The error estimation in the prediction of ultimate drift of RC columns for performance-based earthquake engineering
Ocena napake pri napovedovanju mejnega zamika AB-stebrov
v potresnem inženirstvu
Iztok Peruš1, *, Matjaž Dolšek1
University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova cesta 2,
SI-1000 Ljubljana, Slovenia
Corresponding author. E-mail: iperus@ikpir.fgg.uni-lj.si Received: January 06, 2009	Accepted: July 07, 2009
Abstract: The error in the prediction of the ultimate drift of the reinforced concrete columns is determined by using the CAE method. The results based on the Kolmogorov-Smirnov test indicated that the error of the predicted ultimate drift in columns is mainly normally or log-normally distributed. For the other cases the typical characteristics of the columns were indicated. Their influence will be reduced with extension of the database. Based on the results of the study it was assumed that in the most cases the ultimate drift determined by the CAE method has the coefficient of variation about 0.4. The extensive parametric study performed for the reinforced concrete frame has shown that the influence of different uncertainties, including the uncertainty in prediction of the ultimate drift in columns, is not significant in the range of seismic intensities, which do not cause significant damage in the structure. On the other hand, the uncertainties becomes very important in the range close to the collapse of the structure, since the dispersion in the seismic response parameters is significantly increased and in some cases also the capacity of the structure is decreased if the uncertainties are considered in the analysis.
Izvleček: V okviru opisane študije smo poskušali ugotoviti tipično vrednost napake in njeno porazdelitev pri napovedi kapacitete armiranobetonskih stebrov s CAE-metodo. Pri uporabi testa Kolmogo-
rov-Smirnov smo ugotovili, da se napake napovedi pretežno poraz-deljujejo normalno in/ali log-normalno. Za druge primere, ko to ni tako, smo ugotovili, da imajo stebri tipične lastnosti, katerih vpliv se bo zmanjšal s povečevanjem baze podatkov. Glede na dobljene rezultate smo predpostavili, da lahko v veliki večini primerov s precejšnjo zanesljivostjo pri napovedi mejnega zamika s CAE-metodo pri predpostavki log-normalne porazdelitve upoštevamo koeficient variacije, ki je 0,4. Obširna neelastična parametrična študija štiri-etažne stavbe je pokazala, da lahko vpliv nezanesljivosti na odziv v nekaterih primerih zanemarimo. Po drugi strani pa je določitev kapacitete armiranobetonskih elementov s čim večjo zanesljivostjo zelo pomembna, saj lahko pri matematičnih modelih z visokimi vrednostmi nezanesljivosti pri določitvi začetnih togosti in mejnih rotacij stebrov in prečk opazimo povečanje raztrosa pri potresnem odzivu ter tudi zmanjšanje kapacitete konstrukcije.
Key words: CAE method, error distribution, drift, RC columns, performance-based earthquake engineering (PBEE)
Ključne besede: CAE-metoda, porazdelitev napake, zamik, AB-stebri, potresno inženirstvo
Introduction
The performance of structures in recent catastrophic earthquakes points to the need for improved seismic design approaches capable of achieving explicit determination of seismic risk. The methodology, which successfully treats this problem, is the widely used PEER Center methodology for PBEE (Deierlein, 2004). It is rigorous probabilistic and permits consistent characterization of the inherent uncertainties throughout the process. The seismic risk assessment problem is decomposed into the four basic elements of hazard analysis, structural analysis,
damage analysis and loss estimation. Since the PBEE seeks also to improve quantification of deformation capacity of structural members, the research presented here deals with the prediction of the ultimate drift of reinforced concrete columns failed in flexure, which is expressed in terms of mean or median values and also in terms of dispersion measure, which enables quantification of an error in prediction of the ultimate drift. So far, semi-empirical and empirical approaches are used for determination of the deformation capacity of structural members more or less efficiently. The empirical formulas developed by Fardis and co-workers
(Panagiotakos & Fardis, 2001) have been implemented in Eurocode 8, Part 3 (CEN, 2005). A good overview of the deformation capacity of RC members is provided in FIB (2003). However, capacity of structural elements (i.e. columns and beams of the frame structures) may be predicted also by new approaches, i.e. by using the CAE (Conditional Average Estimator) method. This method, which is based on a special type of multi-dimensional non-parametric regression and represents a kind of probabilistic neural network, was developed by Grabec and Sachse in early nineties and was presented in (Grabec and Sachse, 1997). For successful application of the method, an appropriate database of experimental results is needed. Only recently a more comprehensive databases for RC members become available, such as the databases compiled at the University of Washington (PEER, 2008) and by Fardis and co-workers (Panagiotakos and Fardis, 2001, updated by Fardis and Biskinis, 2003). CAE method has been recently proposed as an alternative ap proa ch to the classical approaches in this field by prediction of the ultimate drift. Members of our research group have extended the CAE method andprepared the sound basis for its use in earthquake engineering (Perus et al., 2006). Note that CAE can be successfully used also in other fields of2 engi-
neering (Pirtovsek-Vecko et al., 2007, 2008).
