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Presently, the modal pushover analysis procedure is extended to multidimensional analysis of structures subjected to multidimensional earthquake excitations. an improved multidimensional modal pushover analysis (IMMPA) method is presented in the paper in order to estimate the response demands of structures subjected to bidirectional earthquake excitations, in which the unidirectional earthquake excitation applied on equivalent SDOF system is replaced by the direct superposition of two components earthquake excitations, and independent analysis in each direction is not required and the application of simplified superposition formulas is avoided. The strength reduction factor spectra based on superposition of earthquake excitations are discussed and compared with the traditional strength reduction factor spectra. The step-by-step procedure is proposed to estimate seismic demands of structures. Two examples are implemented to verify the accuracy of the method, and the results of the examples show that (1) the IMMPA method can be used to estimate the responses of structure subjected to bidirectional earthquake excitations. (2) Along with increase of peak of earthquake acceleration, structural response deviation estimated with the IMMPA method may also increase. (3) Along with increase of the number of total floors of structures, structural response deviation estimated with the IMMPA method may also increase.

In recent years, a challenge for performance based seismic design is to establish an effective and feasible procedure to evaluate structural seismic capacities. The pushover analysis procedure, due to its simplicity and efficiency, is increasingly applied to estimate seismic demands of structures. During past years, the theory and application of pushover analysis procedure have been developed adequately [

A problem for pushover analysis is that the higher mode effects of building structures cannot be considered in the procedure [

The previous studies about pushover analysis are almost based on symmetric building structures and single-directional earthquake excitation. Both theoretical studies and seismic disasters indicate that asymmetric-plan structures with irregular distributions of mass or stiffness are likely to undergo torsional responses coupled with the translational vibrations, and this type of structures is likely to suffer more severe displacement demands at the corner elements under earthquake excitations. In addition, the torsion coupling response will induce the structural space effects that cannot be solved in two-dimensional analysis. So, several research efforts have been made to extend and apply the pushover analysis to asymmetric-plan structures whose inelastic seismic responses are intricate [

For multidimensional MPA, which is applicable to the analysis of asymmetric-plan structures, the modal equivalent single-degree-of-freedom (ESDOF) system is actually subjected to the superposition of bidirectional earthquake excitations. In the current MPA procedures, the response of ESDOF system is calculated according to single-directional earthquake excitation, the

The main objective of this paper is to propose an MPA procedure for evaluating the seismic capacities of asymmetric-plan structures subjected to bidirectional earthquake excitations. An outline of this paper can be expressed as follows. First, the improved procedure is presented based on the assumption of modal equivalent SDOF system which is subjected to the superposition of bidirectional earthquake excitations. Then, the corresponding strength reduction factor spectra are discussed and compared with traditional spectra. Next, the step-by-step procedure is proposed. At the end, two examples are considered to verify the accuracy of the proposed procedure.

Consider an

In the traditional multidimensional MPA procedure, the peak modal responses

If

According to the derivation above, the expression of inertia force vector,

Generally, the response demands of inelastic systems can be obtained by inelastic response spectra, such as strength reduction factor spectra (i.e.,

The yield resisting force expression

On the other hand, for a SDOF system, according to the definitions of strength reduction factor,

During the past years, some researchers have presented convenient

Strength reduction factor

Hard soil site

Intermediate soil site

Soft soil site

The

The spectra of

Hard soil site,

Intermediate soil site,

Soft soil site,

Hard soil site,

Intermediate soil site,

Soft soil site,

Hard soil site,

Intermediate soil site,

Soft soil site,

Hard soil site,

Intermediate soil site,

Soft soil site,

Through the above analysis, the regularity of

Analysis of modification coefficient

Hard soil site,

Intermediate soil site,

Soft soil site,

Hard soil site,

Intermediate soil site,

Soft soil site,

A step-by-step summary of the IMMPA procedure is presented as follows.

