An energy-based method for seismic retrofit of existing frames using hysteretic dampers

Soil Dynamics and Earthquake Engineering 31 (2011) 1385–1396

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An energy-based method for seismic retrofit of existing frames using hysteretic dampers Amadeo Benavent-Climent n Department of Structural Mechanics, University of Granada, Edificio Polite´cnico Fuentenueva, Granada, Spain

a r t i c l e i n f o

abstract

Article history: Received 12 December 2010 Received in revised form 14 May 2011 Accepted 16 May 2011 Available online 14 June 2011

This paper proposes a method for the seismic retrofitting of existing frames by adding hysteretic energy dissipating devices (EDDs). The procedure is based on the energy balance of the structure, and it is used to determine the lateral strength, the lateral stiffness and the energy dissipation capacity of the EDDs needed in each story to achieve prescribed target performance levels for a given earthquake hazard. The performance levels are governed by the maximum lateral displacement. The earthquake hazard is characterized in terms of input energy and several seismological parameters, and further takes into account the proximity of the earthquake to the source. The proposed method deals with the effect of the EDDs explicitly in terms of hysteretic energy, bypassing equivalent viscous damping approximations, and directly quantifies the cumulative damage induced in the EDDs. The validity of the method is assessed numerically through nonlinear dynamic response analyses with near-fault and far-field ground motions, as well as experimentally through dynamic shaking table tests. & 2011 Elsevier Ltd. All rights reserved.

1. Introduction Energy dissipating devices (EDDs), also known as dampers, are giving rise to technology capable of minimizing interstory drift and increasing the earthquake resistance of buildings, to achieve performance-based seismic design. EDDs have been used for a decade, and continue to attract attention in the field of earthquake engineering. A main application, particularly in Europe, is in the seismic rehabilitation of buildings. Mechanisms studied and used for passive energy dissipation include metal yielding, phase transformation of metals, friction sliding, fluid orificing, and deformation of viscoelastic solids or liquids. The EDDs based on the yielding of metals—commonly known as hysteretic dampers—are among the most popular. The present study focuses on the application of hysteretic EEDs to the seismic retrofit of buildings. In order to develop cost-effective retrofit solutions, it is of paramount importance to have practical and sufficiently accurate design procedures, able to define the EDDs that satisfy target building performance levels for a given earthquake hazard. There are important differences between new construction design and seismic retrofit design, using EDDs. When EDDs are used in new structures, the main frame can be designed to contribute to dissipating, through plastic deformations, a portion of the total energy input by the earthquake. This can relax the

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requirements of the EDDs (strength, stiffness, ultimate energy dissipation capacity) and lower their cost. In contrast, most existing structures were built according to past seismic codes and their energy dissipation capacity, if any, is very limited. The seismic retrofit design of such structures can therefore not rely on the contribution of the main frame to dissipate energy; the design of the dampers must guarantee that, under the design earthquake, the main structure remains ‘‘basically’’ within the elastic range. In new construction design of frames with EDD, the members can be easily dimensioned to accommodate the forces transferred by the EDDs, while in seismic retrofit design the strength, the stiffness and the location of the EDDs must be carefully studied in order to control torsion effects and to avoid unexpected failures in the main frame before the dampers attain their maximum strength. The design of new lateral force-resisting elements consisting of EDDs for the seismic retrofit of existing structures must consider important issues such as: (i) the connection of the EDDs with the existing frame; (ii) the compatibility of deformation with the existing lateral force-resisting or gravity load-carrying system; (iii) the extent to which the new system relieves the existing structure of load or deformation at all levels; (iv) the significance of the mass added by the EDDs, and (v) the need for extensive new foundations. The first guidelines for the application of hysteretic EDDs to building rehabilitation are the standards developed by the Federal Emergency Management Agency (FEMA) in Reports 273, 274 [1] and 356 [2], which contain both linear and nonlinear static procedures. In the linear static procedures, the energy dissipation

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afforded by the EDDs is accounted for indirectly—by reducing the design earthquake loads or the response spectrum through a factor that depends on an equivalent effective viscous damping. In the nonlinear static procedures (pushover analysis), the effect of the hysteretic EDDs is either recognized indirectly, by the increase in building stiffness afforded by such devices, or directly, by means of an equivalent effective viscous damping similar to that used in linear static procedures. The linear static procedures oversimplify the highly nonlinear random response of the entire building-device structure, however, and their use is subjected to severe restrictions (e.g. the effective damping afforded by the EDD cannot exceed 30% of critical damping in the fundamental mode). The nonlinear static pushover procedures likewise have deficiencies and theoretical inconsistencies [3,4]. One is that they replace the hysteretic damping provided by the EDDs with an equivalent viscous damping, although there is no physical principle that justifies the existence of a stable relationship between hysteretic energy dissipation of the maximum displacement excursion and equivalent viscous damping, particularly for highly inelastic systems. Moreover, the current design philosophy—basically force-based and displacement-based procedures—considers cyclic degradation only in an implicit manner; it is difficult to include the effect of duration-related cumulative damage [5], which can lead to unsatisfactory performance at levels associated with repairable damage and collapse prevention [6]. In the literature, many experimental and theoretical studies have focused on EDDs and buildings equipped with such devices. Ruiz et al. [7] suggested determining the strength of the EDDs using dynamic step-by-step analyses in a trial and error procedure to attain minimum damage to the structure and maximum nonlinear behavior in the EDDs. Gluck et al. [8] proposed a method of designing supplemental EDDs based on optimal linear control theory. Rivakov and Gluck [9] proposed replacing the hysteretic EDDs with equivalent viscous dampers that dissipate the same energy per cycle, and applying optimal control theory to determine the levels of viscous damping in each story. Uetani et al. [10] developed a computer program based on optimization techniques to determine the required story stiffness of a building with EDDs so that it would exhibit a specified distribution of interstory drifts. Karami et al. [11] proposed a method based on iterative time-history response analyses. Choi and Kim [12] developed an energy-based design method for a particular type of EDD—the buckling-restrained brace—that involves performing series of time-history analyses. Lin et al. [13] developed a directdisplacement-based procedure for nonlinear viscous dampers. Other interesting proposals have been made in recent years with respect to design methodologies for metallic hysteretic dampers [14–16], several of which entail deformation-based design procedures [17–19]. Over the past few decades, the seismic design methodology based on energy first proposed by Housner [20] has been largely accepted [21–24] and has been recently introduced in the Japanese seismic code [25]. This paper proposes an energy-based procedure for the seismic upgrading of existing structures by adding hysteretic EDDs. The procedure is grounded on recent research on the cumulative demand of earthquake ground motions [26]. The intention is to develop a practical design method to be used as a tool for conventional low- to medium-rise structures, without resorting to nonlinear time history analyses. The method determines the lateral stiffness, the lateral strength and the energy dissipation capacity that must be provided by the dampers installed in each story, so that the main existing frame does not exceed a limit of drift beyond which plastic deformations (i.e. structural damage) would take place. Once the required strength, stiffness and energy dissipation capacity of the dampers are determined, an appropriate EDD and a solution for connecting

the dampers to the main frame that satisfies these requirements must be selected. This second part must take into account the performance of the connection, which is an important issue, but lies outside the scope of this paper. The connection of the damper is a big problem in the seismic retrofit design. When RC frame structures are strengthened by using the brace-type dampers, the stress concentration in the connection between the RC beamcolumn joints and the damper devices must be carefully studied to avoid damage on the concrete. Problems can also arise from the need of tension-type connections between the damper and the frame that require drilling for anchorage of tension/shear connections. The connection details must be designed to satisfy the requirement of rigid connection; that is, neither uplift, sliding nor damage occurs under a design working force larger than the maximum force expected on the dampers. It is very important to avoid relative displacements between the main frame and the end of the dampers at the connection, especially in stiff structures. This is a point of paramount importance so that the damper can dissipate energy at small lateral drifts of the main frame, and thus protect efficiently the construction from structural and non-structural damage.

