H. P. Hong1 and P. Hong2

1 Department of Civil and Environmental Eng., University of Western Ontario, N5A 6B9 Canada
2Department of Civil Eng., Aeronautical and Industrial Institute of Nanchang, China

Abstract: Probabilistic assessment of the ductility demand and reliability analysis were carried out for bilinear hysteretic SDOF systems. The assessment considered two sets of strong ground motion records, and was focused on the evaluation of the mean and the coefficient of variation of the ductility demand for a given value of the normalized yield strength. The results indicate that the ductility demand could be modeled as a Frechet (Extreme value type II) variate. Based on the obtained results, empirical equations were provided to predict the mean of the ductility demand for bilinear SDOF systems of different natural vibration periods, damping ratios, and ratios of the post yield stiffness to the initial stiffness. The numerical results show that the coefficient of variation (cov) of the ductility demand can go as high as to about 1.0 depending on the characteristics of the structure. Also, a simple approach was given to estimate the probability of incipient damage and the probability of incipient collapse using the developed probabilistic characterization of the ductility demand. The approach, which could be suitable for carrying out design code calibration analysis, is illustrated numerically.


For a given strong ground motion, the peak responses of a linear or nonlinear single-degree – of-freedom (SDOF) system with and without strength degradation can be carried out using time – step integration methods. The obtained peak responses of SDOF systems can be employed in defining the linear elastic response spectrum and yield response spectra, and/or the ratios between peak linear elastic response and inelastic responses (Chopra 2000). These quantities are relevant for designing and assessing the safety of structures. Its use for the so-called displacement-based design has been discussed by many including Chopra and Geol (2000) and Borzi et al. (2001).

Let Fe denote the minimum strength required for a SDOF system to remain linear elastic during a ground motion, and DE(Tn,£,) denote the peak linear elastic displacement where Tn and

are the natural vibration period and the damping ratio, respectively. If the strength of the structure is less than Fe, the system responds inelastically with yield displacement represented by Dy (Tn,£,p) and peak inelastic displacement represented by Dt (Tn ,£,p), where p represents the displacement ductility factor. Given a set of strong ground motion records, the yield reduction factor Ry, Ry = De (Tn,#)/ Dy (Tn,£,p), and the ratio Rp, Rp = D, (Tn,£p)/ De (Tn,£), can be

calculated. Note that Rp = p/Ry which can be written as Rp = рф where ф is defined as 1/Ry

and is known as the normalized yield strength or the de-amplification factor. Note also that p does not always increase monotonically as Ry decreases and more than one value of Ry could lead to the same ductility demand p. By considering that for a given value of p it is the largest yield strength, hence the largest ф (or smallest Ry) that is relevant for design, an iterative procedure that is described in detail in Chopra (2000) can be employed to evaluate the required ф (or Ry) for a given ductility factor p. Note that the above is equivalent to say that given a value of ф, it is the maximum ductility demand, for all the normalized yield strength less than or equal to the specified value of ф, that is relevant for design. This view is adopted though out this study.


M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 711-721.

© 2006 Springer. Printed in the Netherlands.

Samples of Ry or Rp obtained to meet specified target ductility level are employed to find statistics of the ratios Ry and/or Rp and to develop empirical equations to predict the mean of the Ry and/or Rp as functions of ductility demand p (Veletsos and Newmark 1960, Krawinkler and Nassar 1992, Vidic et al. 1994, Miranda 2000 and Riddell et al. 2002, Hong and Jiang 2004). The means of Ry and/or Rp are employed to scale the design response spectrum or peak linear elastic responses to obtain the design yield strength or the yield responses.

The evaluation of Ry or Rp to meet specified target ductility factor is computationally intensive because the iteration mentioned previously. It is much more efficient to evaluate the ductility demand for a given value of the normalized yield strength because the latter does not require the iteration over the ductility factor. Also, it is noted that rather than develop empirical equations to predict the values of Ry, Rp, or the normalized yield reduction factor ф for given values of p, one may instead develop empirical equations to predict the ductility demand p based on regression analysis conditioned on ф. A regression equation developed to predict the expected normalized yield strength is likely to differ from the one developed to predict the expected ductility (factor) demand. Perhaps, the former may be interpreted as a designer knows the ductility capacity of the structure to be designed and is interested in finding the minimum required yield strength; and the latter may be considered as a designer’s task is to check a new design or evaluate an existing structure with a known yield strength level and is interested in finding what would be the ductility demand due to strong ground motions. Therefore, the latter that seems lacking in the literature is equally relevant as the former. The need for empirical equations to predict the ductility demand may be further justified based on that the uncertainty or variability associated with the ductility capacity is much greater than the yield strength (Nakashima 1997), hence, a designer could have better control on the yield strength level than on the ductility capacity, and a codification should be focused on incorporating the uncertainty in ductility capacity and ductility demand. Note that systematic assessment of the impact of uncertainty in ductility capacity on the structural reliability is not often investigated

In the following, statistics of the ductility demand are evaluated using two sets of strong ground motion records. The evaluation of samples of the ductility demand is carried out for given values of ф. This largely reduces the computing time since iterations over ф to find the ductility factor that matches a specified ductility level are not required. Also, empirical equations for the statistics of p conditioned on the normalized yield strength are presented. In a few cases, comparison of these results to the ones obtained to meet specific ductility level is also given. The evaluation of the ductility demand presented in this study considers several damping ratios and the elastoplastic as well as bilinear hysteretic systems. A very simple method for assessing the reliability of bilinear system by using the developed empirical equations is presented. The method can be used to evaluate the probability of incipient collapse as well as the incipient of damage. Its use is illustrated by numerical examples.