Effects of the Ductility Capacity on the Seismic Performance of Cross-Laminated Timber Structures Equipped with Frictional Isolators

Abstract

In developing countries with high seismic activity, a need exists to construct resilient infrastructure and reduce the housing deficit. Industrialized timber construction and the implementation of seismic isolation interfaces may represent a good alternative to respond to these demands. This paper studies the feasibility of constructing cross-laminated timber (CLT) buildings equipped with frictional pendulum bearings in Chile or similar highly seismic regions. The first part of this study shows a first-order approach for modeling the highly nonlinear behavior of CLT walls using a Smooth Hysteretic Model (SHM). An equivalent model of a base-isolated building was developed using the SHM as well as a physical model of the Friction Pendulum System in order to assess the seismic performance of CLT buildings with frictional isolators. The second part of this research presents and discusses the results of a broad parametric analysis concerning the seismic performance of base-isolated CLT buildings. The seismic assessment was carried out by deriving fragility curves and including the uncertainty linked to the seismic input and the friction coefficient of the isolation system. Constructing lateral resistant systems based on CLT walls presents a feasible alternative for buildings in high seismic hazard areas. Excellent seismic performance is achieved if the superstructure’s is designed with a reduction factor of 1, or if the superstructure’s fundamental period ranges from 0.6 to 0.9 s and is designed with a reduction factor of 2 and ductility capacity of 6 or more. An excellent seismic performance can be obtained for larger reduction factor values if the superstructure has middle to high maximum ductility capacity.

1. Introduction

Cross-laminated timber (CLT) was introduced as a construction alternative in the early 1990s [1]. This construction material is well-established in Europe and is being used more and more in Australasia, North America, and Japan. A CLT element is built by gluing together an uneven number of structural lumber layers. Each layer is oriented at 90° relative to its adjacent layer. In this manner, a solid panel capable of withstanding both vertical and horizontal loads can be formed. CLT has relevant advantages, such as high prefabrication and installation rates and low damage to the site environment. Environmental benefits arise when comparing timber and wood-engineered products with conventional materials used in construction (i.e., steel and reinforced concrete) [2,3]. For instance, lower CO₂-e emission factors, embodied energy, and embodied carbon are linked to timber and wood-engineering products.

Understanding the behavior of CLT buildings subjected to lateral loads is critical to achieve reliable seismic designs. Since CLT elements are stiff in their plane, the connections of CLT structural systems play a crucial role in their nonlinear response. Several experimental studies have been developed to advance the understanding of CLT walls’ monotonic and cyclic behavior [1,4,5,6,7,8]. The nonlinear response of this kind of element is presided by different phenomena: the crushing of wood, the yielding of connectors, and friction between components. The combination of these mechanisms influences the ductility and energy dissipation capacity of the elements. Under increasing cyclic magnitude displacements, elements exhibit wood embedding and increasing clearances of connectors, leading to strength and stiffness degradation, and one of the most important characteristics of CLT wall behavior: pinching. The highly nonlinear response of timber elements has motivated the development of several numerical models to predict their response. Dong et al. developed a uniaxial hysteretic model for dowel-type timber joints, suitable for representing shear connections in timber structures [9]. A simple numerical model to represent the cyclic behavior of wood shear walls was presented by Folz and Filiatrault [10]. This model includes an equivalent nonlinear spring to predict the hysteretic behavior of shear walls. The same authors developed a simple model to predict the response of wood frame structures [11]. The representation of three-dimensional buildings is carried out considering only three degrees of freedom per floor and using two-dimensional planar elements to incorporate the walls. Chacón and Guidons developed the ASPID model [12], a phenomenological-based model to represent the hysteretic response of timber structures. This formulation accounts for pinching, strength and stiffness degradation, and asymmetry.

An important percentage of multistory timber buildings, ranging from 5 to 18 stories, have been constructed in countries with low-to-medium seismic hazards (Finland, Austria, Norway, and the United Kingdom) [13,14,15]. Fewer applications of multistorey CLT structures can be found in countries with high seismic hazards, such as Japan and New Zealand [16,17]. It is likely that one of the most important aspects that restrain the expansion of CLT buildings in active seismic regions is that, under beyond-design seismic loads, their cyclic behavior presents some unknown aspects. Significant studies are being carried out to assess the seismic performance of CLT buildings. Full-scale numerical models have been developed. Bita and Tannert performed a collapse prevention analysis for a mid-rise CLT building [18]. A numerical model of a narrow-paneled CLT building was developed and analyzed by Sato et al. [19]. Aloisio et al. carried out a dynamic identification of an eight-story CLT structure to update a numerical model of a CLT building [20]. A numerical model of a multi-story building braced with a CLT core and perimeter shear walls was developed and studied by Polastri et al. [21]. Demirci et al. studied the drift, shear, and acceleration demands of CLT buildings [22,23]. Full-scale tests of CLT buildings have been conducted. The SOFIE project represents one of the most relevant studies on multi-story CLT structures subjected to 3D shaking table tests [24].

Chile is characterized as one of the most seismically active countries in the world. In February 2010, the country was hit by an M8.8 event (Maule Earthquake) that caused a devastating tsunami, affecting more than 12 million people. The earthquake produced 571 deaths and generated USD 30 billion in economic losses [25]. This high seismic hazard has deterred the development of timber buildings in the Chilean territory, as in several other countries. Chile has another important problem: the housing deficit. The Chilean state has promoted industrialization as an alternative to overcome this problem. Using new materials and construction systems for medium- and high-rise buildings represents alternatives to improve the current scenario. However, constructing conventional (fixed-base) medium-rise timber buildings is more expensive than traditional construction. As mentioned above, the unknown issues about the seismic performance of CLT buildings induce a response modification factor of $R_I=2$ [1], increasing the costs of CLT panels and the anchors needed to resist high-magnitude tensile forces and shear stresses.

Seismic isolation may represent a feasible alternative to achieve a cost-effective design of timber structures in high seismic hazard zones, taking advantage of the benefits of industrialization. The use of isolation interfaces allows an elongation of the natural period of a dynamic system to a range where ground motions have less energy, reducing the adverse effects of the seismic input on superstructures. As shown in Figure 1, the use of seismic isolation, when compared with conventional fixed-base buildings, reduces the magnitude of seismic forces, the superstructure’s absolute accelerations, and the displacement demands. Although seismic isolation is an excellent approach to protect mid-rise buildings and ensure continuous operation (i.e., better seismic performance), relatively few studies have focused on multistory CLT buildings equipped with seismic isolators [26]. Bolvardi et al. studied an inter-story isolation system in tall CLT platform buildings [27]. This strategy differs from conventional seismic isolation since the inter-story isolation system separates the CLT structure into two sections: the top and bottom segments. Another example is the study that proposed the Floor Isolated Re-centering Modular Construction System (FIRMOCS) [28]. This approach differs from traditional seismic isolation, which realizes a single isolation interface between the superstructure and the ground. Medel and Contreras developed resilience-based predictive models of CLT buildings equipped with rubber bearings [29]. This study modeled the superstructure assuming linear behavior and frictional devices did not form the isolation system. The present research varies from the cited studies since it focuses on traditional seismic isolation formed by frictional devices and considers the highly nonlinear behavior of CLT shear walls.

Experimental tests of base-isolated timber structures subjected to large ground motions have recently been conducted [30,31]. In these tests, excellent seismic performance was achieved. Low peak acceleration and story drift ratios were measured, indicating an elastic response of the superstructure.

Based on the above-mentioned issues, this study aims to advance the understanding of the dynamic behavior and seismic performance of CLT buildings equipped with frictional isolators. The investigation is separated into two main parts: (i) the development of a novel, efficient, and simplified model of a base-isolated CLT building and (ii) the evaluation of the seismic performance of base-isolated CLT structures. The first part of this study consists of the adjustment and validation through experimental data of a Smooth Hysteretic Model (SHM), aiming to represent the highly nonlinear response of CLT shear walls. This adjusted and validated numerical formulation is employed to develop a new efficient and simplified model of a CLT building with friction pendular bearings suitable for evaluating its seismic performance. The second part of the research consists of the seismic performance evaluation of CLT structures equipped with frictional pendulum bearings. The assessment involves a broad parametric analysis of superstructures designed to be constructed in Chilean territory or similar highly seismic regions. Seismic performance is assessed through seismic fragility curves. These fragility curves were derived considering the uncertainties linked to the seismic input and the friction coefficient of the isolation system. The second part of the investigation aims to determine the design requirements, in terms of ductility capacity and reduction factors in the design of the superstructure, that lead to excellent seismic performance. While the first part of this research is covered in Section 2 and Section 3, the second is covered in Section 4, Section 5, Section 6, Section 7 and Section 8.

The novelty of this work lies in analyzing and comparing, via incremental dynamic analysis (IDA) and fragility analysis, the seismic behavior of CLT buildings, considering several parameters of the superstructure and the isolation system. In particular, the initial period of the building, the isolated period, the mass distribution ratio, the maximum ductility capacity, and the reduction factor employed during its design are included in the analysis. Additionally, a new efficient and simplified model is presented for the seismic response evaluation of base-isolated timber structures.

