Effects of Column Stiffness Irregularity on the Seismic Response of Bridges in the Longitudinal...

7/30/2019 Effects of Column Stiffness Irregularity on the Seismic Response of Bridges in the Longitudinal Direction

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Effects of Column Stiffness Irregularity on the Seismic Response of Bridges1

in the Longitudinal Direction2

Payam Tehrani1

and Denis Mitchell13

45

Abstract: The longitudinal seismic responses of 4-span continuous bridges designed based on6

the 2006 Canadian Highway Bridge Design Code were studied using elastic response spectrum7

and inelastic time-history analyses. Several boundary conditions including unrestrained8

horizontal movements at the abutments and different abutment stiffnesses were considered in the9

nonlinear analyses. The seismic response of more than 2600 bridges were studied to determine10

the effects of different design and modelling parameters including the effects of different column11

heights, column diameters, and superstructure mass as well as different abutment stiffnesses. The12

bridges were designed using two different force modification factors of 3 and 5. The effects of13

column stiffness ratios on the elastic and inelastic analysis results, maximum ductility demands,14

concentration of ductility demands, and demand to capacity ratios were investigated. The results15

indicate that the seismic response and maximum ductility demands in the longitudinal direction16are influenced by important parameters such as the total stiffness of the substructure, the column17

stiffness ratio and the aspect ratio of the columns.18

19

Key words:Irregular bridges, Column stiffness, Bridge abutment, Inelastic time history analysis,20

CHBDC 2006, NBCC 2010, Seismic response21

22

23

24

Rsum:2526

27 Mots cls28

1 McGill University, Department of Civil Engineering, 817 Sherbrooke St. West, Montreal QC H3A 0C3

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Introduction30

Bridges with significantly different column heights result in considerable concentration of31

seismic demands in the stiffer, shorter columns. In some cases, the deformation demands on the32

short columns can cause failure before the longer, more flexible columns yield. In addition, the33

sequential yielding of the ductile members may result in substantial deviations of the nonlinear34

response predictions from the linear response predictions made with the assumption of a global35

force reduction factor, R. This difference is due to the fact that plastic hinges, which appear first36

in the stiffer columns, may lead to concentrations of unacceptably high ductility demands in37

these columns. Where possible, the stiffness of piers should be adjusted to attempt to achieve38

uniform yield displacements and ductility demands on individual columns. A summary of the39

methods to improve the seismic performance of such bridges is discussed by Tehrani and40

Mitchell (2012).41

Examples of earthquake damage to irregular bridges with different column heights (e.g.,42

failure of the shorter columns), has been reported (Broderick and Elnashai, 1995; Mitchell et al.43

1995; Chen and Duan 2000). The transverse response of bridges with different column heights44

and different superstructure stiffnesses was studied by Tehrani and Mitchell (2012) which45

demonstrated that irregularities due to different column stiffnesses have significant effects on the46

seismic behaviour of bridges. Research is needed to investigate if these effects are also important47

in the longitudinal direction and to evaluate the seismic safety of bridges with column stiffness48

irregularities. In this paper, the effects of different column stiffnesses and stiffness ratios on the49

longitudinal responses of bridges are presented. In addition, the influence of the abutments on the50

seismic response of bridges is investigated. The main parameters in this study include the51

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column heights and diameters, different force modification factors, abutment stiffness and52

strength, and the hysteresis stiffness degradation parameters used in the nonlinear analyses. The53

influence of such parameters on the maximum ductility demands, maximum drift ratios,54

concentrations of ductility demands, ductility demand to capacity ratios, as well as comparisons55

of predictions from the elastic and inelastic analyses, are presented.56

Modelling and analysis of the bridges57

A computer program was developed (Tehrani 2012) to design the columns and to carry out58

moment-curvature analyses to compute the curvatures corresponding to different steel and59

concrete strains used as damage indicators. The displacements, drifts, curvature ductilities, and60

displacement ductility capacities then can be computed for different performance levels. The61

confinement effects in the concrete core were considered using the Mander equation (Mander et62

al. 1988) in the moment-curvature analysis assuming that spiral confinement reinforcement was63

provided in accordance with the provisions of the Canadian Highway Bridge Design Code64

(CHBDC) (CSA 2006). The bilinear idealization of the moment curvature curves and the65

determination of the effective curvature at yield, plastic hinge lengths and strain penetration66

depths for the vertical bars were included in the analyses based on the recommendations by67

Priestley et al. (1996 and 2007). More details are given by Tehrani and Mitchell (2012).68

The modified Takeda hysteresis model (Otani 1981) was used in this study to represent the69

behaviour of the RC columns using Ruaumoko software (Carr 2009). Moment-curvature70

responses and predicted plastic hinge lengths are used as input to the RUAUMOKO program for71

the nonlinear analysis. This model has two main parameters, and , which control the72

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unloading and the reloading stiffness, respectively (see Fig. 1). The parameter is usually in the73

range of 0 to 0.5 and varies between 0 and 0.6. Increasing the parameter decreases the74

unloading stiffness and increasing the parameter increases the reloading stiffness. For bridge75

columns, Priestley et al. (1996 and 2007) recommend using the conservative values of = 0.576

and =0.77

Range of parameters studied78

The seismic responses in the longitudinal direction of 4-span continuous straight bridge79

structures were studied (see Fig. 2). For the bridges studied the superstructure is continuous, the80

columns are hinged at the top and the expansion joints are situated at the abutments. According81

to Priestley et al. (2007), the longitudinal and transverse behaviour may be studied independently82

for straight bridges, such as those considered in this study.83

The bridge structures were designed according to the CHBDC (CSA 2006). CSA S6-06 uses84

R=3 for single columns and R=5 for multiple column bents. Both of these values were used in85

the design of bridges to investigate the influence of the R factor on the response of irregular86

bridges. An importance factor of 1.5 (i.e., I = 1.5 for emergency-route bridges) was used for the87

design and the bridges were located Vancouver assuming a site class C (i.e., 360 Vs30 76088

m/sec).89

In accordance with the minimum and maximum longitudinal column reinforcement ratios in90

