A Nonlinear Static Procedure for the Seismic Design of Symmetrical Irregular Bridges

Displacement-based seismic design methods support the performance-based seismic design philosophy known to be the most advanced seismic design theory.'is paper explores one common type of irregular-continuous bridges and studies the prediction of their elastoplastic displacement demand, based on a new nonlinear static procedure. 'is benefits to achieve the operation of displacement-based seismic design. 'ree irregular-continuous bridges are analyzed to advance the equivalent SDOF system, build the capacity spectrum and the inelastic spectrum, and generate the new nonlinear static analysis. 'e proposed approach is used to simplify the prediction of elastoplastic displacement demand and is validated by parametric analysis. 'e new nonlinear static procedure is also used to achieve the displacement-based seismic design procedure. It is tested by an example to obtain results which show that after several combinations of the capacity spectrum (obtained by a pushover analysis) and the inelastic demand spectrum, the simplified displacement-based seismic design of the common irregular-continuous bridges can be achieved. By this design, the seismic damage on structures is effectively controlled.


Introduction
In recent years, displacement-based seismic design methods have rapidly developed, and the prediction of elastoplastic displacement demand on structures, under seismic action, has become a crucial issue [1]. Although inelastic time history analysis (ITHA) can calculate the elastoplastic displacement demand on structures, recent studies have concentrated on developing simplified methods for related questions due to the complexity of ITHA. In the present time, however, the mature simplified methods are primarily suitable for regular-continuous bridges, e.g., the pushover analysis method under uniform load in AASHTO [1]. When comparing irregular-continuous bridges [2] influenced by modes to the regular-continuous bridges, the simplified prediction methods of elastoplastic displacement demand require further study [3]. Although various of methodologies, such as the modal adaptive nonlinear static procedure (MANSP) [4][5][6], the modal pushover analysis (MPA) [7][8][9][10], and the incremental response spectral analysis (IRSA) [11], have been proposed, further investigation on their calculation accuracy, application scope, and the degree of simplification is needed.
In previous studies, some extremely irregular-continuous bridges were selected to validate the calculation accuracy [12,13]. ose bridges with asymmetry, obviously unequal pier length, and other irregular properties at the same time are common cases in mountain area or other similar areas. Regarding practical continuous bridges in general areas, cases with many regular factors and only few irregular factors [14], leading to the obvious influence of high modes, are referred to as the common irregular-continuous bridges. e irregular behavior comes from the dynamic response of the bridges under investigation despite these bridges having a rather regular geometric layout. In terms of common irregular-continuous bridges, the many rules in their seismic response benefit in finding a proper simplified prediction method of elastoplastic displacement demand.
In addition to the pushover-based analysis methods mentioned above, many other methods have been established for the simplified prediction of elastoplastic displacement demand in irregular-continuous bridges. For instance, Kowalsky proposed a displacement-based seismic design method, grounded in the ideas of equivalent system and equivalent damping ratio, etc. [15][16][17][18]. is proposed method involves the advantages of easy operation. Kappos, Gidaris, and Gkatzogias further improved some aspects of the proposed method [19,20], such as how to combine the damping ratios of structural components to form the damping ratio of bridge systems. e ideas introduced in these methods are worth studying in the research of a new nonlinear static procedure analysis methods.
By taking one common type of irregular-continuous bridges in transverse direction as the object of study, this paper proposes a simplified prediction method of seismic displacement demand. Based on their seismic response characteristics (the equivalent system concept and the basic idea of pushover analysis), this paper also proposes the corresponding displacement-based seismic design procedure.

