International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

A new probabilistic model of fully non-stationary ground motion and its application Liu, Zhangjun; Liu, Wei; Dan, Qingwen Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 A New Probabilistic Model of Fully Non-Stationary Ground Motion and its Application Zhangjun Liu Professor, College of Civil Engineering & Architecture, China Three Gorges University, Yichang, China Wei Liu Graduate Student, College of Civil Engineering & Architecture, China Three Gorges University, Yichang, China Qingwen Dan Graduate Student, College of Civil Engineering & Architecture, China Three Gorges University, Yichang, China ABSTRACT: A numerical simulation scheme is presented that combines the advantages of the spectral representation method (SRM) and the idea of random function to generate fully non-stationary ground motion. In the paper, firstly, a new family of spectral representation method is proposed. Secondly, these standard orthogonal random variables in SRM formula can be defined as the orthogonal function form of a basic random variable, and the original stochastic process can be represented as a function of the basic random variable. Its advantage is proved that fully non-stationary ground motion can be completely represented by a dimension-reduced spectral model with just few elementary random variables through defining the high-dimensional standard orthogonal random variables of classical spectral representation into the low-dimensional orthogonal random function. Finally, it provides an opportunity to incorporate the probability density evolution method (PDEM), and study the stochastic dynamics in engineering.  It is well understood in earthquake engineering community that the rational description and model of random earthquake ground motions underlies the seismic analysis and reliability assessment of engineering structures (Wang and Li, 2011). The systematic development on the study of random seismic ground motion began with the pioneered work contributed by Housner in 1947 (Douglas and Aochi, 2008), who modeled the seismic acceleration as a pulse-structured random process. While the nonlinear random analysis, in practices, of seismic structures involves the transform of earthquake ground motion models from frequency-domain representation to temporal-domain counterpart. The practical demand highly prompts the enthusiasm of researchers on the stochastic simulation of random processes. There has been tens of simulation techniques so far among which, nevertheless, the spectral representation is widely used due to its rigorous mathematical formulation and easier-to-be-implemented algorithm (Liu et al, 2014, 2015). In the paper, a numerical simulation scheme is presented that combines the advantages of the spectral representation method (SRM) and the idea of random function to generate fully non-stationary ground motion. Its advantage is proved that fully non-stationary ground motion can be completely represented by a dimension-reduced spectral model with just few elementary random variables through defining the high-dimensional standard orthogonal random variables of classical spectral representation into 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 the low-dimensional orthogonal random functions. The proposed technique provides an opportunity to employ the probability density evolution method (PDEM) which has been developed by Li and Chen in the past few years to study the stochastic nonlinear responses of general structural systems (Li and Chen, 2009). An example, which deals with a MDOF aqueduct structure subjected to ground motions, is investigated to illustrate the proposed procedure. 1. EVOLUTIONARY POWER SPECTRUM OF FULLY NON-STATIONARY GROUND MOTION In this paper, based on the Kanai-Tajimi power spectrum of stationary ground motion acceleration process, the evolutionary power spectrum (two-side spectrum) of fully non-stationary ground motion acceleration process is given by g2( , ) ( )XS t A t   4 2 2 2g g g022 2 2 2 2g g g( ) 4 ( ) ( ) ( )( ) 4 ( ) ( )t t t S tt t t           (1) where g ( )t and g ( )t  are the frequency and the damping ratio of site soil, respectively; )(0 tS  is the spectral intensity factor; )(tA  is deterministic intensity modulation function. In Eq. (1), )(0 tS  can be expressed as 2P02g gg( )1( ) 2 ( )2 ( )AS tt tt                   (2) where AP denotes the ground motion peak acceleration,   denotes the peak factor. In Eq. (1), the parameters g ( )t  and g ( )t  can be expressed as (Cacciolaand Deodatis, 2011) g 0( ) tt a T  , g 0( ) tt b T          (3) where T is the duration of ground motion, 0 0,  and ba,  are site soil parameters.  