fig. 14. In the experiment, the model was subjected to various earthquake waves in air and in water. And the input waves were modified in order to make sure the waves which loaded on the tank were more accurate. FIGURE 14 Testing Model: (a) pier model in air and (b) pier model in water (color figure available online).
33 5. The Comparison of the Simplified Method and the Shaking Table Experiment The experimental model was analyzed by the simplified method proposed in this article; the results of the calculation and the experiment were compared and are shown in Table 4. TABLE 4 Comparison of natural period and frequency of vibration obtained from the simplified method and the shaking table experiment. In air In water Methods Frequency(Hz) Period(s) Frequency(Hz) Period(s) Experimental Value Calculated Value R (%)
34 5. The Comparison of the Simplified Method and the Shaking Table Experiment The elastoplastic dynamic response time history analysis of the piers in water based on the added mass method Trends are consistent The comparison between the results of experiment and numerical simulation showed that effects of the hydrodynamic pressure to the bridge foundation cannot be ignored based on the added mass method and the shaking table experiment.
35 5. The Comparison of the Simplified Method and the Shaking Table Experiment The dynamic responses of the model in water subjected to sine waves, the Tianjin wave and Takatori wave, were analyzed by using both the simplified method and the experiment. 1 Hz, 3 Hz, 5 Hz, and 7 Hz sine waves were all taken into account. The results of the peak acceleration value along the model are compared in Figs when the peak acceleration value of the sine waves and seismic waves is 100 Gal or 200 Gal. (a) 100Gal (b) 200Gal FIGURE 15 Distribution of the peak of the horizontal acceleration under harmonic loads (color figure available online).
36 5. The Comparison of the Simplified Method and the Shaking Table Experiment According to the comparison, the calculation and experiment results of the dynamic responses of the model in water are almost the same, and there is a certain deviation only in some cases. (a) 100Gal (b) 200Gal FIGURE 16 Distribution of the peak of the horizontal acceleration under the Tianjin Earthquake (color figure available online).
37 5. The Comparison of the Simplified Method and the Shaking Table Experiment The maximum deviations under 100 Gal/7 Hz and 200 Gal/7 Hz sine waves are 15.4% and 10.5%, respectively, the maximum deviations under the Tianjin wave and the Takatori wave are 9.0% and 12.5%, respectively. As the simplified method is accurate enough and makes the calculation easier, it is feasible to apply the simplified method in practical structural analysis. (a) 100Gal (b) 200Gal FIGURE 17 Distribution of the peak of the horizontal acceleration under the Takatori Earthquake (color figure available online).
38 Application In order to study the effect of the hydrodynamic pressure on the dynamic behavior of the whole bridge structure, based on establishing the finite element computation model of the Ushineohashi Bridge(main span 260m), using the added mass method to make the computation. The results showed that hydrodynamic pressure had large effects on the performance of the bridge.
39 曲げモーメント M(tfm) Application The Application requirements of study Taking the performance design of steel arch bridge for example: the structural design of the pier and abutment, including the main beam are closely related to hydrodynamic effect. 部材番号 n 4000 b Mz 曲率 φ(1/m)
40 6. Conclusions From all the results shown in this article, it could be concluded as follows. 1. According to Morison s hydrodynamic theory, the hydrodynamic pressure could be treated as added mass and is composed of inertial force and drag force. 2. The results of the model indicate that water has an obvious effect on the structure. Therefore, it is necessary to study the influence of water on the structures. 3. Comparing the results obtained from these two methods the deviations of the natural vibration periods do not exceed 4%, the displacements and accelerations are almost the same. 4. The simplified method and the experiment led to different results, but the values did not have an obvious deviation. This further demonstrates the validity of the method.
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VPG TRANSDUCERS Load Cells and Weigh Modules Technical Note VPGT-06 Scope Load cells may be used to weigh vessels in various installation configurations. The installation of load cells into a practical
University of Colorado, Boulder CU Scholar Undergraduate Honors Theses Honors Program Spring 2015 Biomechanics of Parkour: The Vertical Wall-Run Technique Integrative Physiology, Peter.Lawson@Colorado.EDU
J. Ocean Eng. Mar. Energy (215) 1:223 236 DOI 1.17/s4722-15-15-5 ERRATUM Erratum to: Effect of seabed slope on the pile behaviour of a fixed offshore platform under lateral forces K. Muthukkumaran K. S.
