Dissipation in sheathing-to-framing connections of light-frame timber shear walls under seismic loads

Engineering Structures 208 (2020) 110246

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Dissipation in sheathing-to-framing connections of light-frame timber shear walls under seismic loads

T

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Giorgia Di Gangia, Cristoforo Demartinob, , Giuseppe Quarantaa, Giorgio Montia,c a

Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Rome, Italy Zhejiang University/University of Il linois at Urbana Champaign Institute, Zhejiang University, Haining, Zhejiang, China c College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, PR China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Light-frame timber shear walls Equivalent viscous damping Multi-objective optimization OpenSees Capacity spectrum method

The present study is concerned with the seismic analysis and design of light-frame timber shear walls, with focus on the energy dissipation ensured by sheathing-to-framing connections under in-plane lateral loads. In this perspective, a suitable parametric finite element model for light-frame timber shear walls is first developed using the software OpenSees. By means of such model, the equivalent viscous damping of the wall is assessed numerically, together with the damping factor adopted within the capacity spectrum method. Furthermore, the optimal layouts of slender (1.2 m × 2.4 m/3.94 ft × 7.88 ft) and squat (2.4 m × 2.4 m/7.88 ft × 7.88 ft) lightframe timber shear walls are found by solving a multi-objective optimization problem in which racking capacity and total material cost are optimized simultaneously (this task is accomplished via simple enumeration because of the low cardinality of the design variables set). So doing, it is shown that the total material cost of the optimal (non-dominated) wall configurations has a strong influence on the racking capacity whereas it does not affect the equivalent viscous damping ratio, which remains almost constant. A novel simplified analytical procedure for predicting the capacity curve of light-frame timber shear walls is finally proposed.

1. Introduction Shear walls are structural members generally used to withstand inplane lateral actions, such as wind or seismic forces. Reinforced concrete or steel shear walls are typically employed within medium- or high-rise buildings [e.g.,1–3]. Conversely, timber shear walls are mainly employed within residential buildings with one or two storeys in Northern Europe, North America and New Zealand [e.g.,4]. Timber shear walls-based constructions are very attractive for several reasons, including aesthetic quality and rapid assembly of the elements. A light-frame timber shear wall is an integrated system mostly employed in platform-frame buildings. A typical configuration of the wall is shown in Fig. 1. Non-structural layers are often included to provide thermal insulation as well as fire and vapor resistance [e.g.,5,6]. In what follows, only the structural part will be considered. This is built by assembling vertical studs and horizontal plates into a hinged frame [7]. The frame is then braced on one or both sides with a sheathing panel, attached by means of metal fasteners (e.g., nails, screws or staples). The uplift of the shear wall subjected to a horizontal force – due to its rigid rotation – is controlled by hold-downs, whereas

its rigid translation is prevented by angle brackets [8,9]. The frame size depends on the size of the panel used to sheathe it. Panels can be made of Oriented Strand Board (OSB), ply-wood, gypsum, Glued Laminated Guadua (GLG) bamboo [11], ply-bamboo, fibreboard, and so on. Commonly, as pointed out by Wang et al. [6], the size of a light-frame timber wall is 1.22 m × 2.44 m (4 ft × 8 ft) or 2.44 m × 2.44 m (8 ft × 8 ft). Frame elements cross-sections are about 38 mm × 89 mm (1.5 in. × 3.5 in.) and 38 mm × 140 mm (1.5 in. × 5.5 in.) for internal and external wall studs, respectively. In Alpine countries, frame elements with cross-section sizes 80 mm × 160 mm (3.15 in. × 6.3 in.), 120 mm × 160 mm (4.7 in. × 6.3 in.), 120 mm × 200 mm (4.7 in. × 7.87 in.), 140 mm × 160 mm (5.5 in. × 6.3 in.) and 160 mm × 200 mm (6.3 in. × 7.87 in.) are also used [12]. The cross-section size of external frame elements is often chosen to fulfill the requirements for thermal insulation (see Fig. 1). Common non-hardened fasteners with smooth thick shank are typically employed for frame connections since they dissipate more energy with respect to fasteners with a ring shank. The sizes mostly used are 6D (diameter 3.05 mm/0.12 in., length 50.8 mm/ 2 in.), 8D (diameter 3.40 mm/0.134 in., length 63.5 mm/2.5 in.) and 10D (diameter 3.76 mm/0.148 in., length 73.2 mm/3 in.), according to

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Corresponding author. E-mail addresses: [email protected] (G. Di Gangi), [email protected] (C. Demartino), [email protected] (G. Quaranta), [email protected] (G. Monti). https://doi.org/10.1016/j.engstruct.2020.110246 Received 1 June 2019; Received in revised form 30 November 2019; Accepted 14 January 2020 0141-0296/ © 2020 Elsevier Ltd. All rights reserved.

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G. Di Gangi, et al.

Nomenclature

fv Gmean hes his hp k fsec Kp sec KSH

List of symbols

α

Stiffness degradation parameter for the shear wall spring element Parameter that identifies the resistance decrement of the αf considered fastener αm Aspect ratio, i.e. height-to-width ratio of the wall = H /(L/ np) , aspect ratio of the wall panel αpm Stiffness degradation parameter for the spring element β η Damping factor γ Coefficient that represents the energy dissipation distribution along the perimeter horizontal plates κ Coefficient that represents the energy dissipation distribution along the perimeter vertical studs Shape parameter depending on the aspect ratio of the wall λ (αpm) panel Experimental force values Fexp Fnum Predicted force values var(Fexp) Variance of the experimental force values θ Vector collecting the model parameters ∼ Available ductility of the fastener μf ∼ EDf Maximum energy that can be dissipated by a single fastener ∼ Ff , Rd Peak strength of the fastener ∼ Ff , ud Ultimate force of the fastener ∼ Fv, Rd Racking load-carrying capacity of the wall ∼ Fv, ud Ultimate global force of the wall ∼ uf , Rd Displacement at the peak strength of the fastener ∼ u v, Rd Displacement at the peak strength of the wall ξ0.05 Inherent viscous damping equal to 5% Equivalent (hysteretic) viscous damping ξeq ξf Equivalent viscous damping of a single fastener Total equivalent viscous damping ξtot bes Base of the external vertical studs Base of the internal vertical studs bis bp Base of the horizontal plates Dmax = β ·Dun Last unloading displacement of the spring element Dun Spring element displacement at peak strength Du Mean characteristic value of the wood elastic modulus E0, mean parallel to grain Dissipated energy under the bilinearized force-displaceEDf ment curve of the fastener ED Energy dissipated in one hysteresis cycle by the structural system Elastic energy under the bilinearized force-displacement Es0f curve of the fastener Available potential energy to failure of the structural Es0 system Intercept strength of the shear wall spring element for the F0 asymptotic line to the envelope curve Ff , Rd Peak strength of the bilinearized force-displacement curve of the fastener fh Number of perimeter horizontal fasteners FI Intercept strength of the spring element for the pinching branch of the hysteretic curve Fv, Rd Peak strength of the bilinearized force-displacement curve of the wall

nbs np ns R1

R2 R3

R4 S0 uf , Rd uf , ud u v, Rd u v, ud μf

μSH AM BWBN c C-LS CSM CUREE EP EPHM F FE H I L LE LS-LS ML N/A N/S nstuds O OSB R S s SAWS sph spv

