Seismic performance of steel MRF building with nonlinear viscous dampers

Abstract

This paper presents an experimental study of the seismic response of a 0.6-scale three-story seismic-resistant building structure consisting of a moment resisting frame (MRF) with reduced beam sections (RBS), and a frame with nonlinear viscous dampers and associated bracing (called the DBF). The emphasis is on assessing the seismic performance for the design basis earthquake (DBE) and maximum considered earthquake (MCE). Three MRF designs were studied, with the MRF designed for 100%, 75%, and 60%, respectively, of the required base shear design strength determined according to ASCE 7-10. The DBF with nonlinear viscous dampers was designed to control the lateral drift demands. Earthquake simulations using ensembles of DBE and MCE ground motions were conducted using the real-time hybrid simulation method. The results show the drift demand and damage that occurs in the MRF under seismic loading. Overall, the results show that a high level of seismic performance can be achieved under DBE and MCE ground motions, even for a building structure designed for as little as 60% of the base shear design strength required by ASCE 7-10 for a structure without dampers.

Graphical abstract

Keywords

seismic response / steel MRF / nonlinear viscous damper / design basis earthquake / real-time hybrid simulation

1 Introduction

Current seismic design provisions documents, such as those in ASCE/SEI 41-06 [1], FEMA P-750 [2], and ASCE/SEI 7-10 [3], include provisions for the use of passive damping devices. The provisions enable engineers to specify performance objectives that are comparable to or better than the objectives for a building with a conventional seismic lateral-force-resisting system. ASCE 7-10 requires building structures with passive damping devices to have a seismic lateral-force-resisting system (SLFRS) with a complete load path, and this SLFRS must be designed for a minimum of 75% of the base shear design strength required for a similar system without damping devices.

Past research related to the experimental study presented in this paper includes: the seismic response of structures with viscous dampers [4–6]; the design and analysis of structures with viscous dampers (e.g., Refs. [7,8]); and the optimal design of viscous dampers for structural systems (e.g., Refs. [9–12]). Additionally, the seismic response of structures with nonlinear viscous dampers has also been studied by others (e.g., Refs. [13–16]). Most of this previous work was analytical or numerical; there have been few large-scale experimental investigations on the seismic performance of structures with nonlinear viscous dampers.

This paper presents an experimental study of the seismic performance of a large-scale multistory steel frame building structure with nonlinear viscous dampers, using real-time hybrid simulation (RTHS). The test structures include a three-story moment resisting frame (MRF), a three-story frame with one nonlinear viscous damper and associated bracing in each story (i.e., the damped-braced frame (DBF)), and the gravity load system with the associated seismic mass and gravity loads. Ensembles of ground motions that represent the design basis earthquake (DBE) and maximum considered earthquake (MCE), respectively, were used as input for the RTHS. This paper presents and assesses the performance of the test structures subjected to these ground motion ensembles. The RTHS were performed at the Lehigh NEES (George E. Brown, Jr. Network for Earthquake Engineering Simulation) Real-Time Multidirectional (RTMD) Earthquake Simulation Facility.

2 Prototype building and test structure

2.1 Prototype building

The prototype building is shown in Fig. 1 and is a three-story, six-bay by six-bay office building assumed to be located on a stiff soil site in Pomona, California. Figure 1(a) shows the plan of the building, which has a symmetric layout. The structural system of the building consists of a primary seismic lateral-force-resisting system (SLFRS), a damping system, and a gravity load system. The primary SLFRS includes eight identical single-bay perimeter moment-resisting frames (MRFs); the damping system includes eight single-bay frames with nonlinear viscous dampers and associated bracing (DBFs); and the gravity load frames are uniformly distributed in the plan. Each MRF has an associated DBF; the MRF and DBF are assumed to work in parallel in one horizontal direction through the action of the floor diaphragm. The seismic area tributary to one MRF and one DBF is one-quarter of the total area of each floor of the building. Figure 1(b) shows section views of the building in the east–west direction. The horizontal movement of the building is restrained at the ground level, and the columns of the MRF and DBF are fixed at their base in the basement.

