Seismic structural reliability using different nonlinear dynamic response surface approximations

Structural Safety 31 (2009) 432–442

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Seismic structural reliability using different nonlinear dynamic response surface approximations Oscar Möller a, Ricardo O. Foschi b,*, Marcelo Rubinstein a, Laura Quiroz a a b

Instituto de Mecánica Aplicada y Estructuras IMAE, Universidad Nacional de Rosario, Riobamba y Berutti, 2000 Rosario, Argentina Department of Civil Engineering, University of British Columbia, 6250 Applied Science Lane, Vancouver, British Columbia, Canada V6T 1Z4

a r t i c l e

i n f o

Article history: Received 11 July 2007 Received in revised form 5 December 2008 Accepted 8 December 2008 Available online 5 February 2009 Keywords: Reliability Response surfaces Neural networks Performance-based design Earthquake engineering

a b s t r a c t Performance-based design in earthquake engineering should be approached as an optimization process for optimum design parameters, in order to achieve satisfactory performance over the service life. Each of the specified performance criteria should be met with a prescribed minimum reliability and, given these constraints, a minimum or optimum total cost may be sought. The reliability estimations involve a nonlinear analysis for the dynamic responses of the structure, calculated via a step by step procedure over the complete earthquake record. This task could be computationally demanding, making unfeasible the direct implementation of a standard Montecarlo simulation. Dynamic responses represented by response surfaces make the simulation and the optimization process much more efficient. This paper presents a comparison between three methods for the implementation of response surfaces: a global approximation of a deterministic database, local interpolation of that database, or using artificial neural networks. The comparison uses, as an example, a 5-story reinforced concrete building. The results show good agreement between the methods and the paper discusses their corresponding advantages and limitations. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The general objective in the design of a structure is to achieve its satisfactory performance, over the service life, with a prescribed minimum reliability for each performance criterion. This requires consideration of the many uncertainties involved, and the quantification of the non-performance probabilities [1,2]. Performancebased design [3,4] has been advanced as a means to obtain a structural design that satisfies the specific performance requirements with prescribed minimum reliabilities and, given these constraints, also achieves a minimum or optimum total cost. The standard approach consists of: (1) the definition of several performance levels associated with different damage states; and (2) specification of design earthquakes (in terms of return periods or exceedence probability in a certain time interval). Then, according to the type and use of the structure, the approach specifies performance objectives that are linked to achieving a given performance level for each design earthquake. In a probability-based approach, the seismic action and the structural capacity are functions of random variables. In particular, the ground motion includes uncertain variables like its peak intensity, the form of the record itself and the duration of the strong * Corresponding author. Tel.: +1 604 822 2560; fax: +1 604 822 6901. E-mail addresses: [email protected] (O. Möller), [email protected] (R.O. Foschi). 0167-4730/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.strusafe.2008.12.001

segment. The dynamic response of interest also depends on the structural geometry and the uncertain characteristics of the material properties, including nonlinear effects leading to hysteretic damping, stiffness or strength degradation. Consideration of the intervening uncertainties permits the evaluation of the ‘‘probability of non-performance or failure”, or the probability of not satisfying (for example, on an annual basis) a given performance limit state. Comparison of the achieved exceedence probabilities with pre-determined minimum targets must be included in the optimization of the design parameters, with the objective of, given those constraints, minimizing the total structural cost. Problems in earthquake engineering are characterized by nonlinear dynamic behavior, and it is not possible to establish an explicit relationship between the input variables and the structural responses. Thus, it is not feasible to write explicit forms for the different performance limit states, and the calculation of failure probabilities must rely on simulation. For each replication in the simulation, however, the calculation of maximum responses requires obtaining the complete nonlinear dynamic history, a process that may be computationally demanding. Furthermore, a large number of replications may be needed to obtain acceptable confidence in probabilities of the order of 102 to 104. On the other hand, it would be desirable to use simulation techniques like the standard Montecarlo [5] because they do not require an assumption about the shape of the failure surface, and the error in the probability estimation is not dependent on that shape.

O. Möller et al. / Structural Safety 31 (2009) 432–442

To mitigate the computational demands, it is useful to implement a strategy by means of which a response surface is used to represent the dynamic analysis results. If the response surface approximation is sufficiently accurate, then the reliability estimations and the performance-based design optimization can be carried out using the surface as a ‘‘substitute” for the actual dynamic analysis. The response surface needs to be adjusted to data, and these have to be developed by a-priori dynamic analyzes for a set of combinations of the intervening variables. Although this requirement may still be computationally demanding, the response surface, given in terms of the design parameters and the remaining intervening variables, allows a quick and efficient reliability estimation and optimization process. The objective in this work is to explore three types of response surface implementations: (1) A polynomial global response surface [5–11]. From a database with a sufficient number of deterministic results obtained with the dynamic nonlinear analysis, a global quadratic surface is adjusted covering the entire range of the variables, subject to particular boundary conditions. (2) A method of local interpolation [12], in which the dynamic response is evaluated by using the discrete results from an a-priori database, with a local interpolation of the results corresponding to those variable combinations closest to the combination desired. (3) A method using artificial neural networks [5,12–14], in which an a-priori database for a set of variable combinations is used to train a neural network to represent each of the dynamic responses. The networks are then used to calculate the responses for any other variable combination, within the variable ranges used during the training. For the comparison presented in this work, the nonlinear dynamic analyzes were carried out with a structural model, DINLI, as described in [10,15]. As an example, a 5-story, two-dimensional reinforced concrete frame was studied. It supported permanent loads and was subjected to seismic excitation likely to occur in the city of Mendoza, Argentina.