2. Background on energy-based design The equation of motion of an inelastic single-degree-of-freedom system (SDOF) subjected to a unidirectional horizontal ground motion can be written as follows: My€ þC y_ þ Q ðyÞ ¼ M z€ g ,

ð1Þ

where M is the mass, C is the damping coefficient, Q(y) is the restoring force, y is the relative displacement, y_ and y€ its first and second derivates with respect to time, and z€ g is the ground _ acceleration. Multiplying (1) by dy ¼ ydt and integrating over the entire duration of the earthquake, i.e. from t ¼0 to t ¼to, the energy balance equation becomes Wk þ Wx þ Ws ¼ E:

ð2Þ R

_ ydt € is the kinetic energy, In the left-hand-side term, Wk ¼ yM R R _ Wx ¼ C y_ 2 dt is the damping energy, and Ws ¼ Q ðyÞydt is the absorbed energy, which is composed of the recoverable elastic strain energy, Wse, and the irrecoverable plastic energy, Wp, i.e. R _ Ws ¼Wse þWp. The right-hand-side term, E ¼ Mz€ g ydt is, by definition, the input energy, which can be expressed in the form of an equivalent velocity VE as rffiffiffiffiffiffi 2E VE ¼ : ð3Þ M Since Wk þWse is the elastic vibrational energy, We, the Eq. (2) can be rewritten as We þ Wp ¼ EWx :

ð4Þ

Further, We þ Wp can also be expressed in the form of an equivalent velocity VD so that We þ Wp ¼

MVD2 : 2

ð5Þ

Eq. (4) holds also for a multi-degree-of-freedom system (MDOF) subjected to a unidirectional horizontal ground motion if the above expression for Wk, Wx, Ws and E is replaced by Wk ¼ R T R R R € _ y_ Mydt, Wx ¼ y_ T Cydt, Ws ¼ y_ T Q dt and E ¼  y_ T Mrz€ g dt, respectively. Here, M is the mass matrix, C the damping matrix € € and Q(t) the restoring force vector; yðtÞ and yðtÞ are the acceleration and velocity vectors relative to the ground respectively; and r represents the displacement vector yðtÞ resulting from a unit support displacement.

A. Benavent-Climent / Soil Dynamics and Earthquake Engineering 31 (2011) 1385–1396

monotonic loading is represented by the elastic-perfectly plastic model shown with bold lines in Fig. 1a. The lateral load– displacement relationship, sQi–di, of a given i-th story accounting only for the EDDs is also assumed to be of the elastic-perfectlyplastic type and is defined by the lateral yield strength sQyi and the initial stiffness ski as shown in Fig. 1b. The lateral shear forceinterstory drift relationship of the entire building-device structure of a i-th story under monotonic loading, Qi  di, is obtained by summing up the forces sustained by each element at a given displacement level di, as shown in Fig. 1c. The goals of the proposed procedure are: (i) to determine the sQyi and ski of the EDDs needed in each story to achieve the required building performance levels for a given earthquake hazard; and (ii) to evaluate the energy dissipation demand on the EDDs. Usually, one of the objectives of the EDDs is to avoid inelastic deformations in the structural elements outside the EDDs, and many researchers consider this objective as one of the basic requirements for a system with EDDs [12,33]. Further, many existing structures were designed according to past seismic codes and their ductility, if any, is very limited. In the case of reinforced concrete frames, it is worth noting that although they do not have a large elastic range, especially if compared with steel frames, there is a range of lateral drift within which the behavior can be assumed to be ‘‘basically’’ elastic and the damage on the main frame would be null or negligible. Accordingly, it is imposed that

On the basis of numerous response analyses, Akiyama [22] concluded that, in general MDOF damped inelastic systems, the total input energy supplied by the earthquake E—and consequently VE—coincides approximately with that of an equivalent elastic SDOF system with mass M equal to the total mass of the MDOF system, and period T equal to that of its first vibration mode. This conclusion has been validated experimentally [23]. In the energy-based seismic design approach, the energy input spectrum in the form of equivalent velocity VE T characterizes the loading effect of the earthquake. Design input energy spectra VE  T [22,27,21] have been proposed in past studies. The cumulative damage of the structure is strongly related to the plastic strain energy Wp. The sum of Wp and the elastic vibrational energy We is what Housner [20] called the energy that damages a structure subjected to seismic action. From Eqs.(2)–(5) it follows that for undamped systems VD ¼VE; otherwise, the difference between VD and VE is the energy dissipated by the inherent damping of the structure. Several empirical expressions have been proposed that allow us to obtain VD from VE [22,27–30]. Moreover, attenuation relationships have been established for use in energy-based seismic design [5] that directly provide Ws—the absorbed energy—for a given earthquake magnitude, source-to-site distance, site class and ductility factor, in terms of an equivalent velocity Va defined by rffiffiffiffiffiffiffiffiffiffi 2WS Va ¼ : ð6Þ M

dmax i r f dyi ,

Although VD T spectra have been recently introduced in some building codes [25], characterization of the design earthquake in terms of energy input spectra VD  T is not as common for professionals as in the form of absolute acceleration spectra Sa T. Energy input spectra in terms of VD  T can be obtained from the Sa  T spectra on the basis of the following considerations: (i) over the range of damping ratios exhibited by actual structures—say less than 0.10—the spectral absolute acceleration Sa of damped elastic SDOF systems is related to the pseudo-velocity spectral response, Spv, by the well known expression Sa ¼ oSpv, where o is the circular frequency. (ii) Hudson [31,32] demonstrated that, except in the case of very long period oscillators, the spectral relative velocity, Sv, differs very little from Spv. (iii) Akiyama [22] showed that Sv provides a good approximation of VD, and validated Housner’s [20] assumption that VD can be taken as equal to Sv for the purposes of earthquake-resistant design.