2. A Smooth Hysteretic Model (SHM) for Modeling CLT Walls

In this section, a brief description of CLT shear walls is presented as systems that withstand seismic lateral loads. The working mechanism is shown, highlighting the main contributors to the wall displacement. A numerical formulation is presented to represent the high nonlinear behavior of this kind of element. This formulation is validated through a comparison with experimental results of three different CLT walls.

2.1. Brief Description of CLT Shear Walls

A seismic lateral resistance system can be realized by connecting panels of CLT employing steel connectors, namely angle brackets and hold-downs, as shown in Figure 2b. When a structure is subjected to a ground motion, lateral forces are transmitted to the building. If the structure is fixed to the ground, these lateral forces can be idealized as increasing concentrated loads acting on the diaphragms (i.e., CLT floors) of the building, as shown in Figure 2a.

The force distribution of CLT walls is a complex phenomenon. This distribution depends on the number and type of connectors, the predominant displacement contributor, and the interaction with adjacent wall panels, among other factors. Figure 2e–h presents the contributors to the shear wall displacement when the element is subjected to a lateral force. These contributors are bending, shear, sliding, and rocking. For an important percentage of shear CLT wall configurations, the displacements generated due to bending and shear are much smaller than the displacement caused by sliding and rocking [32]. Since steel connections exhibit much softer behavior compared to CLT panels, these connections govern the load-carrying characteristics of stiffness, load capacity, and ductility. Figure 2c,d shows a representation of the equilibrium of lateral forces generated in one panel when a sliding or a rocking behavior governs the lateral behavior.

2.2. A Numerical Formulation to Represent the Lateral Response of CLT Walls

In this study, we employ a Smooth Hysteretic Model (SHM) [33] to represent the nonlinear response of CLT walls. These elements exhibit highly nonlinear behavior, including pinching and stiffness/strength degradation. The SHM approach allows the inclusion of these three phenomena.

In Figure 3, we show how to represent the lateral behavior of a CLT shear wall using the SHM. While Figure 3a,b show a CLT wall and the SHM in a nondisplaced configuration, respectively, Figure 3c,d present a CLT wall and the SHM in a displaced configuration. The numerical formulation is based on three elements with different characteristics configured in two systems (see Figure 3b). On the one hand, the first system is composed of a linear spring (Element #1) connecting the lower and upper ends of the CLT wall. This first system is entirely linear. On the other hand, the second system, a nonlinear system, is composed of two springs (Elements #2 and #3) connected in series and connects the lower and upper ends of the CLT wall. The nonlinear behavior of the SHM is concentrated in the second system. While the stiffness/strength degrading characteristics and the elastic/plastic behavior are included using a hysteretic spring (Element #2), the pinching phenomenon is represented through a slip-lock spring (Element #3). The two systems connect the upper and lower ends of the wall in parallel.

The SHM exerts a force $F$, given a displacement $v$, a rate of displacement $\dot{v}$, and a loading history dependent on its hysteretic parameters. Note that the total force $F$ can be split into the force that exerts the linear spring (i.e., $F_{postyield}=a{K}_{0}v$) and the force generated in the hysteretic spring $F^{*}$.

The stiffness of the linear spring (Element #1) can be computed as follows:

$$K_{postyield}=a{K}_0$$

(1)

where $K_0$ is the initial stiffness, considering the contribution of the three springs, and $a$is the post-yield stiffness ratio (i.e., $a=K_{postyield}/K_0)$. Equation (1) is used to represent the post-yield stiffness of a structural element.

The hysteretic spring (Element #2) represents an elastoplastic element, exhibiting a smooth transition between elastic and plastic ranges. If the hysteretic spring includes stiffness degradation, adopted through a pivot rule, its stiffness can be determined according to the following equation:

$$K_{hysteretic}=\left(R_k-a\right)K_0\left(1-{\left|{\displaystyle \frac{F^{*}}{F_y^{*}}}\right|}^N({\eta }_1\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left(F^{*}\dot{v}\right)+{\eta }_2)\right)$$

(2)

where the parameter $N$ controls the transition between elastic and plastic branches, and the parameters ${\eta }_1$ and ${\eta }_2$ control the shape of the unloading curve. The symbol $F_y^{*}$ denotes the yield force developed in the hysteretic spring ($F_y^{*}=\left(1-a\right)F_y$, where $F_y$ represents the total yield force). The parameter $R_k$ allows the inclusion of the stiffness degradation targeting a pivot point located at a distance $\alpha F_y^{*}$ at the opposite side concerning the initial branch, with $\alpha$ being the stiffness degradation parameter. Following the described pivot approach, the parameter $R_k$ can be determined as follows:

$$R_k={\displaystyle \frac{F-\alpha F_y}{K_{0}v+\alpha F_y}}$$

(3)

The SHM also allows the incorporation of strength degradation by reducing the positive or negative initial yield force $F_{y0}^{+/-}$. This degradation is produced due to ductility demand or hysteretic energy dissipated, shown as follows:

$$F_y^{+/-}=F_{y0}^{+/-}\left(1-{\left({\displaystyle \frac{v_{max}^{+/-} }{v_u^{+/-} }}\right)}^{{\displaystyle \frac{1}{{\beta }_1}}}\right)\left(1-{\displaystyle \frac{{\beta }_2}{1-{\beta }_2}}{\displaystyle \frac{H}{H_u}}\right)$$

(4)

where $F_y^{+/-}$ is the current positive or negative yield force, $v_{max}^{+/-}$ is the positive or negative maximum displacements, $v_u^{+/-}$ is the positive or negative ultimate displacement that causes the failure of the element when it is loaded monotonically, $H$ is the hysteretic cumulative energy dissipated, $H_u$ is the energy dissipated that causes the failure of the element, ${\beta }_1$ is the ductility-based strength degradation parameter, and ${\beta }_2$ is the energy-based strength degradation parameter.

The pinching phenomenon can be included by incorporating a slip-lock spring (Element #3) acting in series with the hysteretic spring. The stiffness of this spring can be determined as follows:

$$K_{slip-lock}={\displaystyle \frac{1}{\sqrt{\frac{2}{\pi }} \frac{s}{\sigma F_y^{*}}\exp \left(-\frac{1}{2} {\left(\frac{F^{*}-\lambda F_y^{*}}{\sigma F_y^{*}}\right)}^2\right)}}$$

(5)

where $s=R_s(v_{max}^{+}-v_{max}^{-})$ is the slip length and $R_s$, $\sigma$, and $\lambda$ are parameters representing the pinching phenomenon.

The stiffness of the entire system $K$can be expressed as follows:

$$K=K_{postyield}+{\displaystyle \frac{K_{hysteretic}{K}_{slip-lock}}{K_{hysteretic}+K_{slip-lock}}}$$

(6)

Note that the total force $F$, separated into the linear post-yield force $F_{postyield}$, and the hysteretic force $F^{*}$, can be expressed according to the following equation:

$$F=F_{postyield}+F^{*}=\left(K_{postyield}\right)v+\left({\displaystyle \frac{K_{hysteretic}{K}_{slip-lock}}{K_{hysteretic}+K_{slip-lock}}}\right)v=\left(a{K}_0\right)v+\left({\displaystyle \frac{K_{hysteretic}{K}_{slip-lock}}{K_{hysteretic}+K_{slip-lock}}}\right)v$$

(7)

In total, sixteen parameters can be adjusted to match the force–displacement relationship of a particular tested element. The parameters are as follows: $a$, $K_0$, $N$, ${\eta }_1$, ${\eta }_2$, $\alpha$, $F_{y0}^{+}$, $F_{y0}^{-}$, $v_u^{+}$, $v_u^{-}$, ${\beta }_1$, ${\beta }_2$, $H_u$, $R_s$, $\sigma$, and $\lambda$. One goal of this research is to find the value of these parameters to represent the nonlinear behavior of CLT walls.

