Lei TANG, Wei ZHANG*, Ming-xiao XIE, Zhen YU

1. College of Harbor, Coastal and Offshore Engineering, Hohai University, Nanjing 210098, P. R. China

2. Tianjin Research Institute for Water Transport Engineering, Ministry of Transport, Tianjin 300456, P. R. China

3. Changjiang Waterway Planning, Design and Research Institute, Wuhan 430010, P. R. China

Application of equivalent resistance to simplification of Sutong Bridge piers in tidal river section modeling

Lei TANG1, Wei ZHANG*1, Ming-xiao XIE2, Zhen YU3

1. College of Harbor, Coastal and Offshore Engineering, Hohai University, Nanjing 210098, P. R. China

2. Tianjin Research Institute for Water Transport Engineering, Ministry of Transport, Tianjin 300456, P. R. China

3. Changjiang Waterway Planning, Design and Research Institute, Wuhan 430010, P. R. China

This paper describes some details and procedural steps in the equivalent resistance (E-R) method for simplifying the pier group of the Sutong Bridge, which is located on the tidal reach of the lower Yangtze River, in Jiangsu Province. Using a two-dimensional tidal current numerical model, three different models were established: the non-bridge pier model, original bridge pier model, and simplified bridge pier model. The difference in hydrodynamic parameters, including water level, velocity, and diversion ratio, as well as time efficiency between these three models is discussed in detail. The results show that simplifying the pier group using the E-R method influences the water level and velocity near the piers, but has no influence on the diversion ratio of each cross-section of the Xuliujing reach located in the lower Yangtze River. Furthermore, the simplified bridge pier model takes half the calculation time that the original bridge pier model needs. Thus, it is concluded that the E-R method can be use to simplify bridge piers in tidal river section modeling reasonably and efficiently.

E-R method; tidal river section; pier group; simplification; numerical modeling

1 Introduction

Pier foundations are usually used to bear the loads from upper buildings in cross-river or cross-sea bridge projects, offshore wind farm projects, and muddy coastal harbor projects. The placement of the pier group has a certain effect on water level, velocity, and discharge around the piles. Physical modeling and numerical modeling are the main approaches to investigate the effect (Deng 2007; Cao et al. 2006; Xie et al. 2008; Martin-Vide and Prio 2005). With the development of computer technology and the improvement of numerical calculation methods, many studies have focused on numerical experiments (Wang 2010; Cao et al. 2006; Xie et al. 2008; Qi et al. 2006). Li (2001) pointed out that the domain of the numerical model must be very large compared with the concerned area in order to prevent a project from affecting theopen boundary and/or to obtain the open boundary data easily. However, the pier size is often relatively small, ranging from tens of centimeters to tens of meters. Studying the scope and extent of the impact of piers with such small scales on water flow in a wide calculation domain has been a difficult problem for researchers.

Currently, there are two main solutions to this problem: the direct simulation method and the equivalent simulation method. The direct simulation method treats the pier as an impervious land boundary and determines the outline of the pier by refining the grids around it. However, with the increase in the number of grid cells, and the decrease in grid cell size, the numerical model is time-consuming even if on a high-performance computer. To enable the numerical simulation to execute successfully and ensure the premise of the large-domain hydrodynamic conditions, the equivalent simulation method has often been used in previous studies. There are three kinds of equivalent simulation methods: the local terrain adjustment method (Tang 2002a), local roughness adjustment method (Tang 2002a; Tang 2002b; Cao et al. 2006), and equivalent water-blocking area method (Zhang et al. 2007). Based on analysis of the advantages and disadvantages of the three methods, Xie et al. (2008) has proposed a new generalized method for pile piers, called the equivalent resistance (E-R) method. The superiority of the E-R method has been discussed in terms of water level, flow velocity, and backwater.

In this study, the E-R method was used to simplify the pier group of the Sutong Bridge, located on the downstream reach of the Yangtze River. The differences in water level and velocity in the large domain between the original bridge pier model and the simplified bridge pier model are discussed in detail. The changes of the diversion ratio at each cross-section of the Xuliujing reach of the Yangtze River are also analyzed, and the time efficiency for each model is discussed as well.

2 Numerical modeling

2.1 Governing equations and methodology

A two-dimensional tidal current numerical model was used in this study. The governing equations are composed of the continuity and momentum equations:

wherexandyare the components of the Cartesian coordinate system;tis time;ηis the surface elevation;his the total water depth;uandvare the depth-averaged velocities in thexandydirections, respectively;fis the Coriolis parameter;ρis the density of water;gis thegravitational acceleration;νtis the horizontal turbulent eddy viscosity;τsxandτsyare the surface stresses in thexandydirections, respectively; andτbxandτbyare the bottom stresses in thexandydirections, respectively.