The proposed method requires knowledge of the mass, mi, the lateral yield strength, fQyi, the initial stiffness fki of each story i, and the fundamental period T1 of the existing building (referred to as main structure hereafter). The seismic retrofit strategy proposed consists of adding hysteretic EDDs in each story. The existing structure and the added EDDs are arranged so as to form a dual system consisting of two inelastic springs connected in parallel as shown in Fig. 1. The lateral load–displacement relationship, fQi–di, of a given i-th story of the main structure—without EDDs—under

f Qyi

ð8Þ

where sdyi ( ¼ sQyi /ski) is the yield deformation of the EDDs installed in that story. The maximum lateral force sustained by the main structure, fQmax i, is fQmax i ¼ dmax i  fki. For the buildingdevice structure surviving the earthquake, the plastic strain energy accumulated in the i-th story, Wpi, must not exceed the

Qi Qyi

0.6f Qyi

sQyi i f yi

Qyi ¼ s Qyi þ s dyif ki ,

s Qi

Pushover analysis Idealized curve tan-1fki

ð7Þ

where dmax i is the maximum interstory drift of the entire building-device structure and fdyi ( ¼ fQyi/fki) is the yielding interstory drift of the main structure. The condition imposed by Eq. (7) is intended to avoid the severe degradation effects in the response of the main structure that may arise from stiffness degradation and deterioration as well as shear effects in some types of structures. In structures for which such degradation cannot be avoided, the proposed method is not applicable. Such is the case, for example, of precast structures with slender columns and/or existing reinforced concrete frame systems designed primarily for gravity loads and with high shear degradation. Yet the proposed formulation is indeed applicable to symmetric structures or systems with prevalent translation modes of vibration. If the main existing structure has important irregularities that may place extraordinary displacement demands on elements due to torsional response, measures must be adopted to mitigate torsional effects. One such measure is to locate the EDD so that they can balance the stiffness and make the mass and stiffness centers very close. The lateral yield strength of the entire building-device structure at the i-th story, Qyi, is simply

3. Modeling and design criteria

f Qi

1387

tan-1ski syi

i

i syi

yi

Fig. 1. Idealized interstory drift-shear force curve of each story. (a) Existing (main) structure. (b) Energy dissipating device. (c) Dual system

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ultimate energy dissipation capacity of the EDDs installed in that story, Wui. In turn, Wpi and Wui can be expressed in the form of two non-dimensional coefficients, Zi and Zui, defined by

Zi ¼

Wpi s Qyi s dyi

;

Zui ¼

Wui s Qyi s dyi

,

ð9Þ

thus, the above condition can be written as

Zi r Zui :

ð10Þ

4. Formulation of the method For the sake of convenience, dmax i, Qyi, fQmax i, sQyi and fk1 will be also expressed herein in non-dimensional form by the plastic deformation ratio mi, the shear-force coefficients ai, famax i, sai, and the stiffness ratio w1, respectively, which are defined as follows:

mi ¼

ðdmaxi s dyi Þ ; s dyi

s Qyi

a ¼ PN

s i

k ¼ i mk g

ai ¼ PN

Qyi

k¼i

;

w1 ¼

f k1

keq

:

mk g

;

Qmaxi

f f amaxi ¼ PN

k¼i

mk g

;

ð11Þ

Here N is the total number of stories, g is the acceleration of the gravity, and keq is the stiffness of an equivalent SDOF system of P mass M ¼ mj and period equal to the fundamental period of the main structure, T1, i.e. keq ¼4p2M/T21. 4.1. Stiffness and strength distribution of the EDDs among the stories The ratio between the lateral stiffness provided by the EDDs and the lateral stiffness of the main structure in each story is referred to as Ki ¼

s ki f ki

distribution of sai be that given by Eq. (13) (Appendix B):

a ¼ ai s a1

s i

4.2. Lateral strength to be provided by the EDDs of the first story Once the stiffness ratios Ki and the lateral shear-force coefficient distribution ai are determined, the lateral shear-force coefficient to be provided by the EDDs of the first story, sa1, must be calculated in order to obtain the required lateral shear force coefficient of the EDDs in the other stories sai with Eq. (14). The equations that govern the sa1 required for a given seismic hazard and building performance level are derived next by establishing the energy balance of the structure. When using EDDs, the yield displacement sdyi of the EDDs is made smaller than that of the main structure, fdyi, so that the EDDs begin dissipating energy before the main structure might yield. Moreover, sQyi is commonly smaller than fQyi. As a result, the elastic strain energy stored by the EDDs is commonly negligible in comparison to that of the main structure, and the elastic vibrational energy of the whole building, We, can be approximated from the maximum shear force sustained by the main structure on the first story as follows (refer to Eq. (A.25) and the last paragraph of Appendix A) 2

We ¼

ð13Þ Here x ¼ ði1Þ=N, fkN is the lateral stiffness of the uppermost N-th story of the main structure, whereas TG defines the change of slope of the VD  T bilinear spectra. TG may be called the predominant period of the ground motion [29]. Further, from the definition of sai and Ki given by Eqs. (11) and (12) the following relation must be satisfied so that the

ð14Þ

Of course, instead of Eq. (13), a more refined lateral strength distribution could be used from dynamic analysis by applying a trial and error iterative method. This would not alter the proposed method at all. However, it implies selecting a set of earthquake records, conducting many nonlinear dynamic response analyses, and averaging the distribution derived for each record, which can be a very cumbersome process. Such a process should be used for systems with a degrading type of response under earthquake loading.

ð12Þ

There is no need to make Ki equal in all stories, although this criterion has been used on occasion in the past. The lateral strength distribution of the entire building-device structure, Qyi/Qy1, can be expressed in terms of shear-force coefficients by ai ¼ ai =a1 . The criterion adopted in the proposed method to determine the ai distribution is to attain an even distribution of damage among the EDDs. The damage in the EDDs installed in a given story i can be characterized by the non-dimensional parameter Zi defined by Eq. (9). Past studies [22] showed that the strength distribution ai that makes Zi equal in all stories (Zi ¼ Z) in a low-to- medium rise multi-story building subjected to seismic loads coincides approximately with the maximum shear-force distribution in an equivalent elastic undamped shear strut with similar lateral stiffness distribution along its height. The derivation of the ‘‘exact’’ shear-force coefficient distribution ai for an elastic undamped shear strut subjected to a design earthquake characterized by a bilinear VD  T spectrum is explained in detail in Appendix A. The ‘‘exact’’ solution cannot be expressed with simple equations, but it can be approximated for design purposes with the following expression: " ! ! # k k a T T ai ¼ i ¼ exp 10:02 f 1 0:16 1 x 0:50:05 f 1 0:3 1 x2 : a1 TG TG f kN f kN

Ki ðK1 þ1Þ : K1 ðKi þ1Þ

Mg 2 T12 f amax1 : 2 4p2

ð15Þ

From Eq. (9) and taking into account the coefficients defined in Eq. (11), the plastic strain energy accumulated in the i-th story Wpi can be expressed as follows: Wpi ¼ Zis Qyis dyi ¼ Zi

2 s Qyi s ki

¼ Zis a2i

N X k¼i

!2 mk g

1 s ki

:

ð16Þ

Provided that the strength distribution given by Eq. (13) is adopted, the normalized plastic strain energy Zi can be assumed equal in all stories, i.e. Zi ¼ Z. Thus, taking into account Eq. (12) and using the non-dimensional parameters sai and ai defined above, the total plastic strain energy dissipated by the EDDs of the whole structure, Wp, can be expressed in terms of the plastic strain energy dissipated by the EDDs of the first story, Wp1, by introducing a new ratio g1 ¼Wp/Wp1, which is obtained as follows: 2 3 !2 N N P P 2 4Z a mk g =s ki 5 is i i¼1 k¼i Wp g1 ¼ ¼ Wp1 Z1s a21 M2 g 2 =s k1 (  ) N X mj  ðK1 þ1Þ2 f k1 Ki X ¼ ai , ð17Þ M ðKi þ 1Þ f ki K1 i¼1 thus Wp ¼ g1 Wp1 ¼ g1s Qy1s dy1 Z ¼ ¼

g1s a21 M2 g 2 Z K1 f k1

2 g1s Qy1 Z s k1

g a2 M2 g 2 Z g1s a21 Mg 2 ZT12 ¼ 1s 1 ¼ : K1 w1 keq 4p2 K1 w1

ð18Þ

A. Benavent-Climent / Soil Dynamics and Earthquake Engineering 31 (2011) 1385–1396

Substituting Eqs. (15) and (18) in Eq. (5) gives " 2 # Mg 2 T12 f amax1 MVD2 g1 2 þ : Za ¼ 2 K1 w1 s 1 2 4p2

substituting in Eq. (25) gives ð19Þ

Now, a new parameter ae is introduced. ae is defined as the base shear-force coefficient that the main structure should have in order to absorb by itself—i.e. without EDDs—the amount of input energy MVD2 =2 supplied by the earthquake. The expression for ae can be obtained placing sa1 ¼0 in Eq. (19)—this implies ignoring the EDDs—and solving for famax1( ¼ ae), which gives

ae ¼

2pVD : gT1

ð20Þ

Using Eqs. (20), Eq. (19) can be rewritten as follows: f

a2max1 2

þ

g1 a2 Zs a21 ¼ e : K1 w1 2

ð21Þ

The relation between Zi and mi is a key parameter in seismic design methodologies based on the energy concept, and it has been addressed in different ways in the past [23,34,35]. Based on the results of regression analyses performed with 128 near-fault and 122 far-field earthquake records, Manfredi et al. [26] proposed the following formulae for estimating the equivalent number of plastic cycles neq at the maximum value of plastic excursion that a SDOF system of mass m, elastic period T and yielding force Fy must develop in order to dissipate the total amount of hysteretic energy input by the earthquake rffiffiffiffiffiffiffiffi TNH ð22Þ neq ¼ 1 þ c1 Id ðR1Þc2 : T Here TNH is the initial period of medium period region in the Newmark and Hall [36] spectral representation. R is the reduction factor defined as R ¼mSa/Fy where Sa is the elastic spectral acceleration. Id is a seismological parameter [34] defined by R to 2 € 0 z g dt ð23Þ Id ¼ PGA  PGV where PGA and PGV are the peak ground acceleration and velocity, respectively. In Eq. (22), Manfredi et al. [26] proposed to take c1 ¼0.23, c2 ¼ 0.4 for near-fault earthquakes; and c1 ¼0.18, c2 ¼ 0.6 for far-field earthquakes. In order to apply the Manfredi et al. equation to the proposed method, the multi-story structure is assimilated to an equivalent SDOF system with elastic period T¼T1, mass m¼M and Fy ¼ sQy1 þ fk1  sdy1. Based on this equivalence, while taking into account that Sa is approximately equal to (2p/T)Sv, and that the elastic spectral velocity Sv coincides approximately with VD [5,22], Eq. (22) is rewritten as sffiffiffiffiffiffiffiffi c2 TNH K1 ae 1 : ð24Þ neq ¼ 1 þ c1 Id T1 ðK1 þ 1Þs a1 For the EDDs with elastic-perfectly-plastic characteristics dealt with in this study, neq is, by definition [26]: neq ¼Wpi/ [sQyi(dmax i  sdyi)], which coincides with Zi/mi. In the proposed method, the same neq( ¼ Zi/mi)—given by Eq. (24)—is adopted for all stories. Since Zi was also assumed as constant, Zi ¼ Z, the maximum plastic deformation ratio mi has the same value mi ¼ m ( ¼ Z/neq) in all stories. On the other hand, since the condition given by Eq. (7) was adopted, the maximum base shear-force coefficient of the main structure famax1 is f

amax1 ¼

dmax1 f k1 Mg

f

amax1 ¼

ð25Þ

From the definition of mi ( ¼ m)—Eq. (11)—particularized for the first story, it is obtained that dmax1 ¼ sdy1(m þ1), and

m þ 1Þ

s dy1f k1 ð

Mg

¼

m þ 1Þ

s dy1s k1 ð

K1 Mg

¼

m þ 1Þ

s Qy1 ð

K1 Mg

¼

a m þ1Þ

s 1ð

K1

:

ð26Þ Substituting Eq. (26) in Eq. (21), recalling that m ¼ mi ¼ Z/neq and solving for m gives ffi (sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )   neq g1 2 2neq g1 a2 neq g1 m ¼ K1 þ þ e2  1: ð27Þ w1 K1 w1 w1 s a1 Noting that for the other stories m ¼(dmax i  sdyi)/sdyi, using Eqs. (14) and (27), and solving for dmax i gives the equation that predicts the maximum displacement of a given story i P ai a ðK1 þ 1Þð Nj¼ i mj gÞ dmaxi ¼ s 1 f ki ðKi þ 1Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi (s )  neq g1 2 2neq g1 a2e neq g1 þ þ 2  : ð28Þ w1 K1 w1 w1 s a1

5. The procedure In the proposed method, the characteristics of the existing main structure—without EDDs—to be retrofitted (i.e. mi, fki, fdyi and T1) are assumed to be known data. mi, fki, fdyi and T1 can be estimated by using approximate formulae [22], or by creating a finite element based model and performing a pushover analysis using a triangular lateral load distribution. In the latter option, a fQi–di curve is obtained for each story—the dotted line in Fig. 1a—from which fki and fdyi can be estimated by using the secant stiffness at 60% of the yield strength, as suggested by FEMA 273 [1], and T1 is obtained from an eigenvalue analysis. The goal of the proposed method is to determine the lateral stiffness ski, lateral strength sQyi, and the normalized energy dissipation demand Z of the EDDs to be installed in each story, so that the entire building-device structure satisfies predetermined performance levels defined by the maximum allowed interstory drift, dallow i, for a given earthquake hazard characterized by VD, TG, TNH, Id and the proximity to the source. The basic steps involved in the procedure are as follows. Step 1: Characterize the design earthquake in terms of a bilinear VD  T spectra defined by the maximum demand VDmax and the predominant period TG—i.e. VD ¼TVDmax/TG for ToTG, VD ¼VDmax for TZTG—and the values of the seismological parameters Id, TNH, c1 and c2. Step 2: Prescribe the maximum interstory drift allowed in each story, dallow i, in accordance with the acceptance criteria for building components at the target building rehabilitation performance level. Step 3: Calculate ai for each story i with Eq. (13), ae with Eq. (20) and w1 with Eq. (11). Step 4: Choose a set of values for Ki, and compute g1 with Eq. (17). From i¼1 to i ¼N proceed for each story as follows. Starting with sa1 ¼0, iterate in Eq. (28)—with neq given by Eq. (24)—increasing the values of sa1 until the predicted dmax i gets close to dallow i within an acceptable tolerance (for example, 5% of dallow i). In these iterations, sa1 shall not be larger than the value given by the following expression, so that sdyi r fdyi (see Appendix C) s 1r

a

:

1389

f dyif ki K1 ðKi þ1Þ

ai ðK1 þ 1Þ

N P

:

ð29Þ

mk g

k¼i

If in a given story i it is not possible to find a sa1 that makes

dmax i close enough to dallow i, restart step 4 with larger values

K ¼7, Z ¼5 T1 ¼ 1.02 s K¼3, Z ¼ 22

0.06 0.05 0.04 0.04 0.03 0.03 246 210 222 288 276 270

K¼ 8, Z ¼ 8

0.17 0.14 0.11 0.09 0.08 0.08 656 560 592 768 736 720 (0.62) (0.73) (0.68) (0.81) (0.80) (0.54) 97 77 62 9 8 7 6 5 4 3 2 1

K¼ 4, Z ¼ 61 K ¼11, Z ¼10 T1 ¼ 0.37 s

1.57 (0.57) 2.22 (0.81) 2.24 (0.60)

1067 847 682

0.24 388 0.18 308 0.14 248

0.06 0.04 0.03

T1 ¼ 0.69 s

1.71 2.00 1.87 2.24 2.21 2.04 82 70 74 96 92 90

(kN cm) fki

a

s i

(kN/cm) s ki

a

s i

(kN/cm) s ki

d (cm) (%)

f yi

(kN/cm) f ki

K¼ 4, Z ¼ 35

0.04 0.04 0.03 0.03 0.03 0.02 0.02 0.02 0.02 324 276 284 372 372 372 404 376 424 0.18 0.14 0.12 0.10 0.08 0.08 0.07 0.07 0.06 567 483 497 651 651 651 707 658 742 (0.62) (0.73) (0.68) (0.75) (0.73) (0.71) (0.91) (0.89) (0.52) 1.72 2.01 1.88 2.05 2.01 1.95 2.51 2.46 1.95 81 69 71 93 93 93 101 94 106

a

s i

(kN/cm) s ki

a

s i

(kN/cm) s ki

d (cm) (%)

f yi

(kN/cm) f ki

a

s i

(kN/cm) ski f yi

d (cm) (%)

s ki

(kN/cm)

a

s i

2nd design earthquake (far-field) 1st design earthquake (near-fault) 2nd design earthquake (far-field) 1st design earthquake (near-fault)

Design of the EDDs Story 3-Story frame

Table 1 Properties of the main frames and hysteretic dampers.

To illustrate the application of the proposed method, three reinforced concrete (RC) frames with wide beam-column connections of 3, 6 and 9 stories were retrofitted with hysteretic dampers. The structural configuration and detailing of the main structures corresponded to typical residential buildings located in the moderate-seismicity southern part of Spain, and they were designed according to earlier seismic codes, which required relatively reduced lateral strength and did not prescribe any seismic detailing. Table 1 shows the fki, fdyi obtained from a pushover analysis, and the T1 calculated by an eigenvalue analysis using the initial elastic stiffness of the members. fdyi is expressed in centimeters and as a percentage of the story height (into brackets). All the stories had the same mass mi ¼0.57 kNs2/cm. The story height was also equal in all stories (275 cm) excepting the first story (375 cm). This exceptionally small story height of 275 cm is typical of residential constructions built in Spain during the 70s, 80s and 90s of the 20th century. Two design earthquakes were considered, and for each of them the required EDDs were obtained. The first design earthquake represented a near-fault ground motion—thus c1 ¼0.23, c2 ¼0.4—and it was defined by VDmax ¼45 cm/s, TG ¼0.52 s, TNH ¼0.65 s and Id ¼7.5. The second design earthquake represented a far-field ground motion —thus c1 ¼0.18, c2 ¼0.6— and it was defined by VDmax ¼45 cm/s, TG ¼1 s, TNH ¼1 s and Id ¼16. The value VDmax ¼45 cm/s was determined from input energy spectra proposed for moderate-seismicity regions [27], and it is

6-Story frame

6. Examples of application and numerical validation

Main frame (without EDDs)

Design of the EDDs

9-Story frame

Once the lateral stiffness ski and the lateral strength sQyi of the EDDs to be installed in each story are determined, an appropriate type of hysteretic damper must be chosen. To this end, it is necessary to check that the normalized ultimate energy dissipation capacity of the damper Zui is larger than the demand Zi ( ¼ Z) as indicated by Eq.(10). Z is simply calculated by making sa1 ¼ sa1max in Eq. (24) to obtain neq, substituting this neq and sa1 ¼ sa1max in Eq. (27) to calculate m ( ¼ mi), and recalling that Z ¼neqm. The estimation of Zui for a given type of hysteretic damper is beyond the scope of this paper; yet a procedure is proposed by Benavent-Climent [37]. Finally, it is worth emphasizing that under earthquake excitations, structures undergo torsional as well as lateral motions if their mass and stiffness centers do not coincide. The coupling between lateral and torsional motions can significantly increase the lateral displacements as compared to those experienced by the same structure with a symmetric plan. Existing structures typically exhibit irregularities that may place extraordinary displacement demands on elements due to torsional response. In the proposed method, it is assumed that the EDD are appropriately placed to balance the stiffness, so that the mass and stiffness centers of each story are very close and torsion effects are mitigated.