Number of perimeter vertical fasteners Mean characteristic value of the wood shear modulus Height of the external vertical studs Height of the internal vertical studs Height of the horizontal plates Secant stiffness at the peak strength of the fastener Strength degradation of the spring element Global secant stiffness of the wall related to the sheathingto-framing connections contribution Number of wall braced sides (1 or 2) Number of panels on each side of the wall Number of vertical studs in the analytical procedure Stiffness ratio of the asymptotic line to the spring element envelope curve Stiffness ratio of the descending branch of the spring element envelope curve Stiffness ratio of the unloading branch off the spring element envelope curve Stiffness ratio of the pinching branch for the spring element Initial stiffness of the shear wall spring element Displacement at the peak strength of the bilinearized force-displacement curve of the fastener Ultimate displacement of the fastener Displacement at the peak strength of the bilinearized force-displacement curve of the wall Ultimate displacement of the wall Ductility of the bilinearized force-displacement curve of the fastener Bilinearized ductility of the sheathing-to-framing connections Analytical model Bouc-Wen-Baber-Noori model Parameter that takes into account the aspect ratio of the wall panel Collapse Limit State Capacity Spectrum Method Consortium of Universities for Research in Earthquake Engineering Elasto-plastic Evolutionary parameter hysteretic model Flexible Finite element Wall height Isotropic material model Wall width Linear-elastic Life Safety Limit State Multi-linear Not assigned/not available Not-symmetrical version Number of vertical studs in the FE model Orthotropic material model Oriented Strand Board Rigid Total number of samples Fasteners spacing Seismic analysis of wood-frame structures model Horizontal fasteners spacing Vertical fasteners spacing

intermediate studs (where the spacing is usually twice or thrice), so to prevent buckling of the sheathing panels [7]. The satisfactory performance of light-frame timber shear wall

the Penny system classification used in the United States. The fasteners are placed on the perimeter studs (with spacing typically equal to 50 mm/1.97 in., 75 mm/2.95 in. or 100 mm/3.94 in.) as well as on the 2

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energy in wood shear walls, by comparing the hysteresis loops carried out from experimental tests with numerical predictions. Albeit previous studies have already quantified the damping properties of wood structural systems, there is not yet a simplified closed-form procedure to predict the global response of a light-frame timber shear wall by considering its geometric features and starting from the local properties of a single fastener. Regarding the finite element (FE) analysis, frame members and sheathing panels are commonly modeled with beam-type elements and plane-stress elements, respectively, assuming elastic behavior in compression and elastic-brittle behavior in tension [12]. Sheathing-toframing connections and base connections are usually modeled with non-linear springs [29]. Simplified analytical models have been elaborated to evaluate the response of cross-laminated and light-frame timber shear walls considering rigid frame members for the latter [9,17–21]. Moreover, only few parametrized numerical models have been elaborated up to now [10,30]. Table 1 provides a summary of some available numerical and analytical approaches. Additionally, very few efforts have been spent so far in the search for and the analysis of optimal wall configurations [31]. Within this framework, the present study aims at investigating the dissipation in sheathing-to-framing connections of light-frame timber shear walls under seismic loads, thereby providing new useful results and tools for their analysis and design. Overall, the present paper is organized as follows. First, an original parametric FE model developed within the open-source software OpenSees [52] is illustrated in Section 2. The results of a complete parametric analysis are discussed in Section 3. They are employed to assess, both, ductility and equivalent viscous damping of light-frame timber shear walls, as well as the damping factor adopted in the Capacity Spectrum Method (CSM) [e.g.,53,54]. Section 3 is intended to highlight the influence of each parameter on the global response of a light-frame timber wall. The optimum design of the wall is then addressed in Section 4 by exploiting the developed numerical FE model and taking into account typical geometries in use within timber buildings. The design problem is formulated in such a way to optimize simultaneously racking capacity and total material cost, and the equivalent viscous damping ratio is estimated for all the extracted optimal wall layouts. In this way, a kind of catalogue for designers and manufacturers is obtained, from which the wall configuration that minimizes the total material cost for a required seismic capacity can be directly deduced. An useful evidence is that the optimal (non-dominated) wall configurations have almost the same equivalent viscous damping value. Finally, a novel simplified analytical procedure is illustrated in Section 5 to predict analytically the backbone forcedisplacement curve of light-frame timber walls by considering their geometric features and starting from the properties of a single fastener. The simplified analytical procedure is elaborated and validated on the basis of the numerical results carried out using the developed FE model.

assemblies under earthquake has been confirmed in a considerable amount of previous studies, in which racking resistance, stiffness and ductility have been investigated through experimental tests, numerical simulations and analytical methods. Such a good seismic behavior is basically attributable to the high strength-to-density ratio of the timber and to the remarkable ductility of the connections. Particularly, experimental tests demonstrated that the structural behavior under lateral loads of a light-frame timber shear wall is mainly influenced by its connections, such as sheathing-to-framing joints [8,13,14], base [15] and stud-plate joints [13]. This is because timber has a poor dissipative capacity due to its brittle behavior, unless it is properly reinforced [16]. Conversely, steel connections in light-frame timber walls (i.e., sheathing-to-framing, hold-down and angle-bracket, see Fig. 1) ensure a good amount of energy dissipation and cyclic ductility notwithstanding their significant pinching, strength degradation and softening. This evidence is properly reflected into the numerical models available in the literature, where the global non-linear wall response is related to the load-deformation relationships of the connections [e.g.,9,17–21]. For instance, Polensek and Bastendorff [22] carried out some experimental tests under cyclic loading in order to correlate the global damping and the fastener joint properties. Polensek and Schimel [23] performed dynamic tests on shear walls systems and correlated the vibration amplitudes with the global damping. It was observed that it increases gradually with the wall displacement magnitude whereas it is not substantially influenced by the properties of the timber. Salenikovich [10] and Salenikovich and Dolan [24] tested experimentally 56 OSB sheathed light-frame shear walls with different aspect ratios, observing that also the density of studs at the wall ends affects the global response in terms of deformation capacity. Other studies investigated the influence on the damping of different kinds of joint (glued and nailed joints in [25] or dried joints in [26]) and sheathing panel, also considering the presence of openings [27]. A quasi-linear relationship between equivalent viscous damping and drift (i.e., ratio between top plate displacement and wall height) has been obtained in Filiatrault et al. [28] by means of pushover analyses on 24 wood building configurations. Humbert et al. [13] also studied the dissipated Exterior wall finish Top plate

Insulation material

Vertical stud

2. Finite element modeling

Sheathing panel Hold-down

An original FE model has been developed within the open-source software OpenSees [52] for the analysis and design of light-frame timber shear walls, with focus on the energy dissipation ensured by sheathing-to-framing connections under in-plane lateral loads. Generally, the hysteretic behavior of a light-frame timber shear wall is mainly influenced by sheathing-to-framing connections if other steel connections (i.e., hold-down and angle-bracket) are designed with overstrength [14]. Accordingly, the hold-downs behavior is not considered within the numerical model in the following. Framing system and sheathing panel are modeled as elastic elements and the behavior of the connections are simulated by means of zero-length non-linear elements. This kind of model is widely adopted in literature [4,12,40,42] because it was demonstrated that the global response of the shear wall is mostly related to the non-linear behavior of the sheathing-to-framing connections [17–19], whereas the sheathing

Angle-bracket Fasteners (nails) Sheathing panel Vapor retarder Interior wall finish Fig. 1. Typical configuration of a light-frame timber shear wall with a full attachment (according to the definitions given by Salenikovich [10]), braced on both sides and with non-structural layers for thermal insulation and fire/vapor resistance. 3

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Table 1 Available numerical and analytical approaches (LE: linear-elastic; EP: elasto-plastic; R: rigid; F: flexible; I: isotropic material model; O: orthotropic material model; ML: multi-linear; EPHM: evolutionary parameter hysteretic model [32]; SAWS: seismic analysis of wood-frame structures model [33–35]; BWBN: Bouc-Wen-BaberNoori model [36–38]; AM: analytical model, FEM: finite element model; N/S: not symmetrical version; N/A: not assigned/not available). Reference

