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Engineering Structures 259 (2022) 114183

Available online 6 April 2022 0141-0296/© 2022 Elsevier Ltd. All rights reserved.

Empirical seismic fragility assessment and optimal risk mitigation of building contents

Sereen Majdalaweyh *, Weichiang Pang Glenn Department of Civil Engineering, Clemson University, Lowry Hall, SC 29634, United States

A R T I C L E I N F O

Keywords: Building contents Restrainers Fragility function Loss function Mitigation FEMA P-58

A B S T R A C T

Building contents proved challenging in performance-based earthquake engineering frameworks because of the data scarcity. Despite the content seismic losses in the past two decades, the overall effectiveness of restrainers to contents has not been thoroughly investigated. This paper aims to provide a detailed analysis for assessing content damage, loss, and protection to better understand content risk and performance. In this paper, analytical fragility functions for rigid block-type contents are developed based on nonlinear time history analysis of four dynamic models to represent freestanding and anchored building contents with elastic-brittle restrainers considering multiple failure modes. The effectiveness of restrainers is quantified by the improvement of the fragility median. A simplified expression is proposed to estimate content lognormal fragility parameters based on a wide range of variables’ correlation between contents’ characteristics and restraint strength. The proposed simplified expression is adopted in FEMA P-58 loss assessment framework through a case study of a 4-story light- frame wood office building to estimate content dollar loss. To further investigate content risk mitigation, we used a quantitative loss assessment for multiple mitigation scenarios by anchoring different content types such as heavy, expensive, electrical, furniture, or glassware. It is found that elastic-brittle restrainers are more effective for rigid systems than flexible systems, and decision-makers should design restrainers based on a combination of block-like content and restrainer characteristics for efficient protection and life safety. For the case study, anchoring expensive components was the optimal mitigation scenario, which resulted in a 74% reduction in the average annual losses compared to freestanding contents.

1. Introduction

Seismic performance of structural and nonstructural components has been the subject of research in terms of modeling and design. However, only a scant body of knowledge exists for loss estimation of building contents, despite the fact that contents contribute a large ratio of the building’s total investment cost accounting for 20%, 17%, and 44% of total building cost in typical offices, hotels, and hospitals, respectively [1]. Additionally, building contents contribute significantly to seismic monetary losses even at lower earthquake shaking intensity, where structural losses are negligible. One such notable event is the 1989 Loma Prieta earthquake (Mw = 6.9), which resulted in significant losses due to damage in contents; Two libraries in San Francisco each suffered over a million dollars in damage due to the overturning of bookshelves [2]. Another example is the 2014 South Napa Valley Earthquake (Mw = 6.0); 56% of total inspected buildings suffered significant content damage. It was also found that in some cases, contents governed the overall losses;

the average financial loss from contents was almost double that of the nonstructural component damage. For example, a 2-story reinforced concrete restaurant building suffered damages from the 2014 South Napa Valley earthquake, where the total content loss reached $15,000 from damages to wine bottles and plates [3].

Considering these past observations, it can be concluded that un- derstanding the seismic performance of contents is of substantial value and can lead to loss prevention and improved public safety. One proven strategy for reducing content losses is to protect contents by holding them down using restrainers. The efficiency of a restrainer can be quantified within the seismic performance-based framework where building seismic risk is quantified via negative consequences usually through four steps: (1) probability seismic hazard analysis, (2) building response analysis, (3) damage measure analysis, and (4) loss analysis [4,5]. In this context, damage measure analysis can be represented by fragility function, which gives a probabilistic relation between damage and structural response that can be further assessed in quantifying losses

* Corresponding author. E-mail addresses: [email protected] (S. Majdalaweyh), [email protected] (W. Pang).

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Engineering Structures

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https://doi.org/10.1016/j.engstruct.2022.114183 Received 6 May 2021; Received in revised form 17 February 2022; Accepted 20 March 2022

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using different measures (dollar, casualty, and downtime). The fragility function describes the conditional probability of the building compo- nent (content) entering a specified Damage State (DS) (e.g., minor damage or complete damage). A DS is reached when the Engineering Demand Parameter (EDP), which is the structural response parameter such as Peak Floor Acceleration (PFA), exceeds a predefined threshold.

A broad spectrum of research studies provided freestanding content fragility functions against its major failure modes, namely sliding and overturning; Some research works developed analytical sliding and overturning fragility functions [6-12] using Nonlinear Time History Analysis (NTHA) or so-called Incremental Dynamic Analysis (IDA) [13] where the Intensity Measure (IM) of a set of ground motions is scaled up until failure occurs. However, NTHA is only applicable to a predefined range of deterministic parameters. However, the seismic response of contents relies on many factors (occupancy type, content type, and restrainer condition) [14]. An attempt was made to simplify the NTHA procedure to derive the freestanding fragility function, where some studies provided generalized sliding fragility for standard laboratory equipment [14,19,20]. And other studies developed generalized rocking fragility function based on dimensionless Intensity Measure (IM) [18,19].

Furthermore, to avoid going through NTHA, FEMA P-58 (P58) [14] provided an online database by also known as the Performance Assess- ment Calculation Tool (PACT), which comprises component-based fragility functions and consequence data. The PACT content fragility database was limited to 11 content components: bookcases, cabinets, and desktop computers. This group of components does not reflect all components inside different building occupancy types. P58 content fragility functions are based on American Society of Civil Engineers ASCE 43-05 calculations for nuclear components. However, ASCE 43-05 equations proved inefficient for sliding and overturning fragility func- tions [9,10]. Consequently, all these existing options provide limited guidance for engineers and risk managers to control content losses. Therefore, a generalized, more robust fragility functions database covers

a wide range of commercial building contents is needed. In terms of restrained contents, limited research focused on anchored

contents: Garcia and Soong [7] developed a sliding fragility function for anchored rigid block electrical components. Contento et al. [22] developed dimensionless rocking fragility functions for anchored rigid blocks and found the effectiveness of base isolation and pendulum mass damper based on the probability of failure; however, the effectiveness of base anchorage in this study was lacking.

This paper addresses some of the critical issues: (1) improve content fragility development for both freestanding and anchored contents subjected to multiple failure modes and expand the existing content fragility, and consequence database; (2) provide a thorough walk- through of contents risk assessment approach; (3) discuss the possible and optimal risk mitigation scenarios through contents financial loss disaggregation and integrated loss figures such as loss function and annual average loss.

