# Modal-Based Ground Motion Selection Method for the Nonlinear Response Time History Analysis of Reinforced Concrete Shear Wall Structures

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## Abstract

**:**

## 1. Introduction

## 2. Modal-Based Ground Motion Selection Procedure for RC Shear Wall Structures

#### 2.1. Original Modal-Based Ground Motion Selection Procedure

**u**is story displacements vector,

**k**,

**c**and

**m**stand for the stiffness, damping and mass matrices, respectively, and

**i**is the influence vector. ${\ddot{u}}_{g}\left(t\right)$ is the acceleration history of the input ground motion. As the loading history does not affect the independence of the lateral force, the solution of Equation (1) can be expressed as:

#### 2.2. Proposed Modal-Based Ground Motion Selection Procedure for RC Shear Wall Structures

- (1).
- Apply lateral force whose distribution is $m{\mathbf{\varphi}}_{n}$, where n = 1, 2, to the structure separately, and generate two sets of curves of base shear-roof displacement (${V}_{bn}-{u}_{rn}$). Convert the (${V}_{bn}-{u}_{rn}$) curves to bilinear curves.
- (2).
- Transfer the (${V}_{bn}-{u}_{rn}$) bilinear curves to the (${F}_{sn}/{L}_{n}-{D}_{n}$) pushover curves of the NL-ESDOFs using Equation (7):$$\frac{{F}_{sn}}{{L}_{n}}=\frac{{V}_{bn}}{{M}_{n}^{*}}\text{}\mathrm{and}\text{}{D}_{n}=\frac{{u}_{rn}}{{\Gamma}_{n}{\varphi}_{rn}}$$
- (3).
- The vibration period of NL-ESDOFs can be calculated as:$${T}_{n}^{*}=2\pi \sqrt{\frac{{L}_{n}{D}_{ny}}{{F}_{sny}}}\text{}\mathrm{and}\text{}{D}_{ny}=\frac{{u}_{rny}}{{\Gamma}_{n}{\varphi}_{rn}}$$
- (4).
- Develop the NL-ESDOFs model using the base shear-top displacement relations of (${F}_{sn}/{L}_{n}-{D}_{n}$) curves and ${T}_{n}^{*}$ as the vibration period of NL-ESDOFs.

## 3. Case Study

#### 3.1. Structure Prototype

#### 3.2. Characteristics of the Seed IGMs

_{30}of site soil is the average shear velocity of top 30 m site soil and is assumed to be 400 m/s. Therefore, the site is classified as Class C of NEHRP. Detailed information for constructing the CMS are summarizes in Table 3, and the target CMSs and DSs are shown in Figure 2. It is seen from Figure 2 that CMSs built with the exact moment magnitude, but different conditioning periods are identical.

#### 3.3. IGM Selection for the NLRHA of RC Shear Wall Structures

## 4. Results and Discussions

#### 4.1. Comparison of the Results of MGMS Procedures

#### 4.2. Comparison of Seismic Demands by Different IGM Selection Procedures

#### 4.3. Comparison of the Deviation of the Computed Demands

## 5. Conclusions

- Considering the modal response characteristics of RC shear wall structures, the MGMS procedure for shear wall structures presented led to a more reliable computation of seismic demands than the original version of the MGMS procedure.
- Compared with the CM procedure with uniformly weight value for all the period points in computing difference between the response spectra of IGMs and mean spectra of the set (UW-CM), the CM procedure adopting variable weight value (VW-CM) procedure can ensure a more reliable and reasonable computation of seismic demands of RC shear wall structures.
- Compared with the VW-CM procedure, the presented MGMS procedure could noticeably improve the reliability and reasonability of the computed seismic deformation, including the floor displacement and inter-story drift ratio, and achieve similar reliability and reasonability in calculating the seismically induced force.
- Since the MGMS procedure just requires conducting NLRHA of equivalent single-degree-of-freedom systems, the computational consumption is minor. Taking advantage of high efficiency and great effectiveness in improving the reliability and reasonability of the NLRHA, the MGMS is an excellent supplement to the seismic design codes of practice’s IGM selection procedure for the NLRHA of wall structures.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Nomenclature

