Investigating the effects of combinations of irregularities on seismic ductility demands and mean response for four-span RC bridges considering displacement direction

Abstract

The effects of combinations of different types of irregularities have not been studied in details in the past and current seismic design codes do not address this issue appropriately. In this research, 76 regular and irregular bridges with irregularities in both superstructure and substructure were designed and evaluated to investigate the impact of combinations of irregularities on the seismic ductility demands. The irregularity parameters considered in this study include irregularities in span arrangement, different lengths of columns, different abutments support conditions and different stiffness of superstructure. The bridges were designed and checked according to AASHTO provisions. Inelastic time history analysis was conducted using OpenSees software and ductility demands in bridge columns for different bridge configurations were predicted. Predictions of ductility demands were based on the mean responses obtained using a number of ground motion records. Finally, the effect of considering displacement directions in predicting the mean bridge response (i.e., using different methods for predicting the mean response) for irregular and regular bridges was investigated. The results indicate that the combinations of irregularities can significantly increase the ductility demands in some cases compared to the case of regular bridges.

Keywords

Ductility demand seismic behavior irregular bridge nonlinear time history analysis mean response

1 Introduction

Displacement ductility demands in RC bridge columns can be used to predict the extent of the damage expected when the bridge is subjected to earthquake ground motions at different hazard levels. Larger ductility demands indicate that greater structural damage is expected resulting in a larger probability of failure or exceeding different damage states. Therefore, the ductility demands can be a good measure for describing the seismic behavior of ductile members. Capacities considered for ductile structural members in the irregular bridges may not be sufficient in severe earthquakes. This is because the design of irregular bridges are carried out using elastic analysis methods, while it has been shown in different studies that the use of elastic analysis can underestimate the maximum seismic demands in irregular bridge configurations [1 –15]. Therefore, in practical designs, provided that the probability of the design (severe) earthquake occurrence during the service life of the structure is sufficiently low, the structure is designed for a lower capacity, and instead, arrangements are made for the creation of plastic deformations, proportional to the desired performance level (e.g. life safety), at the predetermined locations in ductile structural elements known as plastic hinges. For assessment of the structural behavior and damage often the concept of ductility is used by seismic codes and guidelines. Ductility is the ability to deform during several cycles of displacements larger than yield displacement so that no significant loss of strength occurs [16]. Therefore, in order to prevent the structural damage or sudden collapse of the structures, the ductility capacity of the seismic resisting members (e.g. bridges columns) should be considered sufficiently larger than the predicted ductility demands obtained from seismic analysis. For the case of high ductility demands, a proper enclosure for the cross section’s core (i.e., confinement) should be considered so that the member can still be able to withstand gravity loads [16]. That is higher ductility demands correspond to lager probabilities of failure. Typically for any structure, design values of structural displacement ductility factor should not exceed 6, unless special studies are carried out to justify them [17]. On the other hand, by using deformation parameters (e.g. ductility demand), it is possible to assess the structure’s performance for a certain level of earthquake or for a certain level of reliability. Because when a structure or a ductile member yields, the force level does not change as much as its deformation. So it can be said that the ductility demand can be a good measure for describing the seismic behavior or expected extent of damage in ductile members especially in the case of irregular structures. This is because for irregular structures subjected to strong ground motions plastic deformations tend to concentrate in the stiffer ductile members [11] and it is expected that all of the ductile members will not contribute significantly to the seismic response of the structure. For example, in bridges with a stiff (short) column, due to the large stiffness of these columns, the tendency to absorb lateral force is greater. Therefore, these members yield earlier and their stiffness decreases. As a result, relatively high ductility demands are imposed on these stiff or short columns in the bridges categorized as irregular [2]. Therefore, such columns should be sufficiently ductile to control and limit the structural damage due to earthquake (consistent with the performance objectives considered in design). Although design regulations express limitations (e.g. the rebar percentage limit, etc) to ensure adequate ductility for RC sections, in the design of RC structures according to the current force-based regulations, strength checking often has a priority to the ductility [18]. On the other hand, the presence of any irregularity in span lengths, in addition to the irregularity in columns stiffness, can make the problem even worse. Thus, it is necessary and important to evaluate seismic behavior of irregular bridges with irregularities in both span lengths and column lengths. This important subject has not been appropriately addressed in the past. Regarding these topics, in this study, the ductility demand parameter of the columns is used to express the results for bridges with different configurations and irregularities in both superstructure and substructure which are designed according to the latest edition of AASHTO [19]. The displacement ductility demand, μ_D, is determined by dividing the column displacement demand (Δ_col) (that is obtained from OpenSees software results) to ideal yield displacement (Δ_y) according to Equation (1). Using the Equations (1) to (4), the yield displacement and the ductility demand for each column are calculated. The plastic hinge lengths and the displacement ductility demands were determined according to the AASHTO [20] and the yield penetration lengths and the yield displacements were calculated according to Ref. [21]. $μ_{D} = \frac{Δ_{column}}{Δ_{yi}}$ (1) $Δ_{yi} = φ_{yi} {(H + L_{sp})}^{2 / 3}$ (2) $L_{p} = 0.08 H + 0.15 f_{ye} d_{bl} ⩾ 0.3 f_{ye} d_{bl}$ (3) $L_{sp} = 0.15 f_{ye} d_{bl}$ (4)

