Risk Assessment of Overturning of Freestanding Non-Structural Building Contents in Buckling-Restrained Braced Frames

Abstract

The increasing demand in structural engineering now extends beyond collapse prevention to encompass business continuity planning (BCP). In response, energy dissipation devices have garnered significant attention for building response control. Among these, buckling-restrained braces (BRBs) are particularly favored due to their stable hysteretic behavior and well-established design provisions. However, BCP also necessitates the prevention of furniture overturning—an area that remains quantitatively underexplored in the context of buckling-restrained braced frames (BRBFs). Addressing this gap, this research designs BRBFs using various design criteria and performs incremental dynamic analysis (IDA) with artificially generated seismic waves. The results are compared with previously developed fragility curves for furniture overturning under different BRB design conditions. The findings demonstrate that the fragility of furniture overturning can be mitigated by a natural frequency shift, which alters the threshold of critical peak floor acceleration. These results, combined with hazard curves obtained from various locations across Japan, quantify the mean annual frequency of furniture overturning. The study reveals that increased floor acceleration in stiffer BRBFs can lead to a 3.8-fold higher risk of furniture overturning compared to frames without BRBs. This heightened risk also arises from the greater hazards at shorter natural periods due to stricter response reduction demands. The probabilistic risk analysis, which integrates fragility and hazard assessments, provides deeper insights into the evaluation of BCP.

1. Introduction

In the face of significant earthquakes occurring globally, appropriate seismic retrofitting is essential. One effective method for collapse prevention is the implementation of seismic retrofitting or strengthening by buckling-restrained braces (BRBs). A BRB consists of a steel core encased in concrete, which is further confined by a steel tube. Typically, BRBs are installed in diagonal, V, or chevron (inverted-V) configurations, as depicted in Figure 1.

Since the invention of the BRB by Kimura et al. [1], extensive research has been conducted to evaluate BRBs at the component level [2,3,4]. These studies have demonstrated that BRBs possess exceptional ductility and energy-dissipation capacity, highlighting their ability to withstand strong earthquakes.

Currently, the design procedure is outlined in the prevailing design specifications [5,6]. The method for calculating the required amount of BRBs was investigated by Kasai et al. [7], and the proposed method is now incorporated into the guidelines of the Architectural Institute of Japan (AIJ) [5]. The necessary design considerations for the main frame have been extensively studied at both the member and frame levels [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31].

Thanks to these major advancements, buildings equipped with BRBs can withstand more significant earthquakes. As illustrated in Figure 2a,b, buckling-restrained braced frames (BRBFs) effectively prevent collapse even during large-scale seismic events, which could otherwise lead to the collapse of moment-resisting frames (MRFs). In recent years, the structural engineering field has increasingly emphasized the importance of ensuring business continuity after a major earthquake, aligning with the concept of Business Continuity Planning (BCP). However, previous research has predominantly focused on collapse prevention [32,33,34,35]. Faced with this need, structural health monitoring and damage detection methods have been emerging and have recently advanced [36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51]. While they effectively contribute to resilience, avoiding any damage that impacts BCP is the best option. Figure 2c illustrates the interior of a room following the 2022 Fukushima-Oki earthquake. The building was equipped with an energy dissipation device. Although the building remained intact and did not collapse, furniture toppled over, leading to a suspension of business operations during the repair work. Particularly after the Great East Japan Earthquake (GEJE) in 2011, researchers have shown increased interest in the damage to non-structural components, such as ceiling failures [52,53].

Regarding furniture overturning, a fundamental study was conducted by Kaneko [54,55]. The essential characteristics of overturning ratios of rigid bodies subjected to large input motions were examined through analytical studies. The results revealed that overturning ratios are influenced by the sizes and shapes of rigid bodies, as well as the levels and predominant frequencies of the input motions. Based on these studies, a general fragility curve was proposed to describe the relationship between overturning ratios and the amplification of input motions.

Subsequent studies have advanced the understanding of the fragility of rocking bodies by considering deformation, nonlinearity, and sliding [56,57,58,59,60,61]. These studies typically derived the moment equilibrium around the center of rotation using the moment of inertia and applied horizontal forces. Additionally, deformation is modeled using a spring and dashpot system at the corners of the rocking bodies to better reflect realistic conditions.

Subsequently, Saito et al. [62] conducted a seismic risk evaluation study considering furniture overturning in mid- and low-rise office buildings. Saito et al. [62] focused on mid- and low-rise steel frame office buildings. The study used the ratio of Q_u/Q_un (the horizontal load-bearing capacity, Q_u, to the required horizontal yield strength, Q_un) as an analytical parameter. Seismic response analysis was used to evaluate response values, and building damage was assessed through Seismic Risk Analysis. Additionally, the impact of furniture within the building was analyzed by determining the overturning ratio. The study demonstrated that by increasing Q_u/Q_un, the accuracy of seismic safety analysis for the building can be enhanced.