The natural frequencies,

For the

Idealize the

Determine

Determine the roof object displacement of

Repeat steps 1–5 for as many modes as required for sufficient accuracy.

Determine the total extreme responses,

In order to clarify how the proposed methodology should be applied, two simple analytical examples are presented. The structures considered are a 10-storey and a 15-storey eccentric reinforced concrete frame buildings, as illustrated in Figure ^{2} for 2th~11th floor; the sectional sizes of columns are 700 × 700 mm^{2} for 1th~3th storey, 600 × 600 mm^{2} for 4th~7th storey, and 500 × 500 mm^{2} for 8th~10th storey. For the 15-storey building, the sectional sizes of beams are 300 × 700 mm^{2} for 2th~5th floor, 300 × 650 mm^{2} for 6th~11th floor, and 300 × 600 mm^{2} for 12th~16th floor; the sectional sizes of columns are 700 × 700 mm^{2} for 1th~4th storey, 650 × 650 mm^{2} for 5th~10th storey, and 500 × 500 mm^{2} for 11th~15th storey. Steel ratios are approximately 1.5% for beam sections and 2% for column sections of the two buildings. Concrete compression strength is selected as 30 MPa for all columns and beams of the structure. The design dead load and live load are, respectively, 6.6 kN/m^{2} (4.7 kN/m^{2}) and 1.0 kN/m^{2} (2.0 kN/m^{2}) for each floor (roof). The damping of the building is modeled by the Rayleigh damping, and damping ratio

Schematic of example buildings.

The bidirectional “Taft” earthquake records (Kern County 1952, 1095 Taft Lincoln School) are selected to verify the accuracy of the presented method. The acceleration peak values of the records are adjusted to 3.1 m/sec^{2} and 4.0 m/sec^{2}, respectively. The nonlinear response time history analyses (NL-RHA) are implemented for the two example building subjected to the selected earthquake record and the time history responses of roof displacements are illustrated in Figure

Displacement of roof of the structural models.

10-story model, ^{2}

10-story model, ^{2}

15-story model, ^{2}

10-story model, ^{2}

10-story model, ^{2}

15-story model, ^{2}

15-story model, ^{2}

15-story model, ^{2}

The method to compare estimation results of IMMPA with exact results of NL-RHA for verifying the procedure is as follows.

The restoring force model of modal equivalent SDOF system is established by modal pushover analysis.

The dynamic response

Corresponding to the eight objective control points given in Section

The IMMPA is implemented and the modal static responses

The demand responses are compared between IMMPA and NL-RHA.

The comparisons of story drift ratios and floor displacements between NL-RHA method and IMMPA method are illustrated in Figure

Comparison of deformation responses between IMMPA and NL-RHA.

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A new IMMPA idea is presented in the paper based on traditional MPA method: superposed bidirectional earthquake excitation acting on modal ESDOF system can be regarded as an unidirectional “combined earthquake excitation”; in pushover procedure, the static force replacing “combined earthquake excitation” is assigned to three components of each floor of a structure based on model of vibration.

In accordance with this idea, in solution of structural displacement response, it is required to input “combined earthquake acceleration” to modal ESDOF system in order to establish

In order to check accuracy of the IMMPA procedure presented in this paper, two structures with mass eccentricity are designed, a bidirectional earthquake ground motion, Taft, is taken as earthquake excitation of the structural system, and deformation distribution of the structure is calculated, respectively, with NL-RHA and IMMPA, and the results are concluded as follows after comparative analysis.

In general, IMMPA method can be used to accurately estimate the distribution feature of plastic deformation of the structure along its floors.

Along with increase of peak of earthquake acceleration, structural response deviation estimated with IMMPA method may also increase.

Along with increase of the number of total floors of structures, structural response deviation estimated with IMMPA method may also increase.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research work was jointly supported by the National Natural Science Foundation of China (Grant no. 51108067), the “973” Project (Grand no. 2011CB013605), and Dalian Nationalities University project (Grant no. DC120101095).