2.3. State-Space Approach to Solve Equations Governing the SHM

In this study, we adopt the state-space approach [34] to calculate the hysteretic response of CLT walls. This approach needs state variables and first-order equations to evaluate their time-dependent evolution. The described SHM can be approached by defining six state variables: $F^{*}$, $H$, $v_{max}^{+}$, $v_{max}^{-}$, $F_y^{+}$, and $F_y^{-}$. These variables can be arranged in the following state vector:

$${\mathit{Z}}_{SHM}={[F^{*},\hspace{1em}\hspace{1em}H,\hspace{1em}\hspace{1em}{v}_{max}^{+},\hspace{1em}\hspace{1em}{v}_{max}^{-},\hspace{1em}\hspace{1em}{F}_y^{+},\hspace{1em}\hspace{1em}{F}_y^{-}]}^T$$

(8)

The next step is to find-first order differential equations to evaluate these variables in rate form. The evolution of the yield force developed in the hysteretic spring $F^{*}$ can be obtained using Equation (7) as follows:

$${\dot{F}}^{\mathit{*}}={\displaystyle \frac{d\left(F^{*}\right)}{dv}} {\displaystyle \frac{dv}{dt}}=K\dot{v}=\left(K_{postyield}+{\displaystyle \frac{K_{hysteretic}{K}_{slip-lock}}{K_{hysteretic}+K_{slip-lock}}}\right)\dot{v}$$

(9)

The incremental form of the hysteretic energy can be expressed as follows:

$$\mathsf{\Delta }H=\left({\displaystyle \frac{F+\left(F+\mathsf{\Delta }F\right)}{2}}\right)\left(\mathsf{\Delta }v-{\displaystyle \frac{\mathsf{\Delta }F}{R_k{K}_0}}\right)$$

(10)

Equation (10) allows to determine an expression for the hysteretic energy in rate form:

$$\dot{H}=F\left(\dot{v}-{\displaystyle \frac{\dot{F}}{R_k{K}_0}}\right)=F\left(1-{\displaystyle \frac{K_{postyield}+R_k{K}_{hysteretic}}{R_k{K}_0}}\right)\dot{v}$$

(11)

The maximum positive or negative displacement in rate form can be expressed as:

$${\dot{v}}_{max}^{+}=\dot{v}U(v-v_{max}^{+})U\left(\dot{v}\right)$$

(12)

$${\dot{v}}_{max}^{-}=\dot{v}U\left(v_{max}^{-}-v\right)\left(1-U\left(\dot{v}\right)\right)$$

(13)

where $U(·)$ represents the Heaviside function. The current positive or negative yield force $F_y^{+/-}$ in rate form can be derived, using Equation (4), as follows:

$$\begin{array}{cc}\hfill {\dot{F}}_y^{+/-}={\displaystyle \frac{d\left(F_y^{+/-}\right)}{dv}} {\displaystyle \frac{dv}{dt}}& =\hfill \\ & =F_{y0}^{+/-}\left[\left(1-{\displaystyle \frac{{\beta }_2}{1-{\beta }_2}}{\displaystyle \frac{H}{H_u}}\right)|\left(-{\displaystyle \frac{1}{\left({\beta }_1{\left(v_u^{+/-}\right)}^{\frac{1}{{\beta }_1 }}\right)}}{\left(v_{max}^{+/-}\right)}^{{\displaystyle \frac{\left(1-{\beta }_1\right)}{{\beta }_1 }}}\right)\right]{\dot{v}}_{max}^{+/-}\\ & +\left(1-{\left({\displaystyle \frac{v_{max}^{+/-} }{v_u^{+/-} }}\right)}^{{\displaystyle \frac{1}{{\beta }_1}}}\right) \left(-{\displaystyle \frac{{\beta }_2}{\left(1-{\beta }_2\right)H_u}}\right)F^{*}\left(1-{\displaystyle \frac{1}{\left(1-a\right)R_k{K}_0}}{\displaystyle \frac{K_{hysteretic}{K}_{slip-lock}}{K_{hysteretic}+K_{slip-lock}}}\right)\dot{v}\end{array}$$

(14)

It is possible to obtain the rate form of the state vector ${\mathit{Z}}_{SHM}$ by using Equations (9) and (11)–(14). In this way, we obtain a first order equation, depending only on the variables contained in the state vector, to solve the equations governing the SHM:

$${\dot{\mathit{Z}}}_{SHM}=\mathit{g}\left({\mathit{Z}}_{SHM}\right)$$

(15)

where $\mathit{g}(·)$ represents a vectorial function that evaluates the rate form of the state variables. It is possible to solve the equation governing the SHM given a set of time–history displacement $v\left(t_i\right)$ and rate of displacement $\dot{v}\left(t_i\right)$for each discrete time value $t_i.$In this study, we solve the vectorial first-order equation represented through Equation (15) using the solver ode23t available in MATLAB version R2024a [35]. The use of the solver needs the evaluation of the rate form of the state vector ${\dot{\mathit{Z}}}_{SHM}$ for each time value. A pseudo-code to evaluate this rate form is presented in Appendix A. The set of time–history displacement and rate of displacement can be imposed during an experimental test, for instance, when a controlled displacement load protocol is used. In this case, the experimental test design includes the definition of the pairs $v\left(t_i\right)-t_i$ and $\dot{v}\left(t_i\right)-t_i$. Additionally, the instrumentation of experimental tests includes devices to record the displacement and displacement rate during the execution of the experiment. This information can be used as input to evaluate the numerical response.

2.4. Model Validation

The SHM presented in the last subsection is adjusted and validated using three experimental tests of CLT walls [6]. These three experimental tests were selected aiming to represent three cases of predominant displacement behavior of the walls [6]: (i) rocking behavior, (ii) rocking–sliding behavior, and (iii) sliding behavior. While

Table 1. Configuration of tested walls and their kinematic behavior.

Test ID	Number of Hold-Downs	Number of Angle Brackets	Number of Screws in Vertical Joints	Vertical Load (kN/m)	Predominant Deformation	Panel Interaction
Test-01	2	2	-	18.5	Sliding	Single
Test-02	2	4	-	18.5	Rocking–sliding	Single
Test-03	4	4	5	18.5	Rocking	Couple

lists the characteristics of the test specimens and indicates their identification name, Figure 4 shows illustrations of the tested CLT walls. The CLT walls were anchored to a steel foundation in all tests using angle brackets and hold-downs. All specimens were realized using five-layered panels with a thickness of 85 mm. While Test-01 and Test-02 consisted of single wall panels with an aspect ratio of 1:1 (dimensions 2.95 $\times$ 2.95 m), Test-03 consisted of two coupled panels with a half-lap joint with an aspect ratio of 1:2 (dimensions 1.48 $\times$ 2.95 m). The three considered tests differ in the number of hold-downs (HTT22 with twelve 4 $\times$ 60 mm Anker annual ring nails) and angle brackets (BMF90 $\times$ 48 $\times$ 3 $\times$ 116 mm with 114 $\times$ 60 mm Anker annual ring nails) employed.

The specimen of Test-01 exhibited a failure of the nails of the angle brackets due to excessive shear, generating all the wall displacement as a consequence of the wall’s sliding. In this case, the steel part of the hold-downs presented a buckling failure. The specimen of Test-02 failed due to a combination of shear force and overturning moments that triggered yielding in the metal connectors. In the case of Test-02, the displacement was generated as a combination of rocking and sliding. The rocking of the panels mainly generated the displacement of the specimen of Test-03. This kinematic behavior caused the yielding of annular ring nails, starting from the outer edges of the wall. In this case, no important deformation of connectors was observed.

An identification process was carried out to find the parameters that represent the hysteretic response of the specimens of Test-01, Test-02, and Test-03. For simplicity, typical values of the parameters that control the shape of the unloading curve were used: ${\eta }_1={\eta }_2=1/2$. The parameters controlling the transition between elastic and plastic branches and the ratio between the post-yield stiffness and initial stiffness were also considered constant for all the cases: $N=1$ and $a=1/100$, respectively. Additionally, the CLT walls are assumed to have a symmetric initial yielding resistance, hence $F_{y0}^{+}=F_{y0}^{-}=F_{y0}$ and $v_u^{+}=v_u^{-}=v_u$.

A multi-linear objective function $\epsilon$($·$) was employed to find the values of the parameters $\mathsf{\Theta }$ that minimize the difference between the numerical and experimental results:

$$\epsilon \left(\mathsf{\Theta }\right)=0.5E_{F,NRMS}+0.5E_{E,NRMS}$$

(16)

where $E_{F,NRMS}$ and $E_{E,NRMS}$ are the Normalized Root Mean Square (NRMS) errors between the numerical and experimental results.

Table 2. Identified parameters of the SHM representing experimental responses.

Test ID	$\mathit{\alpha }$	${\mathit{R}}_{\mathit{s}}$	$\mathit{\sigma }$	$\mathit{\lambda }$	${\mathit{\beta }}_1$	${\mathit{\beta }}_2$	${\mathit{K}}_0$ (kN/mm)	${\mathit{F}}_{\mathit{y}0}$ (kN)	${\mathit{v}}_{\mathit{u}}$ (mm)	${\mathit{H}}_{\mathit{u}}$ (kJ)	${\mathit{v}}_{\mathit{y}}$ (mm)	${\mathit{\mu }}_{\mathit{m}\mathit{a}\mathit{x}}$	${\mathit{R}}_{{\mathit{H}}_{\mathit{u}}}$
Test-01	5	0.35	0.13	0.22	0.30	0.30	10.50	153.3	96	43.3	8	12	5
Test-02	5	0.37	0.21	0.20	0.30	0.30	10.52	136.7	91	62.2	13	7	5
Test-03	8	0.38	0.25	0.15	0.26	0.24	9.00	131.4	94.9	63.3	14.6	7	5

presents the parameters that minimize the objective function for the three experimental specimens.