The seismic design of the prototype building was based on the seismic provisions of ASCE 7-10 and criteria from AISC [17–19]. The MRFs were designed to satisfy the strength criterion of ASCE 7-10, but they do not satisfy the drift criterion of ASCE 7-10. The DBFs with nonlinear viscous dampers were added to control the story drifts. Each MRF was designed for the required base shear design strength determined according to the equivalent lateral force (ELF) procedure in ASCE 7-10 with R = 8, S_DS = 1.0g, S_D1 = 0.6g, and a design period T_des = 0.71 s. The required base shear design strength for one MRF with the accidental torsion effect included was calculated as 11.5% of the effective seismic weight of one-quarter of the building which is tributary to one MRF. The members of the structure are sized to resist this design base shear. The resulting MRF provides 100% of the base shear design strength required by ASCE 7-10, and is referred as D100V in this paper.

To study the effect of designing the MRFs with a base shear design strength less than D100V, but using added damping to control the story drifts, reduced strength designs of the MRFs were developed. The resulting MRFs are referred to as D75V and D60V, where D75V and D60V were designed to resist 75% and 60% of the base shear design demand from the ELF procedure, respectively. Since ASCE 7-10 requires the base shear design strength of the SLRFS to be at least 75% of the base shear design demand from the ELF procedure, D60V is an MRF design which does not satisfy ASCE 7-10. To implement the D100V, the D75V, and the D60V designs in a simple and consistent way in the laboratory, all three structures have the same members (and therefore the same base shear strength), but the tributary seismic weight is increased by 1/3 and 2/3 for the D75V and D60V structures, respectively. The increased tributary seismic weight, 1/3 and 2/3 more than the seismic weight for the D100V structure, respectively, was provided by increasing the plan area of the prototype building structure accordingly (as shown in Fig. 2). The design base shear coefficients for the D75V and D60V structures are the same as for the D100V structure according to ASCE 7-10, so the increased seismic weights for these structures cause the base shear design demand from the ELF procedure for the D75V and D60V structures to be 1/3 and 2/3 more than for the D100V structure, respectively.

2.2 Test structure

The test structure used for the experimental study in this paper, shown in Fig. 3, is scaled from the prototype building using a scale factor l= 0.6 to accommodate laboratory conditions. Due to the symmetry of the prototype building, as noted previously, the test structure consists of a single-bay MRF, an associated single-bay DBF, the gravity load system and associated seismic mass that is tributary to the MRF and DBF. To establish similitude between the prototype building and the test structure, maintaining equal stress and acceleration, time is scaled by \sqrt{\lambda }, and the mass and gravity loads of the building are scaled by {\lambda }^{\mathrm{2}}. The seismic weights (W_s) for the D100V, D75V, and D60V test structures are 2702, 3603, and 4504 kN, respectively. The test structure member sizes were determined by scaling down the member section properties of the prototype building, and then adjusting them as needed to use available steel sections that closely meet the design criteria. In the test structure, only 2/3 of the true height of the basement column of the prototype building is included, and the column is pinned at an assumed inflection point at 1/3 of the true column height from the bottom of the basement.

2.3 Design of reduced beam sections (RBS) of MRF

Reduced beam section (RBS) beam-to-column connections were used in the MRF. The RBS limit the moment that can develop at the face of the column by reducing the moment capacity within the reduced beam section away from the column. Thus, the RBS reduce the possibility of fracturing the beam-to-column CJP welds. The RBS were designed according to AISC [19]. The location and dimensions of the RBS were determined so that 85 to 100 percent of the beam plastic moment (M_pe) is allowed to develop at the face of column (M_f), where M_pe is the beam section plastic moment based on the expected yield stress and M_f is the moment at the face of column. The designs of the RBS are shown in Fig. 4 and summarized in Table 1. With these designs, M_f / M_pe is 88%, 917%, and 91% for the 1st, 2nd, and 3rd floor, respectively. The lateral strength of the MRF is reduced by about 16% by using the RBS.

