Reliability-based overstrength factors of cross-laminated timber shear walls for seismic design

Engineering Structures 228 (2021) 111547

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Reliability-based overstrength factors of cross-laminated timber shear walls for seismic design Angelo Aloisio *, Massimo Fragiacomo Department of Civil, Construction-Architectural and Environmental Engineering, Universit` a degli Studi dell’Aquila, Via G. Gronchi, 18, L’Aquila, 67100 Abruzzo, Italy

A R T I C L E I N F O

A B S T R A C T

Keywords: Overstrength ratios Reliability methods Cross-Laminated Timber buildings Ductility Capacity design

The ductile collapse mechanisms of structures should be less resistant than the brittle mechanisms to ensure a ductile seismic response: in this way, the ductile mechanisms activate before the brittle ones. This sort of chronological law of collapse is obtained in the design phase by providing a proper ”overstrength” to the brittle mechanisms. The realization of overstrength plays a crucial role in the design, and several studies endeavoured to estimate the best overstrength factors, defined as the ratio between the characteristic load-carrying capacity of the non-ductile element and the characteristic load-carrying capacity of the ductile element. In this paper, the conventional definition of overstrength is discussed and compared to a probabilistic definition based on reli­ ability methods. The probabilistic definition of overstrength drives the assessment of the overstrength factors of Cross-Laminated Timber buildings using a sort of indirect approach. The Extended-Energy dependent generalized Bouc-Wen model is used to estimate the nonlinear seismic response of a set of Cross-Laminated Timber shear walls with different ductility. The results are compared with the existing formulations, attempting to draw correlations possibly useful in the design phase.

1. Introduction Timber structures are used in seismic-prone areas from the dawn of time [1,2], but they have undergone a revival of popularity over the last years [3–5]. In particular, when dealing with high-rise structures, CrossLaminatedTimber (CLT) buildings exhibit a more reliable seismic per­ formance than light-frame timber shear walls [6–9]. The numerical modelling of the inelastic response of CLT shear walls is challenging, and there is a need for practice-oriented methods for their seismic design [10–12]. In CLT shear walls with conventional pinching connections, the connections between adjacent panels to transfer in-plane shear (vertical construction joints and horizontal diaphragm joints) are usually realized with self-tapping screws (STS) in different ways: half-lap joints, spline joints with laminated veneer lumber (LVL) or steel splines and butt joints with crossed inclined screws. The connections used at foundation and in-between storeys are generally angle brackets and hold-downs, manufactured to prevent respectively sliding and rocking of the shear walls [13]. Italian and European seismic codes [14,15] require compliance with the capacity design, to assure the development of plastic deformations in the dissipative components before the failure of

the non-dissipative ones. The ductile performance is ensured by providing overstrength to brittle failure mechanisms. However, there is a considerable scatter between overstrength factors in scientific literature. The current research mainly focuses on the calibration of over­ strength factors for seismic design. Still, seismic design is not the sole application of overstrength concepts. They are essential in all those circumstances where failure may occur, and the designer must prevent brittle failures by adopting adequate safety margins. In standard design methods, the chosen safety margins derive from the selection of proper partial safety factors. This research contribution focuses on the lateral response of Cross-Laminated Timber panels and ignores other significant variables which must be considered for more accurate calibration of the safety margins (moisture, duration of the load, e.g.). The authors attempt to overcome the limits of the conventional ap­ proaches followed in deriving the overstrength values. The conventional notion of overstrength leads to a definition in terms of Peak Ground Acceleration (PGA) rather than resistance. This definition is related to the ductility behaviour of the connection, rather than to a specific connection. The approach aims at determining overstrength values of specific structural systems. The authors firstly discuss overstrength by

* Corresponding author. E-mail addresses: [email protected] (A. Aloisio), [email protected] (M. Fragiacomo). https://doi.org/10.1016/j.engstruct.2020.111547 Received 12 March 2020; Received in revised form 23 October 2020; Accepted 5 November 2020 Available online 5 December 2020 0141-0296/© 2020 Elsevier Ltd. All rights reserved.

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Engineering Structures 228 (2021) 111547

definition of overstrength. The Pf,d should be greater than the Pf,b , meaning that there is an overstrength of brittle mechanisms in terms of failure probabilities. However, the need for overstrength in design requires that the overstrength factor is expressed in terms of resistance, but referred to a specific limit state or failure probability. Further, intriguing parallelism exists between ductility and over­ strength. Both concepts can be derived in alternative and nominally equivalent algebraic expressions. Ductility expresses the gap between failure and yielding, overstrength the gap between ductile and brittle mechanisms. To achieve ductility, the ultimate displacement should be higher than the yielding displacement; to achieve overstrength the probability of failure related to ductile mechanisms should be higher than the probability of failure related to brittle mechanisms.

Fig. 1. Representation of a limit state function.

proposing an alternative definition. Then, a simple application to CLT shear walls shows the first results of the method.

3. Overstrength in timber connections

2. Theoretical background: probabilistic assessment vs approximate methods

Few studies investigated the role of overstrength for a ductile design of timber connections. Jorissen and Fragiacomo [21] determined over­ strength values for dowelled connections based on 490 tests. According to [21], overstrength is the ratio between the characteristic load car­ rying capacity Rc,d according to the European design code EC 5 and the 95th-percentile of the test results.