This paper proposes a simplified expression to predict fragility pa- rameters of block-like components. The interest of this work is block-like building contents in commercial buildings. This simplified expression covers a comprehensive content database for a wide range of content characteristics variables and restraint strength. The proposed expression characterizes the lognormal median and dispersion of contents’ fragility function where Peak Horizontal Floor Acceleration is used as an IM. The resulted fragility functions can then be used to quantify the effectiveness of restrainers as compared with freestanding contents in terms of fragility median improvement. The simplified expression is helpful to assess content losses and study the effectiveness of anchorage in the context of losses. Fragility functions are employed in the P58 loss assessment framework through a case study of a 4-story light-frame wood office building. It is worth mentioning that different content types (expensive, heavy, furniture, and fragile) are anchored separately or together to predict the best mitigation scenario.

This paper is organized after the introduction as follows. The first part, “Fragility assessment,” explains the fragility derivation and

Fig. 1. Content representation as a rigid block: (a) rigid block scheme, (b) sliding of the freestanding rigid block, (c) rocking of the freestanding rigid block, (d) brittle elastic behavior of restrainers, (e) sliding of anchored rigid block, (f) rocking of anchored rigid block.

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simplified expression development. The second part, “Loss assessment,” discusses the content consequence functions, losses, and mitigation. Finally, a “Case study” is presented as an application for the proposed methodology.

2. Fragility assessment

Fragility functions can be developed using earthquake reconnais- sance data, experimentally, analytically, or by expert judgment [14]. Due to lack of sufficient data the primary focus of this paper is on analytically developed fragility functions of selected contents defined in terms of the conditional probability of content damage due to multiple failure modes; overturning and excessive sliding. In this study, building contents are modeled as rigid blocks. Contents fragility functions are developed by solving nonlinear dynamic models of rigid blocks with and without restrainers (anchorage devices) considering sliding and rocking. Several past studies [23,24] have shown that Peak Floor Acceleration (PHFA) correlates well with contents’ damage probability. Thus, PHFA is used as an intensity measure for the content fragility curves.

2.1. Dynamic models for rigid blocks with or without restrainers

Housner [25] and Newmark [26] proposed the first freestanding rocking and sliding dynamic model, respectively. Some literature exists on the sliding and rocking dynamic models for freestanding blocks [27- 30]. On the other hand, limited studies focused on the dynamic response of the restrained rigid blocks. Garcia and Soong [31] studied the sliding response of anchored rigid blocks with post-tensioned cables, and the

rocking response of anchored rigid blocks with restrainers was studied by [23,32].

In this paper, contents are modeled as rectangular rigid blocks (Fig. 1a) with a weight (W), a height (h), and a width (b) resting on a horizontal surface, subjected to horizontal and vertical ground motions denoted as ẍg(t) and ÿg(t), respectively. The restrainers are assumed to have an elastic-brittle behavior, as shown in Fig. 1d.

2.1.1. Sliding model A schematic of the freestanding and anchored sliding block is shown

in Fig. 1b and Fig. 1e, respectively. The variable × denotes the relative motion between the center of the content and the supporting surface (e. g. floor). Adopting a surface-block simple coulomb-type friction model, the equation that governs the nonlinear dynamic sliding response of anchored block is as follows [31]:

ẍ(t) + μ [

g + ÿg(t) ]

sgn[ẋ(t) ] + g.Fu W.xu

x(t) = − ẍg(t) (1)

where, ẍ(t), ẋ(t) and x(t) are the relative acceleration, velocity, and displacement response of the rigid block, respectively. Fu is the re- strainer’s ultimate strength, g is the gravitational acceleration, and sgn[ẋ(t) ] is the signum function. The block starts to slide once the normalized inertial force |ẍg(t)| (normalized by the mass of the content)

exceeds the normalized friction force μ [

g +ÿg(t)

]

. Eq. (1) is only valid

before the breakage of the restrainer when the sliding displacement x exceeds the ultimate or rupture displacement (xu). The rupture

displacement can be defined by:

xu = σg ( Teq/2π

)2 (2)

where Teq = 2π ̅̅̅̅̅̅̅̅̅̅̅̅̅ m/Keq

√ , the natural period of the system in the absence

of friction, Keq is the total stiffness of the system, and σ = Fu/W is the strength ratio, which is the ultimate strength of the restrainer normal- ized by the weight of the content. When the restrainer breaks, the equation reduces to the freestanding nonlinear block response (Fig. 1b):

ẍ(t) + μ [

g + ÿg(t) ]

sgn[ẋ(t) ] = − ẍg(t) (3)

According to the normalized equation of motion shown in Eq. (1), the sliding response of an anchored rigid block depends on three vari- ables, namely, μ, σ, and xu, whereas the freestanding sliding response of a rigid block given in Eq. (3) depends only on one variable, namely the friction coefficient, μ.

2.1.2. Rocking model A schematic of the rocking block is shown in Fig. 1c and Fig. 1f. The

block starts to rock when the overturning moment induced by the hor- izontal inertia force exceeds the sum of the restoring moment produced by the weight of the block and the moment due to vertical inertia force. After the initiation of rocking, the block will continue to rock until the rotational angle exceeds the overturning stability angle α which is equal to tan− 1(γ) where γ is the slenderness ratio defined as width to height ratio. The nonlinear dynamic model for rocking motion is given below [33]:

where ϑ̈(t) and ϑ(t) is the angular acceleration and the rotation angle of the block, p is the frequency factor, which is equal to

̅̅̅̅̅̅̅̅̅̅̅̅̅̅ 3g/4R

√ , and R is

the block radius, which is equal to 0.5 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ b2 + h2

√ . sgn[ϑ(t) ] is the signum

function. ϑu is the ultimate or the rupture rotational angle for the restrainer. f (ϑ) is the fracture function defined as follows:

f (θ) = {

1, ϑ(t) ≤ ϑu 0, ϑ(t) ≥ ϑu

}

(5)

According to this model, the rocking response of anchored rigid blocks depends on α or γ, R, σ, and ϑu. The freestanding rocking response of the block can be represented by Eq. (4) when f(ϑ) equals to zero and it depends on α or γ , and R. In order to encounter the conservation of angular momentum, the coefficient of restitution is needed. The maximum coefficient of restitution considered as

( 1 − 32sinα

2 )2

[33].

