Seismic control design for slip hysteretic timber structures based on tuning the equivalent stiffness

Engineering Structures 128 (2016) 199–214

Contents lists available at ScienceDirect

Engineering Structures journal homepage: www.elsevier.com/locate/engstruct

Seismic control design for slip hysteretic timber structures based on tuning the equivalent stiffness Wuchuan Pu a,⇑, Changcheng Liu a, Hang Zhang a, Kazuhiko Kasai b a b

Dept. of Civil Engineering, Wuhan University of Technology, China Urban Disaster Prevention Research Core, Tokyo Institute of Technology, Japan

a r t i c l e

i n f o

Article history: Received 8 August 2015 Revised 19 September 2016 Accepted 20 September 2016

Keywords: Hysteretic damper Seismic control Slip hysteretic model Timber structure

a b s t r a c t In this study, a new methodology for the direct displacement-based seismic control design of timber building structures with an idealized slip-hysteretic model is proposed. The structure under consideration is simulated by the shear beam model, and hysteretic dampers are used to improve the structural equivalent damping ratio and stiffness and thus to reduce the seismic responses. The equivalent damping ratio and stiffness are defined for an inelastic structural system based on steady-state hysteresis, and a spectrum-based response prediction and damper capacity design approaches are proposed for singledegree-of-freedom (SDOF) systems. A procedure to distribute the optimal damper capacity from the SDOF system to a multi-degree-of-freedom (MDOF) system is also presented. The distribution of the dampers is based on the criterion that the equivalent stiffness of the system is compensated by the hysteretic dampers to produce the optimal equivalent stiffness that is proportional to the story shear. Nonlinear time history analyses (NTHA) of four example buildings with assumed parameters and with added hysteretic dampers illustrate that the methodology is useful for the design of slip hysteretic structures. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Timber construction is attractive in architectural design, which has influenced the increased construction of multistory buildings in recent years. A 9-story residential timber building has been reported in the past [1], and much taller buildings are under construction or planned. On the other hand, experimental studies have also shown that traditional timber-framed structures have a very limited energy dissipation capability because of the ‘‘slip” phenomenon (also referred to as the ‘‘pinching” effect) or stiffness degradation in its hysteretic loop [2,3]. The slip phenomenon is accompanied by the degradation of strength and stiffness [4]. These hysteretic characteristics suggest a reduced aseismic performance of timber structures within the elasto-plastic range. Seismic control technology can be employed to increase the energy dissipation capability through supplemental damping devices and thus to improve the seismic performance of structures. The application of energy dissipation devices in structures has been recognized for a decade to be effective at suppressing seismic vibration or damage, as witnessed by extensive use in various types of building structures. In particular, the protection of both ⇑ Corresponding author. E-mail address: [email protected] (W. Pu). http://dx.doi.org/10.1016/j.engstruct.2016.09.041 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved.

the structural members and their contents has been incorporated into design criteria in recent years. Reducing the acceleration response has become as important as reducing the displacement; in such cases, energy dissipation technology is preferable to conventional approaches, which increase the structural member size to match the strength requirements. In Japan, old timber houses, particularly those that were constructed before the release of the so-called ‘‘New seismic code” in 1981, are required to be retrofitted by a certain engineering measure, including energy dissipation technology. Energy dissipation devices are also being adopted extensively in new timber constructions to attract customers in terms of the concept of property protection. In contrast to reinforced concrete (RC) or steel buildings, however, relatively less research has been conducted on the seismic control of timberframed buildings, which may be because timber structures, particularly high-rise buildings, have not been as popular as RC or steel buildings throughout the world in the past. No available manual or standard design procedures currently exist to guide engineering designers to manage the seismic control design of timber buildings. Research regarding the application of dampers in timber structures began in the 1990s, with hysteretic dampers being one of the typical types [5,6]. Filiatrault [7] performed an analytical study on a timber-frame building with added friction dampers, and the

200

W. Pu et al. / Engineering Structures 128 (2016) 199–214

numerical analysis showed that the added dampers could improve pinching hysteresis and thus increase the energy dissipation of the structural system. Much more research has focused on the application of dampers in timber buildings following the 1994 Northridge Earthquake and 1995 Hyogo-ken Nanbu Earthquake, which significantly damaged or collapsed timber-framed buildings. Higgins [8] proposed steel dampers, which only work when subjected to tension, and applied these devices in timber structures. Mizuno and Yamada [9] performed an experimental study on timber-frame structures with friction-type dampers. Lopez-Almansa et al. [10] proposed a seismic protection system, which included an outer steel frame with hysteretic energy dissipators, for timber buildings. The effective operation of a supplemental damping system depends on not only its mechanism and capacity but also its placement in the structure. The damper capacity design for a traditional low-rise timber house building (typically one or two stories) could be determined through a trial-and-error approach; however, such an approach would probably be time consuming when mid-rise buildings are considered. The design methodology for such multistory buildings should be theoretically established to provide engineers with a practical approach to seismic control. Moreover, the unique hysteretic characteristics of timber frames should be fully examined as the design procedure is established. In the recent past, the optimal design of dampers in structure has been one of the major concerns of seismic control, and many studies have been conducted on the subject. Ruiz et al. [11] developed a dynamic step-by-step analysis method that incorporated a trial-and-error procedure to minimize the damage to the structure and record the maximum nonlinear behavior in the hysteretic damper. Gluck et al. [12] developed a method that was based on the optimal linear control theory to determine the design of hysteretic dampers. Uetani et al. [13] developed a computer program that was based on the gradient projection algorithm to optimize the dampers. Benavent [14] developed an energy-based method for the seismic retrofitting of existing buildings with hysteretic dampers. Takewaki [15] proposed a systematic procedure to determine the optimal damper positioning based on the minimization of the transfer function amplitude of inter-story drifts. A transfer function was also employed in the damper distribution method by Aydin et al. [16], in which the transfer function of the base shear force was used as the objective function instead of the displacement of the building. Ashour and Hanson [17] suggested placing dampers at the locations that would maximize the damping ratio of the fundamental mode. Zhang and Soong [18] and Shukla and Datta [19] proposed determining the locations of the dampers by using a sequential optimization procedure. These previous studies provided optimization methods that were based on various aspects, including energy balance, a reliability objective, a genetic algorithm, a transfer function, etc., most of which require relatively significant computational effort and are difficult to use in practice. Over the past several years, displacement-based design (DBD) has become one of the major areas of interest in structural engineering within the framework of performance-based design, and significant efforts have been made to develop practical design procedures for various types of structures. The DBD method is expected to yield more reasonable and accurate design results because this method requires a more comprehensive examination of the variations in the mechanical properties of the structures. Representative research work can be found in Lin et al. [20], Kim and Seo [21], Vargas and Bruneau [22], Sullivan and Lago [23], Martínez et al. [24] etc. Lin et al. [20] proposed a seismic DBD method for new buildings that were equipped with a viscous energy dissipation system by using the linear iterations of the effective stiffness and effective damping ratio. Sullivan and Lago [23] developed a simplified non-iterative means of controlling the systematic damping of the structures by

choosing the proportions of forces that were resisted by the viscous dampers relative to the main structure. Unlike in the abovementioned methods, the stiffness of a system is considered a critical parameter that influences the seismic response, and the displacement profile of a structure is supposed to be greatly affected by the stiffness distribution throughout the height. In this paper, a direct DBD method for the seismic control of slip hysteretic buildings is developed, in which the optimal equivalent stiffness criterion is used. In the proposed methodology, the multistory building is replaced by a SDOF system, which is utilized to determine the total capacity of the damper according to the stiffness and ductility, and the damper capacity for each individual story is derived based on the tuning of the equivalent stiffness of the system. Hysteretic dampers are utilized in this study. The proposed method is oriented toward practical use and is a simpler and less time-consuming alternative to existing methods; thus, this method is expected to be preferred by design engineers. 2. Definition of the mechanical properties of a damped SDOF system 2.1. Basic system description This paper addresses a shear-type structure, in which the bending deformation is far smaller than the shear deformation and is consequently neglected. The structural system consists of two structural sub-systems: the main structure and the supplemental damping system. Fig. 1 presents the constitution of the system schematically. Both sub-systems are condensed, and each is represented by a hysteretic spring (Fig. 1(b)). These spring elements are connected in parallel, which implies that the devices experience the same horizontal deformation. In Fig. 1(b), M denotes the lumped mass of the structural system, and Kf and c denote the stiffness and damping coefficient of the main frame, respectively. Kd denotes the horizontal stiffness of the damper, which can be obtained from the axial stiffness and installation angle of the damper (see Section 5.4). In engineering practices, dampers are linked to the main frame through an elastic brace that is placed in series. The connecting brace and the energy dissipating part are both displacement-dependent components and experience the same force, so their displacements are always in phase; thus, the series system of the connecting brace (Kb) and the energy-dissipating devices (K 0d ) can be represented by a combined spring element (Kd). Once Kd is determined, the corresponding value of K 0d can be obtained by assuming a value of Kb (see Section 5.4). In this work, ‘‘damper” indicates a series system that comprises energydissipation devices and an elastic connecting brace, and its stiffness is denoted by Kd. 2.2. Hysteretic model for the main frame Typical timber structures with nailed connections develop a steady-state hysteretic curve that is very different from the one

F

F

Kd M

M Kd

Kf c Damper

ϕ

Kd′

Kb

(a) Sketch of system (b) Structural model (c) Damper representation Fig. 1. Constitution of the SDOF system.

