Use of empirical equations in estimating reliabilities

Limit state functions and reliability analysis

The developed empirical equations can be employed for the probabilistic evaluation of the performance of a designed or an existing structural system, and for design code calibration. For the evaluation, only two levels, one for the incipient damage (incipient inelastic deformation) and the other for the incipient collapse (Wen 2001), will be considered. Let DR (Tn,£) denote the yield displacement capacity of the structural system, and Z denote the ratio of DR (Tn,£) to De (Tn ,£). In this section, De (Tn, £) represents the annual peak linear elastic displacement. Since the yield displacement capacity DR (Tn,£)less than the seismic demand DE (Tn,£) implies that the structure will at least sustain damage, and the maximum inelastic displacement (capacity) jUrDr (Tn,£) less than the maximum inelastic ductility demand implies that the structure will collapse, the limit state functions for these two performance levels are, gD = Dr (Tn,£)/De (Tn,£) -1 = Z-1, (5)


gc ={MrDr (Tn ,£))/(ф)De (Tn,£))-1, (6)

where gD represents the limit state function of the incipient damage; gc represents the limit state function of the incipient collapse, respectively; Z = DR (Tn,£)/ DE (Tn,£); and pR denotes the ductility capacity of the structural system. To emphasize that the ductility demand p is a function of the normalized yield strength ф, the notation for the ductility demand U in Eq.(6) is replaced by р(ф). Damage occurs if gD is less than zero and, collapse occurs if gC is less than zero.

Note that if Z is greater than unity (i. e., gD > 0) implying that the yield displacement capacity of the structural system is greater than the linear elastic demand caused by the strong ground motions. Note also that since by definition Z = DR(Tn, Z)/ DE(Tn, Z), Z represents the normalized yield strength (i. e., Z = ф), Eq. (6) can be re-written as

gc = Ur / M(0 -1. (7)

The yield strength is usually considered to be normal or lognormally distributed with a cov of about 0.1 to 0.15 (Ellingwood et al. (1980), Nakashima (1997)). Therefore, the yield displacement capacity of the structure can be considered to be lognormally distributed with a cov of 0.15. The uncertainty associated with the ductility capacity UR is much more significant than that associated with the yield strength. According to Nakashima (1997), the cov of UR can vary from about 0.5 to 1.0. In this section it is considered that UR can be modeled as a lognormal variate with a cov with of 0.5. Further, it is considered that the peak linear elastic displacement De (Tn, Z) is lognormally distributed as well with a cov within 1 to 10. This can be justified based on the seismic hazard studies given by Adams and Halchuk (2003).

Based on these adopted probabilistic models, the evaluation of the probability of the incipient damage, PD, is straight forward (Madsen et al. (1986)). It can be calculated from,

Pd = Prob(Z< 1) = F( (1), (8a)


Fz(Z) = ф((іпZ — ln(mc /^^vf))/-^a+Z) (8b)

where m^ = (1 + vE )R / mE, vj = (1 + vR )(1 + vE) — 1, mR and mE denote the means of DR (Tn, Z) and De (Tn, Z), respectively; vR and vE denote the cov of DR(Tn, Z) and DE(Tn, Z), and Ф() is the standard normal probability distribution function. Note that Z is lognromally distributed with mean mZ and cov of vz since DR (Tn, Z) and DE (Tn ,Z) are lognormally distributed.

The probability of the incipient collapse, PC, Pc = Prob(gc < 0), can be evaluated by recursively using the first-order reliability method (FORM) or the simulation techniques. In this study, the simulation technique is employed for the analysis. Note that Pc = Prob(gc < 0) can be expressed as

Pc = Prob( / U(Z) < 1|Z < 1)Prob(Z < 1) = Prob( / v(C) < l|Z < 1)Pd, (9)

since Prob(//R / ji(Z) < l|Z> 1) = 0. The basic steps for evaluating Prob(//R / ji(Z) < 1|Z < 1) by using simulation technique are:

1) Generate a sample of Z according to the updated (or truncated) probability distribution function of Z, Fj (Z)/ PD

2) Find the mean of p(Z) using Eq. (1) with parameters defined in Eqs. (3) and (4), and the cov of p(Z) from figures similar to Figure 3;

3) Using the obtained value in Step 2) define the probability distribution of p(Z), which is considered to be Frechet distributed;

4) Generate samples of pR and p(Z) according to their probability distributions; and check if /uR / ju(Z) is less than or larger than unity;

5) Repeat Steps 1) to 5) to generate enough samples of /uR / /u(Z) and to count number of times that /uR / jU(Z) is less than one for estimating Prob(//R / ju(Z) < 1|Z < 1).