A Common Type of Irregular-Continuous Bridges
Regular-continuous bridges are generally defined as bridges that can be simplified as single-degree-of-freedom (SDOF) systems or as those with dynamic responses controlled by only a decisive fundamental mode. To carry out the simplified seismic design safely, the AASHTO [1] and China's Guidelines for Seismic Design of Highway Bridges (2008) define regular-continuous bridges according to structural characteristics. However, transverse dynamic responses of many actual continuous bridges are controlled by two or more modes. When this occurs, they are referred to as irregular-continuous bridges. Figure 1 shows one kind of the most popular irregular-continuous bridges containing many regular aspects, e.g., nearly symmetric distribution of pier length, nearly equivalent distribution of span length, and many other aspects which satisfy the structural requirements of regular-continuous bridges. However, the transverse dynamic response of these bridges is still controlled by two or more modes. is is due to the influence of the following irregular factors: (1) different pier lengths and a comparatively small stiffness ratio of girder to pier; (2) in order to prevent excessive internal forces and deformations at abutments under earthquakes, bidirectional sliding bearings are set on all abutments, as shown in Figure 1. Meanwhile, if the stiffness ratio of girder to pier is comparatively smaller, the mass of superstructure endured by each pier will be different under transverse seismic actions, even if the pier heights are nearly the same. In Figure 1, the modal shapes for the first and second modes are nearly identical in shape for the three different bridges. e reason is that the bidirectional sliding bearings are set on all abutments, and the stiffness ratio of girder to pier is small as described above, which control the first and second modes of the three different bridges. However, there are some differences in the bending degree of the corresponding vibration modes of the three bridges, since these bridges have different pier distributions. It is noted that shear keys are considered and designed in the transverse direction of abutments to meet the requirements under normal loading and their failure is only permitted under severe earthquakes [13]. e irregular-continuous bridges with relatively regular geometry are the study object of this paper. ree 4 × 40 m typical, common irregular-continuous bridges in Figure 1 have been selected for analysis. e friction coefficient of sliding bearings is equal to 0.02. e section properties of girders and columns are shown in Table 1. Earthquake load adopts the elastic response spectrum for soil profile III in Chinese criteria (JTJ 004-89), as shown in Figure 2(a). According to requirements of this paper, it is transformed in the following two ways: (i) it is converted into the inelastic demand spectrum, which will be used for the simplified prediction method of seismic displacement demand. e detailed procedure will be discussed in the following sections. (ii) To transform the ground motion input of ITHA, seven accelerograms are generated by the Simqke procedure [21]. Results of ITHA are regarded as the benchmark for comparison using the simplified prediction method. Figure 2(b) gives the first accelerogram while other figures are omitted due to similarity.

Characteristics of Seismic Displacement
e seismic displacement of girder is nearly symmetric for the common irregular-continuous bridges, and when induced by the gradual increase of peak ground accelerations (PGAs), its shape changes as the pier's yielding degree increases [22]. Furthermore, the shape will be relatively unchanged after the pier's yielding degree arrives at a given value. In this section, a concept of the equivalent system is used to decompose the girder's seismic displacement into the displacement of the equivalent SDOF system. e coefficient of displacement shape is applied to the foregoing three irregular-continuous bridges to study their characteristics of seismic displacement.

Concept of Equivalent SDOF System.
To operate the pushover analysis and to carry out displacement-based seismic design procedure, it is necessary to first analyze how to transform a multi-degree-of-freedom (MDOF) system of a continuous bridge into a SDOF system, i.e., how to decompose the seismic displacement of the bridge into the displacement equivalent SDOF system and the coefficient of displacement shape.
In the finite element model (FEM), m i , Δ i , F i , and a i are defined to be the ith structure node's mass, displacement, inertia force [23], and acceleration, respectively. e corresponding values of the equivalent SDOF system are denoted by m eq , Δ eq , F eq , and a eq , respectively. e relationship between Δ i and Δ eq , and a i and a eq is supposed as where c i is the coefficient of displacement shape.
2 Shock and Vibration e inertia force of the equivalent system is to be equal to the resultant inertia force of the original system; hence, And thus, the mass of the equivalent system m eq is denoted by According to (2) and (3), it is obtained by Substitute equation (1) into equation (5); hence, Suppose the inertia forces of the equivalent system and the original system to be equal, as follows: Substituting equation (6) into equation (7) obtains Substitute equation (8) into equation (1), and then Substitute equation (9) into equation (4); hence, erefore, the relationship MDOF system being equivalent to the SDOF system is developed with the following characteristics: (1) When a bridge structure is under elastic state, parameters m eq , c i , and Δ eq are only related to the shape of the elastic displacement vector Δ, which is equivalent to mode vector Φ n . Compared to a certain mode in multimode pushover analysis [8], m eq , in equation (10), similar to mode participation mass, c i in equation (9) is similar to the product of the mode participation factor Γ n and the corresponding value ϕ in of the mode vector Φ n , and Δ eq in equation (8) is similar to the response spectrum displacement s d of a certain single mode. (2) When the bridge is under plastic state, m eq , c i , and Δ eq are still related to the shape of the displacement vector Δ. (3) Δ can be decomposed into the product of Δ eq and c i no matter what status the bridge is under, e.g., elastic state or plastic state.