Frame elements

Sheathing panel

Fastener model

Openings

Approach

Software or code

Tuomi and McCutcheon [17] (1978) Easley et al. [39] (1982) Gupta and Kuo [18] (1985) Gupta and Kuo [19] (1987) Gutkowski and Castillo [40](1988) Filiatrault [41](1990) Dolan and Foschi [4](1991) White and Dolan [42](1995) Foliente et al. [37] (2000) Richard et al. [43] (2002) Folz and Filiatrault [44], Folz and Filiatrault [45] (2004) Doudak et al. [46] (2006) Pang et al. [32] (2007) Källsner and Girhammar [7] (2009) Xu and Dolan [38] (2009) Yasumura [29] (2010) Rinaldin et al. [47] (2013) Pintarič and Premrov [48] (2013) Humbert et al. [13] (2014) Vogrinec et al. [49] (2016) Casagrande et al. [21] (2016) Gattesco and Boem [12] (2016) Jayamon et al. [50](2016) Anil et al. [51] (2018)

N/A LE, R LE, F LE, F LE LE, R LE LE, I N/A LE, R R N/A N/A LE, R N/A LE, R, I LE, R, I LE, F, I LE, R LE, F, I LE, R LE, R R EP, O

N/A LE LE LE LE, O LE LE, O LE, O N/A LE, O LE N/A N/A LE N/A LE, O LE, I LE, I I LE, I LE LE, I LE EP, O

N/A N/A ML ML ML ML ML SAWS BWBN SAWS SAWS N/A EPHM ML, LE BWBN ML ML N/A SAWS (N/S) N/A N/A ML SAWS ML

N/A No N/A N/A Yes No No No No Yes Yes Yes N/A No N/A Yes No No No No No No No Yes

AM AM AM AM FEM FEM FEM FEM FEM FEM FEM FEM FEM AM FEM FEM FEM AM FEM AM AM, FEM FEM FEM FEM

– – – – WANELS SWAP SHWALL WALSEIZ BRIANZ N/A CASHEW SAP2000 N/A – Abaqus CASTEM 2000 Abaqus, SAP2000 – Code_Aster – UNITN Abaqus OpenSees Ansys

built types of elements available in OpenSees are used: Elastic Beam Column Element, ShellMITC4 Element, ZeroLength Element and CoupledZeroLength Element. The frame elements are modeled using the Elastic Beam Column Element. The whole sheathing panel is modeled by means of the ShellMITC4 Element, and the mesh size is automatically adjusted based on the fasteners spacing. In order to model the frame joints as perfect hinges [29,43,55], the ZeroLength Element is employed at both ends of the vertical studs, using a low stiffness value for the rotation along the y direction and assigning an infinite value to the remaining degrees of freedom. The assumption of perfect hinge is commonly accepted because the rotational strength of fasteners joint in the lateral direction is relatively low [43]. The behavior of the sheathing-to-framing connections is modeled with non-linear load-slip relationships along the x and z directions. The CoupledZeroLength Element is employed to consider a circular yielding surface, thus preventing the overestimation of fastener stiffness and force under non-linear loading conditions [21,56]. Accordingly, these elements produce a force proportional to the relative displacement of the nodes between shell elements and frame. All these elements are labeled by considering the layer they belong to (first digit) and taking into account their progressive position along x and z directions (next couples of digits), as illustrated in Fig. 2b. Layers 2, 3, 5, 6 and 7 refer to vertical studs, horizontal plates, hinges within the frame, panel (together with its nodes) and sheathing-to-framing connections, respectively. The SAWS Material model is adopted to simulate the mechanical behavior of a single fastener. This phenomenological model, also known as the 10-parameter CUREE (Consortium of Universities for Research in Earthquake Engineering) model, was originally proposed by Foschi [33] and then updated and developed by Folz and Filiatrault [35,44,45], with the addition of the post-capacity degradation [34] (see Fig. 3). It is among the most reliable models to simulate the behavior of a single fastener [57] and is widely used for modeling timber shear walls [58]. Finally, since the frame is double sheathed (i.e., it is braced with one sheathing panel on both sides of the frame, as shown in Fig. 1), the symmetric shell and zero-length elements are included in layers 8 and 9, respectively.

panel mainly develops elastic in-plane shear forces and frame members have a negligible bending [9,18]. It is noteworthy that Anil et al. [51] only assume an elasto-plastic model for sheathing panels and frame elements (see Table 1). No vertical load is applied in the FE model. The vertical load can be neglected because it mainly influences the uplift of the wall [14], the latter basically controlled by the hold-down connections, which are not considered in this study. Additionally, geometrical non-linearities are disregarded [9] and a linear coordinate transformation has been used. It is worth highlighting that, for the drift levels attained by the considered wall configurations, the P − Δ effect has a limited influence on the global response, as verified with the numerical FE model proposed herein and previously confirmed by Jayamon et al. [50]. The model implementation is based on the following geometric parameters: i) panel height and width (H and L ); ii) horizontal and vertical fasteners spacing (sph and spv ); iii) number of vertical studs (nstuds ); iv) frame elements cross-section sizes, namely base and height of external (bes and hes ) and internal (bis and his ) vertical studs crosssection as well as base and height of the horizontal plates cross-section (bp and hp ). The flow chart in Fig. 2a shows the overall approach implemented to build the FE model. Details about the numerical implementation of the model in OpenSees are given hereafter for the sake of completeness and to facilitate the replication of the obtained results. Once the geometrical parameters are defined, the nodes are created on a regular grid according to the horizontal and vertical fasteners spacing (see Fig. 2a). All the nodes are tagged with a label as follows: the first digit denotes the layer they belong to; the two next couples of digits denote the i th (in the x direction) and j th (in the z direction) positions of the nodes on the main grid, respectively. Three layers are employed for the nodes, namely 2, 4 and 6. The first layer is 2, which includes the fictitious nodes used to insert the internal releases between the end of vertical studs and the horizontal plates. The layer 4 includes the perimeter nodes belonging to the frame. The layer 6 includes the nodes belonging to the shell elements. An example of the FE model generated for 5 fasteners in the horizontal direction and 3 fasteners in the vertical one is illustrated in Fig. 2b, where wall width and height are aligned with the x -axis and the z -axis, respectively. The elements are generated in the second step. The following in4

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G. Di Gangi, et al. GEOMETRIC INPUT PARAMETERS Frame elements cross-section sizes: width (b_{}) and height (h_{}) of internal (_{is}) and external (_{es}) studs, horizontal plates (_{p})

Wall size: Height (H), Width (L)

Number of vertical studs (nstuds)

Perimeter fasteners spacing: horizontal (sph), vertical (spv)

2 horizontal plates (top and bottom)

Number of vertical elasticBeamColumn elements

Shell elements mesh size

Number of zero-length elements

Zero-length elements (internal releases)

Number of horizontal elasticBeamColumn elements OUTPUT ELEMENTS (automatically updated)

OUTPUT MODEL

(a) L 3 00 02

4 01 02

3 01 02

4 02 02

2 00 02 6 00 02 6 01 02

4 04 02

3 03 02

6 01 01

6 00 01

2 04 02 6 03 02

6 02 02

2 02 01

2 00 01

4 03 02

3 02 02

2 02 02 6 04 02

2 04 01

4 00 02

6 03 01

6 02 01

4 02 01

6 01 00

6 00 00

6 03 01

6 02 01

2 0 2 00

2 0 0 00

6 00 01 6 01 01

4 04 01

6 04 01

2 0 4 00

4 00 01

6 03 00

6 02 00

H

Legend Perimeter nodes of the frame Fictitious nodes to insert internal releases Nodes at corners of shell elements

Z 2 00 00

6 00 00 6 01 00 3 00 00

6 02 00 3 01 00

2 02 00 6 03 00

2 02 00 3 02 00

Shell elements

6 04 00

2 04 00

3 03 00

Zero-length elements (internal releases, 5 xx zz)

X 4 00 00

4 01 00

4 02 00

Beam elements

4 03 00

4 04 00

Zero-length elements (fasteners, 7 xx zz)

(b) Fig. 2. Graphical illustration of the approach implemented for building the parametric FE model (a) and example of a layout of the FE model developed in OpenSees (b).