CMS | Conditional mean spectrum | i | Influence vector |

CM-UW | Uniformly weighted closest spectra matching procedure | ${L}_{n}$ | Modal excitation factor of mode n |

CM-VW | Variably weighted closest spectra matching procedure | m, c and k | Mass, damping, and stiffness matrices |

DS | design spectrum | ${M}_{n}$ | Generalized mass of mode n |

FCCFD | Frequency contents combination in the frequency domain | ${S}_{aj}\left(T\right)$ | Spectrum acceleration value of motion j at period T |

FCCTD | Frequency contents combination in the time domain | ${S}_{a}^{t}\left(T\right)$ | Spectrum acceleration value of target spectra at period T |

IDR | Inter-story drift ratio | ${S}_{d1}$ | Design spectrum acceleration at 1.0 s |

IGMs | Input ground motions | ${S}_{ds}$ | Design spectrum acceleration at 0.2 s |

MGMS | Modal-based ground motion selection | ${t}_{k}$ | Any time point within the duration of IGM |

MSE | Mean computed weighted mean squared error | ${T}_{j0}$ | duration of motion j |

NL-ESDOF | Nonlinear equivalent single-degree-of-freedom system | ${T}_{L}$ | Long-period transition period |

NLRHA | nonlinear response time history analysis | ${T}_{n}^{*}$ | Vibration period of NL-ESDOF of mode n |

SDOF | Single-degree-of-freedom system | $\widehat{{u}_{jn}}$ | The maximum displacement of NL-ESDOF of mode n under motion j |

u | Displacement vector of floor | ${u}_{jn}\left({t}_{k}\right)$ | Displacement of NL-ESDOF of mode n under motion j at time tk |

${a}_{n}\left(\mathrm{t}\right)$ | Normalized top displacement time history of mode n | ${u}_{rn}$ | Roof displacement of mode n |

$\overline{{d}_{ij}}$ | Benchmark demands ith story for motion set j | ${V}_{bn}$ | The base shear force of mode n |

${d}_{M}^{ij}$ | Seismic demands at ith story calculated using IGMs from selection method M for motion set j | ${\alpha}_{i}$ | Modal mass coefficient of mode i |

${D}_{n}$ | Peak displacement of mode n | ${\beta}_{j}$ | Maximum modal combination factor of motion j |

${d}_{n}$ | Displacement of ESDOFs of mode n | ${\beta}_{jk}$ | Maximum modal combination factor of motion j at time tk |

${D}_{n}\left(t\right)$ | Top displacement time history of the mode n | ${\Gamma}_{n}$ | the modal participating factor of mode n |

${d}_{ny}$ | Yield displacement of ESDOFs of mode n | ${\delta}_{M}^{ij}$ | The relative difference between the benchmark demands of motion set j and demands computed using IGMs from selection method M at the ith story |

${F}_{sn}$ | Restoring force of ESDOFs of mode n | ${\varphi}_{n}$ | Mode shape of mode n |

${F}_{sny}$ | Yield force of ESDOFs of mode n | ${\mathbf{\varphi}}_{rn}$ | Mode shape value at the roof of mode n |

## Appendix A

Record ID | Motion Set 1 | Motion Set 2 | ||||
---|---|---|---|---|---|---|

Scale Factor | RSN No. | Component (deg) | Scale Factor | RSN No. | Component (deg) | |

