Estimating the seismic response of a structure with known dynamic characteristics to a given characterization of the ground motion forms an important part of its overall seismic safety assessment.  Irrespective of the mode of this characterization, the response analysis must account for the uncertainty associated with the time description of a sample of the parent ground motion process, and statistical variations in the peak structural response should be accurately estimated. Since the direct step-by-step time integration of the equations of motion is not practical for an ensemble of accelerograms, and since the direct non stationary modeling of ground motions (as enveloped via modulated stationary processes) is complicated and lacks generalization, Professor M.D. Trifunac and his co-workers formulated response spectrum-based techniques for estimating the probabilistic response in a variety of situations. The focus of their efforts has been on the simplicity and convenience associated with the use of response spectrum ordinates, and on the estimation of second largest, third largest, fourth largest, ... peak responses in addition to the largest peak response. This additional information on higher order peaks has been shown to be quite useful in various damage assessment studies. These techniques can be easily used with the knowledge of the (i) Fourier spectra, (ii) response spectra, and (iii) strong motion duration, of the anticipated ground motion, and can thus be easily integrated into an overall seismic risk analysis.

The use of response spectra in probabilistic estimation of the structural response has followed from the pioneering work of Udwadia and  Trifunac (1974).  They showed how a probabilistic response spectrum can be computed from the Fourier transform of given ground motion by using the statistics of the largest peak of oscillator response. Amini and Trifunac (1981, 1985) first applied these ideas to propose a  response spectrum superposition technique for probabilistic response analysis of simple, fixed-base, shear buildings to translational component of ground motion. Their formulation included the estimation of higher order peak amplitudes, besides the largest peak amplitude. Later, Gupta and Trifunac (1988a) improved the estimation of higher order peak amplitudes significantly by employing the concept of order statistics, and Gupta and Trifunac (1987a) included the effect of modal interaction. Gupta and Trifunac (1987b,c, 1988b) extended these  formulations respectively to the cases of  (i) symmetric building response to torsional component of ground motion, (ii) building response to  simultaneously acting three translational components of ground motion, and (iii) building response to translational and rocking components of ground motion. Gupta and Trifunac (1990a,b,c, 1991a) further generalized  the formulations for rotational components of ground motion by including the effects of phase difference between the translational and rotational components, while Trifunac and Gupta (1991) showed how Pseudo Relative Acceleration can be evaluated approximately.

Gupta and Trifunac (1990)  and Gupta  and Trifunac (1991b) showed how the probabilistic response of flexible-base, shear buildings can be estimated for the excitation comprising of the translational and rotational components of ground motion, when soil-structure interaction effects are included.  Gupta and Trifunac (1993) have used the above ideas to study the contribution of rocking to the total response of multistoried buildings during the 1989 Loma Prieta earthquake.

Recently, Gupta and Trifunac (1996) present a comprehensive review of the state of the art in probabilistic estimation of maxima of structural response, and continue with (1) analysis of nonstationarity of seismic response (Gupta and Trifunac, 1998a,b), (2) improved probabilistic spectrum superposition (Gupta and TRifunac, 1998c), and (3) refinement in statistics of ordered peaks in stationary stochastic processes (Gupta and Trifunac, 1998d,e).

The above cited work, describes the response of a given structure to an earthquake of known size and occurring at a given distance.  Since the Uniform Hazard Spectrum concept cannot be applied directly to a multi-degree-of-freedom structural system to compute Uniform Hazard Envelopes of maximum shear forces, bending moments and inter-storey drifts, we are also developing methods for extension of the Uniform Hazard methodology to the response of multi degree-of-freedom structural systems.  The first examples of this methodology are presented in Gupta (1994) and  Todorovska (1995).
 

