3.1 Nonlinear modeling components
The seismic load resisting systems for designed buildings consisted of a 3-bay perimeter steel moment frame on each side. For incremental dynamic analysis, two-dimensional (2D) models were created in the Open System for Earthquake Engineering Simulation (OpenSees) platform (McKenna et al. 2000). Therefore, only half of the story seismic mass was assigned to the story nodes in the 2D model. The P-\(\varDelta\) effect of gravity frames was also considered through leaning columns. The steel moment frames consist of the elastic beam and column elements with lumped plasticity concentrated at each end. Hence, elasticBeamColumn and zeroLength elements were used to model beam and column elements and plastic hinge rotational springs, respectively. The moment-rotation relationship of the plastic hinge springs was assumed to be deteriorating bilinear hysteretic model that will be explained later. 2% Rayleigh damping was applied to the first mode and the mode with a period of 0.2 s. The schematic view of described nonlinear model is depicted in Fig. 4.
The non-linear behavior of moment frames mainly depends on the nonlinear behavior of plastic hinges. In this study, the Bilin material that simulates the modified Ibarra-Medina-Krawinkler deterioration model with bilinear hysteretic response was used. In general, the hysteretic behavior of Lignos and Krawinkler (2012) springs is defined based on the following three basic rules:
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Backbone (skeleton) curve that defines strength and deformation bounds for an undeteriorated system.
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A set of rules that define the cyclic behavior of a system between the backbone curves which is the bilinear hysteretic model, here.
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A set of rules that define four modes of deterioration including basic strength, post-cap strength, unloading stiffness and accelerated reloading stiffness deteriorations.
According to Fig. 5, the skeleton curve of Bilin material is defined using elastic stiffness (\({K}_{e}\)), effective flexural yield strength (\({M}_{y}\)), peak flexural strength (\({M}_{max}\)), pre-peak plastic rotation (\({\theta }_{p}\)), post-peak plastic rotation (\({\theta }_{pc})\), and residual flexural strength (\({M}_{r}\)). The other model parameter is the deterioration parameter (\(\varLambda\)) which is usually assumed to be the same for all modes of deterioration. The model parameters, including the skeleton curve and deterioration parameters, were generally determined by the empirical equations proposed by Lignos and Krawinkler (2012) obtained from the regression analysis of a large number of test data. However, the revised version of equations from (NIST GCR 2017); was employed in this study. It is necessary to explain that the code of Bilin material, developed by Lignos in the OpenSees platform, is based on the monotonic skeleton curve definition. (NIST GCR 2017); presents recommended skeleton curves for both monotonic and cyclic definitions. Therefore, the monotonic envelope parameters of the report should be considered to employ Bilin from the OpenSees material library.
The applied backbone curve relationships and deterioration relationships are reviewed in the following. In this section, first, the relations for wide-flange beam sections with RBS connections are presented, and then the relations applied for wide-flange column sections are provided.
For beams with RBS, the effective flexural yield strength (\({M}_{y}\)) and the peak flexural strength (\({M}_{max}\)) are calculated from Eqs. (2) and (3), respectively.
$${M}_{y}= \beta {M}_{pe}= \beta {R}_{y}{Z}_{RBS}{F}_{y}$$
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$${M}_{max}=1.1{M}_{y}$$
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where \(\beta =1.1\) is a factor to capture cyclic hardening effects on effective flexural yield strength, \({Z}_{RBS}\) is the plastic modulus of the reduced beam section, \({F}_{y}\) is the nominal yield stress of steel material, \({R}_{y}\) is the ratio of the expected yield stress to the nominal yield stress of that material.
The pre-peak plastic rotation (\({\theta }_{p}\)) and post-peak plastic rotation (\({\theta }_{pc}\)) for beam plastic hinges are computed as follows:
$${\theta }_{p}=0.09{\left(\frac{h}{{t}_{w}}\right)}^{0.3}{\left(\frac{{b}_{f}}{{2t}_{f}}\right)}^{-0.1}{\left(\frac{L}{d}\right)}^{0.1}{\left(\frac{{c}_{unit}d}{533}\right)}^{-0.8}$$
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$${\theta }_{pc}=6.5{\left(\frac{h}{{t}_{w}}\right)}^{-0.5}{\left(\frac{{b}_{f}}{{2t}_{f}}\right)}^{-0.9}$$
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where \(h/{t}_{w}\) and \({b}_{f}/{2t}_{f}\) are the web and flange depth-to-thickness ratios, respectively, \(L/d\) is the shear span-to-depth ratio of the steel beam and \({c}_{unit}=1\) if d is defined in millimeters.