1 | 4.764 | RSN762 | 0 | 4.106 | RSN15 | 21 |

2 | 4.582 | RSN769 | 0 | 3.828 | RSN289 | 0 |

3 | 2.303 | RSN787 | 270 | 4.319 | RSN735 | 0 |

4 | 3.548 | RSN827 | 0 | 3.975 | RSN755 | 195 |

5 | 4.805 | RSN1261 | E | 4.168 | RSN827 | 0 |

6 | 4.790 | RSN1263 | E | 4.620 | RSN1005 | 90 |

7 | 4.415 | RSN1277 | E | 3.888 | RSN1282 | E |

8 | 4.663 | RSN127 | E | 4.179 | RSN1297 | N |

9 | 4.332 | RSN1300 | N | 3.610 | RSN1471 | E |

10 | 2.098 | RSN1484 | E | 2.247 | RSN1541 | E |

11 | 2.348 | RSN1500 | E | 2.619 | RSN1762 | 90 |

12 | 4.396 | RSN1522 | E | 4.536 | RSN1794 | 90 |

13 | 2.326 | RSN1762 | 90 | 1.404 | RSN3748 | 270 |

14 | 3.832 | RSN1794 | 90 | 2.127 | RSN3750 | 270 |

15 | 1.133 | RSN3748 | 70 | 3.357 | RSN3751 | 270 |

16 | 3.021 | RSN3751 | 270 | 4.136 | RSN3757 | 90 |

17 | 3.642 | RSN3757 | 90 | 2.624 | RSN4865 | NS |

18 | 4.190 | RSN4844 | NS | 4.146 | RSN4872 | NS |

19 | 3.803 | RSN4872 | NS | 2.729 | RSN5778 | NS |

20 | 3.756 | RSN6980 | E | 2.862 | RSN5806 | NS |

**Figure A3.**Mean spectra of IGMs selected from motion sets 1 and 2 with different selection procedures.

Record ID | Motion Set 3 | Motion Set 4 | ||||
---|---|---|---|---|---|---|

Scale Factor | RSN No. | Component (deg) | Scale Factor | RSN No. | Component (deg) | |

1 | 4.572 | RSN28 | 0 | 3.666 | RSN796 | 0 |

2 | 3.839 | RSN736 | 137 | 3.656 | RSN827 | 0 |

3 | 3.266 | RSN827 | 0 | 3.281 | RSN832 | 0 |

4 | 3.544 | RSN838 | 0 | 3.976 | RSN1019 | 0 |

5 | 3.552 | RSN1019 | 0 | 3.009 | RSN1083 | 170 |

6 | 4.657 | RSN1029 | 0 | 4.225 | RSN1166 | 180 |

7 | 3.774 | RSN1166 | 180 | 3.127 | RSN1208 | E |

8 | 3.615 | RSN1277 | E | 4.047 | RSN1277 | E |

9 | 4.153 | RSN1293 | N | 4.080 | RSN1280 | E |

10 | 4.606 | RSN1346 | N | 4.649 | RSN1293 | N |

11 | 4.136 | RSN1349 | N | 4.629 | RSN1349 | N |

12 | 4.613 | RSN1436 | E | 3.281 | RSN1471 | E |

13 | 3.340 | RSN1488 | E | 3.738 | RSN1488 | E |

14 | 2.997 | RSN1548 | E | 3.178 | RSN1794 | 90 |

15 | 2.964 | RSN3757 | 90 | 3.318 | RSN3757 | 90 |

16 | 3.370 | RSN4844 | NS | 3.772 | RSN4844 | NS |

17 | 3.741 | RSN4892 | NS | 4.187 | RSN4892 | NS |

18 | 4.702 | RSN5681 | NS | 3.154 | RSN5284 | NS |

19 | 3.061 | RSN6948 | E | 3.426 | RSN6948 | E |

20 | 3.277 | RSN6949 | W | 3.085 | RSN6980 | E |

**Figure A4.**Mean spectra of IGMs selected from motion sets 3 and 4 with different selection procedures.

Record ID | Motion Set 5 | Motion Set 6 | ||||
---|---|---|---|---|---|---|

Scale Factor | RSN No. | Component (deg) | Scale Factor | RSN No. | Component (deg) | |

