Quantifying Analysis Method Uncertainty Sac Method

The SAC Joint Venture1 research program is currently in the process of finalizing a series of seismic design guidelines (FEMA, 2000) which incorporates the Cornell Demand and Capacity Factor Design (DCFD) methodology for the seismic performance evaluation of steel frame buildings. When an approximate analysis method is used to conduct a seismic performance evaluation of a building, the reliability framework must account for the uncertainty in the analysis method. This section describes the method for quantifying analysis method uncertainty developed as part of the research for the SAC Joint Venture research program and is therefore called the SAC Method.

A fundamental component of the Cornell DCFD methodology is the relationship between the building response quantity used to define performance, e.g. maximum interstory drift ration (IDR), and the earthquake ground motion parameter used to define the seismic hazard, e.g. spectral acceleration. Recall from Chapter 5 that Cornell proposed the two-parameter function given in Eq. (5.39) to describe this relationship. When the nonlinear time history analysis method is used to calculate building response quantities, then the variation of the data points about Eq. (5.39), i.e. the conditional

1 SAC is a joint venture of the Structural Engineers Association of California (SEAOC), the Applied Technology Council (ATC), and California Universities for Research in Earthquake Engineering (CUREe).

logarithmic standard deviation, was defined as inherent randomness, or the uncertainty in the building response due to the record-to-record variation in the ground motion. The basis of the SAC Method is that inherent randomness defined in this manner, is the inherent randomness in building response regardless of the analysis method used to estimate building response. In addition, recall from Chapter 4 that the uncertainty in building response calculated using nonlinear static analysis methods was quantified by bias factors that also varied from record-to-record. The basic assumption in the SAC Method is that this record-to-record variation observed in the bias factors is the same record-to-record variation observed in the values of the building response calculated using the nonlinear time history analysis method. Therefore, including both of these measures of uncertainty in the DCFD methodology would be overestimating the total uncertainty in predicting building response. With this assumption, the analysis method uncertainty need only be represented by a median value of the bias factor.

The following is the step-by-step procedure for calculating the median nonlinear static analysis method bias factors according to the SAC Method.

(1) For a given building model, perform nonlinear time history analyses using N earthquake ground motion records corresponding to a given level of seismic hazard, e.g. 10% in 50 years.

(2) Find the median value of the building response quantity of interest, e.g. maximum IDR, using the N values calculated in step (1). The median value is called the Median Nonlinear Time History Demand.

(3) For each of the N earthquake ground motion records, calculate the acceleration response spectrum using a value of damping that corresponds with the level of damping specified in the nonlinear time history analyses performed in Step (1).

(4) Calculate the median acceleration response spectrum from the N response spectra calculated in Step (3).

(5) Using linear regression, fit a smooth response spectrum of the form shown in Figure 4.19, Chapter 4, to the median response spectrum calculated in Step (4).

(6) Perform a nonlinear static pushover analysis on the building as described in Chapter 3. Calculate the target displacement using the smooth median response spectrum created in Step (5), and extract the building response quantities of interest from the pushover analysis results. This value is called the Median Nonlinear Static Demand.

(7) Calculate the median bias factor by taking the ratio of the Median Nonlinear Time History Demand to thew Median Nonlinear Static Demand.

The three Pre-Northridge model welded steel moment resisting frame (WSMF) buildings described in Appendix B were used to calculate the median analysis method bias factors according to the SAC Method. In Chapter 4, the Ml and M2 models of the 3-, 9-, and 20-story buildings were analyzed by nonlinear dynamic time history analysis method using the earthquake ground motion records in the 50%, 10%, and 2% in 50 year earthquake hazard levels. As described in the SAC Method procedure, a median smooth 2% damped response spectrum of the form shown in Figure 4.19, Chapter 4, was constructed for each of the earthquake ground motion records in each earthquake hazard level. The resulting parameters that describe the shape of the smooth spectra are given in Table D. 1. The 2% damped median response spectra and the median smooth spectra for each earthquake hazard level are shown in Figure D.l.