In seismic design, the conventional definition of overstrength derives from the following expression: (1)

γ Rd ⋅Rk,d = Rk,e

where Rk,e is the characteristic load-carrying capacity of the non-ductile element and Rk,d is the characteristic load-carrying capacity of the ductile element [16]. A brittle or non-ductile structural element remains almost elastic until failure. Still, Eq. (1) does not always refer to a comparison between a ductile and non-ductile element: ductility and brittleness may coexist in the same structural element, like in timber connections. The most general definition should refer to collapse mechanisms: the timber connection acts like a structural element, which can exhibit ductile and non-ductile collapse mechanisms. However, Eq. (1) has the following limitation; the Ultimate Limit State (ULS) of a structural element derives from a comparison between load and resis­ tance: Eq. (1) does not consider the load effects. Overstrength should be designed to achieve a ductile behaviour during a certain event (seismic, blast, e.g.); Overstrenght is not a mere property of the structural element but rather the result of a structural performance: overstrength depends on the particular load scenario and the configuration of the structural arrangement. The limit state function, not just the resistance expresses the per­ formance of a structural element. The limit state function, g(X), is the basis of reliability-based calculations. This function expresses the sub­ traction between resistance (R) and load (L) as a function of the basic uncertainty variables (X) describing the loads and resistances, see Fig. 1. The definition of a limit state function is associated with a given failure mechanisms. Conventionally, the resistance and the load of a system are both random variables [17]: they are described by a probability distri­ bution for the resistance and load respectively. Hence, the probabilistic definition of the limit state function leads to the more general definition of probability of failure. The probability of failure may be determined by the following integral [18]: ⎛ ⎞ ⎛ ⎛ ⎞ ⎞ ∫ PF = P⎝g⎝X ⎠⩽0⎠ = fX ⎝X ⎠dX (2)

γ Rd =

Rc,0.95 Rc,d

(3)

where Rc,0.95 signifies the 95th percentile of the connection strength distribution. Fragiacomo and Jorrisen [21] found overstrength values between 1.2 and 1.85. Brühl and Kuhlmann [22], following Eq. (3), proposed γ Rd = 1.85 as overstrength ratio of dowelled connections [7]. Schick et al. [23] proposed γ Rd = 2.20 for connections to non timber elements and γ Rd = 1.65 for nailed timber to timber connections following a procedure based on partial overstrength values. To the authors’ knowledge, the scientific literature on this subject is not exhaustive. It is worthy of mentioning the recent paper by Ottenhaus et al. [24]: the authors presented an analytical method to derive over­ strength factors of dowel-type timber connections following the capacity design. Nevertheless, despite the novelty of the method by [25], the approach does not deal with the structural reliability field. In essence, the existing formulations may suffer from several weaknesses: • the proposed overstrength values are mostly based on the evaluation of tests on connections as if overstrength were just a property of the connection rather than the result of a structural performance; • the overstrength values are very scattered between different typol­ ogies of connections. The national or international standards should balance between synthesis needs and faithfulness to scientific results. The scatter between overstrength values does not facilitate the task of the code writers; • the definition of overstrength should be based on reliability methods, which encompass the probabilistic balance between load and resis­ tance in estimating a structural performance; • so far, the existing researches do not obey the same procedure in assessing overstrength: a standard approach would enable compar­ ison between homogeneous values.

Ω

Conversely, the listed weaknesses denote the limits which may pre­ vent a rigorous approach for timber connections:

where fX (X) is joint probability density function and Ω is the failure domain where g(X)⩽0, Fig. 1 [19,20]. The conventional definition of overstrength should be revised to account two aspects: the load effect and the probabilistic definition of load and resistance. Besides, overstrength is not a rigid mathematical expression, in the same way as the ductility which has numerous alge­ braic formulations [21]. The probability of failure of the ductile mechanism, Pf,d , and the probability of failure of the brittle element, Pf ,b , should drive the

• it is challenging to simulate both ductile and brittle mechanism in the same connection as pairwise and disjointed events: the force­ –displacement law of a connection corresponds to a ductile or brittle mechanism. For instance, if the practitioner uses a certain function to simulate the ductile behaviour of the connection, there is no possi­ bility to observe the brittle failure of the same connection; 2

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Engineering Structures 228 (2021) 111547

threshold. 4.2. Method Four sub-tasks may lead to an approximate estimation of over­ strength: (i) definition of the structural system characterized by a ductile and brittle failure mechanism; (ii) evaluation of the fragility functions associated with the two collapse events: the two failure probabilities associated with the two collapse mechanisms (Pf,b and Pf,d ) are related to an intensity measure (PGA); (iii) assumption of a certain probability of ̂ f ); (iv) failure corresponding to the chosen limit state, see Fig. 2 (Pf = P

the ratio between PGAb and PGAd is a measure of overstrength, see Fig. 2. The proposed definition of overstrength refers to the lateral response of timber structural assemblies. The failure of these structures mainly originates from the failure of the steel connections. The steel connectors provide ductility to the structural systems. So, a ductile response of the structural systems is ensured by the ductile response of the connections. Brittleness may occur if the connections do not activate or are not adequately designed to support the horizontal load. The proposed definition of overstrength only holds when brittle mechanisms are less resistant than ductile ones due to the lack of connections or their improper design. Based on the proposed definition of overstrength, if brittle and ductile mechanisms are equally probable, the overstrength ratio would be one: there is no overstrength. The definition of over­ strength can only refer to the situation where brittle failures are more fragile than ductile ones. Besides, the selection of a failure probability threshold determines the expected gap between the brittle and ductile mechanisms associated with a specific overstrength value.

Fig. 2. Representation of the proposed method.

