Multi-variate seismic fragility assessment of CLT coupled wall systems

Abstract

The cross-laminated timber coupled wall (CLT-CW) system, a recently proposed timber-based structural system, has limited understanding of its seismic performance. The existing research in probabilistic seismic fragility assessment (PSFA) of CLT buildings reveals gap, particularly regarding comprehensive evaluation of CLT-CW systems and the impact of its various design parameters. To fully describe the state of the post-earthquake performance of structures, state-of-the-art studies recommend using multi-variate fragility analysis. Accordingly, this article presents a bi-variate PSFA of CLT-CW systems using two engineering demand parameters: the maximum and residual inter-story drift ratios. For the seismicity of Vancouver, British Columbia, Canada, 11 prototype buildings are evaluated considering different design parameters: coupling ratio, coupling beam shear force profile, CLT wall configuration, building story height, and ductility-related seismic force modification factor. Bi-dimensional numerical models of the systems are developed in OpenSees, and incremental dynamic analyses are performed using 30 ground motion records. Three limit state capacities and three limit state function combinations are utilized to develop probabilistic seismic fragility curves. The fragility curves under the different limit state function combinations are compared, and the effect of the different design parameters is investigated. This study contributes to a deeper understanding of the seismic performance of CLT-CW systems, assisting engineers and researchers in assessing seismic risk and developing seismic-resilient structures.

Keywords

Coupling ratio cross-laminated timber (CLT)engineering demand parameters (EDPs)multi-variate seismic fragility maximum inter-story drift ratio residual inter-story drift ratio

Introduction

Over the past few years, timber-based constructions have experienced a resurgence in popularity due to the continuing urban densification, the need for sustainable construction materials, and the rapid development of high-performance engineered wood products (Milaj et al., 2017). Cross-laminated timber (CLT), a panel-type engineered wood product made of orthogonally stacked layers of dry lumber boards, has been effectively utilized as a floor and wall structural elements in multi-story timber buildings (Izzi et al., 2018; Tannert et al., 2018). Two primary construction methods, platform and balloon framing, are employed for CLT wall systems (Karacabeyli and Gagnon, 2019). In platform construction, the CLT floor panel at each story acts as a platform for erecting structural members at the upper story (Karacabeyli and Gagnon, 2019). While this approach is widely used for low-to-medium rise buildings (Sandoli et al., 2021), it has limitations when applied to taller structures (Chen and Popovski, 2020). In contrast, balloon framing incorporates continuous CLT wall panels that span multiple stories, with CLT floor panels attached to the sides of wall panels at each level (Karacabeyli and Gagnon, 2019). This technique effectively reduce the compression buildup on floor panels and minimize the distribution of connections and fasteners (Shahnewaz et al., 2021; Wang et al., 2023). The CLT coupled wall (CLT-CW) system (Tesfamariam et al., 2021a; Teweldebrhan et al., 2022; Teweldebrhan and Tesfamariam, 2022) is a noteworthy addition to the state-of-the-art knowledge in balloon-type CLT construction. The seismic performance of individual CLT panels and platform-framed CLT shear walls has been widely explored in literature (e.g. Shahnewaz et al., 2020; van de Lindt et al., 2019); however, the seismic research on balloon-framed CLT systems remains incomplete and there remain research topics worthy of investigation (Shahnewaz et al., 2021; Tesfamariam and Teweldebrhan, 2023). Given CLT-CW systems are a relatively new structural solution, with unique structural characteristics and energy dissipation mechanism (Teweldebrhan and Tesfamariam, 2022), their performance under seismic loading, including responses to varying seismic hazard levels and the impact of different design parameters, has not yet been thoroughly investigated. This study aims to address this knowledge gap and provide insights into the seismic performance of CLT-CW systems.

The most common approach to evaluate the seismic performance of a system is through probabilistic seismic fragility assessment (PSFA) (Cornell et al., 2002). Integral to performance-based earthquake engineering methodology, PSFA enables the computation of fragility curves (FCs) that provide the probability of a structure or its components reaching a specific damage state (i.e. exceeding a given limit state) under a given intensity of ground motion (GM). The observed structural damage can be defined in terms of strength, displacement, damage, or energy (Feng et al., 2023). In literature, the maximum inter-story drift ratio (MaxISDR), residual inter-story drift ratio (ResISDR), and peak floor acceleration are of the most common engineering demand parameters (EDPs) employed to develop the seismic fragility of the structural and nonstructural components of structural systems (Hu and Wang, 2021; Tesfamariam and Goda, 2015; Yakhchalian et al., 2021). The traditional PSFA, which typically considers a single EDP, often fails to fully capture the post-earthquake performance of structures (Ataei and Padgett, 2013; Uma et al., 2010). In practice, multiple structural components may contribute to the global structural performance (Mangalathu et al., 2018). Moreover, structural and nonstructural components are vulnerable to different EDPs (De Risi et al., 2019). Besides, when comparing structural or design alternatives, an alternative can exhibit lower MaxISDR but higher ResISDR, indicating good performance during the earthquake but poor post-earthquake functionality and vice versa (Feng et al., 2023). To overcome these limitations, recent studies (e.g. Cimellaro and Reinhorn, 2011; De Risi et al., 2019; Feng et al., 2023; Mangalathu et al., 2018) have proposed PSFA based on multiple EDPs (mEDPs). With mEDPs-based FCs, a comprehensive description of system performance can be obtained to make more informed and direct decisions (Feng et al., 2023).