Ch16Lectures Page 1 Ch16Lectures Thursday, April 16, 2009 12:22 PM Ch16Lectures Page 2 Ch16Lectures Page 3 Ch16Lectures Page 4 The following animation illustrates the interference of two wave pulses travelling
Stability of Pipeline and details of Anchor blocks For Offshore pipeline laying at Gulf of Kutch, Gujarat. Client WELSPUN (I) LTD, Gujarat EPC Contactor PATEL CONSTRUCTION CO. Consultants Prof. R. Sundaravadivelu
Non-Linear Seismic Analysis of Multi-Storey Building Uzair Khan 1 1 M.Tech (Structural & Construction Engineering) Department of Civil Engineering, National Institute of Technology, Jalandhar. Hina Gupta
Truck-mounted Mass Flow Metering for LPG Delivery Itron, Inc. 1310 Emerald Rd. Greenwood, SC 29646 Introduction The mathematician Gustave De Coriolis first described the Coriolis Effect, which is an inertial
Waves Introduction A vibration must be the source of a wave. Waves in turn also cause vibrations. They are intrinsically connected. Waves transmit energy. There are different ways in which waves can be
SECM/15/080 Vertical Uplift Capacity of a Group of Equally Spaced Helical Screw Anchors in Sand S. Mukherjee 1* and Dr. S. Mittal 2 1 Amity University, Noida, India 2 Indian Institute of Technology, Roorkee,
4 th International Conference THE NEW FOOTBRIDGE IN CITY OF CRACOW DYNAMIC DESIGN AND FINAL VERIFICATION IN TEST LOADING Krzysztof ŻÓŁTOWSKI Professor University of Technology Gdansk, PL firstname.lastname@example.org
PAPER 2 THEORY QUESTIONS 1 (a) Water waves are transverse waves. Sound is a longitudinal wave. (i) Describe the difference between transverse waves and longitudinal waves. In your account, draw a diagram
Tutorial 5 Relative equilibrium 1. n open rectangular tank 3m long and 2m wide is filled with water to a depth of 1.5m. Find the slope of the water surface when the tank moves with an acceleration of 5m/s
Tidal streams and tidal stream energy device design This technical article introduces fundamental characteristics of tidal streams and links these to the power production of tidal stream energy devices.
13: Coastal and Ocean Engineering ENGI.8751 Undergraduate Student Forum Faculty of Engineering and Applied Science, Memorial University, St. John s, NL, Canada MARCH 2013 Paper Code. (13 - walsh) The Challenge
ame School Date AP Lab 11.3 Archimedes Principle Explore the Apparatus We ll use the Buoyancy Apparatus in this lab activity. Before starting this activity check to see if there is an introductory video
Static Extended Trailing Edge for Lift Enhancement: Experimental and Computational Studies T. Liu, J. Montefort, W. Liou Western Michigan University Kalamazoo, MI 49008 and Q. Shams NASA Langley Research
Chapter 14 Vibrations and Waves Chapter 14 Vibrations and Waves In this chapter you will: Examine vibrational motion and learn how it relates to waves. Determine how waves transfer energy. Describe wave
UNDERWATER BRIDGE INSPECTION REPORT STRUCTURE NO. 62555B WABASHA STREET SOUTH BOUND (MASA 235 SB) OVER THE MISSISSIPPI RIVER CITY OF ST. PAUL OCTOBER 31, 2012 PREPARED FOR THE MINNESOTA DEPARTMENT OF TRANSPORTATION
Simulator For Performance Prediction Of Reciprocating Considering Various Losses Aditya S. Bawane 1, Dr. V.K. Bhojwani 2, Mitali B. Deshmukh 3 1 (Mechanical Engineering Department, JSCOE, S.P. Pune University,
AP Physics Multiple Choice Practice Fluid Mechanics 1. A cork has weight mg and density 5% of water s density. A string is tied around the cork and attached to the bottom of a water-filled container. The
1 LECTURE OUTLINE CHAPTER 14 Waves and Sound 14-1 Types of Waves 2 A wave: Is a disturbance that propagates from one place to another. 1- Transverse Wave: The displacement of the medium is perpendicular
EFFECT OF CORNER CUTOFFS ON FLOW CHARACTERISTICS AROUND A SQUARE CYLINDER Yoichi Yamagishi 1, Shigeo Kimura 1, Makoto Oki 2 and Chisa Hatayama 3 ABSTRACT It is known that for a square cylinder subjected
CHAPTER 185 SAND TRANSPORT UNDER GROUPING WAVES Shinji Sato 1 Abstract Laboratory experiments as well as numerical modeling were conducted for sand transport under non-breaking grouping waves. Experiments