In the above equations, H is the height of the column, φ_yi is the ideal yield curvature, L_sp is the strain penetration length, f_ye is the expected yield strength of the longitudinal bars, d_bl is the diameter of the longitudinal bars, Δ_column is the displacement demand and L_pis the plastic hinge length.

Also, in the end, the maximum difference between the two averaging methods for predicting nonlinear analysis results has been investigated for the bridges considered in this study. This procedure is proposed by Priestley et al. [21] and is based on the fact that displacement is a vector quantity (i.e., both values and directions are important) and hence the averaging procedure should consider the direction at which the maximum displacements occur. This method enables the determination of realistic values of compression strains in the concrete and tensile strain in steel corresponding to the displacements either in positive or negative directions. In the other words, when a structure has a symmetrical stiffness in both positive and negative direction, considering the direction of the response may not be as important as structures with asymmetric stiffness [21]. On the other hand, there are some structural responses that may be dependent on the direction of responses. For example, the closing and opening of a movement joint in bridge superstructure may be obtained more realistically by considering the direction of responses. It should be noted that in this research, the ductility results are expressed using averaging method based on absolute values.

2 Modeling of bridges

In this research, 76 regular and irregular bridge configurations with irregularities in both the superstructure and substructure have been considered. In the transverse direction half of the bridges studied are restrained and the other half are unrestrained at the abutments. The irregularity parameters considered include irregularities in span arrangement, different lengths of columns, different abutments support conditions and different stiffness of superstructure. The bridges are shown in Table 1 using the symbol $B_{* * *}^{* * * *}$ (Bridge), in which the lower index indicates the substructure members (columns) height from left to right (numbers 1, 2 and 3 stand for 7, 14 and 21 m high columns, respectively) and the higher index indicates superstructure members (spans) length from left to right (numbers 1, 2 and 3 stand for 21, 42 and 63 m length spans, respectively). For example, the bridge $B_{321}^{3122}$ has three columns with heights of 21, 14 and 7 meters and four spans with lengths of 63, 21, 42 and 42 meters (respectively, from left to right). For simplicity the bridges are also shown using the Greek numbers from I to XIX, in which number I refers to regular bridges and the others refer to irregular ones. All bridges are designed according to AASHTO [19] and are controlled based on AASHTO [20] as recommended by AASHTO [19]. To evaluate the seismic behavior of the bridges nonlinear time history analysis has been conducted using OpenSees software. Other parameters such as ductility limitation, the transverse reinforcement ratio, axial force ratio, longitudinal reinforcement ratio, shear design and etc were considered based on the provisions of AASHTO [19, 20]. These parameters are important and can influence the seismic performance of a reinforced concrete column [22, 23]. Details of structural modeling are described below.