References

• Amini, A. and M.D. Trifunac (1981).  Distribution  of  peaks  in  linear earthquake  response,  J.  Eng.  Mech.  Div.,   Proc.   ASCE, 107(EM1)},  207-227.
• Amini, A. and M.D. Trifunac (1985). Statistical  extension  of  response  spectrum superposition,  Soil Dyn. Earthq. Eng.,  4(2), 54-63.
• Gupta, I.D. (1994).  A probabilistic approach for estimating the response of multi-degree-of-freedom structures, Soil Dyn. Earthq. Eng., 13(2), 1994.
• Gupta, I.D. and M.D. Trifunac (1996).  Investigation of Nonstationarity in Stochastic Seismic Response of Structures, Dept. of Civil Eng. Report No. CE 96-01, Univ. Southern California, Los Angeles, California.
• Gupta, I.D. and M.D. Trifunac (1987a).  Order statistics of peaks in earthquake response of multi-degree-of-freedom systems,  Earthq. Eng. Eng. Vib.,  7(4)}, 15-50.
• Gupta, I.D.  and  M.D.  Trifunac  (1987b).  A  note  on  contribution  of  torsional  excitation  to  earthquake  response  of   simple   symmetric  buildings,  Earthq. Eng. Eng. Vib.,  7(3), 27-46.
• Gupta, I.D. and M.D. Trifunac (1987c).  Order statistics of peaks of response to multi-component seismic excitation, Bull. Ind. Soc. Earthq. Tech.,  24(3,4}, 135-159.
• Gupta, I.D. and M.D. Trifunac (1988a).  Order statistics of peaks in earthquake response,  J. Eng. Mech., 114(10), 1605-1627.
• Gupta,  I.D.  and  M.D.  Trifunac  (1988b).  A  note  on  contribution of rocking excitation  to  earthquake  response  of  simple  buildings,  Bull. Ind. Soc. Earthq. Tech.,  25(2), 73-89.
• Gupta, I.D. and M.D. Trifunac (1990). Probabilistic spectrum superposition for response analysis including the effects of soil-structure interaction,  J. Prob. Eng. Mech., 5(1),  9-18.
• Gupta, I.D. and M.D. Trifunac (1998a). Defining equivalent stationary PSDF to account for nonstationarity of earthquake ground motion, with I.D. Gupta, Soil Dyn. Earthq. Eng., 17(2), 89-99.
• Gupta, I.D. and M.D. Trifunac (1998b).  A note on nonstationarity of seismic response of structures, Europ. Earthq. Eng., (in press).
• Gupta, I.D. and M.D. Trifunac (1998c).  An improved probabilistic spectrum superposition,  Soil Dyn. Earthqu. Eng., 17(1), 1-11.
• Gupta, I.D. and M.D. Trifunac (1998d).  A Note on the Statistics of Ordered Peaks in Stationary Stochastic Processes,  Soil Dyn. Earthq. Eng., 17(5), 317-328.
• Gupta, I.D. and M.D. Trifunac (1998e).  Statistics of ordered peaks in the response of nonclassically damped structures, Prob. Eng. Mech.,  (in press).
• Gupta, V.K. and M.D. Trifunac (1990a).  A  note  on  contributions  of  ground  torsion  to  seismic  response  of   symmetric   multistoried buildings,  Earthq. Eng. Eng. Vib., 10(3), 27-40.
• Gupta, V.K.  and  M.D.  Trifunac  (1990b).  Response  of  multistoried  buildings to ground translation and rocking  during  earthquakes,  J. Prob. Eng. Mech., 5(3), 138-145.
• Gupta, V.K.  and  M.D.  Trifunac  (1990c).  Response  of  multistoried buildings to  ground  translation  and  torsion  during  earthquakes, European Earthq. Eng., IV(1), 34-42.
• Gupta, V.K. and M.D. Trifunac (1991a).  Effects of  ground  rocking  on dynamic  response  of  multistoried  buildings  during   earthquakes, Struct. Eng./Earthquake Eng. (JSCE),  8(2), 43-50.
• Gupta,  V.K.  and  M.D.  Trifunac  (1991b). Seismic   response   of  multistoried  buildings  including  the  effects  of  soil-structure interaction, Soil Dyn. Earthq. Eng., 10(8), 414-422.
• Gupta, V.K. and M.D. Trifunac (1993).  A  note  on  the  effects  of ground rocking on the response of buildings during 1989 Loma  Prieta earthquake,  Earthq. Eng. Eng. Vib., 13(2), 12-28.
• Todorovska, M.I. (1995). A note on distribution of amplitudes of peaks in structural response including uncertainties of the exciting ground motion and of the structural model, Soil Dyn.Earthq. Eng., 14 (3), 211-217.
• Trifunac, M.D. and V.K. Gupta (1991).  Pseudo  relative  acceleration    spectrum,  J. Eng. Mech. (ASCE), 117(4), 924-927.
• Udwadia, F.E. and M.D. Trifunac  (1974). Characterization  of  response  spectra through the statistics of oscillator response,  Bull. Seism. Soc.  Amer.,  64(1), 205-219.