2.2. Failure mechanism

Three mutually independent and sequential damage states are considered: (1) DS1: restraint breakage, which results in the need to repair or replace the restraint. However, it is assumed that the content itself is undamaged in Damage State 1 (DS1). The main postulate of the sliding dynamic model is that the restraint breaks when the absolute maximum block displacement |x|max exceeds the rapture displacement xu. In the case of the rocking model, the restraint breaks when the ab- solute maximum rotational angle |ϑ|max exceeds the rapture rotational angle ϑu. (2) DS2: (i) Excessive sliding when the absolute max sliding

ϑ̈(t) = − p2 ⎧ ⎨

⎩ sin[αsgn[ϑ(t) ] − ϑ(t) ]

⎛

⎝1 + ÿg(t)

g

⎞

⎠ + ẍg(t)

g cos[αsgn[ϑ(t) ] − ϑ(t) ] +

3σgsinα2 2bϑup2

sinϑ(t)f (ϑ)

⎫ ⎬

⎭ (4)

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displacement exceeds a threshold displacement (xthreshold). The displacement threshold depends on the type of content as well as the occupancy type. For example, the threshold displacement for bench- mounted equipment is defined as the displacement where the equip- ment slide, fall off, and get damaged. Different studies performed experimental and analytical studies on bench-mounted laboratory equipment [6,8,15,17,34,35]. And it was found that the displacement threshold for bench-mounted laboratory equipment in the range of [5- 10] cm at μ being between [0.2–0.45]. In the case of office contents such as desks, cabinets, and drawers: (1) It may result in tearing off power cords and plugs, which can be crucial for medical equipment, (2) It may result in a linear collision among the contents, and (3) It may result in blockage of egress routes and doors. Accordingly, in this paper, a range of displacement thresholds is selected to represent different scenarios. (ii) Overturning of a rigid block occurs when the absolute ratio of the rotational angle exceeds the overturning stability angle. Table 1 lists the failure mechanism for multiple failure modes and limits states. More details will be discussed in the sections that follow.

2.3. Rigid block variables and response

In this study, a freestanding block-like content and its interaction with the supporting surface is characterized using three parameters: (1) slenderness ratio, γ ranges from slender content (0.1) to square-shape content (1.0) with a step size of 0.1; (2) friction coefficient, μ, ranges from low friction surface (0.05) to high friction surface (0.7) with a step size of 0.05 to represent different types of building contents on different surfaces; and (3) block radius represents the content size, R, ranges from 0.1 to 1.0 m at 0.1 m intervals. Two additional parameters are required to describe the characteristics of a restrained or anchored content: strength ratio σ (restrainer strength to content weight) and natural period Teq. A survey of actual contents and typical anchors indicates that the strength ratio ranges from 0.1 to 10, with a low natural period of 0.05 s (rigid) and 0.2 s (flexible). Variables’ ranges were selected based on a comprehensive survey of content components in different occu- pancy types; this survey is based on systematic data collection taxonomy [1]. These variables are considered mutually independent and summa- rized in Table 2. It is worth mentioning that contents are assumed to be insensitive to acceleration.

These variables are implemented in the aforementioned dynamic models and subjected to a group of ground motions scaled at horizontal PGA from 0.05 to 4 with a step size of 0.05; then, the nonlinear dynamic models are solved numerically using MATLAB R2019 function (ODE45). In this study, FEMA P695 bi-axial 22 pairs of far-field ground motion set from the PEER NGA database are used [36]. In order to remove the unwanted variability from magnitude, site source distance, and others but maintain the inter-record variability, records are normalized by its peak ground velocity.

Fig. 2 shows a sample response of a block representing a three- shelves bookcase with a height of 1 m, width of 0.3 m, and a weight of 0.5 kg resting on vinyl flooring with μ = 0.30. The restraint Fu is 40% higher than the bookcase weight to give a σ of 1.4, giving a Teq of 0.05 s. The bookcase is subjected to the 1994 Northridge earthquake at Bev- erley Hills station, assuming the Peak Ground Acceleration (PGA) is equal to PHFA. Fig. 2a presents the horizontal and vertical ground ac- celeration scaled to 0.3 g. Fig. 2b shows the freestanding and restrained block response. The figure emphasizes the effectiveness of the re- strainers in this case; first, for the sliding response, the restrained

Table 1 Failure mechanism for content multiple failure modes.

Installation Status

Damage State (DS)

Description Limit State

Anchored only DS1 Restraint Breakage due to sliding

|x|max ≥ xu

Restraint Breakage due to rocking

|ϑ|max ≥ ϑu

Free or anchored

DS2 Excessive sliding |x|max ≥ xthreshold Overturning |ϑ|max ≥ ϑ

Table 2 Parameters for freestanding and restrained (anchored) rigid block contents.

Comp. Variables Values range

Rigid block Slenderness ratio, γ [0.10–1.0] Friction coefficient,μ [0.05–0.7] Block radius, R (m) [0.10–1.0]

Restrainer Strength ratio,σ [0.10–10.0] Natural period, Teq (sec.) [0.05,0.2]

Fig. 2. Sliding and rocking response for a 3-shelves bookcase: (a) horizontal and vertical ground acceleration of 1994 Northridge ground motion at Beverly Hills station scaled to 0.3 g; (b) sliding and rocking response of freestanding and restrained bookcase.

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bookcase did not start sliding at 0.3 g while the freestanding bookcase moved for 11 cm. Second, for the rocking response, the restrainer pre- vented the block from overturning.

2.4. Fragility development and fitting

A number of 4.2 million nonlinear dynamic analyses (IDA) are con- ducted to derive content fragility functions for the range of parameters aforementioned in Table 2. The conditional probability of failure is calculated using Monte-Carlo Simulation (MCS) as the ratio of the sim- ulations that failed to the total number of simulations. Fig. 3a shows the IDA response of freestanding bookcase to sliding and rocking, and the corresponding fragility functions based on a prescribed damage state. The sliding fragility function is derived for a sliding threshold of 0.3 m as damage state, and the rocking fragility curve is developed for over- turning damage state. Furthermore, Fig. 3b presents the anchored bookcase IDA response and associated fragility functions. These fragility functions are derived for restraint breakage damage state.

Lognormal distribution fragility fitting is the most common distri- bution for most structural and nonstructural components [37,38]. Therefore, the probability of failure of contents is fitted to lognormal distribution where the parameters of the fragility function (median and dispersion) are derived using the Maximum Likelihood Estimation (MLE).

2.5. Simplified expression development

Content fragility functions can be developed using IDA, but for simplicity and applicability, generalized fragility functions are prefer-

Fig. 3. Bookcase response to different PHFA levels: (a) freestanding bookcase response and fragility functions at sliding threshold of 0.3 m or overturning; (b) anchored bookcase response and fragility functions when the restraint breaks.

Fig. 4. Correlation matrix between fragility curve median and models’ inde- pendent variables.

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able as they represent a wide range of most common contents. The generalized fragility functions are expressed by the median (θ) and dispersion (β) of the lognormal distribution. To understand the rela- tionship between θ(DS2) and the models’ variables (μ, α, R, σ, DM, Teq), the correlation matrix is presented in Fig. 4. In this figure, the variables: μ, α, R, σ, DM, in both freestanding and anchored blocks, show a positive correlation with the fragility median where the correlation factor (ρ) larger than one. However, the natural period shows a negative corre- lation (ρ < 1). The strongest correlation in sliding model is μ. When μ increases, the fragility median will increase, which indicates that there is a lower probability of exceeding the sliding limit. The strongest corre- lation for the rocking model is the overturning stability angle α, such that when α increases, the probability of overturning decreases.