Other parameters are important when designing structures subjected to seismic loads. In this study, three additional parameters, dependent on the parameters that define the SHM response, can provide insight into the behavior of base-isolated timber structures. The first parameter is the yield displacement $v_y$, which can be obtained as:

$$v_y=F_{y0}/K_0$$

(17)

The second parameter is the maximum ductility ${\mu }_{max}$, determined as:

$${\mu }_{max}=v_u/v_y$$

(18)

The last parameter is related to the ratio between the energy dissipated to ultimate displacement when the element is loaded cyclically and when it is loaded monotonically:

$$R_{H_u}=H_u/\left(F_{y0}{v}_u\right)$$

(19)

These three additional parameters are also presented in Table 2.

In Figure 5, Figure 6 and Figure 7, a comparison between the experimental responses and the results obtained using the SHM is presented. On these figures, subfigure (a) shows the evolution of the forces generated on the CLT wall, subfigure (b) depicts the cumulative energy dissipated during the tests, and subfigure (c) presents the hysteretic force–displacement relationship. A good agreement between the numerical and experimental responses is observed in the three analyzed cases.

3. Equivalent Model of a Timber Structure Equipped with Frictional Bearings

This section briefly describes one of the most used frictional bearings: the Friction Pendulum System (FPS) [36]. Additionally, we present an equivalent model of a CLT structure equipped with this device and the equation of motion governing its dynamic behavior.

3.1. Brief Description of the Friction Pendulum System (FPS)

The Friction Pendulum System (FPS) was introduced by Victor Zayas [36]. The device consists of two components: a concave plate with a spherical sliding surface characterized by a radius of curvature $R$ and an articulated slider. The interaction between these components generates a low friction interaction characterized by a friction coefficient ${\mu }_d=\tan \left(\varphi \right)$, with $\varphi$ denoting the friction angle. An illustration of the device is presented in Figure 8a.

The FPS bearing uses its geometry and gravity to achieve conventional seismic isolation: flexible lateral behavior and a stiff vertical response. In Figure 8b, the forces generated in the device are shown when a lateral force $f_a$ and a vertical force $W$, representing the weight of the superstructure, are acting on the bearing. The external forces $f_a$ and $W$ are equilibrated by a reaction force $F_R.$ In this subsection, we analyze the lateral projection of this reaction force. If small displacements are assumed (i.e., small values of the angle $\theta$), the lateral projection of the reaction force can be split into two components: the pendular force $f_p$ and the friction force $f_{\mu }$. The pendular force depends on the vertical load applied on the bearing and the lateral displacement of the articulated slider $q_b$ as follows:

$$f_p={\displaystyle \frac{W}{R}}{q}_b$$

(20)

Note that the term $W/R$ represents the lateral stiffness of the isolator. The friction force when the slider slides on the spherical surface can be determined as follows:

$$f_{\mu }={\mu }_{d}W\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left({\dot{q}}_b\right)$$

(21)

In this manner, the magnitude of the lateral force acting in the bearing, considering small displacements, can be determined as:

$$f_a=f_p+f_{\mu }={\displaystyle \frac{W}{R}}{q}_b+{\mu }_{d}W\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left({\dot{q}}_b\right)$$

(22)

In Figure 8c–e, the normalized force–displacement relationship is presented in terms of normalized pendular force $f_p/W$, normalized friction force $f_{\mu }/W$, and normalized total force ${(f}_p+f_{\mu })/W$.

It is possible to determine the isolated period $T_b$of an FPS bearing, considering the stiffness $K_{FPS}$ and the mass of the system $M_{FPS}$ as follows:

$$T_b=2\pi \sqrt{{\displaystyle \frac{M_{FPS}}{K_{FPS}}}}=2\pi \sqrt{{\displaystyle \frac{W/g}{W/R}}}=2\pi \sqrt{{\displaystyle \frac{R}{g}}}$$

(23)

where $g$is the gravity acceleration. One of the most relevant characteristics of conventional friction pendulum bearings employed to achieve seismic isolation is that the isolation period only depends on the geometry of the device (i.e., the effective radius). This feature is relevant since this device can isolate superstructures or equipment independent of their mass. FPS bearings can be used to achieve seismic isolation of timber structures despite having lighter self-weights compared to reinforced concrete structures.

3.2. Equivalent Model of CLT Structure Equipped with FPS Bearings

In this study, the seismic response of base-isolated timber structures is assessed using a simplified model. A similar model has been used in several studies that conduct parametric analysis of buildings equipped with frictional isolators [37,38,39,40,41,42]. An illustration of the equivalent model is presented in Figure 9. The superstructure is represented by a two-degree-of-freedom system, considering the horizontal displacements of the base and the roof. While the symbol $q_b\left(t\right)$ denotes the base displacement, the symbol $q_s\left(t\right)$ denotes the roof displacement. Hence, the displacement of the superstructure can be computed as $v_s\left(t\right)=q_s\left(t\right)-q_b\left(t\right)$. Two masses are added to the superstructure: the mass of the base $m_b$, and the mass of the roof $m_s$, leading to a total mass of the superstructure of $m_t=m_b+m_s$, and a total self-weight $W=m_{t}g$. The mass distribution ratio is defined as $\gamma =m_s/m_t$. The superstructure’s initial stiffness, damping coefficient, and yield displacement are denoted by the symbols $k_s$, $c_s$, and $v_y$, respectively. The critical damping ratio linked to the superstructure was set as equal to ${\xi }_s=2\%$. This value is used to determine the parameter $c_s$. The force developed in the superstructure $f_s$ is modeled using the SHM. It is possible to determine $f_s$ given a set of displacement $v_s$ and velocity ${\dot{v}}_s$ and define the sixteen parameters of the SHM.

The natural period of the superstructure $T_s$, when considered as a fixed-base structure, depends on the initial stiffness of the building and its mass distribution (i.e., $T_s=2\pi \sqrt{m_s/k_s}$). On the other hand, the isolated period $T_b$, when single curvature isolators are considered, depends only on the radius of curvature of the sliding surface $R$ (i.e., $T_b=2\pi \sqrt{R/g}$).

The isolation system is modeled through an equivalent isolator, employing the physical model of the Friction Pendulum System (FPS) [43]. This model can account for essential modeling features such as large displacements, actual device geometry, sticking and sliding phases, and kinematics constraints.

Elastomeric bearings are commonly used to realize isolation systems. These devices may be subjected to instability due to large displacements and compressive loads caused by gravity and overturning forces [44]. The role of shear modulus on the properties of elastomeric bearings has been studied in devices subjected to combined axial and shear loads [45]. The mentioned instability problems do not apply to pendulum friction bearings; for this reason, despite employing a model capable of accounting for large displacements (i.e., geometric effects), the instability of the isolation devices was not considered in this study.

The force developed in an isolation system formed by frictional pendulum bearings, when the motion of the dynamic system is considered only in one horizontal direction, can be determined using the following expression:

$$f_a(t)=-{\displaystyle \frac{N\left(t\right)}{R}}{q}_b(t)+{\mu }_d(t)\left|N\left(t\right)\right|{\displaystyle \frac{\sqrt{R^2-{q_b}^2\left(t\right)}}{R}}{z}_a(t)$$

(24)

where $f_a$ is the lateral force acting on the isolation system, and $N\left(t\right)$ is the normal force (negative in compression) normally oriented to the sliding surface with a radius of curvature $R$. The symbol ${\mu }_d\left(t\right)$ represents the sliding velocity-dependent friction coefficient developed between the articulated slider and the concave sliding surface of the isolators. In this study, the dependence between the value of the friction coefficient and the velocity of sliding $\dot{s}\left(t\right)$ is considered using the expression provided by Constantinou et al. [46]:

$${\mu }_d\left(t\right)=f_{max}-\left(f_{max}-f_{min}\right)\mathrm{e}\mathrm{x}\mathrm{p}(-r\left|\dot{s}\left(t\right)\right|)$$

(25)

where the parameters $f_{max}$ and $f_{min}$ are the friction coefficients at large and slow sliding velocity, respectively, and $r$ is a constant for a given contact pressure and interface condition. In this study, we employ a constant value of $r=30$ s/m. The symbol $z_a\left(t\right)$ in Equation (21) represents one of the hysteretic parameters of the two-dimensional Bouc–Wen model [47]. This parameter allows us to determine the magnitude of the friction force and the sticking and sliding phases developed in the isolation system. The parameter $z_a\left(t\right)$ can be defined as a state variable. Similarly to the six state variables of the SHM, the evolution of the hysteretic variable associated with the isolation system can be described by a first-order state equation as follows:

$${\dot{z}}_a=g_a\left(z_a\right)={\displaystyle \frac{1}{{\mathsf{\Delta }}_s}}\left(A-(\beta +\gamma \mathrm{s}\mathrm{g}\mathrm{n}({\dot{q}}_b\left(t\right)z_a\left)\right){z_a}^2\right){\dot{q}}_b(t)$$

(26)

where $g_a(·)$ represents the function that evaluates the rate form of the state variable $z_a$; $A,$ $\beta$, and $\gamma$ represent dimensionless constants that define the shape of the hysteretic response; and ${\mathsf{\Delta }}_s$ represents the displacement that triggers the sliding phase.