Table 1
Names of designed bridges and their geometric characteristics

Bridge name ^* Column designation (m) Span designation (m) Bridge

C1 C2 C3 S1 S2 S3 S4 category

$(B_{222}^{2222})$ Bridge I 14 14 14 42 42 42 42 Regular

$(B_{321}^{3113})$ Bridge II 21 14 7 63 21 21 63 Irregular

$(B_{321}^{1331})$ Bridge III 21 14 7 21 63 63 21 Irregular

$(B_{312}^{3113})$ Bridge IV 21 7 14 63 21 21 63 Irregular

$(B_{312}^{1331})$ Bridge V 21 7 14 21 63 63 21 Irregular

$(B_{321}^{2213})$ Bridge VI 21 14 7 42 42 21 63 Irregular

$(B_{312}^{2213})$ Bridge VII 21 7 14 42 42 21 63 Irregular

$(B_{312}^{3122})$ Bridge VIII 21 7 14 63 21 42 42 Irregular

$(B_{321}^{1313})$ Bridge IX 21 14 7 21 63 21 63 Irregular

$(B_{312}^{1313})$ Bridge X 21 7 14 21 63 21 63 Irregular

$(B_{312}^{3131})$ Bridge XI 21 7 14 63 21 63 21 Irregular

$(B_{231}^{3113})$ Bridge XII 14 21 7 63 21 21 63 Irregular

$(B_{231}^{1331})$ Bridge XIII 14 21 7 21 63 63 21 Irregular

$(B_{231}^{2213})$ Bridge XIV 14 21 7 42 42 21 63 Irregular

$(B_{321}^{3122})$ Bridge XV 21 14 7 63 21 42 42 Irregular

$(B_{231}^{3122})$ Bridge XVI 14 21 7 63 21 42 42 Irregular

$(B_{231}^{1313})$ Bridge XVII 14 21 7 21 63 21 63 Irregular

$(B_{321}^{3131})$ Bridge XVIII 21 14 7 63 21 63 21 Irregular

$(B_{231}^{3131})$ Bridge XIX 14 21 7 63 21 63 21 Irregular

Bridge name ^*	Column designation (m)	Span designation (m)	Bridge
$(B_{222}^{2222})$ Bridge I	14	14	14	42	42	42	42	Regular
$(B_{321}^{3113})$ Bridge II	21	14	7	63	21	21	63	Irregular
$(B_{321}^{1331})$ Bridge III	21	14	7	21	63	63	21	Irregular
$(B_{312}^{3113})$ Bridge IV	21	7	14	63	21	21	63	Irregular
$(B_{312}^{1331})$ Bridge V	21	7	14	21	63	63	21	Irregular
$(B_{321}^{2213})$ Bridge VI	21	14	7	42	42	21	63	Irregular
$(B_{312}^{2213})$ Bridge VII	21	7	14	42	42	21	63	Irregular
$(B_{312}^{3122})$ Bridge VIII	21	7	14	63	21	42	42	Irregular
$(B_{321}^{1313})$ Bridge IX	21	14	7	21	63	21	63	Irregular
$(B_{312}^{1313})$ Bridge X	21	7	14	21	63	21	63	Irregular
$(B_{312}^{3131})$ Bridge XI	21	7	14	63	21	63	21	Irregular
$(B_{231}^{3113})$ Bridge XII	14	21	7	63	21	21	63	Irregular
$(B_{231}^{1331})$ Bridge XIII	14	21	7	21	63	63	21	Irregular
$(B_{231}^{2213})$ Bridge XIV	14	21	7	42	42	21	63	Irregular
$(B_{321}^{3122})$ Bridge XV	21	14	7	63	21	42	42	Irregular
$(B_{231}^{3122})$ Bridge XVI	14	21	7	63	21	42	42	Irregular
$(B_{231}^{1313})$ Bridge XVII	14	21	7	21	63	21	63	Irregular
$(B_{321}^{3131})$ Bridge XVIII	21	14	7	63	21	63	21	Irregular
$(B_{231}^{3131})$ Bridge XIX	14	21	7	63	21	63	21	Irregular

*The lower index indicates the substructure members (columns) height from left to right (numbers 1, 2 and 3 stand for 7, 14 and 21 m high columns, respectively) and the higher index shows superstructure members (spans) length from left to right (numbers 1, 2 and 3 stand for 21, 42 and 63 m length spans, respectively).