The probabilistic risk assessment (PRA) framework is well-suited for quantifying the effectiveness of seismic retrofitting in preventing furniture overturning. In a foundational study, Ishikawa et al. [63] proposed a procedure for seismic risk evaluation in buildings, incorporating the results of probabilistic seismic hazard assessment (PSHA), structural fragility analysis, and economic loss estimation. The study produced a seismic risk curve that illustrates the relationship between financial loss and its annual exceedance probability, with case studies validating the proposed methodology. This approach is widely applied globally [64,65,66,67].

PRA typically involves incremental dynamic analysis (IDA), comprehensively outlined by Vamvatsikos and Cornell [68]. IDA is a parametric method developed in various forms to estimate structural performance under seismic loads. It entails applying scaled ground motion records to a structural model across multiple intensity levels, producing response curves parameterized by intensity. Their study established unified terminology, introduced suitable algorithms, and examined the properties of IDA curves for both single- and multi-degree-of-freedom systems. Additionally, it discussed methods for summarizing multi-record IDA results, comparing IDA with conventional static pushover analysis, and evaluating the yield reduction R-factor.

Eads et al. [69] investigated various aspects of calculating the mean annual frequency of collapse and introduced an efficient approach for estimating the sideways collapse risk of structures in seismic regions. The deaggregation of the mean annual frequency of collapse showed that this risk is generally dominated by earthquake ground motion intensities in the lower half of the collapse fragility curve. Their study also quantified the uncertainty in both the collapse fragility curve and the mean annual frequency of collapse based on the number of ground motions used in the analysis. The proposed method significantly reduced the computational effort and the uncertainty associated with these estimates.

Deylami and Mahdavipour [70] critiqued a significant drawback of BRBFs, namely the low post-yield stiffness of steel cores, which results in large residual drifts in a story after earthquakes. Their study examined the seismic demands of low- and mid-rise BRBFs and Dual-BRBFs using Probabilistic Seismic Demand Analysis (PSDA). By comparing demand hazard curves, they concluded that employing BRBFs as a dual system could significantly reduce residual drift demands, improving the resilience of such structures after earthquakes with lower repair costs. Additionally, the study used two nonlinear models—a deteriorating model and a non-deteriorating model—to examine the impact of degradation on Dual-BRBFs’ seismic demands. Their analysis revealed residual deformations are more susceptible to degradation than maximum deformations. However, deterioration is mitigated by the significant stiffness of BRBs, which keeps MRFs within a low range of nonlinearity.

The aforementioned studies primarily concentrated on structural collapse or damage related to structural safety. However, as previously noted, the recent societal demand increasingly emphasizes the importance of maintaining business continuity even after major earthquakes. Despite this, approaches to BCP based on PRA remain limited. In response to this, the present study specifically focuses on applying PRA to the issue of furniture overturning in BRBFs. Additionally, the effectiveness of retrofitting with BRBs for BCP is evaluated analytically. Therefore, building collapse is not assessed in this study, as advancements in BRBF design have already enhanced the structural capacity to prevent collapse effectively.

In this study, the model structure is derived from the design provisions of the AIJ and retrofitted with BRBs according to various design criteria. Realistic seismic waves are generated, reflecting the underlying ground structure and potential fault fracture scenarios. Using these seismic waves, IDA is conducted to establish fragility curves for furniture overturning. Finally, fragility and hazard curves for various locations in Japan are compared, and the mean annual frequency of furniture overturning is computed.

The findings of this research contribute to advancing seismic design by incorporating considerations for BCP. Moreover, the study highlights the importance of adequately securing furniture, even in buildings retrofitted with BRBs. In practical applications, engineers can quantify the risk of furniture overturning and assess its impact on BCP, providing a reasonable criterion for discussions with building owners or tenants. As such, the methodology presented in this research offers a platform for developing more sophisticated design strategies that account for business contingency.

2. Model Structure for Seismic Retrofitting Using Buckling-Restrained Braces

2.1. Outline of MRF Model Structure

This study considers the seismic upgrading of a four-story Japanese office building. The structure in question is a representative model of a steel moment-resisting frame (MRF) as specified in the provisions of the AIJ [5].

The structure represents a typical office building designed in accordance with Japanese regulations, featuring a regular floor plan. Consequently, torsional response is not a significant concern. This regularity makes the building suitable for studying the relationship between retrofitting with BRBs and the risk of furniture overturning.

A drawing of the structure is presented in Figure 3. The building’s floor plan spans 25.6 m in the x direction and 19.2 m in the y direction. The weight distribution for each story is shown in Figure 3. The columns are square hollow section (SHS) tubes, with widths ranging from 350 to 450 mm and wall thicknesses between 16 and 22 mm.

