Seismic design and performance evaluation of self-centering timber moment resisting frames

Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx–xxx

Contents lists available at ScienceDirect

Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn

Seismic design and performance evaluation of self-centering timber moment resisting frames ⁎

Zhan Shu, Zheng Li , Minjuan He, Xiuzhi Zheng, Tingting Wu Department of Structural Engineering, Tongji University, Shanghai 200092, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Self-centering Glulam structure Post-tensioned timber frame Braced timber frame Seismic performance Seismic design Fragility analysis

Motivated by the recent concept of seismic resilient structures and societies, the seismic resilient timber solutions were developed to provide the lateral drift capacity while reducing the structural damage through controlled rocking and self-centering motion. However, very limited studies have evaluated the seismic performance of the post-tensioned self-centering timber frame at the building level, restricting the real-world applications of such structure. This study provides a post-tensioned self-centering timber frame solution for a building hypothetically designed in a high seismic zone in China. Besides, the same architectural design is provided with another comparable structural solution, i.e., a typical post and beam frame system with timber braces, conforming to the design code requirements. Subsequently, the nonlinear numerical models are carefully developed and their crucial component behaviors are calibrated with the existing experimental results. Finally, the seismic performance of the two structural solutions are systematically evaluated. The peak inter-story drift, the residual interstory drift, and the Park-Ang damage index at the system level are investigated. Moreover, the fragility curves are generated demonstrating the self-centering feature of the seismic resilient timber solution. The advantages as well as challenges of such post-tensioned self-centering timber frame structures are clearly captured in the study.

1. Introduction Over recent years, the concept of resilience has been introduced to the field of earthquake engineering as it relates to seismic disaster mitigation, management, and post-earthquake restoration. The severe socio-economic impacts of moderate to major earthquakes in terms of damage/dollars/downtime, has stimulated and facilitated the wider acceptance and implementation of cost-efficient structural damagecontrol technologies. Timber structures are featured by cheaper structural components and faster assembling time, which can be considered as an important type of earthquake resilient structure with quick and low-cost restoration characteristics to the community [1]. New approaches are developed to facilitate the resilience design of timber structures. A few innovative self-centering rocking mechanisms are proposed using post-tensioned steel bars [2], replaceable steel energy dissipaters [3], and slip friction joints [4–8] to improve the seismic performance of timber or timber hybrid structural systems. On the other hand, seismic resilient solutions are often provided at the connection level. A few novel super-elastic shape memory alloy (SMA) components are proposed to serve as the self-centering devices to minimize earthquake induced losses for both timber frames [9] and steel frames [10–12]. ⁎

Considering that the conventionally designed beam-to-column bolted connections with slotted-in steel plates are vulnerable to moderate or major earthquakes, the post-tensioned technique was also implemented to the timber connections to provide seismic resilience. Different from the steel post-tensioned connections that often operates together with the slotted bolted friction plates [13], the post-tensioned timber connections have their unique features. The embryonic work of post-tensioned timber connection was proposed for seismic resisting timber frame with laminated veneer lumber (LVL) for multi-story buildings at the University of Canterbury in New Zealand [14–16]. After the feasibility of the post-tensioned timber concept was validated, full scale internal and external LVL beam-to-column connections were studied experimentally [17]. The tested systems exhibited high levels of ductility, negligible residual deformations and no significant damage of the structural elements. The numerical model was also created to predict the behavior of the beam-column joint connections with various structural configurations. Since wood has a much lower stiffness and strength when loaded perpendicular to the grain, the bearing behavior of the column face against the beam end become a limiting consideration for the load resisting performance of the post-tensioned timber connections. Consequently, several different approaches were proposed to realize the

Corresponding author. E-mail address: [email protected] (Z. Li).

https://doi.org/10.1016/j.soildyn.2018.08.038 Received 30 December 2017; Received in revised form 30 August 2018; Accepted 31 August 2018 0267-7261/ © 2018 Elsevier Ltd. All rights reserved.

Please cite this article as: Shu, Z., Soil Dynamics and Earthquake Engineering, https://doi.org/10.1016/j.soildyn.2018.08.038

Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx–xxx

Z. Shu et al.

post-tensioned design. Firstly, steel angles were proposed to provide the energy dissipation as well as strengthening the beam-to-column joints. A typical 2/3 scale 3-story glulam building with such design was tested on a shaking table test in the lab of University of Basilicata in Potenza, Italy [18,19]. Secondly, orthotropic timber engineered wood materials such as LVL were used to strengthen the joints. Energy dissipation could be provided at the connection level or between the wall and the floor. Timber buildings with such features were tested, designed, and built in New Zealand such as the EXPAN/Structural Timber Innovation Company building on the campus of University of Canterbury [16] and the two-story building of College of Creative Arts on Massey University's Wellington campus. Thirdly, proposed by the researchers at Swiss Federal Institute of Technology in Zurich (ETH), the beam-to-column connection fabricated with spruce glued laminated timber (glulam) was specially reinforced with hardwood made of European ash (i.e. Fraxinus excelsior), strengthening the compression zones perpendicular to grain of the columns as well as the bottom parts of the beams close to the columns. In such design, the only steel element required was the steel tendons providing the re-centering force to the connection. Tests were performed on small clear wood specimens to estimate the stiffness of the beam-to-column interface [20]. The rotational stiffness and the failure mode of the connections were also investigated [21]. In addition, a one-story post-tensioned timber frame structure was constructed and tested at ETH [22]. In order to facilitate such technology in China, the rotational performance of such self-centering connections with relatively low grade glulam was studied, and disc springs were used to mitigate the loss of pretension force in the post-tensioned steel strand [23]. However, the post-tensioned timber frame structures are relatively new to the timber family and have limited numbers of implementations compared with other timber structures around the world. Only finite amount of studies were done to evaluate the performance of post-tensioned timber structures at the system level. Consequently, the current post-tensioned timber structural designs are not well supported by any code or design software. In addition, very few existing studies have compared the performance between the conventionally designed timber structures with a post-tensioned earthquake resilient timber structure. As the trend in performance-based design criteria and objectives towards low-damage design philosophy and technologies are urgent, a system level study of the post-tensioned earthquake resilient timber structures becomes highly necessary. In this study, a prototype post and beam timber structure with lateral braces and its structural resilience improved counterpart using post-tensioned beam-to-column connections are proposed. The seismic behaviors of the two structural solutions are compared systematically. Performances are evaluated for multiple engineering demand parameters (EDP) such as the peak inter-story drift (PD), the residual interstory drift (RD), and a comprehensive structural performance index. Moreover, the seismic resilience capacity of timber structures are demonstrated by fragility analysis. From the provided seismic design and performance evaluation, the benefits and challenges of such self-centering post-tensioned earthquake resilient timber structures are clearly described.