Applying these principles, this study utilizes two EDPs (MaxISDR and ResISDR) to develop non-collapse FCs and investigate the PSFA of CLT-CW systems. The selection of these specific EDPs is driven by several factors. First, their ability to represent two significant aspects of seismic structural performance of the system: deformation and repairability (Feng et al., 2023). Post-earthquake damage investigations have revealed that large deformations are the primary cause of damage in both structural and nonstructural components (FEMA P-58-1, 2018). On the other hand, with the growing attention on resilience of structures, ResISDR has emerged as a vital index that serves as a measure of the repairability or functionality of the structural system after an earthquake event (FEMA P-58-1, 2018). Second, both the EDPs are easily compute-able damage indicators with well-established performance limit state thresholds. To this effect, several researchers have utilized these EDPs for PSFA, particularly in the context of timber-based structural systems (e.g. Luo et al., 2022; Sun et al., 2019). However, in those studies, both EDPs were employed independently, which makes decision indirect, and lacks sufficient consideration of the correlations among different EDPs (Du and Padgett, 2020). In the PSFA of CLT- CW system, both MaxISDR and ResISDR hold importance. A recent study by You et al. (2023) indicates that although MaxISDR is the primary damage indicator for CLT-CW system, ResISDR also plays a significant role from a resilience-focused perspective. Remarkably, nearly 80% of total repair costs of damaged structural components are attributed to the damage of coupling beams due to residual deformation. As a result, the design choices of different design parameters (presented as cases in this study) can be influenced by ResISDR, which can only be captured using mEDP-based PSFA.

The PSFA of the CLT-CW system is assessed for 11 prototype buildings, categorized into six cases to study the effects of different design parameters. The baseline case features a 15-story symmetrical CW system with uniform coupling beam strength, a 30% coupling ratio $(CR)$ , and a ductility-related seismic modification $(R_{d})$ value of 2. While $CR$ indicates the part of the total base moment converted into wall axial forces, $R_{d}$ represents the National Building Code (NBC) seismic base shear reduction factor. As $CR$ value greatly affects the structural behavior of CW systems, Case 2 is introduced and systems with $CR$ = 10, 30, and 50% are examined. Different coupling beam shear profile have been used in CW system, and their effects are examined in Case 3. Three distributions are considered: non-linear based on the continuum medium method (CMM), uniform (average), and steeped distributions. Cases 4 to 6 are considered to represent the practical CW configurations (SCS, double-coupled symmetrical (DCS), and single-coupled unsymmetrical (SCU)) and examine the performance of systems with different $R_{d}$ factors (2, 3, and 4) and building story heights (10, 15, and 20), respectively. A two-dimensional (2D) numerical model of the systems is developed in OpenSees, and incremental dynamic analysis (IDA) is performed using 30 GM records (bi-directional) that represent the seismicity of Vancouver—Canada. Three limit state surfaces (LSSs; linear, circular, and square) are considered for computing the FCs at three non-collapse limit state capacities—immediate occupancy (IO), life safety (LS), and collapse prevention (CP).

Building details

The current study is based on the outcomes of a recently completed research project, funded by the British Columbia Forestry Innovation Investment Ltd., in terms of building information, design details, modeling, and selected GM records (Tesfamariam et al., 2021a; Teweldebrhan et al., 2022; Teweldebrhan and Tesfamariam, 2022). The structural system comprises CLT balloon shear-walls, CLT floor panels, glulam beams and columns, steel coupling beams with replaceable shear-links, and buckling-restrained brace (BRB) hold-downs (Figure 1). In this system, the glulam beams and columns were designed to support the vertical gravity load of the system, and the balloon CLT shear walls, coupling beams, and hold-downs were designed to resist the seismic force. The building was assumed to be located in Vancouver, and the associated snow and seismic loads were determined based on the provisions of the National Building Code of Canada (NBC, 2020), considering a class C soil condition. Analyses were performed using the ETABS program, and all the timber elements were designed following the CSA 086-19 standard (2019). The complete design procedure, along with a numerical example, can be found in Tesfamariam et al. (2021a), Teweldebrhan et al. (2022) and Teweldebrhan and Tesfamariam (2022).

Figure 1.

CLT-CW system components.

A total of 11 different CLT-CW systems are developed to study the effect of five design parameters ( $CR$ , coupling beam shear force profile, wall configuration, $R_{d}$ , and story height) on the performance of the CLT-CW system. The $CR$ value indicates the part of the total base moment converted into wall axial forces by means of the shear forces developed in the coupling beams (Smith and Coull, 1991). Its value has a significant impact on the structural performance of CW systems, and analytically, for a two-wall system, $CR$ is computed as:

CR = \frac{T L_{w}}{M_{\max}} = \frac{T L_{w}}{T L_{w} + \sum M_{wall}} = \frac{L_{w} \sum V_{z, i}}{T L_{w} + \sum M_{wall}},

(1)

where $M_{\max}$ = total overturning moment due to lateral forces, $T$ = axial load induced by the shear forces of the coupling beams, $L_{w}$ = distance between the centroid of the two walls, $M_{wall}$ = reduced base moment of the individual walls, and $\sum V_{z, i}$ = accumulation of coupling beam shears. Studies (e.g. El-Tawil et al., 2010) classified the behavior of CW as low, intermediate, and high (Figure 2a) based on the values of $CR$ . While low $CR$ value offers little structural benefit, higher $CR$ value lead to higher axial forces in wall piers that potentially induce premature crushing failure at the toe of the walls. An intermediate $CR$ value (30% to 45%) are the optimal and recommended choices (El-Tawil et al., 2010). The same $CR$ value can be achieved using different coupling beam shear profile. Using CMM, Smith and Coull (1991) derived a non-linear shear strength demand in the coupling beams (Figure 2b). However, uniform (average) and stepped coupling beam strengths are also utilized in different studies (El-Tawil et al., 2010) (Figure 2a). Yet another important feature of CW system is the CW configurations (the number of walls and their lengths). The three most widely used CWconfigurations are SCS, DCS, and SCU wall systems (Figure 2c).

Figure 2.

CLT-CW systems with different: (a) CR values and strain diagrams, (b) coupling beam shear force profiles, (c) CLT wall configurations, and (d) story heights.