The main girders consist of wide-flange sections with depths ranging from 550 to 800 mm and flange widths of 200 to 300 mm, with plate thicknesses between 9 and 25 mm. All girder-column connections are designed as rigid moment connections. The steel used is SN490, per Japanese standards [71], with a yield stress of 325 N/mm².

2.2. Equivalent Linearization-Based Design of a Buckling-Restrained Brace

For this study, the BRBs are designed based on the principles of equivalent linearization, a method introduced initially by Kasai et al. [7] and currently codified in the Japanese design specifications [5]. Figure 4 illustrates the concept of response reduction of BRBFs. Respective figures indicate the idealized displacement, velocity, and acceleration spectrum. The design primarily aims to reduce the displacement response. This stems from the enhanced stiffness using the added BRBs. Also, the enlargement of the damping factor reduces the response spectrum; thereby, the displacement response further decreases, as presented in Figure 4a.

Using mathematical expressions for the effective vibration period and damping ratio of a building equipped with BRBs, along with idealized seismic response spectra, the peak seismic responses of the system and its local components can be expressed as continuous functions of structural and seismic parameters. These relationships are depicted in “performance curves”.

Figure 5 illustrates the concept of equivalent linearization for a BRB integrated with an elastic frame, represented as an equivalent single-degree-of-freedom (SDOF) system. Figure 5a shows the displacement response spectrum used in this study, where the vertical axis represents the displacement response, denoted as S_e/ω², and the horizontal axis represents the natural period, T. The design spectrum used in this study is derived from Eurocode-8 [72], with the ground type assumed to be Type C and the reference ground acceleration, a_g, set at 3.0 m/s². Additionally, importance factors γ_I of 1.0, 1.2, and 1.4, corresponding to importance classes II, III, and IV, respectively, are applied in the BRB design.

Five scenarios are considered for seismic upgrading:

• θ_t = 1/120 rad with γ_Ia_g = 3.0 m/s² peak ground acceleration (PGA),
• θ_t = 1/150 rad with γ_Ia_g = 3.0 m/s² PGA,
• θ_t = 1/200 rad with γ_Ia_g = 3.0 m/s² PGA,
• θ_t = 1/200 rad with γ_Ia_g = 3.6 m/s² PGA, and
• θ_t = 1/200 rad with γ_Ia_g = 4.2 m/s² PGA.

Here, θ_t represents the target story drift.

Figure 5b illustrates the performance curve, which consists of the displacement reduction ratio (R_d) and the force (or pseudo-acceleration) reduction ratio (R_pa), both expressed relative to the frame’s responses without BRBs. In equivalent linearization, the target story drift θ_t is initially determined as a sufficiently small value to prevent structural damage. The displacement response can be computed using the equivalent SDOF system and response spectra. This response is converted into story drift θ by dividing by the effective height, h_ef, of the equivalent SDOF. The displacement reduction ratio, R_d, is calculated as R_d = θ_t/θ.

The curve depicted in Figure 5b theoretically captures the influence of the stiffness ratio K_a/K_f between the added component (a damper and brace combined in series) and the frame, as well as the ductility demand μ of the added component. These curves incorporate these critical factors and provide the necessary K_a/K_f and μ values to achieve the target drift or displacement reduction ratio R_d. The required yield force of the damper can also be derived from the R_d and force reduction ratio R_a values. The results of this study suggest that the design ductility of the BRB is 4.0, as the performance curve identifies this value as the optimal response reduction point, indicated by the convex portion in Figure 5b.

Once the necessary K_a/K_f value is determined for the SDOF system, it can be translated into the requirements for dampers in a multi-story system under the following assumptions:

• The equivalent damping factor is consistent between the SDOF and multi-story systems.
• The inter-story drift is uniform across all stories and equals the target story drift for the seismic force, calibrated according to the A_i distribution.
• The ductility of the BRB is uniform across all stories.

Here, the A_i distribution represents the vertical seismic force distribution guided by the AIJ [73]. The necessary stiffness of individual stories K_ai is computed using the following equations to meet these criteria.

$$K_{ai}={\displaystyle \frac{Q_i}{h_i}}{\displaystyle \frac{{{\displaystyle \sum }}_{i=1}^N\left(K_{fi}{h_i}^2\right)}{{{\displaystyle \sum }}_{i=1}^N\left(Q_i{h}_i\right)}}\left(\mu +{\displaystyle \frac{K_a}{K_f}}\right)-\mu ·K_{fi}$$

(1)

In this context, Q_i represents the story shear force, h_i denotes the height of the i-th floor, K_fi signifies the story stiffness of the i-th floor, and μ denotes the ductility of the BRBs at the target story drift. Additionally, the required yield axial displacement u_ayi is determined based on the design ductility μ, the story height h_i, and the target story drift θ_t.

$$u_{ayi}={\displaystyle \frac{{\theta }_t}{\mu }}·h_i$$

(2)

Ultimately, the yield axial forces of the respective BRBs, F_ay, can be computed based on the axial stiffness K_ai and the yield axial displacement u_ay. The formulation for F_ay is provided below.

$$F_{ayi}=K_{ai}·u_{ayi}$$

(3)

The calibrated specifications for the BRBs are detailed in

Table 1. Required capacities of BRBs (unit: kN/mm for K_fi and K_ai, kN for F_ai).

