design of riprap for bridge pier groups · 2020. 3. 11. · draft 1 1 design of riprap for bridge...

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Design of riprap for bridge pier groups

Journal: Canadian Journal of Civil Engineering

Manuscript ID cjce-2019-0007.R2

Manuscript Type: Article

Date Submitted by the Author: 18-Jul-2019

Complete List of Authors: Rashno, Emad; Amirkabir University of Technology, civil and Enviornmental Eng.Zarrati, A.R.; Amirkabir University of Technology, Dept. of Civil EngineeringKarimaei Tabarestani, Mojtaba; Shahid Rajaee Teacher Training University,

Keyword: Bridge, pier group, local scour, stable riprap size, riprap extent

Is the invited manuscript for consideration in a Special

Issue? :Not applicable (regular submission)

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Canadian Journal of Civil Engineering

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1 Design of riprap for bridge pier groups

2 Emad Rashno1, Amir Reza Zarrati2 (IAHR Member), Mojtaba Karimaei Tabarestani3*

34 1- Graduated MSc Student, Department of Civil and Environmental Engineering, 5 Amirkabir University of Technology, Tehran, Iran. Email: [email protected] 2- Professor, Department of Civil and Environmental Engineering, Amirkabir University of 8 Technology, Tehran, Iran. Email: [email protected]

10 3- Assistant Professor, Department of Civil Engineering, Shahid Rajaee Teacher Training 11 University, Tehran, Iran. Email: [email protected] *Corresponding Author

13 Abstract: The stable riprap size and the optimized extension of the riprap layer around

14 double and triple piers along the flow direction are studied experimentally. Results showed

15 that the critical riprap failure area and stable riprap size around the first pier remain

16 unchanged with increasing pier spacing. In addition, the largest stable riprap should always

17 be placed in front of the first pier in comparison to the remaining downstream piers.

18 However, by increasing the pier spacing, stable riprap size around the second and third pier

19 increased and approached that around the first pier. A relationship was developed for

20 designing stable riprap size in pier groups. Based on this relationship different riprap sizes are

21 suggested for different zones around the pier group. Experiments showed that the critical

22 zone around the piers only includes a small area and the rest of the riprap extent area can be

23 protected with smaller riprap stones.

24 Keywords: Bridge, pier group, local scour, stable riprap size, riprap extent

25 Introduction

26 The need for bridges with wider decks for serving large volumes of traffic resulted from ever

27 increasing economic growth is well evident (Wang et al. 2016a). In order to construct wide

28 bridges, single piers are replaced by pier groups. The flow forces applied to piers, flow

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29 pattern, and local scouring are different for pier groups, as compared to those single piers

30 (Ataie-Ashtiani and Aslani- Kordkandi, 2012; Wang et al., 2016a; Galan et al., 2019).

31 In the case of two piers along the flow direction, when the piers are adequately close to each

32 other, a more complicated flow pattern develops compared to that of a single pier. When flow

33 in a channel approaches a pier, a stagnation pressure is established on the pier face. As a

34 result of the pressure gradient over the pier face, a down-flow forms that is then diverted to

35 pier sides, forming the so-called horseshoe vortex (Graf and Istiarto 2002; Wang et al. 2016a;

36 Karimaei Tabarestani and Zarrati 2017 and 2019a). The horseshoe vortex and down-flow in

37 pier groups are similarly developed as for a single pier. Due to the differences in flow

38 patterns between a pier group and single pier, local scouring and bed erosion mechanisms are

39 also different. In pier groups aligned with the flow, the upstream pier protects the

40 downstream one, while, the downstream pier scour hole intensifies scouring of the upstream

41 pier. In addition, wake vortices shed from the upstream pier, increase scouring of the

42 downstream pier (Richardson and Davis 1995; Ataie-Ashtiani and Aslani-Kordkandi 2012).