2.4 Nonlinear viscous dampers and design of DBF

Nonlinear viscous dampers were used in the test structure, with one damper in each story. Each damper has a nominal force capacity and stroke capacity of 600 kN and 125 mm, respectively. Tests [20] showed the dampers have a damping coefficient of C_a = 696 kN-s/m and velocity exponent a = 0.44. A pinned beam splice using two bolted structural tees, shown in Figs. 3 and 5, was used in the DBF between the beam and the beam stub to transfer axial and shear forces but to minimize the moment that develops in the beam and column in the beam-to-column connection region. Minimizing this moment minimizes the frame action of the DBF. The design details of the brace-beam-to-column connection of the DBF are given in Dong [20]. The designs of the test structures are summarized in Table 2. Table 3 gives design predictions for the first mode natural frequencies of the test structures, and the DBE story drift ratios. The story drift ratios were calculated using the lateral forces from the ELF procedure acting on linear elastic models of the D100V, D75V, and D60V test structures, including the MRF, gravity load system, and DBF without the dampers, and amplified (using Cd = 5.5, from ASCE 7-10) to account for inelastic response. The story drift is defined as the difference in the floor displacements of adjacent floors and the story drift ratio is calculated as the story drift divided by the story height. The maximum story drift ratios for the D75V and D60V structures without the dampers are greater than the allowable story drift ratio of 2.5% rad. for structures in risk category I or II in ASCE 7-10 and the 2.5% rad limit for the “Life Safety” performance level in ASCE 41-06.

2.5 Design demand predictions for test structure with dampers

The damping added to the test structure by the nonlinear viscous dampers was estimated using an equivalent viscous damping ratio (x_e) as follows [21]:

(1){\xi }_e={\displaystyle \frac{\mathrm{1}}{\mathrm{4}\pi }}{\displaystyle \frac{{\displaystyle \sum D{E}_i}}{{\displaystyle \sum S{E}_i}}},

where SE_i is the maximum strain energy per cycle of harmonic response in story i; DE_i is the energy dissipated by the damper per cycle of harmonic response in story i. {\displaystyle \sum S{E}_i} can be calculated using the lateral force energy (LFE) method [22], as {\displaystyle \sum S{E}_i}=u^T{k}_{t}u, where k_t is the stiffness matrix of the structure, and u is the displacement vector under a pattern of lateral force. {\displaystyle \sum D{E}_i} was calculated as the summation of the area of the damper force-deformation hysteresis loop in each story under the imposed displacements u.

An iterative approach [20] was employed for predicting the equivalent damping ratio and story drift ratio demand of the D100V, D75V, and D60V test structures with the nonlinear viscous dampers: Step (1) calculate the floor displacement response vector, u, using lateral forces from the ELF procedure (with R = 8) acting on a linear elastic model of the MRF and DBF without dampers and using C_d = 5.5 to amplify the displacements to account for inelastic response; Step (2) calculate x_e using Eq. (1) in which the floor displacements from u (amplified to account for inelastic response) represent the predicted deformation amplitude under the DBE; Step (3) determine the damping reduction factor, B₁, according to ASCE 7-10 as a function of the total damping ratio, x_t, which equals the sum of x_e plus the inherent damping ratio of the building (which represents other energy dissipation within the building during low-amplitude dynamic response); Step (4) calculate the floor displacement response of the structure using a linear elastic model of the MRF and DBF without dampers, using C_d = 5.5 to amplify the displacements to account for inelastic response, and using the B₁ factor to account for the added damping; Step (5) iterate Step (1) through Step (4) with an updated floor displacement response until the floor displacement response and equivalent damping ratio converge. Using this procedure, the equivalent damping ratio x_e for the dampers was calculated and added to the inherent damping for the building (which represents energy dissipation within the building during low-amplitude dynamic response and was assumed to be 2%), to determine the DBE and MCE drift demands. The predicted x_e provided by the dampers and story drift design demands for the test structures under the DBE and MCE are shown in Tables 4 and 5. The D60V structure has a smaller x_e and greater story drift ratio demands than the D100V and D75V structures. The maximum story drifts of the structures under the DBE and MCE are less than 2.5%, and therefore, satisfy the “Life Safety” performance level specified in ASCE 41-06, and the drift control provisions of ASCE 7-10.

3 Experimental program

3.1 Test methodology

Using the real-time hybrid simulation (RTHS) method, earthquake simulations with DBE-level ground motions were conducted on the test structure. In an RTHS, the complete structural system is divided into experimental (physical) and analytical (numerical) substructures. The experimental substructure is constructed and subjected to loading in the laboratory, while the analytical substructure is created by developing a numerical model of this part of the structure using computer-based modeling software. The coupling between the experimental and analytical substructures is achieved by maintaining compatibility and equilibrium at the interface between the substructures. Figure 6 shows the substructures used in the RTHS. More details of these RTHS and an assessment of the accuracy of these RTHS can be found in Dong et al. [23].