Fig. 1. Architectural sketch of “Rubik's cube” timber structure.

evacuation [27], and are highly compatible with the post and beam timber structural systems. For example, Xiong and Liu [28] performed the monotonic tests and cyclic tests of a bolted glulam timber frame structure with different kinds of braces such as K-brace, X-brace, and knee-brace. The results showed that the K-braces and the X-braces significantly increased the lateral stiffness of the tested frames. This section first provides an architectural design of a prototype timber building. Then, two structural solutions are selected for the building. The lateral resistance of the first solution is provided by the timber brace system. In addition, the design of the post-tensioned selfcentering timber frame is provided with post-tensioned glulam frame members and the light frame in-fill shear walls. 2.1. Prototype timber building To investigate the seismic performance at the structural level, the timber structural solutions are provided for a building hypothetically designed in Sichuan province, which is a seismic active zone in the central west part of China. The hypothetical 3-dimensional (3D) architectural layout of the building is depicted in Fig. 1. The building is named as the “Rubik's cube” building as each of the 3 stories consists of a 3 bay by 3 bay plane with 6000-mm and 5000-mm span in one direction and three 5000-mm spans along the other direction. The flooring considered for the building consisted of solid timber panels covered by a grout and screed layer followed by architectural finish. The final floor could contain a rooftop sightseeing platform consisting of waterproofing measures and leveling concrete. A glass facade (and balustrade in the case of the roof level) was considered to surround the building. The staircase and the elevators are arranged in the central part of the building. The same architectural design is designed with two different structural systems, i.e. the prototype timber brace (TB) solution and the post-tensioned timber (PTT) solution. This study places the emphasis on the “structural” elements, which are generally sensitive to the structural inter-story drifts. Finishes such as pipelines, doors, windows, trims, cladding, and so on were less considered in this study.

2. Prototype buildings and seismic design procedure The post-tensioned self-centering timber frames are relatively new. Not many existing timber buildings were built with post-tensioned selfcentering features. On the other hand, the conventional light frame timber buildings are commonly seen timber structures. They are usually supported with the traditional plywood sheathed shear walls that are connected with metal fasteners such as nails, screws, rivets or bolts [24–26]. Furthermore, the timber brace system is also a commonly adopted solution to improve the lateral resistance of timber structures. Compared with the frames in-filled with timber shear walls, the braced frames release more architectural space, provide faster occupant

2.2. Structural design with brace system The structural design of the TB building was strictly carried out according to Chinese codes including GB 50005-2003 and GB/T 507082012 that focus on timber structural systems [29,30]. The glulam ranks TCYD21, which is a popular wood product in China whose material properties are shown in Table 1. In the table, fm stands for the flexural strength, fc stands for the compressive strength parallel to the grain, and ft stands for the tensile strength parallel to the grain. The design considered both vertical loads and lateral loads 2

Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx–xxx

Z. Shu et al.

Table 1 Material properties of wood. Glulam

Density (kg/m3)

Elastic modulus (N/mm2)

fm (N/mm2)

fc (N/mm2)

ft (N/mm2)

TCYD21 GL32h

495 500

9500 14,200

21 32

18 32

13 25.6

conforming to GB 50009-2012 [31]. The value of 1.0 dead load plus 0.5 live load was defined as the mass source. The dead load contains two parts. The beams, columns, and the brace elements have a total weight of approximately 420 kN. In addition, the weight densities for the floor, the roof, and the wall loads are summarized in Table 2. The live load is 2.5 kN/m2 at the floor levels and 2.0 kN/m2 on the roof. Finally, the snow load is a constant at 0.25 kN/m2. The lateral loads on the structure are composed of wind load of 0.30 kN/m2 acting on the external walls and earthquake action. The earthquake loads are further introduced in the subsequent part of the paper. To provide lateral resistance for such post and beam timber structure, 24 K-braces were set in the middle bay of each planar frame. Fig. 2(a) shows the plan view with dimensions in the unit of mm. During design, a 3D linear structural model was established with SAP2000 software [32]. The model (shown in Fig. 2(b)) estimates that the first two modes of structure are the lateral motions along two horizontal directions with similar periods around 0.35 s. The third mode, which is the torsional mode, has a period of 0.31 s. In the SAP2000 model, the lateral forces in the structure were completely resisted by the brace system assuming hinged frame joint behaviors. The left portion of Table 3 lists the component design of the three-story timber braced frame. The braces were designed as axially loaded members. The design value of compressive strength and tensile strength parallel to the grain of braces for story 1 and 2 were 1108 kN and 800 kN. The braces on the third story had a compressive strength parallel to grain of 691 kN as well as a tensile strength parallel to grain of 499 kN. The slenderness ratios of the braces from the bottom to the top of the structure were 23.4, 19.9 and 27.4, which were kept within the allowable ratio of 50. The timber braces are less likely to buckle compared with the steel braces, which usually need to be restrained against buckling e.g., [33].

Fig. 2. The plan view and the 3D view of the TB structural solution.

