Seismic performance of HWBBF considering different design methods and structural heights

Abstract

Previous research has shown that using buckling-restrained braces (BRBs) at hinged wall (HW) base (HWBB) can effectively mitigate lateral deformation of steel moment-resisting frames (MRFs) in earthquakes. Force-based and displacement-based design methods have been proposed to design HWBB to strengthen steel MRF and this paper comprehensively compares these two design methods, in terms of design steps, advantages/disadvantages, and structure responses. In addition, this paper investigates the building height below which the HW seismic moment demand can be properly controlled. First, 3-story, 9-story, and 20-story steel MRFs in the SAC project are used as benchmark steel MRFs. Secondly, HWs and HWBBs are designed to strengthen the benchmark steel MRFs using force-based and displacement-based methods, called HWFs and HWBBFs, respectively. Thirdly, nonlinear time history analyses are conducted to compare the structural responses of the MRFs, HWBBFs and HWFs in earthquakes. The results show the following. 1) HW seismic force demands increase as structural height increases, which may lead to uneconomical HW design. The HW seismic moment demand can be properly controlled when the building is lower than nine stories. 2) The displacement-based design method is recommended due to the benefit of identifying unfeasible component dimensions during the design process, as well as better achieving the design target displacement.

Graphical abstract

Keywords

hinged wall / moment-resisting frame / seismic design / displacement-based design / nonlinear time-history analysis

1 Introduction

Shear walls [1,2] and rocking walls [3–6] are used as lateral force resisting systems (LFRSs) for resisting earthquakes. Formation of plastic hinges at shear walls may lead to difficulty in repairing shear walls after earthquakes, and elongation of plastic hinges may induce collapse of a building [2]. Compression toes in rocking walls may undergo localized concrete crushing during rocking of walls in earthquakes [7–9]. A hinged wall (HW) is a wall with a hinge at the base, and the wall section is separate from the ground except the hinge. Under lateral loads, the HW rotates around the hinge, and so formation of plastic hinge and localized damage at toes can be avoided. In addition, an HW panel has large bending stiffness, therefore, HWs can be used to force steel moment-resisting frames (MRFs) to deform uniformly along the height and mitigate damage concentrations in MRFs under earthquakes [10–12]. The structure system consisting of the MRF and HW is termed hinged wall frame (HWF).

To mitigate structural seismic displacements, supplemental energy dissipation devices are often designed to work with HWs [13,14] and other structures [15–17]. Feng et al. [18] developed an earthquake-resilient wall by installing buckling-restrained braces (BRBs) [19,20] at a concrete HW base (HW with BRBs at base, named HWBB) and used the HWBB to strengthen a steel MRF and convert the MRF to a dual system (named HWBBF). HWBB makes use of large earthquake induced rotations that occur at the HW base so that good amount of energy can be dissipated by BRBs in earthquakes, and therefore steel MRF seismic damage is mitigated [18,21]. The BRBs are replaceable after earthquakes. However, connections from BRBs to HW [18,22,23] and connections from HW to MRF [18,24] need to be properly designed. Feng et al. [21] investigated a 20-story MRF structure and concluded that modular HW segments can be used to reduce large seismic moment demands in HWs of high-rise HWBBF. Wang et al. [22] and Jiang et al. [23] conducted experimental studies on HWBB, and the results showed that the HWBB hysteresis curve was fat, revealing no obvious strength or stiffness degradation, implying that the HWBB can stably dissipate seismic energy. HW seismic moment demand increases as structure height increases and can exceed the moment capacity provided by BRBs. To mitigate the HW seismic moment demand, researchers have proposed use of modular HWs [25–30]. However, research on height limits when using a single HW unit is rare. As such, this paper investigates HWBBF seismic responses with different heights to find out the height limit of using a single HW to strengthen steel MRFs. Steel MRFs from the SAC project are often used as benchmark structures for research purposes [31–35]. The analyses use the above ground part of Pre-Northridge design 3-story, 9-story, and 20-story steel MRFs from the SAC project as benchmark structures [36,37].

The HWFs and HWBBFs can be designed using two methods: force-based design method and displacement-based design method. The force-based design method follows the design method that is documented in ASCE 7-10 [38], and uses the design response spectrum. The force-based design method involves stiffness ratios between HW and MRF, and between HW and BRB, these ratios were recommended in Ref. [21]. The displacement-based design method was developed early [39,40]. Many researchers used displacement-based design methods to design structures with HWs and/or BRBs [14,41]. Sun et al. [14] proposed a practical displacement-based seismic design approach to determine parameters of energy dissipator used for HWFs. The minimum wall stiffness and strength were also determined through the proposed design approach. Kalapodis et al. [41] designed BRBs frames using an improved direct displacement-based design method, which uses a multi-degree-of-freedom system instead of a single-degree-of-freedom (SDOF) system. However, a comprehensive comparison between the force-based design and displacement-based design on HWBBF was not found based on authors’ knowledge. The comparisons of the two design methods to HWBBF are worth investigation, so that it is easy for engineers to understand advantages of each design method.