3.2 Substructures and test setup

The analytical substructure, shown in Fig. 6(a), includes the single-bay MRF and the gravity load system which is represented by a lean-on column with the gravity loads and the seismic mass at each floor level. The cross section properties of the lean-on column are equivalent to a summation of the cross section properties of the gravity columns in the seismic tributary area for the test structure. The numerical model for the analytical substructure has a total of 296 degrees of freedom and 91 elements. Nonlinear displacement-based beam-column fiber elements with Gauss-Lobatto quadrature are used to model the members of the MRF. Each column is modeled with 4 elements in each story. Each beam is modeled with 13 elements, including 5 elements for each of the reduced beam section (RBS), 1 element between the RBS, and 1 element between the RBS and column at each end of the beam. For the columns, seven fiber sections (integration points) are used along the length of each element. For the beams, five fiber sections are used along the length of each element. Each fiber section is discretized into 22 fibers, with 12 fibers for the web and 5 fibers for each flange. Panel zone elements are used to model the shear deformation and uniform bending deformation of the panel zones of the MRF. A bilinear uniaxial stress-strain relationship with strain hardening is used for the steel in the MRF. The elastic modulus and yield stress are 200 GPa and 345 MPa, respectively, and the strain hardening ratio (the post-yielding modulus over the elastic modulus) is 0.01. The lean-on column, which represents the gravity load system, is modeled with elastic beam-column elements. The seismic mass is lumped and the gravity load is applied at each floor level on the lean-on column, so that the P-delta effects of the gravity loads are included in the analytical substructure. The lean-on column is connected to the MRF using a rigid diaphragm. The inherent damping of the building (i.e., energy dissipation within the building during low-amplitude dynamic response) is modeled using Rayleigh damping and is included in the analytical substructure. The Rayleigh damping model is based on a 2% damping ratio for the first and second modes of the building.

Figure 6(b) shows the experimental substructure, which is the DBF with a nonlinear viscous damper and associated bracing in each story. As shown in Fig. 7, the DBF experimental substructure is pinned to the base test fixture at each column base, and is horizontally restrained at the ground level by a pair of ground links. The test setup has three servo-hydraulic actuators to impose displacements on the DBF through a loading beam system that is connected to the top flanges of the beams at mid span of the DBF. The loading beam system simulates the floor diaphragm of the building, which transmits the floor-level inertial forces to the top flanges of the beams. A bracing frame provides the out-of-plane (East–west direction) bracing of the DBF, as shown in Fig. 8.

The nonlinear viscous damper was placed between the brace and floor beam in each story of the DBF, as shown in Fig. 9. The damper was pin-connected to the floor beam through a clevis that was welded on the top flange of the floor beam. The damper-brace connection was designed to enable the damper to be connected to the brace with a pin connection. The damper was connected to a load cell through a bolted end plate connection, and the load cell was attached to each of the two side plates using bolts in drilled and tapped holes. Each of the side plates was connected to a damper attachment plate using a threaded-pin connection, in which the pin was turned into a drilled and tapped hole in the side plate passing through the damper attachment plate without threads. The damper attachment plates were welded to the lower gusset plates. This connection enables free rotation of the damper in the plane of the DBF, and the damper force to be measured accurately by the load cell during tests.

4 Ground motions for RTHS

Ground motions were selected for the prototype building site in Pomona, California [20]. The site was selected because the uniform hazard spectrum (UHS) at the DBE level and at the MCE level are consistent with the spectra from ASCE 7-10. For the UHS, the DBE and MCE are defined to have a 10% and 2% probability of exceedance (POE) in 50 years, respectively. For the DBE, an ensemble of 14 ground motion records was selected for the RTHS from the PEER NGA Database [24]. For the MCE an ensemble of 12 ground motions for the D100V and D75V structures and an ensemble of 7 ground motions for the D60V structure was selected from the PEER NGA Database. Due to laboratory constraints, fewer MCE ground motions were used for the D60V structure than for the D100V and D75V structures. Each ground motion record was scaled so that the median spectral accelerations of the ground motion ensemble match the corresponding UHS over a period range of 0.2 to 4.0 s. Details of the process for selecting and scaling of the ground motions can be found in Dong [20]. Information for the ensemble of ground motions are listed in Tables 6 through 8.

