Study of Precision for Structural Responses of Buildings using Advanced Scalar Intensity Measures

1. Earthquake intensity measures (IMs) are used in assessing the seismic performance of structures. They can also be considered as scale factors for recorded ground motions in incremental dynamic analysis. “Sufficiency” and “efficiency”, that will be estimated using linear regression analyses, are two criteria for examining the concluded accuracy of using IMs in the structural performance assessment. In this paper, two advanced scalar IMs, including inelastic spectral displacement (Sdi) and inelastic spectral displacement with a higher-mode factor (IM1I&2E), are investigated. This research is concentrated on the evaluation of efficiency and sufficiency of advanced IMs for near-fault ground motions. For this purpose, by conducting nonlinear time history analyses, maximum inter-story drift ratio () as main structural response is estimated. After estimating IMs, for the assessment of their efficiency and sufficiency with respect to M, R and or, via linear regression analysis, standard deviation of residuals and a parameter called “p-value” are used respectively. Here, M, R and or are moment magnitude, closest distance to the rupture and directivity parameters, respectively. 2. Methodology2.1. Building models and earthquake records: Under the hypothesis of a rigid diaphragm, a 2-D centerline model of each of the ten symmetric and regular reinforced concrete and steel moment resisting framed and dual system frame building models with 3, 9 and 20 stories, which have been designed with ETABS, is created for nonlinear analysis using SAP2000. Each model consists of two exterior and interior frames which are connected together with a rigid link at every floor (as an example, see Fig. 1). For reinforced concrete buildings, the shear walls are modeled as a column at the middle of the corresponding span. Also, in the members of reinforced concrete building models, longitudinal and confinement bars are modeled. Flexural hinges (M3), interactional axial-flexural hinges (P-M3) and axial hinges (P) are assigned at the ends of beams, columns and braces, respectively. The nonlinear analyses take into account P- effects. The hysteretic models that are considered for hinges at the ends of members constitute stiffness and strength deterioration. Mass and stiffness proportional (Rayleigh) damping are assumed, with a damping ratio of 5% specified at the fundamental period (T1) and at a period with modal participating mass ratio more than 90%.In this research, a suite of 31 near-fault ground motion records, which have forward-directivity effects and have been recorded on “stiff soil” or “very dense soil and soft rock”, are selected from the PEER strong motion database [1]. The ranges of moment magnitude (Mw) and closest distance to the rupture (Rclose) of these records are from 6.53 to 6.93 and from 0.07 to 21.68 km, respectively.2.2. Methods and theoryIn this research, the strike-normal component of the ground motion records are considered. For this purpose, by using the values of the strike of the faults, each near-fault ground motion records are rotated to the strike-normal direction using MATLAB.Following the idea and notation of Luco and Cornell [1, 2], definition of IM1I&2E can be expressed as in which refers to the nth mode participating factor for the story in which the first two mode square root of the sum of the squares (SRSS) estimate of occurs. Since yield displacement dy is required for estimating IM1I&2E and Sdi, therefore, first-mode nonlinear static analyses (pushover) are conducted for every model and by means of base shear-roof displacement diagrams, the values of dy are calculated. As expressed in Eq. (1), values of Sd(T1,), Sd(T2,), Sdi(T1,, dy) and modal participation factors are required for estimating IM1I&2E and Sdi. The values of linear and nonlinear spectral displacements for a single-degree-of-freedom model with the values of T1, T2,, and dy are obtained using NONLIN.To estimate for every building model, nonlinear time history analyses are conducted by SAP2000. After performing nonlinear time history analyses and calculating IMs, for the assessment of sufficiency with respect to M, R and or, efficiency and bias of each IM for every building model, linear regression analyses using MATLAB are carried out. The efficiency of IM is measured by the degree of dispersion about the regression fit, and the sufficiency of IM is measured by the amount of statistical independency of the residuals, obtained from the regression of structural demand measure (DM) on IM, regarding M, R and or. In this research, for the evaluation of the efficiency and sufficiency of each IM, a one-parameter linear natural logarithmic regression is utilized. According to the attenuation relationships and equations which have been proposed by references [3] and [4], the employed regression model can be expresses as Where and c are the regression coefficients to be estimated and is the random error in given IM and M, R and or. The standard deviation of, indicated as is a criterion for assessing the efficiency of IM. The use of linear natural logarithmic regression is based on the roughly log-normal distribution of and invariable. If the coefficient c on M, R and or are not statistically significant, which can be computed by the equivalent p-value, then the given IM can be expressed as sufficient. The p-value is defined as the probability of finding an estimate of c at least as large (in absolute value) as that observed if, in fact, the true value of c is 0 [5]. Consequently, a p-value approximately less than 0.05 implies that the calculated coefficient c is statistically significant, and for this reason, IM is insufficient.3. 1) For 3-story reinforced concrete and steel buildings with dual system and moment resisting frames, both of the IMs are sufficient regarding M, R and or. However, due to the high standard deviation, they are not efficient. It’s due to that these buildings are low-rise then higher-mode effects are negligible and their behavior under the considered records are highly nonlinear; so, both of the IMs have acceptable results.2) For 9-story reinforced concrete buildings with dual system frame, despite Sdi has admissible sufficiency with respect to the considered 3 parameters, but it has high bias and standard deviation. The dispersion of the points around the line in Fig. 2 confirms the mentioned result. In addition to sufficiency with respect to M, R and or, IM1I&2E is efficient and it is the best choice for this kind of buildings. Also, it is noticed that the slope of the regression line is negative (as an example, see Fig. 4) and the dispersion of the points around the line for both of the IMs is the same and the value of the error for all of the considered parameters is in a small interval which respectively, show that c has a negative value and p-value is satisfying. However, for 9-story steel buildings with dual system frame, Sdi, despite sufficiency with respect to the 3 parameters, has relatively high bias and standard deviation. Also, IM1I&2E has high bias and standard deviation and does not supply sufficiency with respect to and. Then, because of high nonlinearity of this building under near-fault ground motions, applying Sdi is more appropriate than IM1I&2E.3) Due to high nonlinearity of the building under the records, for 9-story reinforced concrete and steel buildings with moment resisting frame, despite high bias, employing Sdi is more appropriate than IM1I&2E which does not supply sufficiency with respect to R.4) For 20-story reinforced concrete and steel buildings with moment resisting frames, Sdi has high bias and insufficiency with respect to R; but IM1I&2E supply both the efficiency and sufficiency. Then, the use of IM1I&2E for this kind of structures is more appropriate and the reason is low-nonlinearity and higher modes contribution in building’s behavior.4. Based on the results of the current research work, the following conclusions are drawn:1) Both of the IMs supply the sufficiency with respect to M.2) For supplying the sufficiency of with respect to the directivity parameters (or), Sdi is proper.3) For supplying the sufficiency of with respect to R, Sdi is proper.4) For providing less bias and standard deviation for, IM1I&2E is appropriate.5) Eventually, it can be concluded that Sdi is more appropriate than IM1I&2E; however, for increasing the accuracy and supplying efficiency and sufficiency for each case, this IM must be considered with other parameters (e.g. spectral shape parameters) like vector IMs.