Θt		1/120 rad		1/150 rad		1/200 rad		1/200 rad		1/200 rad
γIag		3.0 m/s2		3.0 m/2		3.0 m/s2		3.6 m/s2		4.2 m/s2
Story	Kfi	Kai	Fai	Kai	Fai	Kai	Fai	Kai	Fai	Kai	Fai
R	194.5	—	—	—	—	—	—	90.00	450.2	249.2	1246
4	249.2	—	—	2.800	18.60	186.2	936.9	378.8	1906	631.1	3175
3	323.1	—	—	32.20	216.1	275.2	1385	530.4	2669	864.6	4350
2	211.8	198.5	2253	270.2	2556	475.2	3371	690.5	4898	972.4	6898

. These BRBs are positioned in bay Y2, which is situated at the center of the frame. The design-based shear ratios are 0.47, 0.57, and 0.66 for seismic accelerations of γ_Ia_g = 3.0 m/s², 3.6 m/s², and 4.2 m/s², respectively.

3. Outline of Nonlinear Response History Analyses and Fragility Assessment of Furniture Overturning

3.1. Outline of Nonlinear Response History Analyses

A three-dimensional model was constructed, as detailed in Figure 6. Nonlinear response history analyses (NRHA) were conducted using ABAQUS ver. 2021, a finite element analysis (FEA) software package. For specific settings and element definitions, readers are referred to the ABAQUS manual [74]. The ABAQUS/Standard module is employed for the analyses described. The model utilizes a three-dimensional configuration with encastre constraints applied at the column base. As this research focuses on behavior along the x-axis, out-of-plane (OOP) deformation is constrained by the boundary conditions.

Beams and columns are represented using beam elements (B31), with distributed plasticity modeling adopted. Member yielding is identified based on von Mises stress, considering the sum of stress from both axial force and flexural moment for yield detection. The yield strength is determined to be 325 N/mm². Elastic-perfectly plastic (EPP) hysteresis is applied to each member, with column and beam intersections defined as rigid bodies.

The flexural stiffness of girders is doubled for beams with concrete slabs on both sides and increased by 1.5 times for beams with a half slab in order to model the influence of the concrete slab. The contributions of the concrete slabs to flexural stiffness are calculated according to Japanese design guidelines [75]. The contribution of the concrete slab is influenced by the shear connector performance [76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112]. Also, structural members can originate the buckling in a huge deformation, as summarized in the previous studies [113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139]. However, this research assumes idealized conditions to examine the risk of furniture overturning specifically. Additionally, Rayleigh damping is utilized for each structural member, with primary and secondary damping ratios set at 0.02.

The established model was verified by comparing the natural period and load-displacement relationship with those provided in the AIJ provision [5].

3.2. Fragility Assessment of Furniture Overturning

The fragility of furniture overturning is assessed statistically. Although various approaches exist for evaluating fragility [56,57,58,59,60,61], this research adopts the methodology proposed by Kaneko [54]. Kaneko et al. [54] integrated a theoretical framework with post-earthquake inspections of overturned bodies following several large seismic events. The resulting trend is represented by a continuous function, making it straightforward to apply when determining the threshold acceleration. The application of alternative evaluation methods will be explored in future research.

Based on a previous study [54], the probability of furniture overturning is determined using the following equations. The acceleration that results in a 50% rate of furniture overturning is calculated using Equation (6).

$$R\left(A_f\right)=\alpha ·\phi \left({\displaystyle \frac{ln{A}_f-ln{A}_{R50}}{\zeta }}\right)$$

(4)

$$A_{R50}=\left\{\begin{array}{c}{\displaystyle \frac{B}{H}}·g\left(1+{\displaystyle \frac{B}{H}}\right)\left(F_f\le F_b\right)\\ 10{\displaystyle \frac{B}{\sqrt{H}}}{\left(1+{\displaystyle \frac{B}{H}}\right)}^{2.5}·2\pi F_f\left(F_f>F_b\right)\end{array}\right.$$

(5)

$$F_f={\displaystyle \frac{V_f}{2\pi D_f}}$$

(6)

where φ represents the log-normal distribution with a mean of lnA_R50 and a standard deviation of ζ (where ζ = 0.50). A_f denotes the peak floor acceleration (PFA) response, V_f the peak floor response velocity, D_f the peak floor response displacement, α is the coefficient representing the influence of slip, B is the depth of the furniture, H is the height of the furniture, and g is the gravitational acceleration. In this research, the constants in Equation (5) are determined as α = 0.8 and ζ = 0.50.