5 Experimental response and seismic performance evaluation

5.1 DBE story drifts

The story drift ratio time histories from the RTHS using the HECTOR-11625090 record for the D100V, D75V, and D60V structures are shown in Fig. 10. The figure shows that the D100V, D75V, and D60V structures have a maximum peak story drift ratios of 0.76%, 1.01%, and 1.27% rad. in the second story, respectively. Negligible residual story drifts are seen at the end of the simulations, indicating even the test structures with reduced base shear design strength and with nonlinear viscous dampers under exhibited nearly elastic response under the DBE.

The mean and the coefficient of variation (COV) of the peak story drift ratios from the RTHS using the ensemble of 14 DBE ground motions are summarized in Table 9 for each story of the test structures. The COV for the peak story drift ratio is less than 0.2 for each structure, which indicates relatively narrow dispersion of the peak story drift ratios. The mean peak story drift ratios are smaller than the design predictions (Table 4). Table 10 gives the mean and COV for the residual story drift ratios. The mean residual story drift ratios are quite small (negligible) for all the structures, indicating that the test structures have very little permanent deformation under the DBE ground motions. Overall, the story drifts for each test structure subjected to the DBE ground motions are far below the limit for the “Life Safety” performance level in ASCE 41-06 which requires the peak story drift ratio to be less than 2.5% rad. and the residual story drift ratio to be less than 1.0% rad. The story drifts for the D100V structure are close to the limit for “Immediate Occupancy” performance level in ASCE 41-06 which requires the peak story drift ratio to be less than 0.7% rad. and the residual story drift ratio to be negligible.

The reason for the reduced amount in the design prediction for story drift is associated with the fact that the design model for the DBF assumes pin conditions for the beam-to-column and diagonal bracing-to-column connections. The gusset plates present in the DBF of the test structure provides rotational restraint at the ends of the beam as well as the diagonal bracing, thereby stiffening the DBF and reducing the story drift. Accounting for the effects of the gusset plate connection details by including rotational restraint in the model would improve the design prediction. However, the degree of rotational restraint is not known, and requires additional studies to develop guidelines for selecting the proper amount of rotational restraint of the gusset plate connection to predict story drift.

5.2 MCE story drift demand

The story drift ratio time histories from the RTHS using the 1999 Chi-Chi earthquake TCU055-N record for the D100V, D75V, and D60V structures are shown in Fig. 11. It is observed that the story drift ratio time histories for the D100V and D75V structures are almost zero at the end of the simulations, but residual story drifts are seen at the end of the simulation for the D60V structure. The second story has a larger drift ratio than the first and third stories for each structure.

Table 11 summarizes the statistics for the peak story drift ratios for the D100V and D75V test structures from the RTHS for the 12 ground motions along with the peak story drift ratios for the D60V test structure from the RTHS for the 7 ground motions. The second story peak story drift ratio is larger than the first and third stories for all ground motions. The peak story drift increases for the structures as the base shear design strength decreases. The mean peak story drift ratios for each structure are smaller than the design predictions (Table 4). For the D100V structure, the mean peak story drift ratios are 16%, 17%, and 55% less than the design predictions for the first, second, and third story, respectively. For the D75V structure, the mean peak story drift ratios are11%, 8%, and 20% less than the design predictions for the first, second, and third story, respectively. For the D60V structure, the mean peak story drift ratios are 7%, 6%, and 17% less than the design predictions for the first, second, and third story, respectively. The coefficient of variation (COV) for the peak story drift ratios area maximum of 22% and 21% for the D100V and D75V structures, respectively, and 14% for the D60V structure, which suggests that the dispersion is reduced with the increasing of the peak story drift ratio.

Figure 12 shows the probability of exceedance (POE) for the peak story drift ratio for each test structure. In the figure, the data points correspond to the POE directly from the RTHS peak story drift ratio results. The continuous line is a lognormal distribution function fit to the data points. The lognormal distribution fits the cumulative distribution from the data, indicating that the story drift ratio demands can be assumed to be lognormally distributed. Accordingly, the probability of the peak story drift ratio exceeding the limit for a certain performance level can be estimated. The POE for the “Life Safety” performance level story drift limit (2.5% rad) for the D75V and D60V structures are 4% and 14%, respectively; and the POE for the “Collapse Prevention” performance level story drift (5.0% rad) for all test structures are much less 1%, indicating a very low probability of collapse even for the structures with reduced base shear design strength. For example, for the D60V structure, the 1% POE value of the peak story drift ratio is 2.9% rad, which is much less than the 5.0% rad. limit for the “Collapse Prevention” performance level in ASCE 41-06.