3. Model validation and parametric analysis

framing displacements are used in the following to validate the FE model in which the contribution of hold-down and angle-bracket connections is neglected. For the sake of completeness, it is pointed out that other studies also provide the hysteretic response of the fasteners [e.g.,57] together with the corresponding ones of the shear wall where they are employed [e.g.,35].

In this study, the experimental data provided by Gattesco and Boem [12] are used for the identification of the single fastener and the validation of the shear wall FE model. This experimental study was considered because it provides the global response of the wall and that of the fastener used for the sheathing-to-framing connections. In particular, Gattesco and Boem [12] provide the experimental global response of the wall by decoupling the effects due to the sheathing-toframing, hold-down and angle-bracket connections. The sheathing-to-

3.1. Parametric identification of the single fastener hysteresis model Since the global non-linear behavior of the wall depends on its 5

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behavior of the connections using the SAWS Material model. Therefore, the corresponding estimated parameters have been implemented within the FE model of the light-frame timber shear wall in order to predict the global structural response under later loading (without further model updating). In order to simulate as best as possible the testing conditions, the same displacement increments are adopted (i.e., a horizontal cyclic loading consisting of 14 displacement cycles is applied to the top plate). No vertical load is applied during the analysis. The comparison between experimental data and predicted response is shown in Fig. 5. This comparison substantiates the correctness of the FE model with the possible exception of the softening-type behavior once the maximum capacity is exceeded. 3.3. Numerical evaluation of the response The validated FE model is employed to investigate the energy dissipation ensured by the sheathing-to-framing connections through a complete parametric analysis. This investigation is performed by changing one-parameter-at-atime, starting from a reference configuration having the same geometrical and mechanical characteristics of the specimen labeled PLS8 by Gattesco and Boem [12]. All the numerical analyses are carried out by imposing a horizontal cyclic loading under displacement-controlled conditions to the top plate. Two Limit States (LS) are defined by observing the global behavior of the wall and the local behavior of the fasteners (see Fig. 6), namely the Life Safety Limit State (LS-LS) and the Collapse Limit State (C-LS). First, the displacement of the top plate is increased until the local failure criterion – related to the single fastener – is reached (C-LS). Then, the displacement increment is reversed and one symmetric loop is obtained. The LS-LS corresponds to the global racking capacity, i.e., the peak of the force in the global response of the wall (Fig. 6a). It occurs when almost all fasteners along the perimeter frame elements exceed the displacement corresponding to their own peak force (see Fig. 6b). The displacement (i.e., the relative displacement between sheathing panel and frame elements) corresponding to the peak force of a single fastener Du is equal to 3.8 mm (see Table 2). The racking capacity is reached in the LS-LS. By increasing the global displacement beyond the LS-LS, the last fastener in the frame exceeds the peak force. As a consequence, a global softening-type behavior of the wall occurs because all the fasteners soften. Afterward, the C-LS occurs when the most stressed fastener, usually at the bottom corner, reaches its failure displacement (Fig. 6c). This is

Fig. 3. Description of the SAWS Material model parameters. (Dun : last unloading displacement, which controls the strength degradation in the response; Kp = S0 [(F0/ S0 )/ Dmax ]α ).

connections response [4,17–19,40–42], the proper calibration of the single fastener hysteresis model (see Fig. 3) is a critical task. To this end, the SAWS Material model parameters are identified (Fig. 4) using the experimental data provided by Gattesco and Boem [12] for a fastener (ring nail) with 2.8 mm diameter and 70 mm length (ϕ2.8/70 for short). The parametric identification is performed by means of the Differential Evolution algorithm [e.g.,59–61], which is employed to minimize the following objective function:

f (θ ) =

1 (Fnum − Fexp)T (Fnum − Fexp), S·var(Fexp )

(1)

where θ = {F0, FI , Du , S0, R1, R2 , R3, R 4 , α, β } is the vector collecting the model parameters, Fnum and Fexp are predicted and experimental force values, respectively, S denotes the total number of samples and var(Fexp) is the variance of the experimental force values. The optimization problem is solved by assuming the ranges of values for the involved model parameters reported in Table 2, where the final identified values are also listed. A preliminary sensitivity analysis has been carried out to identify the suitable search space for the model parameters. It can be observed in Fig. 4 that a good agreement between experimental and identified force-displacement curves is achieved. 3.2. Validation of the global numerical model The numerical FE model is first validated using the experimental data provided by Gattesco and Boem [12] for the specimen labeled PLS8. Width and height of the panel are equal to 1800 mm and 2600 mm, respectively. The wall consists of four vertical studs with a cross-section size equal to 140 mm × 160 mm (5.5 in. × 6.3 in.) and two horizontal plates (on the top and the bottom) with a cross-section size equal to 120 mm × 160 mm (4.7 in. × 6.3 in.). It is sheathed with one 15 mm (0.59 in.) thick particle board sheathing panel on each wall side, connected to the timber frame by means of type ϕ2.8/70 ring nails whose spacing is equal to 50 mm (1.97 in.). The mechanical properties of the frame elements are those of the red spruce wood species with strength class C24 in accordance with Table 1 of EN 338:2003 [62], specifically E0, mean = 11 GPa and Gmean = 0.69 GPa . Note that the fastener used in the PLS8 specimen is the same considered in the present study for the simulation of the hysteretic

Fig. 4. Parametric identification of the SAWS Material model for sheathing-toframing connections: comparison between experimental data (from Gattesco and Boem [12]) and numerical predictions after parameters estimation. 6

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Table 2 Identified SAWS Material model parameters using the experimental results by Gattesco and Boem [12] for the fastener (ring nail) type ϕ2.8/70 (refer to Fig. 3 for some details about the model parameters). The search space for the model parameters is also provided. Parameters

F0 [kN]

FI [kN]

Du [mm]

S0 [kN/mm]

R1 [–]

R2 [–]

R3 [–]

R4 [–]

α [–]

β [–]

Identified Lower Bound Upper Bound

1.00 0.24 1.80

0.28 0.056 0.42

3.8 0.764 5.73

1.8 0.26 1.95

0.095 0.048 0.12

−0.13 −0.032 −0.24

2.2 0.578 4.335

0.028 0.008 0.06

0.75 0.16 1.2

1.2 0.24 1.8

parameters that vary with respect to the reference configuration (PLS8 by Gattesco and Boem [12]) during the sensitivity analysis are the following: (i) wall aspect ratio, i.e., height-to-width ratio (αm = H / L ); (ii) horizontal and vertical fasteners spacing; (iii) number of vertical studs; (iv) cross-section size of the frame elements. Fig. 7 presents a summary of the numerical results obtained by varying these parameters. The variation of the aspect ratio (αm = {0.7, 1, 1.4} ) is here obtained by changing the wall width (L = {3.7, 2.6, 1.8} m) by keeping the height constant (H = 2.6 m). As shown in Fig. 7a, the reduction of the aspect ratio leads to an increase of stiffness and racking capacity, together with a diminution of the displacement at C-LS. This is in agreement with the findings by Anil et al. [51], who showed that for αm ≥ 3 the flexural behavior dominates the response whereas, by reducing the aspect ratio, the shear deformation contribution increases, with an increment of stiffness and racking capacity. The increment of the fastener spacing from 50 mm to 100 mm leads to almost halving both the racking capacity and the stiffness of the wall, see Fig. 7b. The racking capacity reduction depends on the smaller number of fasteners on the perimeter frame elements, which makes the overall system more flexible and capable of withstanding lower loads. Fig. 7c shows that the increment of the vertical studs number leads to a slight increment (about 10%) of racking capacity and stiffness. The increment of the overall number of fasteners due to a higher number of vertical studs has negligible effects on the racking capacity. This is because the perimeter fasteners mainly contribute to the overall response whereas the others remain in the elastic range (see Figs. 6b,c). The variation of the frame elements cross-section size is obtained by changing both the vertical studs and horizontal plates. It is noteworthy that the vertical studs cross-section in the reference wall configuration is equal to 140 mm × 160 mm (5.5 in. × 6.3 in.) whereas the horizontal plates cross-section is equal to 120 mm × 160 mm (4.7 in. × 6.3 in.). Two different wall configurations are analyzed, with cross-section sizes equal to 38 mm × 140 mm (1.5 in. × 5.5 in.) and 38 mm × 89 mm (1.5