201

W. Pu et al. / Engineering Structures 128 (2016) 199–214

From the second cycle onward, the loading curve after slip is oriented to the maximum deformation (8 ? 9 ? 10), and this model assumes that this loading stiffness is equal to the secant stiffness of the maximum deformation. The shaded region in Fig. 2 illustrates the steady-state hysteresis under cyclic loading. A slip strength ratio q is defined to quantify the degree of slip in the hysteretic loop. According to Fig. 2(a), q is the ratio of the resistant force at point 8 and the yielding force Fy of the structure. Theoretically, q varies from 0 to 1. The value of q is generally small for actual timber buildings; a value of 0.15 was reported in [28] based on the results of an experimental study on wood panels with nailed connections. The values of q that were obtained from the test results of [29] varied between 0.04 and 0.38, depending on the structural member type and fastener spacing, and most of the values ranged between 0.12 and 0.18. Fig. 2 shows plots of the hysteresis curves for q = 0, q = 0.5 and q = 1. Notably, q = 1 yields a bilinear model, which has been commonly used in structural engineering. Fig. 3 shows an example of the curve fitting of a slip model based on the force-displacement relationship of a timber-framed structure that is subjected to earthquake loading [28].

that is generated in the virgin cycle. This hysteresis is characterized by two phenomena: the slip (or pinching) effect and the memory of the material [25]. The slip effect is characterized by a hysteretic loop that is thinner in the middle than at the ends and is induced by cavities that form around the fasteners. The memory of the material indicates that the load curve somehow depends on the loading history, which will be described later. Hysteresis models that have been previously proposed for timber frames consisted of a combination of the bilinear and slip models, which indicated a relatively simple mechanism and could exactly reproduce the dynamic behavior of a shear wall [26,27]. This hysteresis model is adopted in this study to simulate the mechanical behavior of a timber structure under dynamic forces. In Fig. 2(a), the initial elastic stiffness and post-yielding stiffness are denoted by Kf and pKf, respectively. When a load is applied to the structure beginning from zero deformation, the structure responds elastically (0 ? 1) and experiences plastic deformation after the elastic deformation limit uy is reached (1 ? 2). Then, the structure unloads from the maximum deformation um (2 ? 3) and deforms elastically with stiffness Kf. The slip phenomenon appears first after the elastic deformation is fully restored (3 ? 4). During the first loading cycle, the structure experiences a virgin deformation in the negative direction, and the cavity in the negative direction has not been formed, so the hysteresis curve continues with an elastic stiffness Kf (4 ? 5). After further loading and unloading at the same level, the structure develops a hysteresis in the positive direction (5 ? 6 ? 7 ? 8) that is symmetric to the previous hysteresis in the negative direction.

2.3. Hysteretic model of the damper The mechanisms that are used for energy dissipation include the yielding of metal, frictional sliding, the flow of fluid through an orifice, and the deformation of viscoelastic solids or liquids. Hysteretic dampers that are based on the yielding of a metallic material are among the most popular. Although these hysteretic

F

2(10)

Fm Fy K f qFy 8

7

0

1 pKf 9

4

12

um

uy

u

3(11)

5

6(13)

(a) q=0.5

F

F

pK f Kf

pK f Kf

uy

(b) q=0

um

uy

u

(c) q=1

Fig. 2. Slip hysteretic model with different slip ratios.

um

u

202

W. Pu et al. / Engineering Structures 128 (2016) 199–214

provided at the beginning of the design. Therefore, the damper capacity can be represented in terms of its elastic stiffness and target ductility factor ld (=um/udy). For convenience, the dimensionless stiffness ratio rk and ductility ratio rl are used to denote the damper capacity demand. Equation 1 defines the stiffness ratio rk and ductility ratio rl , where ld and lf are the ductility factors of the damper and the main frame, respectively, at a target displacement um.

F

u

rk ¼ K d =K f ; r l ¼ ld =lf

Experiment Slip model Skeleton curve

Fig. 3. Hysteretic curve of a timber-framed building and the idealized slip model [28].

dampers exhibit some degree of time dependency [30], this factor is commonly ignored, and a much simpler elasto-perfectly plastic model is employed in practical design. Fig. 4(b) shows the hysteretic curve for the damper, which can be defined by the stiffness Kd and yielding deformation udy. The damper and the main structure are arranged in parallel to comprise the damped system, and the force-deformation curve of the damped system can be obtained by summing the forces that are sustained by the damper and the main structure, as shown in Fig. 4.

ð1a; bÞ

Slip hysteretic loops are stated to exhibit a dependence on the loading history. Kasai et al. [35] compared the energy dissipation of slip hysteretic structures that were subjected to different types of ground motions and noted that the difference between the virgin cycle and the following steady-state cycles could be neglected for structures that were subjected to non-pulse-like ground motions but should be considered when estimating the equivalent parameters in the case of pulse-like ground motions. In this paper, an equivalent linearization is developed based on only the steadystate hysteresis. Epf and Epd are used to denote the energy that is dissipated by the main frame and the damper, respectively, within one steady-state cycle; Esf and Esd denote the strain energy of the main frame and the damper, respectively, at the maximum displacement um. These energy functions are formulated as follows:

Epf ¼ K f u2y ðlf  1Þð1  pÞð1 þ qÞ2

ð2aÞ

Epd ¼ 4K d u2dy ðld  1Þ

ð2bÞ

1 K eq;f u2m ; 2

1 K eq;d u2m 2

3. Definition of the mechanical properties of the structural system

Esf ¼

3.1. Hysteretic damping

where Keq,f and Keq,d are the secant stiffness of the main frame and damper, respectively, and are expressed as

The seismic performance assessment of the structural system with added dampers is based on elastic response spectra, so the nonlinear hysteretic characteristics of the structural system are equivalently linearized. The applicability of the equivalent linearization technique has been proven by extensive investigations on various types of structures, such as conventional seismic structures [31], passively controlled structures [32], and isolated structures [33,34]. The equivalent damping ratio and stiffness are key parameters that determine the maximum seismic response of the structure and thus should be formulated appropriately. In the current practice of seismic control design, particularly for the retrofitting of existing buildings, the properties of the main frame are already established, and the main design work involves finding the optimal damper parameters that improve the structural seismic performance to the required level. In a displacement-based design philosophy, the target displacement of the structure, which is also the expected peak deformation of the damper, is also

Ff

1 þ plf  p

lf

;

K eq;d ¼

ð2c; dÞ

Kd

ð3a; bÞ

ld

The most general way to define the damping ratio is to equate the energy that is dissipated in one hysteretic cycle of the inelastic system to that of an equivalent elasto-viscous system [36]. Based on this concept, the damping ratio of the system with added dampers can be expressed as Eq. (4):

n0p ¼

r 2l ðlf  1Þð1  pÞð1 þ qÞ2 þ 4r k ðr l lf  1Þ Epf þ Epd ¼ 4pðEsf þ Esd Þ 2pr 2l lf ð1 þ plf  pÞ þ 2pr k rl lf

F Kd

Kf u

ð4Þ

However, Eq. (4) is only applicable for a structure in a harmonic response with constant vibration amplitude. Eq. (4) likely overestimates the equivalent damping ratio of the system when the structure is subjected to seismic excitations. In such cases, the average damping ratio may be used to consider the variations in

Fd

pK f

(a)

K eq;f ¼ K f

Esd ¼

udy

u

(b)

Fig. 4. Hysteresis of (a) main frame, (b) damper, and (c) system.

udy uy

(c)

u

203

W. Pu et al. / Engineering Structures 128 (2016) 199–214

the amplitude of the random response [37]. By integrating Eq. (4) from 0 to lf (Eq. (5)), the average hysteretic damping ratio np;eq can be written as Eq. (6):

np;eq ¼

np;eq ¼

1

lf 1

lf

Z lf 0

l

n0p ð

l

0 f Þd

0 f

ð5Þ

 ~  c3 þ ~c4 lf ~c ~c ~c þ ~c2 ~c4 ~c5  2 Lnðlf Þ þ 1 3 Ln ~c3 ~c3 ~c4 ~c3 þ ~c4

ð6Þ

where

~c1 ¼ ð1  pÞð1 þ qÞ2 þ 4rk =r l ;

c c 1 nv ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ rk 2 mðK f þ K d Þ 2 mK f

