A CAPACITY SPECTRUM DESIGN APPROACH FOR HYBRID SLIDING-ROCKING 1

POST-TENSIONED SEGMENTAL BRIDGES 2

3

4

5

Sreenivas Madhusudhanan 6

Graduate Student Researcher 7

Department of Civil, Environmental and Architectural Engineering 8

University of Colorado – Boulder 9

Boulder, CO 80309 10

Email: srma6521@colorado.edu 11

12

Petros Sideris, Corresponding Author 13

Assistant Professor 14

Department of Civil, Environmental and Architectural Engineering 15

University of Colorado – Boulder 16

Boulder, CO 80309 17

Tel.: 303-492-4333, Email: petros.sideris@colorado.edu 18

19

20

Word count: 4492 words text + 10 figures x 250 words = 6992 words 21

References: 35 22

23

Submission Date: August 1st, 2014 24

25

26

27

ABSTRACT 28

In this paper, a capacity spectrum design method for the hybrid sliding-rocking (HSR) segmental 29

bridge piers. The HSR bridges consist of unbonded post-tensioning, end rocking joints and 30

intermediate sliding joints (or slip-dominant joints) along the column height. Joint sliding provides 31

energy dissipation with small damage and control of the applied seismic loading. Residual joint 32

sliding is small and restorable after an intense earthquake event. Force-based seismic design of the 33

HSR columns is difficult to apply because representative R-factors are not available and are 34

difficult to estimate. For this reason, displacement-based design methods are investigated. In the 35

proposed method, a pushover curve for the HSR columns is computed analytically. Then, the 36

equivalent viscous damping ratio is computed as function of the column displacement. Eventually, 37

the spectral demand obtained from the FEMA 356 is compared to the spectral capacity obtained 38

from the pushover curve are compared. Iterative application of this method can result in 39

economical designs. A preliminary comparison of the proposed design method with the results of 40

an experimental study is presented. The experimental results compare reasonably well with the 41

results of the design method. 42

43

Keywords: Seismic design, segmental column, displacement-based design, prefabricated 44

elements, accelerated bridge construction, unbonded post-tensioning 45

46

47

48

INTRODUCTION 49

In the U.S., a vast number of bridges, many of which are located in seismic regions, have been 50

classified as structurally deficient or functionally obsolete (1). These bridges are in need of 51

immediate retrofit or replacement. To minimize the downtime and the resulting socio-economic 52

impact of closed bridges, accelerated bridge construction (ABC) techniques have been 53

investigated, mainly in the field of precast concrete segmental construction. 54

Several bridge substructure systems that combine construction rapidity with improved seismic 55

performance have been recently developed. These systems typically consist of prefabricated 56

segmental or monolithic elements post-tensioned together with internal, usually unbonded, post-57

tensioning. Fabrication of these segments/elements takes place off-site, whereas assembly is 58

conducted on-site, resulting in significant erection rapidity. The majority of the substructure 59

systems studied over the years may be divided into: (i) bents with prefabricated monolithic 60

columns (without post-tensioning) connected with the bent cap and the foundation through rigid 61

(or emulative of rigid) connections (2-8), and (ii) bents with segmental or prefabricated monolithic 62

rocking columns incorporating internal unbonded post-tensioning (8-12). While unbonded post-63

tensioning increases considerably the ductility capacity and self-centering capabilities of rocking 64

columns, energy dissipation can be provided by internal partially debonded yielding rebar crossing 65

the rocking joints (13-19), or externally attached yielding links at the rocking joints (11, 19). 66

Concrete compression crushing at the end joints can be controlled through proper confinement, 67

steel jacketing (9, 11), fiber reinforced polymer (FRP) jacketing (12), and use of high performance 68

concrete materials (17, 20). 69

A novel bridge substructure system was recently introduced by Sideris (21) and Sideris et al.(22, 70

23). The proposed system included substructure columns with unbonded post-tensioning, end 71

rocking joints and intermediate sliding joints (or slip-dominant joints) along the height of the 72

columns. Joint sliding provides energy dissipation with small damage and control of the applied 73

seismic loading. Residual joint sliding is small and restorable after an intense earthquake event. 74

The proposed columns are termed hybrid sliding-rocking (HSR) columns. 75

For the seismic design of bridges incorporating HSR columns (or any of the aforementioned 76

substructure systems), force-based methods usually available in current design codes are difficult 77

to apply due to the lack of representative response modification factor, R. In force-based design 78

methods, R-factors indirectly account for energy dissipation properties and ductility capacity of 79

system. Representative R-factors are difficult to estimate, particularly for highly nonlinear 80

systems. A method to estimate representative R-factors for new systems is provided in (24). This 81

method, which primarily focuses on buildings rather than bridges, requires a vast number of 82

nonlinear time history analyses and becomes tedious as the number of design variables increases. 83

