response spectrum-based seismic response of bridge

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Response Spectrum-Based Seismic Response of Bridge Embankments

Journal: Canadian Geotechnical Journal

Manuscript ID cgj-2018-0674.R3

Manuscript Type: Article

Date Submitted by the Author: 19-Nov-2019

Complete List of Authors: Carvajal, Juan-Carlos; Thurber Engineering Ltd Vancouver OfficeFinn, W.D. Liam; The University of British Columbia, Civil EngineeringVentura, Carlos; The University of British Columbia, Civil Engineering

Keyword: embankment, seismic, response spectrum, fundamental period, shear strain

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Canadian Geotechnical Journal

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Response Spectrum-Based Seismic Response of Bridge Embankments 1

2

Juan-Carlos Carvajal, William D. Liam Finn, and Carlos Estuardo Ventura. 3

4

Juan-Carlos Carvajal. Thurber Engineering Ltd., 900 - 1281 West Georgia Street, Vancouver, BC 5

V6E 3J7, Canada. (email: [email protected]) 6

7

William D. Liam Finn. Department of Civil Engineering, The University of British Columbia, 6250 8

Applied Science Lane, Vancouver, BC V6T 1Z4, Canada. (email: [email protected]) 9

10

Carlos Estuardo Ventura. Department of Civil Engineering, The University of British Columbia, 11

6250 Applied Science Lane, Vancouver, BC V6T 1Z4, Canada. (email: [email protected]) 12

13

Corresponding author: 14

Juan-Carlos Carvajal 15

2505 - 2020 Haro Street, Vancouver, BC, V6G 1J3, Canada 16

phone: +1 604 600 9889 17

email: [email protected] 18

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Abstract: A single degree of freedom model is presented for calculating the free-field seismic response 19

of bridge embankments due to horizontal ground shaking using equivalent linear analysis and a design 20

response spectrum. The shear wave velocity profile, base flexibility, 2D shape and damping ratio of the 21

embankment are accounted for in the model. A step-by-step procedure is presented for calculating the 22

effective cyclic shear strain of the embankment, equivalent homogeneous shear modulus and damping 23

ratio, fundamental period of vibration, peak crest acceleration, peak shear stress profile, peak shear strain 24

profile, equivalent linear shear modulus profile and peak relative displacement profile. Model calibration 25

and verification of the proposed procedure is carried out with linear, equivalent linear and nonlinear finite 26

element analysis for embankments with fundamental periods of vibration between 0.1 s and 1.0 s. The 27

proposed model is simple, rational and suitable for practical implementation using spreadsheets for a 28

preliminary design phase of bridge embankments. 29

30

Keywords: embankment, seismic, response spectrum, fundamental period, shear strain.31

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Introduction 32

Strong-motion earthquake data and numerical studies have shown that the free-field seismic response of 33

approach embankments has a significant effect on the response of Integral Abutment Bridges (IAB) due 34

to the lack of expansion joints in the structure, the flexibility of the abutment foundations and the 35

embankment-abutment-structure interaction (Wilson and Tan 1990a; Zang and Makris 2002a; Inel and 36

Aschhim 2004; Kotsoglou and Pantazopoulou 2007, 2009 and Carvajal 2011). Therefore, accurate 37

modeling of the approach embankments is of significant importance for estimating seismic demands in 38

IABs. In this context, the term “free field” refers to the lack of any influence from structural vibrations 39

on the embankment response. 40

41

Early studies for modeling the seismic response of embankments focused on earth dams. Mononobe et 42

al. (1936), Ambraseys (1960) and Seed and Martin (1966) used shear beam theory to calculate periods 43

and mode shapes of vibration of linear, homogeneous, triangular embankments supported on a rigid base. 44

Subsequently, Makdisi and Seed (1977) included the effect of ground motion intensity and nonlinear soil 45

response by means of equivalent linear analysis. The studies concluded the dynamic response of 46

embankment dams is controlled by the characteristics of the base motion, the fundamental period of 47

vibration and the damping ratio. 48

49

The dynamic response of bridge embankments differs from that of embankment dams because of the 50

trapezoidal shape given by the crest width. Wilson and Tan (1990b) calculated the seismic response of 51

homogenous bridge embankments using a single degree of freedom system and linear analysis. The 52

model is simple and agreed reasonably well with the recorded seismic responses. However, its 53

application was limited to the identification of the fundamental period of vibration of embankments 54

instrumented with strong motion sensors. 55

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56

Zhang and Makris (2002b) formulated a multi-modal, equivalent linear model for calculating the seismic 57

response of the embankment crest. The method was validated with strong-motion earthquake records 58

from instrumented embankments and with finite element analysis. The validation indicated the seismic 59

response is strongly controlled by the fundamental period of vibration of the embankment. The model is 60

rational and accurate; however, it requires implementation in a computer program and selection of 61

earthquake records. 62

63

Despite the availability of models for calculating the seismic response of bridge embankments, their 64

implementation is not suitable for practical design procedures because of the complexity of the solutions. 65

In addition, the models are developed for homogenous embankments supported on a rigid base, which 66

represent a small fraction of the cases found in practice. 67

68

This paper presents the analytical development, calibration and verification of a simple model for 69

calculating the free-field seismic response of bridge embankments using shear beam theory and 70

equivalent linear analysis. The model represents the general case of a 2D embankment with a linear shear 71

wave velocity profile supported on a flexible base. The seismic response is simplified using a single 72

degree of freedom system and a design response spectrum. The application of the model is demonstrated 73

with three verification cases. 74

75

The proposed model incorporates the advantages and simplifications of the available bridge embankment 76

models and solves their limitations for easy implementation in practical design procedures. The proposed 77

methodology is an empirical procedure suitable for a preliminary design phase of bridge embankments. 78

79

Proposed Procedure 80

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The proposed procedure uses the response spectrum method applied to a simplified model to approximate 81

the seismic response of bridge embankments. The procedure is divided in two parts: a) calculation of the 82

equivalent linear properties and b) calculation of the peak response quantities. 83

84

The calculation of the equivalent linear properties of the model is described first. A six-step procedure 85

for calculating the equivalent linear shear modulus and damping ratio is developed in this part. Then, a 86

two-step procedure is developed for calculating corrections factors to improve the estimation of the peak 87

response quantities of interest for practical analysis using the proposed model. 88

89

The justification for the development of the different components of the model are discussed in the 90

following sections and the complete step-by-step procedure is summarized. The assumptions in the 91

proposed model and its limitations are also discussed. 92

93

Discretization of the Equation of Motion 94

The seismic response of approach embankments in the free field is similar to that of soil deposits. Hence, 95

the differential equation of motion of a uniform 1D shear beam can be used as an approximation to 96

represent its dynamic response (Eq. 1 and Fig. 1). 97

98

(1) ( ) ( )g(t) (z,t) (z) (z,t)A u u A G u ' ' 0ρ + − = 99

100

In this formulation, z is the vertical coordinate system, t is time, H is the embankment height, ρ is the 101

embankment density, A is the cross-section area, assumed uniform with z, ug(t) is the ground 102

displacement, üg(t) is the ground acceleration or input base motion, u(z,t) and ü(z,t) are the relative 103

displacement and relative acceleration of the embankment with respect to its base, respectively, ut(z,t) = 104

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ug(t) + u(z,t) is the total displacement, üt(z,t) = üg(t) + ü(z,t) is the total acceleration and G(z) is the shear 105

modulus profile. The symbol (') refers to first variation δ/δz. The viscous component of the shear stress 106

τ(z,t) is ignored in Eq. 1 for simplicity in the discretization but included later in the equation of motion of 107

a single degree of freedom system using the damping ratio ξ. 108

109

Equation 2 expresses the Galerkin formulation (Cook et al. 2001) for finding an approximate solution to 110