CSA S6 (2006), a range of ratios between 0.8% and 6.0% was investigated. The transverse steel91

ratios were determined based on the CHBDC 2006 provisions to satisfy the requirements for92

confinement in the plastic hinge regions and to provide factored shear resistances corresponding93

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to the capacity design philosophy. However in most cases the spiral confinement reinforcement94

ratio of 1.2% controlled. The concrete compressive strength and the yield stress of the95

reinforcing bars were taken as 40 and 400 MPa, respectively.96

To investigate the effects of different column heights, different diameters and varying column97

stiffness ratios on the seismic response, a parametric study was carried out. Column heights were98

varied from 7 to 28 m with increments of 3.5 m (i.e., 7 different heights for each column). The99

column diameters were 1.5, 2.0, and 2.5 m and in each configuration the three columns all had100

the same diameters (see Fig. 2). It was assumed that the position of columns along the bridge has101

no effect on the seismic response in the longitudinal direction, since the columns are hinged at102

the top and the stiffness of the superstructure is large in the longitudinal direction. These103

combinations of column heights and diameters resulted in 252 bridges with different column104

arrangements. The superstructure mass was considered as a uniformly distributed load of 200105

kN/m. The effect of increasing the superstructure mass to 300 kN/m on the seismic response was106

also investigated for some cases. Further, different hysteresis parameters, and , were used in107

the structural modelling. In addition, the effects of abutment stiffness and capacity on the seismic108

response of bridges were also considered for different number of piles and different gap lengths109

between the superstructure and the abutment. Rigid links were used to model the superstructure110

depth, while the columns were hinged at the top. A schematic view of the structural models is111

shown in Fig. 3. Considering all of the parameters, more than 2600 bridge structures with112

different geometries, designs and modelling parameters were studied.113

For the inelastic time history analyses, 7 spectrum matched records were used. These artificial114

records were generated using the SIMQKE software (Vanmarcke and Gasparini, 1979 and Carr,115

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2009). For the nonlinear analyses of the bridges, the records were matched to the design spectra116

given in the National Building Code of Canada (NBCC) (NRCC 2010). These spectra117

correspond to 2% probability of exceedance in 50 years, while the design spectrum given in the118

CHBDC (CSA 2006) corresponds to 10% probability of exceedance in 50 years. The 2010119

NBCC spectrum with the hazard level of 2% in 50 years, which will be used in the next edition120

of the CHBDC, was used to evaluate the seismic behaviour of the bridges. For this more121

appropriate probability of occurrence, the effects of irregularity on the seismic response are122

expected to be more pronounced. The bridges should not collapse at this seismic hazard level.123

The resulting average response spectrum of the seven records used along with the design124

spectra for Vancouver based on the CHBDC (CSA 2006) and the NBCC (NRCC 2010) are125

shown in Fig. 4. The average displacement for each bridge was determined using the averaging126

procedure proposed by Priestley et al. (2007) which considered the maximum positive and127

maximum negative displacements predicted for each input record.128

The capacities of the columns were determined assuming that the maximum strains in steel129

bars attained the bar buckling strain limits given by Berry and Eberhard (2007) and the130

maximum concrete compression strain predicted by the Mander et al. (1988) equation modified131

based on the recommendations by Priestley et al. (2007) for the life safety limit state. More132

details concerning the evaluation of ductility capacity of columns using different methods are133

available in Tehrani and Mitchell (2012).134

135

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Modelling the abutments136

Fig. 5 shows the abutment details, with a gap between the backwall and the superstructure137

which is supported on bearing pads. The simplified abutment model developed by Mackie and138

Stojadinovic (2003) and Aviram et al. (2008) was used to study the influence of the abutments139

on the seismic response of the bridges in the longitudinal direction. The longitudinal response is140

a function of the system response including the elastomeric bearing pads, the gap, the abutment141

backwall, the abutment piles, and the soil backfill material. Prior to impact due to gap closure,142

the superstructure forces are transmitted through the elastomeric bearing pads to the abutment,143

and subsequently to the piles and backfill, in a series system. After gap closure, the144

superstructure bears directly on the abutment backwall and mobilizes the full passive backfill145

pressure (Aviram et al. 2008). In the simplified model used the effects of the bearing pads on the146

responses are ignored (i.e., free movements of superstructure before gap closure). However, it147

has been shown that the results from the simplified abutment models in the longitudinal direction148

are in good agreement with those obtained using more detailed models (Aviram et al. 2008).149

The abutment stiffness, Kabt, and its ultimate strength, Pbw, are obtained from Eq. [1] and [2]150

from Caltrans (2006) which are based on a study by Maroney and Chai (1994).151

152

[1] Kabt=Ki wbw (hbw/1.7)153

154[2] Pbw= p Ae (hbw/1.7)155

156

[3] Ae= hbw wbw157

158

where Ki is the initial embankment fill stiffness and is taken as 11500 kN/m/m, p is the159

passive soil pressure taken as 239 kPa and Ae is the effective abutment area, defined as the area160

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which is effective for mobilizing the backfill. The effective backwall area is given by Eq. [3]161

(Caltrans 2006), with hbw and wbw defined in Fig. 5. In this study hbw and wbw were taken as 2.0162

m and 9.0 m, respectively based on the superstructure geometry.163

To model the abutments the bilinear hysteresis loop model with slackness (Carr, 2009), as164

shown in Fig. 6a, was used in the RUAUMOKO software (Carr, 2009). This hysteresis model165

includes a gap and a spring in series which can be used to model the initial gap and the resulting166

stiffness and strength associated with the abutments. An example of the hysteretic behaviour of167

such an element obtained in the analyses is demonstrated in Fig. 6b. As shown, the abutment168

elements only resist compression forces.169

Different abutment stiffnesses and strengths were considered in the structural modelling to170

study the seismic response of bridges in the longitudinal direction including cases with (see Fig.171