In this study, the Soil–Structure Interaction (SII) was not considered. Including the SII can increase the seismic response of base-isolated buildings [48]. This aspect represents a limitation and can affect the results obtained, deteriorating the estimated seismic performance of the analyzed cases.

3.3. Equation of Motion of the Equivalent Model and a Solving Strategy

The equation of motion of the equivalent model of a timber structure equipped with frictional isolators can be expressed as:

$$\mathit{M}\ddot{\mathit{u}}\left(t\right)+\mathit{C}\dot{\mathit{u}}\left(t\right)+{{\mathit{L}}_{\mathit{s}}}^T{f}_s++{{\mathit{L}}_{\mathit{a}}}^T{f}_a=-\mathit{M}\mathit{R}{\ddot{u}}_g$$

(27)

in which $\dot{\mathit{u}}\left(t\right)={\left[{\dot{q}}_b\left(t\right), {\dot{q}}_s\left(t\right)\right]}^T$and $\ddot{\mathit{u}}\left(t\right)={\left[{\ddot{q}}_b\left(t\right), {\ddot{q}}_s\left(t\right)\right]}^T$ are the vectors containing the velocities and accelerations of degrees-of-freedom related to a reference frame fixed to the ground; $\mathit{M}$ and $\mathit{C}$ are the mass and damping matrices of the superstructure; ${\mathit{L}}_{\mathit{s}}$ and ${\mathit{L}}_{\mathit{a}}$ are the kinematic transformation matrices for the superstructure and the isolation system; and $\mathit{R}$ is the input influence vector associated with the ground acceleration in the horizontal direction ${\ddot{u}}_g$.

The equation of motion can be solved by employing a first-order approach [34]. The degrees of freedom of the dynamics system (i.e., $\mathit{u}\left(t\right)={\left[q_b\left(t\right), q_s\left(t\right)\right]}^T$) and the velocities (i.e., $\dot{\mathit{u}}\left(t\right)={\left[{\dot{q}}_b\left(t\right), {\dot{q}}_s\left(t\right)\right]}^T$), are defined as state variables to follow this approach. In this way, the state vector, representing the equivalent model, can be expressed as:

$$\mathit{Z}(t)=\left[\begin{array}{c}\mathit{u}\left(t\right)\\ \dot{\mathit{u}}\left(t\right)\\ \begin{array}{c}{\mathit{Z}}_{SHM}\left(t\right)\\ z_a\left(t\right)\end{array}\end{array}\right]$$

(28)

By employing Equations (15), (22) and (23), it is possible to obtain a vectorial first-order equation depending only on the variables contained on the state vector (Equation (24)) as follows:

$$\dot{\mathit{Z}}\left(t\right)=\left[\begin{array}{c}\dot{\mathit{u}}\left(t\right)\\ \ddot{\mathit{u}}\left(t\right)\\ \begin{array}{c}{\dot{\mathit{Z}}}_{SHM}\left(t\right)\\ {\dot{z}}_a\left(t\right)\end{array}\end{array}\right]=\left[\begin{array}{c}\dot{\mathit{u}}\left(t\right)\\ -{\mathit{M}}^{-1}(\mathit{C}\dot{\mathit{u}}\left(t\right)+{{\mathit{L}}_{\mathit{s}}}^T{f}_s++{{\mathit{L}}_{\mathit{a}}}^T{f}_a+\mathit{M}\mathit{R}{\ddot{u}}_g)\\ \begin{array}{c}\mathit{g}\left({\mathit{Z}}_{SHM}\right(t\left)\right)\\ g_a\left(z_a\right(t\left)\right)\end{array}\end{array}\right]$$

(29)

Equation (25) shows the advantages of using a state-space approach. Since the equation of motion can be expressed as a first-order equation, and all the hysteretic parameters of the superstructure and the isolation systems follow first-order equations to evaluate their evolution, it is possible to solve the entire problem simultaneously. In this study, we numerically solve Equation (25) using the solver ode23t available in the MATLAB environment [35]. Other alternative nonlinear techniques were not considered to represent the lateral behavior of CLT buildings. Other sources of nonlinearities, such as geometric effects in the superstructure, were not considered.

3.4. Time–History Examples

In this study, we employ the equivalent model using the average values of the parameters fitted in Section 2.2 regarding the strength and stiffness degradation and the pinching phenomenon. It is important to mention that the parameters are related to the response of CLT walls and not to the CLT buildings. Hence, employing these values may represent an important limitation. This decision is based on the response of regular superstructures with the same story resistance in height. The dynamic response of this kind of dynamic system is mainly represented by its first mode, a flexible lateral mode in which the superstructure behaves as a rigid body. Under this assumption, the seismic loads are constant in height, leading to maximum shear demand in the first story [49,50]. If the building has the same strength in all stories, the nonlinear response will be concentrated in the CLC walls of the first story. In this case, the use of the fitted parameters is reasonable since, on this story, the elements work in parallel.

In Figure 10 and Figure 11, examples of time–history analyses are presented for two CLT buildings equipped with frictional devices subjected to the Hualañe Hospital record (Maule Earthquake, 2010) [51] for different maximum ductility capacities of ${\mu }_{max}=6$ and 4, respectively. Figure 10 and Figure 11 include the dynamic response in terms of normalized base displacement $q_b\left(t\right)/R$, superstructure displacement $v_s\left(t\right)/v_y$, the evolution of the state-variables representing the normalized hysteretic force $F^{*}/F_{y0},$ and positive and negative yield forces $F_y^{+/-}/F_{y0}$. The hysteretic loops related to the isolation system and superstructure are also included. While the superstructure is characterized by a period of $T_s=0.6$ s, a mass distribution ratio of $\gamma =0.7$, and a yield displacement of $v_y=2.4$ cm, the isolation system is characterized by an isolated period of $T_b=3$ s and a friction coefficient at large velocity of $f_{max}=5\%$.

The time–history analyses were conducted by scaling the ground motion to the Maximum Possible Earthquake (MPE) level, which has a probability of exceedance of 10% in 100 years according to the Chilean code [52]. Concerning the case of a superstructure with a maximum ductility capacity of ${\mu }_{max}=6$ (Figure 10), although the superstructure exhibits nonlinear behavior, exceeding a ductility demand of 2, good seismic behavior is observed. Strength degradation of the superstructure is observed due to ductility demand and dissipated energy (Figure 10c). However, this decrease does not exceed 80% at the end of the ground motion.

A building collapse is detected if the superstructure has a maximum ductility capacity of ${\mu }_{max}=4$ (Figure 11). The collapse is triggered due to an excessive ductility demand. Note that when the normalized displacement exceeds $v_s\left(t\right)/v_y\ge {\mu }_{max}$at time $t=68.3$s (Figure 11c), the positive hysteretic force $F_y^{+}$, representing the remaining resistance of the building, is completely nullified (Figure 11d). The energy dissipated during the ground motion and the ductility demand produce a dramatic stiffness and strength degradation (Figure 11e). This example highlights the importance of the parameter ${\mu }_{max}$on the seismic performance of CLT base-isolated buildings represented through the developed equivalent model.

Table 3. Key performance indicators of the seismic performance of the time–history analyses.

Time–History Example Shown in	Maximum $\mathbf{Ductility} \mathbf{Capacity} {\mathit{\mu }}_{\mathit{m}\mathit{a}\mathit{x}}$	Collapse	$\mathbf{Maximum} \mathbf{Ductility} \mathbf{Demand} \mathit{\mu }$	Positive Strength Degradation ${\mathit{F}}_{\mathit{y}}^{+}/{\mathit{F}}_{\mathit{y}0}$	Negative Strength Degradation ${\mathit{F}}_{\mathit{y}}^{-}/{\mathit{F}}_{\mathit{y}0}$
Figure 10	6.0	No	2.9	0.81	0.82
Figure 11	4.0	Yes	4.0	0.00	0.77

includes comparative information indicating the key performance indicators of the two examples of the presented time–history analyses.

4. Uncertainties Within the Seismic Fragility Assessment

Two sources of uncertainties were considered to assess the seismic fragility of CLT base-isolated structures: (i) the friction coefficient characterizing the contact between the articulated slider and the sliding surface of the devices forming the isolation system and (ii) the earthquake event characteristics. Other sources of uncertainties, such as mechanical and geometrical properties of frictional devices or superstructures, were not considered since they do not produce relevant effects on the statistics of the response parameters when the isolation degree, defined as $I_d=T_b/T_s$, is high [53,54].