43 There are some recent studies in the literature on determining scour depth around group piers.

44 Amini and Solimani (2018) studied the scour depth at pile groups with various pile spacing

45 and arrangements experimentally. They found that the pile spacing variation in-line with the

46 flow has a minor effect on the scour depth and pile spacing perpendicular to the flow gave the

47 largest scour depths. Keshavarzi et al. (2018) investigated the effect of spacing between two

48 piers aligned in the flow direction on the maximum scour depth experimentally. Their results

49 showed that the maximum scour depth at upstream of the front pier occurs when the spacing

50 between the two piers (center-to-center) is 2.5 times the diameter of the pier. Bateni et al.

51 (2019) presented genetic expression programming and multivariate adaptive regression

52 splines to estimate clear-water local scour depth at pile groups using the flow, sediment, and

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53 pile characteristics. Other studies in this field were carried out by Golan et al. (2018); Liang

54 et al. (2018); and Liu et al. (2018).

55 One of the common methods for scour protection at bridge piers is to use riprap around the

56 piers (Lauchlan and Melville 2001; Karimaei Tabarestani and Zarrati 2013 and 2019b).

57 Stable riprap size is the stone size for which the riprap remains stable under a particular

58 hydraulic condition. In stable riprap size studies, the riprap stability number, Nc, is a key

59 parameter (Parola 1993; Richardson and Davis 1995; Karimaei Tabarestani and Zarrati 2013

60 and 2015), which expresses the relationship between flow conditions and riprap properties as

61 defined below:

62 (1) 50

2

dgUN

s

oc

63 where U0 is undisturbed upstream depth-averaged flow velocity, ρ is fluid density, ρs is riprap

64 stone density, g is gravitational acceleration and d50 is median riprap grain size. Parolla

65 (1993) studied round-nosed rectangular as well as cylindrical piers to present a relationship

66 for stable riprap size in terms of Nc. Yoon et al. (1995) modified the Parolla’s method by

67 taking into account flow depth correction coefficients, bed material size, and pier size and

68 presented a new approach. Karimaei Tabarestani and Zarrati (2013) and Karimaei

69 Tabarestani et al. (2015) studied circular and round-nosed rectangular piers with different

70 aspect ratios and skew angles with and without a protective collar to propose a

71 comprehensive design method for stable riprap size. This relationship is expressed as follows:

72 (2) 432185.2 KKKKN c

73 where is a riprap size adjustment factor, where D is round nose rectangular 21501 DdK

74 pier width or circular pier diameter; is a factor to consider the effect of flow 41502 dyK

75 depth; is a factor to consider the effect of rectangular pier alignment, where 233 effDDK

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76 Deff is the projected length of the rectangular pier perpendicular to the flow direction and is 1

77 for a single circular pier; and K4 is a collar adjustment factor, which is equal to 1 when there is

78 no collar protection.

79 Full design of riprap requires designing three parameters: riprap layer thickness, riprap size,

80 and riprap extension area (Chiew and Lim 2000; Lauchlan and Melville 2001; Karimaei

81 Tabarestani and Zarrati 2013). Previous studies showed that 3d50 as the riprap layer thickness

82 can stabilize the riprap layer against winnowing failure (Chiew 1995; Mashahir et al. 2010;

83 Karimaei Tabarestani et al. 2015; Khademghaeiny et al. 2019).

84 On the other hand, considering a suitable extension can stabilize the riprap layer against edge

85 failure. Various researchers have proposed different shapes for riprap extension around a

86 single circular pier (Garde and Raju 2000). Mashahir et al. (2010) presented the necessary

87 riprap extension around rectangular piers. The riprap extension for a pier aligned with the

88 flow is shown in Fig. 1. In this figure, D is the width of the rectangular pier.

89 It will be shown later that such an extension can also be employed for cylindrical piers

90 located in a row. The literature review shows that no study has been reported focusing on the

91 stability of a riprap layer around cylindrical bridge pier groups. Therefore, the main aim of

92 this paper is to investigate stability of riprap around cylindrical pier groups and present an

93 appropriate method to design both a stable riprap size and extension. Since in practice, which

94 piers are not usually located in the transverse direction with a distance less than 10D, here we

95 only consider piers in the longitudinal direction.