In the framework of performance-based design, direct displacement-based design (DDBD) 84

methods (25) provide an attractive alternative. DDBD methods directly account for the nonlinear 85

properties of the system through an equivalent linearization obtained for a target design 86

displacement. Major advantages of the DDBD methods include: (i) Primary design parameters are 87

the deformations which are representative measures of damage, (ii) R-factors are not needed, and 88

(iii) system performance can be evaluated at various hazard levels. 89

Capacity spectrum methods (CSMs) also belong in the framework of DDBD methods, since they 90

share similar concepts. CSMs were originally developed for the seismic performance evaluation 91

of existing structures (26-28). However, they can also be used for the evaluation of new designs. 92

In the present work, a CSM is used in an iterative framework for the design of bridges 93

incorporating HSR substructure columns. 94

CAPACITY SPECTRUM METHOD FOR THE DESIGN OF HSR COLUMNS 95

In the CSMs, the nonlinear hysteretic response of a system is linearized at a given displacement 96

using the system secant properties. The hysteretic energy dissipation is represented in the 97

linearized (secant) system by an equivalent viscous damping ratio. The CSM requires knowledge 98

of the pushover curve of the system up to the collapse point, and knowledge of the energy 99

dissipation at any given displacement. The pushover curve can be transformed to spectral ordinates 100

(spectral acceleration and spectral displacement) which represents the so-called spectral capacity 101

curve. On the other hand, the design response spectrum (spectral acceleration versus period) can 102

be converted to the same spectral ordinates (spectral acceleration and spectral displacement) 103

representing the so-called spectral demand curve. The spectral demand curve – and the design 104

spectrum as well – can be selected to represent any seismic hazard level of interest and can be 105

modified based on the equivalent viscous damping ratio. The point of intersection of the spectral 106

capacity curve with the spectral demand curve (see FIGURE 1) is termed performance point and 107

indicates the performance of the system at a given seismic hazard level. If there is no point of 108

intersection, this means that the system will fail at the given hazard. 109

The HSR columns include several design variables, such as dimensions, material properties, post-110

tensioning forces, and joint sliding properties (number and distribution of sliding joints, sliding 111

amplitude, duct adaptor dimensions, frictional properties). In the proposed CSM, for a set of design 112

parameters, the pushover curve and the equivalent viscous damping ratio are computed 113

analytically. The corresponding spectral capacity curve is then compared with a spectral demand 114

curve obtained from FEMA 356 (29). The design variables can be varied by the design engineer 115

so that the most economical design is obtained. The design procedure will be incorporated into a 116

computer code (30) to facilitate use from design engineers. Except for trial and error values of the 117

design variables, optimal design can be obtained by incorporating the proposed procedure into an 118

optimization framework. 119

120 FIGURE 1: Capacity spectrum method 121

Analytical Derivation of Pushover Curve 122

A reference column is first presented. The equilibrium and compatibility equations for an HSR 123

columns are derived next. These equations are used to compute the pushover curve of a rocking-124

only column and a HSR column. 125

Reference Model 126

A typical HSR-SD column is shown in FIGURE 2 (a). The column consists of a rocking joint at 127

SA

SD

Spectral Demand Curve

Spectral

Capacity Curve

Performance

Point

the bottom and slip-dominant (SD) joints along the column height. The rocking joint incorporate 128

shear keys or high coefficient of friction to prevent sliding at the bottom. The sliding capacity of 129

SD joints is controlled by duct adaptors, as shown in FIGURE 2 (a). The column is subjected to a 130

constant vertical force, Pv, and monotonically increasing horizontal force, Ph. 131

(a)

(b)

FIGURE 2: (a) Typical HSR-SD column, and (b) Concrete stress and strain distribution at the 132 bottom of a typical HSR-SD column. 133

Analysis of a HSR-SD Column 134

The response of a HSR-SD column is controlled by the rocking response at the bottom joint and 135

the sliding at the SD joints. Under monotonic loading, different deformation stages can be 136

considered. The lateral force and displacement of the column at any deformation stage is computed 137

using a set of equilibrium and compatibility equations. The equilibrium at the bottom joint in the 138

vertical direction is given as: 139

v PT cP N N

(1)

where Pv is the applied vertical load, NPT0 is the force of all PT tendons, and Nc is the resultant 140

concrete at the bottom of the joint. 141

The moment equilibrium at the bottom joint is given by: 142

,

m h sl j v c PT

j

h P u P M M (2)

where Mc is the moment resistance due to concrete, MPT,0 is the moment resistance due to the 143

unbonded post-tensioning and Ph is the horizontal force applied at a height hm, where hm is the 144

center of mass of the superstructure. 145

For a given strain c and contact length cr (< d), as shown in FIGURE 2 (b), the concrete axial 146

force including confinement effects can be obtained using the equivalent stress block method by 147

Paulay and Priestley (31) as: 148

'( )( ) c cc cc cc r wN f c b (3)

where bw is the width of the cross-section, αcc and βcc are the stress block parameters obtained from 149

FIGURE 3, f’cc is the strength of the confined concrete obtained from Mander (32) and εcc is 150

