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

Structural system response and reliability analysis under imcomplete earthquake records Comerford, Liam; Jensen, Hector; Beer, Michael; Mayorga, Carlos; Kougioumtzoglou, Ioannis; Kusanovic, Danilo Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Structural System Response and Reliability Analysis underIncomplete Earthquake RecordsLiam ComerfordPhD Candidate, Institute for Risk and Uncertainty, Univ. of Liverpool, Liverpool, UKHector JensenProfessor, Dept. of Civil Engineering, Santa Maria Univ., Valparaiso, ChileMichael BeerProfessor, Institute for Risk and Uncertainty, Univ. of Liverpool, Liverpool, UKCarlos MayorgaMSc Candidate, Dept. of Civil Engineering, Santa Maria Univ., Valparaiso, ChileIoannis KougioumtzoglouAssistant Professor, Dept. of Civil Engineering and Engineering Mechanics, ColumbiaUniv., USADanilo KusanovicResearch Assistant, Dept. of Civil Engineering, Santa Maria Univ., Valparaiso, ChileABSTRACT: This work is concerned with the reliability analysis of structural systems under incompleteearthquake records. An artificial neural network approach is developed and implemented to address theproblem associated with missing data in the context of evolutionary power spectra estimation of under-lying non-stationary stochastic processes. The effectiveness of the proposed approach is investigated bythe reliability analysis of a large structural model under simulated earthquake excitation.1. INTRODUCTIONThe analysis of structures under random dynamicexcitations, such as earthquake loads, requires re-alistic stochastic modeling of the excitations aswell as numerically efficient simulation techniques.Evolutionary power spectra (EPS) provide an ap-pealing model for capturing the statistics and thetime-varying frequency content of the underlyingnon-stationary stochastic processes. Further, theycan be used as a basis for joint time-frequency sys-tem response analysis, or efficient stochastic simu-lation utilizing advanced Monte Carlo techniques.Several approaches exist for estimating EPS basedon time records, among which, wavelet-based ap-proaches appear to have received particular atten-tion. However, for stochastic process model basedMonte Carlo simulations to be reliable, EPS es-timation techniques (including wavelet-based ap-proaches) often require a significant amount of dataand/or some prior knowledge of the underlyingphysics of the process. In general, the more dataon which a model is built the more statisticallyaccurate the simulation is likely to be. However,in many engineering applications large amounts ofdata can be difficult to acquire for several reasons,such as cost, frequency and unpredictability of theevents, and sensor failures. Further, available datacan often be highly limited and irregularly sampled.Hence, the EPS estimation needs to cope with frag-mentary and incomplete data records, i.e. miss-ing data. In this regard a methodology based ona neural network has been developed in order to ad-dress the problem of missing data in EPS estima-tion (Comerford et al. (2014)). The neural network112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015is employed to capture the stochastic pattern in theavailable data in an “average sense”. The trainednetwork, having stored process trends within itsconnection weights, is then exploited for generatingnew data to fill sampling gaps compatible with theunderlying stochastic process. A significant advan-tage of the approach relates to the fact that no priorknowledge of statistics of the underlying processis required. Finally, EPS estimates are derived byutilizing harmonic wavelet based approaches (e.g.Spanos and Failla (2005)).2. MATHEMATICAL FORMULATIONIn the following section the ANN based stochasticprocess simulation procedure is introduced and ex-tended to account for missing data in both station-ary and non-stationary process records. A concisereview on non-stationary stochastic process rep-resentation via harmonic wavelets is included forcompleteness.2.1. ANN based stochastic process simulationAn approach for simulating stationary stochasticprocesses given a short input sample with the aidof artificial neural networks (ANN) has been devel-oped by Beer and Spanos (2009). In this setting, ashort recorded sample is used to train a neural net-work to recognize the pattern to capture the prop-erties of the underlying process. Once trained, thenetwork can then be used to generate more sampleswith similar properties. Although this method doesnot immediately account for gaps in the samples,it offers a basis for simple modification to supportmissing data.2.1.1. Network ArchitectureThe base network architecture has a multi-layerfeed-forward structure and is trained via back-propagation learning, this is a common layout forfunction approximation; see Haykin (1999). Sucha network consists of inputs, connections, neuronsand outputs, with each neuron applying an ‘activa-tion function’ to the sum of the inputs. In this casethe sigmoid function Eq. (1) is chosen and thereforethe output of each neuron is given by Eq. (2).