2.1 Materials modeling

To determine the ductility capacity of the concrete members, a non-linear stress-strain model for confined and non-confined concrete should be used. The model developed by Mander et al. [24] is used in this paper which applies to all sections and all levels of confinement. The reinforced bar type has been selected from AASHTO [20] and the concrete characteristic resistance (non-confined (is 40 MPa. In Tables 2 and 3, the specifications of concrete and reinforcing bars are given.

Table 2
Concrete characteristics

Specific strength (MPa) Poisson’s coefficient Modulus of elasticity (MPa) Weight per volume unit (kN/m³) Material

40 0.2 33900 23 Concrete

Specific strength (MPa)	Poisson’s coefficient	Modulus of elasticity (MPa)	Weight per volume unit (kN/m³)	Material
40	0.2	33900	23	Concrete

Table 3

Reinforcement bar characteristics

Specific strength (MPa)	Poisson’s coefficient	Modulus of elasticity (MPa)	Weight per volume unit (kN/m³)	Material
475	0.3	200000	76.9729	Bar

2.2 Bridge members modeling

Four-span continuous RC bridges with a total length of 168 meters are considered in this study. Columns connection to the base and to the superstructure is fixed and hinged, respectively (Fig. 1) All bridge columns have equal diameters but have different longitudinal reinforcement ratios. The diameter of the columns are 2.25 m and 2.50 m for the case of the superstructure with larger and smaller transverse stiffness (i.e, the transverse moment of inertia of 30 and 60 m⁴), respectively. The superstructure section is made of a pre-stressed single-cell box with a height of 2.4 m [16] and a width of 9.6 meters and 12 meters for smaller and larger transversal stiffness of superstructure, respectively (Fig. 1).

Fig.1

Bridge side view and cross-section of the superstructure and the columns.

2.3 Nonlinear analysis setting

Usually, in bridges, substructures are the lateral seismic resistance system [16]. Regarding this note and considering that aspect ratios of columns are larger than 2.5 and the column behavior is dominated by flexure, the nonlinear modeling in this project is in the form of a concentrated plastic hinge at the base of the column (Fig. 1). The recommendations proposed in Ref. [25] was used to model plastic hinges and based on their recommendations, the ratio of the elastic spring stiffness to the rotational stiffness of beam-column element was considered as 10. Plastic hinge hysteretic behavior is modeled using modified Takeda cycle behavior [26] that has been widely used for nonlinear modeling of RC members. The modified Takeda model consists of two coefficients αand β, which determine the unloading and reloading stiffness, respectively. After verifying the cyclic behavior of columns with available experimental data in Ref. [27] and considering the recommendations in Ref. [21], the values of 0.7 and 0.0 are chosen for α and β, respectively (Fig. 2). In the linear and non-linear analysis, the cracking coefficient was determined using the recommendations by AASHTO [20]. Modeling of masses are applied according to Ref. [28]. According to the AASHTO [20], the number of superstructure nodes should be so that at least a node presents in any ¼ of the length of each span in addition to end nodes. For verification purposes, the structural models were also created in CSiBridge software. The maximum difference in the first mode period between all OpenSees and CSiBridge models was only 3.90%. Rayleigh damping with mass and tangential stiffness matrices [29 –31] are used with damping ratio of 5%. Extensive sensitivity analyses were carried out on all important modeling parameters such as the plastic hinge properties, analysis time steps and damping using a record from the 1994 Northridge Earthquake–Canoga Park Station (NR94cnp) (see Fig. 3 for some examples of such sensitivity analyses). From Fig. 3 it is observed that the structural responses are not significantly sensitive to the parameters used in modeling.

Fig.2

Verification of modelling.

Fig.3

Sensitivity analyses results: (a) the plastic hinge properties; (b) time steps; (c) damping.

