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Displacement-based seismic design methods support the performance-based seismic design philosophy known to be the most advanced seismic design theory. This 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. This benefits to achieve the operation of displacement-based seismic design. Three 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. The proposed approach is used to simplify the prediction of elastoplastic displacement demand and is validated by parametric analysis. The 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.

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 [

In previous studies, some extremely irregular-continuous bridges were selected to validate the calculation accuracy [

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. [

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.

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 [

One type of most popular irregular-continuous bridges.

The irregular-continuous bridges with relatively regular geometry are the study object of this paper. Three 4 × 40 m typical, common irregular-continuous bridges in Figure

Section properties of girder and piers.

Components | Area (m^{2}) | Moment of inertia (m^{4}) | Polar moment of inertia (m^{4}) | Concrete type | Longitudinal reinforcement steel and area ratio |
---|---|---|---|---|---|

Girder | 7 | 40 | 14 | C50 | — |

Column | 2.25 | 0.422 | 0.722 | C30 | HRB335, 0.66% |

Earthquake input: (a) response spectra for soil profile III in Chinese criteria (JTJ 004-89) and generated by Simqke and (b) one accelerogram corresponding to (a).

The 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 [

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),

The 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

According to (

Substitute equation (

Suppose the inertia forces of the equivalent system and the original system to be equal, as follows:

Substituting equation (

Substitute equation (

Substitute equation (

Therefore, the relationship MDOF system being equivalent to the SDOF system is developed with the following characteristics:

When a bridge structure is under elastic state, parameters

When the bridge is under plastic state,

Displacement vector

FEM, for each bridge, is developed by OpenSees program [

The 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 [

The mass

The ratios of mass

The ratio of mass

Equivalent SDOF system: (a) mass, (b) displacement, (c) coefficient of displacement shape of the 051005 bridge, (d) coefficient of displacement shape of the 100510 bridge, and (e) coefficient of displacement shape of the 050505 bridge.

The displacement

The coefficient of displacement shape

When PGA is small and the bridge is in an elastic state, the value of

When PGA is larger and the bridge begins to yield at different degrees, the value of

When PGA is noticeably larger than case (2), the value of

The changing range of

This section gives a simplified prediction procedure of seismic displacement demand. The principle of the procedure is to combine the structural capacity spectrum and the inelastic demand spectrum to estimate the seismic displacement response of structure. The following will discuss each part of the simplified prediction procedure.

The 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. The 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. These forces will be used to build the capacity spectrum of irregular-continuous bridge in this section. This alternative method is simpler than the previous solution.

The 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

The FEM of a bridge is analyzed by the response spectrum analysis to obtain the elastic displacement vector

The bridge is pushed to a certain plastic state under the distribution of forces

The

This 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

When the bridge is pushed by the distribution of forces

Based on the concept of the equivalent system in Section

According to the case analysis in Section

Based on Section

The aforementioned elastic response spectrum is converted as follows:

Figure

Generation procedure of the inelastic demand spectrum.

The inelastic demand spectrum and the capacity spectrum are drawn in the same figure. The capacity spectrum will intersect with different demand spectrums corresponding to different

Seismic displacement demand

Results show that the seismic displacement response of irregular-continuous bridges has two characteristics as PGA increases: ① the displacement

Based on the concept of the equivalent system, the displacement vector

Displacement shape from RSP must reflect the changes of

Taking irregular-continuous bridges in Figure

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. The Chinese response spectrum of soil type III in Figure

The seismic displacement for each seismic level is calculated by ITHA, and the corresponding displacement

Seismic displacements from RSP and ITHA for the same

Comparison of the equivalent SDOF system’s displacement by ITHA and RSP: (a) 051005 bridge, (b) 100510 bridge, and (c) 050505 bridge.

Comparison of seismic displacement by ITHA and RSP: (a) 051005 bridge, (b) 100510 bridge, and (c) 050505 bridge.