1 | 3.2018 | RSN731 | 0 | 3.7418 | RSN731 | 0 |

2 | 2.9568 | RSN832 | 0 | 3.7123 | RSN736 | 137 |

3 | 3.4254 | RSN838 | 0 | 4.8197 | RSN812 | 0 |

4 | 3.9506 | RSN1261 | E | 3.4185 | RSN827 | 0 |

5 | 3.5022 | RSN1277 | E | 3.4556 | RSN832 | 0 |

6 | 4.7352 | RSN1279 | E | 3.0436 | RSN1208 | E |

7 | 4.5283 | RSN1285 | E | 4.617 | RSN1261 | E |

8 | 4.6728 | RSN1335 | E | 4.093 | RSN1277 | E |

9 | 4.0228 | RSN1339 | E | 4.7014 | RSN1339 | E |

10 | 4.3041 | RSN1436 | E | 4.8635 | RSN1431 | E |

11 | 4.5664 | RSN1470 | E | 4.2951 | RSN1466 | E |

12 | 3.3515 | RSN1475 | E | 3.0868 | RSN1471 | E |

13 | 2.9285 | RSN3757 | 90 | 4.3791 | RSN1522 | E |

14 | 4.796 | RSN3994 | 90 | 3.4225 | RSN3757 | 90 |

15 | 3.3816 | RSN4844 | NS | 3.952 | RSN4844 | NS |

16 | 2.9919 | RSN4872 | NS | 3.4966 | RSN4872 | NS |

17 | 3.065 | RSN5284 | NS | 3.582 | RSN5284 | NS |

18 | 4.4291 | RSN5681 | NS | 3.8479 | RSN5776 | NS |

19 | 4.4052 | RSN5796 | NS | 3.7347 | RSN6901 | W |

20 | 3.1957 | RSN6901 | W | 3.3329 | RSN6980 | E |

**Figure A5.**Mean spectra of IGMs selected from motion sets 5 and 6 with different selection procedures.

Record ID | Motion Set 7 | Motion Set 8 | ||||
---|---|---|---|---|---|---|

Scale Factor | RSN No. | Component (deg) | Scale Factor | RSN No. | Component (deg) | |