Eq. 1, where u(z,t) is a proposed solution for the relative displacement of the embankment that must satisfy 111

the boundary conditions of the physical model: a) zero relative displacement at the base u(0,t) = 0 and b) 112

zero shear stress at the crest τ(H,t) = G(H) u'(H,t) = 0. 113

114

(2) ( ) ( )H

g(t) (z,t) (z) (z,t) (z,t)0

A u u A G u ' ' u z 0 ρ + − δ = ∫

where (z,t) (H,t) (z)u u = ψ 115

116

u(H,t) is a dynamic degree of freedom that represents the relative displacement of the crest (z = H) and 117

depends only on the time t. ψ(z) is a dimensionless shape function with bounds ±1 that satisfies the 118

boundary conditions and depends only on the coordinate z. ψ(z) represents a mode shape of vibration of 119

the embankment. 120

121

Carvajal (2011) demonstrated by means of a modal contribution analysis that the fundamental mode 122

shape of vibration captures with a good level of accuracy the dynamic response of the embankment, 123

which is consistent with observations using strong-motion earthquake data from instrumented 124

embankments (Wilson and Tan 1990b; Zhang and Makris 2002b). 125

126

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Equation 3 represents the shape function ψ(z) associated with the first mode of vibration of a uniform and 127

homogeneous 1D shear beam. 128

129

(3) (z)z sin

2 Hπ ψ =

130

131

Using Eq. 3 in the solution of Eq. 2 results in Eq. 4, which is the equation of motion of an undamped 132

single degree of freedom (SDOF) system u(H,t), located at the crest of the embankment. 133

134

(4) (H,t) (H,t) g(t)M u K u = I u+ − 135

where 136

(5) H

2(z)

0

HM A z BL2ρ

= ρ ψ δ =∫ 137

(6) ( )H 2

eq(z) (z) (z)

0

GK A G ' ' z BL

8 Hπ

= − ψ ψ δ =∫ 138

and 139

(7) H

(z)0

2I A z BL H= ρ ψ δ = ρπ∫ 140

141

In these equations, M is the generalized mass, K is the generalized stiffness and I is the generalized load 142

factor associated with the fundamental mode of vibration. A = BL is the horizontal cross-section area 143

where B and L are crest width and length of the embankment in the transverse and longitudinal directions, 144

respectively, and assumed uniform with z. 145

146

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Equation 8 is an equivalent representation of Eq. 4, normalized with respect to M, for a damped SDOF 147

system, but expressed in terms of the fundamental period of vibration T1, the damping ratio ξeq and the 148

scaling factor of the input base motion Fm. 149

150

(8) 2

(H,t) eq (H,t) (H,t) m g(t)1 1

4 2u u u F uT T

π π+ ξ + = −

151

where 152

(9) ( )1 2Teq

T 4H FGρ

= 153

and 154

(10) ( )m 2C4F F F Fα ξ=π

155

156

Several modification factors were added on Eq. 9 and 10 to account for the 2D shape of the embankment 157

(F2T and F2C), the flexibility of the supporting base (Fα) and the damping ratio (Fξ). These are calibrated 158

in this paper using a combination of finite element analysis and analytical approximations. 159

160

Geq and ξeq are the equivalent linear shear modulus and damping ratio, respectively, that represent the 161

seismic response of an equivalent uniform, homogenous, 1D embankment. A procedure is presented in 162

this paper for calculating these properties. 163

164

Response Spectrum Analysis 165

Selection and scaling of earthquake ground motions üg(t) and solution of Eq. 8 is a time-consuming task. 166

Response spectrum analysis, on the other hand, is a cost-efficient method to estimate the peak demands 167

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for a given design earthquake expressed in terms of the spectral acceleration Sa, the period of vibration 168

T and the damping ratio ξ (Chopra 1999). 169

170

Figure 2 shows, for instance, the design response spectrum of a 2475-year return period earthquake on 171

firm ground. This response spectrum represents a solution of Eq. 8 for 0 s ≤ T1 ≤ 2 s, ξeq = 5% and Fm = 172

1. 173

174

Estimation of peak response quantities at any given z in the embankment are obtained using Sa, T1 and 175

ψ(z) (Carvajal 2011). The four peak quantities of interest for seismic analysis are: total crest acceleration 176

üt(H) (Eq. 11), relative displacement profile u(z) (Eq. 12), shear strain profile γ(z) = δu(z)/δz (Eq. 13) and 177

shear stress profile τ(z) = γ(z) Geq (Eq. 14). 178

179

(11) t(H) a m uu S F C≈

180

(12) 2

1(z) a m

T zu sin S F40 2 H

π ≈

181

(13) 2

1(z) a m

4T zcos S F (%)H 2 H

π γ ≈

182

(14) 2

1 eq(z) a m

T G zcos S F C25H 2 H τ

π τ ≈

183

184

The corrections factors Cü and Cτ have been added in the above equations to improve the estimation of 185

the response quantity. These are calibrated further in this paper using 1D equivalent linear analysis. 186

187

Period Modification Factor (F2T) 188

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Figure 3a illustrates a 2D embankment with crest width B, height H and side slope S:1V. The exact 189

solution for calculation of the fundamental period of vibration T2D in the transverse direction of the 190

embankment is in Zhang and Makris (2002b). The solution, however, is complex for implementation in 191

practical design procedures. 192

193

Wilson and Tan (1990b) formulated a simple model for calculating T2D by multiplying the fundamental 194

period of vibration of a uniform 1D embankment, T1D = 4H(ρ/Geq)1/2, with a dimensionless period 195

modification factor F2T as shown in Figure 3b. In this way, both 2D and equivalent uniform 1D 196

embankments have the same fundamental period of vibration: T2D = T1D F2T. Wilson and Tan calibrated 197

F2T only for the transverse direction of the embankment and for few values of the ratio B/H and S using 198

finite element analysis. 199

200

Figure 4 shows a 3D Finite Element (FE) model of an approach embankment used for calibration of F2T 201

in the transverse and the longitudinal direction. The geometric properties of the model are H = L = 10 m, 202

B/H = 0.1 to 16 and S = 1, 1.5, and 2. A database of fundamental periods of vibration T2D in each direction 203

were obtained with the computer program ABAQUS and normalized with respect to T1D. The 204

displacement of the FE model was restrained in the vertical direction to simulate an infinitely long 205

embankment (L/H >> 1) when calculating T2D in the longitudinal direction. 206

207

Figure 5 plots the F2T vs B/H data for the longitudinal and the transverse direction of the embankment. 208

The data show that the 2D shape shortens the period of vibration of the equivalent uniform 1D 209

embankment, 0.7 < F2T < 1, as B/H tends to 0. 210

211

The data in Fig. 5 also implies that the fundamental period of vibration of the embankment is shorter in 212

the transverse direction (Transv.) in comparison to that in the longitudinal direction (Long.). The 213

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difference, however, is minor: ∆F2T < 0.04. The effect of the slope S on F2T is very minor in each direction 214

with an average difference of ∆F2T < 0.01. Therefore, it is assumed for practical purposes that the 215

fundamental period of vibration is equal in both the longitudinal and the transverse directions of the 216

embankment and independent of the slope S. This is consistent with Zhang and Makris’s observations 217

using strong motion earthquake data from instrumented embankments (2002b) and Wilson and Tan’s 218

finite element analyses (1990b). 219

220

Equation 15 provides the proposed period modification factor F2T, obtained by curve fitting the FE data 221

in Fig. 5. 222

223

(15) 2T0.74 0.77 B/HF 1

1 0.75 B/H+

= ≤+

224

225

Crest Response Modification Factor (F2C) 226

The 2D shape of the embankment also modifies the relative crest displacement of the equivalent uniform 227

1D embankment. A simple approximation to quantify the effect is by recalculating the generalized mass 228

M and load factor I with the 2D cross-section area of the embankment A(z) = BL (1 + 2S(H-z)/B) using 229