Some advanced displacement-based design methods have been developed in recent years [42,43]. To make the displacement-based design of HWBBFs easy for engineering applications, this paper uses an elastic displacement spectrum, first to determine the structural fundamental period, and then to complete the design of HWBBFs. This simplified displacement-based design method for HWBBF was proposed in Ref. [18]. As a summary, the paper investigates influences of different building heights and different design methods on the seismic performance of HWFs and HWBBFs, through performing nonlinear time history analyses (NTHAs).

2 Configuration of the HWBB and HWBBF

The HWBB configuration [18,21] and the HWBB lateral deformation profile in earthquakes are shown in Fig.1(a). The HWBB contains an HW with two BRBs at base. In earthquakes, the HW rotates around the hinge support. The shear force is resisted by the hinge support and the flexural moment is resisted by force couples from the two BRBs. The HWBB works as a shear wall before yielding of BRBs. Once the BRBs yield, the HWBB works as an HW with moment resistance and energy dissipation capacity.

Fig.1(b) shows a model of the HWBB. The HWBB is considered as a flexural beam with a rotational spring at the base. This rotational spring represents the two BRBs at HW base and is termed “BRB spring” in this paper. The moment and the rotation of the BRB spring are assumed to be elastic perfectly plastic, which is acceptable for structural level analyses [44]. Calculations of yield moment (M_y,s) and rotational stiffness (k_s) of the BRB spring can be obtained based on Eqs. (1) and (2), respectively [18]. The BRB spring yield rotation (θ_y,s) is calculated by Eq. (3) [18].

(1)M_{\mathrm{y},\mathrm{s}}=f_{\mathrm{y},\mathrm{B}\mathrm{R}\mathrm{B}}{A}_{\mathrm{B}\mathrm{R}\mathrm{B}}B,

(2)k_{\mathrm{s}}=\frac{E_{\mathrm{s}}{A}_{\mathrm{B}\mathrm{R}\mathrm{B}}{B}^2}{2H_{\mathrm{B}\mathrm{R}\mathrm{B}}},

(3){\theta }_{\mathrm{y},\mathrm{s}}=\frac{2H_{\mathrm{B}\mathrm{R}\mathrm{B}}{f}_{\mathrm{y},\mathrm{B}\mathrm{R}\mathrm{B}}}{B{E}_{\mathrm{s}}},

where f_y,BRB, E_s, and A_BRB represent the yield strength, elastic modulus, and cross-section area of the BRB core plate, respectively; B represents the BRB distance at HW base, which is close to HW width; H_BRB and H represent the BRB length and HW heights, respectively.

Fig.1(c) describes the HWBBF configuration and the deformation under lateral loads in earthquakes. The HWBBF in Fig.1(c) comprises one HWBB and one MRF. Rigid links represent that the HWBB and the MRF have the same horizontal displacement at each floor level. The rigid links transmit only the horizontal anchorage force between the HWBB and the MRF [18]. The design performance targets of the HWBBF are as follows. 1) All members of the structure are elastic in small earthquakes to ensure that the structure is free of damage. 2) HWs are elastic in both design basis earthquakes (DBEs) and maximum considered earthquakes (MCEs), in which the elastic HWs can effectively control the MRF lateral deformation profile. 3) Steel MRF members undergo small nonlinearity in both DBE and MCE. 4) BRBs at HW base dissipate most of the seismic energy in both DBE and MCE. Note that since the MRF beams are rigidly connected to columns, it is difficult to prevent MRF from entering inelastic in earthquakes. As such, BRBs can only dissipate most but not all the seismic energy.

3 Design of comparison structures

3.1 Comparison Structures

HWFs, HWBBFs and MRFs for 3-story, 9-story, and 20-story structures are analyzed as comparison structures. 12 comparison structures are investigated, including three benchmark MRFs (MRF-3, MRF-9, and MRF-20), three HWFs (from force-based design) and six HWBBFs (three from force-based design and three from displacement-based design, named HWBBF-FB and HWBBF-DB, respectively).

The plan views of three benchmark MRFs (MRF-20, MRF-9, and MRF-3) are shown in Fig.2(a)–Fig.2(c), respectively. The eastwest (E) is the direction for earthquake excitation, in which there are two identical MRFs serving as LFRSs. Structure geometries, beam sections, column sections and seismic mass of each story are shown in Fig.2(e), Fig.2(g), and Fig.2(i). More information about benchmark MRFs can be found in Refs. [16,17].

Tab.1 shows basic information of the 12 comparison structures. Note that HWBBF refers to both HWBBF-FB and HWBBF-DB if not otherwise specified. Plan views of HWFs and HWBBFs for 20-story, 9-story, and 3-story structures are shown in Fig.2(a)–Fig.2(c), respectively. Two identical HWs or HWBBs are added to work with MRFs in the eastwest direction. Elevation views of the HWs and HWBBs used in the HWFs and HWBBFs are shown in Fig.2(d), Fig.2(f), and Fig.2(h).