Main frame (without EDDs)

Design of the EDDs

for Ki. Once the appropriate sa1 is obtained, keep this value as sa1i ¼ sa1 and proceed with the next story. The parameter sa1i represents the shear-force coefficient required for the EDDs of the first story so that the maximum interstory drift at the i-th story does not exceed the allowed limit dallow i. Step 5: Select the maximum of the sa1i, i.e. sa1max ¼max{sa1i}, which gives the required lateral strength for the EDDs of the first story. Obtain the lateral strength required in the other stories, sai, by making sa1 ¼ sa1max in Eq. (14). Calculate the lateral stiffness ski and the lateral strength sQyi required for the EDDs of each story taking into account that ski ¼Ki fki and P mk g. s Qyi ¼ s ai

1st design earthquake 2nd design earthquake (near-fault) (far-field)

A. Benavent-Climent / Soil Dynamics and Earthquake Engineering 31 (2011) 1385–1396

Main frame (without EDDs)

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associated to a PGA of 0.13g and a 5% damping fraction. This PGA is a typical value prescribed by the current Spanish seismic code for the southern part of Spain. It corresponds to a design earthquake having a probability of exceedance of 10% in 50 years and a return period of 475 years (i.e. 10/50/475). According to FEMA 356 [2], for an enhanced rehabilitation objective, the required building performance level for this earthquake hazard 10/50/475 is Immediate Occupancy (IO). Only light damage is allowed in the IO performance level, and according to FEMA-356 [2] in RC structures this involves limiting the interstory drifts below 1% of the story height. The lateral yield displacement fdyi of the bare frames used in this study ranged between 0.52% and 0.6% in the first story—375 cm height—and 0.57–0.91% in the other stories—275 cm height. In all cases fdyi was below the 1% set for the IO performance level. Accordingly, so as to achieve the aforementioned enhanced rehabilitation objective, the EDDs were designed so that dmax i r fdyi. The same Ki ¼K was adopted in all stories. The resulting lateral stiffness ski, lateral strength sQyi expressed in terms of sai, and energy dissipation capacity in terms of Z required for the EDDs for each design earthquake are indicated in Table 1. It is worth noting that although all structures were seismic-upgraded for the same VD, the near-fault design earthquake demanded EDDs with much larger K and sai (i.e. greater strength) than the far-field design earthquake. To validate the proposed design method, a series of nonlinear dynamic response analyses were carried out with the three structures equipped with EDDs described above. Two sets of acceleration records were considered. The first set included the near-fault records JMA-Kobe (Hyogo-ken nambu earthquake, 1995), El Centro (Imperial Valley earthquake, 1979), Rocca (Ancona earthquake, 1972), Tolmezzo (Friuli earthquake, 1976) and Korinthos (Alkion earthquake, 1981); their Id index ranged between 6.9 and 7.9 with the average Id ¼7.5; the values of TG and TNH for the corresponding VD  T and Newmark–Hall envelope spectra were TG ¼ 0.52 s and TNH ¼0.65 s. The second set included the far-field records Calitri (Campano Lucano earthquake, 1980), SCT (Mexico, 1985), Petrovac (Montenegro earthquake, 1979), Taft (Kern county, 1952) and Montebello (Northridge earthquake, 1994); their Id index ranged between 14 and 17 and the average was Id ¼16; the values of TG and TNH for the corresponding VD T and Newmark–Hall envelope spectra were TG ¼1 s and TNH ¼1 s. All the records were scaled so that the total energy input contributable

9 Story 8

9

Prediction (far-field) Prediction (near-fault) Montebello Petrovac Far-field SCT

to damage expressed in terms of equivalent velocity was VDmax ¼45 cm/s. The frames with EDDs designed for the first and second design earthquakes were subjected to the respective set of acceleration records. The results of the analyses are shown in Fig. 2. The solid and the dashed lines show the maximum interstory drift dmax i —in percentage of the story height— predicted with Eq. (28) for the sa1max that governed the design of the EDDs. The solid lines correspond to the near-fault design earthquake, while the dashed lines correspond to the far-field one. The symbols with thin lines show the results of the time-history response analyses. In all cases the maximum interstory drift obtained from the response analyses remains below the maximum allowed drift in the critical story (i.e. the first story). Also, Fig. 2 shows that Eq. (28) provides a satisfactory upper-bound prediction of the maximum drifts exhibited by the entire building-device structure, and that this prediction is close to the response under the severest earthquake.

7. Experimental validation To experimentally support the proposed method, dynamic tests were conducted on a half-scale model with the 3 m  3 m shaking table of the University of Granada. Fig. 3 gives an overall view. The test structure consisted of a reinforced concrete slab of 125 mm-depth supported on four steel columns made with 80 mm  80 mm  4 mm box-sections (referred to as main structure hereafter), and two brace dampers whose source of energy dissipation was the plastic deformation of steel plates installed between two U-shaped steel sections. The main structure was designed to behave as a strong slab-weak column structure and its total weight was 72.4 kN. The purpose of the tests was to assess the accuracy of the proposed method, which idealizes the existing frame as elastic. In order to use a specimen whose behavior was as close as possible to the model used in the idealization, RHS steel columns were used instead of reinforced concrete columns. Before performing the dynamic tests, the main structure—without the brace dampers—was tested under lateral forces and it was found that: (i) its lateral stiffness was fk¼37.2 kN/cm and (ii) it could sustain lateral displacements beyond 1.8 cm within the elastic range. The lateral stiffness and strength provided by the two brace dampers was obtained from static tests conducted prior to the dynamic experiments with

Story

9

8

8

7

7

6

6

5

5

4

4

3

3

3

2

2

2

7 6 5 4

1

Calitri Taft Tolmezzo Korinthos Near-fault Rocca El Centro JMA-Kobe

δi(%)

0.0 0.2 0.4 0.6 0.8 1.0

1

1391

δi(%) 0.0 0.2 0.4 0.6 0.8 1.0

1

Story

δi(%) 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 2. Comparison between prediction and results of dynamic analyses. (a) 3 story frame, (b) 6 story frame and (c) 9 story frame.

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Added weigth (steel blocks)

Accelerometers

Gauges U-section

Slab

Brace damper Steel plates that plastify (inside) Displacement transducer

Column

Gauges Dynamic actuator

3×3m shaking table Fig. 3. Overall view of the shaking table test.

similar brace dampers, giving sk¼ 80 kN/cm and sQy ¼46 kN, respectively. These values give a shear force coefficient for the EDDs of sa ¼ sQy/Mg¼ 0.64 and a stiffness ratio of K ¼ sk/fk¼2.2. The 1980 Campano Lucano earthquake recorded in Calitri was used as the input motion for the shaking table. The dynamic shaking table test of the main structure with the brace dampers was carried out scaling the earthquake record to PGA¼0.55 g. The maximum lateral displacement of the slab—relative to the table—recorded by the displacement transducers was 1.55 cm. The parameters that characterized the input earthquake were also determined from the measurements provided by the instrumentation during the test. Since the main structure remained elastic, all plastic strain energy consumed by the system, Wp, was dissipated by the brace dampers. Also, at the end of the tests the velocity was nearly 0 and the elastic strain energy stored in the system was very small in comparison to Wp, thus We was negligible. Consequently VD was calculated from the total energy dissipated by the brace dampers, which was estimated from the readings provided by the displacement transducers and the strain gages at both ends of each brace damper, giving VD ¼120 cm/s. The parameters Id, TG and TNH were obtained from the actual acceleration record measured by an accelerometer located on the shaking table during the test, giving pffiffiffiffiffiffiffiffiffi Id ¼18.5, TG ¼0.75 s and TNH ¼0.9 s—the time was scaled by 1=2 to fulfill similarity laws between prototype and test model. By substituting these values in Eq. (28) and adopting for c1 and c2 the values corresponding to a far-field earthquake, the maximum lateral displacement predicted is 1.62 cm, which is very close to the experimental result (1.55 cm). The numerical model used for idealizing the structure tested on the shaking table was a single degree of freedom system, whose restoring force was controlled by two elements connected in parallel: one is the ‘‘flexible’’ element that represents the main structure without dampers, and the other is the ’’stiff’’ element that represents the EDDs. The flexible element was an elastic spring of stiffness fk¼37.2 kN/cm. The stiff element was assumed to follow an elastic-perfectly-plastic behavior (i.e. without strain hardening effects) and it was defined by a lateral yield strength of sQy ¼ 46 kN and an initial stiffness of sk¼80 kN/cm. One of the factors that can justify the higher lateral deformability obtained with the numerical simulation (1.62 cm) as compared to the shaking table tests (1.55 cm) is that in the former the strain hardening effects actually exhibited by the EDDs were neglected.