In this paper the CAE method is presented briefly and corresponding error estimation is addressed. Some examples on the error estimates of the ultimate drift of RC columns are shown and then general recommendations about the use of error estimates in PBEE are given. Practical example of real building demonstrates the proposed estimates.
Cae method for prediction of ultimate DRIFT and error ESTIMATioN
Detailed description of the CAE method from the engineering point of view is given in Perus et al. (2006). The basic equations for determining the ultimate drift of RC columns are:

where
e = 0i
2 N
Z o
(1)
(2)
and
02 =
( 2*)
gg^g eXp
n) w„
-Z
i=1
S - bnl )2
2w2
(3)
2=1
i =1
1
In above equations ô is the estimate of the ultimate drift, 8n is the same output variable corresponding to the n-th model vector in the database, N is the number of model vectors in the database, bnl is the 0-th input variable of Ae n-th madel vector in the database, and bt is the l-th input variable corresponding to the prediction vector. Note that the esch model vector corresponds to the resultf of one experiment from the databaf e. O is the number of input vari-wbles, and defines the dimension of the samp le space. T he Gau ss ian functi on i s uocd for smooth interpolation between the posnts of the model vietors. In this context the width e is called the
n
"sm e othmg" eatam eter shat oornespon -3s to n-th model vector from the daSa-base. M osr case th e sam e wMfii w of
n
tlae Getussionn funcOion is used for all of the input variables. Therefore it is important that the input parametens in ths equation for an are normalized, ganer-ally in te snnge from 0 to f.
An intermedia-e result in the eomputa-tional process is parameter p, define A as
ing parameter w. It helps to detect the possib le le ss accurate predictions (indicated by small values of p) due to the data distribution in the databa se and due to loeal extrapolation outside the dati range.
When the expression for ultimate drift 8 is compared with the expression for the first order moment of the random variaWe X, which corresponds to the mean v alue m
o
E[g]=SZ X T r 1 a = Xr
15-
f o
P =N i An
1 y n=f
14)
It represents a measure of how the influence of aU the mode 1 vectors in theda-tabase is; epread over the sample space and it strongly depends on the smooth-
similarity between the two expressions becomhs egideno. (x(x.) 'ss the proba.il-ity of the random variable X = r. and correspond to eXe —e^hte An wh1th lep onn on the; s ion il arity belween the input nosriables of the prediction vector, and on ihe corresponding input h-riaSles pertineet to the model vec-torr stored m the database. Also, there is evident similarity when the central second order moment of the prob abU -ity distriNution of random variable X, aalled variance, givrn by the expression
N
Ur= Ohn[X]=re 1 a -rnx-a Tr 1X- )6-
i=i
is compared witO the prediction of 1s-c allsd "local stand ard deviation" in the CAE method:
i=i
El =
± k-k f-
(76
n=1
Such interpretation of the CAE equations allows us to estimate the corresponding probability distribution and the median value. Note, that the probability density function is composed of weights An for ascending order of the corresponding values Vn.