Based on the correlation matrix, the median and standard deviation values are fitted into a surface function (second-degree polynomial function) using MATLAB R2019 nonlinear regression function [39]. Table 3 presents the proposed equations for freestanding blocks along with the corresponding R2 and Root Mean Square Error (RMSE). The proposed equations for anchor dynamic models are presented in Ap- pendix A. Based on the problem complexity; the equations are divided into different cases where linear interpolation is valid.

To quantify the accuracy of each equation, a comparison is made between the predicted values (proposed) and the analytical values (IDA based) for the sliding and rocking models in Figs. 5-7. Fig. 5a and b show the values of analytical versus predicted for the sliding and rocking model at both natural periods of 0.05 and 0.20 s, respectively. The first column of both Fig. 5(a and b) presents the median values, and the second column presents the dispersion values. The figures, and R2 values show that the equations give a good prediction to median and dispersion

values. Another importance for the proposed equations is the ability to

quantify the effectiveness of restrainers which assess designers to make a decision on the required restraint capacity for rigid-block components. To study the efficiency of restrainers, the ratio of anchored fragility median to the freestanding fragility median “Effectiveness Ratio” is calculated. When the ratio is less than or equal to one, it means that the restrainers are ineffective. The effectiveness ratio corresponds to different parameters and can be simply derived from the proposed equations.

3. Loss assessment

3.1. Content consequence and loss functions

The content loss function is defined as a function of loss ratio with respect to the spectral acceleration intensity at a given period of vibra- tion with a 5% damping. The loss ratio is the normalized repair cost to the total replacement value. In this paper, the four integrated steps of the Performance Based Earthquake Engineering (PBEE) framework are solved to find the loss function using a MATLAB toolbox called Apoca- lyptic Structural Assessment Program (ASAP) developed and adopted from previous studies [40-42]. The first step is to quantify the seismic hazard for the location of interest using a representative IM. The second step is to quantify the responses of the building under various levels of earthquake hazards. Different building response variables are referred to as the Engineering Demand Parameters (EDP). One way to quantify the EDPs is to construct a structural model and use nonlinear time history analysis to assess the dynamic responses of the building. The third step is to quantify the damage states (DS) of building components, which may include structural, nonstructural, and building contents, using a fragility framework. Finally, find the consequences or losses (monetary loss, casualty, or downtime), where each DS is associated with a consequence function. These four steps can be solved using the total probability theorem, which considers the uncertainty in each step.

ASAP framework follows FEMA P-58 recommendations where an array of loss scenarios is generated using four MCS modules, and each scenario is called a realization. Each realization initiates with checking the ’collapse state’ of the building using the collapse fragility curve obtained from building IDA. If the realization is deemed with building collapse, the content loss is the total content Replacement Cost (RC). If

Table 3 Proposed equations of the median (θ) and dispersion (β) for freestanding block.

Model Equation R2 RMSE

Freestanding sliding

θ = − 0.13 + 2.97μ + 3.25DM − 1.69μ2 + 1.08DMμ − 2.39DM2

0.992 0.0485

β = 0.44 − 1.04μ + 0.76DM + 0.97μ2 − 0.47DMμ − 0.72DM2

0.941 0.0288

Freestanding rocking

θ = 0.79γ + 0.05R + 0.56γ2 + 1.47Rγ − 0.16R2

0.998 0.0307

β = − 0.04 + 0.29γ + 0.74R − 0.19γ2 − 0.26Rγ − 0.26R2

0.937 0.0255

Fig. 5. Predicted and analytical median and dispersion for (a) sliding model equations; (b) rocking model.

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the realization resulted with no collapse, a detailed loss estimation is conducted to find the non-collapse losses. Accordingly, the content loss due to different failure modes, namely sliding and rocking, is estimated based on the fault tree model shown schematically in Fig. 6. This figure illustrates a sample fault tree model used to find content loss; the first two plots are the sliding and rocking fragility functions respectively for both freestanding and anchored components with μ = 0.40, σ = 1.0, γ = 0.6, R = 0.2 m and Teq = 0.05 s. The figure also shows content conse- quence functions for different content categories; furniture, electrical and fragile items, where the x-axis presents the ratio of damage cost to content value.

Based on engineering judgment, the content consequence function is assumed to be normally distributed with damage ratio as the median, and 20% variance in order to define the uncertainty of components’ repair cost. Content damage due to overturning or sliding is relative to the damage state. As explained in this study (see Table 1), it is assumed that each component has three damage states if anchored (DS0, DS1, DS2) and two damage states if freestanding (DS0, DS2). In this study, the damage ratio is the maximum damage the component suffers after an earthquake shaking. In the case of DS1, the damage ratio is the restrainer replacement cost ratio for all anchored components. In the case of DS2, the damage ratio depends on the content type. For example, glassware is expected to have 100% damage if it overturns whereas chairs will not. The damage ratio is assigned to contents: 30% for furniture, 80% for electrical components, and 100% for glassware [43].

Finally, the content non-collapse loss from each failure mode is estimated using MCS, then the maximum of the two modes is

Fig. 6. Fault tree model for content loss.

Fig. 7. Loss functions for all freestanding and all anchored components in the case study using the analytical and predicted fragility parameters.

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considered. Finally, the total content loss is a combination of collapse and non-collapse losses. In addition to the content loss function, one of the most important measures that helps benchmark insurance premiums and decision-makers and is used in this paper is the Average Annual Loss (AAL) which can be derived using the loss and hazard function [44].

4. Case study

A four-story light wood frame office building located in Seattle, Washington, is considered a prototype for this study. The building was designed for American Technology Council ATC 116 project [45,46] following ASCE 7–10 [47] for high seismicity, which refers to locations at the upper limit of seismic design category (SDC) D, referred to as “Dmax” and corresponding to a short-period response acceleration parameter of 1.00 g [47]. The building contains approximately 428 square m (4608 square ft) of office spaces and accommodates six 71 square m (768 square ft) office units or so-called consequence areas. The units are distributed into four levels. The design and plan details are described in [48]. The building is modeled using Timber 3D, a MATLAB- based nonlinear structural analysis software for wood frames developed by [49,50]. Timber 3D is capable of simulating three-dimensional

seismic response. The fundamental period of the building is 0.506 s. IDA is performed in order to generate building EDP’s needed for

performance modeling of the building, i.e. peak inter-story drifts, peak residual drifts, and peak floor accelerations under different intensity measures. The IDA is performed using the aforementioned 22 bi-axial far-field ground motions used for content fragility assessment. Ground motions are scaled according to FEMA P-695 procedure [36].

A total of 113 content objects are included in one consequence area. The normative quantity for each component, i.e., the quantity for each component per unit gross square area, is estimated based on engineering judgment. The quantity value is calculated based on Xactimate 2019 database [51]. Xactimate is a computer software that estimates personal property and emergency repairs. The list of contents and their charac- teristics (aspect ratio, friction coefficient, damage measure, quantity, and cost) is provided in Table B1.