and (iii) a shear sub-element, to describe the distortion due to shear deformation in the element and in the interface elementnode. Rigid element portions at the ends take into account nonnegligible node dimensions. The system’s masses are assumed concentrated at each story level. The structural model also assumes Rayleigh-proportional damping, with a linear combination of the mass and initial stiffness matrices. Gravitational loads are included in the analysis of each plane frame, taking into account their influence in the response parameters by considering their effect on the internal stresses in each bar element and their corresponding plastic behavior in critical crosssections. The nonlinear system of Eq. (1) is solved by direct integration using Newmark’s method, and Newton–Raphson iterations are carried out within each time step to obtain equilibrium between external and internal actions. The solution provides the required response parameters for each plane frame, as required for the subsequent reliability analysis. To formulate performance requirements in terms of accumulated damage, the index proposed in Ref. [16] was used as an example. This localized index DI is a linear combination of the maximum deformation, normalized, and a term that considers the effect of energy dissipation during the cyclic deformation,

DI ¼

R dE /m þ be /u My /u

Fig. 1 shows a three-dimensional structure composed of vertical two-dimensional frames linked, at each story, by in-plane rigid slabs. If u is the global vector of displacement and rotational degrees of freedom, then the dynamic equilibrium equation for the system is:

€ þ C tþDt u_ þ KT Du ¼ tþDt R  t F M tþDt u

ð2Þ

in which /m is the maximum resulting cross-sectional rotation, /u is R the ultimate rotation during a monotonic loading, dE is the energy dissipated due to hysteresis, My is the yielding moment, and be is a constant that depends on the cross-sectional details, and the acting axial and shear forces. The parameter be has substantial variability, and the following mean values have been recommended: be = 0.05 for well-detailed cross-sections, and be = 0.25 for those poorly detailed [16,22,23]. The index DI is calculated at the ends of each bar element, and all the local indices are then combined to produce a global damage index DIES for the complete structure:

DIES ¼ 2. Brief description of the model for deterministic, nonlinear dynamic response analysis

433

X

wi DIi

ð3Þ

P 2 DI DIi wi ¼ P or DIES ¼ P i DIj DIj

ð4Þ

with

in which it has been assumed that each weight wi is proportional to the local index DIi, and satisfying the constraint that the sum of the weights must be 1.0 [24].

ð1Þ

in which M, C, KT are, respectively, the mass, the damping and the € , tþDt u_ are the global acceleration tangent stiffness matrices; tþDt u and velocity vectors at time (t + Dt); (Du) is the global displacement increment between times t and t + Dt; tþDt R is the vector of external actions at time t + Dt; and tF is the vector of internal forces at t. In DINLI, the plane frames are discretized with bar elements that allow the representation of the different mechanisms that contribute to the hysteretic response in the critical regions of a reinforced concrete member [10,11]. For this, each bar element contains three sub-elements: (i) one, elasto-plastic, to represent the elastic behavior of the element and the nonlinear response at its ends, with the length of the end portions depending on the load history; (ii) a connection sub-element, to represent the concentrated relative rotation between the bar element and the node, due to degradation of the bond and slip of the reinforcement;

3. Probabilistic model 3.1. Random variables The dynamic response of a structure to seismic excitation is influenced by many variables, some related to the ground motion demand and others to the structural behavior. All these variables exhibit uncertainty. 3.1.1. Seismic demand The seismic excitation can be introduced in the analysis using either historical or simulated records. Without loss of generality, the second approach is adopted in this paper only for the purpose of the response surface comparisons. Input accelerograms are simulated using the method proposed in Ref. [17],

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Fig. 1. Structural model and degrees of freedom.

aðtÞ ¼ IðtÞ

NFR X

f4SXX ðn Df Þ ½1 þ dS RN  Df g1=2 sinð2p n Df t þ hn Þ

ð5Þ

n¼1

where SXX is the power spectral density function for the process, NFR is the number of sine functions or frequencies included, between 0 and fmax, such that NFR P fmax T, T being the duration of the simulated record. The ordinates of the function SXX are taken to be random, with dS being the coefficient of variation and RN a Standard Normal variable. Df is a frequency step, and hn are random phase angles with a uniform distribution between 0 and 2p. The process thus generated is stationary, but non-stationarity is introduced with the modulation function I(t) defined as follows:

IðtÞ ¼ ðt=T 1 Þd

for 0  t  T 1

IðtÞ ¼ 1

for T 1  t  T 2

IðtÞ ¼ ec ðtT 2 Þ

ð6Þ

for T 2  t  T

in which t is time, T1 and T2 specific times and d and c are constants. For the comparison in this work we use the power spectral density function SXX introduced in [18],