Duct

AdaptorsDuct

Tendon

Ph

Pv

Sliding

Rocking

hp

hm

uh

dPT,1

dPT,2

cr

εc

σc

Νc

d

corresponding strain at f’cc obtained from from Mander (32). 151

Similarly, the moment resistance due to concrete is given as: 152

'( )( )2 2

cc rc cc cc cc r w

cdM f c b

(4)

where d is the total depth of the concrete cross-section. Note that FIGURE 3 permits use of the 153

equivalent stress block method for any strain, εc, and for both the confined and unconfined 154

concrete. 155

When the maximum concrete stress is smaller than 0.5fc’, the stress distribution over the joint can 156

be assumed to be linear. In that case, the total concrete force can be computed by analytical 157

integration as: 158

c

A

N x dA (5)

where σ(x) is the value of the concrete stress at distance x from the reference axis. 159

Similarly, the concrete moment resistance in that case is given as: 160

c

A

M x x dA (6)

For a given concrete compressive strain and as long as joint opening occurs (i.e., cr < d), the 161

rocking rotation at the bottom, θr, can be obtained by the following compatibility equation:. 162

cr r r r

r

l lc

(7)

where ϕr is the rocking curvature at the bottom joint, and lr is the equivalent hinge length. Different 163

researchers have suggested different values for equivalent hinge length. Different values of lr are 164

considered at different deformation stages, such as lr = 5.25 cr for εc = 0.003 (33), and lr = cr for εc 165

= εcu (34), where εcu is the ultimate strain of confined concrete. If no gap opening occurs (i.e., cr = 166

d), then θr = 0. 167

The extension of the i-th tendon which is located at a distance dPT,i from the end compression fiber 168

is given as (21-23): 169

, ,

Due to rockingat bottom Dueto joint sliding

12 1

cos

PT i r PT i r da

j j

u d c h (8)

where, ψj is the tendon deviation angle due to sliding at the j-th joint and is given as: 170

, ,

,tan sgn2

sl j sl b

j sl j

da

u uu

h (9)

where < u > are the Macauley brackets (< u > = u for u > 0 and < u > = 0 otherwise), usl,j is the 171

current sliding at the j-th joint (with |usl,j| ≤ Dda-DPT, where Dda = duct adapter diameter, and DPT 172

= diameter of the PT tendon) and usl,b is the sliding amplitude at which the bearing contact between 173

the duct and the tendon initiates. Note that for all SD joints, usl,b = Dd-Dpt, where Dd is the diameter 174

of the duct. 175

Assuming that all tendons have the same geometric and material properties, the force at the i-th 176

tendon is given as: 177

, , , , , 0 PT PTPT i PTo i PT i PT y PT PT i

PT

E AN N u f A and N

L

(10)

where NPTo,i is the initial post-tensioning force of the i-th tendon, EPT is the Young’s modulus of 178

the tendon, APT is the cross-section area of the tendon, LPT is the length of the tendon, and fPT,y is 179

the yield stress of the tendon. The total force of the unbonded tendons is given as: 180

,PT PT i

i

N N

(11)

The corresponding moment due to the unbonded tendons is given as: 181

, ,2

PT PT i PT i

i

dM d N

(12)

At a given deformation stage, the corresponding lateral displacement is given as: 182

23

,

Displacement due to lateralload Rigidbodyrotation effect

3 2 2 2

p m p p m pp p r

h h h m p m p sl j

j

h h h h h hh h lu P P h h h u

EI EI EI EI

(13)

where hp is the height of the deformable part of the column, E is young’s modulus of the concrete 183

and I is the moment of inertia of the column cross-section. The first term corresponds to the elastic 184

response of the column, the second term corresponds to the rigid body rotation and translation of 185

the cap beam and superstructure as a result of the elastic translation and rotation of the top end of 186

the column, the third term corresponds to the displacement of the center of mass of the 187

superstructure resulting from rigid body rotation due to rocking at the bottom, and the fourth term 188

represents the displacement due to sliding at all SD joints. 189

(a) (b)

FIGURE 3: Stress block parameter for any given strain, εc, after (31) 190

Application to a Rocking Column without Joint Sliding 191

The response of a rocking column is controlled by the response at the bottom joint. Sliding remains 192

zero at all times (i.e., usl,j = 0 and ψj = 0 for all j). Using the stress block method, Eqs. (1) to (13) 193

provide the force vs. displacement for monotonic lateral loading of the column for any selected 194

value of εc. However, it is more instructive to consider the following distinct stages instead of 195

several random points. These distinct stages are: (i) Decompression, (ii) Concrete proportionality 196

limit, (iii) Unconfined concrete strength, and (iv) Ultimate compressive strain of confined 197

concrete. The stress and strain distribution at each stage is shown in FIGURE 4. 198

c'

ccf'

cc cca f

cc rccc

c

cc

c

cc

'