• the simulation of the response of a structural system should consider plasticity and load redistribution after plasticization, which is the true essence of a ductile performance; • the calculated probability of failure depends on the choice of the hazard scenario. Probabilistic Seismic Hazard Analysis (PSHA) con­ sists of five specific sub-tasks, which may waste the generality of the result: identify earthquake sources; characterize the distribution of earthquake magnitudes from each source; characterize the distribu­ tion of source-to-site distances from each source; predict the result­ ing distribution of ground motion intensity; combine previous information to compute the annual rate of exceeding a given ground motion intensity. Besides, the overstrength values should not depend on intensity parameters (e.g. Peak Ground Acceleration) to be effectively useful; • traditional structural design is based on the comparison between capacity and demand, given a particular Ultimate Limit State (ULS). Each ULS refers to structural reliability values. Then overstrength, which directly enters the classical inequality between capacity and demand, should be calibrated to the structural reliability associated with the chosen ULS.

4.3. Discussion The proposed approach attempts to simplify several aspects which make the estimate of overstrength challenging. Failure of structural systems occurs from ductile or brittle mechanisms. They represent a set of pairwise disjoint events whose union is the entire failure domain:

The authors endeavoured to present a straightforward method which could assess overstrength by considering the aspects above, namely: overstrength refers to a structural assembly, not just the connection; overstrength depends on the load effect; overstrength concern specific reliability thresholds associated with the chosen ULS; overstrength does not depend on the intensity values of the seismic scenario.

Pf = Pf ,b + Pf ,d

where Pf is the probability of failure, Pf,b the probability of brittle failure and Pf,d the probability of ductile failure. However, it is complicated to simulate two collapse mechanisms as independent events. When per­ forming numerical simulations, the scholar assigns to the system hys­ teresis laws, specific characteristics which, somehow, drive the failure mechanisms. Conversely, it is straightforward to estimate the failure probabilities given specific collapse mechanisms [19]. The fragility curves relate the conditional probabilities vs an intensity measure, the PGA for instance. The scholar can determine the fragility curves of the same system, by assigning several collapse mechanisms, brittle and ductile. If an absolute displacement indicates the failure condition, the fail­ ure displacement associated with ductile mechanisms is higher than the failure displacement associated with brittle ones. Then, supposing the same hazard scenario, the failure probability of a brittle mechanism Pf,b is higher than the failure probability given a ductile mechanism Pf,d . The higher risk associated with Pf,b represents the actual reason of pre­ venting brittle failure by providing overstrength to brittle mechanisms. Further, the approach attempts to surmount the barrier between brittle and ductile. There is a range of possible mechanisms character­ ized by different ductility ratios [26]: in principle, each one can be considered brittle concerning another. For this reason, the authors considered a CLT wall panel character­ ized by various connection arrangements, from the lowest ductile μ ≈ 1 to the most ductile μ > 4: μ is a measure of the ductility, an adimensional quantity defined as μ = (vu − vy )/vy , where vu and vy are the ultimate and yielding displacements respectively according to [27].

4. A probabilistic approach to overstrength of structural connections 4.1. Background: Force vs PGAs The notion of overstrength recalls that of force. Reliability methods deal with failure probabilities, but the overstrength factor cannot derive from probability values directly. A rigorous approach requires that overstrength, still estimated as the ratio between forces, corresponds to the same reliability threshold. Besides, the need for overstrength mainly arises when dealing with seismic excitation: the force engaging the structural component is mostly proportional to the seismic intensity (PGA). Then, the ratio between forces may be equivalent to a ratio be­ tween accelerations, assuming the same inertia of the system. The PGA associated with the brittle failure PGAb could be considered proportional to the brittle resistance. The PGA associated with the ductile failure PGAd could be considered proportional to the ductile resistance. To achieve a ductile performance, the following should hold, assuming that the two PGA correspond to the same failure probability: γ Rd =

Rk,d PGAd ≡ | Rk,e PGAb Pf =̂P f

(5)

(4)

̂ f is a given failure probability where Pf is the failure probability and P 3

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The CLT configuration with the lowest ductility is chosen as a reference, reproducing the brittle mechanisms. The other CLT configu­ rations are compared to the reference one to estimate the overstrength ratio: given a certain failure probability, the ratio between the PGA of the two comparing fragility curves may give the overstrength ratio: ̂ γ Rd =

PGAd | PGAb Pf =̂P f

Table 1 Wall configuration tested, the applied vertical load and the corresponding yielding vy and failure displacements vu . Test Number

(6)

I.1 I.2 I.3 I.4 II.1 II.2 II.3 II.4 III.1 III.2 III.3 III.4 III.5 III.6 III.7 III.8

Eq. (6) can be derived from the following intuitive considerations. The authors are treating a structural system obtained by assembling several components. The connections represent the ductility sources of the system. A reduced number of connections (2 hold-downs) produce a brittle system (μ ≈ 1). The additional connections increase ductility and reduce the probability of failure. Then it is clear that PGAd > PGAb . The ductile system has an additional number of connections which increase the failure PGA. The proposed method only holds for consistent load, structure and related probability assumptions. Further developments will generalize the procedure to account for all uncertainties sources (e.g. load, model). According to the presented approach, overstrength should depend on ductility: the more the system is ductile, the more overstrength must be ensured to prevent brittle mechanisms. This is reasonable: if the system reliability holds on very ductile mechanisms, the consequences of hav­ ing a brittle mechanism are higher than those concerning a system with low ductility. The higher the ductility margin between mechanisms, within the same structural system, the higher overstrength should be provided to prevent the occurrence of brittle failure. In the next section, the authors estimate overstrength of different typologies of CLT wall panels, and all referred to the lowest ductile: the CLT wall panel with two hold-downs. The presented approach may enhance the literature dispersion on overstrength of timber connections. Overstrenght could be assumed as dependent on the ductility ratio from an empirical regression law. Then, the connection ductility not just the connection typology would drive the design of timber connections.