Eqs. (7) and (8) can describe variations in the inherent damping for an elastic structural system in which the damping coefficient and stiffness are constants. As plastic deformation develops, however, modeling the inherent damping becomes more complicated and some assumptions must be introduced. The stiffness proportional damping model has been utilized to simulate variations in the damping ratio in structures that respond inelastically. In a general form, the stiffness proportional damping ratio can be written as follows:

bK nv ¼ pffiffiffiffiffiffiffiffiffiffiffi 2 mK eq

~c2 ¼ ð1  pÞð1 þ qÞ2 þ 4rk =r 2l ;

~c3 ¼ 2pð1  p þ rk =rl Þ; ~c4 ¼ 2pp; h i l þr k ~c5 ¼ p2r ðr k þ 1ÞLn r1þr  Lnr l : l k

Fig. 5 illustrates a comparison of the damping ratios from Eqs. (4) and (6). The average damping ratio decreases to 40% of that from the steady-state hysteresis at lf = 2. 3.2. Inherent viscous damping In addition to the hysteretic damping that results from the plastic deformation of the hysteretic damper and main frame, the inherent equivalent viscous damping of the structure that results from various mechanisms, such as friction between structural or non-structural members and the opening and closing of microcracks, should be considered. For an elastic structural system, the inherent damping ratio nv is commonly written as

c nv ¼ pffiffiffiffiffiffiffiffiffiffi 2 mK f

ð7Þ

where c is the damping coefficient and m is the structural mass. The installation of the hysteretic damper is expected to increase the structural stiffness while barely influencing the energy-dissipation capability of the elastic main structure. Therefore, the damping ratio of the elastic system with the added damper can be written as Eq. (8). The inherent equivalent viscous damping is reduced because of the increased stiffness.

0.3

1

ξp' (left axis) ξp,eq (left axis) ξp,eq/ξp' (right axis)

0.2

ð9Þ

where Keq is the equivalent stiffness, which determines the equivalent frequency of the inelastic system. The damping coefficient c (=bK) is assumed to be proportional to the stiffness, and b is the scaling factor. The initial elastic stiffness, tangent stiffness and secant stiffness can be used for K in the numerator of Eq. (9), among which the first two are the most popular. If a constant damping coefficient c is required in the analysis, the initial stiffness proportional model could be adopted. However, this model likely overestimates the inherent damping for an inelastic system with significantly degraded stiffness. In published research, such as that of Priestley et al. [38], the inherent damping of inelastic SDOF systems should be modeled with a tangent-stiffness proportional damping coefficient to avoid the exaggeration of the damping ratios because of stiffness degradation. However, the tangent stiffness proportional damping is discontinuous because of the break in the tangent stiffness in the idealized multi-linearization representation of the structural hysteresis. In such situations, the calculation algorithm in the time history analysis (THA) must be carefully chosen; otherwise, a spurious damping force may arise in the THA [39,40]. Based on these findings, the secant stiffness proportional damping model could be advantageous [39]. At present, unfortunately, the secant stiffness proportional model is not included in most structural analysis programs. As an example, Fig. 6 compares the initial stiffness, tangent stiffness and secant stiffness proportional damping models for the assumed parameters of p = 0.2, q = 0.2, rk = 0.3, rl = 4, and lf = 4.

1

0.3 0.2

0.5

0.5

0.1

0.1 0

0 1

2

μf

(a) p=0.2, q=0.2,

0.3

0

=0

0

1

4

3

1

0.2

2

μf

(b) p=0.2, q=1,

0.3

4

3

=0

1

0.2 0.5

0.1

0.5

0.1

0

0 1

4 3 μf (c) p=0.2, q=0.2, =0.2, =3 2

ð8Þ

0

0

1

2

μf

(d) p=0.2, q=1,

3

=0.2, =3

Fig. 5. Comparison of the steady-state and averaged damping ratios.

4

204

W. Pu et al. / Engineering Structures 128 (2016) 199–214

0.1

neq ¼ nv ;eq þ np;eq

ξv

Initial stiffness proportional Tangent stiffness proportional Secant stiffness proportional Equation (10)

0.08 0.06

ð11Þ

3.3. Equivalent stiffness In a common approach, the secant stiffness of an inelastic system is utilized to represent the stiffness of the equivalent linear system, which can be readily obtained as follows:

0.04 0.02

K eq ¼ K eq;f þ K eq;d ¼ K f

μf

0 1

0

2

3

1 þ plf  p

4

lf

þ

rk

!

r l lf

ð12Þ

(a) Damping ratio obtained from the virgin cycle 4. Damper capacity design for an SDOF system 0.1

ξv

4.1. Seismic performance curves

Initial stiffness proportional Tangent stiffness proportional Secant stiffness proportional Equation (10)

0.08 0.06 0.04 0.02

μf

0 0

1

2

3

4

(b) Damping ratio obtained from the steady-state cycle Fig. 6. Comparison of various stiffness proportional damping models.

The inherent damping ratio of the structure without a damper is 0.03, and Fig. 6 shows the damping ratios based on Eq. (9). Fig. 6 (a) and (b) are obtained from the structural stiffness of the virgin cycle and the steady-state cycle, respectively. As seen in the figure, the initial stiffness proportional damping significantly increases as the plastic deformation increases and is generally larger than the values from the other two models. In the tangent stiffness proportional damping model, the inherent viscous damping based on the envelope decreases to almost zero because of the loss of stiffness after the main frame yields; in contrast, the damping ratio based on the steady-state hysteresis increases as lf increases, which results from an increase in the tangent stiffness (Fig. 3(a), points 8–10). Because the tangent stiffness of the main frame after slip is equal to the secant stiffness, the tangent stiffness proportional damping should be equal to the secant stiffness proportional damping for the structural system without the damper. These two types of damping exhibit somewhat different results in a system with added dampers because of the contribution of the dampers to the secant stiffness of the system. In this work, the tangent stiffness proportional model is used in the THA, in which the coefficient b is added only to the stiffness of the main frame in the analysis. To approximate the inherent viscous damping of the equivalent linear system nv ;eq , we use the tangent stiffness proportional damping of the system at the maximum deformation um as follows:

1 þ plf  p nv ;eq ¼ nv ;ef qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lf ð1 þ plf  pÞ þ lrflrk

ð10Þ

where nv ;ef is the initial viscous damping for the elastic main frame. The damping ratio from Eq. (10) is also plotted in Fig. 6 for comparison. The total equivalent damping of the inelastic damped system is estimated by summing the inherent equivalent viscous damping and the hysteretic damping, as shown by Eq. (11):

As will be shown later, the design procedure that is proposed for a MDOF system includes a step to perform a spectrum-based response prediction for an equivalent SDOF system, for which the equivalent parameters can be calculated by using the equations in Section 3. Because more than one design solution exists for a target seismic performance, showing how the seismic performance is affected by the damper properties is beneficial before selecting a specific optimal design. To this end, two non-dimensional response reduction factors are defined, and seismic performance curves are developed in this section. One of the factors is the displacement reduction factor Rd, which quantifies the degree of displacement reduction that results from adding dampers. This value can be obtained by dividing the displacement response of the damped inelastic system by that of the undamped elastic main structure. In view of the randomness of the seismic response and to avoid a large discrepancy in the spectrum from small shifts in the period, the average of the spectral responses of the elastic system and the equivalent elastic system is used to estimate the displacement of the inelastic system. Consequently, Rd is defined as follows:

Rd ¼

DðT eq ; neq Þ þ DðT 0 ; nv 0 Þ 2DðT f ; nv Þ

ð13Þ

where D denotes the spectral displacement, Tf is the natural period of the elastic undamped structure, T0 and nv 0 are the natural period and the viscous damping ratio (Eq. (8)), respectively, of the elastic structure with added dampers, and Teq is the equivalent period of the inelastic structure with dampers. The periods can be calculated by using the following equations:

T0 ¼ Tf

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K f =ðK f þ K d Þ

ð14Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K f =ðK eq;f þ K eq;d Þ

ð15Þ

T eq ¼ T f

In addition to Rd, a pseudo-acceleration reduction factor Rpa is defined. Although the objective of the proposed procedure is to suppress the displacement, showing the acceleration reduction is beneficial to improve the selection of the damper properties. In a similar manner, Rpa is defined as

Rpa ¼

AðT eq ; neq Þ þ AðT 0 ; nv 0 Þ 2AðT f ; nv Þ

ð16Þ

where A denotes the spectral acceleration value. Given the ductility ratio rl and stiffness ratio rk and the calculated values of T0/Tf, Teq/Tf, and neq, the values of Rd and Rpa can be obtained based on the above definitions. Repeating the calculation with different assumed values of rl and rk and plotting Rd vs. Rpa yields the seismic performance curve, which intuitively

205

W. Pu et al. / Engineering Structures 128 (2016) 199–214

describes how the structural response is reduced by the various damper properties. Fig. 7 illustrates examples of performance curves with assumed parameter values. The chosen rl and rk span a wide range to allow a comprehensive examination of the structural performance. The results show that multiple values of rl and rk exist within a limited range that meet a given displacement reduction factor. However, the acceleration reductions may be different among those design solutions, and these reductions could be used as an auxiliary condition for choosing the design parameters. Generally, increasing rk effectively reduces the displacement but probably increases the acceleration, particularly when rl is small. Increasing rl reduces the acceleration, but reducing the acceleration response by increasing rl has a limit; because rk is constant, the displacement is only marginally affected by rl. Fig. 8. Acceleration design spectrum.