The frictional interaction between two contact surfaces is a complex phenomenon. Experimental tests show that the value of the friction coefficient depends on several factors, such as the sliding velocity, the contact pressure, the environmental condition, and the energy dissipated during the sliding, among other factors [30,46,55]. Another source of uncertainty is linked to the coefficient of friction: the wearing of the sliding surface or the contact surface of the slider over time. This wear could increase the value of the friction coefficient. In this study, we link the uncertainties of the friction phenomenon to the friction coefficient at large sliding velocity $f_{max}$, as carried out in previous studies [37,38,56,57]. We adopted a Gaussian Probability Density Function (PDF), truncated at 2% and 8%, and with a mean value of 5% to consider the variability of this parameter. We sampled eight values from this distribution using the Latin Hypercube Sampling (LHS) method [58,59]. The friction coefficient at slow sliding velocity is assumed to be correlated with the friction at large sliding velocity ($f_{min}=f_{max}/2$).

Concerning the uncertainties related to the earthquake source characteristics, an Intensity Measure (IM) was introduced into the fragility analysis to include the variability of seismic input intensity. This study used spectral acceleration at the isolated period $S_a\left(T_b\right)$ as the IM. The uncertainties on the characteristics of the seismic records were included by selecting a set of 41 natural ground motions and considering two isolated periods of $T_b=3$ and $4$ s.

5. Ground Motion Selection

This section presents the ground motions employed in this study and the criteria adopted to perform their selection. Since this study aims to evaluate the seismic performance of base-isolated CLT buildings to be constructed in Chile, only earthquake ground motions from interface and intraslab subduction events in Chile were considered.

The processed acceleration time-series, along with the source, path, and site metadata of these ground motions, were retrieved from the Next Generation Attenuation for Subduction Zones (NGA-Sub) database [51]. The selection process consisted of filtering and preselecting only ground motion records measured in sites compatible with soil class B, according to the Chilean Standard NCh.2745—analysis and design of base-isolated buildings [52]. Hence, only ground motions recorded in stations found in soil class B were considered. A soil with a time-averaged shear wave velocity in the upper 30 m of depth between $500$ m/s ${<V}_{S30}<900$ m/s classifies as soil class B. This soil class is representative of an important area of Santiago (Chile), where a base-isolated building is likely to be constructed. Additional criteria were applied to select the ground motion records: the moment magnitude (M) of the events and the Peak Ground Acceleration (PGA) values had to exceed M5 and 0.05 $g$, respectively. Subsequently, each preselected spectrum derived from each ground motion record $S_{a,rec}$ was scaled using a scale factor ($SF$) to minimize the difference with the target spectrum $S_{a,target}$, considering a period range defined by $T_i\in$[$2.5 \mathrm{s}, 5.5 \mathrm{s}$]. The scaling procedure only consisted of multiplying the entire original record by the scale factor $SF$. As seen in previous studies [60,61,62], the value of that difference was quantified by an index that measures the average spectral deviation, shown as follows:

$${\delta }_{ind}=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{\left({\displaystyle \frac{S_{a,rec}\left(T_i\right)\times SF-S_{a,target}\left(T_i\right)}{S_{a,target}\left(T_i\right)}}\right)}^2}$$

(30)

where $N$ is the number of periods within the range and $T_i$ corresponds to the i-th period within the defined interval. The selected ground motions present ${\delta }_{ind}$ values below 0.9 and scale factors $SF$ between $1$ and 6.4. To ensure the analysis was independent of the horizontal directional characteristics of the earthquake ground motions, the RotD50 spectrum [63] was used for scaling and computing ${\delta }_{ind}$ for each record.

The ground motion selection procedure resulted in a total of 41 records. From these, RotD100 accelerograms were generated for each period of the base-isolated structures considered in this study, providing a basis for the parametric analysis. These accelerograms were scaled such that their spectra matched the target spectrum at the isolated period $T_b$ of each evaluated building. Figure 12 presents the set of selected ground motions records, showing a comparison between their scaled spectra to the target spectrum within the period range of 2.5 to 5.5 s. Figure 13a shows the magnitude–distance distribution of the selected ground motion records, highlighting several large-magnitude earthquakes that occurred in Chile, including the 2010 M8.8 Maule, 2015 M8.3 Illapel, 2014 M8.2 Iquique, 1985 M8 Valparaíso, and interface events, as well as the 2005 M7.8 Tarapacá event, an intraslab earthquake. The closest distance to the fault rupture ($R_{RUP}$) ranged from 25 to 250 km, approximately. Figure 13b shows the PGA-distance distribution of the selected unscaled ground motion records. RotD50 PGA values range between 0.05 and 0.75 g, approximately. Details of the selected ground motions are presented in Appendix B.

6. Parametric Study

In the present work, a broad parametric analysis is conducted to evaluate the effects of the maximum ductility capacity ${\mu }_{max}$on the fragility of timber structures equipped with frictional bearings. Several values related to inelastic and elastic properties of the superstructure were considered. Three initial natural periods were evaluated: $T_s=0.3$, $0.6$, and $0.9$ s. The selection of these values aims to include CLT buildings with different numbers of stories. The natural period increases as the number of stories increases. The period range includes CLT buildings with approximately three to fifteen stories. The effectiveness of seismic isolation is better for increasing values of the isolation degree $I_d=T_b/T_s$ [53,54]. For this reason, increasing the natural period of the superstructure may represent a potential drawback.

Two mass distribution ratios were counted: $\gamma =0.7$ and $0.9$. A larger value of $\gamma$ implies that the mass is concentrated at the first levels of the buildings, hence concentrating the seismic force in those levels. In this way, it is possible to include in the analysis structures that resist the demands of ground motions differently. Four values of maximum ductility were considered: ${\mu }_{max}=v_u/v_y=4$, $6$, $8$, and $10$. As mentioned above, the nonlinear response of CLT shear walls is governed mainly by the characteristics of the connectors. In this manner, higher values of ${\mu }_{max}$indicate connectors, or connectors configurations, exhibit more ductile behavior. According to experimental tests conducted in CLT shear wall systems [6], the number of angle brackets used to connect the panel with CLT floor panels or the foundation plays a crucial role regarding the ductility capacity of the element. For instance, increasing the angle brackets from two to four may increase the ultimate displacement by 47% without an abrupt change in the initial stiffness. Another strategy to achieve the ductile behavior of the CLT wall is to avoid the shear failure of the nails. If the CLT wall behavior is governed by rocking, a desirable behavior since a greater re-center capacity is observed, it is important to prevent the pulling-out of the nails. Hence, the plastic behavior of the connectors should occur in the angle brackets and hold-downs working in tension.

Once the parameters $T_s$, $\gamma$, and ${\mu }_{max}$ are defined for a particular case, it is possible to determine the yield displacement of the superstructure $v_y$ following the criteria of the Chilean Standard concerning base-isolated structures [52]. The yield displacement $v_y$ is linked to a reduction factor $R_I$. In this research, three typical values of reduction factors are employed: $R_I=1.0$, $1.6$, and $2.0$. The range of reduction factors represents commonly used values in real projects. A larger value of $R_I$ is linked to superstructures with large energy dissipation capacity. For each isolated period considered in the parametric analysis, there are 72 different superstructures.

The remaining parameters defining the hysteretic response of the superstructure are taken as the mean value of those obtained by adjusting the SHM with the experimental response of CLT walls. These parameters are linked to the stiffness degradation ($\alpha$), the pinching phenomenon ($R_s$, $\sigma$, and $\lambda$), the strength degradation (${\beta }_1$ and ${\beta }_2$), and the failure due to the dissipated energy ($R_{H_u}$)

Two isolated periods defined the dynamic response of the isolation system: $T_b=3$ and 4 s. Since the isolated period is linked to the radius of curvature of the bearings, two radii were adopted: $R=2.24$ and $3.98$ m. A larger isolated period may reduce the seismic force transmitted by ground motion but increase the base displacement of the isolation system. In this manner, the parametric analysis accounts for two isolation systems working slightly differently. In total, this study contemplates 144 different base-isolated buildings.

7. Design of Base-Isolated Buildings

The design of base-isolated buildings is driven by the criteria of the Chilean Standard NCh.2745—analysis and design of base-isolated buildings [52]. In particular, the resistance of the superstructure $F_{y0}$must be determined to withstand the design seismic load. Since the initial stiffness of buildings $K_0$ is defined, given an initial natural period $T_s$ and mass distribution ratio $\gamma$, the output of the design process is the yield displacement $v_y$ of buildings (i.e., $v_y=F_{y0}/K_0$).

The design of base-isolated buildings can be carried out using different approaches. In this study, we follow two methodologies: the Static Analysis procedure and the Dynamic Analysis procedure. Concerning the Dynamic Analysis procedure, we performed time–history analyses to determine the design values related to the maximum displacement of the superstructure.

The Static Analysis procedure is separated into two steps and uses two friction coefficient values to obtain the design parameters. First, the displacement of the isolation system $D_D$, given the Design Earthquake (DE), which has a probability of exceedance of 10% in 50 years, is estimated using the lower bound of the friction coefficient (2% according to the defined Gaussian PDF). Then, the required maximum displacement of the superstructure $v_{max}$ is determined using the obtained value of $D_D$but considering the upper bound of the friction coefficient (8% according to the defined Gaussian PDF). The yield displacement of the superstructure can be computed considering the reduction factor $R_I$ as follows:

$$v_y={\displaystyle \frac{v_{max}}{R_I}}$$

(31)

Note that a reduction factor of $R_I=1.0$ implies that the expected behavior of the superstructure will be essentially elastic, given the Design Earthquake.