European code requirements as in [35], was selected for the PTT solution. Notice that the ultimate joint rotation under major earthquake was adopted as 0.075 rad. High-strength steel tendon, conforming to Chinese Code for Design of Steel Structures [36], with a maximum tensile strength of 1770 MPa was used to provide post-tension force for the joints. Table 4 gives the story shear and the moment resisting demand of each post-tensioned beam-to-column connection under major seismic hazard level with an inelastic spectrum scaling factor of 2.0 according to [37]. The ultimate moment capacity of the joint was determined using the Modified Monolithic Beam Analogy (MMBA) as proposed in [38]. The MMBA considers a global compatibility condition by assuming that the displacement of a post-tensioned system is equal to that of an equivalently reinforced monolithic connection when subjected to the same lateral load. However, upon decompression, the post-tensioned system will have additional displacement/rotation due to the initiation of the gap between the column face and the beam end. Thus, MMBA has been divided into three regions: before decompression, between decompression, and wood yielding in compression. Fig. 3 gives the sketch of the post-tensioned joint and the MMBA parameters. θcon is the rotation at the beam-column interface; c is the neutral axis depth (the point at which the compression zone between the beam and the column ends); ypt is the distance between the extreme fiber of the beam and the posttensioned element centerline; ys,t is the distance between the extreme bottom fiber and the dissipative tension reinforcing; and ys,c is the distance between the extreme bottom fiber and the posttension element Fig. 4 demonstrates the design flowchart for the post-tensioned selfcentering timber joints. The initialization of the design procedure requires the estimation of the cross-sectional sizes of the beam/column members. It has been noted from previous research that because wood has very low stiffness and strength when loaded perpendicular to the grain, the bearing behavior of the column face against the beam end has to become a limiting consideration for the load-resisting performance of the posttensioned timber connections. It is recommended by Wanninger et al. [20] that the compressive stress on the column face induced by the post-tensioned steel tendon should be less than 3.0 MPa, and the initial post-tension force in the steel tendon (Tpt,initial) can be approximately determined. The decompression moment (MA) can be simply obtained as Tpt,initial× (h/6). Then, according to the ultimate joint

2.3. Structural design with self-centering beam-column joints The design of the lateral load resisting system of the PTT self-centering frame can be divided into two parts: the determination of horizontal actions, and the sizing of beams, columns, post-tensioning elements, and reinforcements in order to resist these actions. The earthquake-induced horizontal action can be determined according to the response spectrum in Chinese Code for Seismic Design of Buildings [34]. Different from the TB building, the column of the PTT solution needs to be stiff enough to “elastically” transfer the moment demand to the beam-to-column joints. Consequently, the sizes of the columns and beams of the PTT solution are larger than those of the TB solution (as shown in Table 3). GL32h glulam, which is in accordance with the Table 2 Floor, roof, and wall loads of the braced model (unit: kN/m2). Floor loads Vitrified tile layer (8 mm–10 mm thickness) Cement mortar layer (20 mm thickness) Fine ballast concrete layer (40 mm thickness) Composite floor of light wood structure

Roof loads 0.55 0.40 1.00 0.54

External wall

Fine ballast concrete layer (40 mm thickness) Cement mortar layer (10 mm thickness) SBS waterproof layer(3 mm thickness) Cement mortar layer (20 mm thickness) XPS insulating layer(40 mm thickness) Fine ballast concrete layer (20 mm thickness) Composite floor of light wood structure

2.49

3

1 0.2 0.1 0.4 0.05 0.5 0.54 2.79

Red cedar cladding Counter batten Wood keel OSB board Gypsum board Others

0.2 0.35 0.11 0.15 0.15 0.03 0.99

Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx–xxx

Z. Shu et al.

Table 3 Cross section dimensions of structural members. TB building

Cross section (mm)

PTT building

Cross section (mm)

Column at floor 1 Column at floor 2 Column at floor 3 (roof) Beam at floor 1 Beam at floor 2 Beam at floor 3 (roof) Brace at floor 1 Brace at floor 2 Brace at floor 3 (roof)

400 × 400 400 × 400 300 × 300 400 × 300 400 × 300 400 × 300 280 × 220 280 × 220 240 × 160

Column at floor 1 Column at floor 2 Column at floor 3 (roof) Beam at floor 1 Beam at floor 2 Beam at floor 3 (roof) Shear wall at floor 1 Shear wall at floor 2 Shear wall at floor 3 (roof)

600 × 600 600 × 600 450 × 450 600 × 400 550 × 350 450 × 300 38 × 140 dimensional lumber sheathed by 12 thickness OSB

are arranged in the middle bay of the timber frame. The in-fill light frame wood shear walls consist of a wood frame made of dimension lumber, and sheathing panels made of oriented strand board (OSB), and the sheathing panels are connected to the wood frame by nails. The wood frame is fabricated with dimension lumber with 38 mm × 140 mm in cross section. The distance between adjacent lumber is 400 mm. The thickness of sheathing panels are 12 mm. In addition, the panels are attached to the framing members with 82 mm long× 3.8 mm diameter spiral nails at 150 mm on center along the edges and at 300 mm on center along the intermediate supports.

Table 4 Story shear and moment demands under major earthquakes. No. of story

Story height (m)

Story shear (kN)

Total moment demand (kN-m)

Moment demand for single connection (kNm)

1 2 3

4.5 3.6 3.6

697.28 427.13 166.33

1743.20 854.25 332.65

290.53 142.38 55.44

rotation (θimp), the neutral axis depth is estimated and the post-tension force in the steel tendon (Tpt) can be updated as Tpt,initial + ΔTpt. According to the deformation mode of the joint (θimp), the increase of the post-tension force in the steel tendon (ΔTpt) and the tensile (C) / compressive (T) forces in the steel angles are obtained. Finally, the sectional equilibrium should be checked and the moment resisting capacity of the joint should be larger than the moment demand, otherwise the cross-sectional size of the beam / column members should be adjusted. Finally, the design parameters of the self-centering beamcolumn joint in this study are summarized in Table 5. It is noted from Table 5 that the ultimate moment capacity of the joints is bigger than the moment demand as in Table 4. However, GB 50011 [32] recommends that in seismic prone zones, structures should be designed with dual lateral load resisting systems to ensure structural robustness under major earthquakes. In this study, light wood frame shear walls are introduced as in-fill walls to increase the lateral resistance of the post-tensioned timber frame system. Such combination has been widely adopted in practical engineering for conventional timber frame systems as discussed in [26]. The in-fill wood shear walls

3. Numerical models 3.1. Calibrated numerical structure model The above study shows that the two directional properties of the building share similar features. In addition, as the diaphragm is relatively rigid and the structural mass is evenly distributed throughout the plan, the lateral torsion mode is not dominating the overall structural behavior. Therefore, 2D models were established in for this study. The simplified models consider the second internal bay along the longer span of structure. The finite element models were developed for the sample structures within the OpenSees software platform [39]. The timber material properties were modeled according to Table 1. The timber beams, columns, and braces were modeled by two different types of elements. The plastic zones were modeled by the “zeroLength” element with the “Pinching4” uniaxial material. A “zeroLength” element defines two nodes at the same location. The nodes could be assigned with multiple

Fig. 3. MMBA parameters of a post-tensioned joint. 4

Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx–xxx

Z. Shu et al.

Fig. 5. Experiments that were used to calibrate the parameters of the numerical models.