This study examines three types of furniture with dimensions specified in

Table 2. Furniture profiles.

Type	B [mm]	H [mm]	Fb [Hz]
I (low)	45.0	120	0.883
II (medium)	45.0	150	0.859
III (tall)	45.0	200	0.814

. The depth of furniture B remains constant while the height H is varied to modify the dimensions. The respective cases are categorized as “low”, “middle”, and “high”. The furniture is assumed to be not anchored to the walls or floors. The frequency of furniture F_b varies depending on the height of the furniture.

$$F_b={\displaystyle \frac{15.6}{\sqrt{H}}}{\left(1+{\displaystyle \frac{B}{H}}\right)}^{-1.5}$$

(7)

The fragility curves calculated using the above equations are presented in Figure 7. In Figure 7, the taller furniture is more likely to overturn at lower floor accelerations.

4. Computation of Artificial Seismic Waves Reflecting Ground Characteristics

The seismic waves are calculated considering the rupture process of faults and the ground structure. Three locations in central Sendai were selected, focusing on the Nagamachi-Rifu Fault (M7.5) located directly beneath the city. Figure 8 illustrates the surface projection of the characterized source model from the National Seismic Hazard Map for Japan [140] and the locations of the two strong motion generation areas [141]. In this analysis, two cases with different rupture initiation points are evaluated. The respective locations are marked by ★.

The evaluation of seismic waveforms follows the methodology outlined in the National Seismic Hazard Map for Japan [140]. First, the waveforms at the engineering bedrock level are predicted using the stochastic waveform synthesis method [142]. The component waveforms are artificially generated using the envelope shape from Boore [143] and random phase. The Q value of the propagation path is given by the following equation [144]:

$$Q=110f^{0.7}$$

(8)

where f represents the frequency. The amplification rate of the deep ground from the seismic bedrock to the engineering bedrock is derived from the 1D structure directly beneath each site, using the subsurface structure model applied in earthquake damage estimation for Sendai [145]. Ultimately, the amplification rate is calibrated under the condition of vertical incidence of S-waves. The response of the surface ground layer above the engineering bedrock is approximated using a modified R-O model [146], which accounts for strain-dependent rigidity and attenuation as used in the earthquake damage simulation for Sendai [145]. Surface acceleration waveforms are then obtained through a fully nonlinear time-domain analysis of total stress, using the seismic motion at the engineering bedrock as the input.

The coordinates of the three locations, along with the PGA and PGV of the calculated waveforms at the surface, are shown in

Table 3. Profile of seismic waves (unit: m/s² for PGA and m/s for PGV).

Site	Longitude	Latitude	Case	Direction	PGA	PGV
Aobayama	38°15′18″ N	140°50′18″ E	Case 1	NS	7.77	0.64
				EW	6.55	0.58
			Case2	NS	10.1	0.65
				EW	11.3	1.64
Katahira	38°15′14″ N	140°52′26″ E	Case 1	NS	4.27	0.51
				EW	4.48	0.43
			Case2	NS	6.23	0.48
				EW	6.42	1.21
Nagamachi	38°13′26″ N	140°52′49″ E	Case 1	NS	5.63	0.54
				EW	5.25	0.52
			Case2	NS	6.71	0.73
				EW	5.89	1.32

. Figure 9 presents the acceleration response spectra with a damping factor of 2%. All evaluation points are located in central Sendai City, near the upper edge of the strong motion generation area on the southern side. The differences observed between the evaluation points are primarily due to variations in ground conditions.

5. Contribution of Seismic Retrofitting to Structural Response Reduction Based on Incremental Dynamic Analysis

The input is standardized based on the acceleration response of the Single-Degree-of-Freedom (SDOF) system with a damping factor of 0.02. The natural periods of the structure for the MRF and BRBFs are derived from eigenvalue analysis. The intervals range from 0.1 G to 3.0 G with a step size of 0.1 G, resulting in a total of 2160 cases. The seismic waves utilized in the IDA are the 12 waves generated in the preceding chapter.

Figure 10 presents the peak story drift for the MRF and BRBFs. The horizontal axis represents S_a(T), and the vertical axis shows the peak story drift. Since the story drift becomes the most significant in the first story, the floor displacement of the second story δ₂ is extracted in Figure 10. The median and 16^th/84^th percentile lines are depicted by bold black lines. Overall, the story drift decreases with more stringent design criteria, and building collapse can be prevented by retrofitting with BRBFs.