SXX ðf Þ ¼ S0

1 þ 4 n2g ðf =fg Þ2 2 2

½1  ðf =fg Þ  þ

4 n2g ðf =fg Þ2

ðf =ff Þ4 ½1  ðf =ff Þ2  2 þ 4 n2f ðf =ff Þ2 ð7Þ

in which S0 is a constant or the power spectral density corresponding to white noise; fg, ng are respectively, the characteristic ground frequency and the ground damping ratio; and ff, nf are parameters for a high-pass filter to attenuate low frequency components. The baseline of the accelerogram thus generated is further corrected to minimize the mean square value of the corresponding velocities. The record is finally scaled to have a peak acceleration aG, a random peak value that could occur at the site in case of an earthquake. The characteristic ground frequency fg is also a random variable that, in general, is correlated with the peak acceleration. For simplicity, and for the specific objective of the comparisons in this

work, such correlation is not taken into account. Thus, the randomness in aG and fg are represented by variables X(1), X(2) and X(3),

aG ¼ Xð1Þ ½1:0 þ Xð2Þ

ð8Þ

fg ¼ Xð3Þ

ð9Þ

The distribution of the acceleration aG will vary with the source of earthquakes affecting the site. In Eq. (8), and according to the usual form of attenuation relationships, X(1) is assumed to obey a lognormal distribution representing the average acceleration at a site. On the other hand, X(2), with a Normal distribution, takes into account the variability over the different sources influencing the site seismicity. Other variables in the ground motion characterization were assumed deterministically related to aG and fg, in accordance with Table 1, while ng = 0.20 fg, ff = 0.25, and nf = 0.60. Fig. 2a and b shows, as an example, a function SXX for fg = 2 Hz and dS = 0.40, and a resulting accelerogram for aG = 600 cm/s2. The parameters for the lognormal distribution of X(1), conditional on the occurrence of an earthquake, can be obtained from the site seismicity, and appropriate attenuation relationships linking the effects of magnitude and epicentral distance. However, here the data used correspond to acceleration levels with different exceedence probabilities in 50 years, for the city of Mendoza and a mean arrival rate m = 0.20 for magnitudes M > 5. These data are shown in Table 2 and are taken from the work done by INPRES

Table 1 Parameters for the generation of simulated records. Parameter

aG 6 350 cm/s2

350 < aG 6 700 cm/s2

aG > 700 cm/s2

T (sec) T1 (sec) T2 (sec) c d NFR fmax (Hz) dS

5.12 0.50 4.00 2.0 2.0 100 12 0.40

10.24 1.50 8.00 1.0 2.0 200 15 0.40

20.48 2.00 16.00 0.7 2.0 300 15 0.40

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O. Möller et al. / Structural Safety 31 (2009) 432–442

b

600

Accel. ( cm/sec 2 )

40000 PSDF ( cm2/sec 3 )

a

30000 20000 10000 0

400 200 0 -200 -400 -600

0

2.5

5

7.5

10

0

2

Frequency (Hz)

4

6

8

10

Time (sec)

Fig. 2. (a) Power spectral density function, fg = 2 Hz, ng = 0.40, ff = 0.25, nf = 0.60, dS = 0.40. (b) Generated accelerogram, aG = 600 cm/s2, T1 = 1.5 s, T2 = 8 s, c = 1, d = 2.

Table 2 Seismicity data for the city of Mendoza, Argentina. 50 year-exceedence probability

Annual exceedence probability

Return period (years)

Peak ground acceleration (g)

Exceedence probability in case of earthquake

Mean Xð1Þ (g)

Coeff. of variation of Xð1Þ

0.50 0.10 0.05

0.01377 0.00210 0.00103

73 475 975

0.27 0.60 0.80

0.06993 0.01054 0.00513

0.0959

1.383

[19] using all the contributing sources for that city. These data were then utilized to obtain first the exceedence probabilities in case of an earthquake, according to Eq. (40), and then using the results to calibrate a lognormal distribution corresponding to X(1) in case of an earthquake. The calibrated mean and coefficient of variation for X(1) are also shown in Table 2. 3.1.2. Characteristics of the structural system The mass assigned to each story is random with a mean M i . Thus, the mass Mi for each story is given as

M i ¼ M i ½1 þ Xð4Þ

ð10Þ

with X(4) a random variable. The masses for all stories are assumed perfectly correlated. Similarly, the position for the mass center of each story, Z MC i , is assumed to be random,

Z MC i

¼ Z MC i þ Xð5Þ

ð11Þ

 MC is the corresponding mean value. in which Z i The random variables associated with the structural capacity are the strength R and the ultimate deformation /u for the end k of each bar element. These quantities are assumed to be perfectly correlated among all bar element ends, and expressed as

 k ½1 þ Xð6Þ / ¼ /  u ½1 þ Xð7Þ Rk ¼ R uk k

ð12Þ

RLIM ¼ RLIM ½1:0 þ dRL Xð8Þ

with a mean value RLIM and coefficient of variation dRL, X(8) being a Standard Normal random variable. Following the SEAOC guidelines [3], the following performance levels and limit states were considered in this work: (a) Performance level: Operational

— Elastic roof displacement G11 ðXÞ ¼ v y ½1 þ 0:1 Xð8Þ  v max ðXÞ G12 ðXÞ ¼ 0:005 ½1 þ 0:1 Xð8Þ  DISTMðXÞ — Damage index for frame G21 ðXÞ ¼ 0:50 ½1 þ 0:1 Xð8Þ  DIðXÞ — Max: local damage index G22 ðXÞ ¼ 0:60 ½1 þ 0:1 Xð8Þ  DILOMðXÞ

ð18Þ

G23 ðXÞ ¼ 0:015 ½1 þ 0:1 Xð8Þ  DISTMðXÞ

ð19Þ

— System; global damage index G24 ðXÞ ¼ 0:40 ½1 þ 0:1 Xð8Þ  DIESðXÞ

ð20Þ

(c) Performance level: Collapse

— System; global damage index

G21 ðXÞ ¼ 0:80 ½1 þ 0:1 Xð8Þ  DIðXÞ

ð21Þ

— Max: local damage index G22 ðXÞ ¼ 1:00 ½1 þ 0:1 Xð8Þ  DILOMðXÞ

ð22Þ

— Inter-story drift G23 ðXÞ ¼ 0:025 ½1 þ 0:1 Xð8Þ  DISTMðXÞ G24 ðXÞ ¼ 0:75 ½1 þ 0:1 Xð8Þ  DIESðXÞ

in which RLIM is the limiting value for the response parameter R(X), which is a function of the set of random variables X(i), i = 1, 7. The limiting value is also considered to be a random variable,