4.1. Simplified expression in loss assessment

The simplified expression explained before is used to develop content fragility parameters for this case study. Both NTHA (analytical) fragility parameters and the predicted fragility parameters are used in the loss assessment framework to find the loss functions. Fig. 7 portrays the loss functions for anchored components as well as freestanding components with respect to Sa level at the building’s fundamental period. It also highlights three main intensity levels, namely: (1) Maximum Considered Earthquake level MCE (2% in 50 years, return period 2475 years), (2) Design Based Earthquake level DBE (10% in 50 years, return period 475 years), and (3) Service Level Earthquake level SLE (50% in 3 years, re- turn period 43 years) for this building.

Examination of the figure reveals that the derived equation and analytical fragility parameters have identical loss functions and further proves the accuracy of the proposed equation. Therefore the proposed equation can be used for developing loss functions for other scenarios. Also, as one can see from the figure, the loss for freestanding compo- nents is higher than anchored components, which aligns with the fact

Fig. 8. Maximum Peak Floor Acceleration (PFA) versus non-collapse content loss ratio at each floor: (a) loss ratio and PFA at SLE level. (b) loss ratio and PFA at DBE level.

Table 4 Mitigation scenario cases.

Mitigation Scenario

Description

Case1 Heavy components are anchored ≥ 45 kg, such as Fridge, plotter, commercial printer, and File cabinets. (19% of total content value)

Case2 Expensive components are anchored ≥ $100, such as desktop computers, monitors, and projectors. (67% of the total cost value)

Case3 Heavy and expensive components are anchored (72% of total content value)

Case4 All components are anchored (100% of total content value)

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that freestanding components are more vulnerable to damage. For instance, the loss ratio for freestanding components at SLE level is 0.3, whereas it is equal to zero for anchored components.

4.2. Mitigation scenarios

For each consequence area, no two contents are equally vulnerable to seismic excitation. Some of them are more vulnerable as compared to others. To reduce the vulnerability of such components, we can anchor them, but it is challenging to decide which components should be anchored for economic and feasibility reasons. Also, it is impractical to anchor all components. In order to find the loss incurred from any mitigation scenario and evaluate the efficiency of anchoring different components, one needs to follow the multi-fold PBEE methodology.

In order to choose the optimal mitigation scenario, we need to look at the freestanding and anchored content loss disaggregation. The most vulnerable contents in this case study can be classified into four cate- gories: (1) heavy components refer to components weigh more than 45 kg such as bookcases, (2) expensive components refer to components that cost more than $100, (3) electrical components, and (4) fragile items refer to glassware. Fig. 8 shows the non-collapse loss ratio of anchored (represented by dark color) and freestanding components (represented by light color) at each floor with respect to the demand distribution defined by Peak Floor Acceleration (PFA). For both figures, the PFA median increases from the ground floor to higher floors, therefore content loss ratio is increases because the content fragility intensity measure is acceleration. The second, third, and fourth floors have almost the same PFA and hence have the same content loss ratio.

Fig. 9. Loss results for all mitigation scenarios: (a) loss function for different scenarios; (b) anchored components loss ratio to freestanding components loss ratio.

Fig. 10. Non-collapse loss at each floor.

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At SLE level (see Fig. 8a), all anchored components have zero loss ratio contrary to the freestanding components. Also, expensive free- standing components hold the highest share in the loss ratio, followed by heavy components. At the DBE level (see Fig. 8b), the expensive and heavy anchored components start to fail and contribute to losses incurred, but they are significantly lower than the freestanding losses. Based on these results, four different mitigation scenario cases can be used in this case study, as listed in Table 4.

The loss function resulted from the detailed PBEE analysis for each mitigation scenario is presented in Fig. 9. The left diagram of this figure presents the loss ratio for each mitigation scenario with respect to the spectral acceleration at the building fundamental period. Additionally, the figure shows the lognormal fitted building collapse fragility. Most losses are incurred in the case of freestanding, and a downward trend can be observed as we go from case1, case2, case3, and finally case 4. This can be attributed to various characteristics like components’ damage (fragility), components’ consequence, and building demand which is the PFA.

The effectiveness of anchorage on losses for each mitigation scenario is quantified using the ratio between the anchored loss ratio to the freestanding loss ratio. The ratio of unity indicates that the anchorage of the associated components did not have any effect on the loss ratio. Fig. 9b presents this ratio with respect to the Sa at the building funda- mental period; it can be noticed that the ratio for all cases increases at higher intensity levels, which means that restrainers are more efficient at lower intensity levels. The effectiveness of restraints decreases at higher intensity levels due to the increase of the probability of building collapse (see Fig. 9a). Hence, higher collapse losses contribute to the failure of components regardless of their installation status (freestanding or anchored). The content non-collapse losses shown in Fig. 10 further emphasize this point. It can be noticed that at intensity levels higher than MCE at all floors, the non-collapse losses decrease until they become negligible when the building probability of collapse exceeds 90%.

Fig. 11 shows the AAL values for all mitigation scenarios. The AAL reduction ratio, which is the ratio between the difference of freestanding and anchored scenario AAL to the anchored scenario AAL, this ratio increases from case 1 through case 4 which proves that restrains help to reduce the annual content seismic loss, for example, the percentage reduction in case2 loss (anchoring expensive components) is 78%.

5. Summary and conclusions

This study developed a set of analytical fragility functions for a wide range of commercial building contents. Four main dynamic models were considered: (1) freestanding sliding rigid block, (2) freestanding rocking rigid block, (3) anchored sliding rigid block with elastic-brittle re- strainers, and (4) anchored rocking rigid block with elastic-brittle re- strainers. For each model, several incremental dynamic analyses (IDA) were performed to analytically derive the fragility functions. The re- strainers generally were found to be improving the performance of blocks under sliding and rocking. The elastic-brittle restrainers’ effec- tiveness varies with respect to the strength to weight ratio, the compo- nent natural period, and rigid block characteristics.

Based on the aforementioned correlation of friction coefficient, aspect ratio, size, strength ratio, and natural period with the fragility parameters, namely the lognormal median and dispersion, this study proposed a simplified expression to estimate fragility parameters for both freestanding and anchored contents. The final fragility expression obtained showed a good fit with the analytical fragility functions.