4.2. Validation of the performance curves 0.2

A NTHA is performed on SDOF structural systems with assumed parameters to verify the validity of the proposed performance curve. Artificial ground motions that are compatible with the design spectrum as regulated in the seismic code [41] are considered in this analysis. The characteristic period Tg (upper limit period of the constant acceleration range) of the design spectrum is set to 0.5 s. Fig. 8 illustrates the acceleration spectra of 4 ground motions that are generated by SeismoMatch [42] accompanied by the target spectrum. Fig. 9 shows an example of the acceleration time history. Usually, estimating the seismic response of a nonlinear system requires an iteration analysis, in which Teq and neq are repeatedly updated until the estimated response converges. To examine the accuracy of the result, the following procedure is utilized in this work: (1) the generated ground motion is scaled so that a target ductility ratio lf of the main frame is achieved in the NTHA and the scaling factor of the ground motion is saved; (2) Teq and neq are calculated based on the target ductility ratio; (3) the spectrum is amplified by the same scaling factor that was obtained in (1), and the response is estimated by using the equivalent parameters that were obtained in (2); and (4) the spectrum-based response and the THA-based response of the nonlinear structural system are compared. Agreement between the two methods implies that the procedure for developing the performance curve is useful, and vice versa. As shown in Table 1, different combinations of values for Tf, q, rl, rk, and lf are considered, and a total of 864 THAs are performed on 216 structural systems. Note that q = 1 is also considered in analysis for purpose of verifying the applicability of performance curves for bilinear hysteretic models, and it should not be under-

Acc. (g)

0.1 0 -0.1

t (s)

-0.2 0

4

8

12

16

20

Fig. 9. Example of artificial acceleration.

Table 1 Parameters of damped SDOF structural systems. Parameters

Values

Tf q rk rl

0.5 s, 1 s, 2 s 0, 0.2, 0.5, 1 0.1, 0.2, 0.4 2, 4 1, 2, 3

lf

stood as a suggestion of using q = 1 for timber structures. Fig. 10 illustrates the ratios of the spectrum-based results and THAbased results. In Fig. 10, each symbol denotes a structural system with a specific parametric combination. The ratios have been averaged over the 4 ground motions, and the corresponding standard deviations of the response ratios are plotted. The comparison shows that the spectrum-based responses match very well with the THA-based responses.

Fig. 7. Examples of performance curves.

206

W. Pu et al. / Engineering Structures 128 (2016) 199–214

Fig. 10. Comparison of THA- and spectrum-based responses of SDOF damped systems.

5. Proposed design procedure for a multistory building

Define the target story drift

5.1. Framework of the design procedure This work is an extension of the approach that was developed for an elastic structure [43], which established the general framework of the design procedure. Both the energy dissipation and the stiffness variation should be comprehensively examined for inelastic structures. In particular, the structural stiffness distribution along the height changes as plastic deformation develops in the structure. Tuning the stiffness of the system by adding dampers accompanied by energy dissipation is desirable. A preferable design approach should consider all these effects to develop a good balance between the main frame and the dampers. A simple and direct DBD methodology for the seismic control of an inelastic multistory structure is presented in this paper. The main structure is assumed to already be determined, and no structural member strength design process is involved. The displacement profile of the structure is taken as a straight line, and the structure is expected to experience an identical maximum story drift h at each level. The use of story drift as a performance index lies in the idea that seismic damage is governed by the story drift limit as typically regulated in seismic codes. For instance, the design manual for traditional timber structures in Japan [44] states a drift limit of 1/120 for the damage-free limit state and 1/30 for the repairable limit state. Fig. 11 shows the process of the proposed method. The story drift limit for the design level of the seismic intensity must be determined prior to the damper’s design. Next, a pushover analysis is performed to obtain the hysteretic curve of each story, from which the idealized bilinear skeleton curve can be constructed by using the energy equivalence principle [45]. The slip strength ratio can be assumed based on existing experimental results (e.g., [28,29]) or obtained from additional tests on substructures. This procedure and formulation could also be applied to structures that can be simulated by the bilinear hysteretic model by setting q = 1. Further experiments should be performed for other types of structures without available hysteretic models. The equivalent SDOF system of the main frame at a prescribed drift limit is subsequently

Pushover analysis to derive the shear beam model with idealized hysteretic slip model

Calculate the parameters of the equivalent SDOF system Plot performance curves for given target story drift Choose design solution (values of damper ductility ratio and stiffness ratio) Calculate the required stiffness of damper for each level of MDOF structure Perform NTHA to check the seismic performance of damped system Fig. 11. Overview of the proposed damper design procedure.

derived based on the MDOF shear beam model with an idealized hysteretic curve. The performance curve can be advantageously used to determine the damper capacity, and the solution is denoted by a combination of rl and rk, which reflect the necessary supplemental stiffness and damping ratio to match the required seismic performance. These design results will be transformed into an MDOF system that is constrained by tuning the equivalent

207

W. Pu et al. / Engineering Structures 128 (2016) 199–214

stiffness throughout the height. The formulation will be shown in detail below. In the final step, a NTHA should be conducted to examine the design. 5.2. Deriving the equivalent SDOF system An equivalent SDOF system has been used extensively in structural design to replace MDOF systems and to simplify the structural design. For instance, the capacity spectrum method that was adopted by ATC-40 [46] and FEMA-274 [47] employed the equivalent SDOF structure to represent the structure’s performance to determine whether the structure can survive earthquakes. Unlike structural performance assessment, the purpose of the pushover analysis within this procedure is to obtain the load-carrying capacity of each story when the drift limit is attained, through which the equivalent stiffness (secant stiffness) and dissipated energy can be calculated. The focus is on the expected contribution of the main frame to the controlled system that responds as required. Therefore, whether the lateral force distribution for the pushover analysis is based only on the fundamental vibration mode of the elastic system at the present stage is irrelevant, although this condition may be an inappropriate representation of the inelastic force distribution in other circumstances [48]. Five parameters must be determined to build the hysteretic slip model of the equivalent SDOF system, which are the height H, elastic stiffness Kf, post-yield stiffness ratio p, ductility ratio lf, and slip strength ratio q. The equivalent SDOF model can be created as follows by assuming the target story drift has been achieved in each level of the main frame structure and that the equivalence of the SDOF and MDOF structure at the target displacement has been maintained. 1. The height of the equivalent SDOF model is obtained by assuming that the vibration mode of the building structure is that of a straight stick: n X mi H2i i¼1

H¼

ð17Þ

n X mi Hi i¼1

where H is the height of the equivalent SDOF model and Hi and mi are the height and mass, respectively, of the i-th level of the MDOF model. 2. The elastic stiffness of the equivalent SDOF model is calculated by equating the strain energy of elastic SDOF and MDOF models that deform with equal story drift angle: n X

Kf ¼

2 K fi hi

i¼1

n X  mi H i i¼1

n X mi H2i

!2

4. Two additional energy equivalence principles are proposed to determine p and lf. The two equivalent systems dissipate the same energy in a steady-state cycle with the target drift amplitude and contain the same strain energy at the target drift. These two conditions can be formulated as Eq. (20), and lf and p can be consequently obtained from Eqs. (21) and (22):

Epf ¼

n n X X Epf ;i ; Esf ¼ Esf ;i i¼1

ð1 þ qÞ2 K f H2 h2  2

lf ¼

n X 2lf Esf ;i

p¼

q¼

n

ð21Þ

i¼1

i¼1



ðlf  1ÞK f H2 h2

1

ð22Þ

lf  1

5.3. Equivalent stiffness tuning of the MDOF system A total of 2n parameters, which are the elastic stiffness and ductility factors of dampers at each story level, must be determined to complete the damper design for an n-story MDOF system. The design for the SDOF system determines the total capacity of the damper, which is given as a combination of the stiffness ratio and ductility factor of the damper. This capacity is allocated to each story level while keeping the equivalence of the controlled SDOF system and the corresponding MDOF system. The added damper provides both energy dissipation and auxiliary stiffness to the system; the stiffness is considered here. The disadvantageous horizontal stiffness distribution of the main frame, particularly in the elasto-plastic range, can be tuned by adding unequal capacities of dampers along the height, thus producing the expected response distribution along the height via the effects of both damping and stiffness. To achieve this goal, damper capacity distribution rules are proposed based on the following criteria. 1. The damping ratios of the controlled MDOF and SDOF systems in a steady-state cycle with target drift amplitude are equal, which is expressed by Eq. (23):