Base shear modification factors (termed $R_{o} R_{d}$ in NBC, over-strength and ductility-related factors, respectively) are used to reduce the base shear demand and allow the structural system to response non-linearly during the design earthquake. For existing structural systems, these values can be read from code standards. For example, for CLT platform-type of shear-walls, CSA 086-19 (2019) proposes an $R_{o} = 1.5$ and $R_{d} = 2$ values. For new system, these values can be estimated by experimental and non-linear numerical methods (Teweldebrhan et al., 2022). Accordingly, the baseline system has 15 stories, two symmetrical CLT walls, uniform coupling beam strengths, $CR$ and $R_{d}$ values of 30% and 2, respectively. The first story height of the building is 3.2 m, while the height of all other stories is 3.0 m. The lengths of the two symmetrical CLT shear walls (each 245 mm thick) and coupling beams are 6 m and 1 m, respectively. The rest 10 prototypes buildings are categorized in to five cases (each with two buildings). While buildings in Cases 2 to 4 are designed to study the effect of different $CR$ values, coupling beam strengths distribution and CW configurations, respectively, systems in Cases 5 and 6 are developed to examine the performance of the system with different $R_{d}$ factors and story heights (Figure 2d). The detailed design information for each of these systems is provided in Table 1.

Table 1.

Case study description

Cases	Description (objective)	CLT-CW system
		Story	$CR$ (%)	$R_{d}$	Wall configuration	CB shear force profile	Code ID.
1	Baseline	15	30	2	SCS	Uniform	BC
2	Study effect of $CR$	15	10	2	SCS	Uniform	CS21
2	Values	15	50	2	SCS	Uniform	CS22
3	Study effect of CB shear	15	30	2	SCS	Stepped	CS31
3	Force profile	15	30	2	SCS	Actual (from CMM)	CS32
4	Study effect of shear wall configuration	15	30	2	DCS	Uniform	CS41
4	Study effect of shear wall configuration	15	30	2	SCU	Uniform	CS42
5	Study effect of $R_{d}$ factor	15	30	3	SCS	Uniform	CS51
5	Study effect of $R_{d}$ factor	15	30	4	SCS	Uniform	CS52
6	Study effect of building height	10	30	2	SCS	Uniform	CS61
6	Study effect of building height	20	30	2	SCS	Uniform	CS62

CLT-CW: cross-laminated timber coupled wall; CR: coupling ratio; SCS: single-coupled symmetrical; CMM: continuum medium method; DCS: double-coupled symmetrical; SCU: single-coupled unsymmetrical.

Seismic hazard and GM selection

Three seismic sources—crustal, in-slab, and interface—significantly contribute to seismic hazards in Vancouver, British Columbia (Halchuk et al., 2014). This study uses a multiple conditional mean spectra (CMS) record selection method to represent the distinct response spectral features of these earthquake sources and their overall seismic hazard contributions (Goda, 2019). The initial inputs of this method include the return period $(T_{R})$ , targeted GM record count, the anchor period $(T_{A})$ , and period range for matching response spectral values ( $T_{\min}$ and $T_{\max}$ ). The latter two are determined from the first and higher mode periods of the structural systems, respectively. Probabilistic seismic hazard analysis (PSHA) is conducted, using an in-house tool developed based on Monte Carlo simulations (Goda and Atkinson, 2011), to provide the necessary site-specific hazard information in terms of magnitude, distance, and seismic hazard contributions. Using these information, CMS are developed for each earthquake types. Moreover, from the PSHA seismic disaggregation results, the required number of GM records for each sources are determined, with records for crustal and in-slab obtained from the PEER-NGA database and interface records obtained from the updated KKiKSK (K-NET, KiK-net, and SK-net in Japan) database (Tesfamariam and Goda, 2015). Upon record selection, the maximum scaling at $T_{A}$ is set to 5, and the matching is done only based on the median with three curves for each earthquake type. Although the confidence interval was not specifically matched, retrospective visual checks were performed by comparing the chosen records’ response spectra spread with each earthquake type’s target CMS confidence interval.

In this study, the first fundamental period of the prototype buildings ranges from 0.7 s to 2.0 s (Table 3). Accordingly, three sets of 30 GM records (bi-directional)—that possess the key features (e.g. magnitude, frequency content, and duration) of the considered seismic sources are selected at $T_{A} = 0.75 s$ , 1.5 s, and 2.0 s to represent the fundamental periods of the prototype buildings. The corresponding distribution of crustal, interface, and in-slab events is given in Table 3. A representative seismic hazard and records parameter for $T_{A} = 2.0 s$ is illustrated in Figure 3. Figure 3a and b shows the PSHA results: seismic hazard and disaggregation for each seismic source for $T_{R} = 2475$ years, respectively. Figure 3c compares the response spectra of the selected GM records with the target spectrum. This figure also provides the statistics of the response spectra of the selected GM records (i.e. median and 16th/84th percentile curves) and the weighted CMS which refers to the combined target response spectrum of the seismic sources. It is important to mention that, although the GM selection method could lead to the selection of various GM records at different intensity measure (IM) levels, the IDA-based FC adopted in this study limits the approach to employing the amplitude scaling method.

Figure 3.

Vancouver’s seismic: (a) hazard curve, (b) disaggregation for spectral acceleration at 2.0 s, and (c) response spectra of the selected records at $T_{A} = 2.0 s$ .

Multi-variate fragility analysis

Demand parameters

As mentioned earlier, two EDPs (MaxISDR and ResISDR) are used to assess the performance of the CLT-CW system. MaxISDR is a direct measure of building’s performance, and it denotes the maximum (absolute) difference in lateral displacement between two consecutive floors divided by the story height. ResISDR, on the other hand, represents the maximum story drift ratio recorded at the end of the GM time history. For decision-making and a proper understanding of the system performance, the relationship between structural demand and capacity should be explored (Powell, 2010). Accordingly, in this study, the demand-to-capacity ratio $(Y = D / C)$ is used to indicate the relative performance of the CLT-CW system under seismic excitation (De Risi et al., 2019). The demand and capacity values represent the outcomes of the structural analyses and the threshold of the performance of the limit states considered, respectively. Three limit states (IO, LS, and CP) are considered and their performance limit state capacity values are summarized in Table 2. It is noteworthy that there are no specific timber-based threshold values, as different research has proposed varying values (e.g. Luo et al., 2022; Sun et al., 2019). Consequently, the authors opted to use the recommendations, for general structures, from NBC (DeVall, 2003) and FEMA P-58-1 (2018) for MaxISDR and ResISDR, respectively.