Eq. 5 and 7 → F2C = (I/M) / (4/π), where I = ρ∫A(z) ψ(z) δz and M = ρ∫A(z) ψ2(z) δz with limits of integration 230

0 and H. 231

232

Equation 16 provides the proposed crest response modification factor, which considers the side slope S, 233

the crest width B and embankment height H. 234

235

(16) 2C0.75 S B/HF0.6 S B/H

+=

+ 236

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237

F2C is plotted in Fig. 6 for S = 1 to 3, which varies from 1.25 for B/H = 0 (triangular embankment) to 1 238

for B/H >> 1 (uniform 1D embankment). F2C increases with S; however, the difference between S = 1 239

and 3 is small (∆F2C < 0.07) and with negligible effect for B/H ≈ 0 and B/H >> 1. 240

241

Linear dynamic analyses were carried out with a 2D finite element model for verification of F2C. The 242

dimensions of the model are H = 10 m, B/H = 0.2 to 4 and S = 1 to 3. Soil density ρ = 2.04 ton/m3, shear 243

modulus G = 20.4 MPa, 5% Rayleigh damping and rigid base condition were assumed in the Finite 244

Element Analyses (FEA). The input base motions consisted of ten earthquake records spectrally matched 245

to the design response spectrum in Fig. 2. The peak relative crest displacement at the center of the model 246

was computed using the computer program PLAXIS 2D. 247

248

Peak relative displacement of the embankment crest was calculated with the proposed model using Eq. 249

12, 9 and 10 with z/H = 1, Fα = 1, Fξ = 1, Sa from Fig. 2, F2T with Eq. 15 and F2C with Eq. 16. 250

251

Figure 7 presents the ratio of the peak relative crest displacement calculated with the proposed model to 252

that obtained with FEA. 253

254

As shown in Figure 7, the proposed model tends to underestimate the crest response, especially for B/H 255

= 0.2 and S = 1. This is mainly the result of the assumed mode shape and the contribution of the higher 256

modes of vibration, which is not considered in F2C. However, the average accuracy of the proposed model 257

is about 0.95 and considered acceptable for design purposes. 258

259

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Peak total crest acceleration was also calculated with both models. The average accuracy of the proposed 260

model decreased to 0.88, which reflects the importance of the contribution of the higher modes of 261

vibration for calculating the acceleration in comparison to the relative displacement (Carvajal 2011). A 262

response correction factor Cü is developed in this paper for improving the estimation of the crest 263

acceleration with the proposed model. 264

265

The influence of the embankment length is not addressed in the calculation of the modification factors 266

F2T and F2C since it is assumed that the length is much greater than the height, L/H >> 1, which is a 267

realistic consideration for bridge embankments. 268

269

Initial Shear Modulus (Gini) 270

The initial seismic response of the embankment is controlled by the small-strain shear modulus profile 271

Gmax(z), usually obtained from the shear wave velocity profile Vs(z) using field testing. Vs(z) is controlled 272

by the soil type, stress history and effective stress state. These conditions generate shear wave velocity 273

profiles with a wide variety of shapes from constant or smooth curves to jagged and step-like graded 274

profiles. 275

276

For practical purposes and simplicity in the implementation of the proposed model, a linear variation of 277

Vs(z) is assumed for representing the initial condition of the embankment. The linearization of the in-situ 278

or estimated Vs profile can be carried out using curve fitting or engineering judgment. 279

280

The Vs(z) profile is calculated with Eq. 17, where Vs-top and Vs-bot are the shear wave velocities at the top 281

and at the bottom of the embankment, respectively. 282

283

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(17) [ ]s(z) s-top sV V 1 (rV 1)(1 z / H)= + − − where s s-bot s-toprV V / V= 284

285

The Vs(z) profile is defined by Vs-top and rVs ≥ 1. Figure 8 plots, for instance, the normalized Vs profiles 286

for the homogeneous case rVs = 1 and for rVs = 2. 287

288

The linear Vs(z) profile represents a quadratic Gmax(z) profile as indicated with Eq. 18, where Gtop is the 289

shear modulus at the top of the embankment. 290

291

(18) [ ]2max(z) top sG G 1 (rV 1)(1 z / H)= + − − where 2top s-topG V= ρ 292

293

The proposed model is a homogeneous-based approximation which requires a single shear modulus value 294

from the Gmax(z) profile to represent the initial dynamic response of the embankment. This modulus, called 295

Gini, is obtained by solving Eq. 6 with G(z) = Gmax(z), which results in a similar expression but with Geq = 296

Gini given with Eq. 19. 297

298

(19) [ ]2ini top s zG G 1 (rV 1) d= + − 299

300

The parameter dz in Eq. 19 represents the normalized effective depth (1 − z/H), measured from the top 301

of the embankment, at which Gmax(z) = Gini and it is approximately equal to 0.7. This value, however, can 302

generate errors up to about 10% in the calculation of the initial period of vibration for rVs ≤ 4. 303

304

In order to improve the estimation of dz, linear dynamic analyses were performed with the computer 305

program Shake2000 using the previous ten spectrally matched earthquake records. The analyses 306

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considered uniform 1D embankments with H = 7 m, 14 m, 28 m, rVs = 1 to 4, T1 = 0.1 s to 1.1 s, 5% 307

damping ratio and rigid base condition. 308

309

Equation 20 provides the calibrated dz parameter. The error in the calculation of the initial period of 310

vibration with the proposed expression is less than 0.2% for rVs = 1 to 4. 311

312

(20) z sd 0.696 0.067 Ln(rV ) 0.68= − ≤ 313

314

Base Modification Factor (Fα) 315

The preceding analyses assumed the embankment is supported on a rigid base. For the general case of a 316

flexible base, stress waves traveling downward in the embankment are partially transmitted to the base, 317

removing energy from the embankment. This is a form of radiation damping and causes a reduction of 318

Sa of the within spectrum in comparison to the rigid base case. In this context, the term “within” refers 319

to the condition in which the ground motion includes the sum of the incident waves and downward 320

propagating waves reflected from the embankment. 321

322

The relative flexibility of the base with respect to the embankment is calculated with the impedance ratio 323

α using Eq. 21 (Kramer 2006), where ρ and Vs-eq are the density and equivalent homogeneous shear wave 324

velocity of the embankment, respectively, and ρbase and Vs-base are the same properties defined previously 325

but for the flexible base. 326

327

(21) eqs-eq

base s-base base s-base

G VV V

ρρα = =

ρ ρ 328

329

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The effect of damping on α is very minor for ξ < 20% and has been ignored for simplicity. In general, 330

the impedance ratio is α < 1 since the shear wave velocity of the supporting base is greater than that of 331

the embankment and ρ ≈ ρbase. 332

333

Figure 9 shows the within response spectra at the base of the embankment for the rigid and flexible base 334

cases. The calculation considered linear dynamic analysis with H = 14 m, rVs = 2, Vs-eq = 112 m/s, T1 = 335

0.5 s, ξ = 5% and one of the earthquake records. The shear wave velocity and impedance ratio for the 336

flexible base case are Vs-base = 560 m/s and α = 0.2, respectively. 337

338

The within response spectrum for the rigid base case in Fig. 9 can be considered equal to the response 339

spectrum of the input ground motion. The response spectrum for the flexible base is a scaled-down 340

version of the rigid base case with a maximum deamplification at T = T1 = 0.5 s. 341

342

The deamplification of the rigid base response spectrum at T1 for a given α is quantified with the 343

impedance-based modification factor Fα defined with Eq. 22. For instance, Fα is about 0.6 at T = 0.5 s in 344

Fig. 9. 345

346

(22) 1

a-flexible

a-rigid ,T

SF 1Sα

α

= ≤

347

348

Linear dynamic analyses were performed to develop a database of α vs Fα values for correlation 349

purposes. The analysis considered the same cases and earthquake records used previously for calibration 350

of dz but with Vs-base = 360 m/s to 1500 m/s. Figure 10 plots the database for uniform 1D embankments 351

with α ≤ 1. 352

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353

Figure 10 shows that Fα reduces as α increases, which means that the deamplification of the rigid base 354

response spectrum at T = T1 increases as the base becomes more flexible: lower Vs-base. 355