5) and without piles. To estimate the stiffness and strength of the piles the empirical pile resistant172

equations given by Goel and Chopra (1997) (see Eqs. [4] and [5]) were used. These equations173

provide an ultimate strength that is assumed to occur at 1 in. (25 mm) displacement. The174

maximum displacement of the piles was taken as 2.4 in. (60 mm). The stiffness of the piles was175

conservatively neglected when deformations exceeded this value, since the abutments and back176

walls are expected to be damaged at deformations higher than this level (Goel and Chopra 1997).177

The combined response of the backfill and piles is determined by the combined response178

predicted by Eqs. [1] to [5].179

[4] Rpile =40 kips / pile = 178 KN/pile180181

[5] Kpile =40 kips /in. = 7000 KN/m182

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Four different gap lengths (i.e., 25, 50, 75, and 150 mm) along with three different pile183

configurations (i.e., 0, 10, and 20 piles) were considered for modelling the influence of the184

abutments. In addition, to study the influence of the abutments on different bridge185

configurations, the column heights were varied from 7 to 28 m for each column with increments186

of 7 m.187

The addition of the backfill contribution without piles provided a significant decrease in the188

ductility demand. The seismic responses were only slightly improved when piles were added to189

the abutments, however the effects of increasing the number of piles were more pronounced for190

the case of smaller gap lengths and for more flexible columns.191

The reductions of ductility demand in the bridge columns are shown in Fig. 7b as a function192

of the ratio of the total stiffness of the columns, Kcols, and the abutment effective stiffness, Keff-abt.193

This reduction is measured with respect to the case when the superstructure has no horizontal194

restraint at the abutments. The abutment effective stiffness, Keff-abt, accounting for the gap195

closure,is defined in Fig. 7a. As this ratio, Kcols / Keff-abt, decreases the influence of the abutments196

in the seismic response becomes more pronounced with up to 80% reduction in the ductility197

demands. This indicates that the seismic response of bridges with flexible columns will be198

affected more significantly by including the abutments in the structural modelling.199

Evaluation of maximum ductility demands200

An important outcome of the nonlinear dynamic analyses is the maximum column ductility201

demands which will be used to assess the seismic performance. The effects of column stiffness202

irregularities on the maximum displacement ductility demands were investigated. For each203

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bridge the ductility demand in the most critical column is reported as the maximum ductility204

demand in the bridge. The results are based on the average responses obtained by means of205

inelastic time history analysis using 7 spectrum matched records.206

The effects of the total stiffness of the columns on the maximum ductility demands obtained207

from nonlinear dynamic analyses are presented in Fig. 8. The stiffness of columns is defined as208

3EIeff/H3

where EIeff is taken as My/y (general yield moment divided by yield curvature)209

determined from moment-curvature analysis, and H is the height of the column. As expected the210

stiffer structures typically attract higher ductility demands in the columns. Another important211

parameter which will influence the maximum column ductility demand is the maximum stiffness212

ratio of the columns (i.e., maximum column stiffness (i.e., stiffness of the shortest column, KS)213

divided by minimum column stiffness (i.e., stiffness of the longest column, KL). Larger column214

stiffness ratios typically result in a concentration of ductility demand in the stiffest column which215

in turn imposes higher ductility demands on this element. The influence of the maximum column216

stiffness ratio, KS/KL, on the maximum ductility demand of columns is depicted in Fig. 9. Since217

both the total stiffness of the columns and the maximum stiffness ratio of columns affect the218

maximum ductility demands, another parameter has been defined as the product of these two219

variables (i.e., total stiffness of columns times maximum stiffness ratio of columns). As220

presented in Fig. 10, this new parameter shows an improved correlation with the maximum221

ductility demands in the columns. The results are presented for two different force modification222

factors of 3 and 5 used in design. As the force modification factor, R, increases, the scatter in the223

maximum predicted ductility demands increases and the maximum ductility demands become224

more sensitive to the product of the total stiffness of the columns and the maximum stiffness225

ratio, as indicated by the slope of the regression lines in Figs. 8 to 10.226

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The correlation between the maximum ductility demand and the product of the total stiffness227

of columns and maximum stiffness ratio is even better, as shown in Fig. 11, when the influence228

of the abutments was included. This is probably due to the fact that the inclusion of the229

abutments in the structural modelling significantly reduces the nonlinear geometric effects due to230

P-Delta effects.231

The results presented in Figs. 8 to 10 were derived assuming the unloading and reloading232

hysteresis parameters of =0.5 and =0 in the modified Takeda hysteresis model. The maximum233

ductility demands for predictions using hysteresis parameters of =0 and =0.6 (i.e., lower234

bound values) and also =0.3 and =0.3 (i.e., close to the average values from tests) were about235

3.0 and 3.2, respectively, while in the case of =0.5 and =0 (i.e., upper bound values) the236

maximum ductility demands were about 3.5. The influence of the hysteresis parameters was237

larger for the bridges with higher values of the parameter total stiffness of columns times the238

max stiffness ratio. However, in general, the influence of the hysteresis parameters on the239

seismic response was not very significant. These effects were even smaller when a force240

modification factor of R=3 was used in design, due to the smaller nonlinear deformations in the241

columns.242

Predictions from inelastic versus elastic analysis243

A study by Tehrani and Mitchell (2012) demonstrated that column stiffness irregularities can244

result in significant deviations of the elastic multi-mode analysis results from the inelastic245

dynamic analysis when the transverse response of bridges were studied. As shown in Fig. 12 the246

differences in the maximum displacement predictions using the elastic and inelastic analyses247