8. Conclusions This paper presents an energy-based design procedure for seismic retrofit of multi-story buildings by installing hysteretic dampers. The method provides the lateral strength, the lateral stiffness and energy dissipation capacity required for the dampers to be installed in each story of the existing structure to achieve a desired building performance level for a given earthquake hazard. The maximum allowed interstory drift controls the target performance level. The earthquake hazard is characterized in terms of energy input, several seismological parameters used in the literature—i.e. the Cosenza and Manfredi adimensional index Id, the predominant period of the earthquake TG, and the initial period of medium period region in the Newmark and Hall spectrum, TNH—and it takes into account the proximity to the source. Under the proposed method, the effect of the hysteretic dampers is recognized directly in terms of hysteretic energy, without having to resort to equivalent viscous damping approximations; further, the cumulative damage induced in the dampers is explicitly evaluated. The proposed method is intended to be applied to low- to medium-rise structures. The application of the method is illustrated with three structures of 3, 6 and 9 stories, and its validity is assessed numerically through nonlinear dynamic response analyses with near-fault and far-field ground motions; and experimentally, by means of dynamic shaking table tests.

Acknowledgments This research was funded by the local government of Spain, ´n, Ciencia y Tecnologı´a (project P07-TEPConsejerı´a de Innovacio ´velop02610) and by the European Union (Fonds Europe´en de De ment Re´gional).

Appendix A The elastic response of a low- to medium-rise multi-story building can be analyzed replacing the structure by a shear strut of height H and using the mode superposition method. The undamped response under a ground motion acceleration z€ g ðtÞis governed by m

  @2 y @ @y G ¼ mz€ g ,  @x @t 2 @x

ðA:1Þ

where x is the height at an arbitrary point; t is the time; y(x,t) is the horizontal displacement; m(x) is the mass, and G(x) is the shear rigidity. Setting the right-hand side of Eq. (A.1) to zero, gives the equation of the free vibration m

  @2 y @ @y G ¼ 0,  @x @t 2 @x

whose solution takes the form of X y¼ qj fj :

ðA:2Þ

ðA:3Þ

Here qj(t) and fj(x) are the time function and the mode function for the j-th mode of vibration, and the summatory S is extended from j ¼1 to N. qj(t) is expressed by qj ðtÞ ¼ aj sinðoj t þ aj Þ,

ðA:4Þ

where the amplitude aj, frequency oj and phase-angle aj are constant for each mode j. The shape function fj(x) is an

A. Benavent-Climent / Soil Dynamics and Earthquake Engineering 31 (2011) 1385–1396

orthogonal function, that is, it satisfies Z H mfi fj dx ¼ 0 if ia j:

ðA:5Þ

o

The function fj(x) can be normalized so that Z

H o

2

mfi dx ¼ M,

ðA:6Þ

where M is the total mass of the system. Substituting Eqs.(A.3) and (A.4) in Eq.(A.2) gives   X  dfj d G ¼ 0, ðA:7Þ qj mo2j fj þ dx dx and to satisfy this equation for arbitrary values of qj the following equation is obtained:   dfj d G ¼ mo2j fj : ðA:8Þ dx dx The solution of Eq. (A.1) takes also the form of Eq. (A.3). Substituting Eq. (A.3) in Eq. (A.1) and taking into account Eq. (A.8) gives X X m fj q€ j þ m o2j fj qj ¼ mz€ g : ðA:9Þ Multiplying both sides of Eq. (A.9) by a given mode function

fs, integrating over the height H, and recalling Eqs. (A.5) and (A.6), the equation that governs the vibration in the s-th mode is: Z H M q€ s þM o2s qs ¼ z€ g mfs dx: ðA:10Þ o

P _ ¼ fj q_ j dt Multiplying the second member of Eq. (A.9) by ydt and integrating over O–H with respect to x and over 0–t0 with respect to t, the total energy input E is expressed by  Z H  Z H Z t0 X X Z t0 E¼ z€ g q_ j mz€ g jj q_ j dt dx ¼  mfj dx dt: o

o

o

o

ðA:11Þ Eq. (A.10) has been already multiplied by fs. Multiplying now the second member of Eq. (A.10) by q_ s dt and integrating over 0–t0 with respect to t, the energy input in a single mode of vibration s, Es, is obtained by Z H  Z t0 z€ g q_ s Es ¼  mfs dx dt: ðA:12Þ o

o

From Eqs.(A.11) and (A.12) it follows that X E¼ Ej :

ðA:13Þ

Now, a new parameter E1j is defined as the energy input in a fictitious undamped SDOF of mass M and frequency oj subjected to the ground motion acceleration z€ g ðtÞ. The equation of motion of this fictitious SDOF system would be M y€ þ Mo2j y ¼ M z€ g (here y indicates a fictitious displacement), and the corresponding input Rt _ For j ¼s, the force represented energy would be E1j ¼  oo M z€ g ydt. RH by the second member of Eq. (A.10) is ð o mfs dxÞ=M times larger than that acting on the fictitious SDOF system (i.e. M z€ g ); therehR i2 H fore, the energy Es given by Eq. (A.12) is ð o mfs dxÞ=M times larger than E1j. Accordingly, recalling Eq. (A.13) this means that !2 RH X mfj dx E¼ E1j o : ðA:14Þ M For convenience, E is now expressed in terms of new coefficients Dj, which satisfy X ðA:15Þ Dj fj ¼ 1:

1393

Multiplying both sides of Eq.(A.15) by mfs and integrating over O–H gives RH mfs dx Ds ¼ o : ðA:16Þ M Raising both sides of Eq. (A.15) to the second power, multiplying by m, integrating over the height 0–H and recalling Eqs.(A.5),(A.6) gives X D2j ¼ 1: ðA:17Þ Substituting Eq. (A.16) in Eq. (A.14), the total energy input E can be rewritten as follows: X E¼ E1j D2j : ðA:18Þ The above defined E1j can be expressed in form of an equivalent velocity V1j(oj) given by E1j ¼