Epf þ Epd ¼ 4pðEsf þ Esd Þ

n X ðEpf ;i þ Epd;i Þ i¼1

4p

n X ðEsf ;i þ Esd;i Þ

ð23Þ

i¼1

ð18Þ

Substituting Eqs. (2a–d) into Eq. (23) yields Eq. (24): n X 2 K di hi ¼ i¼1

n n X X 4p Esf ;i  Epf ;i i¼1

ð24Þ

2. The secant stiffness of the controlled MDOF system that is calculated from the target story drift is proportional to the design story shear, which can be represented by Eq. (25). n X

2

ð19Þ

i¼1

4h2 ðld;i  1Þ  2pnh2 ld;i

Q i hi 1

n X Esf ;i

The equivalent SDOF depends on the target displacement, which determines the stiffness and energy dissipation contribution of the main frame to the final controlled system. Therefore, the above process must be repeated specifically to obtain the equivalent parameters once the target drift is changed.

n¼

3. Section 2.2 showed that q affects the energy dissipation capability of the main frame through a quadratic term, namely, (1 + q)2. The quadratic terms of the MDOF structure are considered as an average to ensure that the energy dissipation of the original MDOF system is more accurately approximated. The q value can be calculated by Eq. (19), where qi is the slip strength ratio for the i-th level of the MDOF structure:

i¼1

ð20a; bÞ !

i¼1

n X Epf ;i

i¼1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX n u u ð1 þ qi Þ2 t

i¼1

ðK eq;fi þ K eq;di Þhi

¼

n X i¼1

Q i hi

i¼1

ð25Þ 2

ðK eq;fi þ K eq;di Þhi

208

W. Pu et al. / Engineering Structures 128 (2016) 199–214

where Qi is the design shear force that is applied on the i-th level. Equation (25) actually represents an equation set that consists of (n  1) independent equations, and the above two conditions generate a total of n equations. 3. For simplicity, the same ductility factor ld,i is adopted for all dampers and is the same as that in the SDOF system:

ld;i ¼ ld ; ði ¼ 1 . . . nÞ

ð26Þ

Eq. (26) represents another equation set that consists of n independent equations. Combining Eqs. (24)–(26) produces a total of 2n independent equations. These equations are used to derive a total of 2n parameters for the dampers, as stated above. Solving the above equations leads to the required elastic stiffness of the damper for the i-th level, as shown by Eq. (27): n n X X 8ld ðld  1Þ Esf ;i  l2d Epf ;i

K d;i ¼

Qi i¼1 i¼1  ld K eqf ;i n X hi  2 2 4h ðld  1Þ  2pnh ld Q i hi

ð27Þ

i¼1

The yielding force of the dampers can be further obtained as

F dy;i ¼ K d;i udy;i ¼ K d;i hhi =ld

ð28Þ

5.4. Deriving the axial parameters of the dampers The stiffness and ductility factors of the dampers that were obtained in Section 5.3 represent their capacity demand in the horizontal direction of a shear model and should be used to derive the axial capacity if diagonal brace type dampers (Fig. 1a) are used in practice. If we assume that the damper of the i-th level is instrumented with an inclination angle of ui, the axial yield deformation b dy;i , and axial stiffness K b d;i can be calculated b dy;i , axial yield force F u by the following equations:

b dy;i ¼ udy;i cos ui u

ð29Þ

b F dy;i ¼ F dy;i = cos ui

ð30Þ

b d;i ¼ K d;i = cos2 u K i

ð31Þ

The damper consists of a connecting brace and energy dissipab 0 of b 0dy;i and stiffness K tion component, so the yield deformation u d;i

multistory timber building with hysteretic dampers as an MDOF system with a lumped mass mi; hysteretic elements that produce forces of Ffi and Fdi for the main structure and damper, respectively; and a dashpot that denotes the damping coefficient ci at the i-th floor. Fig. 12 illustrates the shear model of a 10-story example building. The equation of motion for the system with added dampers is given by

_ _ € g ðtÞ muðtÞ þ cuðtÞ þ Ff ðtÞ þ Fd ðtÞ ¼ m1u

ð34Þ

€ ðtÞ where m and c are the mass and damping matrix, respectively; u _ and uðtÞ are the matrices for the acceleration and velocity, respec€ g ðtÞ is the ground accelertively, of the floors relative to the base; u ation; and 1 is a vector with each element equal to unity. Ff ðtÞ and Fd ðtÞ are the resisting force matrices of the main frame and damper, respectively, which are related to the displacement and the loading history of the system, as described in previous sections. A total of 4 structural models, which are numbered 1–4, are considered. The structural models are all 10 stories high, the story height is equal for all levels (300 cm), and all levels have the same mass of 150 t. Table 2 shows the parameters that were adopted for the models. The stiffness for models 1–3 are purposely designed so that the buildings develop deformations that are concentrated in specific story levels. Model 1 is designed with an elastic stiffness that is proportional to the design shear force and is expected to produce approximately uniform deformations in each story level. Models 2 and 3 contain weak stories in lower levels and upper levels, respectively, and are likely to develop considerable local deformations in these lower levels or upper levels. The stiffness for model 4 is randomly generated within a limited range of 1500–4000 kN/cm. The fundamental periods for all 4 models are 1 s. The values of pi = 0.2, qi = 0.2, and hyi = 0.005 are assigned to models 1–3. For model 4, pi is randomly generated within a range of 0.15–0.3, hyi is generated within 1/200–1/150, and qi is generated within 0.1–0.3. The range of qi is assumed based on the experimental results [28,29] and considered to be practical for timber

F10 c10

m10

Kf10

Kd10

c9

m9 Kf9

Kd9

the energy dissipation component can be further derived as

b d;i = K b b;i Þ b dy;i ð1  K b 0dy;i ¼ u u

ð32Þ

b d;i =ð K b0 ¼ K b b;i K b b;i  K b d;i Þ K d;i

ð33Þ

b b;i is the axial stiffness of the connecting brace at the i-th where K b b;i > K b d;i , so K b0 > K b d;i . In design practice, the stiffness level. Here, K d;i of the connecting brace should be estimated beforehand, and the design parameters for the energy dissipation component can be subsequently derived through the above equations. A large stiffness is generally desired for the connecting brace to improve the efficiency of the energy dissipation. A more detailed discussion regarding the effect of the connecting brace can be found in [45].

F9

Fi

F2

c3

Computer models of a set of example buildings that exhibit typical slip type hysteresis are developed on the program PC-ANSR [49] to verify the proposed methodology. The program models a

Kd3

Kf2

Kd2

Kf1

Kd1

m2 F1

c2

6. Numerical verification 6.1. Structural models

Kf3

m1 c1

Fig. 12. Model of a building with hysteretic dampers.

209

W. Pu et al. / Engineering Structures 128 (2016) 199–214 Table 2 Mechanical parameters of the example buildings. Level

Building no. 1

10 9 8 7 6 5 4 3 2 1

Building no. 2

Building no. 3

Building no. 4

Kfi (kN/cm)

hyi

pi

qi

Kfi (kN/cm)

hyi

pi

qi

Kfi (kN/cm)

hyi

pi

qi

Kfi (kN/cm)

hyi

pi

qi

818 1294 1718 2088 2406 2671 2882 3041 3147 3200

0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005

0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20

0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20

1738 1896 2054 2212 2370 2528 2686 2844 3002 3160

0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005

0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20

0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20

437 874 1311 1748 2185 2622 3059 3496 3933 4370

0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005

0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20

0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20

1703 3662 1690 2100 3345 2125 3987 2663 3312 2348

0.0063 0.0058 0.0061 0.0061 0.0064 0.0063 0.0055 0.0050 0.0054 0.0051

0.176 0.296 0.292 0.190 0.227 0.290 0.215 0.287 0.182 0.241

0.143 0.250 0.262 0.279 0.219 0.274 0.122 0.237 0.125 0.108

structures. All the random numbers are generated through the function of ‘‘Randbetween” built in Excel program. The purpose of using random parameters in model 4 is to obtain a structural model with soft stories randomly distributed along the height. An inherent viscous damping ratio of 0.02 is assumed. The target spectrum in Fig. 8 is employed again, while the peak value is scaled to 1.2 g. A total of 8 ground motions are generated by SeismoMatch [42], and their peak ground accelerations are approximately 0.54 g. Fig. 13 shows the target acceleration spectrum, the acceleration spectrum of each ground motion, and their statistical mean and standard deviation. First, the THA is performed on the structures without added dampers by using the generated ground motions. Fig. 14 shows the peak story drift that is induced by each ground motion and their statistical mean and standard deviation. As expected, the deformation distributions of models 1–3 are governed by the stiffness distribution. Model 1 develops relatively uniform deformation along the height, and the average story drift is approximately 0.01. Model 2 develops comparatively large deformation in the lower levels; in contrast, model 3 develops deformations as large as 0.04 in the upper levels but very small deformations in the lower levels. For model 4, significant deformation occurs in some specific weak stories, including story levels 1, 2, 3 and 5. 6.2. Comparison of seismic control design with different damper parameters According to the responses of the uncontrolled structures, a target drift of 1/150 is considered for seismic control design. The structures are expected to produce a ductility factor of 1.5 for models 1–3 at this level of deformation. The damper design is developed based on the procedure in Fig. 11. The equivalent SDOF model is derived based on the target story drift, and the damper capacity design is developed based on