starts from a close to zero value due to the gap around the bolts. Then, after the full contact between the bolts and the timber, the rotational resistance increases, leading to a semi-rigid situation. In the study, the “Pinching4” uniaxial material model is used to consider such rotational stiffness and the model parameters were accurately calibrated against the existing experimental data [41]. Besides, the performance of the timber braces is heavily influenced by the axial stiffness of the member, which is tested according to Fig. 5(d). For the self-centering joints, the rotational performances were also calculated and verified by the existing studies [19,23]. The experimental investigation of such joints are shown in Fig. 5(e) [16,27]. In this study, as the shear wall is in-filled to a self-centering timber portal frame. The timber shear walls were widely studied in the past. As an example, the parameters of the shear wall model are sampled based on the experimental study by Xiong and Liu [28], which is shown in Fig. 5(b). Besides, the middle portions of these elements were modeled by displacement-based beam/column elements with the uniaxial bilinear flexural material behaviors considering kinematic hardening property. Five-point Gauss-Lobatto integration was assigned to the elements to increase modeling accuracy. Inelastic sections were used to aggregate the axial, the flexural, and the torsional stiffness. Within the OpenSees model, the “corotational” geometric transfer is selected for the beams and columns considering larger deformation against major earthquake ground motions. The self-weight and component behavior of the models were carefully calibrated to represent real structures. The first three modal periods of the 2-dimensional (2D) TB building is 0.33 s, 0.097 s, and 0.049 s. Meanwhile, the first three modal periods of the 2D PTT building is 0.387 s, 0.076 s, and 0.031 s. The first mode of the two structural solutions are shown in Fig. 6. As the column splices were strengthened and the column sizes were enlarged, the first mode of the PTT solution share similar inter-story drifts. For the TB solution, on the other hand, the major deformation is observed over the first two stories.

Fig. 4. Design flowchart for post-tensioned self-centering timber connections. Table 5 Design parameters for the self-centering beam-to-column joints. No. of story

Tpt,initial (kN)

Post-tension tendon area (mm2)

Decompression moment MA (kN-m)

Ultimate moment capacity MB (kN-m)

1 2 3

500 250 200

570 285 285

50.00 22.92 15.00

434.32 178.78 119.69

material properties to represent the force-deformation relationship of the element. The “Pinching4” uniaxial material constructs a uniaxial material that represents a pinched load-deformation response and exhibits stiffness and strength degradation under cyclic loading protocol. It is a 16-parameter model that fully captures the details of the hysteretic performance of a timber structural system [26]. The axial stiffness of the timber braces and the rotational stiffness of all the regular beam-to-column connections (plus the column foot elements) were modeled with such behavior calibrated with the tested results [28]. Fig. 5 shows the test setups of some existing studies. Fig. 5(a) shows a typical timber portal frame with the slotted-in steel plate timber beam-to-column connections [28]. For such connections (shown in Fig. 5(c)), the rotational stiffness is relatively small such that these joints are commonly considered as hinges during design. Nonetheless, such rotational stiffness could slightly influence the system behavior against medium or large earthquakes. Therefore, scholars have started to define and estimate such rotational stiffness [40] (Chapter 12). Usually, the rotational stiffness for slotted-in bolted timber connections 5

Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx–xxx

Z. Shu et al.

occur as the structural damage is low and associated with the selfcentering mechanism. The “post-yielding” stiffness of the PTT frame is dominated by the post-tensioned steel strands and the steel angles. Shown from the pushover analysis, the “post-yielding” behavior is stable over a wide range after the gap opens at the connection level. Furthermore, such design is valid since VDL is larger than the designed base shear. Fig. 6. First mode shapes for two structural solutions.

3.3. Example cases To link the component level behavior with the system level behavior, this section presents two example cases for the sample structures. The earthquakes were chosen such that the larger nonlinearities happen for the two cases. In Fig. 8, the TB frame is excited by the 1971 San Fernando earthquake (magnitude 6.6) with a reverse fault mechanism measured at Pacoima Dam. First, the force-displacement behavior of the column foot connection and the brace body are provided at the bottom of the figure. The moment-rotation curve of the column foot joint shows that such joint provides limited resistance. The axial force and elongation curve of the brace body is a pinched behavior, which is typical for the timber elements due to the interaction between the nails and the wood. In addition, the story shear and inter-story drift curves are provided. By comparing the behaviors of the brace with those of the corresponding story, it could be concluded that majority of the lateral force is resisted by the brace system for such design. Similarly, shown in Fig. 9, the PTT frame is excited by another strong motion, which is the 1989 Loma Prieta earthquake (magnitude 6.9) with a reverse-oblique fault mechanism measured at Los Gatos Lexington Dam. For such system, the story shear force - drift behavior is influenced by a more complex system including the in-fill light wood shear walls, the timber columns, and the beam-to-column self-centering connections. Moreover, as the columns are stiff and their splices are strengthened, the motion of the top story is also influenced by the two lower stories.

Fig. 7. Static monotonic pushover curves for two structural solutions.

3.2. Pushover analyses Static pushover analyses were numerically performed by using a displacement-control loading protocol. The lateral load distribution was calculated based on weight distributions along building height. The 2D monotonic pushover curves for each building model are given in Fig. 7. The points corresponding to the first strength degradation and the achievement of the design-level inter-story drift limit at any level are indicated on each curve. Note that the first strength degradation for TB frame is mainly due to the damage of the braces while the first strength degradation for the PTT frame is caused by the gap opening mechanism of the self-centering beam-to-column connections. One could notice that the TB frame is much stiffer than the PTT frame, which is correct as the PTT frame needs a lower elastic stiffness to initiate the low damage rocking behavior. Table 6 summarizes the basic features of the building specimens. The design base shear (Vd) is equal to 25% of the total design base shear assuming each frame takes 25% of the seismic loads. Specifically, Vd were determined using the modal response spectrum method. In addition, the base shear at first strength degradation point (Vy1), the base shear at full yielding Vy, and the base shear when the codified story drift limit (DL) is achieved in any story VDL are also provided in Table 6. The value of Vy is defined as the shear force on the pushover curve correspond to the yield point obtained at the intersection between the initial stiffness portion of the curves and the post-yield stiffness portion. Furthermore, two different ratios of base shear to seismic weight are also provided in the table, i.e. one for the design base shear and one for the base shear at full yielding. These ratios show that the TB frame is designed to be approximately 15% stronger than the design strength. As the first strength degradation happens when the rocking motion starts at the joint level of the frame, Vy1 and Vy are both smaller than the design base shear. Nonetheless, one would desire such behavior to