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were typically small for response in the longitudinal direction. The differences were generally248

less than 20% with higher differences when the total stiffness of columns was quite low. This is249

probably due to P-Delta effects and the higher dispersions of the response spectra of the ground250

motion records in the longer period range. However, for the bridges with lower total column251

stiffness the maximum ductility demands are typically small (e.g., see Fig. 8). The ratio of the252

displacements obtained from the inelastic and elastic analyses were not significantly affected by253

the maximum stiffness ratio of the columns. The use of R=3 in design also led to similar results254

obtained for the case of R=5.255

The ratios of the maximum displacement demands obtained using the inelastic and elastic256

analyses are presented in Fig. 12a-c for different hysteresis loop parameters. The Takeda257

hysteresis loop model with =0 and =0.6 represents no unloading stiffness degradation and258

small reloading stiffness degradation (i.e., lower bound values), while the choice of =0.5 and259

=0 overestimates the unloading and reloading stiffness degradation (i.e., upper bound values).260

Nevertheless, the effects of using different hysteresis parameters are not significant and the261

differences between the inelastic and elastic results are typically small, as shown in Fig. 12a -c. It262

should be noted that the equal displacement concept is based on bi-linear hysteresis models with263

no stiffness degradation. When stiffness degradation is considered in the nonlinear response,264

somewhat different predictions may be obtained.265

For the cases where the effects of the abutments are included in the nonlinear analysis the266

resulting displacements for the flexible structures will be much smaller (see Fig. 12d) than those267

predicted with the assumption of free longitudinal movements at the ends (see Fig. 12b). The268

elastic responses were computed assuming free movement at the abutments in the longitudinal269

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direction (i.e., roller supports), an assumption typically made in practice for design and analysis.270

Such an assumption is conservative, as is evident by comparing the results in Fig. 12b with those271

shown in Fig. 12d.272

Concentration of seismic demands273

In the longitudinal direction of continuous bridges all of the columns have almost equal274

displacement demands. On the other hand, in the transverse direction the columns will have275

different displacements depending on: a) stiffness and position of the columns; b) the276

superstructure transverse stiffness and c) the abutment restraint conditions.277

The influence of the height ratio of columns, HL/HS, on the maximum to minimum (max/min)278

ductility demands, S/L, for response in the longitudinal direction is shown in Fig. 13 , where S279

and L are the maximum ductility demands in the shortest and longest columns, respectively.280

Increasing the height ratio of the columns leads to higher concentrations of ductility demands in281

a few columns.282

The relationship between the column height ratio and the maximum to minimum ductility283

demands can be derived, assuming that the displacement demand, d, is equal for all columns in284

the longitudinal direction. The maximum and minimum ductility demands can be computed285

using Eq. [6a-b], where y is the displacement at yield, is the displacement ductility demand286

and the subscripts S and L refer to the shortest and longest columns, respectively.287

[6a]( )

dS

y S

=

and [6b]( )

dL

y L

=

288

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The strain penetration depth, Lsp, is much smaller than the column height and can be ignored.289

The displacement at general yielding, y, can be approximated using Eq.[7a-b] for the cantilever290

columns, where HL and HS are the heights of the longest and shortest columns, respectively.291

[7a]

2

( )3

L

y L

H = & [7b]

2

( )3

S

y S

H = 292

293

The yielding curvature, y , is mainly a function of the column diameter and the yield strain294

of the reinforcing bars and can be estimated using Eq. [8] for circular columns (Priestley et al.,295

2007).296

[8]2.25 y

yD

= 297

Since all of the columns in this study have the same diameters for each configuration, it can298

be assumed that the yielding curvature of the columns are almost equal and thus the maximum to299

minimum ductility demand ratio can be estimated using Eq. [9].300

[9]2

( )( ) ( )

( )

y LS SL

L y S S L

DH

H D

= =

301

where (DS/ DL) is the ratio of the column diameter of the shortest column to that of the longest302

column. Eq. 9 shows that if the column heights are different, the only way to avoid concentration303

of ductility in the short column is to makeDLlarger thanDS. The predicted maximum to304

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minimum ductility demands using Eq. [9] are compared with the results from nonlinear analysis305

in Fig. 13.The minor differences are due to the simplifications made in deriving Eq.[9].306

307

Drift ratios308

The maximum drift ratios of columns obtained using nonlinear dynamic analyses are shown309

in Fig. 14a. These maximum drift ratios are generally around 0.5% to 3.5% and typically310

decrease with increasing values of total stiffness of the columns. The drift ratios were relatively311

similar for different R values and hysteretic parameters. The addition of the abutments in the312

structural modelling decreased the maximum drift ratio by about 1.5%, as shown inFig. 14b313

especially for the case of more flexible columns.314

A comparison of the maximum drift ratio versus the maximum ductility demand of the315

columns is shown in Fig. 15a. Drift ratios are widely used in practice (e.g., in codes) for the316

assessment of structural performance. However the drift ratios may not be a good indicator of317

structural damage. As indicated before, the ductility was found to be proportional to ( D) / H2318

(see Eq. [9]). In Fig. 15b another damage indicator is defined as (max D) /HS2

(i.e., max drift319

ratio divided by the aspect ratio). This new parameter shows a significantly improved correlation320

with the column ductility demands, as shown Fig. 15b, compared to Fig. 15a. Therefore this321

parameter is a better indicator of damage, since the ductility demands and the corresponding322

structural damage are proportional to D/H2

rather than 1/H.323

324

325

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Ductility demand versus ductility capacity326

The maximum ductility demands obtained from the nonlinear dynamic analyses were327

compared to the ductility capacities of the columns to evaluate the safety margin of the columns.328

In Fig. 16 the ratio of the maximum ductility demands to the ductility capacities are shown329

versus the maximum stiffness ratio of the columns for different R factors. The ductility capacity330

for each column is a function of the geometric properties and details of reinforcement. These331

ductility capacities were computed for the life safety performance level (i.e., no collapse).332

The ductility demand to ductility capacity ratios from the analyses in the longitudinal333

direction are in the range of 0.2 to about 0.5 when a force modification factor of R=5 was used334

(e.g., see Fig. 16a). In the case of R=3 these ratios were around 0.2 to 0.4 as shown in Fig. 16b.335