Table 2.

Limit state capacities for MaxISDR and ResISDR (DeVall, 2003; FEMA P-58-1, 2018)

Performance limit states	MaxISDR (%)	ResISDR (%)
IO	0.5	0.2
LS	1.5	0.5
CP	2.5	1.0

MaxISDR: maximum inter-story drift ratio; ResISDR: residual inter-story drift ratio; IO: immediate occupancy; LS: life safety; CP: collapse prevention.

Demand parameter combinations

Due to the uncertainty of LSSs, different damage measure combinations are proposed in literature (e.g. Cimellaro and Reinhorn, 2011; De Risi et al., 2019; Feng et al., 2023). Those LSSs are selected to represent a broad spectrum of possible functions and reasonably quantify the post-earthquake performance of a system. The bi-variate combination proposed by Feng et al. (2023) is expressed as the weighted average of the individual EDPs. Whereas the LSSs proposed by Cimellaro and Reinhorn (2011) and De Risi et al. (2019) consider linear, non-linear, dependent, and independent combinations of the individual EDPs. The result of the weighted average method underestimates the probability of exceedance (PE) of a system under a given performance limit state. The latter formulations, on the other hand, result in a greater PE. Accordingly, in this study, three multi-variate LSSs (MvLSSs) proposed by De Risi et al. (2019) are utilized: linear, circular, and square, and their expression is given by (Equations 2–4), respectively. Figure 4 depicts the 3D and 2D views of the LSS for two generic EDPs $Y_{1}$ and $Y_{2}$ .

G (Y) = Y_{1} + Y_{2} - 1

(2)

G (Y) = Y_{1}^{2} + Y_{2}^{2}

(3)

G (Y) = G_{1} (Y) ⋂ G_{2} (Y) = (Y_{1} - 1) ⋂ (Y_{2} - 1)

(4)

where $G (Y, IM)$ is the multi-variate limit state function (MvLSF), $IM$ is the intensity measure, $b$ is the power term that defines the shapes of the LSF, and $N$ is the dimension of the damage measures.

Figure 4.

Limit state surfaces for two EDPs, adapted from De Risi et al. (2019): (a) and (d): linear, (b) and (e): circular, (c) and (f): square.

Fragility curves

The literature proposes different parametric and non-parametric fitting models to develop seismic FCs. Considering its simplicity, good representation, and convenient characteristics, the lognormal cumulative distribution function (CDF) has often been used (Cornell et al. 2002). Accordingly, the lognormal CDF is used, and the method of moments is applied to obtain the parameters of the distribution function at the damage threshold of interest (e.g. IO). These parameters (the mean and standard deviation) are then used to develop the FCs at the specific limit state condition. Analytically, for a single scalar damage measuring index, the resulting FC is described by the following equation (De Risi et al., 2019):

P (D > C / IM = I M_{i}) = P (G (Y) \leq 0 / IM = I M_{i}) = Φ [\frac{\ln (I M_{i}) - \bar{μ}}{\sqrt{β_{IM}^{2} + β_{MDL}^{2}}}]

(5)

where $Φ$ = indicates the standard normal CDF; $I M_{i} = IM$ for the $i^{th}$ GM at the damage threshold of interest ( $i$ ranges from $1$ to $N$ , the number of GM records); $\bar{μ}$ and $β_{IM}$ are median and logarithmic standard deviation of the IMs; and $β_{MDL}$ = epistemic uncertainty in modeling. Assuming fair to medium quality rating of the structural model parameter, $β_{MDL}$ = 0.35 is considered (FEMA P-695, 2009). The values of $\bar{μ}$ and $β_{IM}$ are calculated as:

\bar{μ} = \frac{1}{N} \sum_{i = 1}^{N} \ln (I M_{i})

(6)

β_{IM} = \sqrt{\frac{\sum_{i = 1}^{N} {(\ln (I M_{i}) - \ln (\bar{μ}))}^{2}}{N - 1}}

(7)

However, this study applies the fragility methodology based on the bi-variate damage indices. For the case of an MvLSS, the calculation of IM values corresponding to the onset of a specific limit state is not straightforward. It is necessary to obtain the performance point $(G = 0)$ by interpolating the closest points to the performance point. When the IM is obtained by interpolation, the same procedure (i.e. the method of moments) is applied to predict the FC parameters of the lognormal CDF. De Risi et al. (2019) have provided a detailed formulation and implementation of the calculation.

