Skip to main content
SpringerLink
Log in

Multidimensional performance limit state for probabilistic seismic demand analysis

  • Original Research Paper
  • Published:
Bulletin of Earthquake Engineering Aims and scope Submit manuscript

Abstract

Based on a framework of Probabilistic Seismic Demand Analysis, a nonlinear dynamic model of a reinforced concrete (RC) building was established to obtain a demand hazard curve that considers multidimensional performance limit states (MPLSs), including combinations of peak floor acceleration and interstory drift. A definition of the two limit states is expressed using a generalized MPLSs equation. The peak floor acceleration and the interstory drift were considered to be dependent and were assumed to follow a bidimensional lognormal distribution. The maximum interstory drift and the maximum peak floor acceleration were calculated using Increment Dynamic Analysis and nonlinear time history analysis. The numerical formula for a demand hazard curve of the modelled building was then derived by coupling the bidimensional lognormal distribution with the ground motion hazard curve. The uncertainties involved in MPLSs and structural parameters, as well as the different threshold values for peak floor acceleration, were further considered to determine the sensitivity of demand hazard curves. The analysis results showed that the proposed method can be used to describe the damage performance of various building structures, which are sensitive to multiple response parameters including drift and acceleration. Moreover, it was demonstrated in this study that the demand hazard curves were relatively conservative if the coefficient of variation, the peak floor acceleration threshold, the interaction factor N IDR and added stiffness, were appropriately selected.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Abbreviations

MPLSs:

Multidimensional performance limit states

IDA:

Incremental Dynamic Analysis

PSDA:

Probabilistic Seismic Demand Analysis

PSHA:

Probabilistic seismic hazard analysis

EDP:

Engineering Demand Parameter

PDF:

Probability density function

IDR :

The maximum interstory drift

PFA :

The maximum peak floor acceleration

IM :

Intensity measures

S a (T 1):

Spectral acceleration

idr :

Threshold value of the interstory drift limit state

pfa :

Threshold value of the peak floor acceleration limit state

idr lim :

Interstory drift limit state

Pfa lim :

Peak floor acceleration limit state

\( idr_{{{\text{rand\,lim}},i}} \) :

A random threshold value of the interstory drift limit state at the ith performance level

\( pfa_{{{\text{rand}}\lim ,i}} \) :

A random threshold value for the peak floor acceleration limit state at the ith performance level

\( idr_{{{\text{fixed}}\lim ,i}} \) :

A deterministic threshold value of the interstory drift limit state at the ith performance level

\( pfa_{{{\text{fixed}}\lim ,i}} \) :

A deterministic threshold value for the peak floor acceleration limit state at the ith performance level

idr lim,ij :

A deterministic interstory drift limit state at the ith performance level

pfa lim,j :

The jth fixed peak floor acceleration limit state

ɛ :

Uncertainties of ground motion and epistemic uncertainty

Y :

Uncertainties of structural system characteristics

N IDR , N PFA :

Interaction factor determining the shape of the surface of 2D limit state

Ω:

The one-dimensional failure domain of the structure

D :

The two-dimensional failure domain

\( \mu_{IDR |IM = im} \) :

The log-mean of interstory drift

\( \sigma_{IDR |IM = im} \) :

The log-standard deviation of interstory drift

μ PFA|IM = im :

The log-mean of peak floor acceleration

σ PFA|IM = im :

The log-standard deviation of peak floor acceleration

ρ :

The correlation coefficient

cv:

The coefficient of variation

\( f\left( {IDR,PFA |IM = im} \right) \) :

PDF of a bivariate lognormal distribution

f(IDR):

PDF of the interstory drift

\( f(idr_{{{\text{rand\,lim}},i}} ) \) :

PDF of interstory drift limit state

P(IDR > idr|IM = im):

The hazard fragility function

P(IDR > idr, PFA > pfa|IM = im):

A two-dimensional hazard fragility function

λ IM (im):

Seismic hazard function of the site originated from PSHA

v IDR (idr):

The mean annual frequency of exceeding the interstory drift limit state

v IDR,PFA (idr, pfa):

