AN EMPIRICAL STUDY FOR STATISTICAL EVALUATION OF PEAK RESPONSE DISPLACEMENTS

Year 2019, Volume: 37 Issue: 2, 611 - 624, 01.06.2019

Gülçin Tekin Gülüm Tanırcan

Abstract

In this study, elastic response analyses of single degree of freedom (SDOF) structures under an extensive database of real earthquake excitation are investigated to clarify two particular issues: First is to determine whether the statistical distribution of ordered peaks in response displacement time histories follows Rayleigh distribution as it is a theoretical assumption valid for narrow band response of SDOF system with low damping value. Second is to look for a functional form to define amplitudes of the peaks in response displacement time histories as a function of structural period and response spectral displacement (Sd). Probability density functions (PDFs) were calculated for structural periods between 0.1-5 sec. using order statistics approach. Numerical results support that Rayleigh distribution describes the probabilistic characteristics of ordered peaks in elastic earthquake response adequately. Scale parameters of the distribution were calculated. It was observed that there is a clear logarithmic relationship between the peak number and the normalized peaks’ amplitudes in displacement time histories. Hence, a nonlinear regression model was proposed to define the peaks’ amplitudes in response displacement time histories of a SDOF system with known response Sd values and structural periods. In another words the model is capable of estimating number of times of a Sd that can be repeated by a structure without any damage. It is observed that even in the elastic systems, response values that are quite close to the value of the highest peak displacement are repeated many times. In general, for SDOF systems having natural period of vibration greater than or equal to 0.9 sec., oscillation having 90 % of the value of highest peak is repeated two times under the earthquake excitation. SDOF systems having natural period of vibration vary between 1 and 1.7 sec., ratio between second largest peak and Sd is about 95 %.Those calculations are essential for determining start of low-cycle fatigue behavior particularly for design of column-beam joints of steel structures or bridges.

Keywords

Elastic single degree of freedom systems, earthquake response analysis, regression analysis, order statistics approach, amplitudes of the local maximum values, low-cycle fatigue.

References

• [1] Rice S.O., (1944) Mathematical Analysis of Random Noise, Bell System Technical Journal 23, 282-332.
• [2] Rice S.O., (1945) Mathematical Analysis of Random Noise, Bell System Technical Journal 24, 46-156.
• [3] Cartwright D.E, Longuet-Higgins M.S., (1956) The Statistical Distribution of the Maxima of a Random Function, Proceedings of the Royal Society of London, Series A, 237, 212-232.
• [4] Udwadia F.E, Trifunac M.D., (1974) Characterization of Response Spectra through the Statistics of Oscillator Response, Bulletin of the Seismological Society of America 64, 205-219.
• [5] Yazdani A, Salimi M-R., (2015) Earthquake Response Spectra Estimation of Bilinear Hysteretic Systems Using Random Vibration Theory Method, Earthquakes and Structures 8, 1055-1067.
• [6] Amini A, Trifunac M.D., (1981) Distribution of Peaks in Linear Earthquake Response, Journal of the Engineering Division 107(EM1), 207-227.
• [7] Amini A, Trifunac M.D., (1985) Statistical Extension of Response Spectrum Superposition, International Journal of Soil Dynamics and Earthquake Engineering 4, 54-63.
• [8] Gupta I.D, Trifunac M.D., (1988) Order Statistics of Peaks in Earthquake Response, Journal of Engineering Mechanics 114, 1605-1627.
• [9] Basu B, Gupta V.K., Kundu D., (1996) Ordered Peak Statistics through Digital Simulation, Earthquake Engineering Structural Dynamics 25, 1061-1073.
• [10] Gupta I.D, Trifunac M.D., (1998) A Note on the Statistics of Ordered Peaks in Stationary Stochastic Processes, Soil Dynamics and Earthquake Engineering 17, 317-328.
• [11] NGA (2008) Next Generation Attenuation Web. http://peer.berkeley.edu/products/strong_ground_motion_db.html. Accessed 26 June 2015.
• [12] PEER (1996) Pacific Earthquake Engineering Research Center Web. http://ngawest2.berkeley.edu/. Accessed 26 June 2015.
• [13] Şafak E., (1998) 3D Response Spectra: A Method to Include Duration in Response Spectra, 11th European Conference on Earthquake Engineering, Paris, France.
• [14] Clough R.W, Penzien J., (1993) Dynamics of Structures. 2nd edition, McGraw-Hill, New York.
• [15] Gavin H.P., (2013) The Levenberg-Marquardt Method for Nonlinear Least Squares Curve-Fitting Problems. http://people.duke.edu/~hpgavin/ce281/lm.pdf
• [16] Tekin G, Tanırcan G.B, Şafak E., (2010) Probabilistic Earthquake Response Analysis of Single Degree of Freedom Structures, 14th European Conference on Earthquake Engineering, Ohrid, Macedonia.
• [17] Tekin G., (2010) Probabilistic Earthquake Response Analysis of Single Degree of Freedom Structures, MSc Thesis, Boğaziçi University, Istanbul, Turkey.
• [18] Basu B, Gupta V.K., (1996) Expected Seismic Damage during Narrow-Band Structural Responses, Structural Engineering and Earthquake Engineering 13,119-123.
• [19] Morikawa H, Zerva A., (2007) An Approximate Representation for Statistics of Maximum Responses of One Degree-of-Freedom System, JSCE Annual Meeting, Japan.
• [20] Şafak E., (1988) Analytical Approach to Calculation of Response Spectra from Seismological Models of Ground Motion, Earthquake Engineering and Structural Dynamics 16, 121-134.