ϕ = 11+ e−x(1)y j,n =(1+ e−∑Nj−1m=0 [w j, j−1(n,m)y j−1,m])−1, (2)where y j,n is the output of the n’th neuron in thelayer j, N j−1 is the number of neurons in layer j−1 (plus the bias), w j, j−1 (n,m) is the weight of theconnection from neuron m in layer j−1 to neuron nin layer j and y j−1,m is the output of the m’th neuronin layer j−1.The process of back-propagation learning in-volves incrementally changing the connectionweights w by an amount defined by∆wm,m−1 (n) = ηδmym−1 (n)+α∆wm,m−1 (n−1) ,(3)where ∆wm,m−1 is the change in weight on a con-nection from layer m−1 to m, n is the training cy-cle index, η is the update scale, ym−1 is the outputof the referenced weight dependent neuron and α isa convergence momentum factor. δ is the local gra-dient defined for each layer and is given by Eq. (4)for output layer weights and Eq. (5) for hiddenlayer weights where subscript m denotes the hiddenweight layer and subscript k, the output layer.δk = ek (n)∂vk (n)∂y j (n)(4)δm =∂ym (n)∂vm (n) ∑m+1[δm+1∂vm+1 (n)∂ym (n)](5)The weight changing procedure described byEq. (3) is then repeated multiple times for all avail-able training data.2.1.2. Network Application SchemeGiven a set of sequential inputs corresponding torecorded data points from a process in time, thenetwork is trained to predict the next point in thesequence. The network therefore accepts multipleinputs, passes these inputs through multiple layersof neurons and produces a single output. Duringthe training phase, the next data point in the se-quence is known and therefore the error betweenthis value and the network output can be propagatedback through the network to update the weights.This training procedure is depicted by Figure 1,where each training set contains a small sample of212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 1: ANN input selection for trainingthe whole with the specified number of inputs, plusone to check against the output.The total number of training sets is equal to thelength of the record minus the width of the inputsequence and the output. The order of the train-ing sets is then randomized before each is fed intothe network. Once all have been passed throughthe network, the process is repeated with the setsin a different, random order. This continues untilthe mean error between the network outputs andtrue values is considered to be sufficiently small(several orders of magnitude lower than the processvariance). The output can then be attached to theend of the input sequence and the new input shiftedone step forward in time (discounting the first of thelast input and including the last prediction output)(Figure 2). This process can then be repeated indef-initely to produce a new time series of any length.2.1.3. Network Application with Missing DataAs the network requires inputs to be evenly spacedin time, it is not possible to simply shorten therecorded data by removing unwanted gaps. It isproposed in Comerford et al. (2014) that missingdata points are filled with random values drawnfrom a distribution based on the known data. Thismay be estimated numerically despite the missingdata. With each training cycle the random valuesare re-drawn causing the network to become sensi-tive only to the known data. The procedure is de-picted in Figure 3. As the network inputs have someFigure 2: ANN process prediction procedureFigure 3: Procedure for filling missing data duringtraining on a stationary processrandom elements, the mean network error will notdecrease indefinitely. Instead it will converge onsome mean value with constant variance at whichtime network training is ceased.Two additional modifications to the original net-work application scheme are made when learningnon-stationary processes subject to missing data.Firstly, instead of using a single distribution, esti-mated from the entire process for the missing train-ing data, a windowed distributions are used, cen-tered on each missing data location. Secondly, anadditional input is fed into the network in the non-stationary case. This is simply a linear time in-dex that allows the network to more readily capturetime-dependant features of the