The other two parameters of the skeleton curve are the residual flexural strength (\({M}_{r}\)) and ultimate rotation of the beam sections (\({\theta }_{ult}\)) which are recommended to be assumed \({0.4M}_{y}\) and 0.2 radians, respectively (NIST GCR 2017).
The deterioration parameter (\(\varLambda\)) is extracted from Lignos and Krawinkler (2012) which is also based on the monotonic skeleton curve definition. The deterioration parameter (\(\varLambda\)) for beams with RBS is computed as follows:
$$\varLambda =585{\left(\frac{h}{{t}_{w}}\right)}^{-1.14}{\left(\frac{{b}_{f}}{{2t}_{f}}\right)}^{-0.632}{\left(\frac{{L}_{b}}{{r}_{y}}\right)}^{-0.205}$$
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where \(h/{t}_{w}\) and \({b}_{f}/{2t}_{f}\) are the web and flange depth-to-thickness ratios, \({L}_{b}/{r}_{y}\) is the laterally unbraced length of the steel beam divided by the radius of gyration.
For wide-flange columns, the effective flexural yield strength (\({M}_{y}\)) is calculated as follows:
$${M}_{y}=\left\{\begin{array}{c}1.15Z{R}_{y}{F}_{y}\left(1-{P}_{g}/{P}_{ye}\right) {P}_{g}/{P}_{ye} \le 0.2 \\ 1.15Z{R}_{y}{F}_{y}\left[\frac{9}{8}(1-{P}_{g}/{P}_{ye})\right] {P}_{g}/{P}_{ye}>0.2\end{array}\right.$$
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The factor 1.15 accounts for the effects of cyclic hardening on the hysteretic behavior of wide-flange columns. Z is the plastic modulus of the column sections, \({F}_{y}\) is the nominal yield stress of steel material, \({R}_{y}\) is the yield ratio of the expected yield stress to nominal yield stress for the respective steel material, and (\({P}_{G}/{P}_{ye}\) (is the ratio of the gravity axial load to the expected axial yield strength of the column.
The peak flexural strength (\({M}_{max}\)) is also computed by Eqs. (8) and (9), where \(a\) is a coefficient that defines the hardening ratio ( \({M}_{Max}/{M}_{y}\)).
$$a=12.5 {\left(\frac{h}{{t}_{w}}\right)}^{-0.2}{\left(\frac{{L}_{b}}{{r}_{y}}\right)}^{-0.4}{\left(1-\frac{{P}_{g}}{{P}_{ye}}\right)}^{0.4}\ge 1$$
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The pre-peak plastic rotation (\({\theta }_{p}\)) and post-peak plastic rotation (\({\theta }_{pc}\)) for wide-flange columns are calculated from the following equations.
$${\theta }_{p}=294 {\left(\frac{h}{{t}_{w}}\right)}^{-1.7}{\left(\frac{{L}_{b}}{{r}_{y}}\right)}^{-0.7}{\left(1-\frac{{P}_{g}}{{P}_{ye}}\right)}^{1.6}\le 0.2$$
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$${\theta }_{pc}=90 {\left(\frac{h}{{t}_{w}}\right)}^{-0.8}{\left(\frac{{L}_{b}}{{r}_{y}}\right)}^{-0.8}{\left(1-\frac{{P}_{g}}{{P}_{ye}}\right)}^{2.5}\le 0.3$$
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The residual flexural strength (\({M}_{r}\)) and ultimate rotation (\({\theta }_{ult}\)) of the wide-flange columns is recommended to be considered \((0.5-0.4 {P}_{g}/{P}_{ye}){M}_{y}\) and 0.15 radians, respectively (NIST GCR 2017).