1 | 3.233 | RSN731 | 0 | 4.0338 | RSN731 | 0 |

2 | 3.4122 | RSN812 | 0 | 4.2574 | RSN812 | 0 |

3 | 4.8609 | RSN1232 | E | 3.2824 | RSN827 | 0 |

4 | 3.9131 | RSN1261 | E | 3.6365 | RSN832 | 0 |

5 | 3.2159 | RSN1263 | E | 3.1732 | RSN1208 | E |

6 | 4.0754 | RSN1285 | E | 4.0125 | RSN1263 | E |

7 | 3.7487 | RSN1339 | E | 4.6773 | RSN1339 | E |

8 | 4.3745 | RSN1358 | E | 3.9016 | RSN1464 | E |

9 | 4.6811 | RSN1436 | E | 3.9419 | RSN1465 | E |

10 | 3.1593 | RSN1465 | E | 3.3486 | RSN1467 | E |

11 | 3.2727 | RSN1469 | E | 4.0834 | RSN1469 | E |

12 | 3.9213 | RSN1470 | E | 4.1845 | RSN1473 | E |

13 | 3.3015 | RSN1522 | E | 4.1193 | RSN1522 | E |

14 | 4.1946 | RSN1575 | E | 3.682 | RSN1523 | E |

15 | 4.2539 | RSN1588 | N | 3.8839 | RSN1525 | E |

16 | 4.1117 | RSN3994 | 90 | 2.9612 | RSN3747 | 270 |

17 | 3.1024 | RSN4882 | NS | 3.2803 | RSN4848 | NS |

18 | 3.8213 | RSN5472 | NS | 3.8709 | RSN4882 | NS |

19 | 4.228 | RSN5681 | NS | 4.7679 | RSN5472 | NS |

20 | 4.1963 | RSN5804 | NS | 4.7607 | RSN5783 | NS |

**Figure A6.**Mean spectra of IGMs selected from motion sets 7 and 8 with different selection procedures.

## References

- Katsanos, E.; Sextos, A.; Manolis, G.D. Selection of earthquake ground motion records: A state-of-the-art review from a structural engineering perspective. Soil Dyn. Earthq. Eng.
**2010**, 30, 157–169. [Google Scholar] [CrossRef] - Chieffo, N.; Clementi, F.; Formisano, A.; Lenci, S. Comparative fragility methods for seismic assessment of masonry buildings located in Muccia (Italy). J. Build. Eng.
**2019**, 25, 100813. [Google Scholar] [CrossRef] - Forcellini, D. Numerical simulations of liquefaction on an ordinary building during Italian (20 May 2012) earthquake. Bull. Earthq. Eng.
**2019**, 17, 4797–4823. [Google Scholar] [CrossRef] - Forcellini, D. Soil-structure interaction analyses of shallow-founded structures on a potential-liquefiable soil deposit. Soil Dyn. Earthq. Eng.
**2020**, 133, 106108. [Google Scholar] [CrossRef] - Petridis, C.; Pitilakis, D. Fragility curve modifiers for RC dual buildings to include nonlinear site effects and SSI. Earthq. Spectra
**2020**, 36, 1930–1951. [Google Scholar] [CrossRef] - ASCE. Minimum Design Loads for Buildings and Other Structures (ASCE/SEI 7-17); American Society of Civil Engineering: Reston, VA, USA, 2017. [Google Scholar]
- CEN. ENV 1998-1 Eurocode 8: Design of Structures for Earthquake Resistance—Part 1: General Rules, Seismic Actions and Rules for Buildings; European Committee for Standardisation: Brussels, Belgium, 2004. [Google Scholar]
- Chinese Standard. Code for Seismic Design of Buildings, GB 50011-2010; Chinese Building Press: Beijing, China, 2008. [Google Scholar]
- NZS. Structural Design Actions (NZS 1170.5); Standards New Zealand Technical Committee: Wellington, New Zealand, 2004.
- Baker, J.W.; Cornell, C.A. A vector-valued ground motion intensity measure consisting of spectral acceleration and epsilon. Earthq. Eng. Struct. Dyn.
**2005**, 34, 1193–1217. [Google Scholar] [CrossRef] - Tarbali, K.; Bradley, B.A. The effect of causal parameter bounds in PSHA-based ground motion selection. Earthq. Eng. Struct. Dyn.
**2016**, 45, 1515–1535. [Google Scholar] [CrossRef] - Barbosa, A.R.; Ribeiro, F.L.A.; Neves, L. Influence of earthquake ground-motion duration on damage estimation: Application to steel moment resisting frames. Earthq. Eng. Struct. Dyn.
**2016**, 46, 27–49. [Google Scholar] [CrossRef] - Pan, Y.; Ventura, C.E.; Finn, W.L.; Xiong, H. Effects of ground motion duration on the seismic damage to and collapse capacity of a mid-rise woodframe building. Eng. Struct.
**2019**, 197, 109451. [Google Scholar] [CrossRef] - Tao, D.; Ma, Q.; Li, S.; Xie, Z.; Lin, D.; Li, S. Support Vector Regression for the Relationships between Ground Motion Parameters and Macroseismic Intensity in the Sichuan–Yunnan Region. Appl. Sci.
**2020**, 10, 3086. [Google Scholar] [CrossRef] - Nguyen, V.-Q.; Aaqib, M.; Nguyen, D.-D.; Luat, N.-V.; Park, D. A Site-Specific Response Analysis: A Case Study in Hanoi, Vietnam. Appl. Sci.
**2020**, 10, 3972. [Google Scholar] [CrossRef] - O’Donnell, A.P.; Kurama, Y.C.; Kalkan, E.; Taflanidis, A. Experimental evaluation of four ground-motion scaling methods for dynamic response-history analysis of nonlinear structures. Bull. Earthq. Eng.
**2017**, 15, 1899–1924. [Google Scholar] [CrossRef] - Naeim, F.; Alimoradi, A.; Pezeshk, S. Selection and Scaling of Ground Motion Time Histories for Structural Design Using Genetic Algorithms. Earthq. Spectra
**2004**, 20, 413–426. [Google Scholar] [CrossRef][Green Version] - Al Atik, L.; Abrahamson, N. An Improved Method for Nonstationary Spectral Matching. Earthq. Spectra
**2010**, 26, 601–617. [Google Scholar] [CrossRef][Green Version] - Kayhan, A.H.; Korkmaz, K.A.; Irfanoglu, A. Selecting and scaling real ground motion records using harmony search algorithm. Soil Dyn. Earthq. Eng.