process.It was found in Comerford et al. (2014) that inboth the stationary and non-stationary cases, the312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015ANN method was able to reconstruct time-historieseffectively, capturing the spectral content of theprocess in both time and frequency. However,it also showed that despite capturing the spec-tral trends accurately, the total estimated spectrumpower decreased with number of missing data. Thispresents a problem when utilizing the spectrum forpractical problems such as that addressed in this pa-per. To mitigate the effect of this loss of EPS poweron the estimated structural response magnitude, theEPS is pre-scaled before being applied in stochas-tic simulations. The scale is applied such that thevariance of the ANN realizations are equal to thatof the known data.2.2. Non-stationary power spectrum estimationvia harmonic waveletsRegarding the estimation of non-stationary stochas-tic process power spectra after the ANN real-izations are produced, a time/frequency localizedwavelet basis is used to model the process basedon a framework developed in Nason et al. (2000).In this case, the chosen basis is comprised ofgeneralized harmonic wavelets which have a box-shaped frequency spectrum making them idealfor representing non-stationary harmonic processes(Newland (1994)). The basis is given byΨG(m,n),k(ω) =1(n−m)∆ωe(−iωkTon−m ) (6)for m∆ω ≤ ω ≤ n∆ω , where m,n and k are con-sidered to be positive integers, ∆ω = 2piT0 , and T0 isthe total time duration of the signal under consider-ation. Furthermore, the continuous generalized har-monic wavelet transform (GHWT), which projectsa function f (t) on the wavelet basis, is defined asWG(m,n),k =n−mkT0∫∞−∞f (t)ΨG(m,n),k (t)dt, (7)whereΨG(m,n),k (t) is the time-domain representationof the wavelet. The approach advocates that theEPS SX (ω, t) of the random process X(t) is esti-mated bySX (ω, t) = SX(m,n),k =E(∣∣∣WG(m,n),k [X ]∣∣∣2)(n−m)∆ω, (8)e.g. Spanos and Kougioumtzoglou (2012), whereSX(m,n),k represents the EPS of the process X (t), as-sumed to have a constant value in the intervals[m∆ω,n∆ω] and[kT0n−m ,(k+1)T0n−m]. Thus, the EPS canbe estimated as the ensemble average of the squareof the wavelet coefficients.3. APPLICATION PROBLEMThe objective of the application problem is to eval-uate the performance of the proposed methodologyin treating missing data. In particular, the effect ofestimated spectra on the system reliability of a largestructural model under earthquake excitation is in-vestigated.3.1. Model DescriptionThe structural model shown in Fig. 4 is consid-ered for analysis. It consists of a eight floor three-dimensional reinforced concrete building modelunder stochastic ground acceleration. Materialproperties of the reinforced concrete structure havebeen assumed as follows: Young’s modulus E =2.56×1010 N/m2; Poisson ratio ν = 0.2; and massdensity ρ = 2500 kg/m3. The total mass of the dif-ferent floors is given in Table 1. The height of eachfloor is 3.5 m leading to a total height of 28.0 mfor the structure. The floors are modeled with shellelements with a thickness of 0.2 m. Additionally,beam and column elements are used in the finite el-ement model, which has approximately 102.960 de-grees of freedom. A 5% of critical damping for themodal damping ratios is introduced to the model.Floor 1 2 3 4Mass (ton) 204.7 202.7 198.9 195.6Floor 5 6 7 8Mass (ton) 192.2 188.9 186.7 175.5Table 1: Total mass of the different floorsThe building is excited horizontally by a groundacceleration applied at 35o with respect to the axisx. The ground excitation is modeled as indicated inthe following section. For an improved earthquakeperformance the structural system is enforced withtwelve vibration control devices. In particular, vi-bration control devices composed of a series of412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 4: Isometric view of the finite element modelmetallic U-shaped flexural plates (UFP) are consid-ered in this study (Jensen and Sepulveda (2011)).These devices consist of brace and plate elementswhere the UFP’s are located between the plates.The vibration control devices are connected to thestructure every four floors as indicated in Fig. 4.Each UFP exhibits a one-dimensional hysteretictype of non-linearity modeled by the restoring forcelaw rd(t) = α ke δ (t)+(1−α) keUy z(t), where keis the pre-yield stiffness, Uy is the yield displace-ment, α is the factor which defines the extent towhich the restoring force is linear, z(t) is a dimen-sionless hysteretic variable, and δ (t) is the relativedisplacement between the upper and lower surfacesof the device (in the x or y direction). The auxil-iary variable z(t) satisfies the first-order non-lineardifferential equationz˙(t) = δ˙ (t)[β1− z(t)2[β2 +β3sgn(z(t)δ˙ (t))]](9)where β1, β2 and β3 are dimensionless quantitiesthat characterize the properties of the hysteretic be-havior, sgn(·) is the sign function, and all otherterms have been previously defined. The quanti-ties β1, β1 and β3 correspond to scale, loop fat-ness and loop pinching parameters, respectively.The above characterization of the hysteretic be-havior corresponds to the Bouc-Wen type model(Baber and Wen (1981)).