Figure 11 summarizes the PFA. According to previous studies [54], furniture overturning is associated with PFA. Unlike story drift, PFA cannot be mitigated by more stringent BRB design criteria (see Figure 4c). Increased BRB stiffness, intended to reduce displacement, may result in higher acceleration responses.

Additionally, the occurrence of furniture overturning is related to the building’s equivalent frequency, as considered by Equation (7). The A_R50 differs depending on the balance of the unique frequency of furniture F_b and the characteristic frequency of the building, thereby altering the fragility curve. The threshold acceleration for 50% of furniture overturning is shown in Figure 12. The furniture dimension is low. The median and 16^th/84^th percentile lines are also depicted in the figure. The equivalent frequency is calculated based on the building’s response, precisely the peak floor velocity V_f and peak floor displacement D_f, with the equation provided in Equation (7). Figure 12 demonstrates that the threshold acceleration increases with more stringent BRB design criteria primarily due to smaller peak floor displacements.

Figure 13 illustrates the calculation flow for deriving the fragility curve of furniture overturning. As summarized in Figure 11, the PFA is determined for each story. Simultaneously, the threshold acceleration for a 50% probability of furniture overturning is calculated using the equation provided by Kaneko [54]. The exceedance of this threshold is then evaluated for each ground motion. Finally, the exceedance ratio is computed for the respective spectral accelerations. The fragility curve is obtained by fitting a cumulative distribution function to a log-normal distribution.

Figure 14 illustrates the probability of furniture overturning. The horizontal axis represents the intensity of the input wave, and the vertical axis denotes the probability of exceeding 50% furniture overturning. The plots suggest that the likelihood of overturning decreases with more stringent design criteria. Although PFA can increase with greater BRB stiffness and capacity, the threshold acceleration also rises with more stringent BRB design criteria. Consequently, the fragility curve for θ_t = 1/200 rad (γ_Ia_g = 4.2 m/s²) becomes the smallest in Figure 14.

6. Evaluation of Mean Annual Frequency of Furniture Overturning

To assess the effectiveness of seismic retrofitting in business continuity planning, the mean annual frequency of furniture overturning is determined based on the framework depicted in Figure 15.

This study utilizes the mean annual frequency λ_c for risk evaluation. Calculating λ_c requires two key elements: the seismic hazard curve, which represents the mean annual frequency of exceeding ground motion intensities at a specific site, and the fragility curve for furniture overturning. Ground motion intensity is quantified using an intensity measure (IM), such as S_a(T), the spectral acceleration at the structure’s fundamental period.

The value of λ_c is obtained by integrating the furniture fragility curve with the site-specific seismic hazard curve using the following equation:

$${\lambda }_c={{\displaystyle \int }}_0^{\infty }P\left(C|im\right)·\left|d{\lambda }_{IM}\left(im\right)\right|$$

(9)

where P(C∣IM) represents the probability of furniture overturning under an earthquake with a given ground motion intensity level IM, and λ_IM denotes the mean annual frequency of exceeding the ground motion intensity IM. By multiplying and dividing the right-hand side of Equation (10) by d(IM), the expression for calculating λ_c can be reformulated as the following:

$${\lambda }_c={{\displaystyle \int }}_0^{\infty }P\left(C|im\right)·\left|{\displaystyle \frac{d{\lambda }_{IM}\left(im\right)}{d\left(im\right)}}\right|d\left(im\right)$$

(10)

where dλ_IM(IM)/d(IM) represents the slope of the seismic hazard curve at the site. A closed-form solution for the integral in Equation (11) is usually unavailable, so the integral is typically evaluated through numerical integration. This involves computing the product of the collapse probability conditioned on IM, the slope of the seismic hazard curve at specific IM levels, multiplying by the increment in IM(ΔIM), and summing the results across all IM levels. This procedure is expressed in Equation (12) and visually depicted in Figure 15.

$${\lambda }_c={\displaystyle \sum }_{i=1}^{\infty }P\left(C|i{m}_i\right)·\left|{\displaystyle \frac{d{\lambda }_{IM}\left(i{m}_i\right)}{d\left(im\right)}}\right|·\mathsf{\Delta }im$$

(11)

6.1. Relationship between λ_c and Probability of Collapse

λ_c denotes the mean annual frequency of furniture overturning. Assuming that earthquake occurrences follow a Poisson process over time, the probability of at least one overturning event within a period of t years can be determined using the following equation:

$$P_c\left(in t years\right)=1-exp\left(-{\lambda }_{c}t\right)$$

(12)

Given that λ_c is typically a small value for most buildings, the annual probability of overturning is approximately equal to λ_c, expressed as follows:

$$P_c\left(in 1 years\right)\cong {\lambda }_c$$

(13)

6.2. Deaggregation of λ_c

λ_c alone does not reveal which ground motion intensities contribute most to the collapse risk. To address this, a deaggregation of λ_c provides a valuable method for identifying the relative contributions of various ground motion intensities to the overall collapse risk. This process corresponds to the deaggregation by magnitude, distance, and epsilon (ε) commonly used in PSHA to determine the primary sources contributing to the hazard. The parameter ε quantifies the deviation, in terms of logarithmic standard deviations, between the observed ground spectral acceleration and the spectral acceleration predicted by an attenuation relationship at an arbitrary period.