ð17Þ

— Inter-story drift

3.2. Performance or limit state functions

ð13Þ

ð16Þ

(b) Performance level: Life safety

— Damage index for frame

GðXÞ ¼ RLIM  RðXÞ

ð15Þ

— Inter-story drift

 k and /  u are the corresponding mean values. in which R k The random variables X(4), X(5), X(6) and X(7) are assumed to be Normal, with zero mean and specified standard deviations. The standard deviations of these variables are taken sufficiently small to justify using a Normal distribution, with very small probability of negative values.

The performance functions G(X), or limit states associated with each performance level, can be expressed in general as

ð14Þ

ð23Þ ð24Þ

In Eq. (15) vy is the horizontal yielding displacement at the roof or top story of the frame, as determined from an a-priori push-over analysis, while mmax is the corresponding elastic displacement. In Eqs. (17)–(20), DI is the local damage index for each frame component, DILOM is the maximum local damage index, DISTM is the maximum inter-story drift, and DIES is the global damage index.

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O. Möller et al. / Structural Safety 31 (2009) 432–442

4. Approximating the structural responses The main objective of this work is to compare reliability estimates from different response surface approximations for the dynamic analysis results. As previously mentioned, reliability calculations using Montecarlo simulation is very efficient if the structural responses are obtained from ‘‘substitute approximations” to the dynamic analysis. Three approximation methods were considered as described in the following sections. 4.1. Global response surfaces In this approach the actual response R(x), in general a function of n variables x(i), is approximately represented by an explicit function F(x) as follows,

RðxÞ ffi FðxÞ ¼

n X

wi hi ðxÞ ¼ a þ

i¼1

n X

bi xðiÞ þ

i¼1

n X

ci xðiÞ2

ð25Þ

i¼1

in which hi(x) are polynomial functions of the variables x(i) and wi are constants optimally adjusted when at least 2n + 1 values of R(x) are known. Eq. (25) shows a particular case in which the hi(x) result in a quadratic surface with no interaction terms. The functions hi(x) are defined independently of the combinations x at which R(x) is known, and the approximation depends only on the parameters wi. Thus, the approximation is sensitive to the location of the combinations x. Furthermore, hi(x) are global functions to represent the response over the entire range of R(x), and the approximation could be affected by local variations or errors in the evaluation of R(x). In this work we used a quadratic function as shown in Eq. (25). This function was further constrained to satisfy a specific requirement, that the responses be zero when the peak ground acceleration is zero. Thus, in terms of the variables X(i), the approximation used was

 FðXÞ ¼ aG ða þ

7 X

bi XðiÞ þ

i¼3

þ

7 X

7 X

ci XðiÞ2 Þ þ a2G ða þ

i¼3 2

ci XðiÞ Þ

7 X



bi XðiÞ

i¼3

ð26Þ

i¼3

For different values of aG in a sequence within the range corresponding to the site seismicity, the response R(Xj) can be obtained for different combinations Xj of the remaining variables X(i). For each combination Xj, the dynamic analysis is executed for NS different earthquake records, obtained for NS different sequences of random phases according to Eq. (5). The NS results, RðXj Þk (k = 1, NS), are then used to calculate the mean and the standard deviation of the response, corresponding to the combination Xj, over the NS records: NS 1 X RðXj Þk NS k¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u NS X u 1 ¼t ½RðXj Þk  RðXj Þ2 NS  1 k¼1

RðXj Þ ¼

rRj

dMV

FðXÞ ¼ FðXÞ ½1 þ dMV Xð10Þ þ Xð9Þ

rðXÞ ½1 þ dSD Xð11Þ

ð29Þ

in which X(9), X(10), and X(11) are assumed to be Standard Normal variables representing, respectively, variability over the records, regression error for the mean response and regression error for the standard deviation of the response over the records. 4.2. Local interpolation This method is applied to a previously obtained deterministic response database. A response surface is adjusted locally rather than globally, as in the previous method, and it is then used to interpolate a response value for a given variable combination.  j Þ in Eq. (27), a local surface for the Starting from the results RðX mean values is defined as

FðXÞ ¼ R0 þ aT ðX  X0 Þ þ ðX  X0 ÞT bðX  X0 Þ

ð30Þ

in which R0 is the response available in the database for the combination X0, the closest to the vector X for which the response needs to be interpolated. a is a vector of coefficients associated with linear terms, and b is a matrix of coefficients associated with the quadratic terms in the approximation. This matrix, for simplicity, is assumed to be diagonal. If N is the dimension of the vector X, a total of 2N coefficients must be calculated, requiring at least 2N data from the database in the neighborhood of X0. The coefficients are obtained by least squares, and the accuracy of the approximation is improved the closer X is to the anchor point X0. The procedural steps can be described as follows: (i) The database responses are ordered according to their distance to X, choosing the closest 2N + 1 Xi along with the corresponding responses R(X)i, and identifying the closest as (X0, R0); (ii) Obtain by least squares the coefficients in a and b;  (iii) Use Eq. (30) to calculate the desired mean response FðXÞ. The same procedure is followed with the results rRj , Eq. (27), to construct the local response surface for the standard deviations rðXÞ . The precision of this approach depends on the number of data in the database and their distribution within the range corresponding to the intervening variables. Extrapolation needs to be avoided, and to this end it is necessary to extend the database over a wide range of variable values, particularly for aG, considered the most important variable. It is also recommended to include in the database R(X)i = 0 for aG = 0. Finally, the estimation of the response can be written as

FðXÞ ¼ FðXÞ þ Xð9Þ

rðXÞ

ð31Þ

in which X(9) can be assumed to be a Standard Normal variable.