1.6 S (g) a

Individual results

1.2

Target spectrum

0.8

Mean spectra Mean±standard deviation

0.4

T (s)

0 0

1

2

3

4

Fig. 13. Comparison of artificial ground motion spectra and the target spectrum.

the SDOF model. The obtained mechanical parameters of the 4 equivalent SDOF models are listed in Table 3. As stated previously, the displacement can be reduced by increasing the stiffness or the damping ratio of the structures. Therefore, the damper capacity can be designed with either a combination of large stiffness and small damping ratio or a combination of small stiffness and large damping ratio. Two designs with a ductility ratio rl of 4 (Case 1) and 2 (Case 2) are considered to compare their effects. Table 3 also lists the design results (rl and Kd/Kf) for the equivalent SDOF models. The ductility ratio of Case 1 is twice that of Case 2, but the stiffness ratios for the two cases are very close. This result occurs because the influence of the ductility ratio becomes weaker as the ductility ratio increases, and the response reduction is mainly governed by the stiffness ratio, as illustrated in Fig. 7. Table 4 shows the ductility factor and the stiffness ratio of the dampers on each story level of the MDOF models. The dampers were allocated non-proportionally along the height. Importantly, Eq. (27) may yield a negative damper stiffness for some stories (as shown in Table 4), which means that the equivalent stiffness of those levels is greater than the required value and should be reduced to achieve an optimal distribution. However, adding a negative stiffness to the structure is difficult in practice; consequently, no damper is added to those levels and the stiffness of main structure remains constant. Neglecting the effect of negative damper stiffness indicates that those story levels produced stiffness values that were larger than the theoretical values and are expected to develop deformations that are smaller than the expected target. Fig. 15 shows the story drift of the controlled structures of Case 1. Model 1 has dampers in all the story levels except for the top level, so the equivalent stiffness matches the optimization except at the top level. This configuration creates a very uniform distribution of deformation along the height, as expected. Model 2 shows relatively small deformation in the upper floors because the equivalent stiffness in those floor levels is still larger than the desired values. The deformations in the upper levels in model 3 are controlled by the added dampers very well. However, the lower levels experience larger deformations than those in the uncontrolled case because of changes in the balance of stiffness. Including model 4, the THA results of all the example buildings show that the deformations are well suppressed by the added dampers. Although the story drifts from some individual ground motions exceed the target because of variations in ground motion, the story drift responses are effectively reduced below the target value of 1/150 on average. Moreover, an approximately uniform distribution of story drift along the height is achieved. Fig. 16 shows the story drift of the controlled structures from Case 2. Case 2 also effectively reduces the responses of the structure below the target drift. According to Table 4, Eqs. (3b) and (28), Case 1 and Case 2 produce very close yield strengths and equivalent stiffness values of the dampers for the same target drift. The THA results in both Figs. 15 and 16 demonstrate that the

210

W. Pu et al. / Engineering Structures 128 (2016) 199–214

Mean ± standard deviation

Mean resposne

Target response

10

10

9

9

9

8

8

8

8

7

7

7

7

6 5 4

6 5 4

Story number

10

9

Story number

10

Story number

Story number

Individual results

6 5 4

6 5 4

3

3

3

3

2

2

2

2

1

1 0

0.01

0.02

1 0

0.01

1 0

0.02

0.01

0.02

0.03

0

0.04

0.01

0.02

Story drift (rad)

Story drift (rad)

Story drift (rad)

Story drift (rad)

(a) No.1 model

(b) No.2 model

(c) No.3 model

(d) No.4 model

Fig. 14. Peak story drift of 4 example structural models without dampers.

Table 3 Structural parameters and design results for equivalent SDOF systems. Bldg.

Kf (kN/cm)

H (cm)

p

q

l

hy (rad)

No. No. No. No.

475 499 490 549

2100 2100 2100 2100

0.20 0.20 0.20 0.48

0.200 0.200 0.200 0.203

1.33 1.33 1.33 1.26

0.0050 0.0050 0.0050 0.0053

1 2 3 4

rl, Kd/Kf Case 1

Case 2

Case 3

4, 4, 4, 4,

2, 2, 2, 2,

4, 4, 4, 4,

1.29 1.29 1.29 1.19

1.205 1.205 1.205 1.155

3.545 3.545 3.545 3.420

Table 4 Required ductility and elastic stiffness ratio of the dampers. Design parameter

Level

Building no. 1

Building no. 2

Building no. 3

Building no. 4

Case 1

Case 2

Case 3

Case 1

Case 2

Case 3

Case 1

Case 2

Case 3

Case 1

Case 2

Case 3

Kdi/Kfi

10 9 8 7 6 5 4 3 2 1

(0.52) 0.23 0.54 0.72 0.82 0.89 0.94 0.97 0.98 0.99

(0.32) 0.04 0.19 0.27 0.33 0.36 0.38 0.40 0.40 0.41

0.31 1.23 1.61 1.82 1.95 2.04 2.09 2.13 2.15 2.16

(2.41) (1.04) (0.03) 0.68 1.17 1.47 1.61 1.62 1.53 1.34

(1.24) (0.57) (0.09) 0.26 0.49 0.64 0.71 0.71 0.67 0.58

(2.00) (0.32) 0.91 1.79 2.38 2.74 2.92 2.93 2.82 2.59

2.97 2.61 2.24 1.88 1.52 1.16 0.80 0.44 0.07 (0.29)

1.36 1.19 1.01 0.84 0.66 0.49 0.31 0.14 (0.04) (0.21)

4.58 4.14 3.69 3.25 2.81 2.37 1.92 1.48 1.04 0.60

(2.56) (2.62) 1.40 1.53 (0.28) 3.26 0.38 3.31 2.02 4.84

(1.30) (1.33) 0.64 0.71 (0.19) 1.55 0.14 1.58 0.95 2.34

(2.06) (2.18) 2.76 2.91 0.73 5.07 1.43 4.98 3.40 6.84

ldi

All levels

5.33

2.67

5.33

5.33

2.67

5.33

5.33

2.67

5.33

5.04

2.52

5.04

proposed method can be advantageously employed to optimize the damper capacity and location in a multistory building. 6.3. Effect of earthquake intensity Recent earthquakes, such as 2008 Wenchuan earthquake and 2011 the Great East Japan Earthquake, were as large or larger than the maximum levels of excitation that are considered in the design of typical structures. The randomness of earthquakes means that precisely predicting the intensity of future earthquakes is difficult. Investigating how these structures perform in an earthquake event that is more intense than what is considered in design is very important. For instance, the ground motions in Case 1 are amplified by a factor of 1.5, and THA is conducted again. Fig. 17 plots the obtained maximum story drift for all 4 models. Much larger responses form because of the increase in the seismic intensity, and the dispersion of the responses becomes larger. On the other hand, the vertical

stiffness distribution improves from the unfavorable stiffness distribution of structures without dampers because of compensation from supplemental dampers, and no significant concentration of local deformations is observed. Obviously, the damper capacity and its allocation along the height in Case 1 are no longer optimal for the amplified seismic intensity. The capacity and location of the dampers are redesigned considering the variations in the excitation. Following the previous two cases, this design is numbered Case 3. The same target story drift angle and the same value of rl in Case 1 are adopted. Tables 3 and 4 present the design results for the equivalent SDOF model and the MDOF model, respectively. Compared to Case 1, much larger damper capacity is required in Case 3, and positive damper stiffness is theoretically obtained in some of the story levels where negative damper stiffness was derived in Case 1. Fig. 18 illustrates the responses that are produced by Case 3. The average responses are suppressed under the target value, and uniform distributions of story drifts are generally achieved.