4. Evaluation framework of seismic structural performance 4.1. Selected earthquakes A sufficient number of earthquake records need to be selected for the fragility analyses to obtain conceptually and statistically better building response predictions. The building is designed for the seismic active zone - Sichuan province in China, considering degree 9, group 2, and site II according to the design code GB 50011-2010 [34], the PGA value is 0.40 g and the characteristic period is 0.40 s. 50 strong motions were selected from the historical earthquake records collected from the Pacific Earthquake Engineering Research Center's Next Generation Attenuation (NGA) database. For the details of the earthquake, one could refer to the 25 acceleration pulse-type motions and the first 25 velocity pulse-type motions presented by Tang and Zhang [42]. Most of these ground motions belong to the Maximum considered earthquakes (MCE) associated with a 2% probability of exceedance in 50 years. The average PGA of the 50 ground motions is 0.44 m/s2, which is close to the code specified magnitude. In addition, the average return periods for minor, moderate and major earthquakes are 50, 475, and 2475 years, which are in accordance with 50-year exceedance probabilities of 63%, 10% and 2%, respectively. In China, the acceleration

Table 6 Summary of building frame characteristics. Structure

T1 (sec)

T2 (sec)

W (kN)

Vd (kN)

Vd/W

Vy1 (kN)

Vy (kN)

Vy/W

VDL (kN)

TB PTT

0.329 0.387

0.097 0.076

1099 1145

410 323

0.373 0.312

405 178

471 223

0.429 0.195

649 397

6

Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx–xxx

Z. Shu et al.

Fig. 8. Component and story level responses of TB frame against 1971 San Fernando earthquake.

Fig. 9. Component and story level responses of PTT frame against 1989 Loma Prieta earthquake. 7

Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx–xxx

Z. Shu et al.

about 50%, 45% and 5%, for the first, second, and roof story, respectively. However, the importance factor of the three stories are slightly adjusted according to the structural concerns and the architectural configurations. First, in order to avoid the collapse of the building due to the severe damage at the first story, the importance factor of the first story is slightly increased. In addition, according to the architectural design, there are less human and assets on the second floor, reducing its importance factor. Furthermore, the importance factor of the top level is increased as a rooftop sightseeing platform is designed. Consequently, to calculate the building level PADI, the final array of importance factors for the first, second, and third story is 60%, 25%, and 15%. Nonetheless, the authors found the use of such index limited when evaluating the self-centering timber structures as the structural damage is usually small even when large PD occurs. The post-tensioned design drives back large inter-story drift to the original position and the seismic energy is usually dissipated by the steel angles and the light frame timber shear walls which are cheap and could be easily replaced after a major earthquake event. Although from the IDA approach, a small β value could be usually obtained, the mitigation of the residual drift could not be well captured from such index.

spectra corresponding to the minor (frequent) earthquakes are mainly used to design the structure. The moderate level earthquakes are used for the design of the structural details. And the major (rare) earthquakes are used to check and examine the structural performance once a nonlinear structural model is created. Therefore, the 50 selected historical earthquake excitations were scaled to two different earthquake ensembles (i.e. 50 minor level earthquakes and 50 major level earthquakes). The average PGAs of the two earthquake ensembles equal to 0.14 m/s2 and 0.62 m/s2, according to [34]. As the ground motions are characterized by the user-defined IMs, the choice of IM plays a crucial role in both running the fragility analysis and interpreting simulation results. With the suggestion by Mackie and Stojadinović [43], in a logarithmic reference frame, the linear consistency of the results from probabilistic analysis can be an indicator of the applicability of the IM, which is used to interpret the results. Based on this criterion, the PGA, the PGV, and the Sa(T1) (i.e. the spectral acceleration at the predominant structural period) are generally considered good choices of IM for relating to the EDP measures. This study chooses PGA as the ground motion IM based on their efficiency, practicality, sufficiency and hazard computability compared with other IMs [44]. Detailed comparison of the three IMs will be presented together with the fragility analysis results.

4.3. Fragility methodology 4.2. System level damage index Typically, a fragility curve defines the conditional probability of attaining or exceeding a (or several) specified damage state(s) (DS) for a given set of ground motion intensity measures. The fragility curves are commonly generated by the IDA or the probabilistic seismic demand analysis (PSDA) [49]. This paper adopts the PSDA method, where cloud approach (i.e. using un-scaled earthquake ground motions) was used for nonlinear time history analysis and the regression analysis was applied to obtain the mean and standard deviation for each limit state (LS) by assuming the correlation between the median EDP and an appropriately selected IM following Eq. (2):

The structural level damage indexes were developed and provided in addition to the peak inter-story drift to better evaluate the seismic performances of the proposed structures. For example, the Park-Ang damage model [45] are commonly chosen to evaluate the damage of civil structures caused by seismic loads, which has been used extensively for RC and steel structures in the past two decades. The ParkAng damage index (PADI) incorporates not only the peak structural drift, but also the total dissipated energy, which is expressed by Eq. (1):

PADI =

β Δm + Δu Fey Δu

∫ dE

(1)

EDP = a (IM )b

where PADI could describe the damage intensities of a structural component, a story, or a simplified single degree of freedom system; Δm is the maximum deformation during an external load; Δu is the ultimate deformation under monotonic loading determined experimentally; β is a calibrated parameter to manage the energy portion of the index; Fey is the equivalent yield force of the target; and ∫ dE is the total incremental hysteretic energy absorbed by the target during the earthquake. Assuming a tangent slope decrease, structural collapse could be evaluated by the incremental dynamic analysis (IDA) approach [46,47]. For timber structures, an approach based on the incremental dynamic analysis (IDA) to calibrate the damage model parameter (β value) for wood shear wall was developed and validated by experimental test data. To predict the potential damage to wood frame buildings under future earthquakes, the relationship between calculated damage index and observed building damage is established. Such damage index-based limit state criterion provides a technical basis for the development of performance-based seismic design and evaluation methodologies for wood structures. In this study, PADI is generated at both the component level and the system level to evaluate the seismic performance. At the story level, the ultimate deformation for each floor is considered 5.5% for the TB frame and 7.5% for the PTT frame. Moreover, the overall damage index of a building can be expressed as the sum of the damage index of each story (or major component) of the structure weighted by the corresponding importance factor. For example, for a single-story building, damage of the building is known to be correlated with the absorbed energy [48]. In this case, the energy contribution factor could be used as the importance factor. In this study, the “Rubik's cube” timber building has 3stories. According to the analysis, the energy contribution factors are