Number of Angle Brackets

Number of screws in vertical joints

Vertical load [kN/m]

vy

vu

[mm]

[mm]

2 2 2 2 2 2 2 4 2 2 4 2 2 2 2 2

2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

– – – – 20 20 10 5 2 × 20 2 × 10 2×5 2 × 10 2 × 10 2 × 10 2 × 10 2 × 10

18.5 18.5 9.25 18.5 18.5 18.5 18.5 18.5 18.5 18.5 18.5 18.5 18.5 0 18.5 18.5

10.0 17.0 14.8 23.6 11.0 11.7 13.7 14.6 12.8 17.6 17.4 17.8 16.8 15.3 11.6 15.9

39 57.3 56.6 56.7 73.3 65.4 76 73.7 76.4 76.1 75.7 76.2 79.6 75.2 77.4 77.4

Table 2 Estimation of the CLT wall panels ductility.

5. Application to CLT shear walls This section details the essential tools used by the authors for the estimation of the fragility curves of CLT wall-panels: the constitutive model, the analytical modelling of a one-storey structural archetype, the experimental data used for calibration, a description of the chosen seismic scenario and the method for the estimation of the fragility functions.

Typ.

vy [mm]

vu [mm]

μ = (vu − vy )/vy

I.1 I.2 I.3 I.4 II.1 II.2 II.3 II.4 III.1 III.2 III.3 III.4 III.5 III.6 III.7 III.8

10 17 14.8 23.6 11 11.7 13.7 14.6 12.8 17.6 17.4 17.8 16.8 15.3 11.6 15.9

39 57.3 56.6 56.7 73.3 65.4 76 73.7 76.4 76.1 75.7 76.2 79.6 75.2 77.4 77.4

2.90 2.37 2.82 1.40 5.66 4.59 4.55 4.05 4.97 3.32 3.35 3.28 3.74 3.92 5.67 3.87

where α is the post-to preyield stiffness ratio and k0 the initial stiffness. The evolution of z is determined by an auxiliary ordinary differential equation, which can be written in the following form

5.1. The Extended Energy Dependent Generalized Bouc-Wen (EEGBW) Model

z˙ = x˙ [A − |z|n ψ (x, x˙ , z)]

(9)

where z˙ is the derivative of z with respect to time, A controls the scale of ˙ z) a nonlinear the hysteresis loops and n controls the sharpness, ψ (x, x, function controlling other shape features of the hysteresis loop. The extended ψ function of the EEGBW model is reported in the appendix. The EEGBW hysteresis model possesses all the essential features observed in real structures, which include strength and stiffness degra­ dation and pinching of the successive hysteresis loops.

The Extended Energy Dependent Generalized Bouc-Wen (EEGBW) model [28] reproduces elaborate constitutive laws, in particular, that of CLT wall panels, where the coexistence of different resisting mecha­ nisms makes a direct finite element approach a quite complicated task (rigid rocking of the panel [29,30], plasticization of the compressed fi­ bers, yielding of the hold downs in tension, e.g.). The authors adopted the EEGBW model, which reproduces the hysteretic behaviour of timber structural systems. The equation of motion of the inelastic SDOF system with the Generalized Bouc-Wen model can be expressed as: ( ) m¨x + c˙x + fs x, x˙ , z = − m¨xg (7)

5.2. Analytical modelling of CLT wall A rough analytical model for the prediction of the CLT wall systems behaviour under lateral loads is adopted. Following the forerunner studies of [32–34], the model, depicted in Fig. 4 is chosen. The CLT wall panel (A-B-C-D) is anchored to the foundation (A-D) with hold-downs and angle brackets. A static vertical load is applied on the wall, while a mass m stands on the top of the wall, as shown in Fig. 4. Considering the SDOF hysteretic system in Fig. 4, the equation of motion corresponds ˙ z) term to Eq. (7). All information on the wall is enclosed in the fs (x, x, (the wall configuration, the vertical connections, the static load, e.g.).

where m is the mass, x the displacement, x˙ the derivative of x with ¨ the second derivative of x with respect to time, fs (x, x, ˙ respect to time, x z) the resisting inelastic force, z the auxiliary inelastic variable, c the viscosity coefficient. The modal viscous damping has been set equal to 2% [31]. The resisting force can be written as fs (x, x˙ , z) = αk0 x + (1 − α)k0 z