3.2 Parameters for force-based design

The comparison structures were designed at Los Angeles. The seismic design parameters are shown in Tab.2. Note that response modification factor (R) is not listed for HWFs and HWBBFs in ASCE 7-10 [38]. The R for steel special MRF (R = 8) is selected for the purpose of calculating seismic design forces for HWFs and HWBBFs.

3.3 Design of the HWs and HWBBs for strengthened structures

3.3.1 Force-based design

The force-based design of HWFs and HWBBFs was conducted using modal response spectrum analysis (MRSA), with the square root of the sum of the squares as the modal combination method. The corresponding design spectrum is shown Subsection 4.2.

Fig.3 shows the force-based design process of the HWs and HWBBs in three HWFs and three HWBBF-FBs, in which the section dimension of HWs, the material yield strength, section dimension and length of BRBs (if applicable) are determined. For an existing MRF that needs to be strengthened, the design process is to first determine the wall and BRB sections through predetermined stiffness ratios of MRF-to-wall (λ) and BRB-to-wall (β), respectively [18] (see Steps 1–4 as shown in Fig.3); thereafter, strength demands of the walls and BRB springs using MRSA are calculated and the section dimension and length of the BRBs are estimated (see Steps 5–7 as shown in Fig.3).

In Fig.3, Steps 2–4, 6, and 7 use the following formulas.

(4)C_{\mathrm{f}}=\sum \eta \frac{12E_{\mathrm{s}}{I}_{\mathrm{c}}}{h^2},

(5)\lambda =\sqrt{\frac{C_{\mathrm{f}}{H}^2}{E{I}_{\mathrm{w}}}}\Rightarrow E I_{\mathrm{w}}=\frac{C_{\mathrm{f}}{H}^2}{{\lambda }^2},

(6)\beta =\frac{k_{\mathrm{s}}H}{E{I}_{\mathrm{w}}}\Rightarrow k_{\mathrm{s}}= \frac{E{I}_{\mathrm{w}}\beta }{H},

(7)A_{\mathrm{R}\mathrm{R}\mathrm{B}}=\frac{M_{\mathrm{W},\mathrm{b}}}{B{f}_{\mathrm{y},\mathrm{B}\mathrm{R}\mathrm{B}}},

(8)H_{\mathrm{B}\mathrm{R}\mathrm{B}}=\frac{E_{\mathrm{s}}{A}_{\mathrm{B}\mathrm{R}\mathrm{B}}{B}^2}{2k_{\mathrm{s}}},

where I_c and I_w are the moment of inertia of a single column and the HW, respectively; h represents a story height; η is the correction coefficient for lateral stiffness of columns [18]; Σ represents summation of all columns stiffness at the story; M_W,b is the HWBB moment demand at base from Step 5.

The story at the structural mid-height is used to calculate C_f in Step 2. In Steps 3 and 4, values of λ and β are selected from the appropriate value ranges of 0–2.0 and 0–3.0, respectively, based on the previous study [18]. In Step 5, numerical models for HWBBFs and HWFs are built in SAP2000 [45] based on the MRF configuration, HW section dimensions and BRB spring rotational stiffness from Steps 1, 3, and 4, respectively. The HW design forces are calculated in SAP2000 using MRSA [40]. In Step 6, commonly used steel (f_y,BRB is 235 or 345 MPa) are selected for BRBs. Equations (7) and (8) are reformatted based on Eqs. (1) and (2), respectively.

3.3.2 Displacement-based design

Fig.4 shows the displacement-based design process of the HWBBs in the three HWBBF-DBs. The goal of the displacement-based design is to determine the section dimension of HWs, the material yield strength, section dimension and length of BRBs based on a target displacement and displacement design spectrum. For an existing MRF that needs to be strengthened, the design process should first determine a target displacement for the strengthened structure (u_target) (Steps 1 and 2 shown in Fig.4). The target displacement needs to be less than specified displacement limits in a seismic design code, such as ASCE 7-10 [38]. The target displacement is flexible for adjustment based on requirements of designers or owners. In Step 3, the HWBBF is converted to an equivalent SDOF system using the fundamental mode of the HWBBF. The target displacement of the equivalent SDOF (δ_target) is calculated by Eq. (9), where φ and [M] are the first mode shape vector and mass matrix of the HWBBF, respectively. Since the lateral deformation of HWBBF follows a straight line, an inverted triangle is approximately taken as the first mode shape.

(9){\delta }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}=\frac{{\left\{\phi \right\}}^{\mathrm{T}}[M]\left\{\phi \right\}}{{\left\{\phi \right\}}^{\mathrm{T}}[M]\left\{1\right\}}{u}_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}.

In Step 4, the target period (T_target) is determined based on the δ_target from Step 3 using an elastic displacement design spectrum, which is the average of elastic displacement spectra from 10 earthquake records used for NTHA in this paper. In Step 5, a series of modal analyses are performed on HWBBFs by varying HW section dimension and BRB spring stiffness, to investigate relationships among HWBBF periods, HW section dimensions and BRB spring stiffnesses. The relationships between HWBBF fundamental periods and BRB spring stiffness (k_s) are drawn at different HW section dimensions (affecting EI_w) are named as HWBBF period curves (Fig.5 as an example). In Step 6, the HW section dimensions and the k_s can be determined based on the target period and the HWBBF period curves from Steps 4 and 5. In Step 7, the BRB length (H_BRB) and section dimensions (A_BRB) are designed using Eq. (8), in which E_s, B and k_s are known. In Step 8, other structure parameters such as M_y,s, θ_y,s, λ, and β are calculated using Eqs. (1), (3), (5), and (6).