356

The effect of H, rVs and T1 on Fα for a given α and T1 ≥ 0.2 s is small (∆Fα < 0.1) and well captured with 357

Vs-eq. For T1 = 0.1 s and α ≥ 0.17, the data follow a well defined upper bound. This is the result of the 25 358

Hz low-pass filter in the records which removes the frequency components for T < 0.04 s and leaves the 359

embankment vibrating mainly with the first mode of vibration (T1 = 0.1 s). 360

361

Equations 23 and 24, plotted in Fig. 10, provide the proposed correlation between α and Fα for α ≤ 1. 362

The upper bound of the data was selected for T1 ≥ 0.2 s as a conservative assumption. Calculation of Fα 363

for 0.1 s < T1 < 0.2 s and α > 0.17 is carried out with linear interpolation. 364

365

(23) 0.41F

1 1.05α = + α for T1 = 0.1 s and 0.17 ≤ α ≤ 1 366

(24) 0.8 0.21F

1 2α − α=+ α

for T1 ≥ 0.2 s and α ≤ 1 367

368

Effective Cyclic Shear Strain (γeff) 369

The nonlinear response of the embankment can be approximated with the equivalent linear method (Seed 370

and Idriss 1969), which is based on modulus reduction G/Gmax and damping ξ curves such as the ones 371

plotted in Fig. 11 for sand (Seed and Idriss 1970). Both curves depend on the effective cyclic shear strain 372

γeff mobilized in the soil. 373

374

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The initial condition of the embankment is characterized with Gini (Eq. 19). The degradation of Gini in 375

the proposed model is assumed to be proportional to the peak shear strain profile (Eq. 13), which has a 376

cosine distribution. 377

378

Equation 25 expresses the proposed degraded shear modulus profile Gdeg(z), where Gbot is the mobilized 379

shear modulus at the bottom of the embankment (z = 0) and DG is the maximum modulus degradation 380

ratio. Figure 12 plots, for instance, the normalized Gini and Gdeg profiles of an embankment with Gbot/Gini 381

= 0.4 and DG = 0.6. 382

383

(25) deg(z) ini GzG G 1 D cos

2 H π = −

where botG

ini

GD 1G

= − 384

385

A single shear modulus value from the Gdeg(z) profile is required to characterize the equivalent linear 386

dynamic response of the embankment with the proposed model. That modulus is Geq and it is obtained 387

by solving Eq. 6 with G(z) = Gdeg(z), which results in a similar expression but with Geq given with Eq. 26. 388

389

(26) eq ini G8G G 1 D

3 = − π

390

391

The coordinate z at which Gdeg(z) is equal to Geq is called the effective height zeff. This is obtained with 392

Eq. 27 using Eq. 25 and 26. 393

394

(27) effz 0.36H

≈ 395

396

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Figure 12 illustrates with a vertical dashed line the Geq profile calculated with Eq. 26 for DG = 0.6. The 397

intersection of Geq with Gdeg(z) is at z/H ≈ 0.36, indicated in the figure with a horizontal line. 398

399

The effective cyclic shear strain γeff of the embankment is calculated with Eq. 28 using Eq. 13 and 27. 400

401

(28) 2

1eff (zeff) a m

2.2T0.65 S F (%)H

γ = γ ≈ 402

403

The equivalent linear properties Geq and ξeq of the embankment are obtained from the G/Gmax and ξ 404

curves with γ = γeff as shown by Eq. 29 and 30. 405

406

(29) [ ]eq ini max effG G G Gγ

= 407

(30) [ ]eq effγξ = ξ 408

409

Damping Modification Factor (Fξ) 410

Calculation of the peak response of the embankment using equivalent linear analysis is strongly 411

controlled by damping. The effect of ξeq is included in the proposed model using the damping 412

modification factor Fξ provided with Eq. 31 (CAN/CSA 2014). 413

414

(31) 0.4

eq

oF

−

ξ

ξ = ξ

415

416

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20

In the above equation, ξo is the damping ratio of the design response spectrum, which is usually 5%. The 417

damping modification factor is Fξ ≥ 1 for ξeq ≤ ξo and Fξ ≤ 1 for ξeq ≥ ξo. In this way, the input design 418

response spectrum is scaled up or down depending on γeff and the associated ξeq. 419

420

Equivalent Linear Properties (Geq , ξeq) 421

The calculation of Geq and ξeq depend on γeff and vice versa. Therefore, an iterative procedure is 422

performed to ensure strain compatibility of the properties. The proposed procedure assumes that the 423

entire embankment is characterized by a single set of G/Gmax and ξ curves and it operates as follows: 424

425

Step 1. Specify input data H, B, S, ρ, Vs-top, rVs, G/Gmax and ξ curves, ρbase, Vs-base and design response 426

spectrum (Sa, T, ξo). 427

Step 2. Calculate F2T (Eq. 15), F2C (Eq. 16), Gtop (Eq. 18) and Gini (Eq. 19 and 20). 428

Step 3. Obtain G/Gmax and ξ from the soil curves for γeff = 1x10-4 %. 429

Step 4. Calculate Geq (Eq. 29), ξeq (Eq. 30), T1 (Eq. 9), α (Eq. 21), Fα (Eq. 23 and 24), Fξ (Eq. 31) and 430

Fm (Eq. 10). 431

Step 5. Obtain Sa from the design response spectrum for T1 and calculate γeff(i+1) (Eq. 28). 432

Step 6. Calculate the tolerance (tol) of the convergence with Eq. 32 and evaluate: if tol ≤ 5% → end of 433

procedure; if tol > 5% → repeat steps 3 to 6 with γeff = γeff(i+1). 434

435

(32) (i 1)

eff(i)

efftol 100 1 (%)

+γ= −

γ 436

437

Response Correction Factors (Cü , Cτ) 438

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The proposed model assumes that the fundamental mode shape of vibration of a uniform, homogenous 439

1D shear beam (Eq. 3) captures the seismic response of the embankment. This is a good approximation 440

for short period systems (i.e. T1 < 0.1 s) since the mobilized effective cyclic shear strain and the 441

associated degradation of the initial shear modulus are very small (γeff ∝ T12, see Eq. 28). Therefore, the 442

response of the embankment is almost elastic with a minor modification of the initial mode shape. 443

444

The equivalent linear response, however, modifies the initial mode shape of the embankment due to the 445

degradation of the shear modulus of the soil layers, especially as T1 increases from 0.1 s. In addition, the 446

contribution of the higher modes of vibration becomes important as T1 increases, especially for 447

calculating the peak total crest acceleration (Carvajal 2011). 448

449

In order to improve the accuracy of the proposed model, 1D equivalent linear analyses were performed 450

with the computer program Shake2000 for developing response correction factors. The parametric 451

analysis considered uniform 1D embankments with H = 7 m to 35 m, Vs-top = 150 m/s and 200 m/s, rVs 452

= 1.1 to 3, G/Gmax and ξ curves for sand (Seed and Idriss 1970) and clay with plasticity index of 15% 453

and 30% (Vucetic and Dobry 1991), Vs-base = 560 m/s and 1500 m/s, and the ten spectrally matched 454

earthquake records. The fundamental periods of the embankments varied from 0.1 s to 1 s. 455

456

The calibrated response correction factors for the peak total crest acceleration and peak shear stress 457

profiles are presented in Equations 33 and 34, respectively. 458

459

(33) [ ] 1u 1C A B Ln(T ) −= +

where sA 0.835 0.244 Ln(rV )= − and sB 0.115 0.05 Ln(rV )= − − 460

(34) [ ] 11C 1.25 0.38 T 1.05−

τ = − ≥ 461

462

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The correction factor Cü depends on the fundamental period of vibration (T1) and the distribution of the 463

shear wave velocity (rVs) in the embankment. It varies from Cü = 0.91 for T1 = 0.1 s and rVs = 1 to Cü = 464