Table 12 summarizes the statistics for residual story drift ratios for the D100V and D75V test structures from the RTHS for the ensemble of 12 ground motions, and the residual story drift ratios for the D60V test structure from the RTHS for the ensemble of 7 ground motions. For the D100V structure, the mean residual story drift ratios are quite small (0.1% or less), indicating the D100V structure has very little permanent deformation. The D75V and D60V structures have mean residual story drift ratios less than 0.2%. The relatively large COV for the residual story drift ratio indicates a large variation in the residual story drift ratio as a function of the ground motion, suggesting that the characteristics of the ground motions (the so-called record-to-record variability) significantly affect the residual story drift ratio. The residual story drift ratios for all structures for all MCE level ground motions are less than 0.5% rad which is considered to be the level of residual story drift that impairs operation of the moveable components of a building such as doors, windows, and sliding partitions according to Galambos and Ellingwood [25], Ellingwood [26], and McCormick et al. [27]. These results suggest a high probability of good functional performance of the prototype building with dampers after an MCE level ground motion.

5.3 DBE response of MRF

Yielding in the building during the RTHS was found to occur primarily in the MRF, which was the analytical substructure in the RTHS, as the DBF (experimental substructure with the viscous dampers) remained essentially elastic. Table 13 summarizes the mean peak RBS rotation, the mean residual RBS rotation, the mean peak rotation of the MRF columns, and the mean residual rotation of the MRF columns from the RTHS for the D100V, D75V, and D60V structures. The MRF column rotation is defined as the relative rotation of the column section across a length of two times the column section depth measured from the top flange of the floor beam at each floor level with the assumption that a plastic hinge would form at about one column depth from the top flange of the floor beam. For the D100V structure, the mean peak RBS rotation is 0.35%, 0.21%, and 0.15% rad. for the first, second, and third floor, respectively; the mean peak column rotation is 0.33%, 0.17%, and 0.06% rad. for the first, second, and third story, respectively. The mean residual RBS rotations and mean residual column rotations are negligible, which indicates the RBS and the columns of the D100V structure were essentially elastic under the DBE ground motion.

For the D75V structure, the mean peak RBS rotation is 0.54%, 0.30%, and 0.21% rad. for the first, second, and third floor, respectively; the mean peak column rotation is 0.40%, 0.19%, and 0.09% rad. for the first, second, and third story, respectively. The mean residual RBS rotation is 0.14%, 0.04%, and 0.01% rad. for the first, second, and third floor, respectively. The observable mean residual RBS rotation for the first floor indicates yielding occurred in the RBS in the first floor. The negligible mean residual column rotations indicate the columns of the D75V structure were essentially elastic under the DBE ground motions.

For the D60V structure, the mean peak RBS rotation is 0.77%, 0.46%, and 0.29% rad. for the first, second, and third floor, respectively; the mean peak column rotation is 0.46%, 0.20%, and 0.12% rad. for the first, second, and third story, respectively. The mean residual RBS rotation is 0.14%, 0.10%, and 0.03% rad. for the first, second, and third floor, respectively. Similar to the D75V structure, yielding occurred in the RBS in the first floor of the D60V structure. The columns of the D60V structure were essentially elastic under the DBE ground motions.

5.4 MCE response of MRF

Figure 13 shows the RBS moment-rotation hysteresis for the D60V structure from the RTHS using the NORTHR5082-235 record, which produced the largest story drift ratio demands for the D60V structure among the seven ground motions. The RBS rotation is defined as the relative rotation across the length of the RBS. It is seen that the RBS of the D60V structure yielded significantly during the RTHS with a peak RBS rotation of 2.46%, 1.80%, and 1.54% rad. for the first, second, and third floor, respectively.