The briefly presented CAE method was applied for the prediction of the ultimate drift of the RC columns. The experimental database used in this study is based mainly on the PEER database prepared by the University of Washington. The prediction of the ultimate drift is limited only to the columns which failed in flexure since the limited num-
zahn86u8
tana90u7
suguc15h

to .o
o
(d)
Sl% zho21408
StDev = 0.04 f/a CoV = 0,04 p = 4.30
5
dl%
) -0 9
OS 0.7 ■
o.e ■
0	5 04 0.3 02
01 0^
(e)
StDev = 1.15-|Vo SoV - 0.27 p = 1.13
sak90b6
mug89ah2
Figure 1. Examples of empirical cumulative probability distributions (black line) in predictions of ultimate drift for randomly chosen RC columns from the PEER database and reference normal (red line) and log-normal (green line) distributions. Vertical red line indicates mean value
n
ber of column specimens (so-called model vectors with components b.) in the PEER database failed in shear and therefore the existing database is not yet appropriate for prediction of ultimate drift of columns failed in shear.
By knowing the empirical probability distribution of the sample (RC column), the application of Kolmogorov-Smirnov test (K-S test) can give us the information on the type of known probability distribution. The K-S test is a form of minimum distance estimation used as a nonparametric test of equality of one-dimensional probability distributions. Kolmogorov-Smirnov statistic quantifies a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution (i.e normal and log-normal in our study). Samples are standardised and compared with a standard normal and log-normal distribution, what is equivalent to setting the mean and variance of the reference distribution equal to the sample estimate.
Empirical cumulative probability distributions predicted by CAE method and reference normal and log-normal distributions are shown in Figure 1 for selected columns from the PEER database. The null hypothesis (H0) for each RC column from the PEER database was the assumption that the ultimate drift is distributed normally or log-nor-
mally with expected values of S respectively). The rejected level is at a = 0.01. It turns out that for about 5 2 % of RC columns from and E2a (mx and ¡dx, the PEER datmabase (156 specimens) the null hypothesis can not be rejected (i.e. columns in Figures 1a and 1b). On the other hand, the rest of RC columns for which the n511 hypothesis is rejected (predicted ultimate drift is NOT distributed norm a1ly or log-normally), typically and gs to similar columns with very different drifts (Figure 1c), to columns with large similarity with one column m th2 database (Figure 1d) or to columns with large eimilarity with two or m e columns in the database (Figure 1e) or t1 molumns with small values of p (Figure 1 f). In all these cases relatively large weights An are attributed to them and consequently K-S statistics gets relatively high values which reject the null hypothesid Note, that the existing PEER database is the largest an the most detailed database on RC columns for the time being. It is also known that sample RC c olum ns are not distributed randomly and that the size of the datmbase is still small for more reliaWe analysis. Aathtrs believe that the extended database would solve this problem. Nevertheless, from the engineering point of view, the obtained results indicate that the distribution of ultimate drift, which is predicted by the CAE method, roughly corresponds to the normal or log-normal probability distribution.
The average "local coefficient of variation" (CoV), which is the ratio between the "local standard deviation" and predicted mean value, amounts from zero up to 0.9 in some very rare cases, with an average value of 0.35 and standard deviation of 0.16. Through error and trial procedures it was decided to use a value of 0.4 for CoV in case of assumed log-normal distribution. Furthermore, this value may be considered as a good approximation of CoV in PBEE, especially when it is compared to value of 0.6, obtained by Fardis & Biskins (2003). Note that use of average CoV for prediction of ultimate drifts of RC columns of a building represents a simplification, which can significantly reduce the number of time history analyses (simulations) needed for sufficiently accurate prediction of seismic response parameters, and it does not significantly affects the results, since similar types of columns are usually used in a building.
Application
The aim of the presented example is to demonstrate the influence of some uncertain input variables of the structural model, especially the ultimate drift (rotation) in columns, on the seismic response parameters. For that reason, the relationship between the seismic intensity measure (peak ground acceleration) and the seismic response
parameters (maximum story dirft) was determined for a four storey reinforced concrete frame by using the extended incremental dynamic analysis (extended IDA) (Dolsek, 2009).