'cc

c

fK

f

cc cca rc

Decompression Stage 199

Decompression (FIGURE 4 (a)) is assumed to occur at low stresses (linear response). By setting 200

the stress and strain at the tensile fiber equal to zero and using the expressions for elastic response 201

in Eqs. (3)and (6), the lateral force and displacement are computed. If the resulting stresses are 202

found to exceed 0.5f’c, the “stress-block” method should be used. The system of equations is solved 203

using a Newton-Raphson (N-R) method, where different values of σc are selected until equilibrium 204

and compatibility is satisfied. 205

Concrete Proportionality Limit 206

The proportionality limit refers to the transition from the linear elastic response to the nonlinear 207

response. This transition is assumed to take place at 50% of the nominal concrete compressive 208

strength, f’c. Assuming that the stress equals 0.5f’c at the end compressive fiber (FIGURE 4 (b)), 209

a N-R method is used to select different cr values until equilibrium and compatibility are satisfied. 210

The concrete proportionality limit is usually reached after some opening has occurred. 211

Unconfined Compressive Strength 212

The unconfined compressive strength is reached when the strain at the extreme fiber reaches 0.003. 213

Confinement effects may be neglected in this case (f’c ≈ f’cc, acc ≈ 0.85, βcc ≈ 0.85 – 0.05< f’c (ksi) 214

- 4 ksi>≥0.65). The system of equations is solved using a Newton-Raphson (N-R) method, where 215

different values of cr are selected (for the given εc = 0.003) until equilibrium and compatibility is 216

satisfied. 217

Ultimate Compressive Strain of Confined Concrete 218

The ultimate compressive strain of the confined concrete occurs when the hoop reinforcement 219

fractures. This strain is given as: 220

1.40.004 0.02

'

s yh sm

cu

cc

f

f (14)

where s is the volumetric ratio of transverse steel,

yhf is the yield stress of the hoop reinforcement. 221

εsm is the steel strain at maximum tensile stress (≈ 0.15 for Grade 40 and 0.10 for Grade 60 rebar). 222

The system of equations is solved using a Newton-Raphson (N-R) method, where different values 223

of cr are selected (for the given εc = εcu) until equilibrium and compatibility is satisfied. 224

For a typical rocking column, the pushover curve is presented in FIGURE 5. 225

(a) (b) (c) (d)

FIGURE 4: Stress distribution at various stages: (a) decompression stage, (b) concrete 226 proportionality limit, (c) unconfined concrete strength, and (d) ultimate strain of confined concrete 227

228 FIGURE 5: Force vs. displacement response of rocking-only column. 229

Application to a HSR-SD column 230

For an HSR-SD column, joint sliding initiates (by design) before the strength of the unconfined 231

concrete (εc = 0.003) has been reached at the bottom joint. Also, sliding is designed to reach its 232

maximum at all joints before the ultimate strain of the confined concrete has been reached. 233

Furthermore, for economical designs, the decompression (and/or concrete proportionality limit) 234

occur before sliding initiation (albeit this is not necessary). 235

For an HSR-SD column, the following distinct stages are considered: (i) Decompression, (ii) 236

Concrete Proportionality limit, (iii) Sliding Initiation, (iv) End of sliding at each SD joint, (v) 237

Ultimate strain of confined concrete after sliding has been reached at all SD joints, and (vi) 238

Complete Unloading. 239

The stages of decompression and concrete proportionality limit are evaluated as described for the 240

rocking column. If joint sliding initiates before the concrete proportionality limit stage, the 241

concrete proportionality limit stage is dropped. Also, if joint sliding initiates before the 242

decompression stage, both the decompression stage and the concrete proportionality limit stage 243

are dropped. For evaluation of the force vs. displacement at different stages, Eqs. (1) to (13) are 244

σ<0.5fc’

ε<ε0.5ε =ε0.5

σ=0.5fc’

cr

cr

x

d

d

cr

βcr

x

ε=0.003

σ=αfc’

d-cover

cr

βcr

x

ε=εcu

σ= αf’cc

Decompression

Proportionality Limit

Unconfined Compressive strength

Confined

compressive

strength

Lo

ad

Displacement

considered. 245

Initiation of Joint Sliding 246

Joint sliding initiates when the lateral force, Ph, becomes equal to the shear resistance of the SD 247

joints. For Coulomb friction, this condition becomes: 248

h cP N (15)

where all joints are assumed to have the same friction coefficient and the axial force is assumed to 249

be constant throughout the height of the column. The solution strategy is as follows. A value of 250

the compressive strain, εc, is selected. For this value, the system of equations is solved using a N-251

R method, where different values of cr are selected until equilibrium and compatibility is satisfied. 252

For the resulting values of Ph and Nc, Eq. (15) is examined. If Eq. (15) is not satisfied, then a 253

different value of εc is selected. This strategy continues a value of εc that satisfies Eq. (15) is found. 254

Note that for this stage, usl,j = 0 and ψj = 0 for all j. 255

End of Sliding at each Slip-dominant joint 256

Sliding at the j-th joint ends when the external lateral force, Ph, is in equilibrium with the frictional 257

resistance and the tendon bearing resistance at that joint, as expressed by the following equation: 258

sin( ) h c PT jP N N (16)