1.76 for T1 = 1 s and rVs = 3. The increase in Cü with T1 and rVs is mainly due to the contribution of the 465

higher modes of vibration, which is important for calculating the total acceleration. 466

467

The correction factor for the shear stress profile Cτ depends only on the period of vibration (T1) and 468

varies from Cτ = 1.05 for T1 ≤ 0.7 s to Cτ = 1.15 for T1 = 1 s. 469

470

Peak Shear Strain Profile (γ(z)) 471

The 1D equivalent linear analyses indicated that the peak shear strain profile is not well captured with 472

Eq. 13 for rVs > 1.5. An improvement for estimating γ(z) is presented in Eq. 35, where τ(z) is the corrected 473

peak shear stress profile (Eq. 14), Cγ is the peak shear strain correction factor (Eq. 36) and G(z) is the 474

equivalent linear shear modulus profile, calculated as the product of Gmax (Eq. 18) and the proposed 475

dimensionless function FG(z) for estimating the degradation of Gmax with z (Eq. 37). 476

477

(35) (z) (z)(z)

(z) max(z) G(z)C C

G G Fγ γ

τ τγ = = 478

where 479

(36) 1

1

1 0.3 TC 1.051.22 0.66 Tγ

−= ≥

− 480

and 481

(37) G(z) G m2z 3 zF 1 D cos C cos

2 H 2 H π π = − −

where eqG

ini

G3D 18 G π

= −

482

483

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The response correction factor for the peak shear strain profile Cγ was calibrated with the previous 484

parametric analysis and depends only on the fundamental period of vibration of the embankment. It varies 485

from Cτ = 1.05 for T1 ≤ 0.7 s to Cγ = 1.25 for T1 = 1 s. 486

487

The dimensionless function FG(z) was derived from the Gdeg(z) profile (Eq. 25) and the addition of the 488

contribution of the second mode shape of vibration of the shear strain. Cm2 is the mean participation 489

factor of the second mode, calibrated with the parametric analysis. 490

491

Cm2 is calculated with Eq. 38 and it was calibrated by minimizing the difference between the equivalent 492

linear shear modulus profiles obtained with Shake2000 and the proposed model. 493

494

(38) 2m2 1 1C A B T C T 0= + + ≥ , where sA 0.045 0.312 Ln(rV )= − + , sB 0.425 0.123 Ln(rV )= − and 495

sC 0.256 0.038 Ln(rV )= − − 496

497

The participation factor Cm2 depends on the fundamental period of vibration (T1) and the distribution of 498

the shear wave velocity (rVs) in the embankment. It varies from Cm2 ≈ 0 for T1 = 0.1 s and rVs = 1 to Cm2 499

≈ 0.4 for T1 = 0.5 s and rVs = 3. 500

501

Seismic Response Procedure, Assumptions and Limitations 502

The calculation of the seismic response of the embankment with the proposed model is divided in two 503

main stages: a) calculation of the equivalent linear properties and b) calculation of the peak response 504

quantities. The complete step-by-step procedure is summarized as follows: 505

506

Steps 1 to 6. Calculate the equivalent linear properties (Geq, ξeq) with the previous procedure. 507

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24

Step 7. Calculate the response correction factors Cü (Eq. 33), Cτ (Eq. 34) and Cγ (Eq. 36). 508

Step 8. Calculate the peak response quantities: total crest acceleration ütH (Eq. 11), relative displacement 509

profile u(z) (Eq. 12), shear stress profile τ(z) (Eq. 14) and shear strain profile γ(z) (Eq. 35). 510

511

The assumptions in the proposed model are: a) the seismic response of 1D and 2D embankments is 512

approximated with the fundamental mode shape of vibration of a uniform, homogeneous 1D shear beam, 513

b) the side slope in the transverse direction of the embankment is 1 ≤ S ≤ 3, c) the embankment length in 514

the longitudinal direction is much greater than its height, L/H > 20, d) the shear wave velocity profile is 515

linear with 1 ≤ rVs ≤ 3, e) the fundamental period of vibration of the embankment is 0.1 s ≤ T1 ≤ 1 s, f) 516

T1 is approximately the same in the transverse and longitudinal direction of the embankment, especially 517

for B/H < 10 as per Figure 5, g) the impedance ratio of the embankment-base system is α ≤ 1, h) the 518

nonlinear response is approximated with total stress, equivalent linear analysis, i) the entire embankment 519

is characterised with a single set of G/Gmax and ξ curves and j) the input motion is represented by a design 520

response spectrum. 521

522

Some of the limitations of the model, derived from above assumptions, are: a) stiff embankments 523

supported of soft bases, α > 1, b) embankments with length/height ratio less than 20, c) embankments 524

with variable height in the transverse or longitudinal direction, d) embankments with fixed boundary 525

conditions in the longitudinal direction, i.e. embankment dams, e) effective stress analysis for submerged 526

conditions, and f) fluid-soil interaction analysis, i.e breakwaters. 527

528

The verification of the model is carried out with finite element analysis (FEA) for 1D and 2D 529

embankments using linear, equivalent linear and nonlinear dynamic analysis. The three verification cases 530

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25

use embankments with the same following conditions: height, Vs profile, unit weight, base properties 531

and earthquake records. 532

533

Verification with a 2D Linear Embankment 534

The properties of the 2D embankment are: 14 m height, 12 m crest width, 2H:1V side slope, 20 kN/m3 535

unit weight, 5% damping ratio, linear elastic and shear wave velocity of 133 m/s at the top and 266 m/s 536

at the bottom. The embankment is supported on firm ground with 22 kN/m3 unit weight and 450 m/s 537

shear wave velocity. The seismic hazard is represented by the response spectrum in Fig. 2. 538

539

The verification is carried out with linear dynamic analysis using the computer program PLAXIS 2D. 540

The FE model is divided into twenty layers for discretization of the Vs profile. The input motions consist 541

of the ten earthquake records spectrally matched to the response spectrum in Fig. 2 and applied at the 542

base of the model. 5% Rayleigh damping and compliant base boundary condition with Vs = 450 m/s were 543

considered in the analysis. Total crest acceleration time histories, peak shear stress profiles and peak 544

shear strain profiles were obtained at the center of the FE model for comparison. 545

546

The fundamental period of vibration of the FE model is T1 = 0.21 s. Peak total crest accelerations varied 547

from 0.49 g to 0.95 g with an average of 0.74 g. Figure 13 plots the total crest acceleration time history 548

for earthquake record No 7 for comparison with the proposed model. Figures 14a and 14b plots the mean 549

peak shear stress and peak shear strain profiles obtained from the ten earthquake records. 550

551

The calculation of the seismic response with the proposed model is carried out with the step-by-step 552

procedure as indicated below. Only one iteration is required for linear analysis since the equivalent linear 553

properties are strain independent. 554

Page 25 of 45



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26

555

Step 1. H = 14 m, B = 12 m, S = 2, ρ = 2.04 ton/m3, Vs-top = 133 m/s, rVs = 2, G/Gmax curve = 1, ξ curve 556

= 5%, ρbase = 2.24 ton/m3, Vs-base = 450 m/s and design response spectrum in Fig. 2. 557

Step 2. F2T = 0.85, F2C = 1.15, Gtop = 36 MPa and Gini = 98.1 MPa. 558

Steps 3 to 6 (Table 1). Geq = 98.1 MPa, ξeq = 5%, T1 = 0.22 s and Sa Fm = 0.61 g. 559