Table 14 presents the mean peak RBS rotation, the mean residual RBS rotation, the mean peak rotation of the MRF columns, and the mean residual rotation of the MRF columns from the RTHS for the D100V, D75V, and D60V structures. The MRF column rotation was defined above. It can be seen that the peak RBS rotation in the first floor is larger than in the second and third floors for each structure. The D100V structure has a small mean residual RBS rotation (0.18% and 0.13% rad.) in the RBS of the first and second floors and negligible mean residual RBS rotation (0.04% rad.) in the third floor. The D100V MRF columns have negligible mean residual rotations (less than 0.02% rad.), which indicates yielding did not occur in the MRF columns. For the D75V and D60V structures, relative large mean residual RBS rotations indicate yielding occurred in the RBS of all floors. The MRF columns of the D75V and D60V structures have much larger mean residual rotations in the first story than in the second and third stories, which suggests that the MRF columns in the first story yielded (at the ground-floor level) while the MRF columns in the second and third stories remained elastic for the MCE ground motions.

Figure 14 shows the probability of exceedance (POE) of the peak RBS rotation for each test structure based on the results from the RTHS involving the MCE ground motions. In the figure, the data points correspond to the POE directly from the RTHS peak RBS rotation results. The continuous line is a lognormal distribution function fit to the data points. The lognormal cumulative distribution fits the cumulative distribution from the data, indicating that the peak RBS rotation demands can be assumed to be lognormally distributed. Since the analytical model of the beam-to-column moment connection with an RBS used for the RTHS did not account for deterioration in strength and stiffness of the connection, a direct quantitative assessment of the performance of the connections based on the peak RBS rotation under the MCE is not attainable. Tests of welded steel beam-to-column connections with an RBS (e.g., Refs. [28–30].) and a fragility assessment of these connections [31] show that these connections when designed and constructed according to the prequalification limits and design procedures in AISC [19] can develop story drifts of about 2% rad before deterioration in the strength of the connection (due to damage in the RBS, such as local buckling and fracture) occurs under cyclic loading. Lignos and Krawinkler [32] show that the parameters (story drift ratios) for a deterioration model of these connections are sensitive to the geometry of the connection (ratio of beam web depth over web thickness, ratio of beam flange width over flange thickness, beam depth, shear span over beam depth, and lateral bracing for the RBS). Instead of using the peak story drift ratio, the peak RBS rotation is used as an index in this study to assess the performance of the RBS connection for the MCE ground motions. Based on the lognormal cumulative distribution of the peak RBS rotation, the POE for the 2.0% rad rotation value for each test structure is estimated as 2%, 15%, and 46% for the first floor RBS for the D100V, D75V, and D60V structures, respectively, indicating that flexural strength deterioration in the RBS may be expected for the D75V and D60V structures under the MCE ground motions.

5.5 Column axial force-bending moment response in MRF and DBF

Figures 15 through 17 show the axial force versus bending moment response for the columns in the first story of the D100V, D75V, and D60V structures, respectively, from the RTHS with the selected DBE level ground motion record PTS315. The P-M strength curve of the columns based on the AISC 360-10 [17] strength formulae for beam-column members subjected to flexure and axial forces is also plotted in the figures. The nominal axial and flexural strength of the column section was used in the formulae. The axial force in the MRF columns are shown to be generally in phase with the bending moments. Meanwhile, significant hysteresis loops are shown in the axial force-bending moment response of the DBF columns. The peak axial forces in the DBF columns are much larger than the peak axial forces in the MRF columns at times when the bending moments are at their peaks, and the axial forces in the DBF columns are large at times when the bending moments are small. Figures 18–20 show the axial force versus bending moment response for the columns in the first story of the D100V, D75V, and D60V structure, respectively, during the RTHS with the MCE ground motion record H-BRA315. The P-M strength curve of the columns based on the AISC 360-10 strength formulae for beam-column members subjected to flexure and axial forces is also plotted in the figures. The nominal axial and flexural strength of the column section was used in the formulae. A trend similar to the results for the RTHS DBE were found for the RTHS MCE, where in the latter larger forces developed in the columns due to the higher seismic hazard.