3.4 Design results

3.4.1 Force-based design

Tab.3 shows results from each step in the force-based design. The design base shear should not be less than 85% of the base shear from equivalent lateral force procedure and should not be smaller than a lower limit of 7% of the total structure weight [38]. The design base shear forces of MRF-3, MRF-9, and MRF-20 are 2864, 5322, and 3677 kN, respectively. Tab.3 indicates that MRFs, HWFs, and HWBBF-FBs show similar design base shear forces for the 3-story and 9-story structures, respectively. Larger differences in design base shear forces of the 20-story structures are observed. Note that the area of BRB for HWBBF-FB-9 is larger than HWBBF-FB-20 and M_W,b of HWBBF-FB-9 is smaller than HWBBF-FB-20, the reason for this is that the HW of HWBBF-FB-20 has larger width and results in smaller BRB area per Eq. (7).

3.4.2 Displacement-based design

Tab.4 gives results for each step in the displacement-based design. Fig.5 shows HWBBF period curves for HWBBF-DBs. Fig.5 indicates that at a given B, the structural fundamental period (T₁) lower limit is independent from the k_s. The change in T₁ is negligible after k_s increases to a certain value, implying that a small target structural period may not be achieved only by increasing BRB spring stiffness if the wall itself is flexible. As such, if the preset target displacement is small, it will not be feasible to use a wall with small cross-section. In other words, if the required wall stiffness is too large from the design process, the preset target displacement may be too small and needs to be increased.

3.4.3 Comparison of structures designed using the two methods

The spectra used for the two design methods are correlated with each other. Force-based design uses the design response spectrum from ASCE 7-10 [38] while the displacement-based design uses an average displacement response spectrum. The average displacement spectrum is from the 10 earthquake records used for NTHA, and this average displacement spectrum is correlated to the average acceleration response spectrum from the same 10 earthquake records. In addition, the average acceleration response spectrum matches the design response spectrum (see Fig.8). Thus, the average displacement spectrum is correlated to the design response spectrum.

The force-based design method calculates the force demand based on the design response spectrum and MRSA. The design of structural members is based on pre-determined stiffness ratios and the whole design process is a sequence. All analysis modules are well incorporated into existing commercial software [45]. However, design iterations may be required if the designed structure does not comply to preset target performance.

The displacement-based design method calculates the member stiffness demand based on the displacement response spectrum and modal analyses; engineers are required to perform additional analyses to generate period curves. There is no need to pre-determine stiffness ratio in the displacement-based design. As such, the displacement-based design can avoid iterations caused by using inappropriate stiffness ratios between components. Comparison of Tab.3 and Tab.4 shows that the wall dimension, BRB spring stiffness, BRB yield strength from the displacement-based design are larger than those from the force-based design, implying that 1% target roof drift used in the displacement-based design may not be achieved by a structure using the force-based design, which is also illustrated in Fig.11.

Displacement-based design can reflect inappropriate wall width during the design process. Fig.6 shows different combinations of HW width and BRB spring stiffness to achieve a given target period (T_tar). Fig.6 shows that the required BRB spring stiffness increases as HW width decreases. The required BRB spring stiffness does not vary significantly when the target period is large. Moreover, if the target period is small, it may not be feasible to achieve the target period using flexible HWs (HWs with small width). This indication of unfeasible HW design cannot be reflected in a force-based design.

3.5 Modal properties

Tab.5 shows fundamental periods and effective modal mass ratios of comparison structures. The fundamental periods of HWFs are close to or slightly larger than those of the corresponding MRFs, and the fundamental periods of HWBBFs are smaller than those of the corresponding MRFs. This is consistent with findings from Ref. [21]. The fundamental periods of HWBBF-DBs are smaller than those of corresponding HWBBF-FBs due to stiffer BRBs used in HWBBF-DBs (see k_s values shown in Tab.3 and Tab.4). Tab.5 indicates that the 2nd periods (T₂) and higher modes of HWFs and HWBBFs decrease substantially from that of the corresponding MRF (except for 3-story structures). This is because the existence of HWs makes the structures difficult to deform in higher modes. HWFs and HWBBF-FBs have similar periods in the 2nd and subsequent higher modes, implying that adding BRBs has an insignificant influence on higher modes.

The effective modal mass ratios in the fundamental mode decreases as the structural height increases, indicating that the mass corresponding to higher modes increases with the increase of the structural height. The MRFs fundamental mode effective modal mass ratios are larger than those of HWBBFs. This implies that the masses corresponding to higher modes in HWBBFs is larger than those in the corresponding MRFs.