**2011**, 31, 941–953. [Google Scholar] [CrossRef] - Wang, G. A ground motion selection and modification method capturing response spectrum characteristics and variability of scenario earthquakes. Soil Dyn. Earthq. Eng.
**2011**, 31, 611–625. [Google Scholar] [CrossRef] - Reyes, J.C.; Riaño, A.C.; Kalkan, E.; Quintero, O.A.; Arango, C.M. Assessment of spectrum matching procedure for nonlinear analysis of symmetric- and asymmetric-plan buildings. Eng. Struct.
**2014**, 72, 171–181. [Google Scholar] [CrossRef] - Jiang, W.; Li, B.; Xie, W.-C.; Pandey, M.D. Generate floor response spectra: Part 1. Direct spectra-to-spectra method. Nucl. Eng. Des.
**2015**, 293, 525–546. [Google Scholar] [CrossRef] - Han, S.W.; Ha, S.J. Assessment of ground motion selection criteria specified in current seismic provisions with an accurate selection algorithm. Bull. Earthq. Eng.
**2017**, 15, 4113–4132. [Google Scholar] [CrossRef] - Anajafi, H.; Medina, R.A. Uncertainties in using the spectrum matching technique for generating synthetic ground motions. In Proceedings of the 11th National Conference in Earthquake Engineering, Los Angeles, CA, USA, 25–29 June 2018. [Google Scholar]
- Anajafimarzijarani, H. Improved Seismic Design of Non-Structural Components (NSCs) and Development of Innovative Control Approaches to Enhance the Seismic Performance of Buildings and NSCs. Master’s Thesis, University of New Hampshire, Durham, UK, 2018. [Google Scholar]
- Mergos, P.E.; Sextos, A.G. Selection of earthquake ground motions for multiple objectives using genetic algorithms. Eng. Struct.
**2019**, 187, 414–427. [Google Scholar] [CrossRef] - Baker, J.W.; Lee, C. An Improved Algorithm for Selecting Ground Motions to Match a Conditional Spectrum. J. Earthq. Eng.
**2018**, 22, 708–723. [Google Scholar] [CrossRef] - Zhang, R.; Wang, D.S.; Chen, X.Y.; Li, H.N. Weighted scaling and selecting method of ground motions in time-history analysis considering influence of higher modes. China Civil. Eng. J.
**2019**, 52, 53–68. [Google Scholar] - Chopra, A. Dynamics of Structures: Theory and Applications to Earthquake Engineering, 5th ed.; Pearson Prentice Hall: Upper Saddle River, NJ, USA, 2017. [Google Scholar]
- Liu, Y.; Kuang, J.; Yuen, T.Y. Modal-based ground motion selection procedure for nonlinear response time history analysis of high-rise buildings. Earthq. Eng. Struct. Dyn.
**2019**, 49, 95–110. [Google Scholar] [CrossRef] - Stafford, B.; Coull, A. Tall Building Structures: Analysis and Design; John Wily: New York, NY, USA, 1991. [Google Scholar]
- Moradi, M.J.; Hariri-Ardebili, M.A. Developing a Library of Shear Walls Database and the Neural Network Based Predictive Meta-Model. Appl. Sci.
**2019**, 9, 2562. [Google Scholar] [CrossRef][Green Version] - Baker, J.W. Conditional Mean Spectrum: Tool for Ground-Motion Selection. J. Struct. Eng.
**2011**, 137, 322–331. [Google Scholar] [CrossRef] - FEMA. Improvement of Nonlinear Static Seismic Analysis Procedures: FEMA-440; Federal Emergency Management Agency: Redwood, CA, USA, 2005.
- Huang, K. Continuum MDOF Model for Seismic Analysis of Wall-Frame Structures. Master’s Thesis, Hong Kong University of Science and Technology, Hong Kong, China, 2014. [Google Scholar]
- CSI SV. 18: Integrated Finite Element Analysis and Design of Structures Basic Analysis Reference Manual; Computers and Structures Inc.: Berkeley, CA, USA, 2018. [Google Scholar]
- Ryan, K.L.; Polanco, J. Problems with Rayleigh Damping in Base-Isolated Buildings. J. Struct. Eng.
**2008**, 134, 1780–1784. [Google Scholar] [CrossRef] - Anajafi, H.; Medina, R.A.; Santini-Bell, E. Effects of the improper modeling of viscous damping on the first-mode and higher-mode dominated responses of base-isolated buildings. Earthq. Eng. Struct. Dyn.
**2019**, 49, 51–73. [Google Scholar] [CrossRef] - Kitayama, S.; Constantinou, M.C. Effect of superstructure modeling assumptions on the seismic performance of seismically isolated buildings. Earthq. Eng. Struct. Dyn.
**2021**, 50, 1805–1823. [Google Scholar] [CrossRef] - Chiou, B.S.J.; Darragh, R.; Gregor, N.; Silva, W.J. NGA Project Strong-Motion Database. Earthq. Spectra
**2008**, 24, 23–44. [Google Scholar] [CrossRef][Green Version] - Du, W.; Ning, C.; Wang, G. The effect of amplitude scaling limits on conditional spectrum-based ground motion selection. Earthq. Eng. Struct. Dyn.
**2019**, 48, 1030–1044. [Google Scholar] [CrossRef]