96 Dimensional analysis

97 Considering studies by various researchers such as Parola (1993); Chiew and Lim (2000);

98 Karimaei Tabarestani et al. (2015) and the present study conditions, the parameters affecting

99 riprap stability around cylindrical bridge piers within a pier group can be written as follows:

100 (3) 0,,,,,,,,, 050 KgDGyUdf s

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101 where μ is the dynamic viscosity of the fluid, y is the flow depth, G is the pier spacing, D is

102 the diameter of the cylindrical pier, and Kθ is a coefficient of flow attack angle to the pier

103 group direction (effect of pier skewed angle). These variables include the parameters

104 affecting the flow and riprap characteristics as well as pier dimensions and spacing in the pier

105 groups.

106 Using Buckingham theorem and undertaking dimensional analysis on the variables of Eq. (3),

107 the following dimensionless parameters are obtained:

108 (4)

KDG

Dy

dDdU

Fdg

U s

s

,,,,,50

500

50

0

109 The parameter on the left hand side is riprap stability number (Nc). On the right hand side,

110 is the particle Reynolds number and its contribution to instability for rocks the 500 dU

111 size of riprap, which requires flow at high velocity and turbulence, is negligible (Parola,

112 1993). The next parameter represents the ratio of riprap grain density to flow density, and is

113 constant in the present study. Other parameters including , and are variables 50dD Dy DG

114 in the present study and denote the effects of riprap size, flow depth and pier spacing,

115 respectively. As the direction of the pier group in the present study is along the flow, the final

116 parameter Kθ is also constant.

117 Experimental setup

118 Experiments within the present study were carried out in a horizental flume, 10 m long, 0.73

119 m wide, and 0.60 m deep. Fig. 2 shows a schematice of the experimental flume and its

120 different components. Different measures including perforated steel sheet, clay bricks and

121 steel guiding blades attached to the flume bed were used at the upstream part of the channel

122 to reduce the flow disturbances and produce a nearly uniform approaching flow. In addition,

123 velocity profiles measured by an ADV (Acoustic Doppler Velocitimeter) when the flume bed

124 was fixed showed that the flow was fully developed after 5 m from the flume inlet. The

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125 characteristics of the ADV was: frequency of sampling 25 Hz with a downlooking probe. In

126 all tests, signal to noise ratio of samples was larger than 15 and correlation factor was more

127 than 70% as recommended by the manufacturer. The time of sampling was 120 seconds.

128 The working section with 0.2 m depth and 2 m long was located 5 m downstream from the

129 flume inlet and was in form of a recess below its bed. The recess box was filled with uniform

130 sand, with median size of 0.7 mm and density of 2650 kg/m3. The pier models were installed

131 in the middle of the sediment recess. The tailwater elevation was controlled by a tailgate and

132 flow depth was measured by a point gauge with an accuracy of 0.1 mm. A rectangular sharp-

133 crested weir with a manometer was used to measure the flow discharge at the flume end.

134 Before each experiment, the sand bed was leveled and the flume was carefully filled with

135 water so as not to disturb the erodible bed. The threshold of bed material motion was checked

136 when the piers were not installed. To do this, for a particular flow depth, the discharge was

137 adjusted so that the bed material was at incipient motion by observation. Experiments showed

138 that at incipient of bed material motion, the flow intensity defined as FI = U0/Uc, where Uc is

139 the critical velocity of streambed material calculated from Oliveto and Hager (2002), was

140 0.892. In addition, in this condition the parameter u*/u*c was equal to 0.954, where u* is the

141 shear velocity calculated from the flow depth and energy slope (Sf) at the working section and

142 u*c is the critical shear velocity of streambed material found from the Shields diagram. Table

143 1 shows the values of the different parameters at incipient motion of bed material.

144 Pier groups were built in a linear arrangment including two and three piers in-line. The

145 diamater of each pier was D=40 mm (Fig. 2). This type of arrangment was chosen due to its

146 popularity in engineering design (Ataie-Ashtiani and Beheshti 2006; Wang et al. 2016a and

147 2016b; Rashno et al. 2017). Three typical pier spacings equal to 3D, 4D and 5D were studied

148 in the present work.