Study
Case. Displacement vector Δ of a bridge can be decomposed into the product of Δ eq and c i according to the foregoing concept of the equivalent system. It is used to study the seismic displacement characteristics of three irregular-continuous bridges in Figure 1. FEM, for each bridge, is developed by OpenSees program [24]. e girders, piers, and bearings are simulated by elastic beam, fiber, and zero-length link elements, respectively. e cross section of piers is divided into three parts, including cover concrete, core concrete, and longitudinal bars. e concrete is simulated by concrete07, and the longitudinal bars are simulated by reinforcing steel material with the low-cycle fatigue parameters. e displacement-based fiber elements with adequate integral points are used to calculate the seismic responses. Different zero-length link elements are used to simulate the fixed and sliding bearings, respectively. As for the fixed bearings, the zero-length link element is an elastic link element with a large stiffness and an Structure period T (s) Dynamic magnification factor β Spectrum for soil profile III in Chinese Spectrum 1~7 generated by simkqe Spectrum 1~7 generated by simkqe Spectrum 1~7 generated by simkqe Spectrum 1~7 generated by simkqe Spectrum 1~7 generated by simkqe Spectrum 1~7 generated by simkqe Spectrum 1~7 generated by simkqe assumed large force which is unyielding forever. In terms of the middle and side sliding bearings, the zero-length link element is an elastoplastic link element with a yielding force of 75 kN and 37.5 kN, respectively. e accelerograms corresponding to the response spectrum of soil type III are designated for seismic input. 59 levels of PGA are investigated ranging from 0.02 g to 0.6 g with an interval of 0.01 g [25,26]. e seismic displacement is calculated by ITHA. Parameters m eq , Δ eq , and c i are determined by the concept of the equivalent system in Section 3.1.
e mass m eq is computed by equation (10) and shown in Figure 3(a). As the total mass of each bridge is 3029, 3058, and 3000 tons, respectively, Figure 3(a) shows the following: (1) e ratios of mass m eq to bridge total mass are 94.8%∼96.7%, 85.4%∼98.1%, and 86.5%∼98.1%, respectively. erefore, with the inclusion of the participation masses of each mode, the mass m eq is nearly the same as the bridge total mass.
(2) e ratio of mass m eq to bridge total mass for each bridge increases gradually as PGA increases, thus making m eq closer to the bridge total mass. e displacement Δ eq is then computed by equation (8) and displayed in Figure 3(b). It shows that the displacement Δ eq for each bridge gradually increases as PGA increases. Figure 3(b) also shows that Δ eq of the 051005 bridge and 050505 bridge almost coincides with each other since both trends increase at similar rates. e coefficient of displacement shape c i is then computed by equation (9). c i of the girder points at 0#, 1#, and 2# of each bridge in Figure 1  (1) When PGA is small and the bridge is in an elastic state, the value of c i is stable as PGA increases (2) When PGA is larger and the bridge begins to yield at different degrees, the value of c i changes rapidly as PGA increases (3) When PGA is noticeably larger than case (2), the value of c i changes little and tends to stabilize as PGA increases (4) e changing range of c i at the node 0 of the girder end, i.e., girder point 0 in Figure 1, is relatively smaller when compared to the corresponding value c i in its elastic state

Procedure of Simplified Prediction of Seismic Displacement Demand
is section gives a simplified prediction procedure of seismic displacement demand. e principle of the procedure is to combine the structural capacity spectrum and the inelastic demand spectrum to estimate the seismic displacement response of structure. e following will discuss each part of the simplified prediction procedure.