A point on the λ_c deaggregation curve is retrieved as the product of the overturning probability at a ground motion intensity and the slope of the hazard curve as an intensity function. As indicated in Equation (12), the contribution of a specific (narrow) range of ground motion intensities to λ_c is calculated by multiplying the collapse probability at the midpoint intensity of the range by the slope of the hazard curve at the intensity and by the width of the intensity, ΔIM. As illustrated in Figure 15, the total area under the deaggregation curve equals λ_c. Ground motion intensities with higher values on the deaggregation curve represent more remarkable contributions to λ_c.

Figure 15 demonstrates that ground motion intensities associated with high overturning probabilities do not always contribute significantly to λ_c, as these intensities occur less frequently than those in the lower half of the overturning fragility curve. Consequently, the most significant contribution to λ_c generally comes from intensities within the lower half of the fragility curve, where the collapse probabilities may be smaller. Still, the seismic hazard curve’s slope is typically steeper compared to that for higher intensities.

6.3. Hazard Curve Deaggregation of λ_c

As one of the most earthquake-prone regions on the planet, Japan is chosen as the case study. The PSHA requires a hazard curve in spectral form. The National Research Institute for Earth Science and Disaster Resilience (NIED) provides the Japan Seismic Hazard Information (J-SHIS) system [147,148]. J-SHIS has developed a database of hazard curves based on the response spectrum for exceedance probabilities corresponding to a 50-year return period. These hazard curves are constructed by combining various scenarios involving subduction zones and near-fault earthquakes.

The damping factor provided by J-SHIS is set at 5%, which is generally applicable to reinforced concrete (RC) structures. However, the typical damping factor for steel structures is 2%, as used in the simulations conducted in this research. Therefore, the spectral response data from J-SHIS must be adjusted to reflect the corresponding damping factor. For this conversion, the following equation is provided by AIJ [5].

$$D_h=\sqrt{{\displaystyle \frac{1+25h_0}{1+25h_{eq}}}}$$

(14)

where h₀ is the initial damping factor, and h_eq is the equivalent damping factor. Additionally, the probability for a return period of T_c is converted to the mean annual frequency using the following equation.

$${\lambda }_c=-{\displaystyle \frac{ln\left(1-P_c\right)}{T_c}}$$

(15)

where P_c is the probability of exceedance and T_c designates the time interval where the probability P_c is calculated [149].

Additionally, the hazard curve is provided for specific natural periods: 0.1 s, 0.2 s, 0.3 s, 0.5 s, 1.0 s, 2.0 s, 3.0 s, and 5.0 s. To derive the hazard curve corresponding to the natural periods of the MRF and BRBF systems, the hazard curves are linearly interpolated in log-log space. Since the hazard curves in J-SHIS are derived from empirical equations using arbitrary constants, theoretical interpolation is not possible. Instead, previous studies have customarily interpolated the hazard curve in log-log space [69,150]. Therefore, the interpolation method used in this study follows earlier investigations [69,150].

The target locations are listed in

Table 4. Target locations for risk assessment.

Location	J-SHIS Mesh Code	Longitude	Latitude	Elevation
Hokkaido	6441427814	43.0615 N	141.3547 E	21 m
Miyagi	5740362921	38.2677 N	140.8703 E	46 m
Tokyo	5339452532	35.6885 N	139.6922 E	38 m
Ishikawa	5436657223	36.5615 N	136.6578 E	27 m
Aichi	5236671243	35.1823 N	136.9078 E	10 m
Hyogo	5235012543	34.6906 N	135.1953 E	6 m
Hiroshima	5132436612	34.3844 N	132.4547 E	2 m
Kumamoto	4930156623	32.8031 N	130.7078 E	13 m

, which includes sites selected from various regions across Japan. The hazard curves retrieved from J-SHIS are summarized in Figure 16. The frequency generally becomes greater in a shorter period. The discrepancy varies depending on the region, and Aichi exhibits a relatively huge gap among the natural period.