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149 Three different materials with median size of d50 =3.26 mm (S1), 6.87 mm (S2), and 9.07 mm

150 (S3) were used as riprap stones, with D/d50 in a range of 4.4 to 12.3. Table 2 summarizes the

151 properties of these materials. To avoid winnowing failure, the riprap layer thickness was

152 selected as 3d50 (Chiew 1995). In addition, a screen with a sieve size of 0.5 mm was used

153 between the bed and riprap material as a filter.

154 The present study was divided into two stages. In the first stage, in order to determine stable

155 riprap size, riprap material was spread all over the whole erodible surface to prevent edge

156 failure of the riprap layer. In this way, it was possible to test high discharges without the

157 danger of finer bed material being washed away. In each experiment, the stable riprap size to

158 prevent shear failure around each pier in the pier group was determined. Lachlan and

159 Melville (2001) listed various criteria for shear failure of riprap stones. In the present work, it

160 was observed that movement of a few stones from a certain place eventually led to movement

161 of more and more stones around that area. Therefore, the movement of at least 5 riprap stones

162 in 15 min was considered as the failure in the present work.

163 In each experiment, for a known discharge, the tailwater depth was fixed for 15 min, and if

164 riprap stones did not move, the depth was decreased gradually by approximately 5 mm and

165 the experiment continued for another 15 min. This procedure continued until instability

166 (shear failure) in the riprap layer was observed around the first, and subsequently the second

167 and third (if available) pier in the pier group. Therefore, in each experiment, with a particular

168 flow discharge (Q) and riprap size (S1 to S3), the downstream flow depths for riprap

169 instability around each pier (Hi is for i-th pier) were recorded. Tables 3 and 4 show the range

170 of the effective parameters in the first stage of the present study. More than 50 experiments

171 were conducted to investigate the stable riprap size around pier groups including 2 or 3 piers

172 in linear arrangement. As is shown in these tables, the range of the flow intensity parameter

173 in the present study was 0.89≤FI≤2.01.

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174 In the second stage of experiments, the extension of the riprap layer was determined. In

175 addition, many tests were conducted to examine possibility of using smaller riprap for areas

176 away from higher flow forces. Since, a circular pier group arrangement covers a rectangular

177 area, the extension of riprap suggested by Mashahir et al. (2010) for riprap layer around a

178 rectangular pier (Fig. 1) was employed and tested. The pier group was assumed similar to a

179 rectangular shape pier with a length equal to the distance between the front side of the first

180 pier and trailing side of the last pier, and a width equal to the pier diameter. To ensure the

181 stability of the riprap layer in this stage, experiments were conducted for 10 hours at the near

182 threshold of bed material motion (FI=0.892). After 10 hours if no riprap stone was removed

183 and the scour hole around the riprap layer was less than one riprap size, that layer was

184 considered as stable with no shear or edge failure. It should be noticed that only a small area

185 around each pier was exposed to maximum flow forces, while smaller riprap size can be

186 placed elsewhere. On this basis, first the smallest riprap (S1) was placed within the entire

187 extent area as shown in Fig. 1. In the next step, the failed and damaged areas were replaced

188 by coarser riprap S3 (the stable riprap size as will be explained in the following sections). The

189 last step involved optimizing the extension of coarser riprap area following a trial-and-error

190 approach.

191 Results of Experiments and Analysis

192 Critical Regions of Riprap Failure around a Pier

193 Owing to different conditions in each test, the failure of the riprap layer occurred at different

194 critical regions around each pier. The critical region is exposed to maximum flow forces

195 around a pier and the first movement of stones or instability of riprap layer occurs in this

196 region (Karimaei Tabarestani and Zarrati 2013 and 2015). It was observed that the location of

197 critical regions around each pier depends on the relative pier spacing parameter (G/D). Based

198 on experimental observations, for the first pier similar to a single pier which was presented by

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199 Karimaei Tabarestani and Zarrati (2013), the critical region is at an angle of 45° to the flow

200 direction and remains constant by increasing the pier spacing (Fig. 3). The movement of

201 riprap stones at this location was under the action of flow forces at the high-shear stress zone

202 of diverging flow at the upstream side of the pier. Observations also indicated that by

203 increasing the pier spacing, critical regions around the downstream piers shifted from the

204 sides of the pier toward upstream side of the pier and became similar to the first pier, i.e. at

205 45° to the flow direction. Critical regions around the piers will be discussed in more details in

206 the following sections.