Capacity Spectrum.
e transformation from seismic dynamic loading to static loading and the transformation from the MDOF system to the SDOF system must be studied in order to estimate the seismic displacement of the continuous bridge. In regard to studying the transformation from the MDOF system to the SDOF system, two main methods exist. One solution is the same as the multimode pushover analysis method, in which mode decomposition is executed and each mode refers to a single SDOF system. It can directly use the pushover analysis in theory. Because each important mode is used to determine the distribution of forces for the pushover analysis separately, this method is complex in practice. It also requires several pushover processes. e alternative method treats a continuous bridge as, approximately, a single SDOF system. It is pushed by reasonable distribution of forces, which have been indirectly adopted in the equivalent linear method. ese forces will be used to build the capacity spectrum of irregular-continuous bridge in this section. is alternative method is simpler than the previous solution.
e relationship between the MDOF system and its equivalent SDOF system can be linked by the concept of the equivalent system according to the discussion in Section 3. Based on the above analysis, the following steps are used to obtain the capacity spectrum: (1) e FEM of a bridge is analyzed by the response spectrum analysis to obtain the elastic displacement vector Δ (2) e bridge is pushed to a certain plastic state under the distribution of forces mΔ, and the V b − u r curve is obtained, where V b is the summation of shear force at the bottom of each pier and u r is the displacement of reference point, and m is the mass matrix (3) e V b − u r curve is then transformed into the S a − S d curve by assigning S a � V b /m eq and S d � u r /c i is process of the pushover analysis method is referred to as the pushover analysis method based on response spectrum loads. For short, it is referenced to as RSP. Its basic idea comes from the N2 method [27] and the FEMA pushover method [8], and some similar methods have been used for bridge structures [28,29].
When the bridge is pushed by the distribution of forces mΔ, the position of displacement reference point requires further discussion in this method. When the bridge is under elastic state, the displacement shape obtained by pushover analysis is nearly consistent with the shape of elastic displacement vector Δ. S d � u r /c i of different displacement reference points is nearly the same with each other, and the corresponding S a − S d curve is irrelevant to the position of the displacement reference point. However, when the bridge enters into plastic state, Shock and Vibration the displacement shape (obtained by pushover analysis) and the shape of elastic displacement vector Δ (obtained by response spectrum analysis) become more and more inconsistent.
erefore, S d � u r /c i of different displacement reference points, is not the same. e corresponding S a − S d curve is also different for various positions of displacement reference points.
Based on the concept of the equivalent system in Section 3, vector Δ can be decomposed into the product of Δ eq and c i . In pushover analysis, u r can be expressed as u r � c i S d , in which S d is corresponding to Δ eq of an equivalent system from a physics concept. If the displacement vector Δ obtained by pushover analysis is required to be equal to the results from ITHA when S d � Δ eq , the coefficient of displacement shape c i of the two methods must be the same. In the pushover analysis, c i is constantly changing, creating difficulty in tracking the complexity of its transformation.
us, simplified measures are needed.
According to the case analysis in Section 3, the changing range of c i at the point 0 of the girder end is relatively smaller when compared to the corresponding value c i in its elastic state. Hence, the change of c i at the point 0 of the girder end  under seismic actions is omitted, and the corresponding c i is assumed and set to be always equal to the value of elastic state. erefore, the girder point 0 is chosen as the displacement reference point, and the S a − S d curve of bridge structure can be obtained through the formula S d � u r /c i , in which u r and c i are all the corresponding values of the girder point 0.

Inelastic Demand Spectrum.
Based on Section 2, the elastic response spectrum should be converted into the inelastic demand spectrum, used by the simplified prediction method of seismic displacement demand. e conversion can use C, the ratio of displacement demand of the elastoplastic model to that of its elastic counterpart for one SDOF system subjected to the same earthquake. Many researchers have investigated C to simplify the estimation of seismic displacement demand of a structure [30,31], and C used here adopts Miranda's equation shown as follows [32]: where T is the period of SDOF and μ is its displacement ductility demand. e aforementioned elastic response spectrum is converted as follows: where S d and S a are, respectively, the displacement value and acceleration value of the elastic response spectrum. S u and S ay are, respectively, the displacement value and acceleration value of the inelastic response spectrum. Figure 4 shows how to construct the inelastic demand spectrum based on the aforementioned equations. erefore, the inelastic demand spectrum and the aforementioned capacity spectrum can be applied to the S a − S d coordinate system to obtain the modal displacement response S d [33].

Prediction of Seismic Displacement.
e inelastic demand spectrum and the capacity spectrum are drawn in the same figure. e capacity spectrum will intersect with different demand spectrums corresponding to different μ values, which are the displacement ductility demand factor. Different S d of the intersection points will then also be obtained. Denote μ ′ as S d /S dy , where S dy is the spectrum value of yield-point displacement and μ ′ � 1 when S d is in the elastic region. e S d of the intersection point corresponding to μ ≈ μ ′ , where μ is the displacement ductility demand in Figure 4 and μ ′ � S d /S dy in the capacity spectrum, is the seismic displacement demand of the equivalent SDOF system. S d is equivalent to Δ eq in equation (8).
Seismic displacement demand Δ i of each node in its original structure needs to be reversely solved by using equation (1) after obtaining the S d or Δ eq of the equivalent SDOF system in theory. As to further simply the prediction of displacement demand in practice, it adopts the actual pushover displacement vector u, corresponding to S d , as the seismic displacement demand Δ i of each node in the bridge system.

Verification Case of Simplified Prediction Procedure
Results show that the seismic displacement response of irregular-continuous bridges has two characteristics as PGA increases: ① the displacement Δ eq of the equivalent SDOF system increases gradually and ② the coefficient c i of displacement shape is constantly changing. e two characteristics above should be reflected when judging if a simplified prediction method can correctly predict the seismic displacement response of irregular-continuous bridges. In this part, RSP is applied to three irregularcontinuous bridges in Figure 1 to verify the effectiveness of the simplified prediction method proposed in Section 4.

