Combination rules for steel buildings under seismic loading: MDOF vs SDOF systems

2.1. Structural models

The 3- and 9- story buildings considered in the SAC steel project [13], are used in this study to address the issues raised earlier. They will be denoted hereafter as Models 1 and 2, respectively. The beams and columns sections are given in Table 1. Additional information for the models can be obtained from the report [13]. In this study, the steel buildings are modeled as complex MDOF systems. The accuracy of the rules is also studied for equivalent SDOF systems. One equivalent SDOF model is considered for each steel building. They will be denoted hereafter as Model 1E and Model 2E. The natural period, damping ratio and the yielding strength are selected to be the same for the SAC and the equivalent SDOF models.

2.2. Earthquake loading

To study the responses of the models comprehensively and to make meaningful conclusions, they are excited by twenty recorded earthquake motions in time domain with different frequency contents, recorded at the following stations: Fun Valley, Reservoir 361; Convict Creek; Cerro Prieto; Parkfield, Joaquin Canyon; Olympia Hwy Test Lab; Utilities Bldg, Long Beach; El centro, California; Centerville Beach, Naval Facility; Gilroy Array Sta No 4; Olympia Hwy Test Lab; Castaic-Old Ridge Route; Long Valley Dam; El Centro-Imp Vall Dist; Palo Alto; UCSB Goleta FF; Parkfield Fault Zone 14; Chihuahua; Canoga Park, Santa Susana; Ferndale, California; Indio, Jackson Road. The predominant periods of the earthquakes vary from 0.11 to 0.66 sec. The earthquake time histories were obtained from the Data Sets of the National Strong Motion Program (NSMP) of the United States Geological Surveys (USGS).

2.3. Loading cases

In order to meet the objectives of the study, several seismic and harmonic load cases need to be considered. Recorded horizontal time histories will be denoted as normal components. When they are transformed to uncorrelated components following the procedure suggested by Penzien and Watabe (1975) they will be denoted as principal components. The symbols $X_n$ and $Y_n$ will indicate that the structures are excited by the normal components, and $X_p$ and $Y_p$ will indicate that the principal components are used instead. Hence, the notation ($X_n$, $Y_n$) indicates that the structure is excited by the first and second normal components applied simultaneously to the N-S and E-W directions of the structure, respectively. Similarly, the notation (0, $X_p$) indicates that the structure is excited by only the first principal component acting along the E-W direction. The following particular load cases are considered:

Case 1: ($X_n$, $Y_n$); Case 2: ($Y_n$, $X_n$); Case 3a: ($X_n$, 0); Case 3b: (0, $Y_n$); Case 4a: ($Y_n$, 0) and Case 4b: (0, $X_n$). Similarly, another four cases of analysis are considered when the principal components are applied, they are Case 5: ($X_p$, $Y_p$); Case 6: ($Y_p$, $X_p$); Case 7a: ($X_p$, 0); Case 7b: (0, $Y_p$); Case 8a: ($Y_p$, 0) and Case 8b: (0, $X_p$). Thus, for two structures, twenty earthquakes, eight cases, and considering the responses to be elastic and inelastic, a total of 640 analyses of complex MDOF structures under seismic loading are required. For any response parameter (axial loads or base shear), the reference response for normal components, denoted hereafter as $R_n$, is considered to be the maximum response of Cases 1 and 2. Similarly, the reference response for the principal components, $R_p$, is considered to be the maximum response of Cases 5 and 6.

3.1. Accuracy of the rules

According to loading Case 3 and the 30 % combination rule, two possible combined responses can be calculated for normal components; they are ${{}_{}{}^{3}X}_n+{0.3}^3{Y}_n$ and ${0.3}^3{X}_n{+}^3{Y}_n$, where ${{}_{}{}^{3}X}_n$ and ${{}_{}{}^{3}Y}_n$ are defined as the responses produced for Cases 3a and 3b, respectively. The larger of the two combined responses when normalized with respect to the reference response ($R_n$) defined earlier will give a random variable defined as ${{}_{}{}^{3}R}_{n,30}$. Following exactly the same procedure for Load Case 4, ${{}_{}{}^{4}R}_{n30}$ can be calculated. The combination of both cases (${{}_{}{}^{3}R}_{n,30}$ and ${{}_{}{}^{4}R}_{n,30}$) and 20 earthquakes give a total of 40 sample points, which will be denoted by the random variable $R_{n,30}$. Considering principal components and excitations given by load Cases 7 and 8, 40 sample points are similarly generated, it will be denoted as the random variable $R_{p,30}$. Plots for the $R_{n,30}$ and $R_{p,30}$ parameters for each earthquake are developed, but are not shown, only the fundamental statistics are given below. The statistics of $R_{n,30}$ and $R_{p,30}$ are summarized in Columns 3 through 6 of Table 2. The results clearly indicate that, for the case of axial load, on an average basis, the 30 % combination rule underestimates the combined axial load by about 10 % and that the uncertainty associated with the estimation is too large in some cases. However, for base shear, unlike the case of axial load, both rules reasonably overestimate the combined response, the overestimation is about 10 %. The uncertainty in the estimation is much larger for axial load than for base shear. The observations made for the normal components are essentially identical to that of principal components. It should be noted that interior and exterior columns oriented in the north and south direction of the perimeter frames were considered.

Normalized response parameter similar to those of the 30 % rule are also estimated the SRSS rule. The corresponding random variables are denoted as $R_{n,SRSS}$ and $R_{p,SRSS}$ for normal and principal components, respectively. The statistics are summarized in columns 7 through 10 of Table 2. The major observations made for the 30 % rule for axial loads and base shear also apply to this rule. These results indicate that for complex MDOF systems, there is a certain degree of correlation between the effects of individual components of earthquakes, even for the case of uncorrelated components. Statistic similar to those of elastic behavior are also developed for inelastic behavior but are not shown. All the observation made for elastic behavior essentially remain the same for inelastic behavior. The only additional observation is that the uncertainty in the prediction significantly increases for axial load.

3.2. Correlation between individual effects

The basic assumption of the SRSS rule is that there is no correlation between the horizontal components. It is implicitly assumed that if there is no correlation between the accelerograms, the corresponding effects will also be uncorrelated. The degree of correlation between the individual effects of the horizontal components and the effect of correlation on the accuracy of the rules are now discussed. The correlation coefficients ($\rho$) are estimated for Models 1 and 2, for normal and principal components, for elastic and inelastic behavior and for collinear (axial load) and non-collinear (base shear) response parameters. However, the results are not given, they are only conceptually discussed. It is found that normally recorded components may be highly correlated and that the ρ valuessignificantly vary from one earthquake to another and from one element to another. Most of the values can be considered negligible (smaller than 0.25). For many cases however, the correlation is significant. Values of $\rho$ larger than 0.5 are observed in many cases. It is observed that the rules are not always inaccurate in the estimation of the combined response for large values of $\rho$. On the other hand, small values of the coefficients are not always related to an accurate estimation of combined response. The implication of this is that there may be other factors that influence the accuracy of the combination rules.