211

W. Pu et al. / Engineering Structures 128 (2016) 199–214

Mean ± standard deviation

Mean resposne

Target response

10

10

9

9

9

9

8

8

8

8

7

7

7

7

6 5 4

6 5 4

Story number

10

Story number

10

Story number

Story number

Individual results

6 5 4

6 5 4

3

3

3

3

2

2

2

2

1

1

1

0

0.01

0

0.02

0.01

1 0

0.02

0.01

0

0.02

0.01

Story drift (rad)

Story drift (rad)

Story drift (rad)

Story drift (rad)

(a) No.1 model

(b) No.2 model

(c) No.3 model

(d) No.4 model

0.02

Fig. 15. Peak story drift of 4 example structural models with designed dampers (rl = 4).

Mean resposne

Mean ± standard deviation

Target response

10

10

9

9

9

9

8

8

8

8

7

7

7

7

6 5 4

6 5 4

Story number

10

Story number

10

Story number

Story number

Individual results

6 5 4

6 5 4

3

3

3

3

2

2

2

2

0

0.01

0

0.02

1

1

1

1

Story drift (rad)

(a) No.1 model

0.01

0

0.02

0.01

0

0.02

0.01

Story drift (rad)

Story drift (rad)

Story drift (rad)

(b) No.2 model

(c) No.3 model

(d) No.4 model

0.02

Fig. 16. Peak story drift of 4 example structural models with designed dampers (rl = 2).

Mean ± standard deviation

Mean resposne

Target response 10

9

9

9

9

8

8

8

8

7

7

7

7

6 5 4

6 5 4

Story number

10

Story number

10

Story number

Story number

Individual results 10

6 5 4

6 5 4

3

3

3

3

2

2

2

2

0

0.01

Story drift (rad)

(a) No.1 model

0.02

1

1

1

1

0

0.01

0.02

0

0.01

0.02

0

0.01

Story drift (rad)

Story drift (rad)

Story drift (rad)

(b) No.2 model

(c) No.3 model

(d) No.4 model

Fig. 17. Peak story drift of example buildings derived from Case 1 design and subjected to amplifed ground motions.

0.02

212

W. Pu et al. / Engineering Structures 128 (2016) 199–214

Mean ± standard deviation

Mean resposne

Target response

10

10

9

9

9

8

8

8

8

7

7

7

7

6 5 4

6 5 4

Story number

10

9

Story number

10

Story number

Story number

Individual results

6 5 4

6 5 4

3

3

3

3

2

2

2

2

1

1

1

0

0.01

0.02

0

0.01

0.02

1 0

0.01

0.02

0

0.01

Story drift (rad)

Story drift (rad)

Story drift (rad)

Story drift (rad)

(a) No.1 model

(b) No.2 model

(c) No.3 model

(d) No.4 model

0.02

Fig. 18. Peak story drift of example buildings derived from Case 3 design and subjected to amplifed ground motions.

The numerical analysis confirms that the design procedure can adjust the damper capacity depending on the seismic intensity.

6.4. Remarks on the design method The above analysis has demonstrated that the proposed method can be effectively used to control the seismic response of slip hysteretic timber buildings. The method is directly displacementbased, and no repeated design or numerical analysis on the MDOF models is required. The shear model-based design and analysis save a significant amount of computation effort. These merits make the method much easier to use and more practice-oriented compared to existing methods. Moreover, this method can be extended to other structural materials by employing different hysteretic models of structures. When applying the proposed method in practice, various combinations of design conditions can exist. This paper cannot perform accuracy verification to cover all potential design cases in practice. A few factors that may affect the design are noted here for a more comprehensive understanding of the design process by practical engineers. The target of the design method is to achieve a uniform distribution of story drift along the height. This concept is an underlying assumption that the displacement is governed by the fundamental mode. This assumption is generally reasonable for most current timber buildings, whose heights are very limited; therefore, the proposed method can lead to a satisfactory design. For high-rise or very flexible buildings, however, the effect of higher modes may be significant, and design methods that are based on the fundamental mode may not match the target performance. In such cases, a step to modify a first mode-based design is necessary by considering the effect of higher modes. In the proposed method, the damper capacity is determined so that a shear-force proportional distribution of equivalent stiffness can be achieved, which indicates that the lateral seismic load pattern influences the final design. The lateral load patterns specified in codes differ from one another. Consequently, the application of different seismic load patterns may lead to variations in the design result. In addition, high damping spectra are used in the damper capacity design of the SDOF model. High damping spectra are usually derived from the code’s specified design spectrum (usually derived from a 5% damping ratio) through a damping modification

factor. The damping modification factor varies between codes and significantly depends on the characteristics of the ground motions [50]. Understanding the effect of the damping modification factor would be beneficial for practical designers. Otherwise, a further comparison study is recommended to verify its effect.

7. Conclusions In this study, a design methodology is provided for determining the damper capacity and its optimal distribution along the height of a multistory timber building. A slip hysteretic model that can reproduce the pinching effect is used to simulate the dynamic behavior of timber structure, and a slip strength ratio q is defined to quantify the degree of pinching effect. The hysteretic type damper that is simulated by elasto-perfectly plastic model is considered. Timber buildings are denoted by shear beam models. Within the proposed design framework, the main focus of this study is to: (1) derive the equivalent linearization method for slip hysteretic structures added with hysteretic dampers; (2) develop the performance curves that visualize the seismic control effect of hysteretic dampers; (3) present a procedure to distribute the optimal damper capacity from the SDOF system to a MDOF system. The design formulas are theoretically derived and seismic control design and THAs are performed on four example buildings to verify the effectiveness of the method. The study leads to the following observations: (1). The equivalent damping ratio and stiffness are derived theoretically based on the steady-state response of SDOF timber structure with added hysteretic damper, and the averaged damping ratio is obtained explicitly to consider the randomness of the vibration amplitude induced by earthquake excitations. It is observed that the averaged damping ratios can be as small as half of those derived from the steady-state hysteresis of the maximum deformation, depending on the mechanical parameters and ductility level in consideration. The THAs performed on a large number of SDOF structures demonstrate that the derived equivalent parameters can be effectively used to predict the response of slip hysteretic timber structures added with hysteretic dampers. (2). The performance curves are developed to visualize the response control effect of dampers, from which multiple design solutions can be chosen and compared. The perfor-

W. Pu et al. / Engineering Structures 128 (2016) 199–214

(3).

(4).

(5).

(6).

mance curves lead to the observations that increasing the stiffness ratio of damper effectively reduces the displacement but probably increases the acceleration, particularly when the ductility ratio of damper is small, increasing the ductility ratio reduces the acceleration, but the effect is limited. A method to allocate the damper capacity from an equivalent SDOF system to the corresponding MDOF system is proposed based on the concept of tuning the vertical distribution of equivalent stiffness. The design formulas for dampers are explicitly obtained. For a given target performance, there exist multiple design solutions that are characterized by the combination of stiffness ratio and ductility ratio of dampers. It is observed from the THA performed on the example buildings that the method is able to lead to a uniformly distributed story drift along the height and limit story drift to a target level. The method can also adjust the damper capacity depending on the variation of seismic intensity or structural properties. Depending on the design conditions, the possible influence of higher mode effect, lateral load pattern, and damping modification factor of spectra on the design solutions should be pay attention to. The previous experimental studies [28,29] have suggested that the values of q are generally small (0.04–0.38), while it depends on many factors such as timber material, fastener spacing and connection type. In this paper, relatively small values of q within the range derived from the experiments are considered for MDOF models. On the other hand, q = 1 is considered in the numerical analysis on SDOF systems for purpose of verifying the applicability of equivalent linearization formulas to bilinear models, and it should not be considered as a suggestion of using q = 1 for timber structures. In design practice, slip strength ratio q should be carefully determined because it determines the hysteretic shape and thus the energy dissipation of timber structure. If there is no available data that have been derived from similar conditions, further experiments should be performed to obtain reliable values of q. No iteration processes are required in the proposed methodology, and the design can be realized by using common data processing programs (e.g., Excel) as long as the hysteretic parameters, including q, of the structure are given. The method is a simpler and less time-consuming alternative to existing methods and is oriented toward practical use.

Acknowledgements Grateful thanks are due to the reviewers and the editor for their careful work and thoughtful suggestions that have helped improve this paper substantially. The authors also thank Dr. Kazuhiro Matsuda of the Tokyo Institute of Technology for providing useful comments on the slip hysteretic model. Financial support from the National Natural Science Foundation of China (Grant No. 51208405), the Natural Science Foundation of Hubei Province of China (Grant No. 2015CFB489), and the Open Foundation of National Engineering Laboratory for High Speed Railway Construction (No. HSR2013021) are gratefully acknowledged. References [1] Waugh A, Wells M, Lindegar M. Tall timber buildings: a application of solid timber constructions in multi-story buildings. Proceedings of the international convention of society of wood science and technology and United Nations economic commission for Europe – Timber Committee 2010: Paper SD-5. Geneva, Switzerland.