**Figure 3.**Comparison of the benchmark seismic demands and seismic demands computed by IGMs selected by MGMS procedures for motion set 6.

**Figure 8.**Deviations between the seismic demands from different IGM selection procedures and benchmark demands.

NO. | Mode | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | ||||||

T (s) | $\mathbf{\alpha}$ | T (s) | $\mathbf{\alpha}$ | T (s) | $\mathbf{\alpha}$ | T (s) | $\mathbf{\alpha}$ | T (s) | $\mathbf{\alpha}$ | |

W1 | 0.625 | 0.680 | 0.127 | 0.200 | 0.061 | 0.045 | 0.047 | 0.009 | 0.041 | 0.016 |

W2 | 1.830 | 0.642 | 0.341 | 0.205 | 0.145 | 0.060 | 0.090 | 0.027 | 0.066 | 0.015 |

W3 | 2.392 | 0.630 | 0.436 | 0.200 | 0.182 | 0.064 | 0.110 | 0.031 | 0.079 | 0.020 |

Spectrum No. | S_{ds} (g) | S_{d1} (g) | T_{L} (s) |
---|---|---|---|

DS 1 | 1.00 | 0.75 | 12 |

DS 2 | 1.20 | 0.85 | 10 |

_{ds}and S

_{d1}are spectral accelerations at 0.2 s and 1 s, respectively. T

_{L}is the transition period of the long-period.

Conditional Mean Spectrum | Conditioning Period | Moment Magnitude | Distance to Rupture Plane | V_{s30} |
---|---|---|---|---|

W1 CMS 1 | 0.625 s | 8.5 | ≥12 km | 400 m/s |

W2 CMS 1 | 1.830 s | |||

W3 CMS 1 | 2.391 s | |||

W1 CMS 2 | 0.625 s | 9.0 | ||

W2 CMS 2 | 1.830 s | |||

W3 CMS 2 | 2.391 s |

Motion Set | Target Spectrum | Scaling Factor | The Period Range for Minimizing the MSE | Structures Adopted for |
---|---|---|---|---|

1 | DS 1 | 1.0–5.0 | 1.0 s–5.0 s | W1, W2, W3 |

2 | DS 2 | 1.0–5.0 | 1.0 s–5.0 s | W1, W2, W3 |

3 | W1 CMS 1 | 3.0–5.0 | 0.125 s–1.250 s | W1 |

4 | W1 CMS 2 | 3.0–5.0 | 0.125 s–1.250 s | W1 |

5 | W2 CMS 1 | 3.0–5.0 | 0.366 s–3.660 s | W2 |

6 | W2 CMS 2 | 3.0–5.0 | 0.366 s–3.660 s | W2 |

7 | W3 CMS 1 | 3.0–5.0 | 0.478 s–4.781 s | W3 |

8 | W3 CMS 2 | 3.0–5.0 | 0.478 s–4.781 s | W3 |

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**MDPI and ACS Style**

Liu, Y. Modal-Based Ground Motion Selection Method for the Nonlinear Response Time History Analysis of Reinforced Concrete Shear Wall Structures. *Appl. Sci.* **2021**, *11*, 8230.
https://doi.org/10.3390/app11178230

**AMA Style**

Liu Y. Modal-Based Ground Motion Selection Method for the Nonlinear Response Time History Analysis of Reinforced Concrete Shear Wall Structures. *Applied Sciences*. 2021; 11(17):8230.
https://doi.org/10.3390/app11178230

**Chicago/Turabian Style**

Liu, Yang. 2021. "Modal-Based Ground Motion Selection Method for the Nonlinear Response Time History Analysis of Reinforced Concrete Shear Wall Structures" *Applied Sciences* 11, no. 17: 8230.
https://doi.org/10.3390/app11178230