According to Figure

As a whole,

The difference between

Based on Figure

In general, as for the same displacement of the equivalent SDOF system, seismic displacement from RSP is close to the one from ITHA. This indirectly shows that the displacement shape from RSP can reflect the changes of

The difference between seismic displacement from RSP and that from ITHA becomes more obvious as a whole, as PGA increases.

Results from Figures

As to evaluate the prediction errors of the simplified prediction method in detail, the Chinese response spectrum of soil type III in Figure

As for the 051005 bridge, taking PGA of

Analysis process of the simplified prediction method corresponding to PGA = 0.2 g.

In Figure

The comparison of seismic displacement calculated by the simplified prediction method using RSP and that by ITHA, under five PGA levels of

Comparison of seismic displacement by the ITHA and simplified prediction method using RSP: (a) 051005 bridge, (b) 100510 bridge, and (c) 050505 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

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

The 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.

Three 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

Changing parameters of girder and piers.

Member type | Variables | Parameter values |
---|---|---|

Lateral moment of inertia (m^{4}) | 20, 40, 80, and 160 | |

Polar moment of inertia (m^{4}) | 7, 14, 28, and 56 | |

Section area (m^{2}) | 3.5, 7, 14, and 28 | |

Single span length (m) | 20, 40, 80, and 160 | |

Section area (m^{2}) | 1.0 m × 1.0 m, 1.5 m × 1.5 m, 2.0 m × 2.0 m, and 2.5 m × 2.5 m | |

Area ratio of longitudinal reinforcement | 0.4%, 0.8%, 1.2%, and 1.6% | |

Height distribution of piers | Pier2# varies as 5 m, 10 m, and 15 m, while pier1# equals to pier 3# and varies as 5 m, 10 m, 15 m, and 20 m synchronously |

Based on Table

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

As for each bridge model, the simplified prediction method using RSP and ITHA are used to calculate its seismic displacement, respectively. The ratios of the displacement values of the girder points 0, 1, 2, 3, and 4 in Figure

Ratio of seismic displacement of the simplified prediction method using RSP to that of ITHA.

According to Figure

The 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. This can effectively control the structure’s seismic damage. This procedure has been achieved by using an ITHA method but consumes too long computing time [

Displacement-based seismic design process using a nonlinear static method.

Irregular-continuous bridges can be designed according to two design levels of

As for the design level of small earthquake

In terms of the design level of large earthquake

Under the design level of large earthquake

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

The structure is pushed by the response spectrum load distribution, and the curvature of the most dangerous section of the first yielding pier is monitored. The general displacement

The corresponding general displacement

The capacity coefficient

If

Under other conditions, a new scheme should be chosen

Check of design scheme.

The bridge pier should be redesigned if the former scheme is not satisfactory, i.e., the case (2) in Section

Therefore, the total inertial force of the new scheme after all the piers yield is

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

The sections above are repeated. The scheme that satisfies the requirement of

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.

The known conditions are as follows:

The first bridge with a total mass 2912t of the superstructure in Figure

Earthquake load adopts the response spectrum for soil profile III in Chinese criteria (JTJ 004-89) as shown in Figure

Note that the pier cross section and the reinforcement are unknown and need further design based on the displacement-based seismic design procedure.

The 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

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

Check of design scheme: (a) preliminary scheme and (b) new scheme.

The demand spectrum of the

From Figure

The combination of the capacity spectrum and the demand spectrum of the new scheme is shown in Figure

To check the validity of the design result, the final scheme is calculated by ITHA. The accelerograms in Section

Check of design result: (a) seismic displacement calculated by RSP and ITHA and (b) curvature of the pier base section.

Figure

Figure

The check results show that the seismic design result is proper and correct.

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. Thus, conclusions include the following:

The 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.

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.

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 irregular-continuous bridges with similar characteristics of those used in the case study and those used for the parametric analysis. Those 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 four-span 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 [

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

The authors declare that they have no conflicts of interest.

This paper was supported by the National Natural Science Foundation of China under grant nos. 51778635 and 51778630, the Natural Science Foundation of Hunan Province under grant no. 2019JJ40386, and the Innovation-Driven Plan in Central South University under grant no. 20200017050004. These financial supports are gratefully acknowledged.