ð27Þ

The results for all aG are then used to adjust a response surface for  the mean, FðXÞ, using Eq. (26) and least squares. The difference or  regression error between the actual RðXÞ and the approximate  FðXÞ is quantified with the standard deviation of the relative error [11]:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u NE u 1 X RðXj Þ  FðXj Þ t ¼ NE  1 j¼1 FðXj Þ

in which NE is the number of response evaluations with the dynamic analysis. Similarly, the values of rRj are used to adjust a response surface for the standard deviation rðXÞ over the earthquake records, with relative error standard deviation dSD as in Eq. (28). Finally, the explicit approximating function can be written as

ð28Þ

4.3. Artificial neural networks Neural networks are one of different strategies within Artificial Intelligence [5] and are used to represent general input–output relationships. A neural network is a computational tool that includes a number of interconnected elements or ‘‘neurons”. Information is received by each neuron from those in previous layers of the network, is processed within the neuron and passed forward to neurons in subsequent layers until the output layer is reached. The connection between each pair of neurons is associated with a weight, and these weights constitute the set of parameters that

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O. Möller et al. / Structural Safety 31 (2009) 432–442

must be optimized, during the network training, to best represent the known input–output data. The general expression for this algorithm is

RðXÞ ffi FðXÞ ¼ h

J X

wkj h

N X

!! wji XðiÞ

ð32Þ

i¼0

k¼0

in which h(x) is the function, internal to each neuron, which processes incoming into outgoing information. A sigmoid function, Eq. (35), is usually applied for this purpose. A schematic representation of a network is shown in Fig. 3. In this work, a neural network is used to represent the relationship between the input variables and the mean structural response over the set of earthquake motions. Similarly, another neural network is trained to represent the corresponding standard deviation. The neural network approach has the following characteristics: (i) it is adaptable, as the weights obtained during the training permit a good representation of local variations in the data; (ii) it is flexible, as a new sets of data can be easily accommodated by retraining (‘‘learning”) and (iii) the execution of the neural network algorithm is fast and efficient, facilitating the use in simulations for reliability evaluation and performance-based design optimization. Fig. 3 shows a network with three layers, one for the input with I + 1 neurons, one intermediate or hidden with J + 1 neurons, and finally an output layer with K neurons. Let X pi be the pth-datum for the ith-neuron in the input layer, Ipj the pth-entry datum for the jth-neuron in the hidden layer, and Hpj the pth-output datum for the jth-neuron in the hidden layer. Further, let Ipk be the pth-entry datum for the kth-neuron in the output layer, and Y pk the pthoutput from the kth-neuron in the output layer, the algorithm in Eq. (32) implies that

Ipj ¼

I X

wji X pi þ wj0

Hpj ¼ hðIpj Þ

¼

J X

1XX p ðY k  T pk Þ2 2 p k

wkj Hpj

þ wk0

Y pk

¼

hðIpk Þ

rer

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 NE  u 1 X Yk  Tk ¼t NE  1 k¼1 Tk

1:0 ð1 þ expðxÞÞ

ð35Þ

ð37Þ

Finally, using the trained networks, the response is approximated by

FðXÞ ¼ FðXÞ ½1 þ re MV Xð10Þ þ Xð9Þ

rðXÞ ½1 þ re SD Xð11Þ

ð38Þ

in which X(9), X(10) and X(11) are assumed to be Standard Normal variables as previously described. 5. Calculation of the non-performance probability The probability of non-performance, for each of the limit states considered in a given performance level, is equivalent to evaluating the probability of the corresponding event G(X) < 0. This probability results from the multiple integral over the failure domain G(X) < 0,

Z

Z ...

fX ðxÞdx

ð39Þ

GðXÞ0

ð34Þ

in which wji are the weights connecting the ith-neuron in the input layer with the jth-neuron in the hidden layer, and wkj are the weights connecting the jth hidden neuron with the kth in the output layer. The transfer or processing function h(.) is usually given by the sigmoid,

ð36Þ

 The data Rðxj Þ are used to train a network for the mean values, FðXÞ, and the data for rRj are similarly used to train a corresponding network rðXÞ for the standard deviation. In each case, the dispersion around the network predictions is quantified by the standard deviation of the relative error:

Pf ¼ P½GðXÞ  0 ¼

j

hðxÞ ¼

E¼

ð33Þ

i

Ipk

The weights wji, wkj are optimized in a least squares sense [13] considering the result T pk in a database obtained with the dynamic analysis and the approximation Y pk obtained with the network for the pth data. The minimization algorithm (‘‘back propagation”) searches for the optimum weights to minimize the objective E,

in which fX(x) is the joint probability density function of the intervening random variables. In this work, the integral is approximately estimated via standard Montecarlo simulation, using 106 replications, a task which becomes feasible through the use of any of the three response surface methods described previously. The exceedence probabilities are thus obtained in case of a seismic event for each of the limit states within a performance level, as defined in Eq. (15) through (24). For each performance level, the corresponding total exceedence probability PfE is calculated, regarding all limit states as components of a series system. Finally, the annual probability of exceedence for a given performance level is obtained considering the occurrence of earthquakes as a Poisson process with a mean arrival rate m:

Pf ¼ 1  expðm PfE Þ

ð40Þ

a probability that, for convenience, can also be expressed in terms of an associated reliability index b,

b ¼ U1 ðPf Þ

ð41Þ

6. Application to a 5-story building 6.1. Structural data

Fig. 3. Neural network architecture.