2.4 The loads

The total superstructure dead load is 135 kN/m and 165 kN/m for the case of the superstructure with smaller and larger transverse stiffness, respectively. The earthquake load has also been introduced to the structure in horizontal loads based on the modification factor (i.e. R). This coefficient is considered 5 for the longitudinal direction and 3 for the transverse direction. According to AASHTO [20], since the value of spectral acceleration at the period of 1 Sec is more than 0.5 g, the bridges are in the seismic group D. In nonlinear dynamic analysis, P-Δ effects are considered for columns, and longitudinal and transverse components of ground motion records are applied simultaneously to the bridges. The considered records are a far-field type with distance of larger than 10 km from fault according to FEMA [32]. The selected records in this study are in soil types C and D $(185 \frac{m}{s} \sim < ν_{s} < \sim 760 \frac{m}{s})$ according to AASHTO [19]. In Table 4, the specifications of the records used are presented. The records scaling has been carried out according to ASCE 07 provisions [33] (i.e., the number of records required for nonlinear time history analysis, period range for scaling and also the acceptable error in matching). Figure 4 shows the average SRSS spectrum of the records versus the 1.3 times the design spectrum. In this research a unique scale factor was determined in such a way that the selected scaled records satisfy all code provisions regarding ground motion scaling for all of the bridge models considered in this research. The value of the scale factor was 1.82 in this research. The use of a unique scale factor (i.e. 1.82 in this research) has two advantages for the aim of this paper; first, due to the large number of bridges studied in this research and the relatively high volume of analyses, the probability of errors is reduced and the speed of analysis is increased, and second, the results of the analyses will be independent of the effects of scale factors. Therefore, the records are selected according to the mentioned descriptions from a pool of records including 300 records. It is noted that using a unique scale factor is not an obligation based on the ASCE 07 provisions and any scaling method that satisfies the provisions may be used.

Table 4
Characteristics of the records used in nonlinear time history analysis

# Name* Year PGA_max**(g) Mechanism Magnitude Scale Factor

1 Imperial Valley 1979 0.38 Strike slip 6.53 1.82

2 Victoria 1980 0.65 Strike slip 6.33 1.82

3 Loma Prieta, San Jose - Santa Teresa Hills 1989 0.28 Reverse oblique 6.93 1.82

4 Loma Prieta, UCSC Lick Observatory 1989 0.46 Reverse oblique 6.93 1.82

5 Northridge 1994 0.26 Reverse 6.69 1.82

6 Chi-Chi 1999 0.51 Reverse oblique 7.62 1.82

7 Manjil 1990 0.52 Strike slip 7.37 1.82

8 Hector Mine 1999 0.33 Strike slip 7.13 1.82

9 Iwate, IWT010 2008 0.29 Reverse 6.90 1.82

10 Iwate, Ichinoseki Maikawa 2008 0.39 Reverse 6.90 1.82

11 Darfield 2010 0.36 Strike slip 7.00 1.82

#	Name*	Year	PGA_max**(g)	Mechanism	Magnitude	Scale Factor
1	Imperial Valley	1979	0.38	Strike slip	6.53	1.82
2	Victoria	1980	0.65	Strike slip	6.33	1.82
3	Loma Prieta, San Jose - Santa Teresa Hills	1989	0.28	Reverse oblique	6.93	1.82
4	Loma Prieta, UCSC Lick Observatory	1989	0.46	Reverse oblique	6.93	1.82
5	Northridge	1994	0.26	Reverse	6.69	1.82
6	Chi-Chi	1999	0.51	Reverse oblique	7.62	1.82
7	Manjil	1990	0.52	Strike slip	7.37	1.82
8	Hector Mine	1999	0.33	Strike slip	7.13	1.82
9	Iwate, IWT010	2008	0.29	Reverse	6.90	1.82
10	Iwate, Ichinoseki Maikawa	2008	0.39	Reverse	6.90	1.82
11	Darfield	2010	0.36	Strike slip	7.00	1.82

*Records are available in PEER website (https://ngawest2.berkeley.edu). **Peak ground acceleration.