Numerical examples

The formulations given in previous section are illustrated by simple numerical examples. For the analysis, it is assumed that the structure can be modeled as a SDOF system with natural vibration period equal to 0.5 (sec). Consider that the ratio between the yield displacement capacity of a bilinear hysteretic system mR and the displacement corresponding to the design earthquake load for a linear elastic SDOF system, DEN (Tn,0), фп, is known. фп takes into account factors such as the ductility-related force reduction and the mean resistance is greater than the factored resistance. The yield displacement capacity is considered to be lognormally distributed with cov of 0.15. Further, consider that фп equals 0.5; the mean of the ductility capacity, m^R, equals 4; the ratio of the post yield stiffness to the initial stiffness у takes the value of 0, 0.01 or 0.05; and DEN (Tn, 0) equals 475-year return period value of the peak linear elastic

displacement demand DE (Tn, 0) that is considered to be lognormally distributed. The cov of

De (Tn, 0) , vE, that equals 0.8, 2 and 10 are considered. The small and large values of vE were used to represent approximately the seismic hazard conditions for, respectively, the west and the east of Canada Adams and Halchuck (2003).

Based on these considerations, it can be shown that mz is given by,


In Eq. (10), Ф 1(^) denotes the inverse of the normal probability distribution function. Substituting Eq. (10) into Eq. (8) gives PD equal to 3.44x 10 2, 1.10x 10 2, and 5.74 x 10-3 for vE equal to 0.8, 2 and 10, respectively.

To simplify the evaluation of PC, the numerical analysis was carried out by considering that the cov of p(Z) is independent of Z. The effect of this assumption on the estimated probability will be investigated by comparing the results obtained for the cov of p(Z) equal to 0.4 and 0.8, which are shown in Table 2. The results shown in the table suggest that the probability of incipient collapse is not very sensitive to the assumed cov of the seismic demand (i. e., cov of De (Tn,0), vE) nor to the assumed cov of p(Z). Also, it is noted that that the obtained probability of incipient collapse is insensitive to the considered post-yield stiffness. This may be explained by noting that a structure with Tn = 0.5 and for a normalized yield strength (i. e., фп) around 0.5 the expected ductility demand for у = 0 does not differ significantly from that for у = 5%.

It should be noted that no attempt is made in this study to carry out a design code calibration excise (Madsen et al. (1986)). However, it is noteworthy that given the mean ductility capacity of structure m^R and the nominal or factored design earthquake demand DEN (Tn,0), the formulation and procedure given in this study can be used to calibrate the required resistance factor for the yield displacement (or strength) фп such that use of the factor in design will leads to the designed structures to meet a specified target reliability level.

Table 2. Estimated probability of incipient collapse

cov of p(Z)

































A statistical analysis of the ductility demand was carried out for bilinear hysteretic SDOF systems. The analysis results indicate that the ductility demand can be modeled as a Frechet (Extreme value type II) variate for given values of the normalized yield strength.

The mean of the ductility demand and the normalized yield strength when plotted in a logarithmic paper, follow approximately a straight line for SDOF systems having the same initial natural vibration period. This observation leads to a simple empirical equation in predicting the expected ductility demand. Model parameters for the proposed empirical predicting models were obtained for different natural vibration periods, damping ratios, and ratios of the post yield stiffness to the initial stiffness. The coefficient of variation (cov) of the ductility demand can go as high as to about 1 depending on the characteristics of the structure.

Using the developed probabilistic characterization of the ductility demand, a simple approach to estimate the probability of incipient damage or incipient collapse was given. Numerical results suggest that an accurate empirical predicting model for the cov of the ductility demand may not be necessary since sensitivity analysis results indicate that the variation of this cov on the probability of incipient collapse is not very significant.