(NIST GCR 2017) did not report any relationship for the deterioration parameter (\({\Lambda }\)) of the wide-flange columns. Therefore, the deterioration parameter was extracted from Lignos et al. (2019) which is based on the tests of wide-flange steel columns. The empirical relation calibrated to compute the reference energy dissipation capacity is presented in Eq. (12).
$$\varLambda =\left\{\begin{array}{c}25000{\left(\frac{h}{{t}_{w}}\right)}^{-2.14}{\left(\frac{{L}_{b}}{{r}_{y}}\right)}^{-0.53}{\left(1-\frac{{P}_{g}}{{P}_{ye}}\right)}^{4.92}\le 0.3 \frac{{P}_{g}}{{P}_{ye}} \le 0.35 \\ 268000{\left(\frac{h}{{t}_{w}}\right)}^{-2.30}{\left(\frac{{L}_{b}}{{r}_{y}}\right)}^{-1.3}{\left(1-\frac{{P}_{g}}{{P}_{ye}}\right)}^{1.19}\le 0.3 \frac{{P}_{g}}{{P}_{ye}}>0.35\end{array} \right.$$
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3.2 Simplified Frames
In recent years, using simplified models has received more attention due to the requirements of mass dynamic analysis in the field of earthquake engineering. Among different simplified models, the discrete shear building model and the continuous shear beam model seem to be the oldest models for the simulation of the moment frame systems.
Although the shear models are considered to be appropriate for the simulation of shear-type structures, there are some criticisms of the models. First of all, the shear models lead to significant errors in estimating the seismic response, when the flexural deformation is not ignorable. Second, most of the elements of the stiffness matrix for the shear building are equal to zero which is not proportional to the stiffness matrix of the moment frame. In other words, springs in the shear building only connect the lateral degrees of freedom at adjacent stories. In the end, the shear model only predicts the global displacements of the building (e.g. story drifts) that should be converted into the local deformations (e.g. plastic hinge rotations) and this is another source of error.
Nevertheless, due to its simplicity, the shear models are yet one of the most popular models among many researchers. For example, in recent years, Joyner and Sasani (2020) applied the shear beam model to demonstrate the influence of design variables on building seismic performance and enhancing community resilience. It is noteworthy that their study entails over 60,000 nonlinear time history analyses. In addition, an updated shear-type model, named “Stick-IT” model, has been recently presented and applied to predict seismic demands and economic losses (e.g. d'Aragona et al. 2020, 2021, 2022).
The second category of simplified models for moment frame systems is simplified frames. For example, some researchers used an unnamed single-bay frame as a representative of an n-bay frame to perform mass nonlinear time history analysis (e.g. Lignos et al. 2015; Park and Medina 2007). However, the classic model for simplified frames, presented on the basis of structural analysis, is the “Fish-Bone” family model. The Japanese “Fish-Bone” model can be claimed to be the oldest simplified frame that engineers and researchers have used for nonlinear time history analysis. The ancient Fish-Bone model has been used for several decades by Japanese researchers. But it was not until 2002 that it was introduced with a slight modification to English-language journals and named as “Generic Frame” model (Nakashima et al. 2002). Khaloo and Khosravi (2013) proposed the “Modified Fish-Bone” model to consider the flexural deformation of moment frames and the correct nonlinear behavior of beam plastic hinges. Soleimani et al. (2019) illustrated that the Modified Fish-Bone model confronts challenges in the RC moment frames due to the different positive and negative moment capacities of the beams, and the peak-oriented hysteretic model of the beam plastic hinges. To overcome these problems, they proposed a one-bay frame model called the “Substitute Frame” model. In the following, simplified frame models were also developed for irregular structures (Haghighat and Sharifi 2018; Hosseini et al 2021; Soleimani and Hamidi 2021). In recent years, simplified frames have been widely used in different fields of earthquake and structural engineering. For example, they can be employed to perform IDA and fragility analysis, assess the collapse capacity of structures, and predict seismic damage and economic loss. In addition, they are useful for extensive parametric analysis and optimization analysis in which a large number of nonlinear time history analyses are required. For steel moment frames, both Substitute Frame and Modified Fish-Bone models predict the structural response with almost the same accuracy. Therefore, the “Substitute Frame” model is used in this study which is more convenient for modeling. The general scheme of the Substitute Frame is given in Fig. 4. It also demonstrates how to calculate the Substitute Frame parameters. To know the accuracy and speed of the Substitute Frame model, refer to Soleimani et al. (2022).