Fig.4

Average spectrum of records matched to target spectrum of the code.

3 Results and discussion

3.1 Ductility demand

Figures 5 and 6 show the ductility demands of the bridge columns evaluated in this study in both longitudinal and transverse directions. In the ductility range between 2 and 3 concrete spalling occurs [16]. For this range of ductility demand and extent of damage, the columns are repairable after the occurrence of an earthquake. In this research, this range of ductility demands will be used to define cover spalling damage state.

Fig.5

Longitudinal ductility demand of columns in restrained and unrestrained bridges: (a) superstructure with smaller transversal stiffness; (b) superstructure with larger transversal stiffness.

Fig.6

Transversal ductility demand of columns in restrained and unrestrained bridges: (a) superstructure with smaller transversal stiffness; (b) superstructure with larger transversal stiffness.

According to Figs. 5 and 6, for the case of regular bridges (i.e., Bridge I cases) ductility demands do not exceed the cover spalling damage state for any column of the bridges. This performance is appropriate in terms of the uniform seismic behavior and failure rate among all columns. In irregular bridges, almost in all short columns, the ductility demands in transverse direction are in the cover spalling range, although in a few cases, the ductility demands have exceeded the above range (e.g. Bridge XIV $(B_{231}^{2213}))$ . On the other hand, the ductility demands for short columns in longitudinal direction is approximately 3 to 4. That is, these columns have entered a phase of damage beyond the spalling of the cover concrete. Of course, this behavior is predictable since due to the equal displacement rule and the modification factor considered in design (5 and 3, respectively, in a longitudinal and transverse direction), the stiff (short) columns in the longitudinal direction are expected to enter a significant nonlinear behavior phase (i.e. larger ductility demands). However, almost none of the long columns have entered the nonlinear phase (i.e., ductility demands are about 1 or lower according to Figs. 5 and 6). In the case of medium-length columns, in some irregular bridges these columns entered the range of non-linear behavior, while others do not. Nevertheless, none of the medium-length columns have exceeded the lower limit of the cover spalling zone (i.e., 2) and only minor non-linear behavior is expected in these columns.

The ductility demands in bridges with restrained abutments are typically lower than those in bridges with unrestrained support condition at abutments (Figs. 5 and 6). This is because fixity conditions at the abutments will limit the displacements of columns. Similar behavior can also be seen in the case of two different transversal stiffness of superstructure. In this case, although there are some exceptions (e.g. Bridge II $(B_{321}^{3113})$ and Bridge XII $(B_{231}^{3113}))$ , as the transversal stiffness of superstructure increases, the ductility demands typically decrease (Fig. 7). This is because a stiffer superstructure will tend to cause more uniform deformations along the bridge in the transverse direction that will somewhat limit the maximum deformation demands of the critical columns. On the other hand, increasing the transversal stiffness of superstructure has no significant effects on the ductility demands in longitudinal direction (Fig. 7) because the longitudinal stiffness of superstructure is not changed significantly.

Fig.7

The ratio of ductility demand of columns for superstructure with larger transversal stiffness to ductility demand of columns for superstructure with smaller transversal stiffness for restrained and unrestrained bridges in: (a) longitudinal direction; (b) transverse direction.

The distribution of ductility demands among bridge columns may be of interest because it shows the participation of ductile earthquake resisting systems in seismic response when the structure is subjected to an earthquake. The ratio of the maximum ductility demand to the average ductility demand was used in this research to investigate the distribution of ductility demands among different bridge columns (Fig. 8). From Fig. 8, it can be seen that the range of this ratio for restrained bridges are almost 1.25 to 2.60 and for unrestrained bridges varies from 1.00 to 2.25. The results indicate that the ductility demand distribution is not uniform for bridges with different configurations. Displacement based design methods can be used to solve this problem as recommended by Priestley et al. [21]. As far as the period is of concern, for restrained bridges, the distribution of the ductility demands become much more uniform when it exceeds almost 0.8 Sec. while for unrestrained bridges, the ratio of the maximum ductility demand to the average ductility demand is relatively low when the period is less than almost 3 Sec. Although the results presented are obtained for the cases considered in this study and more researches may be needed, they demonstrate how irregularity can affect the distribution of ductility demands among structural members.