Number of Hold Downs

(8) 4

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• Wall configuration III (tests III.1-III-8): two coupled wall panel with a laminated veneer lumber (LVL) spline joint (180 × 27 mm LVL strip) and dimensions b = 1.48 m and h = 2.95 m for each panel (1:2 aspect ratio). All the wall specimens had equal five-layered panel build-up (17-1717-17-17) with 85 mm thickness in total. The numbers in bold represent the thickness values of vertical board layers, while the numbers in normal font denote horizontal board layers thicknesses. Different connector layouts were used and different vertical load levels were applied on the walls, spanning between 0 and 18.5kN. The same types of metal connectors, screws, and nails as those in the three-story SOFIE building were used [36]. The wall panels were anchored to the steel foundation with hold-downs type HTT22 [ETA-11/0086(ETA 2011a)] with twelve 4 × 60mm Anker annual ring nails [EN 14592 (CEN 2008)], and angle brackets type BMF90 × 48 × 3 × 116mm [ETA-07/0055 (ETA 2009)] with 114 × 60mm Anker annual ring nails. In most of the coupled wall tests, self-tapping screws ∅8 × 100mm [ETA-11/0030 (ETA 2011b)] were used in the vertical joints, except for Test III.7 and III.8 where fully threaded screws ∅8 × 200mm [ETA-11/0030 (ETA 2011b)] connected to the adjacent wall panels through the vertical joints. Force–displacement data, measured in point C Fig. 3, are collected for the estimation of the EEGBW parameters. The response in point C represents the behaviour of an ideal storey supported by the shear wall and can be taken as a synthetical descriptor of the lateral response of the considered structural system modelled as an SDOF oscillator. The opti­ mum EEGBW model is estimated for each of the 16 different wall con­ figurations using an Ordinary Least-Squares operator [37,38] fully detailed by [28]. The model requires adaptations whenever any con­ struction detail changes, see Fig. 5. The authors solved the equations which describe the hysteretic models of the considered structures in the Matlab environment by numerical integration; The experimental data (evaluation point C) have been utilized to estimate the parameters of the Bouc-Wen model.

Fig. 3. Definition of the yielding and ultimate displacement from a the back­ bone curve of an experimental cyclic response.

5.4. Truncated incremental dynamic analyses The response of the structure is investigated using nonlinear dynamic analyses (NLDA) referring to the observed Italian seismic scenarios since 1972. The entire set of Italian earthquake records with PGA > 0.4 g is used in the analysis: the list of the 43 earthquakes is fully reported in the Appendix. The use of the PGA rather than the peak spectral acceleration (PSA) and their spectral shapes descends from the Italian Seismic Code; It classifies the seismic risk of the Italian territory according to the PGA levels. The chosen PGA (0.4 g) is not a categoric threshold, sharply separating destructive earthquakes from low intensity ones. However, since in the Italian territory it is observed that PGA exceeding 0.4–0.5 g are related to severe damages to the structures, 0.4 g has been arbitrarily chosen as a ”filter” for selecting a suite of Italian severe ground motion records, downloaded from the Italian strong motion database ITACA [39]. The authors have preferred to focus on the sole fragility function estimations rather then dealing with the estimate of CLT shear walls structural reliability [40] from probabilistic seismic hazard analysis (PSHA). The quantification of the uncertainty about the location, size and resulting shaking intensity of future Italian earthquakes is still very discussed, especially after the last 10 years (2009–2019) destructive earthquakes of L’Aquila (2009), Central Italy (2016), Montereale (2017) and Ischia (2017). The NLDA were carried out with increasing PGA levels, starting from a 10% fraction of the actual PGA, up to the recorded PGA (Truncated Incremental Dynamic Analysis TIDA), with increasing steps of 5% (19 analysis for each NLDA). Differently from the Incremental Dynamic Analysis (IDA), the truncated one do not need to be scaled to large IM to produce collapse, thus avoiding two possible criticisms raised against the IDA: (i) the IDA are computationally expensive, (ii) it is questionable

Fig. 4. SDOF Idealization of a CLT wall panel through a SDOF mechani­ cal model.

The sole explicit terms in Eq. (7) are the mass m and the coefficient of dynamic viscosity c. A global model of the wall is obtained, represen­ tative of the particular configuration under test. The mass values used in the simulation correspond to the inertia of the vertical load in Table 1. (see Table 2). 5.3. Experimental Data Cyclic wall tests were performed at IVALSA Trees and Timber Institute, San Michele allAdige (Italy). The CLT walls were anchored to a steel foundation with hold-downs and angle brackets. The test setup is described by [35]: 16 cyclic tests in total, see Table 1. Different connection layouts were used with the aim to investigate how they affect the entire wall behaviour and, possibly, optimize the structural perfor­ mance. Three wall configurations were investigated: • Wall configuration I (tests I.1-I.4): single wall panels with di­ mensions b = 2.95 m and h = 2.95 m (1:1 aspect ratio); • Wall configuration II (tests II.1-II.4): two coupled wall panels with a half-lap joint (50 mm overlapping) and dimensions b = 1.48 m and h = 2.95 m for each panel (1:2 aspect ratio);

5

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Fig. 5. Superposition of experimental and simulated hysteresis curves using the EEGBW model (see Table 6), where R stands for Real and S for Simulated cy­ clic response.

6

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Fig. 6. (a) Fragility functions of the 16 CLT configurations; (b) The failure probability threshold intersects the fragility function.

whether scaling typical moderate-intensity measure (IM) ground mo­ tions up to extreme-IM levels is an accurate way to represent shaking associated with real occurrences of such large-IM levels. The total number of performed TIDA was therefore 817. The dynamic equilibrium equations were integrated with a time step of 0.001 s and an equivalent viscous damping of 2%, as suggested by [36], was adopted in the analysis. The failure condition is defined in terms of ultimate displace­ ment according to the results of the experimental tests used to calibrate the EEGBW model, Table 1.

