Model Details for Nonlinear Dynamic Analysis

In the present work, OpenSees (2016) software is utilized to develop nonlinear dynamic models and to preform nonlinear time series analysis for the buildings without dampers and with FVDs. The columns of the frames are modeled using the nonlinear force-based fiber elements that contains distributed plasticity to account for the moment-axial force interaction effect. According to the study of Newell an Uang (2006), deep columns with low- slenderness flanges and webs do not buckle and experience cyclic deterioration under large drifts. Hence, the heavy columns of the designed buildings in this study are not expected to undergo local bucking and the cyclic strength and the stiffness deterioration are therefore not considered for modeling the columns in the frames. The fibers of the column elements are assumed to experience bilinear elastoplastic stress-strain behavior while the ‘Steel01 Material’ in OpenSees (2016) is used to define the fiber element with a 0.002 strain-hardening ratio.

The beams of frames are modeled as elastic elements while two zero-length plastic flexural hinges are located at both ends of the beams. Based on the rules decribed by the Modified Ibarra-Krawinkler Deterioration Model (Lignos and Krawinkler 2011, Lignos et al. 2011), analytical rotational springs which exhibit bilinear hysteretic behavior are used to represent these zero length plastic hinges in the beams. This phenomenological model of can be described by a monotonic backbone curve with a defined reference boundary in terms of the hysteretic behavior undergone by the rotational springs. A set of rules regarding the nonlinear behavior within the strength and the deformation bounds of the springs are established by this deterioration model. A bilinear hysteretic response can be specifically characterized by three cyclic deterioration modes that are the basic yield strength deterioration,

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the post-capping strength deterioration, and the unloading/reloading stiffness deterioration. As illustrated in Figure 6.6, the backbone curve regarding the three modes in the Modified Ibarra-Krawinkler Deterioration Model is defined by three strength parameters and four deformation parameters which are:

𝑀_𝑦 = effective yield strength; 𝑀𝑐 = capping strength;

𝑀𝑟 = residual strength = 𝜅 ∙ 𝑀𝑦 (𝜅 is residual strength ratio);

𝜃_𝑦 = yield rotation;

𝜃_𝑝 = pre-capping plastic rotation; 𝜃𝑝𝑐 = post-capping plastic rotation;

𝜃_𝑢 = ultimate rotation capacity;

A detailed description for determining the parameters of the modified IK model or constructing the ‘Bilin Material’ in the OpenSees (2016) software refers to the study of Lignos and Krawinkler (2011).

Figure 6.6 The illustration of Modified IK Deterioration Model provided by Lignos et al. (2011)

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For modeling the panel zones of the MRFs, Krawinkler model (Krawinkler 1978) are used to simulate the nonlinear behavior of beam-column joints. As presented in Figure 6.7, this phenomenological model is described by four connected rigid links with four compound nodes at the corners and four single nodes at the middle. The stiffness and strength of the panel zone web are simulated by a rotational spring placed at the compound node located at the upper left corner. The column flange bending resistance is represented by an analytical rotational spring placed at the compound node located at the lower right corner. The compound nodes located at the upper right and lower left corner are considered as true flexural hinges and they are set to have no stiffness. It is summed up to be twelve nodes to represent a single Krawinkler model. Each corner of the model utilizes two nodes (equal to one compound node) to constrain x-y and rotational degrees of freedom. A detailed description for the numerical equations determined the required properties of the panel zone refers to the study of Krawinkler (1978).

Figure 6.7 The illustration of Krawinkler model

As discussed in Section 6.1.5, the linear FVDs defined in this study can be modeled as simple linear viscous dashpots using ‘zerolength’ element in OpenSees (2016) software. Additionally, the damper limit states, potentially

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occur when the piston of the damper reaches its stroke limit due to the seismic response, are not considered for the nonlinear model established for collapse simulation. This assumption is identified to be very important for evaluating the collapse performance of the frames with FVDs (Miyamoto et al. 2010). It should also be noted that the stroke limit of typical dampers is around ± 100mm while the strokes of FVDs can be extensible to ±900mm based on customer request in the market (Taylor Devices 2017). Hence, with an extended stroke limit, the FVDs presented in this work do reach its limit states even the buildings undergo a huge drift under the collapse state.

As mentioned in Section 6.1.5, stiff sections of chevron braces are designed to support the dampers and it is assumed that the braces are strong enough to resist the maximum damper forces and avoid buckling. Therefore, the diagonal braces are modeled with elastic truss elements (OpenSees 2016) with confidence.

Elastic beam column elements are used to the model the ‘lean-on’ column with assigned seismic storey mass in each floor. Considering the diaphragm effect, truss elements (OpenSees 2016) are utilized to constrain the x direction displacement of the nodes in the beams to the node in the ‘lean-on’ column at the same floor level.

The Rayleigh damping for the MRFs is defined with 3% damping ratio at first mode and second mode to account for the designed 3% inherent critical damping (Chopra 1995). The Newton-Raphson algorithm set with tangent stiffness and the Newmark method defined with constant accelerations are utilized to solve the numerical dynamic equations while calculating the seismic response of the MRF model.

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6.3 Ground Motions for Nonlinear Dynamic Analysis