Fig. 4 shows a five-story structure with four vertical plane frames. W is the weight at each story, m is the corresponding mass and ICM is the rotational moment of inertia with respect to the center of mass. These frames were designed for the high seismicity zone corresponding to the city of Mendoza, following the guidelines for

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O. Möller et al. / Structural Safety 31 (2009) 432–442

Fig. 4. Five-story building used in the application.

aseismic construction in Argentina [25]. The frames are equal to each other, and include vertical wide elements with dimensions 20  200 cm, linked with shallower 20  60 cm horizontal beams for stories 1 and 2, changing to 20  50 cm for stories 3 through 5. Global response parameters obtained from a push-over were: yielding displacement maximum capacity V0y = 450 KN, vy = 6.87 cm, and initial stiffness K = 65.5 KN/cm. The seismic excitation was assumed acting along the global X-direction, with characteristics corresponding to the microzonation of Great Mendoza, Argentina [19]. The intervening random variables and their probabilistic description are shown in Table 3. 6.2. Nonlinear dynamic analysis A deterministic response database was obtained using the nonlinear dynamic analysis DINLI [15] for a set of variable combinations. These were chosen from the following values, accommodating the statistical ranges shown in Table 3: 2

Xð1Þð1 þ Xð2ÞÞ ¼ aG ¼ 0; 150; 250; 400; 600; 800; 1200 cm=s : Xð3Þ

¼ fg ¼ 2:0; 2:5; 3:0 Hz:

Xð4Þ

¼ dm RN ¼ 0:45; 0; 0:45

Xð5Þ

¼ rZCM RN ¼ 150; 0; 150 cm

Xð6Þ

¼ dR RN ¼ 0:45; 0; 0:45

Xð7Þ

¼ d/u RN ¼ 0:60; 0; 0:60

in which dm, dR, d/u are, respectively, the coefficients of variation of the masses, strength and ultimate curvature, rZCM is the standard deviation for the location of the center of mass, and RN represents the sequence of standard deviations used to generate the

values for the corresponding variable. Values for variables X(3) to X(7), as shown, were taken at their means and at +3 and 3 times their standard deviations. Taking all possible combinations into account, a total of 1458 are developed for the structural analysis, and are augmented to 1701 when the zero responses corresponding to aG = 0 are included. This provides a sufficient number of responses, either when adjusting a global response surface or when using local interpolation or training neural networks, as the combinations adequately cover the variable ranges. For each combination, five sequences of random phases were used to generate NS = 5 earthquake records. The data far exceeds the number of parameters required for adjusting a global response surface. In this case, the parameters were obtained using all the results and a least squares minimization. The structural responses of interest for implementation in the performance functions, Eqs. (15) through (24), were: (1) The maximum horizontal displacement at the top of the building, vmax(X); (2) The local damage index DI(X); (3) The maximum local damage index DILOM(X); and the maximum inter-story drift DISTM(X). (4) For the entire building, the global damage index DIES(X).

6.3. Response approximation The data from the different combinations were used to determine the coefficients of the global response surfaces and to train the neural networks.

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O. Möller et al. / Structural Safety 31 (2009) 432–442 Table 3 Random variables and statistics. Variable

 Mean X

Type

X(1) X(2) X(3) X(4) X(5) X(6) X(7) X(8) X(9) X(10) X(11)

Lognormal Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal

94 cm/s 0 2.50 Hz 0 0 0 0 0 0 0 0

Std. dev. rX 2

130 cm/s 0.25 0.375 Hz 0.15 50 cm 0.15 0.20 1 1 1 1

As an example, Figs. 5 and 6 show the agreement between true responses and those calculated with the global response surfaces, as well as the corresponding standard deviation of the relative error in the representation. Similarly, Figs. 7 and 8 show the agreement for the neural networks approach. These last two figures show a smaller dispersion about the line of perfect agreement, confirming for this example the neural networks’ flexibility and increased adaptability.

Definition

2

aG = X(1)  (1.0 + X(2)) fg = X(3)  i ð1:0 þ Xð4ÞÞ Mi ¼ M CMi þ Xð5Þ Z CMi ¼ Z Rk ¼ Rk ð1:0 þ Xð6ÞÞ /uk ¼ /uk ð1:0 þ Xð7ÞÞ RLIM ¼ RLIMð1:0 þ dRL Xð8ÞÞ FðXÞ ¼ FðXÞ ð1:0 þ dMV Xð10ÞÞ þ Xð9Þ rðXÞ ð1:0 þ dSD Xð11ÞÞ

utilizing the software RELAN [20]. The probabilities PfE, conditional on the seismic event having occurred, can then be expressed in terms of associated reliability indices, and these are shown in Table 4. Finally, the annual failure probabilities can be evaluated, and these are shown in Table 5 for earthquakes with a mean arrival rate m = 0.20. 6.5. Analysis of the results

6.4. Reliability indices As described, Montecarlo simulation was used with each of the three approximation methods to obtain the failure probabilities

With the standard Montecarlo simulation, with N = 106 replications, the coefficients of variation of the obtained probabilities were, in all cases, 62%. The computational time was very small

0.01 Response surface

Response surface

0.04 0.03 0.02 0.01

δ MV = 0.16791

δ SD = 0.54928

0.008 0.006 0.004 0.002

0

0 0

0.01

0.02

0.03

0

0.04

0.002 0.004 0.006 0.008 0.01 Standard deviation of DISTM

Mean value of DISTM

Fig. 5. Maximum inter-story drift, using global response surfaces.