207 Stable riprap Size

208 Tables 3 and 4 present a summary of the experimental results obtained for stable riprap size in

209 2 piers and 3 piers arrangements at different pier spacing and different flow intensities. Present

210 experimental results indicate that the Karimaei Tabarestani and Zarrati (2013) and Karimaei

211 Tabarestani et al. (2015) equation (Eq. 2) is in good agreement with the experimental data for

212 stable riprap size at the first pier. Therefore, the riprap size extracted from Eq. (2) was used for

213 the critical area of the upstream pier. Tables 3 and 4 further contain the values of stability

214 number (Nci is for i-th pier), the ratio of and which are the ratio 122 ccp NNK 133 ccp NNK

215 of riprap stability number of the second and third pier in the group to that of the first pier, as

216 well as flow intensity parameter (FIi is for i-th pier) in the riprap failure condition.

217 Results showed that the variation of and is not considerable with FI and so an average 2PK 3PK

218 value denoted by and was calculated for them in each FI (see Tables 3 and 4). Fig. 4 2PK 3PK

219 and 5 show the variations of and for different pier spacing and riprap stone sizes. 2PK 3PK

220 Accordingly, with increasing the riprap size and relative pier spacing parameter (G/D), 2PK

221 and decrease towards 1 (similar stable riprap size for all piers). For example, for G = 3D, 3PK

222 by increasing d50 from 3.26 mm to 9.07 mm (about 2.8 times), and decrease for 2PK 3PK

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223 about 28% and 32%, respectively (stable riprap size of piers becomes closer to each other). In

224 fact, high shear forces at the two sides of all of the piers is the main flow structure to move

225 larger riprap stones (affected less by the upstream piers). On the other hand, a combination of

226 shear flow forces and wake vortices destabilize smaller riprap stones around downstream

227 piers. It can also be concluded that at larger G/D with the reduction of the protective effect,

228 downstream piers are exposed to direct attack of the flow, thereby tending toward further

229 independence. Therefore, the riprap stability number around the second or third pier (Nc2 or

230 Nc3) approaches that around the first pier (Nc1).

231 To calculate stable riprap size for downstream piers, the same equation as Eq. (2) was used

232 with adding a correction factor (K5) for the second and third piers. For the second pier

233 and for the third pier . This correction factor was determined by 25 pKK 35 pKK

234 nonlinearly curve fitting to the data reported in Tables 3 and 4 in a manner that all designs are

235 in the safe margin. Therefore, Eq. (2) can be rewritten as:

236 (5) 5432185.2 KKKKKN c

237 where K5 is pier group adjustment factor which can be calculated as:

238 (6)

pierThirddy

dD

DG

pierSecondDy

dD

DG

pierFirst

K

049.0

50

345.0

50

176.0

035.0259.0

50

225.0

5

982.0

086.1

1

239 Fig. 6 shows the comparison between results of Eq. (6) and the corresponding experimental

240 data. As is shown in this figure all of the data points are in upper side of design line (safe

241 margin) which shows the acceptable accuracy of Eq. (6). The validity range of Eq. (6) is for

242 , and .5.125.4 50 dD 53 DG 42 Dy

243 Extent of Riprap Layer around the Piers in the pier group

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244 As mentioned before, an extension similar to a rectangular pier was considered for the

245 circular pier group (Fig. 1). In this stage, the optimal configuration of the riprap extent layer

246 with different sizes around a pier group was obtained following a trial-and-error approach.