Numerical model

A finite element (FE) model for a 2D system was developed using Open System for Earthquake Engineering Simulation (McKenna et al., 2000). In this FE, framework the model is formed by assembling the structural components: CLT shear-wall elements, hold-downs, and coupling beams (Figure 5). The CLT wall panels are modeled using ElasticIsotropic material and quad elements (Demirci et al., 2018). To model the continuity of the CLT wall panels (for balloon systems), ZeroLength vertical and horizontal spring elements are utilized using uniaxialMaterial elastic materials (Teweldebrhan et al., 2022). The two ends and central “fuse” of the coupling beams are modeled using OpenSees elasticBeamColumn elements as rigid offsets and Steel01 UniaxialMaterial ZeroLength non-linear vertical springs, respectively (Ji and Molina Hutt, 2020; Zona et al., 2018). To satisfy the high axial demand, BRB hold-downs are utilized, and OpenSees Steel01 uniaxialMaterial with ZeroLength element is used (Tesfamariam et al., 2021b). The contact between the CLT wall and the base is modeled using OpenSees uniaxial elastic notension (ENT) material. A large elastic stiffness value is assigned to the ENT spring under compression. Besides, a leaning column is introduced and modeled as an elasticBeamColumn element to account for the P-delta effect. Elastic UniaxialMaterial and truss elements are used to link the CLT shear walls and leaning column and transfer the P-delta effect. The reader can refer to Tesfamariam et al. (2021a) for the detailed modeling and parameters of the CLT-CW system. Using the developed OpenSees numerical model, modal analyses are performed, and the first three fundamental periods of the CLT-CW systems are obtained (Table 3). Note that, in all the prototype CLT-CW buildings, large difference is exhibited between mode 1 and mode 2. This is attributed to the fact that the lateral displacement of the CLT-CW system is dominated by rocking, which is consistent with the deformation objective of the CSA standard (Teweldebrhan and Tesfamariam, 2022).

Figure 5.

Numerical model: (a) CLT-CW system, (b) hysteric response of coupling beam, and (c) hysteric response of BRB hold-down.

Table 3.

Fundamental periods of the prototype buildings and GM selection parameters

Code ID.	Fundamental periods (s)			GM selection period parameters (s)			GM record numbers
	$1 st$	$2 nd$	$3 rd$	$T_{A}$	$T_{\min}$	$T_{\max}$	Crustal	Interface	Inslab
BC	1.401	0.309	0.139	1.50	0.10	2.50	3	14	13
CS21	1.527	0.353	0.155	1.50	0.10	2.50	3	14	13
CS22	1.370	0.294	0.132	1.50	0.10	2.50	3	14	13
CS31	1.409	0.316	0.139	1.50	0.10	2.50	3	14	13
CS32	1.406	0.317	0.139	1.50	0.10	2.50	3	14	13
CS41	1.310	0.315	0.148	1.50	0.10	2.50	3	14	13
CS42	2.004	0.401	0.172	2.00	0.10	4.00	7	13	10
CS51	1.437	0.324	0.145	1.50	0.10	2.50	3	14	13
CS52	1.470	0.336	0.149	1.50	0.10	2.50	3	14	13
CS61	0.702	0.171	0.081	0.75	0.05	1.50	3	9	18
CS62	2.026	0.422	0.185	2.00	0.10	4.00	7	13	10

GM: ground motion.

Result and discussion

Incremental dynamic analysis

IDA is a series of dynamic analyses performed by repeatedly scaling a GM record until a specific damage on the structure is attained (Vamvatsikos and Cornell, 2002). The resulting curve relates the scaled IM values with one or more structural EDPs or, alternatively, damage measures. Due to the dominance of the first mode, in this study, the 5%-damped spectral acceleration at the fundamental period of the system $(S_{a} (T_{1}))$ is considered as the IM, and the MaxISDR and ResISDR are considered as EDPs. Typical 3D IDA curves for the baseline are shown in Figure 6. The figure shows the IDA curves and the performance points at different performance limit states (IO, LS, and CP) considering the linear (Figure 6a), circular (Figure 6b), and square (Figure 6c) LSSs. The systems first behave elastically, then the slope falls as the $S_{a} (T_{1})$ level increases, indicating that the structure has suffered significant drift damage. Moreover, although the systems are subjected to significantly higher MaxISDR, the ResISDR remains well below 1%, a threshold value to protect against excessive post-earthquake deformations (FEMA P-58-1, 2018). It is worth noting that the distributions of the performance points are different for the different LSSs. A significant difference can be seen between the linear and the other two LSSs. Figure 6 shows the IDA only for the baseline; other cases presented similar features.

Figure 6.

3D IDA curves for the baseline system considering: (a) linear, (b) circular, and (c) square LSSs.

The 2D representation of the same IDAs, in terms of Ys, is shown in Figure 7. The vertical and horizontal axes of the figure represent the normalized ratio between the demand and capacity of MaxISDR and ResISDR, respectively. Three important notes can be made from Figure 7. First, for the case of the square LSSs, the proposed procedure coincident with the classic procedure adopted for the case of sEDP-based fragility analysis (based on the MaxISDR EDP only). Second, when the $D / C$ for MaxISDR reaches 1, the corresponding $D / C$ for ResISDR remains below 0.5 for all the performance limit states indicating that MaxISDR will be the prime EDP that influences the FC. Finally, performance points obtained by the circular and square LSSs are close to one another, which will be noted clearly on the FCs presented in the subsequent result section.

Figure 7.

Two-dimensional IDA curves for the baseline.

Bi-variate EDP-based FCs

As discussed in the previous sections, the seismic fragility of the designed CLT-CW systems is examined under IO, LS, and CP limit states. The performance points are first estimated considering the three limit state functions. Once the $S_{a} (T_{1})$ at the damage threshold of interest (e.g. IO) is obtained for all the GMs and the limit state functions, the logarithmic moments of the distribution of $S_{a}$ are computed. The resulting non-collapse FCs for the baseline are shown in Figure 8. Figure 8a to c illustrates the mEDP-based FC (mEDP FC) and the sEDP-based (MaxISDR only) FC (sEDP FC) at the IO, LS, and CP performance limit states, respectively. Three important points can be noticed from these figures. First, the resulting mEDP FCs for the circular and square LSSs overlap one over the other (Figure 8a to c), and as expected, the mEDP FC for the linear LSS shifts to the left to yield higher PE indicating that linear LSS is highly sensitive to ResISDR (Figure 8a to c). Second, the FCs at the LS and CP limit states (Figure 8b and c) have larger standard deviations, and this corresponds to the fact that the damage data are largely dispersed in the non-linear response zone of the system. However, in the case of IO (Figure 8a), where there is a moderate earthquake intensity, and the system response is almost linear, the FC has a smaller standard deviation. Third, the difference between the mEDP FC based on linear, and the other two LSSs (circular and square) increases significantly (at CP comparing to IO, Figure 8c and Figure 8a, respectively) as the system’s non-linearity increases. Moreover, minor differences start to occur between the mEDP FC on the circular and square LSSs (Figure 8b and c). Overall, MaxISDR is found to govern the mEDP FC with a marginal impact from ResISDR. As all the performance points are intercepted by the MaxISDR (Figure 7), the mEDP FC obtained from the square LSS overlaps with the one developed considering the MaxISDR only (i.e. sEDP FC). Nonetheless, the use of the mEDP FC enables us to fully capture the post-earthquake performance of structures considering the two important EDPs. The developed mEDP FC provides a comprehensive description of the overall fragility of the system to make more informed and direct decisions. Specifically, as timber structures are prone to residual deformations due to the slip in connections that can potentially contribute to the global performance of the system, the mEDP FC can give us more complete and descriptive results. This observation is further explained in the subsequent sections while discussing the effect of different design factors on the fragility of the CLT-CW system. It is worth noting that even if Figure 8 shows the result for the baseline, similar observations can be noted for the other case studies.