where P(C|IM = x) is the probability that a ground motion with IM = x will cause the structure to collapse; Φ is the standard cumulative normal distribution function (CDF); θ is the median of the fragility function (the IM level with 50% probability of collapse); and β is the standard devi­ ation of ln IM (sometimes referred to as the dispersion of IM). The choice of normal probability distribution function in combination with the parameter estimation does influence the quantitative assessment of the overstrength factor. The following qualitative explanation coarsely ex­ plains the choice of the lognormal function: the probability density function (PDF) of the lognormal distribution is pushed toward the mean as the variance decreases, becoming narrower and taller. Conversely, the PDF spreads out away from the mean as the variance increases, becoming broader and shallower. The lognormal PDF expresses the fact that the failure probability decreases at a lower rate in correspondence of the values beyond the mean of the distribution. The parameters θ and β derive from the Maximum Likelihood esti­ mation (MLE). Specifically, the parameters are obtained by maximizing the logarithm of the following likelihood function: { } ] [ / )) ] [ ( ( m [ ∑ ln IM j θ ̂ + n − m ln 1 lnΦ θ, ̂ β = arg min̂ ̂ θ, β β j=1 ( )] ln(IM max /θ) (11) − Φ β

5.5. Estimation of the fragility functions A fragility function specifies a structures probability of collapse, or some other limit state of interest, as a function of some ground motion IM [41] which, in this paper, is the PGA itself. For a given set of ground motion and dynamic structural analysis results, the occurrence or non occurrence of collapse is defined by the exceeding of a given displace­ ment, which was identified from experimental tests [35], Table 1. Following a conventional approach in the field of structural reliability, a lognormal cumulative distribution function is used for fitting the fragility function from data collected from NLDA [41]: ( ) ( ) ln(x/θ) P C|IM = x = Φ (10) β

where

­

7

∧

denotes an estimated parameter, Φ() the standard normal dis

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Fig. 7, using an Ordinary Least Squares operator [37]:

Table 3 Safety requirements for the Ultimate Limit State specified as formal yearly probability of failure Pf obtained from the choice of safety classes and specific failure modes according to the ISO standard on General Principles on Reliability for Structures [42] and the Eurocode EN 1990 Basis for Structural Design [43]. Safety class

Failure type I (Ductile)

Low safety class

Pf ⩽10−

3

Pf ⩽10−

4

Pf ⩽10−

5

Pf ⩽10−

4

Pf ⩽10−

5

Pf ⩽10−

6

Pf ⩽10−

5

Pf ⩽10−

6

Pf ⩽10−

7

Normal safety class High safety class

Failure type II (Ductile)

γ Rd = c1 + c2 μn

(12)

where c1 , c2 and n are the unknown parameters. An almost quadratic relation correlates overstrength values and ductility: the n exponent ranges between 1.8 and 2.2. The overstrength values increase as the probability of failure decreases: very low failure probabilities corre­ spond to severe ULS, Table 3. Higher overstrength factors are a guar­ antee against non-ductile and likely catastrophic consequences. The γ Rd do not exceed 2 for ductility values approximately lower than 4, see Fig. 8. For higher values of ductility, the γ Rd blows up quadratically. The validity of the correlations is limited to the ductility interval 2–5. Lower values refer to almost non-ductile systems where there is no need for overstrength. Higher values cannot be extrapolated, given the absence of experimental data. Fig. 7 is a synthesis of the results: the correlation curves assosiated with the considered failure probabilities are plotted together. The results are in great accordance with the main outcomes on this topic, see Table 4. Still, the values in Table 4 derive from experi­ mental cyclic tests on connections. However, the ductility of the struc­ tural system is not the sum of the ductilities of the single connections of the assembly. The overstrength factors are used for the capacity design of the single connection to ensure the ductile behaviour of the entire structural system. There is a dichotomy between the calibration of overstrength factors and the design phase. The calibration would require the modeling of the CLT system, while the design phase focuses on the capacity design of the single connection. The design phase is consistent if the overstrength factors are properly calibrated on the entire struc­ tural assembly. Despite the difficulties in calibrating overstrength, there is great accordance between the values obtained by [45,47,25,49,50] and those illustrated in Fig. 8. However, the values in Table 4 do not refer to a particular limit state or failure probability. Overstrength is not an absolute value, but it depends on particular reliability threshold. This

Failure type III (Brittle)

tribution PDF, n the number of ground motion used in the analysis, m the number of m ground motions that caused collapse at IM levels lower than IMmax , ϕ() the normal cumulative density function (CDF). The TIDA obtained from all the earthquake recordings are merged yielding the failure probabilities at each PGA step: the number of failures at each step of the TIDAs are summed and divided for the number of the selected earthquake scenarios. The parameters of the lognormal distribution are estimated by fitting these values [9]. 6. Overstrength ratios given Ultimate Limit State Fig. 6(a) shows the fragility functions of the 16 CLT wall configu­ rations. Then, four failure probability thresholds intersecate the curves like in Fig. 6(b): 1⋅10− 6 , 1⋅10− 5 , 1⋅10− 4 , 1⋅10− 3 , see Table 3. The PGA corresponding to the chosen limit state are referred to the PGA of the CLT panel with two hold-downs. The overstrength values obtained from the PGAs are quite dispersed: a further regression model is needed to assess the univocal correspondence between ductility values and over­ strength. The following function fits the overstrength values, reported in

Fig. 7. (a)-(d) Correlations between overstrength values and ductility given four failure probability thresholds. 8

A. Aloisio and M. Fragiacomo

Engineering Structures 228 (2021) 111547

Fig. 8. Overstrength factors for the design of CLT connections characterized by increasing ductility values and failure probabilities.

of the considered structural assemblies. Overstrength is assumed to be the ratio between the Peak Ground Acceleration associated with the brittle and ductile failure mechanisms given a certain failure probability threshold. The main advantage of the methods stands in the attempt to derive a reliability-based definition of overstrength for timber connec­ tions. The proposed definition of overstrength does not include any correspondence with the definition of partial safety factors and is limited to seismic design. This research contribution focuses on the lateral response of Cross-Laminated Timber panels and ignores other significant variables which must be considered for more accurate calibration of the safety margins (moisture, duration of the load, e.g.). Additionally, it centres on the ductile performance of the single connection, neglecting the global plastic response of the structural assembly. Several scholars [45,47,25,49,50] calibrated overstrength factors of Cross-Laminated timber connections based on experimental tests on connections [21]. The obtained values, ranging from 1.5 to 2.5, nearly confirm the results of the proposed simplified approach or more systematic experimental and numerical investigations [53–56]. The overstrength factors are not absolute values but depend on the considered limit state to achieve an optimum balance between the risk of failure and additional costs. The authors will aim at enhancing the proposed approach by bestowing a more general definition of overstrength based on reliability methods, valid for a wider class of structural assemblies and failure mechanisms.