247 Fig. 7 shows the optimal riprap extent around the first, the second and the third pier (if

248 available) in different pier spacing. The optimal configuration of the riprap extent includes

249 critical regions with coarser riprap size (Zone #1) and regions which can be covered with the

250 finer tested riprap size (Zone #2). The critical region for the first pier in all conditions is the

251 same. For either of second or third pier (if available), the critical regions can be assumed to

252 be two 90-degree sectors of a circle with a diameter of 3D on either side of the pier. The

253 degree between upstream boundary of critical regions and flow direction was 45°, 30°, and

254 15° when the pier spacing was 3D, 4D, and 5D, respectively. To make the construction easier

255 and safer, we suggest a full circle for the riprap layer with a diameter of 3D for all piers

256 spacing around the second and third pier (if available) (Fig. 7). Experiments showed that 𝑁𝐶

257 for Zone #2 could be 2.5 times greater than for Zone #1 (65% decrease in riprap size). 𝑁𝐶

258 In the case of 3 piers layout and G = 4D, for instance, from the whole area of riprap extent

259 which is about 84 times the pier cross section area, only about 24% is critical (Zone #1) (in

260 full circle form) and should be covered with the stable riprap size calculated from eq. (5) and

261 the remaining 76% area (Zone #2) can be covered with the fine-grained riprap which can be

262 about 65% finer than designed riprap around the first pier. Area ratio of Zone #1 and 2 for all

263 tested spacing is given in Table 5.

264 Summary and Conclusions

265 In the present work, stable riprap extent and size around circular piers in a pier group with 2

266 and 3 piers aligned with the flow direction were studied experimentally. More than 50 sets of

267 experiments were conducted with different spacing, flow intensities and riprap stone sizes.

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268 In the first stage, stable riprap size around each pier in a pier group was studied. Results

269 showed that stable riprap size at the first pier is similar to the value presented by Karimaei

270 Tabarestani and Zarrati (2013) method. However, due to the protective effect of the upstream

271 pier, the stable riprap size around the downstream piers can be reduced. In addition, when the

272 distance between piers increased, the stable riprap size around downstream piers increased

273 and tended to the same size with stable riprap size around first pier. Based on the

274 experimental data, a new correction factor was employed for the Karimaei Tabarestani and

275 Zarrati (2013) and Karimaei Tabarestani et al. (2015) method for the design of stable riprap

276 around downstream piers.

277 In the second stage, the necessary riprap extension around the pier group was studied. The

278 critical regions for riprap failure around each pier in the pier group were detected. The results

279 showed that the riprap sizes calculated by design equation are only needed in small region

280 around each pier (the critical region).

281 References

282 Amini, A. & Solaimani, N. 2018. The effects of uniform and nonuniform pile spacing

283 variations on local scour at pile groups. Marine Georesources & Geotechnology, 36(7):

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285 Ataie-Ashtiani, B., & Beheshti, A. A. 2006. Experimental investigation of clear-water local

286 scour at pile groups. Journal of Hydraulic Engineering, 132(10): 1100-1104.

287 Ataie-Ashtiani, B., & Aslani-Kordkandi, A. 2012. Flow field around side-by-side piers with

288 and without a scour hole. European Journal of Mechanics-B/Fluids, 36: 152-166.

289 Bateni, S. M., Vosoughifar, H. R., Truce, B. & Jeng, D. S. 2019. Estimation of clear-water

290 local scour at pile groups using genetic expression programming and multivariate

291 adaptive regression splines. Journal of Waterway, Port, Coastal, Ocean Engineering.

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293 Chiew, Y. M. 1995. Mechanics of riprap failure at bridge piers. Journal of Hydraulic

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344 around three piers in a tandem arrangement. Qualitative analysis and pressure

345 measurements. China Technological Sciences, 59(6): 888-896.

346 Wang, H., Tang, H., Liu, Q., & Wang, Y. 2016 b. Local scouring around twin bridge piers in

347 open channel flows. Journal of Hydraulic Engineering, 142(9): 06016008.

348 Yoon, T. H., Yoon, S. B., & Yoon, K. S. 1995. Design of riprap for scour protection around

349 bridge piers. In Proceedings of the International Association for Hydraulic Research (Vol.

350 1, pp. 105-110). Local Organization Committee of the XXV Congress.

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373 Figures Captions

374 Fig. 1. Extent of riprap layer for rectangular bridge piers from Mashahir et al. (2010) .

375 Fig. 2. Schematic view of experimental laboratory and pier group arrangement in the present

376 study.