Characteristics of RSP.
Based on the concept of the equivalent system, the displacement vector Δ can be decomposed into the product of Δ eq and c i . If RSP correctly predicts the seismic displacement response of irregularcontinuous bridges, it must have the following characteristics: (1) S d from RSP must be almost consistent with Δ eq from ITHA (2) Displacement shape from RSP must reflect the changes of c i from ITHA Taking irregular-continuous bridges in Figure 1 as an example, the seismic displacement is solved by RSP and ITHA, respectively. ey are compared with each other to verify RSP's validity. e detailed processes are as follows: (1) FEM of each bridge is built in OpenSees program, in which elastic beam element, fiber element, and nonlinear link element are used to simulate the girder, the piers, and the bearings. e Chinese response spectrum of soil type III in Figure 2(a) and the corresponding accelerograms in Figure 2(b) are chosen as the earthquake input. PGA is divided into 59 levels from 0.02 g to 0.6 g by intervals of 0.01 g. (2) e seismic displacement for each seismic level is calculated by ITHA, and the corresponding displacement Δ eq of the equivalent SDOF system is obtained by equation (8). (3) S d of the equivalent SDOF system is calculated by RSP for each seismic level, and the corresponding pushover displacement vector u is adopted as the seismic displacement of the bridge. (4) S d from RSP and Δ eq from ITHA are compared as shown in Figure 5. (5) Seismic displacements from RSP and ITHA for the same S d or Δ eq are compared as shown in Figure 6.
Shock and Vibration 7 According to Figure 5, some conclusions are obtained as follows: (1) As a whole, S d calculated by RSP is close to Δ eq by ITHA (2) e difference between S d and Δ eq becomes more and more obvious as PGA increases, and S d , calculated by RSP, is larger Based on Figure 6, some conclusions are obtained as follows: (1) In general, as for the same displacement of the equivalent SDOF system, seismic displacement from RSP is close to the one from ITHA. is indirectly shows that the displacement shape from RSP can reflect the changes of c i from ITHA based on equation (9). (2) e difference between seismic displacement from RSP and that from ITHA becomes more obvious as a whole, as PGA increases.
Results from Figures 5 and 6 show that the simplified prediction method, proposed in Section 4, can be used to predict seismic displacement for the irregular-continuous bridges of the case study.
As to evaluate the prediction errors of the simplified prediction method in detail, the Chinese response spectrum of soil type III in Figure 2(a) and the corresponding accelerograms in Figure 2(b) are chosen as the earthquake input for the irregular bridges in Figure 1, and PGA adopts 0.1 g, 0.2 g, 0.4 g, 0.8 g, and 1.6 g, respectively.
e corresponding results are shown in the following sections.

Case 1: 051005 Bridge.
As for the 051005 bridge, taking PGA of a � 0.2 g, for example, the procedure for seismic displacement prediction is described in detail, shown in Figure 7.
In Figure 7, the girder end point 0 is chosen as the displacement reference point. e capacity spectrum is obtained by pushing the bridge under the response spectrum load distribution, in which the spectrum value of yield-point displacement is S dy � 0.021 m. e values of S d for the three intersection points of the capacity spectrum curve and three demand spectrum curves with μ �1.0, 1. show that the seismic displacement of the simplified prediction method is close to that of ITHA. e comparison of seismic displacement calculated by the simplified prediction method using RSP and that by ITHA, under five PGA levels of a � 0.1 g, 0.2 g, 0.4 g, 0.8 g, and 1.6 g, is shown in Figure 8(a). For each PGA level, the seismic displacement calculated by the simplified prediction method using RSP is close to that of ITHA. Even for the PGA level of a � 1.6 g, the maximum relative error of the seismic displacement of the simplified prediction method using RSP is only 16% when compared to that of ITHA. is can satisfy the engineering application. It is meaningless for the PGA level of a � 1.6 g, since most bridges will not suffer such a strong earthquake. Such a case is only used to identify the accuracy of the simplified prediction method using RSP.

Case 2: 100510 Bridge.
As for the 100510 bridge, the comparison of seismic displacement calculated by the simplified prediction method using RSP and that by ITHA is shown in Figure 8(b). In terms of the PGA level of a � 0.1 g, 0.2 g, 0.4 g, and 0.8 g, the ratio of seismic displacement of the simplified prediction method using RSP to that of ITHA ranges from 85% to 118%, which can meet the requirement of the engineering application. At a PGA level of a � 1.6 g, the ratio of seismic displacement of the simplified prediction method using RSP to that of ITHA ranges from 75% to 130%, which shows that as PGA increases, the relative error of the seismic displacement of the simplified prediction method using RSP increases when compared to that of ITHA.