In order to describe the bank of the river reasonably, an unstructured triangle grid generation technology was used. The finite volume method was used to solve the equations above. The moving boundary technique was used to reproduce the phenomena of wetting and drying at the intertidal zones. Some detailed information about thenumerical techniques, calibrations, and verifications of the model can be found in Hu and Tan (1995) and Tan (1998).

2.2 Study area and Sutong Bridge

The numerical model domain is located in the lower Yangtze River in Jiangsu Province, which contains plenty of shoals, such as Tongzhou Shoal, Langshan Shoal, Xintonghai Shoal, and Baimao Shoal. The model has three open boundaries, and its length and area are about 77.5 km and 668.5 km2, respectively. The upstream open boundary is at the Tiansheng Harbor (at a latitude of 32°01′N and longitude of 120°45′E) in Nantong City, and the other two downstream open boundaries reach the Qinglong Harbor (at a latitude of 31°51′N and longitude of 121°14′E) and the Yanlin Tidal Station (at a latitude of 31°35′N and longitude of 121°15′E), which are in the north and south branches of the Yangtze River, respectively. The research object is the pier group of the Sutong Bridge located on the tidal river section. There are 72 piers with different sizes. The minimum pier size is 6 m × 15 m, and the maximum is 48.1 m × 113.7 m. Detailed information is given in Fig. 1.

Fig. 1 Sketch of study area (Elevation system: 1985 national height datum of China, unit: m)

2.3 Calibrations and verifications

The spring tide from October 27, 2007 at 10 a.m. to October 28, 2007 at 1 p.m. was selectedas the representative tide in this study. The numerical model domain located at the tidal river reach of the Yangtze River estuary is influenced by tidal flow and runoff flow together. In order to describe the flow movement reasonably and accurately, the open boundary conditions were provided by the in situ measured data from several local tidal stations. The model time step was 30 s, and the Manning’s roughness coefficient as a function of water depth ranged from 0.012 5 to 0.02. Some detailed analysis on the calibrations and verifications of the model can be found in Zhang et al. (2008).

3 Simplification of pier group

3.1 Concept of simplification

In a flow simulation model, coarser grids can satisfy the accuracy requirement of the hydrodynamic simulation. However, when small-size hydraulic structures such as the bridge piers in this study are considered, the mesh refinement method is used to describe the structures’ contours. With the increase of the number of the grid cells, and especially the rapid decrease of the size of the grid cells, the numerical simulation consumes a lot of time. Therefore, it is really challenging to deal with the balance between grid cell size and time consumption in these situations. In order to enhance calculation efficiency and ensure the consistency of hydrodynamic conditions of the large domain, it is necessary to simplify the pier group by means of special methods. The E-R method was used to simplify the pier group in this study.

The approximate schemes obtained by means of the E-R method are considered feasible. However, simplified scenarios must be selected from the point of view of a real project. Hence, some details should be given attention to in the simplification procedure: (1) In order to meet the requirements of navigation and conveyance capacity of the pier cross-section, the design of the bridge usually contains the design of main navigable spans, auxiliary navigable spans, and a general flow section; the piers of the main and auxiliary navigable spans are very important, so their sizes and horizontal locations must remain unchanged. (2) When using the E-R method to simplify a bridge pier group, the pier size is enlarged, the number of piers is decreased, and the horizontal locations of piers are rearranged. As the randomness of rearrangement, there are lots of scenarios for selection of the horizontal locations of piers. The optimum design should keep piers’ center line the same as that in the original design. (3) The flow in the tidal river is reversing current due to the rising tide and falling tide, so the flow resistance during both the flood tide period and the ebb tide period should be checked. The time-averaged flow resistance of the ultimate simplified scenario must equal that of the original design during both the flood and ebb tide periods. The simplified piers were arranged on the center line of the original piers by means of the equivalent distance distribution method in this study.