Step 7. Cü = 1.12, Cτ = 1.05 and Cγ = 1.05. 560

Step 8. ütH = 0.68 g, u(z) (Eq. 12), τ(z) (Fig. 14a) and γ(z) (Fig. 14b). 561

562

The fundamental period of the embankment obtained with the proposed model is T1 = 0.22 s, which is 563

slightly longer than that obtained with the FE model (T1 = 0.21 s). 564

565

The peak total crest acceleration is 0.68 g, which is about 8% lower than the mean acceleration obtained 566

with FEA (0.74 g). Figure 13 includes the total crest acceleration time history calculated with the 567

proposed model using the computer program SeismoSignal with earthquake record No 7, T1 = 0.22 s and 568

ξ = 5 %. The record was scaled to match the peak total acceleration ütH = 0.68 g for consistency with the 569

procedure. As shown in Fig. 13, the response of the model is generally consistent with that obtained with 570

FEA. 571

572

Peak shear stress and shear strain profiles with the proposed model are plotted in Fig. 14a and 14b, 573

respectively. As shown in the figures, the profiles are also consistent with the mean profiles obtained 574

with FEA. The proposed model is also capable of capturing the reduction of the shear strain in the lower 575

part of the embankment (Fig. 14b) as a result of the increase of the shear modulus Gmax. 576

577

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27

The verification indicates that the proposed modification factors F2T, F2C and Fα provide a good 578

approximation for considering the 2D shape and the base flexibility effects in the proposed model. 579

Likewise, peak shear stress and peak shear strain profiles are also estimated with a good level of accuracy 580

with the proposed model in comparison to the finite element analysis. 581

582

Verification with a 1D Equivalent Linear Embankment 583

The properties of the uniform 1D embankment are the same as those of the 2D embankment but with two 584

main differences: the crest width is much greater than the height (i.e. B > 20H) and the shear modulus 585

and damping curves correspond to those for sand included in Fig. 11. 586

587

The verification is carried out with equivalent linear dynamic analysis using the computer program 588

Shake2000. The FE model is divided into seventy layers for discretization of the Vs profile. The ten 589

spectrally matched earthquake records are applied at the base of the model as outcrop motions. The shear 590

wave velocity of the base is Vs-base = 450 m/s. Total crest acceleration time histories, peak shear stress 591

profiles, equivalent linear shear modulus profiles, damping profiles and peak shear strain profiles are 592

obtained from the FE model for comparison. 593

594

The fundamental period of the FE model varied from 0.43 s to 0.48 s with an average of T1 = 0.45 s. 595

Peak total crest accelerations varied from 0.44 g to 0.62 g with an average of üt(H) = 0.54 g. Figure 15 596

plots the total crest acceleration time history for earthquake record No 7 for comparison with the proposed 597

model. 598

599

Maximum shear modulus Gmax and mean equivalent linear shear modulus Gmean profiles obtained with 600

the FE model are plotted in Fig. 16a. The difference between the two profiles is due to the nonlinear 601

Page 27 of 45



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28

response of the soil layers. The maximum degradation of Gmax is 70% at z ≈ 7.5 m. Mean damping ratio 602

profile is plotted in Figure 16b. The maximum damping ratio is ξ ≈ 12% at z ≈ 7.5 m, consistent with the 603

location of the maximum degradation of Gmax. 604

605

Mean peak shear stress and peak shear strain profiles obtained with the FE model are plotted in Figures 606

17a and 17b, respectively. It is interesting to note that the shear stress profiles of the 1D embankment 607

(Fig. 17a) and the 2D embankment (Fig. 14a) are very similar, even though one is based on equivalent 608

linear analysis and other is based on linear analysis. On the other hand, the shear strain profile obtained 609

with equivalent linear analysis (17b) differs considerably from that obtained with linear analysis (Fig. 610

14b). The significant increase in the shear strain with the equivalent linear model is mainly the result of 611

the nonlinear response of the soil layers. 612

613


procedure as follows. 615

616

Step 1. H = 14 m, B = 20H, S = 2, ρ = 2.04 ton/m3, Vs-top = 133 m/s, rVs = 2, G/Gmax and ξ curves in Fig. 617

11, ρbase = 2.24 ton/m3, Vs-base = 450 m/s and design response spectrum in Fig. 2. 618

Step 2. F2T = 1, F2C ≈ 1, Gtop = 36 MPa and Gini = 98.1 MPa. 619

Steps 3 to 6 (Table 2). Geq = 30 MPa, ξeq = 11.8%, T1 = 0.46 s and Sa Fm = 0.43 g. 620

Step 7. Cü = 1.28, Cτ = 1.05 and Cγ = 1.05. 621

Step 8. ütH = 0.55, u(z) (Eq. 12), τ(z) (Fig. 17a) and γ(z) (Fig. 17b). 622

623

The seismic response procedure converged in three iterations (Table 2). The equivalent linear properties 624

are Geq = 30 MPa and ξeq = 11.8 %. The period of vibration elongates from 0.26 s to 0.46 s as a result of 625

Page 28 of 45



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29

the degradation of the shear modulus Geq from 98.1 MPa to 30 MPa. The total scaling factor of the design 626

response spectrum Fm decreases from 1.85 to 0.53 due mainly to the increase in ξeq from 0.3% to 11.8%. 627

628

The fundamental period of the embankment obtained with the proposed model is T1 = 0.46 s, which is 629

slightly longer than the mean period obtained with FEA (T1 = 0.45 s). 630

631

The peak total crest acceleration calculated with the proposed model is üt(H) = 0.55 g, which is about 2% 632

higher than the mean acceleration obtained with FEA (0.54 g). Figure 15 includes the total crest 633

acceleration time history calculated with the proposed model using the computer program SeismoSignal 634

with earthquake record No 7, T1 = 0.46 s and ξ = 11.8 %. The record was scaled to match the peak total 635

acceleration of 0.55 g for consistency with the procedure. As shown in Fig. 15, the response of the model 636

is generally consistent with that obtained with FEA. The difference between the two responses is due 637

mainly to the small participation of the second mode of vibration in the FE model, which is observed in 638

the figure as a short-period, short-amplitude signal superimposed onto the main response. 639

640

The equivalent linear, homogeneous shear modulus Geq = 30 MPa is plotted in Fig. 16a, which is located 641

between the bounds of the Gmean profile of the FE model. The equivalent linear shear modulus profile 642

G(z) of the proposed model is calculated as the product of the Gmax profile (Eq. 18) and the function FG 643

(Eq. 37 and 38): G(z) = Gmax(z) FG(z). As shown in the figure, the proposed G profile is consistent with that 644

obtained with FEA, especially for z < 8 m. The differences between both profiles for z > 8 m is mainly 645

due to the assumed second mode shape in the proposed model, which is based on a homogenous shear 646

modulus. 647

648

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30

The equivalent homogeneous damping ratio ξeq = 11.8% is plotted in Fig. 16b, which is very similar to 649

the maximum damping ratio obtained with FEA (ξmax = 12%). 650

651

Peak shear stress and shear strain profiles obtained with the proposed model are plotted in Fig. 17a and 652

17b, respectively. As shown in Fig. 17a, the proposed model captures with a good level of accuracy the 653

peak shear stress obtained with FEA. 654

655

The peak shear strain profile of the proposed model is consistent with that of the FEA profile, especially 656

for capturing the reduction of the strain for z < 6 m (Fig. 17b). The maximum strain is overestimated by 657

14% with the proposed model, which is considered acceptable for geotechnical design purposes. The 658

FEA calculated the maximum shear strain at zmax ≈ 8 m while the proposed model calculated it at zmax ≈ 659

6.5 m. This difference in zmax is mainly due to the assumed second mode of vibration for calculating FG(z). 660

661

Verification with a 2D Nonlinear Embankment 662

Physical properties, layer thickness, compliant base conditions and earthquake records for the 2D 663

nonlinear embankment are the same as those of the 2D linear embankment. The soil behavior in the finite 664

element model is represented with the HSsmall model, which considers stress and strain dependent 665

stiffness degradation, hysteretic damping, yielding, and strain hardening (Schanz et al. 1999). Contrary 666

to the linear and equivalent linear models, the HSsmall model can simulate plastic straining. 667