It is assumed that the sliding ends at one joint before sliding begins at the next joint. 259

For the computation of the total post-tensioning force, the tendon extension due to sliding at the 260

current joint and joints that have experienced joint sliding earlier is considered. The solution 261

strategy is as follows. For the first joint, usl,j (with j = 1) is set equal to its maximum value, while 262

usl,j =0 for all other joints. Then, a value of the compressive strain, εc, is selected. For this value, 263

the system of equations is solved using a N-R method, where different values of cr are tried until 264

equilibrium and compatibility is satisfied. For the resulting values of Ph, Nc, and NPT, Eq. (16) is 265

examined. If Eq. (16) is not satisfied, then a different value of εc is selected. This strategy continues 266

a value of εc that satisfies Eq. (16) is identified. For the end of sliding at the second joint, usl,j at 267

both the first and second joint (j = 1 and 2) receives its maximum value, while sliding is set to zero 268

in the rest of the SD joints. Then, a value of the compressive strain, εc, is identified for which Eq. 269

(16) is satisfied. The same procedure is repeated until sliding at all joints is completed. 270

Ultimate Strain of Confined Concrete after Sliding Completion at all Slip-dominant Joints 271

This response stage is similar to the corresponding stage of the rocking column. However, it 272

considered that the sliding, usl,j, has been completed at all SD joints contributing to the tendon 273

extension and the equilibrium and compatibility equations. The system of equations is solved using 274

a Newton-Raphson (N-R) method, where different values of cr are selected, for the given εc = εcu 275

and for usl,j = max at all SD joints) until equilibrium and compatibility is satisfied. 276

Complete Unloading 277

The damage to the column due to rocking at the bottom is assumed small and is neglected. Thus, 278

upon removal of the horizontal load, Ph, residual rocking is zero. However, residual joint sliding 279

cannot be neglected because the dowel effect of the tendons is not sufficient to provide complete 280

sliding self-centering. The residual sliding will be the same at all joints and can be obtained when 281

the following condition is satisfied: 282

sin c PT jN N (17)

This condition was obtained from Eq. (16) by setting Ph = 0 and considering reversal of the friction 283

force. In order to find the residual displacement, the value of usl,j is varied. As a result, ψj is varied 284

(through Eq.(9)) and NPT is varied (through Eq. (8)). The value of usl,j (same for all j) satisfying 285

the equilibrium will be the residual joint sliding when the horizontal load is completely removed 286

(Ph = 0). For unloading after sliding has been completed for some of the joints, the same approach 287

is followed. However, in the computation of NPT (through Eqs. (8) and (10)), only the joints that 288

have completed their sliding are included. 289

A pushover curve for a typical HSR-SD column is shown in FIGURE 6. The area enclosed by the 290

loading and unloading curve represents that energy dissipated due to joint sliding. 291

292 FIGURE 6: Load vs. displacement curve of the member considering sliding. 293

294

Estimation of Equivalent Viscous Damping 295

The equivalent viscous damping ratio can be defined as: 296

0eq hyst

(18)

where ξo is the damping ratio of the elastic system (usually, 2-5%) and ξhyst is the hysteretic viscous 297

damping ratio representing the hysteretic energy dissipation of the system. The hysteretic viscous 298

damping ratio is estimated at a given displacement, uh, as: 299

1

4

Dhyst

S

E

E

(19)

where ED is the energy dissipated at a single cycle of displacement amplitude, uh, and ES is the 300

(secant) linear elastic energy at that displacement, uh. Energy dissipation from rocking at the 301

bottom and damage of the concrete is neglected. Thus, ED solely incorporates the effect of joint 302

sliding. The computation of ED and ES for unloading before and after completion of sliding at all 303

joints is graphically presented in FIGURE 7 (a) and (b), respectively. Considering that FIGURE 7 304

includes only 1/4 of the entire cyclic pushover, it is assumed, in the interest of simplicity, that the 305

hysteretic energy shown in this figure accounts for 1/4 of the total hysteretic energy dissipation, 306

ED. The variation of energy dissipated, (secant) energy stored and damping ratio as function of the 307

displacement, uh, is shown in FIGURE 8. Note that once sliding ends at all joints, the energy is 308

assumed to have completely dissipated, and it remains constant till the member reaches its ultimate 309

stage. 310

Sliding

Initiation

Joint1

Joint2

Joint3

Joint4

Joint5

Unloading

Decompression

(a) (b)

FIGURE 7: Graphical representation of hysteretic energy, ED, and linear elastic energy, ES, : (a) 311 Reversal before maximum sliding is reached at all joints, and (b) Reversal after maximum sliding is 312 reached at all joints. 313

314 FIGURE 8: Hysteretic energy dissipation, linear elastic energy and damping ratio as function of the 315 displacement amplitude. 316