3.2 Steps for simplification

There are lots of piers in a cross-river bridge project, and in most cases the in situhydrological data such as velocity and direction cannot be collected easily. Hydrodynamic numerical modeling, which is an economical and widely used method, is effective for extending the hydrological data, and can be used to provide the hydrodynamic conditions at the locations of bridge piers. Hence, a sufficiently calibrated numerical model without the bridge pier group should be established before carrying out the process of simplification of the bridge piers. When the hydrological data are obtained, the simplification can be executed. Eq. (4) was used to calcultate the water flow resistance (MTPRC 1998):

whereis the water flow resistance;Vis average velocity, defined asis the drag coefficient; andAis the effective area of the pier exposed to the current. Steps in the simplification of bridge piers are as follows:

(1) The formula for the drag coefficientCwwas described in Deng (2007), and the relative parameters in the formula can be selected from theCriteria for Load in Harbor Project(MTPRC 1998). The hourly water flow resistance for each original pier is calculated, and the time-averaged flow resistances for each original pier during the flood tide, ebb tide, and whole tide periods are calculated and marked asfD1,fD2, andfD3, respectively.

(2) The bridge pier group should be divided into several groups for simplification. Taking the pier group of the Sutong Bridge as an example, the pier group is divided into two groups by the main pier 3 and main pier 6: the northern pier group and the southern pier group.

(3) Then, taking the northern pier group as an example, the time-averaged flow resistances of the northern pier group during the flood tide, ebb tide, and whole tide periods are calculated and marked asFD1,FD2, andFD3, respectively.

(4) The size of a simplified pier (taking a square pile as an example) is estimated. We assume thatVin Eq. (4) is a velocity at the center line of the northern pier group, which is obtained from the numerical model without bridge piers, and its location is temporarily assumed at the center line of the northern pier group. The time-average flow resistances of a simplified pier during the flood tide, ebb tide, and whole tide periods are calculated and denoted asandrespectively. Then, the number of the simplified piers for the northern pier group can be obtained by, wherei= 1, 2, 3. The size of a simplified pier should be repeatedly adjusted untiln1,n2, andn3tend toward one integer. This integer will be considered the number of the simplified piers for the northern pier group.

(5) The simplified piers are arranged on the center line of the original northern piers by means of the equivalent distance distribution method. Then, the horizontal coordinate for each simplified northern pier can be obtained.

(6) The hydrodynamic conditions at the simplified northern piers are extracted from the numerical model without bridge piers. The time-averaged flow resistances of the simplified pier group during the flood tide, ebb tide, and whole tide periods are calculated and denoted asandrespectively.

(7) Then,is compared withFDi. Ifdoes not equalFDi, the process returns to the fourth step untilFDiequals

The simplified method for the southern pier group is the same as that for the northern pier group. The results are shown in Table 1. The size of the simplified piers is much larger than the size of the original piers. The number of the original piers is 69, excluding the three piers located in the northern shallow water zone. The number of the simplified piers is 20, which is about 30% of that of the original piers.

Table 1 Parameters for original and simplified piers

3.3 Analysis of resistance after simplification

The total average resistance is selected to identify the equivalence of the resistance before and after simplification, which is defined as the sum of the average resistance of all the single piers at the cross-section. The total average resistances during the flood tide, ebb tide, and whole tide periods before and after simplification are listed in Table 2. The results illustrate that, under the same tidal condition, the relative errors of the total average resistance between the original and simplified total pier groups during the flood tide, ebb tide and the whole tide periods are about 5.7%, 0.9%, and 3.4%, respectively. From the perspective of approximately equal resistance, the sizes of simplified piers are feasible for modeling.

To further illustrate the rationality of selecting the total average resistance as the identifying factor, the relative errors of the total resistance at the maximum flood tide, maximum ebb tide, and high tidal level were analyzed. The analysis results show that the relative errors of the total resistance at the maximum flood tide and maximum ebb tide are in accordance with the relative errors of the total average resistance during the flood tide and ebb tide periods, respectively, and the relative error of the total resistance at the high tidal level is in accordance with the relative error of the total average resistance during the whole tide period.This simplification scheme is a comprehensive result which meets equal resistance requirement not only at the high tidal level but also at the maximum flood tide and maximum ebb tide.

Table 2 Total average resistances of original and simplified pier groups and relative errors

The grids around the piers of the non-bridge pier model, original bridge pier model, and simplified bridge pier model are shown in Fig. 2.

Fig. 2 Schematic diagram of grids around piers

4 Discussion of effect of bridge pier group simplification

To analyze the effect of bridge pier group simplification by the E-R method, hydrodynamic parameters (water level, velocity, and diversion ratio) and time efficiency were addressed. When the parameterCwin Eq. (4) is constant, the flow resistance is proportional to the product ofAand the square ofV. For a steady flow, the flow resistance is only related with velocity, while for the unsteady flow in the tidal river reach, the variation of velocity and the upstream-face area with water level causes flow resistance to fluctuate. Process curves of tidal level, velocity, and flow resistance are shown in Fig. 3, which indicates that the maximum flow resistance does not occur at the high tidal level but at the time of maximum velocity, and the influence of piers on water flow is at its maximum at the same time. Therefore, hydrodynamic conditions at the maximum flood tide and maximum ebb tide were used to check the simplification effect of the E-R method in this study.