668

The analysis is carried out using the computer program PLAXIS 2D. The stiffness properties of the 669

HSsmall model for each layer are calculated from the shear wave velocity profile. The stress coefficient 670

for sand is m = 0.5 and the input reference pressure is calculated at the middepth of each layer. The 671

assumed strength properties are: cohesion C = 5 kPa, friction angle φ = 45°, and dilation angle ψ = 15°. 672

Page 30 of 45



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The small strain parameter is γ0.7 = 2.1x10-4, obtained from the modulus reduction curve for G/Gmax = 673

0.72 (Fig. 11). 5% Rayleigh damping is assumed for compensating the underestimation of hysteric 674

damping in unloading/reloading at low levels of strain (Brinkgreve et al. 2007). 675

676

Total crest acceleration, mean peak shear stress and mean peak shear strain profiles were obtained at the 677

center of the finite element model. 678

679


procedure as follows. 681

682

Step 1. H = 14 m, B = 12 m, S = 2, ρ = 2.04 ton/m3, Vs-top = 133 m/s, rVs = 2, G/Gmax and ξ curves in 683

Fig. 11, ρbase = 2.24 ton/m3, Vs-base = 450 m/s and design response spectrum in Fig. 2. 684

Step 2. F2T = 0.85, F2C = 1.15, Gtop = 36 MPa and Gini = 98.1 MPa. 685

Steps 3 to 6 (Table 3). Geq = 33.5 MPa, ξeq = 10.7%, T1 = 0.37 s and Sa Fm = 0.53 g. 686

Step 7. Cü = 1.23, Cτ = 1.05 and Cγ = 1.05. 687

Step 8. ütH = 0.65, u(z) (Eq. 12), τ(z) (Eq. 14) and γ(z) (Eq. 35). 688

689

The seismic response procedure converged in three iterations (Table 3). The equivalent linear properties 690

are Geq = 33.5 MPa and ξeq = 10.7 %. The period of vibration elongates from 0.22 s to 0.37 s as a result 691

of the degradation of the shear modulus. The peak total crest acceleration is 0.65 g. 692

693

Figure 18 plots the total crest acceleration obtained with the finite element model (FEA) and the proposed 694

model for earthquake record No 7. In general, coherence is observed between both records. The time 695

history of the FE model includes an important participation of short period components. 696

Page 31 of 45



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32

697

The response spectra and the spectral transfer functions of the acceleration time histories of the FE model 698

indicates that the response is controlled by three periods of vibration: 0.23 s, 0.29 s and 0.39 s. The first 699

and third periods are very similar to the initial (0.22 s) and equivalent linear (0.37 s) fundamental periods 700

of vibration obtained with the proposed model, respectively (Table 3). On the other hand, the time history 701

of the proposed model is controlled by the equivalent linear period of vibration T1 = 0.37 s. 702

703

The peak total crest acceleration of the FE model varied from 0.44 g to 0.73 g with a mean value of 0.55 704

g. The peak acceleration obtained with the proposed model is 0.65 g. The overestimation of the peak 705

acceleration with the proposed model is due to the lack of hysteric damping, developed by plastic 706

straining. Therefore, the proposed equivalent linear model provides conservative estimates for peak 707

accelerations, especially for very strong shaking. 708

709

Mean peak shear stress and peak shear strain profiles obtained with the FE model (C = 5 kPa, φ = 40° 710

and ψ = 10°) and the proposed model are plotted in Fig. 19a and 19b, respectively. 711

712

The proposed model captures with a good level of accuracy the mean peak shear profile and it tends to 713

overestimate the response at the lower half of the embankment due to the lack of yielding and plastic 714

straining (Fig. 19a). On the other hand, the peak shear strain profile of the proposed model underestimates 715

the shear strain of the FE model in about 34% (Fig. 19b) as a result of the lack of yielding and plastic 716

straining. However, the proposed model captures the reduction of the shear strain in the lower half of the 717

embankment and follows the general shape of the profile of the FE model. 718

719

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33

An additional analysis was carried out with the FE model using the same strength properties (φ = 40° 720

and ψ = 10°) but with C = 100 kPa. The mean peak shear stress and shear strain profiles at the center of 721

the model are plotted in figures 19a and 19b (C = 100 kPa), respectively. 722

723

The increase in the cohesion increases the shear strength in the FE model. As a result, the mean peak 724

shear stress profile increases and almost matches the peak shear stress profile of the proposed model (Fig. 725

19a), which assumes no limitation in the shear strength. The effect of a higher shear strength reduces the 726

plastic deformations in the peak shear strain profile of the FE model (Fig. 19b). The underestimation of 727

the average strain with the proposed model for this case is about 18%. 728

729

The 1D and 2D verification studies indicate that the proposed model can capture with a good level of 730

accuracy the response of the embankment, especially the fundamental period of vibration and the peak 731

shear stress profile. For strong levels of shaking, the model may overestimate the peak total crest 732

acceleration and underestimate the peak shear strain profile in comparison to nonlinear analyses that 733

include shear yielding and plastic hardening. 734

735

Conclusions 736

This paper presents a simple, yet effective, model for calculating the equivalent linear seismic response 737

of bridge embankments in the free field using a design response spectrum. 738

739

The proposed model is based on the fundamental mode shape of vibration of a uniform, homogenous 1D 740

shear beam and considers a linear shear wave velocity profile. The effects of base flexibility, 2D shape 741

and damping ratio on the embankment response are approximated with simple modification factors. An 742

expression for calculating the effective cyclic shear strain is derived for equivalent linear analysis. 743

Page 33 of 45



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34

744

A step-by-step procedure is developed for easy implementation of the proposed model in practical design 745

procedures. The fundamental period of vibration, equivalent linear properties Geq and ξeq, peak total crest 746

acceleration, equivalent linear shear modulus profile, peak shear stress profile, peak shear strain profile 747

and peak relative displacement profile can be easily estimated using a design response spectrum. 748

749

Parametric analyses were carried out using equivalent linear dynamic analysis for 1D embankments with 750

fundamental periods of vibration between 0.1 s to 1 s. Correction factors were derived from the analyses 751

for improving the peak response quantity estimators. 752

753

Verification of the model was carried out with finite element analysis for 1D and 2D embankments using 754

linear, equivalent linear, and nonlinear soil models. The verification demonstrated that the proposed 755

model is robust and provides a good approximation of the seismic response in comparison to the finite 756

element simulations, especially for determination of the fundamental period of vibration and the peak 757

shear stress profile. 758

759

The verification with nonlinear analysis using the HSsmall soil model indicated that the proposed 760

procedure tends to overestimate the peak total crest acceleration and to underestimate the peak shear 761

strain profile, especially for high intensity motions. 762

763

The proposed model and methodology are based on an empirical procedure suitable for a preliminary 764

design phase of bridge embankments using a spreadsheet. 765

766

Acknowledgements 767

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35

This research project was partially founded by the British Columbia Ministry of Transportation and 768

Infrastructure (MoTI) under the Professional Partnership Program and by a Discovery Grant from the 769

Natural Sciences and Engineering Research Council of Canada (NSERC), both awarded to the third 770

author. 771

772

The authors would like to thank the reviewers of the Canadian Geotechnical Journal and Dr. Alex Sy of 773

Klohn Crippen Berger for their detailed review and comments to improve the final version of the paper. 774

775

References 776

Ambraseys, N. N. 1960. On the shear response of a two-dimensional truncated wedge subjected to an 777

arbitrary disturbance. Bulletin of the Seismological Society of America, 50(1): 45-56. 778

Brinkgreve, R.B.J., Kappert, M.H., and Bonnier, P.G. 2007. Hysteretic damping in a small-strain 779

stiffness model. Numerical Models in Geomechanics. Taylor & Francis Group, London. 780