Figure 17a illustrates the mean annual frequency for low-height furniture. Overall, the MRF demonstrates the highest mean annual frequency except in Aichi. The greatest frequency is observed in the case of θ_t = 1/200 rad (γ_Ia_g = 3.6 m/s²). The hazard curve in Figure 16 shows that Aichi has a higher probability of occurrence in a shorter natural period. Due to the seismic retrofitting principle using BRB, the natural period shortens. As Aichi’s seismic motion characteristics cause more significant acceleration responses in buildings with shorter natural periods, the computed mean annual frequency can exceed that of buildings without seismic retrofitting.

Additionally, the risk of furniture overturning at θ_t = 1/200 rad (γ_Ia_g = 3.0 m/s²) and θ_t = 1/200 rad (γ_Ia_g = 4.2 m/s²) is nearly identical in most cases. As shown in Figure 11, the acceleration response increases in θ_t = 1/200 rad (γ_Ia_g = 4.2 m/s²) due to greater stiffness, while the threshold acceleration also increases. Consequently, the probability of furniture overturning is lowest in θ_t = 1/200 rad (γ_Ia_g = 4.2 m/s²) according to the fragility curve in Figure 14. Meanwhile, the probability of exceedance in acceleration response tends to be more pronounced at shorter natural periods. As previously mentioned, the mean annual frequency is calculated as the product of the fragility curve and the derivative of the hazard curve. Thus, the effectiveness of seismic retrofitting with BRB depends on the balance between the reduction in fragility and the resonance with the seismic wave.

Figure 17b presents the case for medium-height furniture, where the risk of overturning is nearly the same for MRF and θ_t = 1/120 rad. According to the fragility curve in Figure 17b, fragility is reduced through seismic retrofitting. However, the spectral response increases due to the shorter natural period, resulting in nearly identical computed mean annual frequencies. The lowest frequency is observed in θ_t = 1/200 rad (γ_Ia_g = 4.2 m/s²) across all locations, with the second lowest in θ_t = 1/200 rad (γ_Ia_g = 3.0 m/s²). The second-most stringent design criterion of θ_t = 1/200 rad (γ_Ia_g = 3.6 m/s²) exceed the aforementioned two cases and is highest in Aichi.

Figure 17c displays the mean annual frequency for tall furniture. The difference between retrofitting cases is less pronounced compared to low and medium-height furniture. The magnitude relation remains consistent with Figure 17b. However, the lowest mean annual frequency is achieved by MRF, owing to a more significant hazard curve at shorter natural periods. The mean annual frequency reaches 3.8 times greater in θ_t = 1/200 rad (γ_Ia_g = 3.6 m/s²) compared to the MRF.

6.4. Discussion

Seismic retrofitting with BRBs effectively reduces displacement response and prevents building collapse. The design procedure has become straightforward due to the use of equivalent linearization. This method begins with determining the target story drift, making displacement response the primary concern. However, the risk of furniture overturning can increase as the building’s natural period shortens due to the added stiffness from the BRBs. In particular, the mean annual frequency increases when the spectral response at shorter periods becomes more significant. The proposed method quantifies this risk, enabling engineers to consider BCP when designing BRBFs for seismic retrofitting.

This study assumes that the furniture is not anchored. The fragility of furniture overturning can be mitigated through proper anchorage. Even with the installation of seismic damping devices, careful attention is required to maintain business continuity.

7. Conclusions

This study applies probabilistic risk assessment (PRA) to the issue of furniture overturning. Buckling-restrained braced frames (BRBFs) are designed using various design criteria based on the moment-resisting frame (MRF) provisions. Incremental dynamic analysis (IDA) is performed using artificial seismic waves that reflect realistic ground conditions. The resulting fragility curve for furniture overturning is then compared with hazard curves across Japan. The findings are summarized as follows:

(1)

Peak story drift is effectively mitigated with more stringent BRB design criteria. However, when the target story drift is reduced, peak floor acceleration can increase in BRBFs compared to MRFs.

(2)

The fragility of furniture overturning decreases with more stringent design criteria due to an increased critical acceleration threshold.

(3)

The mean annual frequency of acceleration response spectra generally increases with shorter natural periods. Consequently, the mean annual frequency of furniture overturning can be higher in BRBFs compared to MRFs.

(4)

The risk of furniture overturning is influenced by the balance between the building’s performance and the specific hazard characteristics at the location. Particular attention should be given to the shape of the hazard curve, especially in cases where short-period vibrations are prominent.

(5)

The calculation flow outlined in this research provides engineers and researchers with a clear methodology to quantify the risk of furniture overturning in a precise and explicit manner.

The model structure used in this research is a four-story building. In future research, it will be essential to assess the potential risk of furniture overturning in high-rise structures to extend safety warnings to the broader community. Future studies will explore the application of this methodology to buildings with varying aspect ratios and address additional business continuity planning (BCP) concerns, such as roof collapses. A more comprehensive investigation will be presented in subsequent papers.