Similar results were obtained when the transverse response of similar bridges were studied336

(Tehrani and Mitchell 2012). Using a lower force modification factor of R=3 did not337

significantly increase the safety margins as shown in Fig. 16b. It is noted that the minimum338

reinforcement ratio of 0.8% governed the design in some cases.339

As the maximum stiffness ratio of the columns increases, the range of the maximum demand340

to capacity ratios obtained increases as well. For example in Fig. 16a when the maximum341stiffness ratios are less than about 5.0 the maximum demand to capacity ratios are less than342

around 0.3. For maximum stiffness ratios between 5.0 to 10.0 the maximum demand to capacity343

ratio is around 0.4 and exceeding the stiffness ratio of 10.0 can lead to demand to capacity ratios344

of about 0.5.345

When lower stiffness degradations were considered in the modelling (i.e., =0.3 and =0.3),346

the maximum demand to capacity ratios decreased to around 0.42 for the case of R=5. The347

influence of hysteresis parameters was more pronounced for bridges with higher column stiffness348

ratios, possibly due to the higher nonlinear deformations and the concentration of nonlinear349

demands on the columns. When the superstructure mass was increased to 300 kN/m the effects350

of the column stiffness ratios became even more important. Including the influence of the351

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abutments in the structural modelling reduced the maximum demand to capacity ratios to about352

0.35 as shown in Fig. 17.353

Normalized ductility demand354

To improve the correlation of the results a new parameter is defined as the normalized355

ductility demand which is simply calculated by dividing the maximum ductility demands by the356

minimum aspect ratio of the columns (i.e., HS/D). It was observed that the use of the average357

stiffness ratio of all of the columns in lieu of the maximum stiffness ratios can also slightly358

reduce the scatter in the results. The average stiffness ratio was computed as (Have/HS)3

where359

Have is the average height of all columns. The normalized ductility demand and normalized360

demand to capacity ratios are presented in Fig. 18a and b. As can be seen by introducing these361

parameters the correlation has been significantly improved compared to the results presented in362

Figs. 8 to 10. Hence, the overall seismic response and the maximum ductility demands in the363

longitudinal direction are controlled by at least three important parameters including the total364

stiffness of the substructure, the stiffness ratio of the columns, and the minimum aspect ratio of365

the columns.366

Demand to capacity ratios considering transverse and longitudinal responses367

A combination of orthogonal seismic displacement demands are often used to approximately368

account for the directional uncertainty of earthquake motions and the simultaneous occurrence of369

earthquake effects in the two perpendicular horizontal directions. Based on the AASHTO guide370

specifications (AASHTO 2009) the seismic displacements resulting from analyses in the two371

perpendicular directions can be combined. The seismic demand displacements can be obtained372

by adding 100% of the seismic displacements resulting from the analysis in one direction to 30%373

of the seismic displacements resulting from the analysis in the perpendicular direction and vice-374

versa to form two independent cases (AASHTO 2009).375

The transverse responses of similar bridges were studied by Tehrani and Mitchell (2012). To376

estimate the resulting displacement ductility demands due to bidirectional ground motions the377

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resulting displacements from the analysis in the longitudinal and transverse directions were378

combined using the 100% / 30% rule stated above.379

The resulting displacement ductility demand to capacity ratios are shown in Fig. 19. The380

ductility capacities of the columns were computed based on the Life Safety performance level381

(i.e., collapse prevention) according to the recommendations of Priestley et al. (2007). The382

beneficial effects from the abutments in the longitudinal direction were conservatively neglected.383

The resulting demand to capacity ratios are generally less than 0.7 considering the combination384

of maximum ductility demands from transverse and longitudinal directions. The use of the SSRS385

combination rule also resulted in similar predictions with ductility demands being about 5%386

larger on average. As can be seen in Fig. 19, the demand to capacity ratios are less than 0.5 for387

the majority of cases. However as the maximum stiffness ratio of columns exceeds about 8.0 the388

demand to capacity ratios are increased by 40%.389

Conclusions390

Bridges with different configurations were designed based on the 2006 Canadian Highway391

Bridge Design Code (CHBDC). Non-linear time history analyses were used to predict the392

longitudinal seismic responses of these bridges using 7 spectrum-matched records. The393

conclusions from this study are summarized as follows:394

(1) The seismic response and the maximum ductility demands in the longitudinal395direction are controlled by the total stiffness of the substructure, the stiffness ratio of the396

columns, and the minimum aspect ratio of the columns. Seismic ductility demands in the397

longitudinal direction were correlated with the product of the total stiffness of the columns398

and the maximum stiffness ratio of the columns. This indicates that the ductility demands in399

bridge columns increase as the structural stiffness and stiffness irregularity increases.400

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(2) It was demonstrated that the concentration of ductility demands increases401significantly with an increase in the column stiffness ratio, KS/KL or column height ratio,402

HS/HL.403

(3) The influence of the abutments on the longitudinal seismic responses of bridges404was studied. Up to an 80% decrease in seismic ductility demands were observed when the405

abutments were considered in the structural models. The reduction of ductility demand406

correlates with the ratio of the total stiffness of the columns and the effective stiffness of the407

abutments. The influence of the abutments was more pronounced for the bridges with more408

flexible columns and stiffer abutments.409

(4) A dimensionless parameter was defined as( D) / HS2 (i.e.,drift ratio divided by410column aspect ratio) which provided an improved indicator of the structural damage411

compared to the conventional drift ratio (i.e., / H ). It was also demonstrated that412

normalizing the maximum ductility demands by the minimum aspect ratio of the columns413

significantly reduced the dispersions in the results.414

(5) The seismic ductility demand to ductility capacity ratios were estimated for the415combination of the seismic responses in the longitudinal and transverse directions. It was416

observed that the demand to capacity ratios were lower than 0.7 with the majority of the cases417

having values less than 0.5. These ratios decreased, when the influence of the abutments were418

considered in the seismic response. The range of demand to capacity ratios was quite high419

which indicate uneven safety margins for different bridges. Exceeding the maximum stiffness420

ratio of about 5.0 to 8.0 resulted in much larger demand to capacity ratios.421

(6) CSA S6-06 requires elastic dynamic analysis for an emergency-route bridge in422seismic performance zones 2 and higher if the bridge is irregular. This study indicates that the423

ge 19 of 44


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elastic dynamic analysis is appropriate for irregular bridges in the longitudinal direction.424