4 Finite element models and ground motions for nonlinear time history analyses

4.1 Finite element models

The structures investigated in this paper are modeled in OpenSees [46]. The modeling of 20 story structures in this paper is the same as was used in Ref. [21]. The same modeling method is used for 9-story and 3-story structures except small modeling difference due to different beam to column connection details. In MRF-20, however, all MRF beams are rigidly connected to MRF columns. The rightmost MRF beams are pin connected to MRF columns at both ends in MRF-3. The difference can be observed in Fig.2(a) and Fig.2(c). Fig.7 shows a numerical model of HWBBF-9 for explanation purpose. Half of the structure is modeled in two-dimensional (2D) space. The HWs and the BRB spring are modeled with ElasticTimoshenkoBeam elements and a ZeroLength element, respectively. Leaning columns are modeled to capture the structural gravity load induced P-Delta effect. The leaning columns are modeled with ElasticBeamColumn elements using large axial and bending stiffnesses. The steel MRF members are modeled with ForceBeamColumn elements. The MRF beams that are pinned to columns are modeled by coupling the beam end nodes to the corresponding column nodes in both translational degrees-of-freedom (X and Y directions as shown in Fig.7). Truss elements with large axial stiffness are used to connect the MRF and leaning columns. The moments at both ends of the leaning column at each story are released by using ZeroLength elements with very weak rotational stiffness, to eliminate the bending stiffness contribution from the leaning columns to the steel MRF.

The BRB and MRF member material is Steel01 and Steel4, respectively. 1.5 times the yield strength of MRF members is used as the ultimate strength [21,47]. The strain hardening ratios for MRF members and BRBs are 5% and 0% [48], respectively. 2% Rayleigh damping is calculated based on the fundamental and 3rd modes of each structural model and is used in NTHAs. Newmark method is used as the integrator object in NTHAs.

Uniformly distributed loads tributed to the MRF are applied at all beams. The gravity loads from the GLRS are applied at leaning columns. The beam loads and leaning column loads for structures with different floor levels are shown in Tab.6.

4.2 Ground motions

Ten earthquakes used in NTHA are selected from the far-field ground motions listed in FEMA P695 [49]. The accelerations at MCE are 1.5 times those at DBE. The accelerations of ten earthquakes are scaled for the purpose of matching the average response spectrum to the design spectrum for the period from 0.15 to 6.6 s, based on the requirement of ASCE 7-10 [38], since T₁ for 12 structures (see Tab.5) under investigation vary from 0.76 to 4.40 s. The earthquake scaling information is shown in Tab.7, and other earthquake details can be found in Ref. [21]. Fig.8 compares the design and average acceleration response spectra at DBE level. Fig.9 gives the average displacement response spectrum at DBE level. T₁ and T₂ of each structure (from SAP2000 [45]) are shown using vertical solid and dashed lines respectively. Fig.9 shows that the displacement at the fundamental mode is larger than that at higher modes.

5 Nonlinear dynamic time-history analyses

5.1 Lateral displacements

The average lateral displacement envelopes of comparison structures are shown in Fig.10. The MRFs, HWFs, HWBBF-FBs, and HWBBF-DBs are shown in black, thin blue, thick blue and red, respectively, along building height (applicable to other figures unless notified). The lateral displacement envelopes of HWF and HWBBF are close to straight lines. HWBBFs show smaller lateral displacements than those of HWFs, because of the energy dissipated in BRB springs. The HWFs and MRFs have similar structural lateral displacement envelopes for 3-story and 9-story structures, for the following reasons. 1) the fundamental periods of MRFs and HWFs are similar for 3-story and 9-story structures. 2) the MRF shear type deformation shape is not obvious in MRF-3 and MRF-9. MRF-20 shows shear type lateral displacement envelope, and the maximum roof displacement is smaller than that of HWF-20. The HWBBF-DBs show smaller lateral displacement than those of HWBBF-FBs, which is probably due to larger energy dissipation capacity in BRBs of HWBBF-DBs (see Fig.19(a)).

5.2 Inter-story drift

Fig.11 gives average inter-story drift envelopes at DBE and MCE. Fig.11 indicates that the inter-story drifts are more uniformly distributed in HWFs and HWBBFs than in MRFs, for 9-story and 20-story structures, since the HW is stiffer than the MRF. Inter-story drift envelopes of HWFs are close to the average value of the inter-story drift enveoples along the building height of corresponding MRFs. The maximum inter-story drifts of HWFs are smaller than those of the corresponding MRFs; this phenomenon is more obvious in taller structures. HWBBFs have smaller maximum inter-story drifts than those of HWFs because the BRBs at the HW base can dissipate seismic energy in earthquakes. The reduction of the maximum inter-story drift is about 50% from MRFs to HWBBFs in both DBE and MCE. Therefore, BRBs are effective in reducing structural lateral deformation in earthquakes and this reduction is independent of structure height.

HWBBF-DBs show about 1% inter-story drift for 3-story, 9-story, and 20-story structures, while HWBBF-FBs show larger inter-story drift than that of HWBBF-DBs in DBE. The displacement-based design uses 1% structure roof drift as the target displacement. As such, the displacement-based design method is shown to properly control the building lateral deformation in earthquakes in comparison with the force-based design method.