Estimation of Performance Point using FEMA 356 317

The performance point for a given structure is obtained by the intersection of the spectral demand 318

curve and the spectral capacity curve. The damping ratio at the displacement represented by the 319

point of intersection should be the same with the damping ratio used for the derivation of the 320

spectral demand curve. According to the selected solution approach, a displacement, uh, is first 321

selected. The damping ratio at that displacement is then obtained (from FIGURE 8). For this 322

damping ratio, the spectral demand curve is obtained. From the intersection of the spectral capacity 323

curve with the spectral demand curve, a new displacement value of uh is obtained. If this value 324

equals the original value, this is the performance point. If not, a different value of uh is selected 325

and the process is restarted. If there is no point of intersection for all values of uh, the system 326

spectral demand is larger than the spectral capacity. 327

Spectra demands for the DE (10% in 50 years) and MCE (2% in 50 years) events (35) are 328

considered in this study. 329

Graphical representation of the computation of the performance point has already been shown in 330

FIGURE 1. 331

Sliding

Initiation

Joint1

Joint2

Joint3

Joint4

Joint5

Unloading

Decompression

ES

ED/4

Sliding

Initiation

Joint1

Joint2

Joint3

Joint4

Joint5

Unloading

Decompression

ES

ED/4

Sliding initiation

Joint1 Joint2

Joint3 Joint4

Joint5

EVALUATION OF THE DESIGN METHOD 332

A preliminary evaluation of the design method against the experimental data of a shake table 333

testing program with a large-scale single-span bridge specimen (21-23) is presented herein. The 334

specimen included a box-girder segmental superstructure and two single-column piers (see 335

FIGURE 9 (a)). Each pier included a five-segment HSR-SD column with hollow square cross 336

section and a cap beam of trapezoidal solid section (see FIGURE 9 (b) and (c)). The PT system 337

included eight straight internal unbonded tendons of diameter of 0.6 inch and initial post-338

tensioning of 20 kips. The ducts had interior diameter of 0.9 inches. The duct adaptors had interior 339

diameter Dda = 1.375 inches and height hda =1.5 inches. A thin layer of silicon material was applied 340

at the interface of all pier joints to achieve a target coefficient of friction in the range of 0.08 to 341

0.1. For loading in the lateral direction, hm = 158 inches, while hp = 120 inches. 342

The specimen was subjected to several motions of various intensities. For the test considered 343

herein, the lateral and vertical component of a motion (ID No. 5 per FEMA P695 (24) – Delta 344

station, owned by UNAM / UCSD) recorded during the 1979 Imperial Valley earthquake scaled 345

to the MCE seismic hazard level (see FIGURE 10 (a)), as described in Sideris (2012). The 346

hysteretic response for the east and west pier is shown in FIGURE 10 (b). The peak displacement 347

reached approximately 6.5 inches. 348

The spectral demand was represented by the FEMA P695 acceleration response spectrum which 349

is shown in Eq. (20). 350

0

1 1

0

0

1

1

52 0.4 , 0

/ S,

0.2

,

XS

S S

S X s XSXSA S

s S

XS

TS for T T

B T

T S B BSS for T T T and

B T T

Sfor T T

B T

(20)

For the DE hazard, SXS = 1.49 g and SX1 = 0.86 g, whereas for the MCE hazard, SXS = 2.24 g and 351

SX1 = 1.29 g. Also, the reduction coefficients BS and B1 due to damping were obtained from FEMA 352

356 (29). For the MCE seismic hazard, the system performance is at 7.6 inches, which is fairly 353

close to the obtained response. 354

(a) (b) (c)

FIGURE 9: (a) Photo of the test specimen, (b) substructure column, and (c) column cross-section 355

Suspension

Safety System

Pier

Cap Beam

Foundation

Block

WEST SIDE EAST SIDE

Superstructure

(a)

(c)

(b)

FIGURE 10: (a) Acceleration response spectra (5% damped) compared to the MCE and DE 356 hazard, (b) Hysteretic response of the east and west pier in the lateral direction, and (c) Capacity 357 spectrum performance. 358

SUMMARY AND CONCLUSIONS 359

In this paper, a capacity spectrum method for the seismic design of hybrid sliding-rocking (HSR) 360

post-tensioned segmental bridges is presented. The method includes analytical derivation of the 361

pushover curve including the effects of rocking at the bottom of the HSR column as well as the 362

sliding at the intermediate joints along the column height. The equivalent viscous damping ratio 363

including the effects of damping from the initially elastic system and the energy dissipated due to 364

joint sliding is computed as a function of the system displacement. The spectral demand is obtained 365

from the FEMA 356 acceleration response spectrum and the equivalent viscous damping ratio. 366

The spectral capacity is obtained from the pushover curve. The point of intersection between the 367

spectral demand and capacity is the performance point of the structure. By varying the system 368

geometric and material properties (in an iterative manner), economical designs can be obtained. 369