0.8

1.6

Response surface

Response surface

2

1.2 0.8 0.4

δSD = 0.54210

0.6

0.4 0.2

δMV = 0.35710

0

0 0

0.4

0.8

1.2

1.6

Mean value of DILOM

2

0

0.2

0.4

0.6

Standard deviation of DILOM

Fig. 6. Maximum local damage index, using global response surfaces.

0.8

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O. Möller et al. / Structural Safety 31 (2009) 432–442

0.01 0.008

0.03

Neural network

Neural network

0.04

0.02 0.01

0.006 0.004 0.002

σεr MV = 0.11854 0

σεr SD = 0.29007

0 0

0.01

0.02

0.03

0

0.04

0.002 0.004 0.006 0.008 0.01 Standard deviation of DISTM

Mean value of DISTM

Fig. 7. Maximum inter-story drift, using neural networks.

for both the global response surface and the neural networks approaches. Local interpolation was relatively more computational demanding, given the added task, for each replication, of ranking the database in terms of closeness to the desired input. The reliability indices bE in Table 4 show that, for each performance level, the limit state with the lowest index corresponds to the inter-story drift. However, the indices for all the limit states within a performance level are relatively close to each other (within 15%), showing the importance of considering all limit states simultaneously, as in a series system, to obtain the probability of failure for the corresponding level of performance.

The reliability indices b corresponding to the annual failure probabilities, shown in Table 5, are quite consistent between the different methods of calculation, with maximum differences of the order of 9% for the operational level, and 5% for the life safety and collapse levels. The smaller reliability indices were always obtained with local interpolation, perhaps because this method is able to capture local response peaks that are smoothed over by either a global response surface or a neural network representation. However, as already discussed, local interpolation requires a denser database and demands more computational effort. Even though the neural

0.8

2

σεr SD = 0.50902 Neural network

Neural network

1.6 1.2 0.8 0.4

0.6

0.4 0.2

σεr MV = 0.11672

0

0 0

0.4

0.8

1.2

1.6

2

0

Mean value of DILOM

0.2

0.4

0.6

0.8

Standard deviation of DILOM

Fig. 8. Maximum local damage index, using neural networks.

Table 4 Reliability indices bE conditional on a seismic event having occurred. Performance level

Limit state

Response surface

Local interpolation

Neural networks

PfE

bE

PfE

bE

PfE

bE

Operational

G11 G12

0.89173  101 0.12571

1.346 1.147

0.99800  101 0.16440

1.283 0.977

0.78676  101 0.10947

1.414 1.229

Life safety

G21 G22 G23 G24

0.85550  102 0.15174  101 0.17788  101 0.11700  101

2.384 2.166 2.102 2.267

0.80479  102 0.15751  101 0.21883  101 0.12301  101

2.407 2.151 2.016 2.248

0.64047  102 0.13055  101 0.15100  101 0.96565  102

2.489 2.225 2.167 2.339

Collapse

G31 G32 G33 G34

0.37430  102 0.56380  102 0.55100  102 0.40750  102

2.674 2.534 2.542 2.646

0.33013  102 0.61092  102 0.66680  102 0.36005  102

2.716 2.506 2.475 2.687

0.24951  102 0.42434  102 0.45253  102 0.26076  102

2.808 2.632 2.610 2.793

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O. Möller et al. / Structural Safety 31 (2009) 432–442 Table 5 Annual probability of failure, all limit states within a performance level. Performance level

Method

PfE

bE

Pf

b 1

Operational

Response surface Local interpolation Neural networks

0.12157 0.16440 0.10985

1.147 0.977 1.227

0.24021  10 0.32345  101 0.21730  101

1.997 1.847 2.019

Life safety

Response surface Local interpolation Neural networks

0.20601  101 0.25533  101 0.17661  101

2.041 1.951 2.105

0.41117  102 0.50936  102 0.35260  102

2.643 2.569 2.694

Collapse

Response surface Local interpolation Neural networks

0.68380  102 0.84584  102 0.54316  102

2.466 2.389 2.547

0.13667  102 0.16902  102 0.10857  102

2.996 2.931 3.066

networks were able to offer a better representation of the database in comparison with the global response surface, with less dispersion, when this dispersion is considered in the calculation of the response F(X), as in Eq. (29), or (38), the resulting reliability indices from those two approaches were very similar. Although it is not the aim of this work to discuss target values for failure probabilities for each performance level, it is useful to compare the results obtained with those suggested for annual targets in Ref. [21]. – Operational level: Pftarget ¼ 2  102 ! b ¼ 2:054. – Life safety: Pftarget ¼ 2  103 ! b ¼ 2:878. – Collapse: Pftarget ¼ 2  104 ! b ¼ 3:540: It can be concluded, then, that the structure analyzed here approximately meets the target for the operational level but it is somewhat lacking for life safety or collapse. 7. Conclusions