In addition to the three Pre-Northridge buildings, Foutch (2000) designed 3-, 9-, and 20-story buildings that follow the requirements of the 1997 NEHRP Provisions (BSSC, 1997) for office buildings located in Los Angeles, California on a stiff soil building site. Since these design requirements were published after the 1994 Northridge earthquake, these buildings are called the Post-Northridge model WSMF buildings in this research. The building footprints, story heights, and gravity load carrying systems of the Post-Northridge buildings are the same as that of the Pre-Northridge buildings. For each building size, the lateral system was designed four times, each utilizing a different steel shape size, W14x, W24x, W30x, or W36x, for the frame columns. In addition, the buildings were each designed using two different levels of base shear. Foutch called these building designs the Lower and Upper Bound building designs. The Lower Bound buildings were designed using the 1997 NEHRP specified base shear at the fundamental natural building period calculated from an analytical model of the building. The Upper Bound buildings were designed using the 1997 NEHRP specified base shear at the building period, calculated using the building period equation specified in the 1997 NEHRP provisions, which generally results in a building period that is less than the building period estimated from an analytical model of the building. Consequently, the Upper Bound buildings were designed using a base shear larger than that used to design the Lower Bound buildings, and therefore results in a stiffer and stronger building design. Note that due to the lower bound base shear requirement in the 1997 NEHRP Provisions, the design base shear for the Lower and Upper Bound designs of the 20-story building is the same. Therefore, no distinction is made in this research between the Lower and Upper Bound designs for the 20-story building. Foutch (2000) provides a complete description of the Post-Northridge building designs and the analytical models of the buildings.

The M2 model of the Post-Northridge buildings were analyzed by the nonlinear dynamic time history analysis method using the 50%, 10%, and 2% in 50 year earthquake ground motion records. The Post-Northridge buildings were analyzed using damping levels different from the Pre-Northridge buildings. The Rayleigh damping coefficients were selected such that there is 4.3%, 3.6%, and 2.3% damping in the fundamental mode for the 3-, 9-, and 20-story buildings, respectively. Following the SAC Method procedure, a smooth median response spectrum was constructed for each earthquake hazard level and each level of damping. The resulting parameters that describe the shape of the median smooth spectra are given in Tables D.2, D.3, and D.4 for values of damping equal to 4.3%, 3.6%, and 2.3%, respectively. The median response spectra and the median smooth spectra for each earthquake hazard level and each level of damping are shown in Figures D.2 through D.4.

Following the SAC Method procedure, the median bias factors for maximum interstory drift demand were calculated for the Coefficient Method (CM) and the Capacity Spectrum Method (CSM). These bias factors are presented in Table D.5 for the

The following observations can be made regarding the median bias factors for maximum

IDR demand calculated using the CM and CSM forms of the nonlinear static analysis method.

• The median CSM bias factors are consistently larger than the median CM bias factors and are always greater than one. This indicates that the CSM generally under-predicts the building response calculated using the nonlinear time history analysis method. In general, the median CSM bias factors are approximately 50% greater than the median CM bias factors.

• The values of the median bias factors are dependent on the level of seismic hazard. As the level of seismic hazard increases, the median bias factors tend to decrease. The dependency on the level of seismic hazard is not as pronounced for the 3-story building as it is for the 9- and 20-story buildings.

• The median bias factors for the Pre-Northridge buildings show that for estimating maximum IDR there is no clear advantage in performing the building analysis using the more detailed M2 analytical model. In fact, in some cases, the median bias factors reduce when the Ml model is used, indicating that the Ml model produces more conservative estimates of the maximum IDR This same observation was made from the statistics of the record-to-record bias factors given in Chapter 4.

• In Chapter 4, the statistics of the record-to-record bias factors indicated that bias factors were typically independent of the load pattern. The median bias factors calculated in this section show that for the 9- and 20-story buildings, the Uniform load pattern results in smaller values of the median bias factors. This indicates that the Uniform load pattern produces conservative estimates of the maximum EDR demand compared to the Modal load pattern for taller buildings. For the 3-story building, the bias factors for the CM show no dependence on the load pattern, while the bias factors for the CSM show that the Modal load pattern results in slightly smaller values.

• The median bias factors for the Post-Northridge buildings are generally independent of the steel shape sizes used for the frame columns in the lateral system. In addition, there is no major difference between the median bias factors calculated for the Lower and Upper bound building designs.

• Little difference is observed between the CM median bias factors calculated for the Pre- and Post-Northridge buildings. For the CSM, Pre-Northridge building median bias factors are larger than the Post-Northridge building bias factors for the 9- and 20-story buildings.

The previously discussed observations about the median bias factors along with the results presented in Tables D.5 through D.8 were used to provide recommended values of the median bias factors for use in the Cornell DCFD methodology. These recommendations are presented in Table D.9 for the nonlinear static analysis method where the target displacements are calculated using the CM and CSM. The recommended median bias factors are provided for lowrise (based on the 3-story building results), midrise (based on the 9-story building results), and highrise (based on the 20-story building results) buildings and are dependent on the level of seismic hazard.

pansd (mc)

Pansd(Mc)

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Figure D. 1 Median 2% damped response spectra and smooth response spectra

Seismic Eurocode

Figure D.2 Median 4.3% damped response spectra and smooth response spectra

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Figure D.3 Median 3.6% damped response spectra and smooth response spectra

Eurocode Site Amplification

Figure D.4 Median 2.3% damped response spectra and smooth response spectra

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