The proposed method was applied to the design of a large scale single span bridge specimen 370

subjected to a set of shake table tests. The results of the proposed design method matched 371

reasonably well the results of the experimental program. 372

373

REFERENCES 374

1 ASCE's Report Card for America's Infrastructure (2013). "Report Card for America's 375

Infrastructure, Transportation - Bridges." 376

<http://www.infrastructurereportcard.org/a/#p/bridges/overview >. (2013). 377

2 Hieber, D. G., Wacker, J. M., Eberhard, M. O., Stanton, J. F. "Precast Concrete Systems for 378

Rapid Construction of Bridges." Washington State Transportation Center, University of 379

Washington, Seattle, Washington, 2005. 380

3 Pang, J. B., Eberhard, M. O., Stanton, J. F. "Large-Bar Connection for Precast Bridge Bents in 381

Seismic Regions." Journal of Bridge Engineering, 2009;15(3):231-239. 382

4 Stanton, J., Eberhard, M. "Accelerating Bridge Construction in Regions of High Seismicity." 383

Proceedings of the Special International Workshop on Seismic Connection Details for 384

Segmental Bridge Construction - Technical Report MCEER-09-0012, Multidisciplinary Center 385

for Earthquake Engineering Research, Buffalo, NY, U.S.A., Seattle, Washington, 2009. 386

5 Steuck, K. P., Eberhard, M. O., Stanton, J. F. "Anchorage of Large-Diameter Reinforcing Bars 387

SA

(g)

0.0

1.5

3.0

4.5

0 0.2 0.4 0.6 0.8 1

SA

(g

)

T (sec)

DE (AASHTO 2007)

MCE

Lateral Dir. - West Table

Lateral Dir. - East Table

T (sec)

SD

SA

-5 0 5-40

-20

0

20

40

Relat. Displ. ucb

X

(in)

Base S

hear

(kip

s)

Pier Hysteresis in the X-Direction

East

West

-5 0 5-40

-20

0

20

40

Interface Sliding uX (in)

Inte

rface S

hear

(kip

s)

Interface Hysteresis in the X-Direction

East

West

-5 0 5-40

-20

0

20

40

Relat. Displ. ucb

Y

(in)

Base S

hear

(kip

s)

Pier Hysteresis in the Y-Direction

East

West

-5 0 5-40

-20

0

20

40

Interface Sliding uY (in)

Inte

rface S

hear

(kip

s)

Interface Hysteresis in the Y-Direction

East

West

Displacement (in)

-5 0 5-40

-20

0

20

40

Relat. Displ. ucb

X

(in)

Base S

hear

(kip

s)

Pier Hysteresis in the X-Direction

East

West

-5 0 5-40

-20

0

20

40

Interface Sliding uX (in)

Inte

rface S

hear

(kip

s) Interface Hysteresis in the X-Direction

East

West

-5 0 5-40

-20

0

20

40

Relat. Displ. ucb

Y

(in)

Base S

hear

(kip

s)

Pier Hysteresis in the Y-Direction

East

West

-5 0 5-40

-20

0

20

40

Interface Sliding uY (in)

Inte

rface S

hear

(kip

s) Interface Hysteresis in the Y-Direction

East

West

in Ducts." ACI Structural Journal, 2009;106(4):506-513. 388

6 Matsumoto, E. E., Waggoner, M. C., Kreger, M. E., Vogel, J., Wolf, L. "Development of a 389

Precast Concrete Bent-Cap System." PCI Journal, 2008;53(3):74-99. 390

7 Matsumoto, E. K., ME; Waggoner, MC; Sumen, G. "Grouted Connection Tests in 391

Development of Precast Bent Cap System." Transportation Research Record, 2002;(1814):55-392

64. 393

8 Restrepo, J. I., Tobolski, M. J., Matsumoto, E. E. "NCHRP Report 681: Development of a 394

Precast Bent Cap System for Seismic Regions." NCHRP 12-74, National Cooperative 395

Highway Research Program (NCHRP), Transportation Research Board of National 396

Academies, 2011. 397

9 Hewes, J. T. "Seismic Tests on Precast Segmental Concrete Columns with Unbonded 398

Tendons." Bridge Structures: Assessment, Design and Construction, 2007;3(3):215 - 227. 399

10 Hewes, J. T., Priestley, M. J. N. "Seismic Design and Performance of Precast Concrete 400

Segmental Bridge Columns." Department of Structural Engineering, University of California, 401

San Diego, La Jolla, California 92093-0085, 2002. 402

11 Chou, C. C., Chen, Y. C. "Cyclic tests of post-tensioned precast CFT segmental bridge 403

columns with unbonded strands." Earthquake Engineering & Structural Dynamics, 404

2006;35(2):159-175. 405

12 ElGawady, M. A., Dawood, H. M. "Analysis of Segmental Piers consisted of Concrete Filled 406

FRP tubes." Engineering Structures, 2012;38:142-152. 407

13 Ou, Y.-C., Tsai, M. S., Chang, K. C., Lee, G. C. "Cyclic behavior of precast segmental concrete 408

bridge columns with high performance or conventional steel reinforcing bars as energy 409

dissipation bars." Earthquake Engineering & Structural Dynamics, 2010;39(11):1181-1198. 410