However, nonlinear dynamic analysis would be required for such irregular bridges in the425

transverse direction in order to accurately predict the displacement envelope and the ductility426

demands (Tehrani and Mitchell 2012).427

Acknowledgements428

The financial support provided by the Natural Sciences and Engineering Research Council of429

Canada for the Canadian Seismic Research Network is gratefully acknowledged.430

References431

AASHTO 2009, Guide specifications for LRFD seismic bridge design., Subcommittee T-3 for432

Seismic Effects on Bridges, American Association of State Highway and Transportation433

Officials, Washington D.C.434

Aviram, A., Mackie, K.R. and Stojadinovic, B. 2008. Effect of abutment modeling on the435

seismic response of bridge structures. Earthquake Eng & Eng Vibration, 7(4): 395-402.436

Berry, M.P. and Eberhard, M.O. 2007. Performance modeling strategies for modern reinforced437

concrete bridge columns. PEER-2007/07, Pacific Earthquake Engineering Research Center,438

University of California- Berkeley, Berkeley, Calif..439

Broderick, B.M., and Elnashai, A.S. 1995. Analysis of the failure of Interstate 10 freeway ramp440

during the Northridge earthquake of 17 January 1994, Earthquake Engineering & Structural441

Dynamics, 24(2): 189-209.442

Page 20


21/44

Caltrans. 2006. Seismic Design Criteria. California Department of Transportation, California.443

CSA. 2006. CAN/CSA-S6-06 Canadian Highway Bridge Design Code and commentary.444

Canadian Standards Association, Mississauga, ON.445

Carr, A. 2009. RUAUMOKO, a computer program for Inelastic Dynamic Analysis. Department446

of Civil Engineering, University of Canterbury, New Zealand.447

Chen, W.F. and Duan, L. 2000. Bridge Engineering Handbook, CRC Press LLC.448

Goel, R.K. and Chopra, A. 1997. Evaluation of bridge abutment capacity and stiffness during449

earthquakes, Earthquake Spectra, 13(1): 1-23.450

Mackie, K.R. and Stojadinovic, B. 2003. Seismic Demands for Performance-Based Design of451

Bridges. PEER-2003/16, Pacific Earthquake Engineering Research Center, University of452

California- Berkeley, Berkeley, Calif.453

Mander, J.B, Priestley, M.J.N., and Park, R. 1988. Theoretical stress-strain model for confined454

concrete. ASCE Journal of Structural Engineering, 114(8): 1804-1826.455

Maroney, B.H. and Chai, Y.H. 1994. Seismic Design and Retrofitting of Reinforced Concrete456

Bridges. Proceedings of 2nd International Workshop, Earthquake Commission of New457

Zealand, Queenstown, New Zealand.458

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Mitchell, D., Bruneau, M., Williams, M., Anderson, D.L., Saatcioglu, M., and Sexsmith, R.G.459

1995. Performance of bridges in the 1994 Northridge earthquake, Can. J. Civ. Eng. 22(2):460

415-427.461

NRCC. 2010. National Building Code of Canada, Associate Committee on the National Building462

Code, Ottawa, ON.463

Otani, S. 1981. Hysteresis model of reinforced concrete for earthquake response analysis. Journal464

of Fac. of Eng., Univ. of Tokyo, Series B, XXXVI-11 (2): 407-441.465

Priestley, M.J.N., Seible, F., and Calvi, G.M. 1996. Seismic design and retrofit of bridges, John466

Wiley and Sons, New York.467

Priestley, M.J.N., Calvi, G.M. and Kowalsky, M.J. 2007. Direct displacement based design of468

structures, Pavia, Italy.469

Tehrani, P. and Mitchell, D. 2012. Effects of column and superstructure stiffness on the seismic470

response of bridges in the transverse direction, Canadian. Journal of Civil Engineering (in471

press).472

Tehrani, P. 2012. Seismic analysis and behaviour of bridges, Ph.D. thesis, Dept. of Civil Eng.,473

McGill University, Montreal, QC.474

Vanmarcke, E. H. and Gasparini, D.A. 1976. Simulated earthquake motions compatible with475

prescribed response spectra, Research report R76-4, Dept. of Civil Engineering,476

Massachusetts Inst. of Technology, Cambridge.477

Page 22


23/44

List of figures:478

Fig. 1. Modified Takeda hysteresis loop (adapted from Carr (2009))479

Fig. 2. Bridge properties480

Fig. 3. Structural modelling of the bridges481

Fig. 4. Average response spectrum for 7 records used for inelastic time history analysis matching482

the 2010 NBCC spectrum (2% in 50 years)483

Fig. 5. Schematic view of the seat-type abutment and its components484

Fig. 6. Hysteresis modelling of the abutments: a) general model with gap and nonlinear spring485

(Carr 2009) b) typical nonlinear response of abutment forces from analysis486

Fig. 7. Influence of abutment stiffness: a) effective abutment stiffness (Caltrans 2006); b) the487

influence of the ratio of the total stiffness of columns to effective abutment stiffness on the488

maximum ductility demands489

Fig. 8. Effects of total stiffness of columns on the maximum displacement ductility demands: a)490