377 Fig. 3. Schematic position of critical regions in different pier spacing (a) G=3D, (b) G=4D,

378 (c) G=5D.

379 Fig. 4. Value of for different pier spacing and different riprap stone size.2PK


381 Fig. 6. Comparison between Eq. 6 and experimental data.

382 Fig. 7. Details of optimum extent of riprap layer for different distance between piers: (a) 2

383 piers layout (b) 3 piers layout.

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397 Fig. 1. Extent of riprap layer for rectangular bridge piers from Mashahir et al. (2010) .

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416 Fig. 2. Schematic view of experimental laboratory and pier group arrangement in the present

417 study.418

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435 Fig. 3. Schematic position of critical regions in different pier spacing (a) G=3D, (b) G=4D,

436 (c) G=5D.

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490 Fig. 6. Comparison between Eq. 6 and experimental data.

491492

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505 Fig. 7. Details of optimum extent of riprap layer for different distance between piers: (a) 2

506 piers layout (b) 3 piers layout.

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517 Table 1. The values of different parameters at incipient motion of the bed materialParameter y (m) U0 (m/s) Uc (m/s) Sf u* (m/s) u*c (m/s) U0/Uc u*/u*c

Value 0.13 0.316 0.354 0.0003 0.0177 0.0186 0.892 0.954518

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542 Table 2. Characteristic of the riprap and bed material.

Material d50 (mm) σg

Bed material 0.71 1.29S1 3.26 1.27S2 6.87 1.31S3 9.07 1.22

543544

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565 Table 3. Result of Experiments for 2 piers layout.

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TestNo. G d50

Q (m3.s-1)

H1 (mm) FI1

H2(mm) FI2 Nc1 Nc2 Kp2 2PK

1 3D S1 0.0302 94 1.25 83 1.44 3.33 4.80 1.442 3D S1 0.0331 109 1.16 89 1.46 3.24 4.94 1.523 3D S1 0.0365 119 1.15 95 1.49 3.34 5.24 1.56

1.51

4 3D S2 0.039 117 1.26 98 1.54 1.89 2.18 1.155 3D S2 0.0442 134 1.22 111 1.52 1.68 1.89 1.126 3D S2 0.049 152 1.18 121 1.53 1.47 1.82 1.23

1.17

7 3D S3 0.04 91 1.72 86 1.83 2.46 2.76 1.128 3D S3 0.047 110 1.63 105 1.72 2.32 2.56 1.099 3D S3 0.054 125 1.62 121 1.68 2.38 2.54 1.06

1.09

10 4D S1 0.03 95 1.22 84 1.41 3.23 4.48 1.3811 4D S1 0.033 105 1.2 90 1.43 3.22 4.56 1.4112 4D S1 0.036 113 1.21 100 1.39 3.30 4.70 1.42

1.40

13 4D S2 0.0391 116 1.27 104 1.44 1.89 2.12 1.1214 4D S2 0.0442 134 1.22 114 1.47 1.70 1.81 1.0615 4D S2 0.0489 151 1.18 130 1.4 1.46 1.66 1.13

1.10

16 4D S3 0.0398 91 1.71 87 1.8 2.45 2.67 1.0917 4D S3 0.0471 109 1.65 106 1.7 2.34 2.51 1.0718 4D S3 0.0539 123 1.65 121 1.68 2.33 2.53 1.08

1.08

19 5D S1 0.03 93 1.28 88 1.34 3.20 4.13 1.2920 5D S1 0.033 102 1.24 94 1.37 3.30 4.10 1.2421 5D S1 0.036 110 1.25 101 1.37 3.36 4.09 1.21

1.25

22 5D S2 0.0382 115 1.26 105 1.39 1.79 1.88 1.0523 5D S2 0.0441 130 1.27 120 1.39 1.87 1.99 1.0624 5D S2 0.0492 146 1.24 132 1.39 1.50 1.63 1.08

1.06

25 5D S3 0.0402 93 1.68 89 1.77 2.49 2.60 1.0526 5D S3 0.0468 106 1.69 106 1.69 2.35 2.47 1.0427 5D S3 0.0539 123 1.65 118 1.73 2.33 2.45 1.04

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576 Table 4. Results of experiments for 3 piers layout.