Fig.8

The ratio of maximum ductility demand to average ductility demand of columns in: (a) restrained bridges; (b) unrestrained bridges.

3.2 Averaging nonlinear analysis result

The differences between the two methods of averaging nonlinear analysis results are investigated. Priestley et al. [21] referred to two methods for predicting mean structural response based on absolute values and based on the maximum positive and negative values. In the averaging method based on absolute values, the maximum absolute values for each earthquake record in each direction are recorded without considering the sign of this displacement and after calculating the mean values, the response is obtained for each principal direction. On the other hand, in the averaging method considering the sign of displacements, the maximum positive and negative values for each earthquake record in each direction are predicted separately and then these values are averaged based on their sign and the larger of the mean positive or negative value is used for each principal direction.

In Figs. 9 and 10, the differences between the averaging methods based on absolute values and maximum positive and negative values are presented for the seismic responses in the longitudinal and transverse directions, respectively. According to Figs. 9 and 10, the differences for the case of regular bridges are typically below 6% under any of the conditions examined in this study (i.e., different abutments support condition in the transverse direction and different transversal stiffness of superstructure).

Fig.9

The difference of the absolute method from the maximum positive and negative method in the longitudinal direction.

Fig.10

The difference of the absolute method from the maximum positive and negative method in transverse direction.

For the case of irregular bridges in the longitudinal direction (Fig. 9), the difference is typically around 10% to 15% (in some cases it is close to 20%, such as the bridge VIII $(B_{312}^{3122})$ in the case of bridges with stiffer superstructure and restrained abutment conditions). In the transverse direction, according to Fig. 10, this difference is about 5% for most of the restrained bridges and about 10% for the not-restrained bridges. In other words, the transverse response of the bridges studied was less sensitive to the averaging methods used. Moreover, with increasing transversal stiffness of superstructure, the differences between the two methods of averaging in the longitudinal direction for all unrestrained bridges and for most of restrained bridges always increase. While in the transverse direction, with increasing transversal stiffness of superstructure, the difference between the two averaging methods always increases for the case of restrained bridges. In Fig. 11, presents a comparison of the displacement envelopes obtained using inelastic analysis for two restrained and unrestrained bridge based on absolute values and based on the maximum positive and negative values as well as envelope obtained using elastic multimode analysis.

Fig.11

Comparison of the displacement envelopes obtained using inelastic analysis based on absolute values and based on the maximum positive and negative values in bridges with superstructure with smaller transversal stiffness for: (a) restrained Bridge XIIII $(B_{231}^{2213})$ ; (b) unrestrained Bridge XV $(B_{321}^{3122})$ .

Based on the results presented it is observed that for regular bridges the choice of averaging methods has a very minor effect on the predictions of the mean response of the bridge. This is expected as for the regular bridges the responses in different positive and negative directions tend to be more symmetrical. However, for the irregular bridges the response in different positive and negative directions tend to be somewhat unsymmetrical and therefore in such cases considering the direction of the displacements can have more important impact on the overall predictions of the mean response. More research is needed on this subject for different bridge configurations and different types of irregularities.

4 Conclusions

The effects of combinations of different types of irregularities have not been studied in details in the past and current seismic design codes do not address this issue. To investigate the effects of combinations of irregularities on seismic demands in this research the seismic performance of 76 regular and irregular bridges with combinations of irregularities in the infrastructure and substructure were investigated. The irregularity parameters considered included irregularities in span arrangement, different lengths of columns, different abutments support conditions and different superstructure stiffness. The bridges were designed according to AASHTO [19] and were controlled based on AASHTO guidelines [20] as recommended by AASHTO [19]. Nonlinear time-history analyses were conducted using OpenSees software and maximum ductility demands of bridge columns were predicted. Also, in the end, the maximum difference between the two averaging methods of nonlinear results, which are presented by Priestley et al. [21], has been investigated for bridges. The results obtained based on the cases studied in this research are as follows:

For the case of regular bridges studied all bridge columns contributed in the overall seismic behavior of the bridge and therefore the distribution of ductility demands among bridge columns were more uniform. For regular bridges, the maximum displacement ductility demands did not exceed 2.0 in both longitudinal and transverse directions indicating that the extent of damage in bridge columns do not exceed the cover spalling damage state.