Case 3: 050505 Bridge.
As for the 050505 bridge, the comparison of seismic displacement calculated by the simplified prediction method using RSP and that by ITHA is shown in Figure 8(c). In terms of the PGA level of a � 0.1 g, 0.2 g, 0.4 g, and 0.8 g, the ratio of seismic displacement of the simplified prediction method using RSP to that of ITHA ranges from 85% to 119%, which can meet the requirement of engineering application. At a PGA level of a � 1.6 g, the ratio of seismic displacement of the simplified prediction method using RSP to that of ITHA ranges from 98% to 130%, which shows that as PGA increases, the relative error of the seismic displacement of the simplified prediction method using RSP increases when compared to that of ITHA.

Parametric Analysis of Calculation Accuracy of Simplified Prediction Procedure
e results from the foregoing three cases show that the simplified prediction method using RSP is a good predictor of the seismic displacement of irregular-continuous bridges. However, just like other simplified methods, it still is a semitheoretical and semiempirical method. Some assumptions are adopted in the theoretical analysis; therefore, it is not enough to verify the efficiency of the simplified prediction method using RSP based on only three cases. Carrying out more parametric analyses is necessary to ensure the validity of the simplified prediction method using RSP before applying its theories to simplified displacement-based seismic design of irregular-continuous bridges.

Bridge Structure and Seismic Input.
ree cases of continuous bridges are identified as the reference of analysis, whose geometry shapes and section properties of girders and piers are shown in Figure 1 and Table 1, respectively. Based on the three cases, some parameters are changed to produce more combinations as shown in Table 2. e combination rule changes one parameter by keeping the other parameters the same. As the three cases are the simplified model of the true bridges, the new models of Table 2, obtained by changing only one parameter, are reasonable to include many practical bridges. ey can be used for numerical simulation.   Based on Table 2 and to satisfy the study requirements of this paper, a majority of cases are obtained with 69 symmetrical bridges, selected as the study object of the parametric analysis.
When earthquake load is concerned, the simplified prediction method using RSP and ITHA adopt the inelastic demand spectrum and seven accelerograms, respectively, which are all corresponding to the elastic response spectrum as shown in Figure 2(a), and PGA adopts 0.1 g, 0.2 g, 0.4 g, and 0.8 g, respectively.

Numerical Results.
As for each bridge model, the simplified prediction method using RSP and ITHA are used to calculate its seismic displacement, respectively. e ratios of the displacement values of the girder points 0, 1, 2, 3, and 4 in Figure 1 calculated from RSP to that of ITHA are shown in Figure 9.
According to Figure 9, when compared to the results of ITHA, the simplified prediction method using RSP can obtain the reasonable and conservative seismic displacement. e average values of these ratios are 1.03, 1.05, 1.09, and 1.15 when PGA � 0.1 g, 0.2 g, 0.4 g, and 0.8 g, respectively. e relative error of the simplified prediction method using RSP increases as PGA increases.

Procedure of Simplified Displacement-Based
Seismic Design e displacement is the soul in the whole procedure of the displacement-based seismic design method to keep the balance between target displacement and seismic displacement demand. is can effectively control the structure's seismic damage. is procedure has been achieved by using an ITHA method but consumes too long computing time [34]. e simplified prediction method using RSP simplifies the calculation of seismic displacement demand of bridges and saves the computing time. is section will discuss how to apply the simplified prediction method using RSP to the displacement-based seismic design of the irregular-continuous bridges, especially for equilibrium iteration of target displacement and seismic displacement demand ( Figure 10).

Target Displacement.
Irregular-continuous bridges can be designed according to two design levels of E1 and E2: (1) As for the design level of small earthquake E1, main parts of the structure only require little damage, i.e., the maximum section curvature φ E1 of main ductile members should be less than the corresponding yield curvature φ y . e force-based seismic design can then be applied, but this is not the topic of this paper.
(2) In terms of the design level of large earthquake E2, the structure can have severe damage without collapsing or causing other fatal damage, i.e., the maximum section curvature φ E2 of main ductile members should be larger than the corresponding yield curvature φ y and not exceed the permitted limit curvature φ u . e displacement-based seismic design can then be used, and this is the topic of this paper.
Under the design level of large earthquake E2, the displacement-based seismic design using a nonlinear static method will be proposed on the irregular-continuous bridges in this section and the following sections. First, how to obtain the target displacement of the irregular-continuous bridges is listed as follows: (1) FEM of the bridge is built with experience-guided pier size and reinforcement arrangement, which is also achieved by the force-based seismic design under the design level of small earthquake E1. It is seen as the preliminary scheme of the design level of large earthquake E2, which will be continuously optimized in the following process. e FEM is used to obtain the response spectrum load distribution and carry out the following pushover analysis. (2) e structure is pushed by the response spectrum load distribution, and the curvature of the most dangerous section of the first yielding pier is monitored. e general displacement u r , yielding displacement Δ y , and ultimate displacement Δ u of the whole bridge system, represented by the girder point 0 in Figure 1, are obtained when the monitored curvature reaches φ y and φ u , respectively.
(3) e corresponding general displacement S d , yield displacement S dy , and limit displacement S du of the capacity spectrum are calculated according to S d � u r /c i , S dy � Δ y /c i , and S du � Δ u /c i , respectively. c i refers to the coefficient of displacement shape using the girder point 0 in Figure 1 and equation (9) for the elastic state of the bridge.