TestNo. G d50

Q m3.s-1

H1(mm) FI1

H2(mm) FI2

H3(mm) FI3 Nc1 Nc2 Nc3 Kp2 2PK Kp3 3pK

28 3D S1 0.0301 94 1.24 82 1.45 - - 3.20 4.76 - 1.48 -29 3D S1 0.0331 102 1.25 88 1.48 82 1.6 3.26 5.03 5.75 1.54 1.7630 3D S1 0.0362 111 1.24 96 1.46 89 1.59 3.30 5.05 5.88 1.53 1.52 1.78 1.7731 3D S2 0.0394 115 1.3 100 1.52 92 1.67 2.00 2.15 2.45 1.07 1.2232 3D S2 0.044 129 1.27 110 1.52 100 1.7 1.87 1.98 2.28 1.06 1.2233 3D S2 0.049 144 1.25 120 1.54 111 1.68 1.46 1.68 2.03 1.14 1.09 1.38 1.2734 3D S3 0.041 94 1.7 87 1.85 81 2.01 2.65 2.97 3.27 1.12 1.2335 3D S3 0.0471 108 1.67 103 1.76 93 1.97 2.43 2.67 2.97 1.09 1.2236 3D S3 0.0543 123 1.66 115 1.79 105 1.98 2.49 2.66 2.93 1.06 1.09 1.17 1.2137 4D S1 0.0299 95 1.22 84 1.4 - - 3.26 4.50 - 1.38 -38 4D S1 0.033 104 1.22 93 1.38 84 1.55 3.24 4.57 5.23 1.41 1.6139 4D S1 0.0361 119 1.15 100 1.39 93 1.53 3.27 4.92 5.47 1.50 1.43 1.67 1.6440 4D S2 0.0384 114 1.3 100 1.48 92 1.63 1.78 1.97 2.20 1.10 1.2341 4D S2 0.043 124 1.3 110 1.49 102 1.62 1.66 1.78 2.11 1.07 1.2742 4D S2 0.0496 143 1.28 128 1.45 114 1.63 1.56 1.75 1.97 1.11 1.09 1.26 1.2543 4D S3 0.0401 93 1.68 90 1.74 83 1.91 2.48 2.71 2.98 1.09 1.2044 4D S3 0.047 108 1.66 105 1.72 100 1.81 2.33 2.50 2.82 1.07 1.2145 4D S3 0.054 124 1.63 121 1.68 108 1.91 2.42 2.54 2.91 1.05 1.07 1.20 1.2046 5D S1 0.03 95 1.23 88 1.34 81 1.47 3.23 4.11 4.87 1.27 1.5047 5D S1 0.0331 103 1.24 95 1.35 86 1.52 3.27 4.31 5.14 1.31 1.5648 5D S1 0.036 113 1.21 102 1.36 93 1.51 3.24 4.42 5.32 1.36 1.31 1.64 1.5749 5D S2 0.0391 115 1.29 105 1.43 97 1.56 1.94 2.08 2.32 1.07 1.1950 5D S2 0.0442 130 1.27 120 1.39 110 1.53 1.88 1.94 2.13 1.02 1.1351 5D S2 0.0493 140 1.3 130 1.42 120 1.55 1.52 1.68 1.84 1.10 1.06 1.21 1.1852 5D S3 0.0391 91 1.68 88 1.74 85 1.82 2.21 2.41 2.58 1.09 1.1653 5D S3 0.0473 110 1.64 104 1.75 101 1.8 2.36 2.43 2.64 1.02 1.1154 5D S3 0.0533 122 1.64 117 1.72 113 1.79 2.28 2.36 2.60 1.03 1.05 1.14 1.14

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589 Table 5. Details of riprap layer extension area

Area ratio of riprap extent (%)Layout G

Zone #1 Zone #23D 23 774D 20 802 piers5D 18 823D 28 724D 24 763 piers5D 21 79

590

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design of riprap for bridge pier groups · 2020. 3. 11. · draft 1 1 design of riprap for bridge...

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