The mean annual frequency of exceedance that includes two limit states

\( v_{IDR} (idr_{{{\text{rand\,lim}},i}} ) \) :

The mean annual frequency of exceeding the random interstory drift limit state

v IDR,PFA (idr lim,ij , pfa lim,j ):

The mean annual frequency of exceeding ith fixed interstory drift limit state when the effect of jth acceleration threshold is considered

\( v_{IDR,PFA} (idr_{{{\text{rand}}\lim ,i}} ,pfa_{{{\text{rand}}\lim ,i}} ) \) :

The mean annual frequency of exceeding two randomly independent/interdependent limit states

\( P[idr < IDR(Y = y,IM,\varepsilon ) ] \) :

The probability of structural damage in the design working life

\( P[idr < IDR(Y = y,IM,\varepsilon ),pfa ] \) :

The probability of structural damage that reflects two performance limit states

\( P[idr_{\lim ij} < IDR(Y = y,IM,\varepsilon ),pfa_{\lim ,j} ] \) :

The probability of structural damage when the effect of jth acceleration threshold is considered

References

  • ATC58-2 (2006). Preliminary evaluation of methods for defining performance, FEMA, Washington D.C of USA

  • Baker JW (2010) Conditional mean spectrum: tool for ground-motion selection. J Struct Eng 137(3):322–331

    Article  Google Scholar 

  • Baker JW, Cornell AC (2005) A vector-valued ground motion intensity measure consisting of spectral acceleration and epsilon. Earthquake Eng Struct Dynam 34(10):1193–1217

    Article  Google Scholar 

  • Barron-Corverra R (2000) Spectral evaluation of seismic fragility of structures. PhD Dissertation, State University of New York at Buffalo

  • Cimellaro GP (2007) Simultaneous stiffness–damping optimization of structures with respect to acceleration, displacement and base shear. Eng Struct 29(11):2853–2870

    Article  Google Scholar 

  • Cimellaro GP, Reinhorn A (2010) Multidimensional performance limit state for hazard fragility functions. Journal of engineering mechanics 137(1):47–60

    Article  Google Scholar 

  • Cimellaro GP, Reinhorn AM, Bruneau M, Rutenberg A (2006) Multi-dimensional fragility of structures: formulation and evaluation, Multidisciplinary Center for Earthquake Engineering Research, University at Buffalo, State University of New York

  • Cimellaro GP, Lavan O, Reinhorn AM (2009) Design of passive systems for control of inelastic structures. Earthquake Eng Struct Dynam 38(6):783–804

    Article  Google Scholar 

  • Cornell CA (1968) Engineering seismic risk analysis. Bull Seismol Soc Am 58(5):1583–1606

    Google Scholar 

  • Cornell CA, Krawinkler H (2000) Progress and challenges in seismic performance assessment. PEER Center News 3(2):1–3

    Google Scholar 

  • Deierlein GG, Krawinkler H, Cornell CA (2003) A framework for performance-based earthquake engineering. In: Pacific conference on earthquake engineering, pp 1-8

  • FEMA (2006) 445. Next-Generation Performance-Based Seismic Design Guidelines Program Plan for New and Existing Buildings. Prepared by ATC for FEMA, Washington DC

    Google Scholar 

  • GB50011 (2010) Code for seismic design of buildings. China Architecture Industry Press, Beijing of China

    Google Scholar 

  • Ghobarah A (2001) Performance-based design in earthquake engineering: state of development. Eng Struct 23(8):878–884

    Article  Google Scholar 

  • Jalayer F (2003) Direct probabilistic seismic analysis: Implementing non-linear dynamic assessments. PhD Dissertation., Stanford University, Ann Arbor

  • JGJ3 (2010) Technical specification for concrete structures of tall building. China Architecture Industry Press, Beijing of China

    Google Scholar 

  • Kiureghian AD (2005) Non-ergodicity and PEER’s framework formula. Earthquake Eng Struct Dynam 34(13):1643–1652

    Article  Google Scholar 

  • Li Y, Han J, Tian Q, Chen W, Zhao S (2009) Study on influence of infilled walls on seismic performance of RC frame structures. Journal of Earthquake Engineering and Engineering Vibration 3:008

