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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
29
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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
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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
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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
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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
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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
Fig. 11. Effects of columns total stiffness times max stiffness ratio on the maximum496
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
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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
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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
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542
Fig. 2. Bridge properties543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
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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)
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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
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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
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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)
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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
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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
Maximumductilitydemand
Total stiffness of columns (kN/m)
a) b)
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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
Maximumductilitydemand
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
Maximumductilitydemand
KS/KL
a) b)
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687
Fig. 10. Effects of columns total stiffness times max stiffness ratio on the maximum688
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)
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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
Maximumductilitydemand
Total stiffness of columns x KS/KL (kN/m)
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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
Total stiffness of columns (kN/m)
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
Total stiffness of columns (kN/m)
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
Total stiffness of columns (kN/m)
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
Total stiffness of columns (kN/m)
d)
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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]
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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
Total stiffness of columns (kN/m)
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
1000 10000 100000 1000000
Drift
Total stiffness of columns (kN/m)
a) b)
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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)
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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)
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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
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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
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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
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