The Dynamic Analysis procedure can be followed by performing nonlinear time–history analyses. The nonlinearities are concentrated in the isolation system, representing the friction nature of pendulum bearings and assuming a linear superstructure behavior. Forty-one time–history analyses were conducted for each base-isolated building, applying all the selected ground motion records. Each record was scaled to match the spectral pseudo-acceleration at the isolated period $S_a\left(T_b\right)$ linked to the DE. The required maximum displacement of the superstructure $v_{max}$ was determined using the average value of the 41 responses obtained for each ground motion applied. Since the dynamic response is highly dependent on the value of the friction coefficient, two values were employed, namely the lower and upper bound values of the Gaussian PDF, representing the distributions of this parameter.

In

Table 4. Required maximum displacement of buildings with an isolated period of $T_b = 3$s.

		Static Analysis		Time–History Analysis
	${\mathit{T}}_{\mathit{s}}$	${\mathit{D}}_{\mathit{D}}$	${\mathit{v}}_{\mathit{m}\mathit{a}\mathit{x}}$	${\mathit{D}}_{\mathit{D}}$	${\mathit{v}}_{\mathit{m}\mathit{a}\mathit{x}}$*	${\mathit{v}}_{\mathit{m}\mathit{a}\mathit{x}}$**
$\mathit{\gamma }$	(sec)	(cm)	(cm)	(cm)	(cm)	(cm)
0.7	0.3	20.6	0.6	20.5	0.4	0.8
	0.6	20.6	2.4	20.1	1.8	3.1
	0.9	20.6	5.4	19.7	3.8	5.9
0.9	0.3	20.6	0.6	20.4	0.4	0.6
	0.6	20.6	2.4	20.0	1.6	2.2
	0.9	20.6	5.4	19.4	3.5	4.7

$v_{max}$^*: obtained using   $f_{max}=2\mathrm{\%}$; $v_{max}$^**: obtained using   $f_{max}=8\mathrm{\%}$.

and

Table 5. Required maximum displacement of buildings with an isolated period of $T_b = 4$s.

		Static Analysis		Time–History Analysis
	${\mathit{T}}_{\mathit{s}}$	${\mathit{D}}_{\mathit{D}}$	${\mathit{v}}_{\mathit{m}\mathit{a}\mathit{x}}$	${\mathit{D}}_{\mathit{D}}$	${\mathit{v}}_{\mathit{m}\mathit{a}\mathit{x}}$*	${\mathit{v}}_{\mathit{m}\mathit{a}\mathit{x}}$**
$\mathit{\gamma }$	(sec)	(cm)	(cm)	(cm)	(cm)	(cm)
0.7	0.3	14.6	0.4	18.0	0.3	0.8
	0.6	14.6	1.7	17.8	1.3	3.0
	0.9	14.6	3.9	17.6	2.7	5.9
0.9	0.3	14.6	0.4	18.0	0.3	0.5
	0.6	14.6	1.7	17.9	1.0	2.1
	0.9	14.6	3.9	17.3	2.3	4.4

$v_{max}$^*: obtained using   $f_{max}=2\mathrm{\%}$; $v_{max}$^**: obtained using   $f_{max}=8\mathrm{\%}$.

, the required maximum displacement of the superstructure, obtained following both design procedures, is presented for isolated periods of $T_b=3$ and 4 s, respectively. The yield displacement $v_y$, characterizing the nonlinear response of the superstructure, is determined using the maximum magnitude of the three values of $v_{max}$ obtained (one from the Static Analysis and two from the Dynamic Analysis) and applying the reduction factor $R_I$ according to Equation (27).

8. Seismic Fragility of CLT Structures Equipped with Pendulum Frictional Bearings

This section presents the seismic performance of the 144 base-isolated CLT structures defined in the parametric analysis in terms of fragility curves. One Limit State (LS) is assessed: the probability of exceeding a particular maximum ductility capacity value ${\mu }_{max}$ defining the nonlinear behavior of the considered superstructures. The maximum ductility capacity limits considered were 4, 6, 8, and 10.

8.1. Incremental Dynamic Analyses (IDAs)

Performing Incremental Dynamic Analyses (IDAs) is the first step to determining the seismic fragility of structures [64]. Conducting IDAs allows us to evaluate the dynamic response of structures for increasing IM levels. Twelve values of the IM (i.e., $S_a\left(T_b\right)$) were considered for the analyses. These values are linearly spaced, ranging from $0.1 \times S_a{\left(T_b\right)}_{DE}$ to $2.0 \times S_a{\left(T_b\right)}_{DE}$, where $S_a{\left(T_b\right)}_{DE}$ is the spectral acceleration at the isolated period for the Design Earthquake level.

The seismic performance of the CLT base-isolated structures is characterized by the maximum values of the Engineering Demand Parameters (EDPs) of interest. In this study, we analyze one EDP: the maximum ductility demand of the superstructure defined as

$$\mu =v_{s,max}/v_y$$

(32)

where $v_{s,max}={\left|v_s\left(t\right)\right|}_{max}$ is the maximum displacement of the superstructure. The selected response parameter is assumed to follow a lognormal distribution. In this way, estimating the structural response in terms of different percentile levels is possible. We fitted the lognormal distribution by estimating the sample lognormal mean $mea{n}_{ln}\left(EDP\right)$ and the lognormal standard deviation of the sample ${\sigma }_{ln}\left(EDP\right)$ using the maximum likelihood estimation method and without considering the collapses due to excessive dissipated energy or ductility demand. Several parametric studies have employed this probabilistic approach [37,39,42,49,50,56,65,66].

One IDA consists of 6560 simulations, using the 41 selected seismic records scaled to the 20 IM values and combined with the 8 samples of $f_{max}$. The differential equation of motion (Equation (25)) has been repeatedly solved using the ode23t solver available in the MATLAB environment [35].

8.2. Fragility Analyses Results and Discussion

The seismic performance of CLT base-isolated structures can be evaluated by deriving fragility curves. The seismic fragility is described as the probability $P_c$ of exceeding a Limit State (LS) at each defined IM level. In this study, we define the LS threshold as the maximum ductility capacity of the superstructure ${\mu }_{max}$. The probability of exceeding the LS at each IM level is established by fitting a lognormal complementary cumulative distribution function (CDF). The collapse and non-collapse results were considered using the total probability theorem:

$$P_c=P\left(EDP>y \right|{ S}_a\left(T_b\right)=x)=(1-F_{EDP|IM=im}\left(L{S}_{EDP}\right)){\displaystyle \frac{N_{not-collapse}}{N}}+\left(1-{\displaystyle \frac{N_{not-collapse}}{N}}\right)$$

(33)

where $N$ is the total number of analyses at each IM level, and $N_{not-collapse}$ is the number of numerical simulations without collapse. The collapse is reached if the dissipated energy exceeds the maximum capacity of the superstructure or if the ductility demand surpasses the maximum ductility capacity.

The fragility curves representing the seismic performance of all the analyzed CLT base-isolated structures are shown in Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19. Each figure presents the results in subfigures related to different values of superstructure period $T_s$ and mass distribution ratio $\gamma$. While each column contains the results related to a specific value of $T_s$, each row contains the results related to a different value of $\gamma$. In general, better seismic performance is achieved by increasing the values of $T_s$. This observation is consistent with other studies [37,38,57]. This improvement is generated due to a larger yield displacement for increasing natural period. This previous aspect may seem counterintuitive since seismic isolation is more effective for larger isolation degrees $I_d=T_b/T_s$. However, this trend applies when base-isolated structures are compared with conventional fixed-base buildings. As expected, a larger maximum ductility demand ${\mu }_{max}$ leads to lower seismic fragility.

In Figure 14, Figure 15 and Figure 16, the seismic fragility curves related to $T_b=3$ s are presented for increasing values of the reduction factor $R_I$. As shown in Figure 14, if the superstructure is designed using a reduction factor of $R_I=1$, a satisfactory seismic performance (i.e., a very low probability of exceeding ${\mu }_{max}$for the Design Earthquake level) is achieved for almost all cases. The only two cases that do not present good seismic performance are related to a stiff structure with a period of $T_s=0.3$ s and a low ductility capacity of ${\mu }_{max}=4$ (Figure 14a,d). A good seismic performance is achieved for stiff structures if the superstructure exhibits high ductility capacity (i.e., ${\mu }_{max}\ge 6$). It is important to note that all cases with${ R}_I=1$ and ${\mu }_{max}\ge 8$ present an excellent seismic performance, even for the Maximum Possible Earthquake (MPE) level, defined as 1.2 times the Design Earthquake (DE) level, according to the Chilean Standard NCh.2745 [52]. Superstructures with a period $T_s\ge 0.6$ s and designed using $R_I=1$ present good performance even for low ductility capacity (i.e., ${\mu }_{max}=4$).