Table 4 Comparison of overstrength factors derived for some typical connections for CLT structures [44]. Connection/Fastener Nails loaded parallel to face lamination [45,46] Nails loaded perpendicular to face lamination [45,46] 5x50 STS loaded parallel to face lamination [47] 5x75 STS loaded parallel to face lamination [47] 5x50 STS loaded perpendicular to face lamination [47] 5x75 STS loaded perpendicular to face lamination [47] Dowels [25] Panel-to-panel surface spline connections with nails (3”) [48] Panel-to-panel surface spline connections with screws (8”; 6”; 4”) [48] Panel to panel joints with screws (half-lap joint) [49] Panel to panel joints with screws (LVL joint) [49] Hold-down in tension [50] Hold down in shear [50] Angle bracket in tension [50] Angle bracket in shear [50]

γRd 2.04 2.59 2.38 2.50 2.37 2.67 1.95 1.42 2.69; 2.56; 2.62 1.79 2.08 3.38 3.38 3.44 1.97

CRediT authorship contribution statement Angelo Aloisio: Conceptualization, Methodology, Software, Data curation, Writing - review & editing. Massimo Fragiacomo: Concep­ tualization, Supervision.

aspect is not secondary: a proper design should balance the risk of failure and the risk of additional costs [51,52].

Declaration of Competing Interest

7. Conclusions

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

The paper presents a reliability-based definition of the overstrength factor for the seismic design. The experimental cyclic response of a set of Cross-Laminated Timber panels characterized by increasing ductility supports the results of this research. The increasing ductility of the panels derives from the use of different arrangements of the connections. The panel with the lowest ductility is considered as the brittle one compared to the other panels with higher ductility values. The definition of overstrength originates from the estimate of the fragility functions in terms of conditional failure probabilities and Peak Ground Acceleration

Acknowledgments The authors thank Prof. Jochen K¨ ohler for his kind support and useful suggestions. The authors are also appreciative to the anonymous reviewers for their valuable suggestions that significantly helped improve the scientific quality of the manuscript.

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Engineering Structures 228 (2021) 111547

Appendix A. Appendix A.1. Definition of the shape function Hereafter follows the mathematical characterization of the shape function ψ , mentioned in Eq. (9). ( ) () ( ) ( ) ( )(θ+ ) ( )(θ− )] ψ E = θ[β1 (∊)sgn(˙x z) + β2 (∊)sgn(˙xx) + β3 (∊)sgn(xz) + β4 (∊)sgn(˙x) + +β5 ∊ sgn z + β6 ∊ sgn x + β7 ∊ + β8 ∊ θ θ { θ = 0, if {(x ∧ x˙ ∧ z) > 0 ∨ (x ∧ x˙ ∧ z < 0)} ∧ |x(t)| < q|xu (t)|; 1, otherwise

(13) (14)

{ θ+ =

1,

if (x ∧ x˙ ∧ z > 0) ∧ |x(t)| < q|xu (t)|;

0, otherwise

(15)

1,

if (x ∧ x˙ ∧ z < 0) ∧ |x(t)| < q|xu (t)|;

0, otherwise

(16)

{ θ− =

where βi∈[1−

6]

are the Song shape-control parameters [57], β7 is the pinching shape parameter, active when x > 0, x˙ > 0 and z > 0, β8 is the pinching

shape parameter, active when x < 0, x˙ < 0 and z < 0, θ, θ+ , θ− are sign functions time-history dependent, q is a fraction of the pinching level xu , 0 < q < 1 and xu (t) is the time-dependent pinching level in terms of displacement. It is the maximum of the displacement function x(t) within the interval t ∈ [0, t) Strength and stiffness degradation is accounted by setting the βs parameters as linear functions of the dissipated hysteretic energy ∊, as follows: ( ) { } βj ∊ = βj0 + kβj ∊, j ∈ 1 − 8 (17) The dissipated energy ∊ is approximated by: ⎛ ⎞ T ∫ ⎝ ˙ ∊ = 1 − α⎠k uzdt

(18)

0

A.2. List of earthquake recordings Fig. 9 Table 5

Fig. 9. Response spectra of the considered earthquakes in terms of pseudo-acceleration and pseudo-displacement.

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Engineering Structures 228 (2021) 111547

Table 5 List of real earthquake records No.