    Google Scholar 

  • Lin L, Naumoski N, Saatcioglu M, Foo S (2011) Improved intensity measures for probabilistic seismic demand analysis. part 2: application of the improved intensity measures. Can J Civ Eng 38(1):89–99

    Article  Google Scholar 

  • Ming WY, Zheng HS (2010) Two New Kinds of Monte-Carlo Methods for Solving Numerical Integration. Math Practice Theory 10:027

    Google Scholar 

  • Moehle J, Deierlein GG (2004) A framework methodology for performance-based earthquake engineering. In: 13th World conference on earthquake engineering, pp 3812–3814

  • Porter KA (2003) An overview of PEER’s performance-based earthquake engineering methodology. In: Proceedings of 9th international conference on applications of statistics and probability in civil engineering, San Francisco, California

  • Priestley MJN (2000) Performance based seismic design. Bulletin of the New Zealand society for earthquake engineering 3(3):325–346

    Google Scholar 

  • Rahman S, Heqin X (2004) A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probab Eng Mech 19(4):393–408

    Article  Google Scholar 

  • Reinhorn A, Barron-Corverra R, Ayala A (2001) Spectral evaluation of seismic fragility of structures. In: Proc., Proceedings ICOSSAR

  • Shome N (1999) Probabilistic seismic demand analysis of nonlinear structures. 9924607 Ph.D., Stanford University, Ann Arbor

  • Sun F, Pan Z (2011) The extrapolation algorithms for numerical triple integrals. Numerical Mathematics A Journal of Chinese Universities 4:010

    Google Scholar 

  • Sun HB, Wu ZY, Liu XX (2013) Multidimensional performance limit state for structural fragility estimation. Eng Mech 30(5):147–152

    Google Scholar 

  • Tothong P (2007) Probabilisitic seismic demand analysis using advanced ground motion intensity measures, attenuation relationships, and near-fault effects. PhD Dissertation., Stanford University, Ann Arbor

  • Tothong P, Cornell CA (2006) Application of nonlinear static analyses to probabilistic seismic demand analysis. In: Proceedings of 8th US National Conference on Earthquake Engineering 2006, April 18, 2006–April 22, 2006, Earthquake Engineering Research Institute, pp 4373–4382

  • Vamvatsikos D, Cornell CA (2002) Incremental dynamic analysis. Earthquake Eng Struct Dynam 31(3):491–514

    Article  Google Scholar 

  • Vamvatsikos D, Cornell CA (2004) Applied incremental dynamic analysis. Earthquake Spectra 20(2):523–553

    Article  Google Scholar 

  • Wang QA, Wu ZY, Liu SK (2012) Seismic fragility analysis of highway bridges considering multi-dimensional performance limit state. Earthquake Engineering and Engineering Vibration 11(2):185–193

    Article  Google Scholar 

  • Wu QY, Zhu HP, Fan J, Zeng ZH (2012) Seismic performance assessment on some frame structure. J Vib Shock 31(15):158–164

    Google Scholar 

  • Yun S-Y, Hamburger RO, Cornell CA, Foutch DA (2002) Seismic performance evaluation for steel moment frames. J Struct Eng 128(4):534–545

    Article  Google Scholar 

  • Zeng ZH, Fan J, Yu QQ (2012) Performance-based probabilistic seismic demand analysis of bridge structures. Eng Mech 29(3):156–162

    Google Scholar 

Download references

Acknowledgments

The research support from the National Natural Science Foundation of China (51278420) and Graduate Starting Seed Fund of Northwestern Polytechnical University (Z2014114) were greatly appreciated.

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Northwestern Ploytechnical University, Dist. Chang’an, Xi’an, 710129, China

    Xiao Xiao Liu, Zi-Yan Wu & Feng Liang

Authors
  1. Xiao Xiao Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zi-Yan Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Feng Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiao Xiao Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, X.X., Wu, ZY. & Liang, F. Multidimensional performance limit state for probabilistic seismic demand analysis. Bull Earthquake Eng 14, 3389–3408 (2016). https://doi.org/10.1007/s10518-016-0013-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10518-016-0013-6

Keywords

Search

Navigation