Check of Preliminary Scheme.
e capacity coefficient μ c is calculated by μ c � S du /S dy . e coefficient μ dE 2 corresponding to the inelastic demand spectrum of E2 design level is set to be μ dE 2 � μ c . When the capacity spectrum and the inelastic demand spectrum are drawn in the same figure, as shown in Figure 11, the actual seismic displacement of E2 design level situates at S d2 ∼ S du . On this basis, there are two possibilities: (1) If S du ≈ S d2 , the preliminary scheme will be satisfactory for E2 design level (2) Under other conditions, a new scheme should be chosen

New Scheme.
e bridge pier should be redesigned if the former scheme is not satisfactory, i.e., the case (2) in Section 7.2. Specify S an � S an2 + (S au − S a2 ), in which all the piers yield when S au is arrived since the structure is pushed until S a does not dramatically increase. In fact, all the piers will not yield at the same time under a special ground motion if   Figure 9: Ratio of seismic displacement of the simplified prediction method using RSP to that of ITHA. the piers have different length. However, when the ground motion continuously increases, the different piers will gradually enter the yield state such as the capacity spectrum in Figure 11. Finally, all the piers yield if the ground motion is large enough, and this state corresponds to S au on the capacity spectrum in Figure 11. e state that all the piers yield can help to distribute the resultant force to each pier in the following process. erefore, the total inertial force of the new scheme after all the piers yield is F g � m g S an . F g includes all of the shear force at piers and abutments. e sum of shear force at each pier can be denoted by F p � F g − F a , and F p is expressed by equation (13), where F a refers to the sum of shear force at abutments, and F pn is the shear force of the n pier.
In many cases, bridge piers are often designed with the same cross section and the same reinforcement ratio. A principle of the same yield bending moment of each pier can be followed to distribute F p and calculate the yield bending moment M y of each pier as shown in the former expression of equation (14), where h n is the length of the n pier. If the bridge piers are designed with different cross sections or different reinforcement ratios, other special but simple relations can be written as shown in the latter expression of equation (14). e yield bending moment M y calculated by equation (14) can be used to design the new cross section and reinforcement of piers, or other special relations 7.4. Final Scheme. e sections above are repeated. e scheme that satisfies the requirement of S du ≈ S d2 is the final scheme, because the limit displacement S du of the capacity spectrum line and the inelastic demand spectrum line has the same ductility coefficient and the two lines just intersect at the point of S du . After the piers are designed based on equations (13) and (14), other detailed designs of the stirrup of piers, the foundation, and the bearing can then be executed under the principle of capacity protection, which is not the topic of this paper.

Verification Case of the Simplified Displacement-Based Seismic Design Procedure
As to better describe the procedure of the foregoing displacement-based seismic design, a relatively simple irregular-continuous bridge is selected to carry out the displacement-based seismic design. It is then further checked by ITHA.

Introduction of Case.
e known conditions are as follows: (1) e first bridge with a total mass 2912t of the superstructure in Figure 1 is selected as the design case (2) Earthquake load adopts the response spectrum for soil profile III in Chinese criteria (JTJ 004-89) as shown in Figure 2( The elastic shape is used as the lateral force mode for pushover analysis: determine target displacement, capacity spectrum, and demand spectrum under E2 earthquake Note that the pier cross section and the reinforcement are unknown and need further design based on the displacement-based seismic design procedure.