−0.02 −0.01 0 0.01 0.02−4−3−2−101234Relative Displacement [m]Restoring Force [Tonf]Figure 5: Typical displacement-restoring force curve ofone of the U-shaped flexural platesThe following values for the dissipation modelparameters are used in this case: ke = 2.5× 106N/m;Uy = 5×10−3m; α = 0.1; β1 = 1.5; β2 = 0.5;and β3 = 0.5. A typical displacement-restoringforce curve of one of the U-shaped flexural platesunder seismic load is shown in Figure 5.3.2. Excitation ModelA non-stationary ground acceleration process de-fined in terms of the Clough-Penzien power spec-trum is used to generate synthetic ground motionsin this application problem. Based on these records,a number of scenarios with respect to the magnitudeof missing data are constructed.3.2.1. Simulated DataThe non-stationary ground acceleration processis characterized as Sa(t,ω) = h2(t,Mm,r)Sa(ω),where h(·) is an envelope function of timeand Sa(·) is the Clough-Penzien power spectrum(Clough and Penzien (1975)). The envelope func-tion suggested in Saragoni and Hart (1974) is con-sidered in the present formulation. Such function isgiven byh(t,Mm,r) = a1(ttn)a2· exp(−a3 ·ttn)(10)where the parameters a1, a2 and a3 are defined as512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015a1 =( eλ)a2, a2 =−λ ln(η)1+λ · (ln(λ )−1), a3 =a2λ(11)The envelope function has a peak equal to unityat t = λ · tn and equal to η at t = tn where tn =2Tn. The parameter Tn, which corresponds to thethe duration of ground motion, can be expressedas a sum of a path dependent and source depen-dent component (Boore (2003)). More specifi-cally, Tn = 0.05√r2 + r2z + 0.5fa where r is the epi-central distance, rz is the pseudo-depth given bylog(rz) = 0.15 ·Mm − 0.05, where Mm is the mo-ment magnitude, and fa is the so-called corner fre-quency defined as log( fa) = 2.181 − 0.496 · Mm(Atkinson and Silva (2000)). The values λ = 0.2,η = 0.05, r = 25 km, and Mm = 7.0 are consideredin the present study. On the other hand, the Clough-Penzien power spectrum is characterized asSa(ω) = S0 ·ω4(ω2f −ω2)2 +4ξ 2f ω2f ω2ω4g +4ξ 2g ω2g ω2(ω2g −ω2)2 +4ξ 2g ω2g ω2(12)where S0 is the amplitude of the white noisebedrock excitation spectrum, ω f , ξ f , ωg, andξg are parameters related to soil conditions.The values S0 = 0.02[m2/s3], ω f = 1.5[rad/s],ξ f = 0.6, ωg = 15.0[rad/s], and ξg = 0.6 areused in the current application. These val-ues correspond to ground motions of regularintensity on firm soil conditions. The syn-thetic ground motions are generated in thetime domain by a superposition of harmonicwaves as a(t) = h(t,Mm,r) ∑Nωk=1√2Sa(ωk)∆ω(Ak cos(ωk · t)+Bk sin(ωk · t)), where ∆ω =ωc/Nω is the band-width that each harmonicrepresents, ωc = 108 rad/s is the cutoff frequencyconsidered for the spectrum Sa(ω), Nω = 64 isthe number of frequencies in the frequency range(0,ωc), ωk = (k − 0.5)∆ω ,k = 1, ...,Nω , and Akand Bk are independent Gaussian random variableswith zero means and unit variances. The syntheticground motions are discretized at time intervalsequal to ∆t = 0.0293 s with a total duration of 30.0s given samples of length equal to 1024. A samplerealization is shown in Fig. 6.0 5 10 15 20 25 30−0.8−0.6−0.4− [s]Ground Acceleration [m/s2 ]Figure 6: Sample realization of a ground motion ob-tained from the Clough-Penzien power spectrum3.2.2. Simulated Missing DataFor illustration purposes the following strategy isused to remove the data from the samples previ-ously generated. In this strategy the data is removedat ten different intervals located at random posi-tions. In particular three cases, corresponding to10%, 20% and 30% of the data removed, are con-sidered in the present study. Figure 7 shows theestimated spectra obtained by the proposed methodfor the cases of no-missing data, 10% of missingdata at 10 interval locations, 20% of missing dataat 10 interval locations, and 30% of missing data at10 interval locations. To evaluate the effectivenessof the proposed method in handling missing data,the estimated spectra corresponding to the missingdata cases are compared