For the case of irregular bridges studied the maximum ductility demands predicted were much larger than those obtained for the regular bridges, while similar R factors were used for the design of all bridges. The maximum ductility demands in irregular bridges were typically up to 100% larger than those predicted for regular bridges in both transverse and longitudinal directions (i.e., increase form around 1.5 to around 3 in the transverse direction and increase from around 2 to around 4 in the longitudinal direction). In irregular bridges the maximum ductility demands concentrated in the shortest columns, while the other bridge columns did not significantly contribute to nonlinear behavior of the bridges.

In irregular bridges, almost none of the long columns entered the nonlinear phase. In the case of middle-length columns, some columns entered minor non-linear behavior stage, while others do not. However, the extent of damage in none of the middle-length columns exceeded the lower limit of the cover spalling damage state (i.e. ductility demand of 2).

The ductility demands in bridges with restrained transversal abutments were lower than those in bridges with unrestrained abutments. Also as the transversal stiffness of superstructure increased, the ductility demand decreased.

The results indicate that the distribution of ductility demands is not uniform for bridges with different configurations. Displacement based design methods can be used to solve this problem as recommended by Priestley et al. [21].

The use of two different methods for predicting the mean response of the bridges subjected to a number of ground motion records was investigated. The difference between the mean values obtained based on the average of absolute values and average of the maximum positive and negative values for the case of the regular bridges did not exceed around 5%. While in the case of irregular bridges much larger differences observed between the two averaging methods. In the longitudinal direction, the difference is generally about 10% to 15% and in the transverse direction, this difference is about 5% for the restrained bridges and about 10% for the unrestrained bridges.

Due to the unsymmetrical and more complex behavior of irregular bridges, the predicted mean responses are somewhat more sensitive to the methods used for predicting the mean values and therefore care should be taken in seismic evaluation of irregular bridges.

The evaluations presented in this paper were based on the bridges designed based on current force-based seismic design codes. Although AASHTO [20] guidelines attempts to verify that the capacity of structural elements is larger than seismic demands using some displacement-based approaches, the design procedure for the design of bridges are still based on the force-based AASHTO [19] provisions. The problems discussed in this paper are due to the application of similar force modification factors, R, for the case of bridges with different configurations. When force-based design is applied for the case of irregular bridges with different column heights or different span lengths using R factors, the stiffest elements (i.e., typically shorter columns) attract larger seismic forces that require the use of larger reinforcement ratios in such elements. The larger reinforcement ratios in such columns cause two problems. First, it will again increase the stiffness of such members and causing them to attract even larger seismic forces that will make the situation even worse. Second, when the longitudinal reinforcement ratios of columns are increased their ductility capacity decreases. The concentration of large ductility demands observed in the critical short columns and accounting for lower ductility capacity of such columns due to large reinforcement ratios can decrease the safety margins against exceeding different damage states for the case of irregular bridges. To alleviate the problems discussed in this paper for the case of irregular bridges, the use of displacement-based design approaches are recommended (e.g., Ref. [21]).

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Investigating the effects of combinations of irregularities on seismic ductility demands and mean response for four-span RC bridges considering displacement direction

Abstract

Keywords

1 Introduction

Table 2 Concrete characteristics Specific strength (MPa) Poisson’s coefficient Modulus of elasticity (MPa) Weight per volume unit (kN/m3) Material 40 0.2 33900 23 Concrete

3.1 Ductility demand

References

Table 2
Concrete characteristics

Specific strength (MPa) Poisson’s coefficient Modulus of elasticity (MPa) Weight per volume unit (kN/m³) Material

40 0.2 33900 23 Concrete