(2)

where the parameters a and b are regression coefficients obtained from the response data of nonlinear time history analyses. By taking the logarithmic of above equation, one arrives at Eq. (3):

ln (EDP ) = lna + bln (IM )

(3)

The provided equation is a linear regressed equation between ln (EDP) and ln(IM) based on the calculated response distribution. The regression analyses aims at the smallest standard deviation ξEDP | IM , which could be estimated by Eq. (4): n

ξEDP | IM =

∑i = 1 [ln (EDPi ) − (lna + bln (IMi )]2 n−2

(4)

Subsequently, a capacity model uses the EDPs or functions of EDPs to derive the damage index (DI) that can be compared with the LSs correspondent to various dictated DSs. For this study, the inter-story EDPs and the PADI were selected as the structural level damage indices. By further assuming a lognormal distribution of EDP at a given IM, the fragility functions (i.e. the conditional probability of reaching a certain damage state for a given IM) could be written as Eq. (5):

ln (LS ) − ln (aIM b) ⎞ ⎡ ⎤ p ⎢DI ≥ LS|IM⎥ = 1 − Φ ⎛⎜ ⎟ ξEDP | IM ⎝ ⎠ ⎣ ⎦

(5)

where ξEDP|IM is the standard deviation of the logarithmic distribution computed from Eq. (4) and Φ(•) is the standard normal distribution function. The fragility function could be easily realized as it follows the standard cumulative normal distribution. 8

Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx–xxx

Z. Shu et al.

Fig. 10. Average responses against major and minor earthquakes for two structural solutions.

5. Seismic responses of two structural systems Fig. 11. Linear regression of logarithmic IM to EDP correlations.

5.1. Average responses against major and minor earthquakes frames could be concluded from the figure, i.e. they avoid damage concentration, completely eliminate the RD, and provide fast and cheap post-earthquake structural restoring capacity.

The overall seismic responses of the two structural solutions are first presented in Fig. 10, where the peak inter-story drift, the residual interstory drift, and the PADI for each story are plotted along the elevation of the building. The individual responses of the minor earthquakes (scale factor 0.32) and the major earthquakes (scale factor 1.40) are presented with the average values highlighted. Inspecting the inter-story drifts of the two frames, it could be seen that the PDs are lumped at the first two stories for the TB frame and are more evenly distributed throughout the elevation of the PTT frame due to the strong column settings. The average responses of the PTT frame (i.e. 0.9% against the major earthquakes) are smaller than those of the TB frame (i.e. 1.15% against the major earthquakes). In addition, the individual cases show that the extreme PD against the major earthquakes for the PTT frame just satisfied the design story limit of 2.5% whereas the TB frame exceeded the limit and reached 3%. Meanwhile, the PTT frame almost eliminated the RDs against all the earthquakes at all the story levels. For the TB frame, however, the RDs are much larger with an average value of 0.43% and the largest value exceeding 2% at the first story against the major earthquakes. The PADI for each stories generally matches with the PD plots to some extent. It could be noticed that for a few extreme events, the PADIs at the first story reached 1.0 for the TB frame, indicating the complete damaged case at the first story, which is undesirable. For the PTT frame, although it is more likely to initiate the rocking motion at the connection level, the PADI for all the cases are below the complete damaged threshold. The presented results show that the post-tensioned self-centering timber frames provide resilience structural solutions. However, they are not “damage-free” systems: steel angles and light-frame shear walls need to be examined and replaced upon damage after a major earthquake. Furthermore, as such systems usually have weaker lateral resistance over smaller inter-story drift ranges, larger lateral responses might occur against minor earthquakes. Comprehensive considerations shall also include the protection of nonstructural components, which are sometimes acceleration sensitive or vulnerable to earthquakes within small levels of lateral drifts. Finally, advantages of the PTT

5.2. Linear regressions of probabilistic seismic demand analysis The linear fit of the coefficients for the fragility analysis of the TB frame are first provided in Fig. 11 on the logarithmic IM-EDP curves. Shown from the figure, the RD curves have the largest standard deviations compared with the other two curves. In addition, it is proved from the figure that the PGA is a better IM for the sample structures as the corresponding standard deviations are smaller among the three selected IMs. 5.3. Peak and residual drifts Subsequently, the PD fragility curves for the TB frame (left) and the PTT frame (right) is provided in Fig. 12(a). Four damage levels are selected for the PD fragility analyses, i.e. 0.15%, 0.5%, 1%, and 2.5%. The first PD damage state is the state when the story structural members just yielded for the TB frame (i.e. p[PD ≥ LSy1|PGA]). In addition, the last damage state corresponds to the code drift limit state (i.e. p [PD ≥ LSDL|PGA]). It could be noticed that the PTT frame is more likely to exceed the first damage state due to the weaker initial stiffness shown from above sections. However, the exceeding probability of the other three states are smaller than the TB frame. This is because in the PTT frame, the strong columns re-distributed the earthquake forces to the three stories. While setting RD as DI, the PSDA approach needs to be slightly modified as some RD values are near 0, which generates huge errors during the logarithmic regression procedure. The updated approach to generate the RD fragility curves include two steps: Step 1, filter out the cases when the single story exceeds the first yielding point. The residual drifts are all zero for all the filtered cases. Step 2, perform the linear regression analysis, which is more accurate for the yielded cases (i.e. p [RD ≥ DSi,y|PGA] where stands for the ith DS for the yielded cases). Then, the RD fragility is generated using the conditional probability 9

Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx–xxx

Z. Shu et al.

Fig. 12. Fragility curves for the peak and residual inter-story drifts.