Fig. 5. Comparison between experimental data and predicted force-displacement response for the wall specimen PLS8 in Gattesco and Boem [12].

assumed equal to 8.5 mm based on the experimental response of the fastener (ring nail) type ϕ2.8/70 by Gattesco and Boem [12] and corresponds to a 65% strength degradation (see Fig. 4). In the C-LS condition, the most stressed fastener is able to dissipate its maximum available energy, whereas all other fasteners dissipate only a fraction of it because they undergo a lower displacement than their failure displacement. However, the energy dissipation attains its maximum value since most of the fasteners experience plastic deformations. 3.3.1. Results of the sensitivity analysis The role of the geometric parameters on the overall response of a light-frame timber shear wall is now discussed. In particular, the

Fig. 6. Global force-displacement curve of the reference wall configuration (a). Displacement field of the fasteners (the arrows are proportional to the relative displacements of each fastener, with a scale factor equal to 100) at LS-LS (b) and C-LS (c) for the reference wall configuration (black thick arrow: fastener displacement smaller than the one corresponding to peak force; blue arrow: fastener displacement exceeding the peak force; red arrow: fastener displacement at failure). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). 7

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Fig. 7. Influence of aspect ratio (a), horizontal and vertical fasteners spacing (b), number of vertical studs (c) and frame elements cross-section size (d) on the overall force-displacement response of the wall.

calculate the reduced design seismic actions – so as to take into account the energy dissipation ensured by the structure [64] – by means of the so-called damping correction factor η , which is computed as follows [65]:

in. × 3.5 in.), which are commonly employed in practice [6]. Fig. 7d highlights that, the smaller the frame elements cross-section size, the smaller the racking capacity and stiffness of the wall. 3.3.2. Total equivalent viscous damping and racking capacity The force-displacement curves in Fig. 7 are considered to estimate total equivalent viscous damping and racking capacity of the wall. The total equivalent viscous damping, corresponding to the C-LS condition, is computed as follows [63]:

ξtot = ξ0.05 + ξeq = 0.05 +

ED , 4πEs0

η=

10 . 5 + ξtot

(3)

Equivalent viscous damping, total equivalent viscous damping, damping factor and racking capacity are summarized in Fig. 8. For the reference configuration (specimen PLS8 in [12]), ξeq is about 23% , thus resulting in η = 0.55. Fig. 8a highlights a quasi-linear relationship between ξeq and drift. This is in agreement with the study by Filiatrault et al. [28]. The equivalent viscous damping ξeq strongly depends on the aspect ratio, thereby confirming the results by Salenikovich [10]. The smaller the aspect ratio, the larger ξeq given a drift value, but the drift

(2)

where ξ0.05 (equal to 5% ) is the inherent viscous damping and ξeq is the equivalent viscous damping. This latter is calculated using the energy dissipated in a single cycle ED and the elastic strain energy in a half cycle Es0 . The total equivalent viscous damping ξtot can be used to 8

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Fig. 8. Equivalent viscous damping as function of the drift (i.e., ratio between top plate displacement and wall height) for the different wall configurations (a). The stars indicate the drift corresponding to LS-LS whereas the circles indicate the drift corresponding to C-LS. Equivalent viscous damping, corresponding damping factor (for C-LS condition) and racking capacity by varying the input parameters with respect to the reference wall configuration, marked with an asterisk (b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

however, that the recommendation provided by Jayamon et al. [66] refers to a common practice rather than numerical/experimental evidences. Conversely, the reduction of the racking capacity by increasing the fasteners spacing is significant (see Fig. 8b and 7b). On the other hand, a slight growth of the racking capacity is observed by increasing the number of vertical studs (see Fig. 8b), with moderate influence on ξeq for a given drift value (see Fig. 8a). Moreover, it is found that the larger the cross-section size of the frame elements, the larger the racking capacity and ξeq corresponding to the C-LS (see Fig. 8b). The trend observed for ξeq is reflected on η, which is related to ξeq through Eq. (3), as reported in Fig. 8b.

corresponding to the C-LS strongly decreases (see Fig. 8a). As a consequence, ξeq corresponding to the C-LS condition is almost the same for any aspect ratio (see Fig. 8b). Conversely, the higher the aspect ratio, the lower the racking capacity (see also Fig. 7a). It should be highlighted that the drift corresponding to the C-LS is only influenced by the aspect ratio whereas it is practically the same (about 1% variation) when varying the other input parameters (see Fig. 8a). An almost negligible increment of ξeq for a given drift value is observed in Fig. 8a by increasing the fasteners spacing. This conclusion is not in agreement with the indications provided by Jayamon et al. [66] as regards the connection details that can enhance the damping. It should be noted, 9

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Finally, it is useful to compare the ξeq values reported in this study with other outcomes available within the current literature. In particular, Salenikovich [10], Salenikovich and Dolan [24], Chou [26], Filiatrault et al. [28] provided ranges of damping values compatible with findings reported herein. The damping ratio value (about 26% of critical) of the considered fastener is in agreement with the experimental results obtained by Koliou et al. [58] for powder actuated fasteners (25% to 30% of critical one). These additional evidences further confirm the reliability of the numerical predictions obtained via FE analysis.

Table 3 Material unit costs of the wall elements. Frame elements

Sheathing panels

Fasteners

[Euro/kg]

[Euro/m2 ] 6

[Euro/fastener]

6

0.01

here intended to maximize the racking capacity while minimizing the total cost. To this end, the racking capacity has been evaluated using the developed FE model whereas the cost refers to the Italian market of timber frame elements, sheathing panels and fasteners commonly used in platform-frame constructions. A summary of the unit costs is reported in Table 3. These unit costs refer to materials only and are representative to the time when this research work has been carried out. Manufacturing and installation costs are not considered because the authors were not able to find reliable data and workmanship costs are not easily estimable because they can be subjected to large variations. The multi-objective optimum design of two walls is considered. The wall sizes are 1.2 m × 2.4 m (3.9 ft × 7.88 ft) and 2.4 m × 2.4 m (7.88 ft × 7.88 ft), which are rather common configurations in practice. The design space is defined as follows: (i) fasteners spacing (sph and spv ) equal to 50 mm, 75 mm or 100 mm (1.97 in., 2.95 in. or 3.94 in.); (ii) number of vertical studs (nstuds ) equal to 3, 4 or 5; (iii) base of the external and internal vertical studs cross-section (bes , bis ) equal to 38 mm or 140 mm (1.5 in. or 5.5 in.); (iv) base of the horizontal plates (bp ) equal to 38 mm or 120 mm (1.5 in. or 4.7 in.); (v) height of vertical studs cross-section (hes , his ) as well as height of horizontal plates crosssection (hp ) equal to 89 mm, 140 mm or 160 mm (3.5 in., 5.5 in. or 6.3 in.). Moreover, hes , his and hp are required to be equal to each other. Note that the maximum cost of the wall is obtained by maximizing simultaneously number of vertical studs, cross-section size of the frame elements and number of fasteners. This multi-objective optimization problem is addressed by exploring the full set of wall configurations that define the discrete design space and extracting a relevant set of non-dominated solutions only (this is a trivial but viable approach for the present problem because of the reduced number of design alternatives). The number of fasteners has a significant effect on the racking capacity (see Fig. 7b and 8b) while the influence on the total material cost is negligible (see Table 3). Therefore, the best fasteners spacing is the