5.3 Floor acceleration

Fig.12 shows the structural maximum normalized accelerations. For MRF-9 and MRF-20, the maximum floor accelerations are smaller than PGAs, this is probably because the yielding of MRF components results in the lengthening of MRF periods, resulting in smaller acceleration responses. This phenomenon is also shown in Ref. [50].

HWFs and HWBBFs show larger maximum floor accelerations than those of MRFs. This is because after adding an HW to an MRF, the structural period of the 2nd mode decreases (see Tab.5) and the corresponding spectrum acceleration increases (see Fig.8). Previous research showed that higher modes had a large effect on structural acceleration and the contributions from higher modes are not affected by the yielding of structural members [51]. As such, the floor acceleration increment by adding HW is due to the increased contributions of higher modes. In addition, the acceleration envelopes from HWFs, HWBBFs are similar to each other, especially for taller structures, implying that using HW or HWBB does not make large differences to structure floor acceleration responses.

5.4 Structure overturning moment and story shear

Average values of maximum structural overturning moments (OT moments) are shown in Fig.13 with solid lines, and design moment of each structure is shown in dashed lines (this also applies to Fig.15 and Fig.18). The OT moments at upper levels of HWFs and HWBBFs are significantly larger than those of MRFs for 9-story and 20-story structures, while the difference is not such large in design moments for the MRFs, HWFs, and HWBBFs when these have the same height. This indicates that adding an HW increases the higher mode effects, which is similar to is the behavior shown in Fig.12. In addition, the OT moment increment from MRFs to HWFs and HWBBFs increases substantially as structure height increases, indicating increase in higher mode effects as structural height increases. The higher mode effects are not obvious for 3-story structures. Note that the effective modal mass ratios in the second and third modes of the HWF-9 are smaller than those of the MRF-9 (Tab.5), however the higher mode effects in the HWF-9 are larger than those of the MRF-9. This implies that larger effective modal mass ratio does not represent larger higher mode effects in dynamic analyses. The higher mode effects also depend on the spectrum acceleration at different modes, as is mentioned in Ref. [52]. HWBBF-DBs show larger OT moment demands than HWBBF-FBs, especially for 3-story and 9-story structures.

Floor inertial forces resultant effective height (H^{\text{*}}) is used to represent the height of the resultant force of floor inertial forces at each time step and can be calculated using Eq. (10) [53].

(10)H^{\text{*}}=\frac{\sum h_i\left|F_i\right|}{\sum \left|F_i\right|},

where F_i represents the floor inertial force at the ith floor. h_i represents the distance from the ith floor to the ground. H^{\text{*}} should be close to 2/3H if the floor inertial forces form an inverted triangular shape. The floor inertial forces show opposite values along the structural height and do not follow the inverted triangular shape in higher modes. H^{\text{*}} decreases as higher mode effects contributes more in NTHAs. For each NTHA, H^{\text{*}} is calculated at each time step if the base shear is larger than half of the maximum base shear [53]. Average value of all calculated H^{\text{*}} values for each NTHA is used for the H^{\text{*}} of the whole NTHA. Then the average value of H^{\text{*}} from 10 NTHAs is used as the H^{\text{*}} for each structure. Fig.14 shows that H^{\text{*}}/H decreases as the structure height increases, implying larger higher mode effects in taller structures. Fig.14 shows that H^{\text{*}}/H is smaller for MRFs than for HWFs or HWBBFs, for 9-story and 20-story structures, indicating larger higher mode effects after the MRFs are strengthened by HWs or HWBBs. These results of higher mode effects are consistent with those as shown in Fig.12 and Fig.13.

5.5 HW moment and shear demand and corresponding design forces

Fig.15 shows that HW moments are larger than the corresponding design moments at different structures. The seismic moment demand deviates more from the HW design as the structure height increases. As such, taller structure HW needs to resist a larger moment in order to be elastic in DBE and MCE level earthquakes. HWBBF-DBs experience larger HW moment demand than that of HWBBF-FBs for 3-story and 9-story structures. This is because the BRBs in HWBBF-DBs have larger yield strength than those in HWBBF-FBs, and this induces larger moment demand on HWs. HWBBF-DB-20 undergoes similar maximum HW moment demands to that of HWBBF-FB-20 despite using a stronger BRB than that of HWBBF-FB-20. This is because the higher mode effects in both structures are similar, and the largest HW demand occurs at upper floor rather than the wall base.

The HW moment demands increase significantly as HW height increases. This is because the HW moment is proportional to the square of the HW height. The moment demand in HWF-20 is larger than that in HWBBF-20, indicating that BRBs help in reducing HW moment demands. However, Fig.15(a)–Fig.15(e) show that HWBBF has larger moment demand than HWF for shorter HWs, like HWF-3 and HWF-9, implying that adding BRBs increases the seismic moment demands for shorter HWs. This HWBB height dependent “conflict” is due to BRB strength at the HWBB base as well as structural height, as is explained in Fig.16 and Tab.8.