CAN/CSA. 2014. S6-14 Canadian Highway Bridge Design Code. 781

Carvajal, J.C. 2011. Seismic embankment-abutment-structure interaction of integral abutment bridges. 782

PhD thesis, Department of Civil Engineering, University of British Columbia, Vancouver, Canada. 783

Available from http://hdl.handle.net/2429/35577 . 784

Chopra, A. 1999. Dynamics of structures: theory and applications to earthquake engineering. Prentice-785

Hall, India. 786

Cook, R., Malkus, D., Plesha, M., and Witt, R. 2001. Concepts and applications of finite element analysis. 787

John Wiley & Sons, Singapore. 788

Inel, M., and Aschheim, M. 2004. Seismic design of columns of short bridges accounting for 789

embankment flexibility. Journal of Structural Engineering, American Society of Civil Engineers, 790

130(10): 1515-1528. 791

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http://hdl.handle.net/2429/35577

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Kotsoglou, A., and Pantazopoulou, S. 2007. Bridge-embankment interaction under transverse ground 792

excitation. Earthquake Engineering and Structural Dynamics, 36: 1719-1740. 793

Kotsoglou, A., and Pantazopoulou, S. 2009. Assessment and modeling of embankment participation in 794

the seismic response of integral abutment bridges. Bulletin of Earthquake Engineering, 7: 343-361. 795

Kramer, A. 2006. Geotechnical Earthquake Engineering. Prentice Hall, USA. 796

Makdisi, F. I., and Seed, H. B. 1977. A Simplified procedure for estimating earthquake-induced 797

deformations in dams and embankments. Earthquake Engineering Research Center. Report No. 798

UCB/EERC-77/19. University of California, Berkeley. 799

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Congress on Large Dams, Washington, D.C., Vol. IV. 801

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Draft

1

Table 1. Calculation of Geq and eq for the 2D linear embankmenti G/Gmax Geq eq T1 Sa Vs-eq F F Fm Sa Fm eff tol

MPa % s g m/s g % %1 1 98.1 5 0.22 0.88 219 0.44 0.47 1 0.69 0.61 0.044 -

i: iteration

Table 2. Calculation of Geq and eq for the 1D equivalent linear embankmenti G/Gmax Geq eq T1 Sa Vs-eq F F Fm Sa Fm eff tol

MPa % s g m/s g % %1 1 98.1 0.3 0.26 0.89 219 0.44 0.47 3.08 1.85 1.64 0.165 -2 0.288 28.3 12.4 0.48 0.80 118 0.24 0.60 0.70 0.53 0.42 0.148 10.43 0.306 30.0 11.8 0.46 0.81 121 0.25 0.59 0.71 0.53 0.43 0.141 4.2

i: iteration

Table 3. Calculation of Geq and eq for the 2D nonlinear embankmenti G/Gmax Geq eq T1 Sa Vs-eq F F Fm Sa Fm eff tol

MPa % s g m/s g % %1 1 98.1 0.3 0.22 0.88 219 0.44 0.47 3.08 2.12 1.87 0.136 -2 0.319 31.3 11.4 0.39 0.85 124 0.25 0.59 0.72 0.61 0.52 0.119 12.83 0.342 33.5 10.7 0.37 0.85 128 0.26 0.58 0.74 0.62 0.53 0.114 4.5

i: iteration

Page 38 of 45



Draft

1

H

z

ug(t)

u(z,t)

ut(H,t)

Figure 1. One-dimensional shear beam model

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Sa

(g)

T (s)

Figure 2. 5% damped design response spectrum

B

T2D1

S

T1D F2TH

(a) (b)

Figure 3. (a) 2D embankment and (b) equivalent uniform 1D embankment

Page 39 of 45



Draft

2

B

L

H

Figure 4. 3D FE model for calibration of F2T

0.7

0.8

0.9

1

0 2 4 6 8 10

F2

T

B / H

Long. S = 1 Long. S = 1.5 Long. S = 2 Transv. S = 1 Transv. S = 1.5 Transv. S = 2 Fitted curve

Figure 5. Period modification factors obtained from FE analysis

1

1.05

1.1

1.15

1.2

1.25

0 2 4 6 8

F2

C

B / H

S = 3 S = 2 S = 1

Figure 6. Crest response modification factor

Page 40 of 45



Draft

3

0.5

0.6

0.7

0.8

0.9

1

1.1

0.2 1 2 4

Rel

. D

isp.

Rat

io

B / H

S = 1 S = 2 S = 3

Figure 7. Relative crest displacement ratio: Model / FEA

0

0.2

0.4

0.6

0.8

1

0 1 2

z /

H

Vs / Vs-top

rVs = 1

rVs = 2

Figure 8. Shear wave velocity profile

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Sa

(g)

T (s)

Rigid base

Flexible base

Figure 9. Within response spectra for � = 0.2 and T1 = 0.5 s

Page 41 of 45



Draft

4

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1

F�

�

FE data

-

-T1 � 0.2 s

T1 = 0.1 s and � � 0.17

Figure 10. Modification factor for flexible base for � � 1

0

4

8

12

16

20

0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1

�(%

)

G /

Gm

ax

� (%)

G/Gmax��

Figure 11. Modulus reduction and damping curves for sand

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

z /

H

G / Gini

Gini

Gdeg

Geq

Figure 12. Degraded shear modulus profile for DG = 0.6

Page 42 of 45



Draft

5

-0.8

-0.4

0

0.4

0.8

22 24 26 28 30 32

üt (H

,t)(g

)

Time (s)

FEA Model

Figure 13. Total crest acceleration of the 2D linear embankment for earthquake record No 7

0

2

4

6

8

10

12

14

0 30 60 90

z(m

)

� (kPa)

FEA - mean

Model

0

2

4

6

8

10

12

14

0 0.08 0.16 0.24 0.32

� (%)

(a) (b)

Figure 14. (a) peak shear stress and (b) peak shear strain profiles of the 2D linear embankment

-0.6

-0.3

0

0.3

0.6

22 24 26 28 30 32 34 36 38

üt (H

,t)(g

)

Time (s)

FEA Model

Figure 15. Total crest acceleration of the 1D equivalent linear embankment for earthquake record No 7

Page 43 of 45



Draft

6

0

2

4

6

8

10

12

14

0 5 10 15

� (%)

FEA -

Model -

0

2

4

6

8

10

12

14

0 50 100 150

z(m

)

G (MPa)

Gmax

FEA -

Model -

Model - G

Geq

Gmean

�eq

�mean

(a) (b)

Figure 16. (a) shear modulus and (b) damping ratio profiles of the 1D equivalent linear embankment

0

2

4

6

8

10

12

14

0 30 60 90

z(m

)

� (kPa)

FEA - mean

Model

0

2

4

6

8

10

12

14

0 0.08 0.16 0.24 0.32

� (%)

(a) (b)

Figure 17. (a) peak shear stress and (b) peak shear strain profiles of the 1D equivalent linear

embankment

Page 44 of 45



Draft

7

-0.8

-0.4

0

0.4

0.8

17 19 21 23 25 27 29 31

üt (H

,t)(g

)

Time (s)

FEA Model

-0.8

-0.4

0

0.4

0.8

31 33 35 37 39 41 43 45

üt (H

,t)(g

)

Time (s)

FEA Model

Figure 18. Total crest acceleration of the 2D nonlinear embankment for earthquake record No 7

0

2

4

6

8

10

12

14

0 30 60 90

z(m

)

� (kPa)

Model

FEA - C = 5 kPa

FEA - C = 100 kPa

0

2

4

6

8

10

12

14

0 0.08 0.16 0.24 0.32

� (%)

(a) (b)

Figure 19. (a) peak shear stress and (b) peak shear strain profiles of the 2D nonlinear embankment

Page 45 of 45



response spectrum-based seismic response of bridge

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