14 Wang, J. C., Ou, Y. C., Chang, K. C., Lee, G. C. "Large-scale seismic tests of tall concrete 411

bridge columns with precast segmental construction." Earthquake Engineering & Structural 412

Dynamics, 2008;37(12):1449-1465. 413

15 Ou, Y.-C., Wang, P.-H., Tsai, M.-S., Chang, K.-C., Lee, G. C. "Large-Scale Experimental 414

Study of Precast Segmental Unbonded Posttensioned Concrete Bridge Columns for Seismic 415

Regions." Journal of Structural Engineering, 2010;136(3):255-264. 416

16 Ou, Y.-C., Chiewanichakorn, M., Aref, A. J., Lee, G. C. "Seismic Performance of Segmental 417

Precast Unbonded Posttensioned Concrete Bridge Columns." Journal of Structural 418

Engineering, 2007;133(11):1636-1647. 419

17 Motaref, S., Saiidi, M., Sanders, D. "Shake Table Studies of Energy Dissipating Segmental 420

Bridge Columns." Journal of Bridge Engineering, 2013;TBD(TBD):TBD. 421

18 Roh, H., Reinhorn, A. M. "Hysteretic Behavior of Precast Segmental Bridge Piers with 422

Superelastic Shape Memory Alloy Bars." Engineering Structures, 2010;32(10):3394-3403. 423

19 Marriott, D., Pampanin, S., Palermo, A. "Quasi-static and Pseudo-dynamic Testing of 424

Unbonded Post-tensioned Rocking Bridge Piers with External Replaceable Dissipaters." 425

Earthquake Engineering & Structural Dynamics, 2009;38(3):331-354. 426

20 Billington, S. L., Yoon, J. K. "Cyclic Response of Unbonded Posttensioned Precast Columns 427

with Ductile Fiber-Reinforced Concrete." Journal of Bridge Engineering, 2004;9(4):353-363. 428

21 Sideris, P. "Seismic Analysis and Design of Precast Concrete Segmental Bridges." Ph.D. 429

Dissertation, State University of New York at Buffalo, Buffalo, NY, U.S.A. 2012. 430

22 Sideris, P., Aref, A. J., Filiatrault, A. "Large-scale Seismic Testing of a Hybrid Sliding-431

Rocking Post-Tensioned Segmental Bridge System." Journal of Structural Engineering, 432

2014;140(6):04014025. 433

23 Sideris, P., Aref, A. J., Filiatrault, A. "Quasi-Static Cyclic Testing of a Large-Scale Hybrid 434

Sliding-Rocking Segmental Column with Slip-Dominant Joints." Journal of Bridge 435

Engineering (In press, available online), 2014. 436

24 FEMA P695. "Quantification of Building Seismic Performance Factors." Federal Emergency 437

Management Agency, Washington, D.C., 2009. 438

25 Priestley, M. J. N., Calvi, G. M., Kowalsky, M. J. Displacement-Based Seismic Design of 439

Structures, IUSS Press, Pavia, Italy, 2007. 440

26 Fajfar, P. "Capacity Spectrum Method based on Inelastic Demand Spectra." Earthquake 441

Engineering & Structural Dynamics, 1999;28(9):979-993. 442

27 Freeman, S. A. "The Capacity Spectrum Method as a Tool for Seismic Design." Proc., 443

Proceedings of the 11th European conference on earthquake engineering 6-11. 444

28 Freeman, S. A. "Review of the Development of the Capacity Spectrum Method." ISET Journal 445

of Earthquake Technology, 2004;41(1):1-13. 446

29 FEMA 356. Prestandard and Commentary for the Seismic Rehabilitation of Buildings Federal 447

Emergency Management Agency, Washington, D.C., 2000. 448

30 The Mathworks, I. "MATLAB." The Mathworks, Inc., 2014. 449

31 Paulay, T., Priestley, M. J. N. Seismic Design of Reinforced Concrete and Masonry Buildings, 450

John Wiley & Sons, Inc, 1992. 451

32 Mander, J. B., Priestley, M. J. N., Park, R. "Theoretical Stress-Strain Model for Confined 452

Concrete." Journal of Structural Engineering, 1988;114(8):1804-1826. 453

33 Harajli, M. "Proposed Modification of AASHTO-LRFD for Computing Stress in Unbonded 454

Tendons at Ultimate." Journal of Bridge Engineering, 2011;16(6):828-838. 455

34 Restrepo, J., Rahman, A. "Seismic Performance of Self-Centering Structural Walls 456

Incorporating Energy Dissipators." Journal of Structural Engineering, 2007;133(11):1560-457

1570. 458

35 ASCE/SEI 7-10. " Minimum Design Loads for Buildings and Others Structures." American 459

Society of Civil Engineers (ASCE), Reston, VA, 2010. 460

461