5.4. Park-Ang damage index

function expressed by Eq. (6).

p [RD ≥ DSi |PGA] = p [RD ≥ DSi, y |PGA]⋅p [RD ≥ DSy1 |PGA]

(6)

The PADI fragility curves of the TB and PTT frame are provided in Fig. 13. It is shown that the PTT frame is more likely to exceed the four defined levels of PADI. Although the average PD in the first story of the PTT frame is smaller than that of the TB frame, the overall damage is considered larger as the re-distribution of the seismic force increased the average story drifts of the second and third stories. It is proved again from the fragility curves that the PADI as a system level damage indicator got limitations when evaluating the self-centering timber structures. The structural damage associated with the PTT frame shall not be determined only by the PD and the total dissipated energy, but

The RD fragility curves are provided in Fig. 12(b). It could be easily noticed that the PTT frame (on the right) almost eliminated the interstory residual drifts while the prototype TB frame (on the left) according to current design approaches end up with more significant numbers of RD. This is the main beauty of the PTT solution as zero residual drift significantly reduces the seismic disaster, as well as facilitates the post-earthquake restoration process.

Fig. 13. PADI fragility curves for two timber solutions. 10

Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx–xxx

Z. Shu et al.

also by the RD, the reparability, and other resilience related features.

[3] Kramer A, Barbosa AR, Sinha A. Performance of steel energy dissipators connected to cross-laminated timber wall panels subjected to tension and cyclic loading. J Struct Eng 2016;142(4):E4015013. [4] Hashemi A, Masoudnia R, Quenneville P. Seismic performance of hybrid self-centring steel-timber rocking core walls with slip friction connections. J Constr Steel Res 2016;126:201–13. [5] Li Z, He M, Dong H, Shu Z, Wang X. Friction performance assessment of non-asbestos organic (NAO) composite-to-steel interface and polytetrafluoroethylene (PTFE) composite-to-steel interface: experimental evaluation and application in seismic resistant structures. Constr Build Mater 2018;174:272–83. [6] Li Z, Dong H, Wang X, He M. Experimental and numerical investigations into seismic performance of timber-steel hybrid structure with supplemental dampers. Eng Struct 2017;151:33–43. [7] Li Z, He M, Wang X, Li M. Seismic performance assessment of steel frame infilled with prefabricated wood shear walls. J Constr Steel Res 2018;140:62–73. [8] He M, Luo Q, Li Z, Dong H, Li M. Seismic performance evaluation of timber-steel hybrid structure through large-scale shaking table tests. Eng Struct 2018;175:483–500. [9] Huang H, Chang W. Seismic resilience timber connection-adoption of shape memory alloy tubes as dowels. Struct Control Health Monit 2017;24:e1980. [10] Fang C, Yam MCH, Lam ACC, Xie L. Cyclic performance of extended end-plate connections equipped with shape memory alloy bolts. J Constr Steel Res 2014;94:122–36. [11] Fang C, Wang W, He C, Chen YY. Self-centring behaviour of steel and steel-concrete composite connections equipped with NiTi SMA bolts. Eng Struct 2017;150:390–408. [12] Jia L, Li R, Xiang P, Zhou D, Dong Y. Resilient steel frames installed with selfcentering dual-steel buckling-restrained brace. J Constr Steel Res 2018;149:95–104. [13] Shu Z, Ma R, He M. Comprehending the ductile behavior of slotted bolted connections. Struct Des Tall Spec Build 2017;26(3):e1309. [14] Palermo A, Pampanin S, Fragiacomo M, Buchanan A, Deam B. Innovative seismic solutions for multi-storey LVL timber buildings. In: Proceedings of the 9th world conference on timber rngineering; Portland, USA; 2006. [15] Buchanan A, Deam B, Fragiacomo M, Pampanin S, Palermo A. Multi-storey prestressed timber buildings in New Zealand. Struct Eng Int 2008;18(2):166–73. [16] Newcombe MP, Pampanin S, Buchanan A, Palermo A. Section analysis and cyclic behavior of post-tensioned jointed ductile connections for multi-story timber buildings. J Earthq Eng 2008;12(1):83–110. [17] Iqbal A, Pampanin S, Buchanan AH. Seismic performance of full-scale post-tensioned timber beam-column connections. J Earthq Eng 2016;20(3):383–405. [18] Smith T, Pampanin S, Di Cesare A, Ponzo FC, Simonetti M, Nigro D, Carradine D. Shaking table testing of a multi-storey post-tensioned timber building. In: Proceedings of New Zealand society for earthquake engineering (NZSEE) conference; Wairakei, New Zealand; 2014. [19] Di Cesare A, Ponzo FC, Nigro D, Pampanin S, Smith T. Shaking table testing of posttensioned timber frame building with passive energy dissipation systems. B Earthq Eng 2017;15(10):4475–98. [20] Wanninger F, Frangi A, Steiger R. Bearing stiffness in wood-to-wood compression joints. Eng Struct 2015;101:631–40. [21] Wanninger F, Frangi A. Experimental and analytical analysis of a post-tensioned timber connection under gravity loads. Eng Struct 2014;70:117–29. [22] Leyder C, Chatzi E, Wanninger F, Frangi A. Dynamic response of an innovative hybrid structure in hardwood. Proc Inst Civ Eng Constr Mater 2015;168(3):132–43. [23] Li Z, He M, Wang K. Hysteretic performance of self-centering glulam beam-tocolumn connections. J Struct Eng 2018;144(5):04018031. [24] Seaders P, Miller TH, Gupta R. Performance of partially and fully anchored woodframe shear walls under earthquake loads. For Prod J 2009;59(5):42–52. [25] van de Lindt JW, Pei S, Pang W, Shirazi SMH. Collapse testing and analysis of a light-frame wood garage wall. J Struct Eng 2011;138(4):492–501. [26] Shu Z, He M, Li Z. Estimating inelastic drift demands for wood portal frame systems considering accumulated damages. Struct Des Tall Spec Build 2017;2:e1440. [27] Li S, Yu X, Zhang Y, Zhai C. A numerical simulation strategy on occupant evacuation behaviors and casualty prediction in building during earthquakes. Phys A: Stat Mech Appl 2018;490:1238–50. [28] Xiong H, Liu Y. Experimental study of the lateral resistance of bolted glulam timber post and beam structural systems. J Struct Eng 2016;142(4):E4014002. [29] GB 50005, Code for Design of Timber Structures. Ministry of Construction of the People’s Republic of China, Beijing, China (in Chinese); 2003. [30] GB 50708, Technical code of glued laminated timber structures. Ministry of housing and urban‐rural development & general administration of quality supervision, inspection and Quarantine, People’s Republic of China. Beijing, China (in Chinese); 2012. [31] GB 50009, Load code for the design of building structures. Ministry of housing and urban‐rural development & general administration of quality supervision, inspection and Quarantine, People’s Republic of China. Beijing, China (in Chinese); 2012. [32] SAP. Computers and Structures, inc; 2000. [33] Qu Z, Kishiki S, Sakata H, Wada A, Maida Y. Subassemblage cyclic loading test of RC frame with buckling restrained braces in zigzag configuration. Earthq Eng Struct Dyn 2013;42:1087–102. [34] GB 50011, Code for seismic design of buildings. Ministry of housing and urban-rural development of the People’s Republic of China (MOHURD), Beijing, China. (in Chinese); 2010. [35] CEN/TC 124, Timber structures – glued laminated timber and glued solid timber – [FprEN 14080-2011]. Paris, France; 2011. [36] GB 50017, Code for design of steel structures. National standard of the People’s Republic of China (NSPRC), Beijing, China. (in Chinese); 2003.