Figure 8.

Non-collapse baseline fragility curves: (a) IO, (b) LS, (c) CP, and (d) combined equivalent FCs.

From the mEDP FCs shown in Figure 8a to c, two extreme cases are noted. The circular and square LSSs almost yield the same FCs as the sEDP FC, whereas the linear LSS tends to capture the effect of ResISDR at the highest level. However, these LSSs are uncertain (Cimellaro and Reinhorn, 2011; De Risi et al., 2019). Thus, a further FC is developed by combining the effect of these three LSSs. Each mEDP FC was constructed by the cumulative lognormal distribution of the 60 (= the number of GMs) performance points. Accordingly, 180 performance points from the three the LSSs are combined, and their cumulative lognormal distribution is obtained. The same procedure is repeated for each of the performance limit states to yield Figure 8d. These FCs represent the combined effect of the considered LSSs and hence, named as equivalent FC (De Risi et al., 2019). Three points can be noted from the figure. First, the combined mEDP FC aggregates the effect of the original LSSs and gives an intermediate result or FCs. Second, the combined mEDP FC results in higher PE and considers the impact of both the EDPs (MaxISDR and ResISDR) to provide a complete picture of the system performance with respect to the chosen EDPs. Third, the difference between the combined mEDP FCs and the sEDP FCs increases as the system goes from the linear (IO) to the non-linear (CP) zone. Moreover, the dispersion increases from IO to CP. Generally, the equivalent mEDP FCs limit the uncertainty of each of the defined LSSs and provide a combined result. Thus, the discussion on the result of the rest of the case studies will be based on the combined mEDP FCs and sEDP FCs. Quantitatively, the PE and the observed differences are summarized in Table 4.

Table 4.

PE (%) and the change in PE (%) of the mEDP FCs (with respect to the sEDP FC) of the baseline study at different performance limit states

Performance limit state	sEDP	mEDP FCs
	sEDP	Linear		Circular		Square		Combined
	PE	PE	$Δ$ in PE	PE	$Δ$ in PE	PE	$Δ$ in PE	PE	$Δ$ in PE
IO	79.76	84.6	6.07	80.04	0.35	79.76	0	81.52	2.21
LS	7.98	13.89	74.06	8.82	10.53	7.98	0	10.15	27.19
CP	0.83	1.83	120.48	0.88	6.02	0.83	0	1.14	37.35

PE: probability of exceedance; mEDP: multiple engineering demand parameter; FC: fragility curves; IO: immediate occupancy; LS: life safety; CP: collapse prevention.

Effect of coupling ratio

This section discusses explicitly the effect of $CR$ on the multi-variate FCs of the CLT-CW system. In addition to the baseline, two CLT-CW systems were designed by changing the $CR$ values to 10% and 50% and keeping all the other design details constant. Figure 9 illustrates the combined mEDP FCs and sEDP FCs at different performance limit states for the three systems. Three observations can be noted from the figure. First, the combined mEDP FCs and sEDP FCs coincide at IO and show significant differences at other performance limit states (especially for the system with $CR$ = 50%). Second, at the same performance limit state (e.g. CP), the variation between mEDP FCs and sEDP FCs is different. The system designed with $CR$ = 10% shows small differences in FCs, whereas the system designed with $CR$ = 50% shows significant differences between the FCs. This observation matches the remark given in Tesfamariam et al. (2021a), where systems designed with higher $CR$ values exhibit higher ResISDR. The system with a lower $CR$ value (10%) has a lower ResISDR value; thus, a slight difference exists between the FCs. This observation can only be captured by using mEDP FCs and is the limitation of the traditional sEDP FC. Third, the overall performance of the system with the higher $CR$ value is better by exhibiting a lower PE (80.85%, 7.19%, and 0.75% at IO, LS, and CP, respectively) under a given hazard level than those with lower $CR$ values (87.87%, 20.7%, and 2.34% at IO, LS, and CP, respectively). This contributes to the fact that systems with higher $CR$ values have a greater coupling beam strength and higher reduction in the seismic demand of the individual walls and hence, are stronger. The relative difference in the seismic performance of the systems is more pronounced for the LS and CP performance limit states, where the systems exhibit non-linearity. It is worth mentioning that the seismic performances of systems designed with $CR$ values of 30% and 50% are close. However, there is a significant difference between the systems designed with $CR$ values of 10% and 30%. Thus, as the systems with higher $CR$ values are also susceptible to crushing failure, the system designed with $CR$ = 30% would be optimum, less vulnerable than the system with $CR$ = 10% and has almost the same performance as those with $CR$ = 50% (Teweldebrhan et al., 2022).

Figure 9.

Effect of coupling ratio on the non-collapse fragility curves at: (a) IO, (b) LS, and (c) CP.