3.4. Further comments about the aspect ratio Because of the restrictions about the available sizes, the walls with low aspect ratio are braced by using more than one sheathing panel on each side. In practice, therefore, walls having (αm = {0.7, 1} ) are realized with two sheathing panels joined to a common field stud with a butt joint. In order to quantify the difference in the overall response, the parametric FE model has been modified to reflect such condition. Assuming αm = 0.7 and 5 studs as illustrative example, it is shown in Fig. 9a that the racking capacity of the wall with single or double sheathing panel is almost the same. On the other hand, when two separate panels are employed, a reduction of the global secant stiffness at the peak strength is observed, which is due to the reduction of the panels inertia with respect to in-plane actions (see Fig. 9b) [67]. This result is consistent with the outcomes of past studies by He et al. [68] observing that the wider the sheathing panel, the lower the displacement at the ultimate load and the more abrupt is the post-peak behavior in terms of strength degradation rate (see Fig. 7a). 4. Optimum design The design of the light-frame timber shear walls is formulated as a multi-objective optimization problem. Such a kind of problem admits multiple optimal solutions (also known as non-dominated solutions), each one representing a compromise between the selected conflicting optimization criteria. The solution of a multi-objective optimization problem is non-dominated (or Pareto optimal) if there is no other point that improves at least one objective function without detriment to another one [70,71]. Specifically, the optimum design of light-frame timber shear walls is

Fig. 9. Force-displacement response considering single and double sheathing panel on each wall side (a). Deformed configuration of the wall with double sheathing panel on each side in [mm] (b). The deformed configuration is plotted using the software STKO [69]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). 10

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intermediate studs increases for wall configurations with the same cross-section size of the frame elements. This is because the fasteners placed on the intermediate studs do not contribute significantly to the energy dissipation but increase the racking capacity. As a final note, it is interesting to report that in common practice it is possible to modify connection details to enhance damping (i.e., the energy dissipation). Possible modifications are discussed by Jayamon et al. [66]. However, other connection details are not considered in the present study and the fasteners described in Section 3.1 only have been investigated.

lowest one (50 mm/1.97 in.) because it leads to the maximum racking capacity with a negligible increment of material costs. Accordingly, a fasteners spacing equal to 50 mm (1.97 in.) is considered in the following. Figs. 10a,b show the Pareto front of the non-dominated solutions for the slender wall (1.2 m × 2.4 m/3.9 ft × 7.88 ft) and the squat wall (2.4 m × 2.4 m/7.88 ft × 7.88 ft), respectively. They are conceived to be used as a catalogue for designers and manufactures: for a required racking capacity (which, in turn, depends on the seismic demand), it is possible to find immediately the configuration that ensures the minimum total material cost. A major finding emerges from these results. In fact, it is observed that the wall configurations representing the optimal trade-off between total material cost and capacity have similar equivalent viscous damping ratios. From a different perspective, it implies that the equivalent viscous damping ratio of an optimal wall is almost the same regardless its total material cost because the latter affects the racking capacity only. Once again, it is observed that the equivalent viscous damping slightly decreases when the number of

5. Simplified analytical assessment The simplified analytical procedure here proposed for the assessment of light-frame timber shear walls starts from the definition of the design local parameters of a single fastener. These, in turn, are employed to determine the global parameters of the wall (corresponding to relevant Limit States). The case of a single sheathing panel for each side

Fig. 10. Optimal configurations of slender (a) and squat (b) light-frame timber shear walls in terms of racking capacity and dimensionless material cost (i.e., ratio between optimal material cost and maximum material cost). The following data are also provided for each optimal solution: equivalent viscous damping ξeq ; miniature of the optimal wall configuration showing the corresponding number of vertical studs and the fasteners spacing; bar chart identifying the dimensionless optimal size of the frame elements (i.e., ratio between optimal and maximum value of the considered geometrical variable). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). 11

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of the wall will be addressed. The proposed procedure is newly developed, even if it rests on the Eurocode 5 [72, eq. (9.21)] for the estimation of the racking capacity and adopts the model by Casagrande et al. [21, eq. (15)] for the computation of the global stiffness. The proposed procedure benefits of the numerical results carried out via FE analysis for the approximate quantification of the energy dissipation due to sheathing-to-framing connections and for the definitive validation. A guided step-by-step numerical application of the proposed analytical procedure is presented in Appendix A.

condition can be computed using Eq. (2). For this condition, the energy dissipation attains its maximum value (see Fig. 8a). As it has been remarked previously, in such a condition, the most stressed fastener reaches the failure displacement and is able to dissipate its maximum available energy. Conversely, the remaining fasteners do not because their own failure displacement is not reached yet. As a consequence, Es0 (see Eq. (2)) is computed as the sum of the elastic energies of the fasteners placed on the perimeter vertical studs and on the horizontal plates, that is:

5.1. Fastener behavior

Es0 = Es0f (fv + fh ),

where Es0f is the elastic energy of a single fastener (see Eq. (8)), fv is the number of vertical fasteners and fh is the number of horizontal fasteners, respectively. The energy ED dissipated by the wall in a single cycle can be estimated as the sum of the energies dissipated by all the boundary fasteners because, as previously noted, the energy dissipation ensured by fasteners placed on intermediate studs is negligible. Therefore:

The response of a fastener is simplified with a softening-type curve defined by four parameters (Fig. 11), namely: (i) secant stiffness at the ∼ peak strength k fsec ; (ii) peak strength Ff , Rd ; (iii) ultimate displacement uf , ud ; iv) resistance decrement αf . The displacement at the peak strength ∼ uf , Rd = Ff , Rd / k fsec whereas the force at the ultimate displacement is thus ∼ ∼ ∼ uf , ud is Ff , ud = αf ·Ff , Rd . It is expedient to bi-linearize the constitutive law up to failure by imposing the same dissipated energy and assuming the same secant stiffness. The dissipated energy corresponding to the original response curve for two half symmetric parts of the whole cycle (i.e., twice the area under the original response curve in the first quadrant, see Fig. 11) can be calculated as:

∼2 ⎧ 1 ⎡ F f , Rd ⎤⎫ ∼ ∼ ∼ EDf = 2· ⎢ sec + (αf + 1) Ff , Rd ·(uf , ud − uf , Rd )⎥ ⎨ 2 ⎢ kf ⎬ ⎥ ⎦⎭ ⎩ ⎣ ∼2 F f , Rd μf − αf ], = sec [(αf + 1)·∼ kf

ED = EDf (κfv + γfh ),

(4)

(5)

On the other hand, the dissipated energy corresponding to the bilinearized response curve for two half symmetric parts of the whole cycle (i.e., twice the area under the bi-linearized diagram in the first quadrant, see Fig. 11) is: 2 1 F f , Rd ⎞ ⎛ . EDf = 2·⎜Ff , Rd ·uf , ud − 2 k fsec ⎟ ⎝ ⎠

(11)

where κ and γ are correction factors expressed as the ratio between the actual energy dissipated by the boundary fasteners and the available dissipating energy. The meaning of κ and γ is highlighted in Fig. 12. When κ and γ are assumed to be equal to 1, Eq. (11) provides the maximum energy dissipated as if all the fasteners attained their failure displacement simultaneously. These correction factors have been calibrated numerically using the developed FE model, carrying out simulations on walls with different aspect ratios (i.e., αm equal to 0.7, 0.8, 1, 1.4 and 2), keeping constant the cross-section size of the frame elements. As it was previously discussed, adding intermediate vertical studs and/or increasing the fasteners spacing do not affect the energy dissipated by the boundary fasteners. Fig. 13a illustrates the energy dissipation due to fasteners along the frame elements for three different aspect ratios of the wall. The results of the calibration of κ and γ are reported in Fig. 13b. For a fully closed-form assessment, the coefficients κ and γ can be defined using the following simplified formulations:

μf is the available ductility of the fastener defined as: where ∼ k fsec ·uf , ud uf , ud ∼ μf = ∼ . = ∼ uf , Rd Ff , Rd

(10)

κ = min(αm; 1),

(12a)

(6)

By equating Eqs. (4) and (6), the fastener equivalent peak strength Ff , Rd can be found as:

Ff , Rd = k fsec ·uf , ud −

∼2 (k fsec )2 ·uf2, ud − F f , Rd [(αf + 1)·∼ μf − αf ] .