6. Conclusions The paradigm shift towards low-damage seismic design philosophy and technologies is an urgent one. For the earthquake resilient timber structures, though the concept has been proposed by many researchers, very few studies have evaluated the seismic performance of the posttensioned timber frames at the building level. To facilitate more future implementations for the earthquake resilient timber structures, this paper presents detailed insights on the seismic design and performance of the low-damage post-tensioned self-centering timber frame structures. A prototype architecture with 3 stories and 3 bays was hypothetically designed in a high seismic zone in China. The post-tensioned self-centering timber frame structural solution was provided for the building whose self-centering connections were designed by a procedure clearly provided in the paper. Meanwhile, the same architectural design was provided with another comparable structural solution, which is a typical post and beam frame system enhanced with timber K-braces. Subsequently, the nonlinear numerical models were carefully created to represent the behavior of the real structures. The component behaviors such as the beam-to-column connections, the column foot joints, the axial behaviors of the brace system, and the self-centering connections were calibrated with the existing experimental results. Then, the static monotonic pushover analyses showed that the yield strength of the timber braced frame were approximately 15% stronger than the design strength. Meanwhile, the post-tensioned timber frame had a more stable behavior after the gap opens at the connection level. Finally, the seismic performance of the two structural solutions were systematically evaluated. The peak inter-story drift, the residual inter-story drift, and the Park-Ang damage index at the system level were investigated for the two structural solutions. Average responses as well as the fragility analyses were demonstrated in the study. The advantages of such post-tensioned self-centering timber frame structures were clearly captured as they could avoid damage concentration, eliminate the inter-story residual drifts completely, and provide fast and cheap post-earthquake structural restoring capacity. Challenges were also captured throughout the study. First, the design of the post-tensioned timber frames is slightly more complicated compared with existing timber structural solutions, especially given the current situation that no code or design software fully supports such design. Second, such self-centering timber structures are not “damagefree” systems: steel angles and light-frame shear walls still need to be fixed or replaced after a major earthquake. Furthermore, as such systems usually have weaker lateral resistance over smaller inter-story drift ranges, larger lateral responses might occur against minor earthquakes. Protection of nonstructural components might become necessary as they might be vulnerable to earthquakes within small inter-story drift levels. Acknowledgements The authors gratefully acknowledge National Natural Science Foundation of China (Grant No. 51608376 and Grant No. 51878476) and Chenguang Program (Grant No. 16CG18) by Shanghai Education Development Foundation and Shanghai Municipal Education Commission for supporting this research. References [1] Li Z, Zhou R, He M, Sun X. Modern timber construction technology and engineering applications in China. Proc Inst Civ Eng 2018. https://doi.org/10.1680/jcien.18. 00024. [2] Akbas T, Sause R, Ricles JM, Ganey R, Berman J, Loftus S, Dolan JD, Pei S, Lindt JWVD, Blomgren HE. Analytical and experimental lateral-load response of selfcentering posttensioned CLT walls. J Struct Eng 2017;143(6):04017019.

11

Soil Dynamics and Earthquake Engineering xxx (xxxx) xxx–xxx

Z. Shu et al.

[37] Sarti F, Palermo A, Pampanin S, Berman J. Determination of the seismic performance factors for post-tensioned rocking timber wall systems. Earthq Eng Struct Dyn 2017;46:181–200. [38] Tobias S. Post-tensioned timber frames with supplemental damping devices [Doctoral thesis]. Christchurch, New Zealand: University of Canterbury; 2014. [39] OpenSees (open system for earthquake engineering simulation), pacific earthquake engineering research (PEER) Center, Universityof California, Berkeley, CA, USA; 2017. [40] Porteous J, Kermani A. Structural timber design to eurocode 5. Blackwell publishing; 2007. [41] He M, Liu H. Comparison of glulam post-to-beam connections reinforced by two different dowel-type fasteners. Constr Build Mater 2015;99:99–108. [42] Tang Y, Zhang J. Response spectrum-oriented pulse identification and magnitude scaling of forward directivity pulses in near-fault ground motions. Soil Dyn Earthq Eng 2011;31:59–76. [43] Mackie KR, Stojadinović B. Fragility basis for California highway overpass bridge

[44]

[45] [46] [47] [48] [49]

12

seismic decision making. PEER Report 05/02, Pacific earthquake engineering research center. Berkeley, CA, USA: University of California; 2005. Padgett JE, Nielson BG, DesRoches R. Selection of optimal intensity measures in probabilistic seismic demand models of highway bridge portfolios. Earthq Eng Struct Dyn 2008;37(5):711–25. Park YJ, Ang HS, Wen YK. Damage-limiting aseismic design of buildings. Earthq Spectra 1987;3(1):1–26. Vamvatsikos D, Cornell CA. Incremental dynamic analysis. Earthq Eng Struct Dyn 2002;31(3):491–514. Shu Z, Li S, Gao M, Yuan Z. Development of seismic collapse capacity spectra for structures with deteriorating properties. Earthq Struct 2017;12(3):297–307. Liang H, Wen YK, Foliente G. Damage modeling and damage limit state criterion for wood-frame buildings subjected to seismic loads. J Struct Eng ASCE 2011;137:41–8. Zhang J, Huo Y. Evaluating effectiveness and optimum design of isolation devices for highway bridges using the fragility function method. Eng Struct 2009;31(8):1648–60.