Effect of coupling beam shear profiles

For the same $CR$ value, different coupling beam shear profiles are possible. El-Tawil et al. (2010) reviewed the existing studies on this topic. As stated earlier, three distribution profiles (Figure 2b) are considered in this study to investigate the effect of the coupling beam shear profile. In addition to the baseline (designed with a uniform profile), two systems were designed by considering an actual and steeped profile, keeping all the other design details constant. Figure 10 illustrates the effect of these shear profiles on the FCs of the system. As can be seen from the figure, all the systems exhibit almost the same performance (especially at IO). However, a small and important difference is noted between the system designed with the uniform profile and the others (e.g. at CP). The systems designed with the uniform profile are provided with lower strengths (than the required) at higher stories (Figure 2b), and thus, higher responses and residual deformations are observed on the coupling beam responses (Teweldebrhan and Tesfamariam, 2022). This observation is also noted in Figure 10c where the mEDP FCs for the uniform system exhibit a lower performance (higher PE), which the sEDP FC would not capture. The actual profile leads to better performance (with PE = 81.11%, 9.5%, and 0.94% at IO, LS, and CP, respectively); however, it is difficult from a constructability point of view. Thus, without affecting the performance of the system, the stepped shear force profile (with PE = 81.07%, 9.59%, and 0.95% at IO, LS, and CP, respectively) would be a good design choice, especially at higher $CR$ values where the profile is highly non-linear (Harries and McNeice, 2006).

Figure 10.

Effect of shear force profile on the non-collapse fragility curves at: (a) IO, (b) LS, and (c) CP.

Effect of wall configuration

Three CW configurations (Figure 2c)—SCS, DCS, and SCU are considered to examine the effect of different wall configurations on the fragility of the CLT-CW system. Detailed design and model information of those systems can be found in Teweldebrhan et al. (2022). Figure 11 illustrates the effect of these wall configurations on the FCs of the system. As can be seen from the figure, CLT-CW systems with the SCU wall system are vulnerable to intense earthquake excitations. SCS and DCS have better performances at all the performance limit state levels. Moreover, it is interesting to note that at the IO level (Figure 11a), DCS has a higher performance than SCS (PE = 78.94%); however, the system exhibits a lower performance at the LS and CP limit states (PE = 11.78% and 1.23% at LS and CP, respectively). This is attributed to the fact that at a relatively lower GM intensity, the coupling beams in SCS (system with single coupling, four hold-downs, and higher wall flexural stiffnesses) yield earlier than those at DCS (system with double coupling, six hold-downs, and relatively lower wall flexural stiffnesses) and thus exhibits a lower performance. However, as the GM level increases, the coupling beam at DCS yields, their coupling action degrades, and the contribution of flexural walls activates to improve the performance of the SCS wall system. In summary, it is observed that system with SCU exhibits higher fragility or PE (97.37%, 27.75%, and 3.37% at IO, LS, and CP, respectively). In an ideal condition where a designer chooses wall configurations, SCU and DCS are recommended. The performance of the different wall configurations under different $CR$ values, story heights, and $R_{d}$ factors are presented in Teweldebrhan et al. (2022).

Figure 11.

Effect of wall configurations on the non-collapse fragility curves at: (a) IO, (b) LS, and (c) CP.

Effect of ductility-related seismic modification factor

The CLT-CW system is a recently proposed tall timber building. For those new systems to be introduced in the building codes, their seismic performance must be evaluated considering different seismic force modification $(R_{o} R_{d})$ factors. Using an over-strength $(R_{o})$ factor of 1.5, two additional systems with $R_{d} = 3$ and 4 are investigated. The same design details (story height, $CR$ value, wall configuration, and shear force profile) are considered as the baseline which was designed with $R_{d} = 2$ . Figure 12 shows the effect of $R_{d}$ factors on the FCs of the system. As expected, the system designed with $R_{d} = 4$ factor exhibits a lower performance with PE values of 86.03%, 13.59%, and 2.48% at the IO, LS, and CP performance limit state levels, respectively. It is worth noting that there is a greater difference between the performance of the systems designed with $R_{d} = 3$ and 4 compared to those designed with $R_{d} = 2$ and 3. The PE values for $R_{d} = 3$ are 82.33%, 12.32%, and 1.23% at IO, LS, and CP, respectively.

Figure 12.

Effect of ductility-related factors on the non-collapse fragility curves at: (a) IO, (b) LS, and (c) CP.

Effect of building height

With the prevailing building height restriction in high-rise timber buildings, it is important to study the effect of story height on the fragility of the proposed system. Accordingly, two additional systems (10 and 20 stories) are designed with the same design details as the baseline. Figure 13 illustrates the effect of the different story heights on the performance of the system. As the CLT-CW systems are built taller and more slender, their seismic performances become lower. That is, the system with a lower story (e.g. 10) exhibits better seismic performance than the one with a higher level (e.g. 20). Relatively, the seismic performances of the 15 and 20-story systems are closer compared to the 10-story system (Table 5). The performance of the 10-story, by far better than the higher buildings. It is also noted that, relatively, there is a larger difference between the mEDP FCs and sEDP FCs in the 10-story than in the other story levels.

Figure 13.

Effect of building height on the non-collapse fragility curves at: (a) IO, (b) LS, and (c) CP.

Table 5.