For the intensity modulation function )(tA  is given as ( ) exp 1dt tA t c c                       (4) where c  denotes the time of ground motion peak acceleration, d is the parameter controlling the shape of )(tA .  2. THE SPECTRAL REPRESENTATION AND RANDOM FUNCTIONS METHOD OF NON-STATIONARY PROCESSES In general, we may assume that g ( )X t is a non-stationary ground motion process with zero mean, the spectral representation can be expressed (Liu, 2015) gg 1( ) 2 ( , )NX kkX t S t       cos( ) sin( )k k k kt X t Y        (5) whichk k    and  denotes the discrete frequency step; N  denotes the truncated number; g ( , )XS t is the two-side evolutionary power spectrum density function. In Eq. (5), these standard orthogonal random variables { , }k kX Y ( 1,2, , )k N must satisfy the basic conditions as follow     0k kE X E Y ,   0k mE X Y      (6a)    k m k m kmE X X E Y Y          (6b) where  E   denotes the mathematical expectation, km is the Kronecker symbol. For the standard orthogonal random variables { , }k kX Y ( 1,2, , )k N  in Eq. (5), the random function is suggested to represent the standard orthogonal random variables by Liu (2014). Firstly, suppose two orthonormal random 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 variablesnX and nY are functions of the basic random variable  , respectively, namely 2cos( / 4)nX n  , 1,2, ,n N    (7b) 2sin( + / 4)nY n  , 1,2, ,n N   (7b) where the basic random variable  satisfy uniform distribution in the interval [0, 2 )π . It is easy to prove that the orthonormal random variables { , }n nX Y 1,2, ,( )n N defined by Eq.  (7) can satisfy the basic conditions of Eq. (6). Secondly, the orthonormal random variables { , }n nX Y ( 1,2, , )n N need be converted to the standard orthogonal random variables { , }k kX Y ( 1,2, , )k N in Eq. (5) by deterministic one-to-one mapping way.  3. SIMULATION OF FULLY NON-STATIONARY GROUND MOTION In the above evolutionary power spectrum, the parameters’ values of the evolutionary power spectrum are shown in Table 1. Moreover, the ground motion peak acceleration P = 196Acm/s2.  Table 1: Parameter values. 10 /s  0  1/ sa   b  24.94 0.67 8 0.1 sc   d    / sT  5.5 1.8 3.0 20  In the spectral representation and random function method of fully non-stationary ground motion, the basic random variables  is uniformly discretized in the interval [0, 2 )π , and the discrete points’ number sel = 610n. Taking each discrete point value i 1( 1,2, , ) sei n  into Eq. (7) to generate a set of discrete values of orthonormal random variables { , }n nX Y ( 1,2, , )n N , and convert { , }n nX Y  into the standard orthogonal random variables { , }k kX Y ( 1,2, , )k N in Eq. (5) by deterministic one-to-one mapping way. So, the representative history time for fully non-stationary ground motion acceleration process can be generated. The representative time histories are shown in Figure 1.   0 5 10 15 20-200- 50-100-50050100150200250Time [s]Seismic acceleration [cm/s2] Figure 1: Generated representative function of non-stationary ground motion acceleration  Comparison between the mean and standard deviation by the 610 representative time histories and the targets are shown in Figure 2. It is shown that the samples’ ensemble values are consistent with the target values.   