Anchoring different content types will affect the resulting content dollar loss. It is not reasonable to anchor all contents in commercial buildings. Therefore, decision-makers should choose their mitigation scenario carefully to get an optimal solution between the cost of anchors and the cost of repairs. A set of office contents was chosen in this study to evaluate restrainer effectiveness on economic losses. Office contents were classified into four main categories: (1) heavy (more than 45 kg), (2) expensive components (more than $100), (3) furniture, and (4) fragile items, taking into consideration that a component can follow multiple categories. Based on this classification, multiple scenarios were studied through a detailed loss assessment framework on a four-story light-frame wood office building to find the optimal scenario. Key re- sults and conclusions pertaining to the case study are as follows:

• Seismic loss mitigation due to restrainers decreases with an increase in intensity levels because at higher intensity levels the collapse loss is more dominant.

• Anchoring expensive components resulted in 80%, 64%, and 43% loss reduction at SLE, DBE, and MCE levels, respectively. However, anchoring heavy components resulted in an almost 20% reduction of loss ratio at the same levels. Therefore, anchoring expensive com- ponents like desktop computers, printers, and monitors mitigates more dollar losses.

• The annual average loss (AAL) values show that anchoring expensive components will result in a 79% reduction in AAL. In other words, AAL went down from 1.44% ($5800) for all freestanding components to 0.33% ($1350).

Finally, the proposed simplified expression used for both restrainer design and loss estimation will facilitate the decision on the optimal mitigation scenario. These simplified expressions offer flexibility to decision-makers to develop numerous fragility functions for a wide range of freestanding and anchored contents. The extension of studying contents’ damage and loss assessment in different occupancy types using the proposed model is straightforward.

CRediT authorship contribution statement

Sereen Majdalaweyh: Conceptualization, Methodology, Formal analysis, Writing – original draft, Visualization. Weichiang Pang: Writing – review & editing, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 11. Average Annual Loss (AAL) for different mitigation scenarios and freestanding components, AAL reduction percentage at each scenario.

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Acknowledgement

This research did not receive any specific grant from funding

agencies in the public, commercial, or not-for-profit sectors.

Appendix A. . Proposed equations of the median (θ) and dispersion (β) for anchored block

The median θ and dispersion β of anchored block sliding fragility function are represented by the second-order polynomial equation:

θ, β = a0 + a1 μ + a2σ + a3μ2 + a4 μσ + a5 σ2 (A1)

The parameters, R2, and RMSE of this equation are listed in Table A1 and Table A2. Five sets of equations represented for five Damage Measures (DM) for each equivalent period (Teq): (1) Ultimate displacement (xu), (2) 0.05 m, (3) 0.1 m, (4) 0.3 m, (5) 0.5 m.

The median θ and dispersion β of anchored block rocking fragility function is represented by second-order polynomial equation:

θ, β = b0 + b1 γ + b2σ + b3γ2 + b4 γσ + b5σ2 (A2)

In equation A(2), γ should be greater than 0.1 for all cases and greater than 0.2 at R of 1.0 m and Teq of 0.05 s. The parameters, R2, and RMSE of this equation are listed in Table A3 and Table A4. Three sets of equations represented for each DM and each equivalent period (Teq) at different block size (R): (1) R = 0.1 m, (2) R = 0.4 m, (3) R = 0.8 m.

Table A3 The parameters, R2, and RMSE of the median θ and despersion β of anchored block rocking fragility function at Teq of 0.05 sec.

DM Radius (m) Teq = 0.05sec b0 b1 b2 b3 b4 b5 R2 RMSE

ϑu R = 0.1 θ 0.221 − 0.194 − 0.278 0.875 1.823 − 0.019 0.973 0.251 β 0.023 0.466 0.015 − 0.255 0.004 0.002 0.471 0.093

ϑu R = 0.4 θ − 0.122 − 0.181 0.193 0.446 1.661 − 0.070 0.954 0.321 β 0.472 − 0.227 − 0.082 0.277 − 0.007 0.011 0.128 0.116

ϑu R = 0.8 θ − 0.089 − 0.608 0.102 0.304 1.721 − 0.064 0.957 0.300 β 0.494 0.089 − 0.040 − 0.075 − 0.020 0.005 0.265 0.051

α R = 0.1 θ 0.237 0.051 − 0.342 0.792 1.792 − 0.011 0.975 0.241 β 0.048 0.085 0.064 0.047 0.019 − 0.004 0.895 0.033

α R = 0.4 θ − 0.004 1.081 − 0.097 0.489 1.316 − 0.021 0.968 0.244 β 0.288 0.147 − 0.009 − 0.169 0.001 0.003 0.177 0.060

α R = 0.8 θ − 0.056 2.064 − 0.168 − 0.035 1.050 0.000 0.972 0.211 β 0.461 0.050 − 0.026 − 0.190 − 0.013 0.004 0.513 0.048

Table A1 The parameters, R2, and RMSE of the median θ and despersion β of anchored block sliding fragility function at Teq of 0.05 sec.

DM Teq = 0.05 sec. a0 a1 a2 a3 a4 a5 R2 RMSE

xu θ − 0.098 0.974 0.984 − 0.426 − 0.136 − 0.040 0.998 0.053 β 0.169 − 0.201 0.008 0.222 0.003 0.004 0.792 0.029

0.05 θ 0.026 1.707 0.785 − 0.464 − 0.288 − 0.006 0.993 0.099 β 0.171 0.026 − 0.029 0.031 − 0.012 0.010 0.630 0.035

0.10 θ 0.108 2.119 0.695 − 0.607 − 0.341 0.008 0.991 0.109 β 0.240 0.014 − 0.063 − 0.043 0.002 0.013 0.526 0.038

0.30 θ 0.442 3.076 0.454 − 1.141 − 0.365 0.036 0.988 0.111 β 0.459 − 0.449 − 0.126 0.244 0.063 0.017 0.717 0.036

0.5 θ 0.831 3.268 0.314 − 1.534 − 0.264 0.044 0.983 0.120 β 0.533 − 0.577 − 0.135 0.205 0.103 0.015 0.773 0.038

Table A2 The parameters, R2, and RMSE of the median θ and despersion β of anchored block sliding fragility function at Teq of 0.20 sec.

DM (m)

Teq = 0.20 sec. a0 a1 a2 a3 a4 a5 R2 RMSE

xu θ − 0.037 1.554 0.420 − 0.826 0.173 − 0.016 0.998 0.034 β 0.188 − 0.427 0.068 0.421 − 0.028 − 0.005 0.942 0.015

0.05 θ 0.050 2.155 0.354 − 1.029 0.064 − 0.011 0.986 0.078 β 0.246 − 0.307 0.013 0.349 − 0.041 0.002 0.681 0.022

0.10 θ 0.193 2.485 0.209 − 1.137 0.034 0.015 0.993 0.056 β 0.316 − 0.353 − 0.020 0.322 − 0.024 0.006 0.699 0.021

0.30 θ 0.535 3.363 0.078 − 1.673 − 0.009 0.026 0.992 0.049 β 0.511 − 0.744 − 0.073 0.541 0.033 0.008 0.885 0.022

0.5 θ 0.884 3.614 0.016 − 2.115 0.076 0.024 0.985 0.063 β 0.560 − 0.757 − 0.074 0.377 0.067 0.005 0.875 0.030

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Appendix B

See Table B1.

