Probabilistic Treatment of the Capacity

As has been presented previously, the limit state displacement capacity (∆Lsi) of each building class can be defined as a function of the fundamental period (TLsi), the geometrical properties of the building, and the mechanical properties of the

construction materials. Similarly, the limit state period (TLsi) of each building class can be defined as a function of the height (or number of storeys), the geometrical properties of the building, and the mechanical properties of the construction materials. The uncertainty in ∆Lsi and in TLsi is accounted for by constructing a vector of parameters that collects their mean values and standard deviations. By assigning probability distributions to each parameter, FORM can be used to find both the cumulative distribution function (CDF) of the limit state displacement capacity, conditioned to a period, and the CDF of the limit state period, which are then combined to create the joint probability density function of capacity.

3.2.1 Probabilistic Modelling of Geometrical Properties

A given building class within a selected urban area may comprise a large number of structures that present the same number of storeys and failure mode, but that feature varying geometrical properties (e.g., beam height, beam length, column depth, column/storey height), due to the diverse architectural and loading constraints that drove their original design and construction. Since such variability does affect in a significant manner the results of loss assessment studies (see Glaister and Pinho, 2003), it is duly accounted for in the current method by means of the probabilistic modelling described below.

Clearly, one could argue that by carrying out a detailed inspection of the building stock, such variability could be significantly reduced (in the limit, if all buildings were to be examined, it could be wholly eliminated), however at a prohibitive cost in terms of necessary field surveys and modelling requirements (vulnerability would then be effectively assessed on a case-by-case basis). This epistemic component of the geometrical variability of reinforced concrete members has been modelled in the present work by means of normal or log-normal probability distribution functions, derived from European building stock data, as described in Crowley et al. (2004).

3.2.2 Probabilistic Modelling of Reinforcing Bar Yield Strain

Mirza and MacGregor (1979) have suggested that once a probabilistic distribution for yield strength has been found, it can be divided by a deterministic value of the modulus of elasticity, which features a very low coefficient of variation, to produce the distribution of the yield strains. These two researchers have also concluded, through a series of experimental parametric studies, that a normal distribution would accurately represent the variability of reinforcement bars’ yield strength, in the vicinity of the mean, whilst a beta distribution correlated well over the whole range of data. The coefficient of variation in the yield strength was found to be between 8% - 12% when data were taken from different bar sizes from many sources. More recently, the Probabilistic Model Code (JCSS, 2001) has also suggested that a normal distribution can be adopted to model the yield strength of steel. Therefore, a normal distribution for the steel yield strength (and subsequently yield strain) has been adopted in the current work.

The main difficulty in assigning a probability distribution to the yield strength of the steel used in a group of buildings, however, is the possibility that different grades have been used, which would lead to a distribution with multiple peaks and troughs (see Crowley et al., 2004). One approach to solve this problem could be to calculate the probability of failure for the building class given each possible steel grade, using the normal distribution to model the dispersion for each grade, and then to compute a weighted average of failure, knowing or judging the use of each steel grade within the building class. The validity of such an approach would become questionable, however, if different steel grades were often used within individual buildings.

3.2.3 Probabilistic Modelling of Limit States Threshold Parameters

Dymiotis et al. (1999) have studied the seismic reliability of RC frames using interstorey drift to define the serviceability and ultimate structural limit states. They have found that a lognormal distribution may be used to describe the variability in interstorey drift for both limit states. Therefore, the variability in non-structural limit states, defined in this work as a function of interstorey drift limiting values, will be represented by means of lognormal distributions, using the mean drift ratios that have been suggested by Crowley et al. (2004).

Kappos et al. (1999), on the other hand, report the ultimate concrete strain reached in 48 tests of very well-confined RC members. A simple statistical analysis of this data shows that it would appear that in the case of limit state sectional strains a lognormal distribution is also able to describe the variability of these parameters. Hence, and since for the structural limit states it is the sectional steel and concrete strains that define respective boundaries, it would appear that a lognormal distribution may also be applied to describe the variability in these limit state parameters. Again, the mean values suggested by Crowley et al. (2004) are employed, in tandem with assumed coefficients of variation.

3.2.4 Probabilistic Modelling of Scatter in Empirical Relationships

A number of empirical relationships have been used to derive the functions of displacement capacity and period that have been presented in Section 2. These include empirical expressions for the plastic hinge length members and the yield curvature of RC members, all of which are discussed in Glaister and Pinho (2003), and an additional empirical parameter employed in the formula derived by Crowley and Pinho (2004) to relate the height of a building to its yield period. All of the aforementioned relationships rely on empirical coefficients to relate one set of structural properties to another, as for example the coefficient of 0.1 in the yield period vs. height equation, Ty = 0.1HT. The mean value and standard deviation of these coefficients have been taken from the studies carried out to derive those formulae, with a normal distribution being used to model the respective dispersion.

4. CONCLUSIONS

Owing to its transparency, theoretical accuracy and computational efficiency, the procedure presented herein is particularly suitable for loss estimation studies. The definition of the displacement capacity is transparent as one may use any chosen number of storeys, geometrical, material or limit state threshold properties in the equations and adapt these easily for use in any part of the world. The conceptual soundness of the methodology has been preliminarily examined by Crowley et al. (2004) through a comparison of vulnerability curves derived using this procedure and those provided in HAZUS; the curves derived using the proposed method led to more realistic vulnerability models which appear to be consistent with field observations following destructive earthquakes. Finally, the large decrease in computational effort required for earthquake loss estimations for scenario events due to the direct consideration of the ground motion uncertainty is also a significant advantage of the proposed methodology.

The above effectively means that the method does cater for rigorous, scenario- based approaches that can be applied to large areas within a reasonable timescale. In this manner, it will be possible for iterative loss assessment studies to be carried out for a given urban area under events with varying return periods and assuming different levels of building stock vulnerability, considering the effects, along with respective costs, of different design code requirements and/or structural upgrading policies. The above could provide politicians, planners and code drafters with quantitative information to inform and guide their decisions, thus allowing the calibration of local regulations for optimum balance between societal investment and public risk, rather than being based on pre-selected return periods whose basis is somewhat arbitrary.

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