Name

Event date

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

Norcia (PG) Norcia (PG) Accumuli (RI) Medolla (MO) Norcia (PG) Medolla (MO) Serravalle di Chienti (MC) Castelsantangelo sul Nera (MC) L’Aquila (AQ) L’Aquila (AQ) Castelsantangelo sul Nera (MC) Norcia (PG) Gemona (UD) Castelsantangelo sul Nera (MC) Montereale (AQ) Novi (MO) Gualdo Tadino (PG) Ancona (AN) Amatrice (RI) Norcia (PG) Castelsantangelo sul Nera (MC) Visso (MC) Norcia (PG) Visso (MC) Norcia (PG) Monte Cavallo (MC) Medolla (MO) Serravalle di Chienti (MC) Medolla (MO) Norcia (PG) Monte Cavallo (MC) San Possidonio (MO) L’Aquila (AQ) Norcia (PG) Norcia (PG) Medolla (MO) Medolla (MO) L’Aquila (AQ) Nocera Umbra (PG) Nocera Umbra (PG) Ancona (AN) Medolla (MO) Ancona (AN)

2016–10-30 2016–10-30 2016–08-24 2012–05-29 2016–10-30 2012–05-29 1997–10-16 2016–10-26 2009-04–07 2009-04–06 2016–10-26 2016–10-30 1976–09-15 2016–10-26 2017–01-18 2012–06-12 1998-04–03 1972–06-14 2017–01-18 2016–10-30 2016–10-26 2016–10-26 2016–10-30 2016–10-26 2016–10-30 1997–10-06 2012–05-29 1997–09-26 2012–05-29 2016–09-03 1997–10-06 2012–06-03 2009-04–06 2016–10-30 2016–10-30 2012–05-29 2016–10-30 2009-04–06 1997–10-11 1997–10-11 1972–06-14 2012–05-29 1972–06-21

06:40:18 06:40:18 01:36:32 07:00:02 06:40:18 07:00:02 12:00:30 19:18:06 17:47:37 01:32:40 19:18:06 06:40:18 03:15:18 19:18:06 10:14:12 01:48:36 07:26:36 18:55:46 10:25:26 06:40:18 17:10:36 19:18:06 06:40:18 19:18:06 06:40:18 23:24:51 07:00:02 09:40:24 07:00:02 01:34:12 23:24:51 19:20:43 01:32:40 06:40:18 06:40:18 07:00:02 06:40:18 01:32:40 03:20:56 03:20:56 21:01:30 07:00:02 15:06:44

Depth [km]

Mw

ML

PGA [g]

9.20 9.20 8.10 8.07 9.20 8.07 0.90 8.70 17.10 8.30 7.50 9.20 6.80 7.50 9.10 8.28 9.65 3.00 8.90 9.20 8.70 7.50 9.20 7.50 9.20 5.50 8.07 5.70 8.07 8.90 5.50 8.66 8.30 9.20 9.20 8.07 9.20 8.30 4.30 4.30 21.00 8.07 4.00

6.5 6.5 6.0 6.0 6.5 6.0 4.3 5.4 5.5 6.1 5.9 6.5 5.9 5.9 5.5 3.9 5.1

6.1 6.1 6.0 5.8 6.1 5.8 4.5 5.4 5.4 5.9 5.9 6.1 6.1 5.9 5.4 4.9 4.9 4.7 5.3 6.1 5.4 5.9 6.1 5.9 6.1 5.4 5.8 5.8 5.8 4.3 5.4 5.1 5.9 6.1 6.1 5.8 6.1 5.9 3.7 3.7 4.2 5.8 4.0

0.95 0.89 0.87 0.86 0.80 0.73 0.73 0.72 0.66 0.66 0.65 0.65 0.64 0.62 0.59 0.58 0.58 0.56 0.56 0.56 0.56 0.55 0.55 0.54 0.53 0.53 0.50 0.50 0.50 0.50 0.49 0.49 0.49 0.49 0.48 0.46 0.45 0.44 0.44 0.43 0.43 0.42 0.41

5.4 6.5 5.4 5.9 6.5 5.9 6.5 5.4 6.0 6.0 6.0 4.2 5.4 4.9 6.1 6.5 6.5 6.0 6.5 6.1

6.0

A.3. Parameters of the EEGBW model Table 6

Table 6 Table of Identified βs parameters n

β1

β2

β3

β4

β5

β6

β7

β8

I.1 I.2 I.3 I.4 II.1 II.2 II.3 II.4 III.1 III.2 III.3 III.4 III.5 III.6 III.7 III.8

52.55 32.79 40.88 21.24 − 13.2 − 44.6 48.51 − 44.8 − 62.7 34.7 26.96 47.64 18.28 22.32 − 5.36 19.15

7.42 28.15 29.16 39.83 55.56 86.59 − 8.71 77.82 108.5 10.68 10.81 − 16.27 11.47 16.43 18.84 14.4

− 8.34 − 28.53 − 16.14 13.95 − 23 − 36.5 − 4.8 − 29.6 − 46 − 2.07 − 26.2 16.61 − 4.88 − 55.1 − 11.4 − 0.77

7.18 2.53 6.34 11.13 − 0.83 − 0.84 − 7.45 1.21 8.09 3.1 − 3.61 3.43 − 1.19 1.86 − 0.67 1.79

10.65 − 0.64 10.37 − 33.17 − 18.5 − 7.07 14.59 50.98 − 25.3 − 6.69 14.07 − 18.6 5.01 − 10 2.73 − 9.29

29.04 4.04 − 12.75 23.72 6.79 − 1.45 − 1.4 − 33.37 18.54 13.56 − 16.35 15.21 − 10.39 0.98 − 5.21 4.73

− 3.64 30.65 36.81 83.88 17.81 13.03 38.84 22.06 18.43 48.46 20 44.35 31.37 − 6.06 16.5 36.91

20.10 22.65 34.55 33.26 37.6 33.3 24.1 18.6 18.4 34 15.3 51.6 26.9 − 0.18 12.7 31.7

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