R=3, =0.5 and =0; b) R=5, =0.5 and =0491

Fig. 9. Effects of maximum column stiffness ratio on the maximum displacement ductility492

demands: a) R=3, =0.5 and =0; b) R=5, =0.5 and =0.493

ge 23 of 44


24/44

Fig. 10. Effects of columns total stiffness times max stiffness ratio on the maximum494

displacement ductility demands: a) R=3, =0.5 and =0; b) R=5, =0.5 and =0.495


displacement ductility demands considering the influence of abutments (Gap=50 mm, R=5,497

=0.5 and =0)498

Fig. 12. Effects of total stiffness of columns on the ratio of the displacements obtained using499

inelastic and elastic analysis for R=5 and: a) =0 and =0.6; b) =0.5 and =0; c) =0.3 and500

=0.3; d) considering the influence of the abutments on seismic response (no piles, gap=50 mm,501

=0.5 and =0)502

Fig. 13. Effects of maximum column height ratio on the Max/Min ductility ratio, S/L (R=5,503

=0.5 and =0) and predictions using Eq. [9].504

Fig. 14. Maximum drift ratio of columns for: a) R=5 and Takeda hysteresis model with =0.5505

and =0; b) considering the influence of the abutments in nonlinear response (no piles, gap=50506

mm, R=5, =0.5 and =0)507

Fig. 15. Maximum drift ratio versus maximum ductility demand using R=5, =0.5 and =0 for:508

a) maximum drift ratio versus maximum ductility demand; b) ( D) / H2

versus maximum509

ductility demand510

Fig. 16. Maximum ductility demand to ductility capacity ratios obtained for: a) R=5, =0.5 and511

=0; b) R=3, =0.5 and =0. .512

Page 24


25/44

Fig. 17. Maximum ductility demand to ductility capacity ratios obtained when abutment effects513

were considered in modelling (Gap=50 mm, R=5, =0.5 and =0)514

Fig. 18. Analysis results using R=5 and Takeda hysteresis model with =0.5 and =0 for: a)515

normalized ductility demands; b) normalized demand to capacity ratios516

Fig. 19. Ductility demand to ductility capacity ratios for different bridge configurations517

considering transverse and longitudinal responses based on the 100%/30% rule for: a) Bridges518

with restrained transverse movements; b) Bridges with unrestrained transverse movements519

520

ge 25 of 44


26/44

521522

Fig. 1. Modified Takeda hysteresis loop (adapted from Carr (2009))523524

525

526

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

Page 26


27/44

542

Fig. 2. Bridge properties543

544

545

546

547

548

549

550

551

552

553

554

555

556

557

558

559

ge 27 of 44


28/44

560

Fig. 3. Structural modelling of the bridges561

562

563

564

565

566

567

568

569

570

571

572

573

574

575

576

577

578

579

580

H1

H2H3

Rigid links Nodes (lumped mass)

1/2 Superstructure depth

Column( lumped plasticity model)

Abutment model Superstructure (elastic element)

Page 28


29/44

581

Fig. 4. Average response spectrum for 7 records used for inelastic time history analysis matching582

the 2010 NBCC spectrum (2% in 50 years)583

584

585

586

587

588

589

590

591

592

593

594

595

596

597

598

599

600

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4

Spectralacceleration(

g)

Period (Sec)

Average Spectrum

NBCC spectrum

CHBDC Spectrum

ge 29 of 44


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601602

Fig. 5. Schematic view of the seat-type abutment and its components603604

605

606

607

608

609

610

611

612

613

614

615

616

617

618

619

Stem wall

Shear key Wbw

hbw

Gap Bridge deck

Back wall

Bearing

wbw

hbw

Wing wall

Back wall Stem wall

Piles

Shear key

Gap Bridge deck

Back wall

Bearing pad

Superstructure

Page 30


31/44

620Fig. 6. Hysteresis modelling of the abutments: a) general model with gap and nonlinear spring (Carr621

2009) b) typical nonlinear response of abutment forces from analysis622

623

624

625

626

627

628

629

630

631

632

633

634

635

636

-3000

-2500

-2000

-1500

-1000

-500

0

500

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Force(

KN)

Displacement (m)

Fy-

Fy+

KoKo

F

d

Ko

Gap+

Gap-

Ko

b)a)

ge 31 of 44


32/44

637638

Fig. 7. Influence of abutment stiffness: a) effective abutment stiffness (Caltrans 2006); b) the639

influence of the ratio of the total stiffness of columns to effective abutment stiffness on the640

maximum ductility demands641642

643

644

645

646

647

648

649

650

651

652

653

Pbw

Force

Deflection

Kabt

Keff-abt

Gap

a)

y = -0.131ln(x) + 0.196

R = 0.85

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0.01 0.1 1 10 100

%Reductionof

ductilitydemand

Kcols/Keff-abt

b)

Keff-abt

Page 32


33/44

a)654Fig. 8. Effects of total stiffness of columns on the maximum displacement ductility demands: a)655

R=3, =0.5 and =0; b) R=5, =0.5 and =0656657

658

659

660

661

662

663

664

665

666

667

668

669

y = 0.19ln(x) - 0.10

R = 0.49

0.0

0.5

1.01.5

2.0

2.5

3.0

3.5

4.0

1000 10000 100000 1000000

Maximumductilitydemand

Total stiffness of columns (kN/m)

y = 0.33ln(x) - 1.07

R = 0.39

0.0

0.5

1.01.5

2.0

2.5

3.0

3.5

4.0

1000 10000 100000 1000000



a) b)

ge 33 of 44


34/44

670

Fig. 9. Effects of maximum column stiffness ratio on the maximum displacement ductility671

demands: a) R=3, =0.5 and =0; b) R=5, =0.5 and =0.672673

674

675

676

677

678

679

680

681

682

683

684

685

686

y = 0.18ln(x) + 1.47R = 0.58

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

1 10 100 1000


KS/KL

y = 0.38 ln(x) + 1.50

R = 0.62

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

1 10 100 1000


KS/KL

a) b)