Fig.16 schematically shows decomposition of HW moment in HWBBFs. Fig.16(a) shows the moment distribution along HWBB height when BRBs yield at the base in the HWBBF. The moment shown in Fig.16(a) can be decomposed into two parts: (1) HW moment caused by the BRBs (see Fig.16(b)) in an HWBB; (2) HW moment caused by the MRF lateral deformation in an HWF (see Fig.16(c)). NTHAs using EQ1 are used as examples. Tab.8 lists the following data. (1) The maximum HW moment demand in HWBBFs and HWFs (named M0 and M2, respectively) and HWBB base moment demand (M1) from NTHAs using EQ1. (2) The maximum roof displacements for HWBBFs and HWFs (named U0 and U2 respectively) from NTHAs using EQ1. The intensity of EQ1 for the NTHA on HWF was adjusted to make the HWF roof displacement match the corresponding HWBBF roof displacement, since this HWF comes from the HWBBF by removing BRBs at HWBB base. Tab.8 shows that M1 is significantly smaller than M2 for HWBBF-20. M1 is smaller than M2 for HWBBF-9 but the difference is not as large as that for HWBBF-20. M1 is much larger than M2 for HWBBF-3. Tab.8 also indicates that the contributions to HW moment demand caused by MRF lateral deformation are larger than those from BRB in tall HWBBFs, and vice versa in short HWBBFs. As such, a tall HWBBF shows smaller HW moment demand than that for a tall HWF due to the following. 1) The MRF lateral deformation contributes more to the total HW moment demand. 2) Using BRBs can mitigate the structure displacement in earthquakes (see Fig.10 and Fig.11). For short HWBBFs, though M2 may decrease due to the reduced structural displacement through the usage of BRBs, the reduction of M2 may not be comparable to that of M1. Thus, a short HWBBF shows larger HW moment demand than that of the corresponding HWF. As such, stronger BRBs induce larger moment demands in HWBBs in shorter HWBBFs, while the BRB strength does not have a large influence on HWBB demands in taller HWBBFs.

Fig.17 shows ratios (HW moment magnification factors) between the HW maximum moment (M_max) and HW base BRB spring moment (M_y,s) for HWBBFs with different heights. Fig.17 also shows that M_max/M_y,s increases as building height increases, implying that the maximum moment occurs at an upper floor. In addition, M_max/M_y,s increases significantly from HWBBF-DBs to corresponding HWBBF-FBs, implying that the BRBs in force-based design is weaker than those in displacement-based design. The BRB spring moment depends on HW section width and BRB yield strength. M_max/M_y,s is smaller in DBE than in MCE since larger earthquakes induce larger moment demands. The moment magnification factors are 1 for HWBBF-3, for MCE and DBE. The moment magnification factors are close to 1 for HWBBF-DB-9 and larger than 1 for HWBBF-FB-9. The moment magnification factors are larger than 1 for HWBBF-20, independent of design methods. As such, the design of the HW needs to consider possible moment magnification from BRB spring moment. In other words, the maximum HW moment demand may be larger than the BRB spring moment for a tall HW. A 9-story HW is close to the threshold for consideration of possible moment magnification based on the results in this paper. Moment magnification may need to be considered when an HW reaches or is larger than 9-story. If the design of HW becomes uneconomical for achieving the design target of keeping the HW elastic in DBE and MCE, usage of modular HW can be considered [21].

Fig.18 shows the HW shear force demands. Similar to what can be seen in Fig.15, the HW shear demands are larger than the design shear. The difference between the shear demand and design shear increases as the structure height increases. The 9-story and 20-story structures show similar maximum HW shear force for HWFs and HWBBFs, implying that the mid-rise and high-rise structures have similar HW shear demands in HWFs and HWBBFs. In addition, HWBBF-DBs show larger shear force than HWBBF-FBs for 3-story and 9-story structures, which is similar to what is shown in Fig.15.

5.6 Buckling-restrained braces spring responses

Fig.19 shows average values of HWBB base energy dissipation and HWBB base accumulated plastic rotation. The dissipated energy at HW base increases as the structure height increases. The accumulated HW base plastic rotation decreases as the structure height increases. The reason why taller HWBB dissipates the most energy while undergoing smaller accumulated plastic rotation, is because taller HWBB has larger BRB spring yield strength. HWBB-DBs have larger dissipated energy and smaller accumulated plastic rotation than that of HWBB-FBs due to stronger BRBs used in HWBB-DBs, implying less potential to undergo low-cycle fatigue failure.

Fig.20 shows the BRB spring rotation based on EQ1 for HWBBF-FBs. HWBBF-FB-20 shows the fewest number of large rotation cycles and HWBBF-FB-3 shows the most number of large rotation cycles. This result indicates that shorter HWBB undergoes more large rotation cycles, which results in larger accumulated plastic rotation as shown in Fig.19(b).