Fig. 3 Curves of tidal level, velocity, and flow resistance on October 27, 2007

4.1 Water level

Four obversation cross-sections (A, B, C, and D) about 3 km long were set at the northern pier group, the main pier 4, main navigable spans, and the southern pier group, respectively. One hundred observation points were arranged in each section. To describe the water surface curve around piers, observation points were installed at intervals of 20 m in the range of about 1.8 km near piers; after that, observation points with intervals of 100 m were installed. Detailed arrangements of the observation cross-sections and points are given in Fig. 1. Water surface curves of each cross-section at the maxmium flood tide and maximum ebb tide are shown in Fig. 4 and Fig. 5, respectively. The error of water level at each cross-section in Fig. 4 and Fig. 5 is the difference between the simplified bridge pier model and original bridge pier model.

Fig. 4 Water surface profiles at maximum flood tide

Fig. 5 Water surface profiles at maximum ebb tide

The following features can be observed from Fig. 4 and Fig. 5: (1) The impact of piers on the water level reaches a long distance from the piers, but is numerically very small, and only few centimeters in magnitude. (2) Both of the original bridge pier model and simplified bridge pier model can simulate the phenomena of the upstream backwater and downstream waterfall around piers at the maxmium flood tide and maximum ebbtide, where the concepts of upstream and downstream are relevant to the flow direction. (3) The water surface curves of the simplified bridge pier model are in good agreement with those of the original bridge pier model at the maximum flood tide and maximum ebb tide, because the error is within the order of magnitude of 10-3m. However, errors of the water level between the simplified bridge pier model and the original bridge pier model are slightly large near piers and at the observation cross-section D. The former is mainly due to the change of the pier size and arrangement after simplification. The latter is the result of the increase of the whole tide average resistance by nearly 3% for the southern pier group after simplification.

Overall, water level calculation results of the simplified bridge pier model are basically identical with those of the original bridge pier model. The E-R method is considered to be effective and the large-scale water level field will not be distorted.

4.2 Velocity

Velocity changes at the maxmium flood tide and maximum ebb tide before and after simplification were analyzed to study the influence of pier simplification by the E-R method on the velocity, including the change rate of velocity and influence distance. The change rate of velocity between the simplified and original bridge pier models can be calculated by the following equation:

whereαis the change rate of velocity before and after simplification,VAis the maximum velocity after simplification, andVPis the maximum velocity before simplification.

Contour maps of the change rate of velocity at the maxmium flood tide and maximum ebb tide are shown in Fig. 6. The influence distanceL, defined as the distance from the pier axis to the contour of the change rate of velocity, is presented in Table 3.

Fig. 6 Change rate of velocity

Table 3 Values of influence distanceLm

The results show the following: (1) For both during the flood tide and ebb tide periods, the contours over the 5% change rate of velocity are concentrated in a small range around piers; the distance of influence is relatively small. During the flood tide period, the maximum influence distance with the envelope of the 5% change rate of velocity is 16 times the simplified pier’s width at the north side and 13 times the simplified pier’s width at the south side; during the ebb tide period, the maximum influence distance is 18 times the simplified pier’s width at the north side and 17 times at the south side. (2) Likewise, whether during the flood tide period or the ebb tide period, the maximum change rate of velocity in the main navigable spans is almost 0. That is to say, there is no difference in the maximum velocity in the main navigable spans between the original and simplified bridge pier models. (3) Whether during the flood tide period or the ebb tide period, the envelope of the 2% change rate of velocity around piers has a wide range corresponding to its long influence distance. During the flood tide period, the maximum influence distance with the envelope of the 2% change rate of velocity is 23 times the simplified pier’s width at the north side and 24 times the simplified piers’ width at the south side; during the ebb tide period, the maximum influence distance is 43 times the simplified pier’s width at the north side and 31 times the simplified pier’s width at the south side.

The conclusions above illustrate that the simplified pier group has a slight influence on the velocity with only small distortion in a small range around the pier group, which is consistent with its influence on the water level. Therefore, simplifying the bridge pier group using the E-R method cannot result in velocity field distortion in a wide domain.