Design Procedure.
e preliminary pier scheme can be obtained by the conceptual design, the experience-guided design, or the force-based seismic design under the design level of small earthquake E1; however, this is not the topic of this paper. In this section, the cross section of the preliminary pier is assumed and given by 1.2 m × 1.2 m with a longitudinal reinforcement ratio of 1.2%.
FEM of the above bridge is the preliminary scheme, built in OpenSees program. According to the material strain capacity, the curvature information of the pier section is φ y � 0.00273 rad/m and φ u � 0.0394rad/m, and the latter of which corresponds to a collapse prevention state but has a safety factor of 2.0 according to Chinese criteria. When the whole bridge structure is pushed by the response spectrum load distribution, the curvature of the most dangerous section of the first yield 5 m pier and the displacement of the girder point 0 in Figure 1 are monitored. e displacement of Δ y and Δ u of the whole bridge system, represented by the displacement of the girder point 0 in Figure 1, is obtained when the monitored curvature reaches φ y and φ u , respectively. e corresponding displacement information of the capacity spectrum is S dy � Δ y /c i � 0.0213 m, S du � Δ u /c i � 0.1077 m, and μ c � S du /S dy � 5.06. e capacity spectrum represents the global measures of ductility, because it is obtained by pushing the whole bridge structure. It also represents the local measures of ductility, because it monitors the most strained 5 m pier and puts the corresponding indexes S dy and S du in Figure 12.
e demand spectrum of the E2 design level is built based on the assumption of μ dE 2 � μ c , and it corresponds to a collapse prevention state of the global measures of ductility controlled by the 5 m pier. e combination of the capacity spectrum and demand spectrum is shown in Figure 12(a). Because S du > S d2 in Figure 12(a) being as well as that in Figure 11, the preliminary scheme is so safe that it needs to decrease the pier cross section or the longitudinal reinforcement ratio.
From Figure  12(a), S an2 + (S au − S a2 ) � 0.8759 + (1.8671 − 1.1560) � 1.5869 m/s 2 , i.e., S an � 1.5869 m/s 2 for the new scheme, and the corresponding total inertia force of the new scheme is F g � 2912 × 1.5869 � 4621 kN. Note that piers almost support the total inertia force, since the abutment bearing is bidirectional sliding only taking a small amount of inertia force. According to the equal yield moment principle, the shear force F pn of three piers is 1852.183 kN, 916.635 kN, and 1852.183 kN, respectively, based on equations (13) and (14), and the yield moment M y of each pier is 5171.033 kN·m. erefore, in the new scheme in Figure 12(b), the cross section of the pier remains unchanged and the longitudinal reinforcement ratio decreases to 0.866% based on the pier yield moment of M y � 5171.033 kN · m. e combination of the capacity spectrum and the demand spectrum of the new scheme is shown in Figure 12(b). e result shows S du ≈ S d2 , which implies that the capacity spectrum line and the demand spectrum line just intersect at the point of S du and satisfies the requirement of seismic design. Consequently, the scheme can be chosen as the final one.

Check of Design Result.
To check the validity of the design result, the final scheme is calculated by ITHA. e accelerograms in Section 2 are chosen as the seismic input, and PGA adopts 0.4 g. e seismic displacement calculated by the simplified prediction method using RSP and ITHA is shown in Figure 13(a). e check of target curvature of the pier base section is shown in Figure 13(b). Figure 13(a) shows that the seismic displacement from the simplified prediction method using RSP is close to that from ITHA. Figure 13(b) shows that the base section curvatures of the two short piers reach the limit value, and the base section curvature of the long pier is much less than the limit value. erefore, the seismic design of the final scheme is controlled by the short pier's deformation capacity of E2 design level.
e check results show that the seismic design result is proper and correct.

Conclusion
By taking one common type of irregular-continuous bridges with quasi-regular geometry, the building procedures of the capacity spectrum and the demand spectrum are discussed. As a result, the simplified displacement-based seismic design procedure is advanced.
us, conclusions include the following: (1) e pushover curve resulted from a pushover analysis can be selected as the capacity spectrum of one common type of irregular-continuous bridges.
In the pushover analysis, the girder end point 0 is selected as the displacement reference point, and the displacement shape from the response spectrum analysis is used to determine the load distribution. (2) By combining the capacity spectrum and the inelastic demand spectrum, the seismic displacement demand can be properly predicted for one common type of irregular-continuous bridges.
(3) After several iterations of the combination of the capacity spectrum and the inelastic demand spectrum, the simplified displacement-based seismic design of one common type of irregular-continuous bridges can be achieved.
It is noted that the above proposed nonlinear static procedure is only applicable for the common irregularcontinuous bridges with similar characteristics of those used in the case study and those used for the parametric analysis.
ose bridges have many regular factors and only few irregular factors, leading to the obvious influence of high modes. And the higher mode effects are mild for the fourspan bridges, which improves the accuracy of the conventional force-based single-load pattern pushover analysis. It needs further investigation whether the above proposed nonlinear static procedure extends beyond to what was presented for the designed bridge in this paper [35,36]. Furthermore, the above proposed nonlinear static procedure is a little complex, such as using a FEM model to help analysis. It needs investigation about how to further simply the proposed nonlinear static procedure in the future.

Data Availability
e data used to support the findings of this study are included within the article. e data include the structural parameters, ground motion inputs, calculation methods, and calculation results.

Conflicts of Interest
e authors declare that they have no conflicts of interest.