Table 15 presents the axial forces at the times when the bending moments are at their peak values in the MRF and DBF columns of the test structures from the RTHS with the DBE and MCE ground motions. Considering the results from the RTHS with the DBE ground motions, for the D100V structure the mean axial force in the MRF columns is 0.47, 0.27, and 0.10 times the mean axial force in the DBF columns for the first, second, and third story, respectively; for the D75V structure, the mean axial force in the MRF columns is 0.51, 0.32, and 0.12 times the mean axial force in the DBF columns for the first, second, and third story, respectively; and for the D60V structure, the mean axial force in the MRF columns is 0.53, 0.35, and 0.15 times the mean axial force in the DBF columns for the first, second, and third story, respectively. For the MCE ground motions the mean axial force in the MRF columns is 0.47, 0.32, and 0.14 times the mean axial force in the DBF columns for the first, second, and third story of the D100V structure, respectively; for the D75V structure, the mean axial force in the MRF columns is 0.46, 0.31, and 0.15 times the mean axial force in the DBF columns for the first, second, and third story, respectively; and for the D60V structure, the mean axial force in the MRF columns is 0.47, 0.33, and 0.16 times the mean axial force in the DBF columns for the first, second, and third story, respectively. The partially in-phase behavior of the damper forces with story drifts causes the column axial force to be large at times of peak bending moment; and, as a result, the DBF column axial forces are much larger than the MRF column axial forces, which must be accounted for in the design of structures with nonlinear viscous dampers. A procedure to account for the in-phase behavior of damper forces in the design of a DBF and MRF is given in Dong [20]. The procedure is based on using an equivalent linear elastic-viscous model for the dampers in association with an elastic-static analysis procedure.

6 Summary and conclusions

This paper presented an experimental study of the response of a 0.6-scale three-story steel frame building structure under the design basis earthquake (DBE). The test structure consists of a single-bay MRF, a single-bay frame with nonlinear viscous dampers and associated bracing (DBF), and the gravity load system and seismic mass tributary to the MRF and DBF. A test structure with a full strength MRF design (i.e., D100V) and test structures with reduced strength (75% and 60%) MRF designs (i.e., D75V and D60V) were studied. Earthquake simulations using an ensemble of DBE and MCE ground motions were conducted using real-time hybrid simulations (RTHS). For the RTHS, the MRF and the gravity load system were modeled numerically as the analytical substructure, while the DBF was tested in the laboratory as the experimental substructure.

The mean of the maximum peak story drift ratios for the D100V, D75V, and D60V test structures from the RTHS involving DBE ground motions are 0.76%, 0.98%, and 1.17% rad., which are 66%, 63%, and 60% less than the design predictions for the test structures without dampers, respectively. The test structures remained elastic with very little residual story drift. The peak story drift ratio demands under MCE ground motions were found to be lognormally distributed. The probability of the peak story drift ratio exceeding the “Life Safety” performance level story drift ratio limit (2.5% rad.) is (less than) 1%, 4%, and 14% for the D100V, D75V, and D60V structures, respectively. The probability of the peak story drift ratio exceeding the “Collapse Prevention” performance level story drift ratio limit (5.0% rad.) is much less than 1% for all three test structures. Although the residual story drift ratios are affected by the characteristics of the ground motions (record-to-record variability), the residual story drift ratios for all three test structures for all MCE level ground motions are less than 0.5% rad., which is a residual drift at which the operation of moveable components of a building such as doors, windows, and sliding partitions may be impaired. The results show that using a reduced strength design for the primary seismic lateral-force-resisting system along with nonlinear viscous dampers has the potential to reduce the story drift demand under both the DBE and MCE.

Under the MCE ground motions, significant yielding was observed in the RBS in the first floor of the D100V structure, in the RBS in all floors of the D75V and D60V structures, and in the MRF columns in the first story of the D75V and D60V structures. The peak RBS rotations were found to be lognormally distributed. The probability of the peak RBS rotation in the first floor exceeding 2% rad. under the MCE is 2%, 15%, and 46% for the D100V, D75V, and D60V structures, respectively.

The in-phase component of the damper forces contribute significantly to the axial force in the DBF columns at the times of peak bending moment, so the axial forces in the DBF columns are much larger than the peak axial forces in the MRF columns at the times of peak bending moments. This phenomena must be accounted for in the design of structures with nonlinear viscous dampers.

In conclusion, the experimental study of the steel MRF test structures with nonlinear viscous dampers under DBE and MCE ground motions showed that an MRF building structure with nonlinear viscous dampers can be designed with a reduced MRF strength level and still achieve high performance between the “Immediate Occupancy” performance level and the “Life Safety” performance level under DBE ground motions. In addition, the MRF building structure with nonlinear viscous dampers designed with a reduced MRF strength level can also achieve a high level of performance under MCE ground motions, with a low probability of collapse based on the peak lateral drift response and a high probability of good post-earthquake functional performance based on the small residual lateral drifts.

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