The four-storey reinforced concrete frame had been designed to reproduce the design practice in southern European countries about forty to fifty years ago and pseudo-dynamically tested in full scale at ELSA Laboratory (Figure 2) (Carvalho & Coelho, 2001). However, the frame can also be typical of buildings built more recently, but without the application of capacity design principles (especially the strong column - weak beam concept), and without up-to-date detailing. The elevation and typical reinforcement in the columns of the four storey frame are presented in Figure 3. The design base shear coefficient amounted to 0.08. In the design, concrete of quality C16/20 and smooth steel bars of class Fe B22k (according to Italian standards) were adopted (Carvalho & Coelho, 2001). Later the strength of material was measured since the pseudo-dynamic tests were performed for the structure. The mean strength of the concrete amounted to 16 MPa, that is less than adopted in the design (fcm for C16/20 is 24 MPa), and the mean yield strength of the steel amounted to 343.4 MPa.
Beam and column flexural behaviour was modelled by one-component
lumped plasticity elements, composed of an elastic beam and two inelastic rotational hinges (defined by the moment-rotation relationship). The element based on the assumption of an inflexion point at the midpoint of the element was employed in nonlinear static and dynamic analyses.
The schematic moment-rotation relationship of the inelastic rotational hinge is shown in Figure 4a. The yield (Y) and the maximum (M) moment in the columns were calculated taking into account the axial forces due to the vertical loading on the frame. The effective beam width of 75 cm and 125 cm were determined according to the Eurocode 2 (CEN, 2004) procedure for the short and long beams, respectively. The characteristic rotations, which describe the moment-rotation envelope of a plastic hinge, were determined according to
the procedure described by Fajfar et al. (2006). The ultimate rotation 0 in
Figure 2. The four-storey reinforced concrete frame building which was tested at ELSA Laboratory
Figure 3. View and typical reinforcement of the columns of the reinforced concrete frame
0	0 /rad	0 /rad
a) schematic	b) first set of structural models c) second set of structural models
Figure 4. The moment-rotation relationship of column plastic hinge: a) schematic representation, b) column C at second storey for first set of structural models and c) column C at second storey for second set of structural models
the columns at the near collapse (NC) limit state (see Figure 4a), which corresponds to a 20 % reduction in the maximum moment, was estimated by means of the CAE method (Perus et al., 2006). For the beams, the EC8-3 (CEN, 2005) formulas were used, the parameter yel being assumed to be equal to 1.0. Due to the absence of seismic detailing, the ultimate rotations were multiplied by a factor of 0.85 (CEN, 2005).
The extended IDA analysis was performed for two sets of structural models, which reflected different sources of uncertainty. Each set consisted of 20 structural models, which were determined based on the Latin Hypercube Sampling method employed in the extended IDA (Dolsek, 2009). In the first set of structural models the following sources of uncertainties were considered in the analysis: mass, strength of
the concrete and that of the reinforcing steel, effective slab width and damping, whereas in the second set of structural models also the model for determining the initial stiffness and ultimate rotation in the plastic hinges of the beams and columns was adopted uncertain. All the input random variables considered for the determination of the set of structural models were assumed to be uncorrelated. The statistical characteristics of the input random variables are presented in Table 1.
The initial stiffness and ultimate rotation in the plastic hinges of the beams and columns was considered deterministic in the first set of structural models, while in the second set all input random variables were considered for determination of the set of structural models. In addition to two sets of structural models, which reflect epistemic uncertainties,
the deterministic structural model was also used for determination of the relationship between the peak ground acceleration and the maximum storey drift. In order to demonstrate the difference between the structural models used in analysis, the moment-rotation relationship of plastic hinge in the column C at second storey is presented for the two set of structural models (Figure 4b and
4c) and compared to the moment-rotation relationship of deterministic model. It can be observed that the dispersion in the moment-rotation relationship is significantly increased in the case of the second set of structural models, since in this case the initial stiffness and ultimate rotation of beams and columns are considered as random variables which have high coefficient of variation.