5.7 Energy dissipation

Fig.21 shows average values of dissipated energy in MRF and BRBs. The dissipated energy in MRF does not vary substantially between MRF and HWF when no BRB is deployed to strengthen structures. The total dissipated energy in MRFs, HWFs, and HWBBFs does not vary significantly in DBE for structures with different heights and in MCE for shorter structures. However, the dissipated energy in MRF decreases substantially in HWBBFs. The energy dissipated in BRB in HWBBF-FB is smaller than that in HWBBF-DB for all structures though BRBs in HWBBF-DBs may undergo smaller maximum rotation; this is because BRBs in HWBBF-DBs are designed with larger strength. As such, the BRB spring dissipated energy depends on both accumulated plastic rotation and BRB spring yield strength.

The total energy dissipations in 9-story and 20-story structures are similar. The 9-story structure has fewer structural components than the 20-story structures, and similar total dissipated energy implies larger energy dissipation in each component of the 9-story structures. As such, the 9-story structures may experience more severe damage than the 20-story structures. The 3-story structures have significantly fewer structural components than that in the 9-story and 20-story structures, therefore, less total dissipated energy is observed in 3-story structures.

Fig.22 shows the average value of MRF dissipated energy along the structure height. The figure indicates that MRFs dissipate larger energy at lower stories for 9-story and 20-story structures. For HWFs, the distribution of the dissipated MRF energy is generally similar to that of MRFs. The dissipated energy at the 1st floor in HWF-3 and HWF-9 is larger than that in the corresponding MRF, implying that using only HW to strengthen MRF may not reduce the energy dissipated in the MRF. HWBBFs show much smaller dissipated MRF energy at each floor level than that in MRFs and HWFs; this is because large seismic energy is dissipated in BRBs at the base (see Fig.21). HWBBF-DBs show smaller dissipated MRF energy at each floor than that of HWBBF-FBs for all structures, especially for 3-story and 9-story structures. This is for two reasons as follows. 1) The HWs in HWBBF-DBs are stiffer than those in HWBBF-FBs; as such, the HWs in HWBBF-DBs can provide larger constraints to MRFs and the MRFs undergo smaller inter-story drift and therefore smaller dissipated energy. 2) BRBs in HWBBF-DBs dissipate larger energy than those in HWBBF-FBs. Fig.11 shows that the maximum structural drift in HWBBF-FB is about half of the HWF. The large reduction in MRF dissipated energy shown in Fig.22 implies that the MRFs in HWBBF-FBs undergo a smaller number of cycles of large MRF drift than that occurs in HWFs. This is verified by the MRF 1st story inter-story drift time history in Fig.23. Fig.23 also shows that BRBs help to reduce the HWF post-peak responses.

6 Conclusions

HW or HWBB are proposed to strengthen existing MRFS which are termed HWF or HWBBF in this paper. In general, HWBBFs may be designed using both force-based and displacement-based design methods. NTHAS are used to investigate the influence on HWBBF seismic performances from two design methods and three structural heights. The following conclusions can be drawn.

1) The displacement response of the HWBBF designed by the proposed displacement-based design method is close to the preset design target displacement, implying that the proposed displacement-based design method can properly control the structure displacement response in comparison to the force-based design method. In addition, the displacement-based design method can help engineers identify inappropriate HW cross-section dimension and unpractical small target displacement during the design process.

2) The MRFs in HWBBF-DBs show smaller maximum inter-story drift and dissipate less energy than that occur in HWBBF-FBs, indicating that MRFs are better protected when a displacement-based design is used to strengthen existing MRFs. This conclusion is valid based on the analyses in this paper, mainly because the target displacement used in the displacement-based design produces stronger and stiffer HWBBs with larger energy dissipation capacity. This is associated with the following points. (1) HWs in HWBBF-DBs are stiffer than those in HWBBF-FBs, so HWs in HWBBF-DBs provide larger constraints to corresponding MRFs. (2) HWBBs in HWBBF-DBs dissipate more energy than that in HWBBF-FBs.

3) The HW maximum moment demand increases as structure height increases due to larger contributions from higher mode effects in taller structures. Based on the design target of 1% roof drift in DBE, nine seems like a threshold of number of stories for consideration of no HW moment magnification from the BRB moment capacity at the HWBB base. Engineers may not need to worry about larger than BRB spring moment for designing HWBBs, for an MRF lower than nine-story. However, this number of stories may be subjected to change if the MRF setup varies. If a structure is taller than 9-story, modular HWBB [21,25–30] may be recommended as one option to reduce HW moment if the HW design turns out to be uneconomical.

4) The BRB spring accumulated plastic rotation decreases as the structural height increases, and the BRB spring dissipated energy increases as the structural height increases. This is because the yield moment of BRB springs increases as HW width increases. The BRBs in shorter HWBB are subjected to more accumulated plastic rotation, which may be more vulnerable to damage in earthquakes. Increasing BRB spring yield strength is helpful to reduce the HWBB accumulated plastic rotation.

As a summary, the HW moment demands in HWBBFs significantly increase as the structural height increases, which may induce an uneconomical HW design. The displacement-based design method is recommended for design of HWBBFs since the method may help engineers to identify unfeasible component dimensions during the design process, and to better achieve the design target displacement for structures. However, this study is based on the steel MRFs from the SAC project; the conclusions may vary depending on the design of HWBBF and prototype MRF.

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