PE (%) and the change in PE (%) of the mEDP FCs (with respect to the sEDP FC) of the baseline study at different performance limit states

Code ID.	Performance limit state	sEDP FC		Combined mEDP FCs		Performance
Code ID.	Performance limit state	PE	$Δ$ in PE	PE	$Δ$ in PE	Performance
CS21	IO	87.59	9.82	87.87	7.79	Reduced
	LS	19.91	149.5	20.7	103.94	Reduced
	CP	1.68	102.41	2.34	105.26	Reduced
CS22	IO	78.54	−1.53	80.85	−0.82	Improved
	LS	5.44	−31.83	7.19	−29.16	Improved
	CP	0.5	−39.76	0.75	−34.21	Improved
CS31	IO	79.07	−0.87	81.07	−0.55	Improved
	LS	7.04	−11.78	9.59	−5.52	Improved
	CP	0.73	−12.05	0.95	−16.67	Improved
CS32	IO	79.15	−0.76	81.11	−0.5	Improved
	LS	6.96	−12.78	9.5	−6.4	Improved
	CP	0.72	−13.25	0.94	−17.54	Improved
CS41	IO	76.36	−4.26	78.94	−3.16	Improved
	LS	9.96	24.81	11.78	16.06	Reduced
	CP	1.07	28.92	1.23	7.89	Reduced
CS42	IO	96.65	21.18	97.37	19.44	Reduced
	LS	22.28	179.2	27.75	173.4	Reduced
	CP	2.09	151.81	3.37	195.61	Reduced
CS51	IO	80.05	0.36	82.33	0.99	Reduced
	LS	10.82	35.59	12.32	21.38	Reduced
	CP	1.2	44.58	1.23	7.89	Reduced
CS52	IO	85.1	6.7	86.03	5.53	Reduced
	LS	13.3	66.67	13.59	33.89	Reduced
	CP	2.48	198.8	2.48	117.54	Reduced
CS61	IO	52.72	−33.9	54.78	−32.8	Improved
	LS	1.81	−77.32	2.79	−72.51	Improved
	CP	0.25	−69.88	0.29	−74.56	Improved
CS62	IO	82.71	3.7	84.71	3.91	Reduced
	LS	9.26	16.04	10.41	2.56	Reduced
	CP	0.84	0.6	1.15	0.88	Reduced

PE: probability of exceedance; mEDP: multiple engineering demand parameter; FC: fragility curves; IO: immediate occupancy; LS: life safety; CP: collapse prevention.

Summary

In the preceding section, the effect of the different design parameters on the fragility of the CLT-CW system is discussed. The parameters that enhance or worsen the performance of the system are identified. Table 5 provides the change in the PE of the different cases with respect to the PE of the baseline. The values in the table are both in terms of the sEDP and mEDP FCs. The value of PE for sEDP and mEDP are different and hence, the percentage change in the performance of the system. The values based on mEDP FCs consider the effect of both the EDPs and the different LSSs. Design parameters: $CR$ , wall configurations, and $R_{d}$ factors are identified to significantly affect the fragility of the system.

Conclusion

Timber-based constructions have experienced a resurgence in popularity over the past few years. One novel contribution to the state-of-the-art knowledge in CLT construction is the recently proposed CLT-CW system. When new structural systems are proposed, it is vital to study their seismic fragility. Seismic fragility assessment of structural systems requires the selection of appropriate EDPs. To capture the entire state of the post-earthquake performance of structures, state-of-the-art studies recommend the use of multi-variate probabilistic seismic fragility analysis. In this article, two EDPs—MaxISDR and ResISDR—are chosen, and multi-EDP-based FCs (mEDP FCs) are developed for 11 prototype buildings. The 2D numerical model of the systems is developed in OpenSees, and 3D IDA is performed using 30 (bi-directional) GM records that represent the seismicity of Vancouver, Canada. Three performance limit state levels—IO, LS, and CP—and three different combinations of LSSs, namely linear, circular, and square, are utilized to develop probabilistic seismic FCs. The 11 prototypes were categorized into six cases based on different design parameters. For each case, mEDP FCs and sEDP FCs were developed and the following conclusions were drawn:

For the 15-story baseline CLT-CW system, the resulting mEDP FCs for the circular and square LSSs overlap one over the other and the mEDP FC for the linear yields higher probability of exceedance (PE). This difference increases significantly at CP where the system exhibits non-linearity. Generally, for this system, the MaxISDR is found to govern the mEDP FCs with a marginal impact from ResISDR.

To prevent biases against the employed LLSs, a combined FC is developed, incorporating the effect of all three LSSs. The resulting equivalent mEDP FCs represent the combined effect of the considered LSSs and can capture the effect of both EDPs on the non-collapse FCs of the systems.

The system with a higher $CR$ value (50%) exhibits a lower PE under a given hazard level than those with lower $CR$ values (e.g. 10%). The relative difference in the seismic performance of the systems designed with $CR = 30 %$ and 50% is small. As systems with higher $CR$ are also likely to undergo crushing failure, the system designed with $CR = 30 %$ would be optimum.

The system with a uniform coupling beam shear force profile exhibits a lower performance (even though the difference is small). The actual profile (based on the CMM) leads to better performance. Nevertheless, a steeped profile with the same performance and ease of constructability would be a good design choice under an ideal design conditions.

The system with a SCU wall is shown to be more vulnerable seismically. The performance can be increased by using SCS and DCS CLT-CW systems.

The systems with higher story height and seismic modification factor exhibit lower performances than those with lower story height and seismic modification factors.

In summary, the use of mEDP FC provides more comprehensive descriptions of the fragility of the system, considering the two most important EDPs. Limitations of the sEDP FCs are noted in the results obtained. The effect of the different LSSs and their combined fragility analysis is observed. Moreover, the design parameters that enhance the seismic performance of the proposed systems are identified. This study, however, has some limitations. The only sources of non-linearity are the coupling beams and hold-downs. Degradation is not considered, the CLT to hold-down connections and CLT to coupling beam connections were modeled as rigid connections. Moreover, the CLT shear walls are modeled as single walls. The stated limitation warrants further investigations.

Footnotes

Authors’s Note

Solomon Tesfamariam is now affiliated to Department of Civil and Environmental Engineering, University of Waterloo.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded to the last author by the British Columbia Forestry Innovation Investment’s (FII) Wood First Program and the Natural Science Engineering Research Council of Canada Discovery Grant (grant no. RGPIN-2019-05013).

ORCID iDs

Katsuichiro Goda

Raffaele De Risi

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