(7)

The equivalent viscous damping of the fastener at the C-LS can be computed using the bi-linear law as follows:

ξf =

EDf 4πEs0f

2 1 F f , Rd ⎞ ⎟ k fsec

2·⎛Ff , Rd ·uf , ud − 2 = ⎝ Ff , Rd·uf , ud 4π 2 ⎜

(

)

⎠ =

uf , ud −

1 Ff , Rd 2 k fsec

πuf , ud

=

1⎛ 1 1− π⎜ 2μf ⎝

⎞ ⎟, ⎠ (8)

where μf is the available ductility using the bi-linear model (calculated as in Eq. (5)):

μf =

uf , ud uf , Rd

=

k fsec ·uf , ud Ff , Rd

Fig. 11. Parameters of the simplified response curve for the fastener. The dashed blue line denotes the envelope of the experimental response of the fastener (ring nail) type ϕ2.8/70 obtained by Gattesco and Boem [12]. The solid black line denotes the simplified experimental envelope. The dash-dot red line is the equivalent bi-linearized system whereas the dashed black line indicates the elastic secant response at the failure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

(9)

5.2. Wall behavior: Total equivalent viscous damping Once the fastener equivalent viscous damping is defined (see Eq. (8)), the total equivalent viscous damping of a wall for the C-LS 12

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have a lower secant stiffness with respect to the one at the peak strength because in the softening part of the constitutive law. The value of the fastener secant stiffness at the peak strength k fsec – which is employed to compute the wall secant stiffness – is reduced by 20% (i.e., 0.80k fsec ) to fit the numerical results provided by the parametric analyses. This reduction can be considered as the average, for all fasteners, of the ratio between the fastener secant stiffness at the peak strength and the fastener secant stiffness at LS-LS. As a result, the global secant stiffness at peak strength (LS-LS) is computed as follows: sec KSH =

nbs ·0.8·k fsec

( )·λ (α s L

,

pm )

(16)

where λ is a shape coefficient depending on αpm [9,21]:

λ = 0.81 + 1.85·αpm.

(17)

Finally, the displacement at the peak strength can be determined by combining Eqs. (14) and (16) as:

∼ Fv, Rd ∼ u v, Rd = sec . KSH Fig. 12. Energy dissipation along vertical studs and horizontal plates, together with the definitions of the correction factors κ and γ . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

5.3.2. Analytical evaluation at the Collapse Limit State The ultimate global force of the wall is determined as in [72], in which:

L ∼ Fv, ud = nbs ·Ff , Rd ·c·⎡ + (ns − 3) ⎤, ⎣s ⎦

1 γ = min ⎛ ; 0.8⎞. ⎝ αm ⎠ ⎜

(19)

⎟

(12b)

where Ff , Rd is the fastener equivalent peak strength (see Eq. (7)) and the terms (ns − 3) is used in the same way as in Eq. (14). Using an approach similar to the bi-linearization of the fastener response, it is possible to calculate the ultimate global displacement of the shear wall. The peak force of the bi-linearized global curve is approximated as the average between the racking capacity at LS-LS (Eq. (14)) and the ultimate global force at C-LS (Eq. (19)):

Finally, the equivalent viscous damping of the shear wall (see Eq. (2)) is computed using Eqs. (8), (10) and (11) as follows:

ξeq =

(18)

EDf (κfv + γfh ) 4π·Es0f (fv + fh )

= ξf

(κfv + γfh ) (fv + fh )

.

(13)

5.3. Wall behavior: Backbone force-displacement curve

Fv, Rd =

The prediction of the backbone force-displacement curve of a lightframe timber shear wall requires the calculation of the characteristic points at, both, LS-LS and C-LS. 5.3.1. Analytical evaluation at the Life Safety Limit State The racking capacity of the wall at LS-LS can be computed using a modified version of the formulation given in [21,72]:

Fv, Rd . sec KSH

(21)

In particular, by means of Eq. (9), the ultimate global displacement can be expressed as:

u v, ud = u v, Rd ·μSH , (14)

(22)

where μSH is the available ductility using the bi-linear model linked to the sheathing-to-framing connections contribution. Thus, the equivalent viscous damping of the shear wall at C-LS (see Eq. (2)) can be computed using the same formulation given in Eq. (8):

where nbs is the number of wall braced sides (i.e., number of sides with ∼ a sheathing panel), Ff , Rd is the fastener peak strength, ns ≥ 3 is the number of vertical studs, s is the fasteners spacing. The modification (ns − 3) was included into Eq. (9.21) of Eurocode 5 [72] to better predict the increment of the racking capacity when the number of vertical studs is higher than 3. Furthermore, c is defined in [21] as a function of the wall panel aspect ratio αpm = H /(L/ np) where np is the number of panels on each side of the wall) as:

αpm < 2 ⎛1 αpm c = ⎜ 2 2 ≤ αpm < 4 . ⎜ αpm ≥ 4 ⎝0

(20)

The associated displacement is defined as follows, assuming the same secant stiffness (see Eq. (16)):

u v, Rd =

L ∼ ∼ Fv, Rd = nbs ·Ff , Rd ·c·⎡ + (ns − 3) ⎤, ⎣s ⎦

∼ ∼ Fv, Rd + Fv, ud . 2

ξeq =

1⎛ 1 ⎞ ⎜1 − ⎟ π⎝ 2μSH ⎠

(23)

from which μSH can be calculated by substituting Eqs. (8) and (13):

μSH =

1 = 2(1 − πξeq )

1 ⎛⎡ 1 ⎛ 2⎡ ⎢1 − π ⎜ ⎢ π ⎝1 − ⎝⎣ ⎣

. 1 2μf

⎞ ⎤ (κfv + γfh ) ⎟⎞ ⎤ ⎥ (fv + fh ) ⎠ ⎥ ⎠⎦ ⎦

(24)

(15) 5.4. Validation of the analytical procedure

The global secant stiffness of a wall at peak strength is calculated using the analytical expression by Casagrande et al. [21], but considering the local secant one of a single fastener, corresponding to its peak strength. However, at the global peak strength (LS-LS), some of the fasteners exceeded the local peak strength. This implies that they

The case studies already considered in Section 3 are now analyzed to validate the proposed simplified procedure. In detail, given the reference configuration PLS8 in [12], the validation is performed by 13

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Fig. 13. Energy dissipation (left, αm = 2 ; middle, αm = 1; right, αm = 0.7 ) along the perimeter frame elements for different aspect ratios of the wall (a) (the coordinates are in mm). Calibration of κ and γ for different aspect ratios (b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

Fig. 14. Comparison of the backbone force-displacement curve predicted by means of FE analysis (black line) and simplified analytical procedure (red line), by varying one at time the aspect ratio (equal to 0.7, 1 or 1.4), the fasteners spacing (equal to 50 mm/1.97 in., 75 mm/2.95 in. or 100 mm/3.94 in.) and the number of vertical studs (equal to 3, 4 or 5) with respect to the reference configuration PLS8 in [12]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). 14

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values of ξeq given a drift value, but the drift corresponding to the C-LS strongly decreases. Conversely, the higher the aspect ratio, the lower the racking capacity. The drift corresponding to the C-LS is only influenced by the aspect ratio, whereas it is practically the same (about 1% variation) when varying the other input parameters. The quasilinear relationship between ξeq and drift (i.e., ratio between top plate displacement and wall height, see Fig. 1) found in [28] and the strong correlation between the equivalent viscous damping ξeq and the aspect ratio observed in [10] are confirmed. The multi-objective optimum design of light-frame slender and squat timber shear walls has been also addressed, in the attempt to find the best configurations that maximize the raking capacity while reducing the total material cost. In doing so, it has been found that the optimum (non-dominated) wall layouts have similar total equivalent viscous damping regardless the total material cost, which basically affects the racking capacity only. Finally, a simplified analytical procedure has been presented in order to predict the backbone force-displacement curve of light-frame timber shear walls, by considering its geometric features and starting from the local properties of a single fastener. The comparison between the results carried out through a FE analysis have confirmed the reliability of the proposed simplified analytical procedure.