References

[1] Taghavi S, Miranda E. Response assessment of nonstructural building elements. Pacific Earthquake Engineering Research Center; 2003.

[2] FEMA E-74. Earthquake hazard mitigation for nonstructural elements; 2005. [3] FEMA P-1024. Performance of buildings and nonstructural components in the 2014

South Napa Earthquake; 2015. [4] Moehle JP, Deierlein GG. A framework methodology for performance-based

earthquake engineering. In: Proceedings of the 13th World Conference on Earthquake Engineering; 2004. p. 3812–4.

[5] Yang TY, Moehle J, Stojadinovic B, Der Kiureghian A. Seismic performance evaluation of facilities: Methodology and implementation. J Struct Eng 2009;135: 1146–54. https://doi.org/10.1061/(ASCE)0733-9445(2009)135:10(1146).

[6] Chong WH, Soong TT. MCEER-00-0005: Sliding fragility of unrestrained equipment in critical facilities. 2000. Doi: 00-0005.

[7] Lopez Garcia D, Soong TT. Sliding fragility of block-type nonstructural components. Part 1: Unrestrained components. Earthquake Eng Struct Dyn 2003; 32:111–29. https://doi.org/10.1002/eqe.217.

[8] Chaudhuri SR, Hutchinson TC. Fragility of bench-mounted equipment considering uncertain parameters. J Struct Eng 2006;132:884–98. https://doi.org/10.1061/ (asce)0733-9445(2006)132:6(884).

[9] Zhu ZY, Soong TT. Toppling fragility of unrestrained equipment. Earthquake Spectra 1998;14(4):695–711.

[10] Konstantinidis D, Nikfar F. Seismic response of sliding equipment and contents in base-isolated buildings subjected to broadband ground motions. Earthquake Eng Struct Dyn 2015;44:865–87. https://doi.org/10.1002/eqe.2490.

[11] Linde S. Rocking response of slender freestanding building contents in fixed-base and base-isolated buildings. McMaster University; 2016.

[12] Petrone C, Di Sarno L, Magliulo G, Cosenza E. Numerical modelling and fragility assessment of typical freestanding building contents. Bull Earthq Eng 2016;15: 1609–33. https://doi.org/10.1007/s10518-016-0034-1.

Table A4 The parameters, R2, and RMSE of the median θ and despersion β of anchored block rocking fragility function at Teq of 0.20 sec.

DM Radius (m) Teq = 0.20sec b0 b1 b2 b3 b4 b5 R2 RMSE

ϑu R = 0.1 θ 0.146 0.295 − 0.039 0.638 0.437 − 0.008 0.985 0.083 β − 0.020 0.279 0.057 − 0.090 0.044 − 0.007 0.924 0.031

ϑu R = 0.5 θ 0.129 0.505 − 0.097 0.447 0.552 − 0.002 0.993 0.070 β 0.103 − 0.102 0.088 0.193 0.015 − 0.008 0.928 0.028

ϑu R = 1.0 θ 0.065 0.450 − 0.044 0.476 0.569 − 0.009 0.985 0.102 β 0.265 − 0.171 0.015 0.182 0.022 − 0.001 0.258 0.093

α R = 0.1 θ 0.148 0.328 − 0.051 0.743 0.413 − 0.005 0.987 0.080 β − 0.012 0.329 0.050 − 0.153 0.047 − 0.006 0.924 0.029

α R = 0.5 θ 0.103 1.118 − 0.135 0.760 0.448 0.005 0.992 0.084 β 0.315 0.038 − 0.019 − 0.064 0.030 0.002 0.409 0.037

α R = 1.0 θ 0.085 1.549 − 0.125 0.784 0.443 0.002 0.988 0.110 β 0.518 − 0.256 − 0.040 0.057 0.047 0.002 0.533 0.042

Table B1 Content and restrainers characteristics and cost in the case study.

No. Component Aspect Ratio Radius (m)

Friction Coeff.

Restraint Capacity/ Weight

Component Cost per m2 ($)

Restraint Cost per m2 ($)

1 Beverage dispenser 0.3 0.4 0.3 11 0.27 0.08 2 Coffee machine 0.8 0.2 0.3 2 0.71 0.28 3 Under counter refrigerator 0.6 0.5 0.3 6 1.58 0.08 4 Desktop computer 0.3 0.2 0.3 2 5.33 0.84 5 Plotter 0.6 0.5 0.3 3 8.67 0.08 6 Laptop 0.6 0.2 0.3 10 5.33 0.28 7 Monitor 0.4 0.3 0.3 7 4.00 0.84 8 Photo Printer 0.4 0.3 0.3 3 1.07 0.09 9 LCD 20–24 in 0.4 0.2 0.3 2 2.00 0.09 10 Camcorder digital camera 1.5 0.1 0.3 17 4.07 0.09 11 Countertop microwave 0.7 0.2 0.3 2 0.93 0.19 12 Printer and Fax machine 1.0 0.3 0.3 10 0.80 0.24 13 Stand-alone projection screen 0.2 0.8 0.3 44 4.00 0.08 14 Projector 3.0 0.1 0.3 10 13.33 0.09 15 Scanner 1.0 0.1 0.3 6 1.13 0.09 16 Conference telephone 4.8 0.2 0.3 25 2.67 0.09 17 Laser Jet printer 1.2 0.3 0.3 24 1.93 0.16 18 Commercial grade printer 0.5 0.6 0.3 8 26.67 0.08 19 Office jet printer 2.7 0.2 0.3 2 0.67 0.09 20 DVD Drive 3.0 0.1 0.3 50 0.27 0.28 21 Big Glass 1.3 0.1 0.3 100 0.04 0.84 22 File cabinets 0.3 0.5 0.3 4 1.53 0.72 23 Bookcases 0.3 0.5 0.3 7 0.87 0.48 24 Vase 0.3 0.2 0.3 100 0.24 0.37 25 Coffee pot 0.9 0.1 0.3 100 0.19 0.19 26 Chair 0.5 0.6 0.3 29 1.13 2.20 27 Client Seating 1.0 0.7 0.3 3 1.20 0.44 28 Desk 0.8 0.5 0.3 3 0.93 0.44 29 Mug 1.3 0.1 0.3 100 0.09 0.84 30 Conference table 1.7 0.7 0.3 2 2.93 0.15 31 Work center 0.8 0.6 0.3 5 1.60 0.88

S. Majdalaweyh and W. Pang

Engineering Structures 259 (2022) 114183

13

[13] Vamvatsikos D, Cornell CA. Applied incremental dynamic analysis. Earthquake Spectra 2004;20:523–53. https://doi.org/10.1193/1.1737737.