4.3 Diversion ratio

In order to study the impact of the bridge piers on the diversion ratio of each river branch, cross-sections for discharge measurement were set at the east branch of the Langshan Shoal (EBLS), west branch of the Langshan Shoal (WBLS), north branch of the Baimao Shoal (NBBS), south branch of the Baimao Shoal (SBBS), north branch of the Yangtze River (NBYR), and south branch of the Yangtze River (SBYR) as shown in Fig. 1. In previous research, it was concluded that the ebb tide played a leading role in shaping the river bed (Sun and Ruan 1988; Wu et al. 2006; Du et al. 2007). Therefore, the diversion ratio during the stable period of the ebb tide was considered an analysis datum in this study, as specified in Table 4.

Table 4 Diversion ratio of each river branch %

The results show the following: (1) Comparing the non-bridge pier model with the original bridge pier model, it can be found that the bridge piers affects the diversion ratio of each river branch in the Xuliujing reach to a certain extent. The diversion ratio of WBLS is affected slightly more, by about 1.5%, and the diversion ratio of the other branches are influenced to a smaller extent, by about 0.5%. It can be concluded that numerial models for this region must take into account the pier group of the Sutong Bridge. (2) Comparing the with the simplified pier group model, there is no difference in the diversion ratio of each branch between the original and simplified bridge pier models. That is to say, simplifying the bridge piers with the E-R method cannot result in distortion of the diversion ratio of each river branch.

Additionally, according to sections 4.1 and 4.2 above, both the water level and velocity around the bridge piers show slight differences between the original and simplified bridge pier models. The discharge at the Sutong Bridge pier cross-section is examined as well. The results are listed in Table 5, where cross-section A1represents the transect between the main pier 4 and the north land bank, cross-section A2represents the transect between the main pier 4 and main pier 5, and cross-section A3represents the transect between the main pier 5 and the south land boundary. The change rates of discharge between the original and simplified bridge pier models at the pier cross-sections are small, as shown by Table 5. For example, during the flood tide period, the change rates of discharge at cross-sections A1, A2, and A3are 1.0%, –0.2%, and–1.1%, respectively. In other words, using the E-R method of dealing with a small-size piergroup will not cause largely change of the discharge capacity. The same conclusion can be easily obtained for the ebb tide period as well.

Table 5 Diversion ratios at pier cross-sections and change rates of discharge between original and simplified bridge pier models %

4.4 Time efficiency

For the simplified model, a certain degree of accuracy is often exchanged for substantial computing time. Therefore, it is necessary to analyze computing efficiency of the model simplified by the E-R method. Grid parameters and computation time in this study are shown in Table 6.

Table 6 Parameters of grids and calculation time

In this study, the finite volume method was used to numerically solve the model. The calculation time depended on both the minimum grid cell size and overall performance of the computer. As shown in Table 6, with the E-R method, the simplified bridge pier model takes half the calculation time that the original bridge pier model needs.

5 Conclusions

In this paper, taking the pier group of the Sutong Bridge as an example, some details and procedural steps in the E-R method for simplifying bridge piers were described. Using the two-dimensional tidal current numerical model, three different models were established: the non-bridge pier model, original bridge pier model, and simplified bridge pier model. The effect of bridge pier simplification was analyzed in terms of water level, velocity, diversion ratio, and time efficiency. The following conclusions are drawn:

(1) The water surface curves of the simplified bridge pier model were in good agreement with those of the original bridge pier model in the large domain. Only small errors existed near the piers. The contour line of the 5% change rate of velocity was concentrated in a small range with a small influence distance, while the contour line of the 2% change rate of velocity was distributed in a relatively larger range than that of the 5% change rate of velocity.

(2) There was no difference in the diversion ratio of each branch between the original and simplified bridge pier models. The simplified bridge pier model takes half the calculation timethat the original bridge pier model needs.

(3) Applying the E-R method to the simplification of the Sutong Bridge piers is entirely feasible, and the E-R method is promising in other engineering applications.

Acknowledgements

The authors would like to give sincere thanks to the Traffic Bureau of Haimen City in Jiangsu Province, P. R. China for funding this work and providing invaluable field measurement data.

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(Edited by Yan LEI)

This work was supported by the Innovation Project of Graduate Education in Jiangsu Province during 2011 (Grant No. CXZZ11_0449) and the Research Plan Project of Transportation Science in Jiangsu Province (Grant No. 20100714-30HDKY001-2).

*Corresponding author (e-mail:zhangweihhu@vip.sina.com)

Received Aug. 1, 2011; accepted Dec. 6, 2011