varying one at time the aspect ratio (equal to 0.7, 1 or 1.4), the fasteners spacing (equal to 50 mm/1.97 in., 75 mm/2.95 in. or 100 mm/ 3.94 in.) and the number of vertical studs (equal to 3, 4 or 5). The results of this validation are illustrated in Fig. 14, where a satisfactory agreement is found between analytical predictions of the force-displacement response and numerical results obtained by means of FE analysis. In particular, the discrepancy between the racking capacity obtained analytically (in agreement with the Eurocode 5 [72]) and numerically is about 3%. Moreover, the proposed procedure predicts rather well the response for all the light-frame timber shear wall configurations, often with a reasonable conservative estimate of the ultimate displacement (10% — 15% less than the maximum value carried out by means of the FE analysis). The ultimate displacement is predicted by exploiting the estimation of the equivalent viscous damping according to the simplified analytical procedure (see Eqs. (22) and (24)). The ultimate displacement obtained with the procedure described herein is about 1% lower than the value computed according to ASCE 41-17 [73] and IBC-2006 [74] (neglecting the contribution associated to the hold-downs). It can also be concluded that the proposed simplified approach is able to predict the equivalent viscous damping in a reasonably accurate way. As a final note, it is worthy highlighting that the comparison was extended to different case studies (also considering different panel sizes), and a similar good agreement was found.

Credit authorship contribution statement

6. Conclusions

Giorgia Di Gangi, Cristoforo Demartino, Giuseppe Quaranta: Conceptualization, Methodology, Software, Writing - review & editing. Giorgio Monti: Conceptualization, Methodology, Writing - review & editing.

A comprehensive framework for the analysis and design of lightframe timber shear walls has been presented in this work, with emphasis on the energy dissipation ensured by the sheathing-to-framing connections under seismic loads. An original parametric FE model has been initially developed by means of the open-source software OpenSees. This numerical model has been thus exploited for a large parametric analysis intended to quantify the effects of geometric input parameters on the overall response of the wall in terms of racking capacity and equivalent viscous damping. The total number of fasteners is the parameter that mainly affect the global response of a light-frame timber shear wall in terms of racking capacity and stiffness. Such a number can increase by reducing the fasteners spacing or the aspect ratio or, alternatively, by increasing the number of vertical studs. Conversely, remarkable effects on the equivalent viscous damping have been observed by varying the cross-section size of frame elements, due to the kinematic compatibility conditions between the shear-type behavior of the frame and that of the sheathing panel (which also rigidly rotates with respect to the frame). The higher the relative displacements, the higher the energy dissipation of fasteners due to their plastic deformation. Small values of the aspect ratio leads to large

Acknowledgements The authors would like to thank Prof. Natalino Gattesco and Dr. Ingrid Boem (University of Trieste, Italy) for having shared the data of the experimental tests reported in Gattesco and Boem [12]. The feedback received by the anonymous reviewers is also gratefully acknowledged. Giorgia di Gangi has contributed to the finalization of the present work during a study period at the Nanjing Tech University (P. R. China). She would like to thank Prof. Yan Xiao (now at Zhejiang University/University of Illinois at Urbana Champaign Institute, Zhejiang, P. R. China) and other faculty members for their warm hospitality. This work has been partially supported by the Zhejiang University/University of Illinois at Urbana-Champaign Institute, P. R. China.

Appendix A A guided step-by-step numerical application of the proposed analytical procedure is presented to facilitate its implementation. The considered case study refers to the specimen labeled PLS8 in [12] (see Section 3.2 for details). According to the proposed procedure, the parameters of the approximated bi-linear force-displacement model of the fastener must be first defined. Fasteners (ring nail) type ϕ2.8/70 are adopted in the considered wall. The experimental data provided by Gattesco and Boem [12] (see Fig. 5) allows to define the following relevant parameters for the fastener:

• secant stiffness∼at the peak strength k • peak strength F = 1.66[kN]; • ultimate displacement u = 8.5[mm]; • resistance decrement α = 0.35.

sec f

= 0.44[kN/mm];

f , Rd

f , ud

f

∼ uf , Rd = Ff , Rd / k fsec = 3.8[mm] whereas the force at the ultimate displacement is Accordingly, the displacement at the peak strength is ∼ ∼ ∼ ∼ ∼ Ff , ud = Ff , Rd ·0.35 = 0.58[kN]. The ductility of the fastener is μf = uf , ud / uf , Rd = 2.24 . The peak strength of the bi-linearized curve is evaluated using Eq. (7) as: Ff , Rd = k fsec ·uf , ud −

∼2 (k fsec )2 ·uf2, ud − F f , Rd [(αf + 1)·∼ μf − αf ] = 1.18[kN]. 15

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The bi-linearized ductility of the fastener is evaluated using Eq. (9):

μf =

k fsec ·uf , ud

= 3.15.

Ff , Rd

The equivalent viscous damping of the fastener is calculated as shown in Eq. (8), thus obtaining:

ξf =

1⎛ 1 ⎞ 1− = 0.267. π⎜ 2μf ⎟ ⎝ ⎠

Using Eq. (15), it is found that c = 1 for αpm = 1.4 (L = 1800mm , H = 2600mm and np = 1). The total number of perimeter vertical fasteners is

fv = [(H / spv )·2·nbs] = 208, whereas the total number of perimeter horizontal fasteners is

fh = {[(L/ sph) + 1]·2·nbs} = 148. The parameter κ and γ can be calculated using Eq. (12):

κ = min(αm; 1) = 1, γ = min

(

1 ; αm

)

0.8 = 0.69.

The shape coefficient is estimated by means of Eq. (17):

λ = 0.810 + (1.85·αpm) = 3.48. The equivalent viscous damping is given by Eq. (13), thus resulting:

ξeq = ξf ·

κfv + γfh fv + fh

= 0.23.

Eq. (14) provides the racking capacity as follows: ∼ ∼ Fv, Rd = nbs ·Ff , Rd ·c·fh, t = 126.16[kN], where fh, t = [L/ sph + (ns − 3)]. The global secant stiffness at the peak strength is calculated using Eq. (16): sec KSH =

nbs ·0.8·k fsec

( )·λ (α s L

= 7.24[kN/mm].

pm )

The displacement at the peak strength is estimated by means of Eq. (18): ∼ Fv, Rd ∼ u v, Rd = sec = 17.43[mm]. KSH The ultimate strength of the wall is evaluated through Eq. (19) as: ∼ Fv, ud = nbs ·Ff , Rd ·c·fh, t = 89.5[kN], The peak strength of the bi-linearized curve is derived from Eq. (20) as follows: ∼ ∼ Fv, Rd + Fv, ud Fv, Rd = = 107.8[kN]. 2 The displacement at the peak strength of the bi-linearized curve is evaluated using Eq. (21):

Fig. A15. Backbone force-displacement curve of the reference timber light-frame shear wall: comparison among experimental data [12], numerical results obtained by means of the developed FE model and analytical predictions carried out from the proposed simplified analytical procedure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

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u v, Rd =

Fv, Rd = 14.9[mm]. sec KSH

The bi-linearized available ductility is calculated through Eq. (24):

μSH =

1 = 1.89. 2(1 − πξeq)

Finally, Eq. (20) is employed to estimate the ultimate displacement of the wall as follows:

u v, ud = u v, Rd ·μSH = 28[mm] Fig. A15 illustrates the comparison between experimental data, numerical results obtained by means of the developed FE model and analytical predictions carried out from the proposed simplified analytical procedure. Appendix B. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.engstruct.2020.110246.

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