Table 1. The statistical characteristic of the input random variables
Name		Mean or Median*	CoV	Distribution
Mass 1st storey	m1	46 t	0.1	normal
Mass 2nd storey	m2	46 t	0.1	normal
Mass 3rd storey	m3	46 t	0.1	normal
Mass 4th storey	m4	40 t	0.1	normal
Concrete strength	f J cm	16 MPa	0.2	normal
Steel strength	fy	343.6 MPa	0.05	log-normal
Effective slab width	beff	75 cm or 125 cm	0.2	normal
Damping		2 %	0.4	normal
Initial stiffness of the columns	© y,c	1-computed	0.36	log-normal
Initial stiffness of the beams	©b	1-computed	0.36	log-normal
Ultimate rotation of the columns	© u,c	1-computed	0.4	log-normal
Ultimate rotation of the beams	© b u,b	1-computed	0.6	log-normal
*mean is shown for normal distribution and median for log-normal distribution
Figure 5. Maximum drift as a function of peak ground acceleration. Results are presented for two ground motion records and for a) first and b) second set of structural model
Two ground motion records were selected from the European strong motion database (Ambraseys, 2000) aiming to demonstrate the influence of the uncertainties on the seismic response parameters. Both ground motion records were recorded on stiff soil during the Montenegro earthquake in 1979. IDA analysis was performed for each ground motion record. The peak ground accel-
eration, corresponding to instability, was determined with tolerance of 0.005 g. Selected results of IDA analysis are presented in Figure 5. In addition to so called IDA and capacity points, which, respectively, represents the maximum storey drift of one nonlinear dynamic analysis for a given ground motion record and the peak ground acceleration, which corresponds to the global
Table 2. Peak ground acceleration capacity of deterministic model, median peak ground acceleration capacity and its dispersion for probabilistic model with and without the probabilistic ultimate rotation in columns (for two ground motion records)
Ground motion record	Deterministic model	Probabilistic model			
		First set of struct. models		Second set of struct. models	
	ag,C/g	ag,C/g	Pg,C	ag,C/g	Pg,C
196x	1.775	1.715	0.197	1.637	0.403
197y	0.473	0.474	0.100	0.477	0.255
dynamic instability of the structure, the IDA curves of the deterministic model and the summarized IDA curves (median ± o) of the first and second set of structural models are also presented.
The results in Figure 5 indicate that the influence of uncertainties on the seismic response parameters can be neglected if the peak ground acceleration is much less than the peak ground acceleration which causes the global dynamic instability of the structure. For this range of peak ground acceleration the summarized IDA curves based on the first and second set of structural models are practically the same in comparison to the IDA curve which is determined for the deterministic model. However, uncertainties can reduce the peak ground acceleration, which corresponds to global dynamic instability of the structure, what can be observed for the ground motion record 196xa. In
this case the median capacity in terms of peak ground acceleration is reduced for about 3 % and 8 % if compared to that of the deterministic model. For the other ground motion record the capacity is practically the same for both sets of structural models and also for the deterministic model (Table 2).
In addition, the significant increase in the dispersion of the seismic response parameters can be observed for the second set of structural models since high uncertainties were used in determination of the initial stiffness and ultimate rotation of columns and beams. The dispersion in peak ground acceleration, which corresponds to global dynamic instability, is increased for about 100 % in the case of second set of structural model if compared to dispersion calculated from the first set of structural models, and it is also significantly dependent on selected ground motion records (Table 2).
Conclusions
The PBEE seeks to improve quantification of deformation capacity of structural members. Therefore, the research presented here deals with the prediction of ultimate drift with special consideration on dispersion measure. The CAE method was applied for this purpose to the RC columns which fails in flexure. It was found that empirical probability distribution of ultimate drift of RC columns roughly corresponds to normal and/or log-normal distribution. Moreover, an average value of 0.4 for CoV in the case of assumed log-normal distribution is proposed to be used in PBEE. It should be noted that the use of average CoV for prediction of ultimate drifts of RC columns of a building represents a simplification. Namely, it does not significantly affect the results, since similar types of columns are usually used in a building. However, uncertainty in prediction of ultimate drift with the CAE method is reduced if compared with procedure proposed by Fardis & Biskins (2003). The reduced uncertainty can significantly reduce the number of time history analyses (simulations) needed for sufficiently accurate prediction of seismic response parameters.