Page 34


35/44

687


displacement ductility demands: a) R=3, =0.5 and =0; b) R=5, =0.5 and =0.689690

691

692

693

694

695

696

697

698

699

700

701

702

y = 0.118ln(x) + 0.37

R = 0.69

0.0

0.5

1.01.5

2.0

2.5

3.0

3.5

4.0

1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08

Maximum

ductilitydemand

Total stiffness of columns X KS/KL (kN/m)

a) y = 0.236ln(x) - 0.62R = 0.66

0.0

0.5

1.01.5

2.0

2.5

3.0

3.5

4.0

1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08

Maximum

ductilitydemand

Total stiffness of columns X KS/KL (kN/m)

b)

ge 35 of 44


36/44

703Fig. 11. Effects of columns total stiffness times max stiffness ratio on the maximum704

displacement ductility demands considering the influence of abutments (Gap=50 mm, R=5,705

=0.5 and =0)706707

708

709

710

711

712

713

714

715

716

717

718

719

720

y = 0.3 ln(x) - 2.1

R = 0.84

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08


Total stiffness of columns x KS/KL (kN/m)

Page 36


37/44

721

Fig. 12. Effects of total stiffness of columns on the ratio of the displacements obtained using722

inelastic and elastic analysis for R=5 and: a) =0 and =0.6; b) =0.5 and =0; c) =0.3 and723

=0.3; d) considering the influence of the abutments on seismic response (no piles, gap=50 mm,724

=0.5 and =0)725

726

727

728

729

730

731

732

733

734

735

736

737

738

739740

741

742

743

744

745

746

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.0E+04 1.0E+05 1.0E+06Inelastic/Ela

sticdisplacement


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.0E+04 1.0E+05 1.0E+06Inelastic/El

asticdisplacement


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.0E+04 1.0E+05 1.0E+06Inelastic/Elasticdisplacement


a) b)

c)

0.0

0.20.4

0.6

0.8

1.0

1.2

1.4

1.0E+04 1.0E+05 1.0E+06Inelastic/

Elasticdisplacement


d)

ge 37 of 44


38/44

747Fig. 13. Effects of maximum column height ratio on the Max/Min ductility ratio, S/L (R=5,748

=0.5 and =0) and predictions using Eq.[9].749

750

751

752

753

754

755

756

757

758

759

760

761

762

763

764

765

766

767

0

2

4

68

10

12

14

16

18

1.0 1.5 2.0 2.5 3.0 3.5 4.0

S/L

HL/HS

Results from analyses

Eq. [9]

Page 38


39/44

768Fig. 14. Maximum drift ratio of columns for: a) R=5 and Takeda hysteresis model with =0.5769

and =0; b) considering the influence of the abutments in nonlinear response (no piles, gap=50770

mm, R=5, =0.5 and =0)771

772

773

774

775

776

777

778

779

780

781

782

783

784

785

786

787

788

789

790

791

792

793

794

795

796

797

798

799

800

801

802

803

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

1000 10000 100000 1000000

Drift


0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

1000 10000 100000 1000000

Drift


a) b)

ge 39 of 44


40/44

804

Fig. 15. Maximum drift ratio versus maximum ductility demand using R=5, =0.5 and =0 for:805

a) maximum drift ratio versus maximum ductility demand; b) ( D) / H2

versus maximum806

ductility demand807808

809

810

811

812

813

814

815

816

817

818

819

820

821

822

823

824

825

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0 1 2 3 4

(maxD)/HS2

Maximum ductility demand

0%

1%

2%

3%

4%

0 1 2 3 4

Maximumdriftratio

Maximum ductility demand

a) b)

Page 40


41/44

826

Fig. 16. Maximum ductility demand to ductility capacity ratios obtained for: a) R=5, =0.5 and827

=0; b) R=3, =0.5 and =0.828

829

830

831

832

833

834

835

836

837

838

839

840

841

842

843

844

845

846

847

848

849

850

851

852

853

854

855

856

857

858

859

860

861

0.0

0.1

0.2

0.3

0.4

0.5

1.0E+00 1.0E+01 1.0E+02

Max(Demand/Capacity)ratio

KS/KL

a)

0.0

0.1

0.2

0.3

0.4

0.5

1.0E+00 1.0E+01 1.0E+02

Max(Dema

nd/Capacity)ratio

KS/KL

b)

ge 41 of 44


42/44

862Fig. 17. Maximum ductility demand to ductility capacity ratios obtained when abutment effects863

were considered in modelling (Gap=50 mm, R=5, =0.5 and =0)864

865

866

867

868

869

870

871

872

873

874

875

876

877

878

879

880

881

882

0.0

0.1

0.2

0.3

0.4

0.5

1.0E+00 1.0E+01 1.0E+02

Max(Deman

d/Capacity)ratio

KS/KL

Page 42


43/44

883

Fig. 18. Analysis results using R=5 and Takeda hysteresis model with =0.5 and =0 for: a)884

normalized ductility demands; b) normalized demand to capacity ratios885886

887

888

889

890

891

892

893

894

895

896

897

898

899

900

901

902

y = 0.152ln(x) - 1.27

R = 0.92

0.1

0.5

5.0

1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07

(Maxductil

itydemand)/(HS/D)

Total stiffness of columns xaverage stiffness ratio (kN/m)

a)

y = 0.017 ln(x) - 0.138

R = 0.93

0.01

0.10

1.00

1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07

Max(Demand/Capacity)/(HS/D)

Total stiffness of columns x average stiffness ratio (kN/m)

b)

ge 43 of 44


44/44

903Fig. 19. Ductility demand to ductility capacity ratios for different bridge configurations904

considering transverse and longitudinal responses based on the 100%/30% rule for: a) Bridges905

with restrained transverse movements; b) Bridges with unrestrained transverse movements906907

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.0E+00 1.0E+01 1.0E+02

Max(Dem

and/Capacity)ratio

Max / Min stiffness

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.0E+00 1.0E+01 1.0E+02

Max(Dem

and/Capacity)ratio

Max / Min stiffness

b)a)

KS/KL KS/KL

Page 44

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