Figure 15 presents the seismic fragility curves related to $T_b=3$ s and $R_I=1.6$. An important increase in fragility is observed, especially regarding low ductility capacity (i.e., ${\mu }_{max}=4$). For stiff superstructures (Figure 15a,d), an unsatisfactory seismic performance is observed for medium to low ductility capacity (i.e., ${\mu }_{max}\le 8$). On the other hand, if the superstructure exhibits a large ductility capacity, good seismic performance is achieved. Seismic performance improves for larger superstructure periods $T_s$ and a high mass distribution ratio $\gamma$. For superstructure with periods of $T_s=0.6$and $T_s=0.9$ s (Figure 15b,c,e,f), a ductility capacity of ${\mu }_{max}\ge 6$ leads to an excellent seismic performance despite using a reduction factor in the superstructure design larger than the unity.

Figure 16 presents the seismic fragility curves related to $T_b=3$ s and $R_I=2$. Stiff superstructures (Figure 16a,d) do not present satisfactory seismic performance, even for large ductility capacity. This observation suggests that using a high reduction factor is not advisable if the fixed-base superstructure exhibits a low natural period. Inadequate performance is observed in all the base-isolated structures with low ductility capacity (i.e., ${\mu }_{max}=4$). Hence, a large ductility capacity must be provided if the superstructure is designed employing a reduction factor of $R_I=2$. An excellent seismic performance is achieved for superstructures with $T_s=0.6$ and $T_s=0.9$ and a ductility capacity ${\mu }_{max}\ge 6$. The reduction factor $R_I=2$ represents the maximum value allowable for designing a base-isolated structure in Chile. This value is employed to design resistant steel and reinforced concrete systems based on frames and shear walls. Thus, CLT superstructures, designed to exhibit large ductility capacity and equipped with frictional isolators, may be a feasible alternative to constructing mid-rise timber structures in a high-hazard area such as Chile. Note that the identified parameters related to the ${\mu }_{max}$of tested walls are larger than 6. Furthermore, experimental campaigns have found that CLT walls typically have ductility capacities larger than 6 [4,5].

In Figure 17, Figure 18 and Figure 19, the seismic fragility curves related to $T_b=4$ s are presented for increasing values of the reduction factor $R_I$. Similar results are observed for cases associated with $T_b=3$ s (Figure 14, Figure 15 and Figure 16). The larger the value of $T_s$, $\gamma$, and ${\mu }_{max}$, the better seismic performance is detected. When comparing CLT base-isolated structures with isolated periods of $T_b=4$ and $3$ s, slightly better seismic performance is achieved for more flexible isolation interfaces

In Figure 17, the results regarding an isolated period of $T_b=4$ s, and a reduction factor of $R_I=1.0$ are shown. Suppose a reduction factor of $R_I=1.0$ is employed to design superstructures with a natural period of $T_s=0.6$ and $0.9$ s. In that case, an excellent seismic performance is achieved for all the considered cases of maximum ductility capacity ${\mu }_{max}$. For stiffer superstructures with $T_s=0.3$ s, a maximum ductility capacity ${\mu }_{max}\ge 8$ is required to achieve a satisfactory seismic performance.

Figure 18 presents the seismic fragility curves related to $T_b=4$ s and $R_I=1.6$. Similarly to cases associated with an isolated period of $T_b=3$ s (Figure 15), an important rise in the probabilities of exceeding the LS threshold is observed. Again, for superstructures with $T_s=0.3$ (Figure 18a,d) an unsatisfactory seismic performance is observed for medium to low ductility capacity (i.e., ${\mu }_{max}\le 6$).

Figure 19 presents the seismic fragility curves related to $T_b=4$ s and $R_I=2$. Similarly to cases with $T_b=3$s (Figure 16), superstructures with a short natural period $T_s$(Figure 19a,d) do not present satisfactory seismic performance. Hence, even for high isolated periods $T_b$, using a high reduction factor is not advisable if the fixed-base superstructure exhibits high initial stiffness. Again, non-satisfactory seismic performance is observed in all cases associated with low ductility capacity (i.e., ${\mu }_{max}=4$). A good seismic performance can be achieved if the superstructure is provided with a large ductility capacity, even when the largest reduction factor allowable by the Chilean code is employed.

Considering all the results obtained regarding fragility curves as a tool for evaluating the seismic performance of CLT structures equipped with friction pendulum bearings, the main findings are as follows:

• Better seismic performance is achieved for more flexible superstructures ($T_s\ge 0.6$ s) due to a larger yield displacement required during the design process, a slightly better seismic performance is achieved for larger mass distribution ratios$(\gamma =0.9$) and larger isolated period ($T_b=4$ s).
• If the superstructure is designed to behave elastically at the Design Earthquake level ($R_I=1.0$), an excellent seismic performance is observed in non-stiff structures ($T_s\ge 0.6$ s) or if the superstructure has a middle to high maximum ductility capacity (${\mu }_{max}\ge 6$).
• For rigid CLT superstructures ($T_s=0.3$ s), designs with $R_I=1.0$ and a ductility capacity of ${\mu }_{max}\ge 6$ are required to achieve acceptable seismic performance.
• For flexible CLT superstructures ($T_s\ge 0.6$ s), designs with $R_I=2.0$ and a ductility capacity of ${\mu }_{max}\ge 6$ are necessary to achieve acceptable seismic performance.
• For larger values of the reduction factor $(R_I\ge 1.6)$, the role of the maximum ductility capacity ${\mu }_{max}$ is crucial, especially for a stiff superstructure ($T_s=0.3$s).
• A reduction factor of $R_I=2$, the maximum value allowable according to the Chilean code of base-isolated buildings, is not recommended for stiff structures ($T_s=0.3$ s) or for superstructures with low maximum ductility capacity (${\mu }_{max}\le 6$).

9. Conclusions

This paper evaluates the seismic performance of cross-laminated timber (CLT) buildings equipped with friction pendulum bearings. The construction of CLT buildings in high seismic hazard areas has been a challenge for the field. This study shows that using seismic protection systems, such as seismic isolation, is a good approach to responding to this issue in Chile.

The first part of the study focused on accurately modeling CLT walls as a lateral-resistant system for seismic loads using a Smooth Hysteretic Model (SHM). This model is suitable to represent important nonlinear phenomena observed on these elements: stiffness and strength degradation, pinching, and failure due to excessive dissipated energy and ductility demand. The numerical formulation was validated using experimental data. The parameters of the SHM were fitted, achieving a good agreement with experimental responses of CLT walls presenting different displacement behaviors: rocking, rocking-sliding, and sliding. Taking advantage of the first-order formulation of the SHM, an equivalent simplified model of a CLT building equipped with frictional isolators was developed. A first-order strategy was presented to numerically determine the nonlinear response of this dynamic system subjected to ground motions. The model, representing a CLT building equipped with frictional isolators, was an efficient tool for evaluating its seismic performance and conducting the second part of this research.

The second part of this investigation presented a broad parametric analysis considering different parameters of the superstructure (initial period $T_s$, mass distribution ratio $\gamma$, design reduction factor $R_I$, and maximum ductility capacity ${\mu }_{max}$) and the isolation level (isolated period $T_b$). In total, 144 base-isolated CLT buildings were studied, and their seismic performance in terms of fragility curves was assessed. The results of this investigation suggest that using frictional isolators is a feasible option for constructing CLT buildings in high seismic hazard zones like Chile. The role of the maximum ductility capacity of the superstructure is a key aspect of the seismic performance of base-isolated CLT building. The larger this parameter, the better the seismic performance achieved. For increasing values of the reduction factor employed during the design process of the superstructure, a larger ductility capacity must be provided to ensure an acceptable response. Considering the limitation of the equivalent model and the assumptions taken, it is reasonable to suggest that base-isolated CLT buildings, designed with a reduction factor of $R_I=1.0$ at the MPE level, will exhibit excellent seismic performance.

The constructed fragility curves can be used to conduct preliminary designs of CLT buildings with seismic isolation interfaces. Since these curves give probabilistic information concerning the seismic performance of buildings, this information can guide designers to achieve different performance goals, depending on the needs of particular projects.

This investigation has limitations that can be overcome in future work. Only one soil class was considered. Studying the seismic performance of base-isolated CLT buildings located in different soil conditions can give insight into the role of the soil on the dynamic response of this kind of dynamic system. Furthermore, the Soil–Structure Interaction can also be studied to assess the effect of considering the soil in the dynamic response. A simplified model was used to obtain the results of this investigation. This aspect helps perform a broad parametric analysis but may represent a significant limitation since the local response of the superstructure or the isolation system is not considered. Developing more complex models is suggested for future investigations. Finally, the incorporation of real-time health monitoring techniques is an aspect that can be explored to improve the long-term reliability of CTL shear walls and friction pendulum bearings.