0 5 10 15 20-2-101020Time [s]Mean [cm/s2] 610 samplesTarget0 5 10 15 20020406080Time [s]Std.D [cm/s2]610 samplesTarget Figure 2: Comparison between mean and standard deviation from samples ensemble and from the target 4. APPLICATION EXAMPLE Recently, the probability density evolution method (PDEM) has made significant progress in stochastic dynamics of structures (Li and Chen, 2009). In this paper, combining the proposed probabilistic model of fully non-stationary 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 ground motion and PDEM, stochastic seismic response analysis and dynamic reliability evaluation on engineering structures can be implemented. In order to illuminate the practicability of proposed probabilistic model of fully non-stationary ground motion, an aqueduct structure as shown in Figure 3 subjected to stochastic ground motion is investigated. The total length of aqueduct structure is 440m, and the length of single span is 40.0m. The cross section of aqueduct body is a single rectangular sink with clear width of the bottom being 8.4m and the height of the aqueduct body being 6.4m.   Figure 3: The finite element model of aqueduct structure  In employing the PDEM, stochastic seismic response analysis and dynamic reliability evaluation for the aqueduct displacement of the mid-span node is carried out to capture the probability information, as shown in Figure 4-7. Pictured in Figure 4 are the mean and the standard deviation. Figure 5 shows the typical PDF at certain instants of time. As shown in Figure 6 and 7, where Figure 6 is the surface constructed by the PDF at different instants of time, while Figure 7 is the contours of the surface, respectively. From the Figure 4-7, the probability density function (PDF) has the typical evolutionary characteristics. 0 5 10 15 20-505x 10-4Time[s]Mean[m]0 5 10 15 200246x 10-3Time[s]Std.D[m] Figure 4: Mean values and standard deviations -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02020406080100120140Displacement[m]PDFPDF at 3 sPDF at 6 sPDF at 9 s Figure 5: Typical probability density function at different time instants  Figure 6: Probability density function evolution surface Time[s]Displacement[m]9 9.5 10 10.5 11 11.5 12-0.02-0.015-0.01-0.0050.0050.010.0150.02 Figure 7: Contours of the PDF surface 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 Based on the equivalent extreme-value event (Li and Chen, 2007), under the control criteria on displacement of the mid-span node, the cumulative distribution function (CDF) can be captured, as shown in the Figure 8. Actually, the CDF is the dynamic reliability of aqueduct structure. 0 0.005 0.01 0.015 0.02 0.025 0.0300.10.20.30.40.50.60.70.80.91Displacement[m]CDF Figure 8: The dynamic reliability of aqueduct structure 5. CONCLUSIONS Rational representation and models of random engineering excitations underlie the stochastic dynamics of structures. A random function and spectral representation scheme are proposed in this paper. The updated scheme is used for the simulation of seismic acceleration processes. Employing the above representation of ground motion, the PDEM can be implemented to capture the instantaneous PDF of responses for general MDOF structure. An aqueduct structure subjected to stochastic seismic excitation is studied to illustrate its applications, showing the effectiveness of the proposed method. 6. ACKNOWLEDGMENTS The authors gratefully acknowledge the National Natural Science Foundation of China (Grant No. 51278282). 7. REFERENCES Cacciola P, Deodatis G. (2011). “A method for generating fully non-stationary and spectrum-compatible ground motion vector processes”. Soil Dynamics and Earthquake Engineering, 31, 351-360. Douglas J, Aochi H. (2008). “A survey of techniques for predicting earthquake ground motions for engineering purposes”. Surveys inGeophysics, 29(3), 187-220. Li J, Chen J.B. (2009). “Stochastic Dynamics of Structures .´ Singapore: John Wiley & Sons. Li J, Chen J.B, Fan W.L. (2007). “The equivalent extreme-value event and evaluation of the structural system reliability”. Structural Safety, 29(2), 112-131. Liu Z.J, Fang X. (2013). “Simulation of stationary ground motion with random functions and spectral representation”. Journal of Vibration and Shock, 32(24), 6-10. (In Chinese) Liu Z.J, Zeng B, Wu L.Q. (2015). “Simulation of non-stationary ground motion by spectral representation and random functions”. Journal of Vibration Engineering, 28(4). (In Chinese) Wang D, Li J. (2011). “Physical random function model of ground motions for engineering purposes”. Science China Technological Sciences, 54(1), 175-182. 

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