213

[2] Shenton III Harry W, Dinehart David W, Elliott Timothy E. Stiffness and energy degradation of wood frame shear walls. Can J Civ Eng 1998;25:412–23. [3] Rinaldin G, Fragiacomo M. Non-linear simulation of shaking-table tests on 3and 7-storey X-Lam timber buildings. Eng Struct 2016;113:133–48. [4] Rainer JH, Karacabeyli E. Performance of wood-framed construction in earthquakes. Proceedings of the 12th WCEE, New Zealand. Paper No. 2454; 2000. [5] Xu LY, Nie X, Fan JS. Cyclic behaviour of low-yield-point steel shear panel dampers. Eng Struct 2016;126:391–404. [6] Donaire-Ávilaa J, Mollaiolib F, Lucchinib A, Benavent-Climent A. Intensity measures for the seismic response prediction of mid-rise buildings with hysteretic dampers. Eng Struct 2015;102:278–95. [7] Filiatrault A. Analytical predictions of the seismic response of friction damped timber shear walls. Earthquake Eng Struct Dynam 1990;19:259–73. [8] Higgins C. Hysteretic dampers for wood frame shear walls. In: Chang PC, editor. Proceeding of the 2001 structures congress and exposition. Washington, DC, US: ASCE; 2001. [9] Mizuno K, Yamada K. Development of the shear wall with friction mechanism for wooden houses. AIJ Tokai chapter architectural research meeting 2009;47:177–8 [in Japanese]. [10] Lopez-Almansa F, Segues E, Cantalapiedra IR. A new steel framing system for seismic protection of timber platform frame buildings. Implementation with hysteretic energy dissipators. Earthq Eng Struct Dynam 2014;44(8):1181–202. [11] Ruiz SE, Urrego E, Silva FL. Influence of the spatial distribution of energydissipating bracing elements on the seismic response of multistory frames. Earthq Eng Struct Dynam 1995;24:1511–25. [12] Gluck N, Reinhorn AM, Gluch J, Levy R. Design of supplemental dampers for control of structures. J Struct Eng 1996;122:1394–9. [13] Uetani H, Tuji M, Takewaki I. Application of an optimum design method to practical building frames with viscous dampers and hysteretic dampers. Eng Struct 2003;25:579–92. [14] Benavent CA. An energy-based method for seismic retrofit of existing frames using hysteretic dampers. Soil Dynam Earthq Eng 2011;31:1385–96. [15] Takewaki I. Optimal damper placement for planar building frames using transfer functions. Struct Multidisc Optim 2000;20:280–7. [16] Aydin E, Boduroglu MH, Guney D. Optimal damper distribution for seismic rehabilitation of planar building structures. Eng Struct 2007;29(2):176–85. [17] Ashour SA, Hanson RD. Elastic seismic response of buildings with supplemental damping Report No. UMCE 87-01. Ann Arbor, MI: Department of Civil Engineering, University of Michigan; 1987. [18] Zhang RH, Soong TT. Seismic design of viscoelastic dampers for structural applications. J Struct Eng 1992;118(5):1375–92. [19] Shukla AK, Datta TK. Optimal use of viscoelastic dampers in building frames for seismic response. J Struct Eng 1999;125(4):401–9. [20] Lin YY, Tsai MH, Hwang JS, Chang KC. Direct displacement-based design for building with passive energy dissipation systems. Eng Struct 2003;25:25–37. [21] Kim JK, Seo YL. Seismic design of low-rise steel frames with bucklingrestrained braces. Eng Struct 2004;26:543–51. [22] Vargas R, Bruneau M. Seismic design of multi-story buildings with metallic structural fuses. In: Proceedings of the eighth U.S. national conference on earthquake engineering, San Francisco, California. [23] Sullivan TJ, Lago A. Towards a simplified direct DBD procedure for the seismic design of moment resisting frames with viscous dampers. Eng Struct 2012;35:140–8. [24] Martínez CA, Curadelli O, Compagnoni ME. Optimal placement of nonlinear hysteretic dampers on planar structures under seismic excitation. Eng Struct 2014;65:89–98. [25] Loss Cristiano. Displacement-based seismic design of timber structures Doctoral thesis. Italy: University of Trento; 2014. [26] Araki Y, Koshihara M, Ohashi Y, et al. A study on hysteresis model for earthquake response analysis of timber structures. J Struct Constr Eng 2004;579:79–85 [in Japanese]. [27] Isoda H, Kawai N. Hysteresis model of walls on Japanese conventional construction: study on seismic behavior of wooden construction. J Struct Constr Eng 2007;616:157–63 [in Japanese]. [28] Sakata H, Kasai K, Wada A, et al. Shaking table tests of wood frames with velocity-dependent dampers. J Struct Constr Eng 2007;615:161–8 [in Japanese]. [29] Bahmani P, Van de Lindt JW, Gershfeld M, et al. Experimental seismic behavior of a full-scale four-story soft-story wood-frame building with retrofits. I: Building design, retrofit methodology, and numerical validation. J Struct Eng 2015;142(4):E4014003. [30] Soong TT, Dargush GF. Passive energy dissipation systems in structural engineering. John Wiley & Sons; 1997. [31] Dwairi HM, Kowalsky MJ, Nau JM. Equivalent damping in support of direct displacement-based design. J Earthq Eng 2007;11:512–30. [32] Mazza F. Nonlinear seismic analysis of unsymmetric-plan structures retrofitted by hysteretic damped braces. Bull Earthq Eng 2016;14:1311–31. [33] Dicleli M, Buddaram S. Equivalent linear analysis of seismic-isolated bridges subjected to near-fault ground motions with forward rupture directivity effect. Eng Struct 2007;29(1):21–32. [34] Matsagar VA, Jangid RS. Influence of isolator characteristics on the response of base-isolated structures. Eng Struct 2004;26(12):1735–49. [35] Kasai K, Pu W, Nishihara K. Peak response evaluation method for sliphysteretic structure controlled by visco-elastic damper. Proceedings of sixth international conference on urban earthquake engineering. Tokyo, Japan; 2009.

214

W. Pu et al. / Engineering Structures 128 (2016) 199–214

[36] Chopra AK. Dynamics of structures: theory and application to earthquake engineering. 3rd ed.. Englewood Cliffs, NJ: Prentice Hall; 2007. p. 103. [37] Newmark NM, Rosenblueth E. Fundamentals of earthquake engineering. Prentice-Hall Inc.; 1971. [38] Priestley MJN, Calvi GM, Kowalsky MJ. Displacement-based seismic design of structures. Pavia, Italy: IUSS Press; 2007. [39] Yanagawa Y, Sekiguchi Y, Usami M. A analytical study of viscous damping in proportion to momentary stiffness. Summaries of technical papers of annual meeting (B2, Structure II); 2004. p. 1001–2 [in Japanese]. [40] Jehel P, Leger P, Ibrahimbegovic A. Initial vs. tangent stiffness-based Rayleigh damping in inelastic time history seismic analyses. Earthq Eng Struct Dynam 2013;42(4):581–602. [41] Code for seismic design of buildings. National Standard of the People’s Republic of China (GB50011-2010); 2010. [42] SeismoMatch. . [43] Kasai K, Fu Y, Watanabe A. Passive control systems for seismic damage mitigation. J Struct Eng 1998;124(5):501–12. [44] AIJ. Recommendation for structural calculation of traditional wood buildings by calculation of response and limit strength. Architectural Institute of Japan; 2002 [in Japanese].

[45] Japan Society of Seismic Isolation. Design and construction manual for passive controlled structures. 2nd ed.; 2005. [46] ATC. Recommended methodology for seismic evaluation and retrofit of existing concrete buildings 1996. Report no. ATC-40, Applied Technology Council, Redwood City, California; 1996. [47] FEMA. NEHRP commentary on the guidelines for the seismic rehabilitation of buildings. Washington DC: FEMA-274; 1997. [48] Calvi GM, Sullivan TJ. Development of a model code for direct displacement based seismic design. Atti di Linea IV, Convegno Finale del progetto RELUIS, 1– 3 Aprile, Napoli, Italia; 2009. [49] Maison Bruce F. PC-ANSR: a computer program for nonlinear structural analysis [CP/OL]. The earthquake engineering online archive. California: Berkeley: NISEE e-Library; 1992. . [50] Pu W, Kasai K, Kabando EK, et al. Evaluation of the damping modification factor for structures subjected to near-fault ground motions. Bull Earthq Eng 2016;14(6). 1519-154.