The influence of uncertainties on the seismic response parameters was dem-
onstrated with an example of four storey RC frame building. The results indicate that the influence of uncertainties on the median value of seismic response parameters can be neglected if the peak ground acceleration is much less than the peak ground acceleration which causes the global dynamic instability of the structure. On the other hand, the increase in the dispersion of the seismic response parameters can be observed for the structural models with high uncertainties in determination of the initial stiffness and ultimate rotation of columns and beams. Therefore, it is important to determine the deformation capacity (i.e. ultimate drift) of RC structural members with predictive models which gives the lowest uncertainties and consequently more accurate prediction of seismic risk. Using the CAE method, as demonstrated in this study, represents a step toward this goal.
Acknowledgements
The results presented in this paper are based on work supported by the Slovenian Research Agency.
References
Ambraseys, N., smith, P., Bernardi, R., Ri-naldis, D., cotton, F., Berge-Thierry,
C. (2000): Dissemination of European Strong-Motion Data. CD-ROM collection. European Council, Environment and Climate Research Programme.
carvalho, E. c., coelho, E. (Editors) (2001): Seismic assessment, strengthening and repair of structures. ECO-EST2-ICONS Report No. 2, European Commission - "Training and Mobility of Researchers" Programme.
CEN. Eurocode 2 (2004): Design of concrete structures - Part 1-1: General rules and rules for buildings. EN 1992-1-1, European Committee for Standardisation, Brussels, December 2004.
CEN. Eurocode 8 (2004): Design of structures for earthquake resistance. Part 1: General rules, seismic action and rules for buildings, Brussels, December
2004.
CEN. Eurocode 8 (2005): Design of structures for earthquake resistance. Part 3: Strengthening and repair of buildings. EN 1998-3, European Committee for Standardisation, Brussels, March
2005.
Deierlein, G. G. (2004): Overview of a comprehensive framework for earthquake performance assessment, PEER Report 2004/05 Performance-Based Seismic Design - Concepts and Implementation (Eds. Fajfar and Krawinkler).
Dolsek, m. (2009): Incremental dynamic analysis with consideration of model-
ling uncertainties. Earthquake Engineering and Structural Dynamics; Vol. 38,pp.805-825.
Fajfar, P., Dolsek, M., Marusic, D., Stra-tan, A. (2006): Pre- and post-test mathematical modelling of a plan-asymmetric reinforced concrete frame building. Earthquake Engineering and Structural Dynamics; Vol. 35, pp. 1359-1379.
Fardis, M. N., Biskins, D. E. (2003): Deformation capacity of RC members, as controlled by flexure or shear. Performance-based engineering for earthquake resistant reinforced concrete structures: A volume honoring Shunsuke Otani. Kabeyasawa T, Shi-ohara H, (eds.), University of Tokyo, pp.511-530.
FIB (2003): Displacement-based seismic design of reinforced concrete buildings. State-of-the-Art Report Prepared by Task Group 7.2, Bulletin 25, (Federation international du beton).
Grabec, I., Sachse, W. (1997): Synergetics of Measurement, Prediction and Control, Springer-Verlag.
Panagiotakos, T. B., Fardis, M. N. (2001): Deformations of reinforced concrete members at yielding and ultimate. ACI Structural Journal; Vol. 98, No. 2, pp. 135-148.
PEER Reinforced Concrete Column Test Database [online]. University of Washington [cited 20. 10. 2008].
Accessible on Internet: http://www. ce.washington.edu/~peera1/.
Perus, I., Poljansek, K., Fajfar, P. (2006): Flexural deformation capacity of rectangular RC columns determined by the CAE method. Earthquake Engineering and Structural Dynamics; Vol. 35, No. 12, pp. 1453-1470.
Pirtovsek-Vecko, T., Kugler, G., Fajfar, P., Fazarinc, M., Perus, I., Tercelj, M.
(2007): Hot forming of AISI D2 steel. RMZ - Materials and Geoenviro-nment; Vol. 54, No. 1, pp. 1-14.
Pirtovsek-Vecko, T., Fazarinc, M., Ku-gler, G., Tercelj, M. (2008): Increasing of hot deformability of tool steels - preliminary results. RMZ - Materials and Geoenvironment; Vol. 55, No. 2, pp. 147-161.