with the spectrum obtainedfor the no-missing data case (target spectrum). It isobserved that the method produces non-stationaryprocesses that fit very well with the shape of thetarget spectrum over time as well as over the fre-quency domain. Based on these results it is seenthat the spectra are reconstructed with high accu-racy.3.3. Effect on System ReliabilityIn this section the effectiveness of the proposedmethod on the reliability assessment of the struc-tural system is investigated. For this purpose the612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 7: Estimated spectra for different cases of miss-ing datasynthetic ground excitation is generated in the timedomain from the estimated spectra obtained in theprevious section. The failure event F is formulatedas a first passage problem during the time of anal-ysis. The structural responses to be controlled arethe interstory drift displacements (in the x or y di-rection) over 48 control points. The control pointsare located over the height of the structure at fivedifferent corners and at the center of mass. Some ofthese points, represented as dots, are shown in Fig.(4). The failure event is defined as F > 1, whereF = maxt∈[0,T ]maxi=1,...,48{| δixmax(t) |δ∗ ,| δiymax(t) |δ∗ } (13)where δixmax(t) denotes the maximum interstorydrift in the x direction at the control point i, i =1, ...,48, δiymax(t) denotes the maximum interstorydrift in the y direction at the control point i, i =1, ...,48, and δ ∗ is the corresponding acceptable re-sponse level. Note that there are 96 response func-tions associated with the failure event. It is alsonoted that the corresponding reliability problem isa high dimensional problem since there are 128 ran-dom variables involved in the problem (number ofrandom variables that characterize the excitation).The failure probabilities in terms of the thresh-old levels are estimated by an advanced simula-tion method called Subset simulation (Au and Beck(2001); Zuev K.M. and Katafygiotis (2012)). Thismethod is the most widely applicable simulationtechnique because it is not based on any geomet-rical assumption about the topology of the failuredomain. In fact, validation calculations have shownthat subset simulation can be applied efficiently toa wide range of dynamical systems including gen-eral linear and non-linear systems (Au and Wang(2014)). Figure 8 shows the failure probability interms of the threshold level for the cases of no-missing data, 10% of missing data at 10 intervallocations, 20% of missing data at 10 interval loca-tions, and 30% of missing data at 10 interval lo-cations. It is seen that the reliability curves cor-responding to the missing data cases almost coin-cide with the reliability curve of the no-missing datacase (target curve). Thus, the effectiveness and use-fulness of the proposed methodology in the contextof reliability analysis is apparent.4. CONCLUSIONSA framework for reliability analysis of structuralsystems under incomplete earthquake records hasbeen presented. Numerical results have shownthat the reliability assessment with missing datais in very good agrement with the results derivedusing the estimated no-missing data power spec-712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 1 2 3 4 5 610−410−310−210−1100Threshold Level [m]ProbabilityofFailure  x 10−2No missing data case10 % missing data20 % missing data30 % missing dataFigure 8: Failure probability in terms of the thresholdlevel for different estimated spectratrum. Based on these preliminary results it is con-cluded that the proposed methodology is potentiallyan effective tool for performing reliability analy-sis of real-size structures under incomplete earth-quake records. The effectiveness of the proposedapproach in the context of different missing datacharacteristics (individual data points and/or datasequence) and amount of missing data as well asin the presence of real records with missing data isleft for future research efforts.5. REFERENCESAtkinson, G. and Silva, W. (2000). “Stochastic mod-eling of california ground motions..” Bulletin of theSeismological Society of America, 90(2), 255–274.Au, S. and Beck, J. (2001). “Estimation of small fail-ure probabilities in high dimensions by subset simu-lation..” Probabilistic Engineering Mechanics, 16(4),263–277.Au, S. and Wang, Y. (2014). “Engineering risk assess-ment with subset simulation..” John Wiley & Sons, 92,283–296.Baber, T. and Wen, Y. (1981). “Random vibration hys-teretic, degrading systems..” Journal of EngineeringMechanics, 107(6), 1069–1087.Beer, M. and Spanos, P. D. 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