 The use of response databases, obtained a-priori for combinations of the random variables, and their representation via response surfaces, greatly facilitates the calculation of reliability of structures which exhibit nonlinear behavior under dynamic, seismic excitation. Three methods for the representation of the discrete databases have been compared: a global response surface, using the database with local interpolation, and representing the database with neural networks.  The major computational task is the development of the discrete database, requiring the use of the nonlinear dynamic analysis for many combinations of the intervening variables. Having developed the database, however, subsequent computation of failure probabilities for different performance levels or limit states can proceed in a very efficient manner, even using standard Montecarlo simulation with a large number of replications. This has the advantage of being able to implement such simulation, without the inherent limitations of other methods (e.g., FORM) which introduce errors due to the nonlinearity of the limit state functions.  The analysis of a five-story building has shown the applicability of these methodologies, with good agreement among the results obtained with each of the representation methods.  Local interpolation, which may be able to better capture local features of the responses, may result in lower (more conservative) reliability estimates.  Either neural networks or global responses surfaces, with appropriate constraint conditions, are capable of approximating well the true responses, and are very efficient in regard to computational time for simulations. Neural networks appear as very useful tools, more flexible and adaptable than global response

surfaces, and are the subject of continuing research by the authors into their applications to seismic reliability and structural optimization.

Acknowledgements This work was supported by the research project ‘‘Confiabilidad de sistemas estructurales bajo solicitaciones dinámicas”, 19/I 202 of the Universidad Nacional de Rosario, Argentina, and by a Grant to the second author from the Natural Sciences and Engineering Research Council of Canada: ‘‘Neural networks for reliability and performance-based design in earthquake engineering”, RGPIN 5882-04. Both supports are gratefully acknowledged. References [1] Melchers RE. Structural reliability – analysis and prediction. John Wiley and Sons; 1987. [2] Thoft Christensen P, Baker MJ. Structural reliability – theory and applications. Springer Verlag; 1982. [3] SEAOC Vision 2000 Committee. Performance based seismic engineering of buildings. Sacramento, California, USA: Structural Engineers Association of California; 1995. [4] FEMA 1997. NEHRP Guidelines for the seismic rehabilitation of buildings. Report 273, Buildings Seismic Safety Council. [5] Hurtado J. Structural reliability – statistical learning perspectives. Lecture Notes in Applied and Computational Mechanics 2004;17 (Springer Verlag). [6] Wong FS. Slope reliability and response surface method. J Geotech Eng ASCE 1985;111:32–53. [7] Faravelli L. Response surface approach for reliability analysis. J Eng Mech ASCE 1989;115:2763–81. [8] Bucher CG, Bourgund U. A fast and efficient response surface approach for structural reliability problems. Struct Saf 1990;7(1):57–66. [9] Kim S-H, Na S-W. Response surface method using vector projected sampling points. Structural Safety 1997;19:3–19. [10] Möller O. Metodología para evaluación de la probabilidad de falla de estructuras sismorresistentes y calibración de códigos, Tesis de Doctorado en Ingeniería (PhD Dissertation). 2000 Rosario, Argentina: Universidad Nacional de Rosario; 2001. [11] Möller O, Foschi R. Reliability evaluation in seismic design. Earthquake Spectra 2003;19(3):579–603. [12] Foschi R, Li H, Zhang J. Reliability and performance-based design: a computational approach and applications. Structural safety 2002;24:205–18. [13] Zhang J. Performance-based seismic design using designed experiments and neural networks, PhD Thesis. Canada: Department of Civil Engineering, University of British Columbia; 2003. [14] Zhang J, Foschi RO. Performance-based design and seismic reliability analysis using designed experiment and neural networks. Probabilistic Engineering Mechanics 2004;19:259–67. [15] Möller O, Rubinstein M, Cóceres H. Combinación de planos sismorresistentes para análisis dinámico no lineal de estructuras espaciales. Mecánica Computacional AMCA 2003;XXII:997–1011. [16] Park YJ, Ang AH-S. Mechanistic seismic damage model for reinforced concrete. J. Struct. Eng. ASCE 1985;111(ST4):722–39. [17] Shinozuka M, Sato Y. Simulation of nonstationary random processes. ASCE J. Eng. Mech. 1967;93(1):11–40. [18] Clough RW, Penzien J. Dynamics of structures. Mc Graw Hill; 1975. [19] INPRES. Microzonificación Sísmica del Gran Mendoza, Publicación Técnica No. 19 (Microzonation for the city of Mendoza, Technical Publication No. 19). San Juan, Argentina: Instituto Nacional de Prevención Sísmica (National Institute for Seismic Prevention); 1995 ().

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O. Möller et al. / Structural Safety 31 (2009) 432–442

[20] Foschi RO, Li H, Zhang J, Folz B, Yao F. Software RELAN: Reliability analysis. Vancouver, Canada: Department of Civil Engineering, University of British Columbia; 2005. [21] Paulay T, Priestley MJN. Seismic design of reinforced concrete and masonry buildings. John Wiley and Sons, Inc.; 1992. [22] Cosenza E, Manfredi G, Ramasco R. The use of damage functionals in earthquake engineering: a comparison between different methods. Earthquake Engineering and Structural Mechanics 1993;22:855–68.

[23] Hindi R, Sexsmith R. A proposed damage model for RC bridge columns under cyclic loading. Earthquake Spectra 2001;17(2):261–90. [24] Bracci JM, Reinhorn AM, Mander JB, Kunnath SK. Deterministic model for seismic damage evaluation of RC structures. Buffalo, NY: National Center for Earthquake Engineering Research, SUNY; 1989 (Tech. Report NCEER 89-0033). [25] INPRES-CIRSOC 103. Code for aseismic construction for Argentina, Part II. Reinforced Concrete; 1991.