[14] Pujols JCG, Ryan KL. Development of generalized fragility functions for seismically induced content disruption. Earthquake Spectra 2016;32:1303–24. https://doi. org/10.1193/081814EQS130M.

[15] Hutchinson TC, Ray CS. Simplified expression for seismic fragility estimation of sliding-dominated equipment and contents. Earthquake Spectra 2006;22:709–32. https://doi.org/10.1193/1.2220637.

[17] Konstantinidis D, Makris N. Experimental and analytical studies on the response of 1/4-scale models of freestanding laboratory equipment subjected to strong earthquake shaking. Bull Earthq Eng 2010;8:1457–77. https://doi.org/10.1007/ s10518-010-9192-8.

[18] Dimitrakopoulos EG, Paraskeva T. Dimensionless fragility curves for rocking response to near-fault excitations. Earthquake Eng Struct Dyn 2015;22:2015–33. https://doi.org/10.1002/eqe.

[19] Purvance M, Anooshehpoor A, Brune J. Freestanding block overturning fragilities: Numerical simulation and experimental validation. Earthquake Eng Struct Dyn 2008;37:791–808. https://doi.org/10.1002/eqe.

[20] Dar A, Konstantinidis D, El-Dakhakhni WW. Evaluation of ASCE 43–05 seismic design criteria for rocking objects in nuclear facilities. J Struct Eng 2016;142: 04016110. https://doi.org/10.1061/(asce)st.1943-541x.0001581.

[22] Contento A, Gardoni P, Di Egidio A, De Leo A. Probabilistic models to assess the seismic safety of rigid block-like elements and the effectiveness of two safety devices. J Struct Eng 2019;145:04019133. https://doi.org/10.1061/(ASCE) ST.1943-541X.0002431.

[23] Makris N, Zhang J. Rocking response and overturning of anchored equipment under seismic excitations; 1999.

[24] Soong T, Garcia DL. Seismic vulnerability and protection of nonstructural components. Proc the Joint NCREE/JRC workshop on International collaboration on earthquake disaster mitigation research. 2003.

[25] Housner G. The behavior of inverted pendulum structures during earthquakes. Bull Seismol Soc Am 1963;53:403–17.

[26] Newmark NM. Effects of earthquakes on damns and embankments. Fifth Rankine Lecture 1965.

[27] Ishiyama Y. Motions of rigid bodies and criteria for overturning by earthquake excitations. Earthquake Eng Struct Dyn 1982;10(5):635–50.

[28] Pompei A, Scalia A, Sumbatyan MA. Dynamics of rigid block due to horizental ground motion. J Eng Mech 1998;124:713–7.

[29] Andreaus U, Casini P. Rocking-sliding of a rigid block: friction influence on free motion. Eng Trans 1998.

[30] Shao Y, Tung CC. Seismic response of temporary structures. Nucl Eng Des 1998;181 (1-3):267–74.

[31] Lopez Garcia D, Soong TT. Sliding fragility of block-type nonstructural components. Part 2: Restrained components. Earthquake Eng Struct Dyn 2003;32: 111–29. https://doi.org/10.1002/eqe.217.

[32] Makris N, Eeri M, Black CJ, Eeri M. Uplifting and Overturning of Equipment Anchored to a Base Foundation. Earthq Spectra 2002;18:631–61. https://doi.org/ 10.1193/1.1515730.

[33] Makris N, Zhang J. Rocking Response of Anchored Blocks under Pulse-Type Motions. J Eng Mech 2001;127(5):484–93.

[34] Comerio MC. PEER testbed study on a laboratory building: exercising seismic performance assessment; 2005. doi: https://doi.org/10.1007/s00586-007-0483-y.

[35] Konstantinidis D, Makris N. Experimental and analytical studies on the response of freestanding laboratory equipment to eathquake shaking. Earthquake Eng Struct Dyn 2009;38:679–98. https://doi.org/10.1002/eqe.

[36] Applied Technology Council. FEMA P-695: Quantification of building seismic performance factors. CA, USA: Redwood City; 2009.

[37] Porter K, Kennedy R, Bachman R. Creating fragility functions for performance- based earthquake engineering. Earthq Spectra 2007;23:471–89. https://doi.org/ 10.1193/1.2720892.

[38] Baker JW. Efficient analytical fragility function fitting using dynamic structural analysis. Earthq Spectra 2015;31(1):579–99.

[39] MathWorks. MATLAB 2019. [40] Safiey A. Performance-based seismic loss estimation of buildings. Clemson

University; 2020. [41] Safiey A, Pang W, Majdalaweyh S. Rethinking treatment of irreparability in the

context of performance-based earthquake engineering 2019. doi: 10.1061/ 9780784482223.023.

[42] Safiey A, Ziaei E, Gu M, Pang W, Rokneddin K, Javanbarg M. Influence of irreparability fragility on seismic vulnerability assessment of buildings. Eleventh US National Conference on Earthquake Engineering 2018;19103.

[43] Reinoso E, Jaimes MA, Esteva L. Seismic vulnerability of an inventory of overturning objects. J Earthquake Eng 2010;14:1008–21. https://doi.org/ 10.1080/13632460903527971.

[44] Applied Technology Council. FEMA P-58-1: Seismic performance assessment of buildings, vol. 1; 2018.

[45] Applied Technology Council (ATC). Solutions to the issues of short-period building seismic performance; 2018.

[46] Kircher C, Pang W, Ziaei E, Filiatrault A, Heintz J. Solution to the short-period building performance paradox. Eleventh U.S. National Conference on Earthquake Engineering, Los Angeles, California; 2018.

[47] ASCE/SEI 7-10. Minimum design loads for buildings and other structures; 2010. [48] Ghehnavieh EZ. Seismic analysis of light-frame wood building with a soft-story

deficiency. Clemson University; 2017. [49] Pang W, Asce AM, Masood S, Shirazi H, Asce SM. Corotational model for cyclic

analysis of light-frame wood shear walls and diaphragms. J Struct Eng 2012;139: 1303–17. https://doi.org/10.1061/(ASCE)ST.1943-541X.0000595.

[50] Pang W, Ghehnavieh EZ, Filiatrault A. A 3D model for collapse analysis of soft- story light-frame wood buildings. World Conference on Timber Engineering WCTE, Auckland New Zealand; 2012.

[51] Xactware. Xactimate 28; 2017.

Further reading

[16] Konstantinidis D, Makris N. Experiemental and analytical studies on the response of freestanding laboratory equipment to earthquake shaking. Earthquake Eng Struct Dyn 2009;38:827–48. https://doi.org/10.1002/eqe.

[21] Chidiac E. Evaluation of seismic design criteria for sliding objects in nuclear facilities. McMaster University; 2016.

S. Majdalaweyh and W. Pang