2 MV1j

2

,

ðA:19Þ

and then the total input energy E given by Eq. (A.18) can be expressed by P M ðV1j Dj Þ2 E¼ ðA:20Þ 2 For t Zt0, the system is oscillating in an undamped free oscillation, thus the term V1j Dj represents the maximum velocity of the system vibrating in mode j. Since the maximum absolute acceleration can be obtained multiplying the maximum velocity by the frequency oj, the maximum force at height x is ojV1jDjf(x)m(x) and the corresponding shear force Q^ j is given by Z H mfj dx: ðA:21Þ Q^ j ðxÞ ¼ oj V1j Dj x

Combining the Q^ j of each vibration mode with the rule of the square root of sum of the squares, the maximum total shear force at height x normalized by the weight over height x and expressed in terms of a shear-force coefficient a^ ðxÞ, is expressed by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RH P P ^2 ðoj V1j Dj x mfj dxÞ2 Q j ðxÞ a^ ðxÞ ¼ R H ¼ , ðA:22Þ RH g x mdx g x mdx making x ¼0 in Eq. (A.22) and recalling Eq. (A.16), the base shear force coefficient is given by qP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðoj V1j D2j Þ2 : ðA:23Þ a^ ð0Þ ¼ g Further, E and a^ ð0Þ can be expressed as a fraction of the corresponding values in a fictitious SDOF system of mass M and frequency equal to that of the first mode o1 of the real MDOF system, subjected to the ground motion z€ g ðtÞ by two new coefficients e and a as follows:

a^ ð0Þ ¼ a

o1 V1 g

;

E¼e

MV12 , 2

ðA:24Þ

where V1 is the input energy, in form of equivalent velocity, in the fictitious SDOF system of frequency o1. In an undamped elastic system, E coincides with the elastic vibration energy We for t Zt0. Thus, since o1 ¼2p/T1 and using E and a^ ð0Þ given by Eq. (A.24), We ( ¼E) is expressed by We ¼ e

Mg 2 T12  e  a^ ð0Þ2 M a^ ð0Þ2 g 2 : ¼ 2 2 2 a2 4p2 2a o1

ðA:25Þ

Based on the above equations, the response of a system of uniform mass m(x)¼ m and linearly changing shear rigidity G(x)¼Go þ(GT Go)(x/H)—here Go and GT are values of G at the bottom and top part of the building—subjected to an earthquake is

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A. Benavent-Climent / Soil Dynamics and Earthquake Engineering 31 (2011) 1385–1396

1.0

x/H T1 = 1 2 3 TG

0.8

5 1.0 Calculated Approximate

0.6

T1 1 = TG

5

3

2

0.8 0.6

Go =1 GT

0.4

x/H

Go =2 GT

0.4

0.2

0.2

0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

1.0

x/H

T1 1 = TG

2

3

5 1.0

0.8 0.6 0.4

T1 1 = TG

x/H

2

3

5

0.8 0.6

Go =4 GT

Go = 10 GT

0.4

0.2

0.2

0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Fig. A1. Calculated and approximated distributions of shear-force coefficient for: (a) Go/GT ¼ 1; (b) Go/GT ¼2; (c) Go/GT ¼4; (d) Go/GT ¼10.

obtained next. The earthquake is characterized by a bilinear input energy spectra expressed in terms of equivalent velocity VD  T by for T oTG : VD ¼ VDmax T=TG ,

ðA:26aÞ

for T ZTG : VD ¼ VDmax ,

ðA:26bÞ

where VDmax is the maximum VD in the spectra and TG is the predominant period of the ground motion. Substituting G(x)¼ Go þ(GT Go)(x/H) in Eq. (A.2) and performing a variable transformation X¼Go þ(GT  Go)(x/H), Eq .(A.2) relates to Bessel’s equation and the oi and fi(x) can be obtained through numerical calculations. Substituting these fi(x) in Eq. (A.16) allows one to calculate the Dj’s. With the frequencies oi obtained above, the corresponding V1j are calculated from the VD  T spectra defined by Eq. (A.26a) and (A.26b). Substituting oj, fj(x) and these Dj’s, V1j’s in Eqs. (A.20), (A.22) and (A.23), the values of E, a^ ðxÞ and a^ ð0Þ are derived. From these a^ ðxÞ and a^ ð0Þ, the exact shear-force coefficient distribution aðxÞ ¼ a^ ðxÞ=a^ ð0Þ is obtained. The distribution aðxÞ calculated in this way is shown in Fig. A1 with solid lines for different values of T1/TG and Go/GT. For simplicity, in this study these solid lines have been approximated with the dashed lines given by Eq. (13). In the proposed Eq. (13), Go/GT has been replaced by fk1/fkN for lumped-mass systems. Finally, the values of parameters a and e in Eq. (24), which relate a^ ð0Þ and E with the corresponding values in a fictitious SDOF of mass M and period T1, were computed for realistic values of T1/TG from 1 to 5 and Go/GT from 1 to 10. It was obtained that e remains very close to 1—the mean is 0.87 and the standard deviation 0.059. Recalling Eq. (A.24), this indicates that the total energy input in the system E is governed by the first mode of vibration. Also, e/a2 was found to be slightly above 1—the mean is 1.33 and the standard deviation 0.045. Taking e/a2 ¼1 for simplicity in Eq. (A.25) yields Eq. (15), where a^ ð0Þ has been renamed as famax1 for a lumped-mass model. Eq. (15) provides a safe-side— and not over-conservative—estimation of We from the base shear force coefficient of the building.

Appendix B Taking into account Eq. (12) and the ratios defined in Eq. (11), it follows from Eqs. (8)   k 1 Qyi ¼ s Qyi þ s dyif ki ¼ s Qyi þ s dyi s i ¼ s Qyi 1 þ : ðB:1Þ Ki Ki PN Dividing Eq. (B.1) by k ¼ i mk g and recalling the ratios defined in Eq. (11), the first and last terms of Eq. (B.1) give   1 ai ¼ s ai 1 þ , ðB:2Þ Ki which, particularized for the first story, yields   1 a1 ¼ s a1 1 þ : K1

ðB:3Þ

Dividing Eq. (B.2) by Eq. (B.3), the distribution of the shear force coefficients is obtained

a K1 ðKi þ 1Þ ai : ¼ s i a1 s a1 Ki ðK1 þ1Þ

ðB:4Þ

In the proposed method, the distribution ai/a1 is made equal to ai defined given by Eq. (13), therefore making ai =a1 ¼ ai in Eq. (B.4) and solving for sai gives Eq. (14).

Appendix C From the coefficients defined in Eq. (11) and recalling Eq. (14),

d is expressed by

s yi

s dyi

¼

s Qyi s ki

a

s i

¼

N P

mk g

k¼i

Ki f ki

N P mk g Ki ðK1 þ 1Þ k ¼ i ¼ ai s a1 , K1 ðKi þ 1Þ Ki f ki

ðC:1Þ

substituting this expression in sdyi r fdyi and solving for sa1 gives Eq. (29).

A. Benavent-Climent / Soil Dynamics and Earthquake Engineering 31 (2011) 1385–1396

Appendix D. List of symbols

Qyi

a

R r

C C c1 c2 Dj E e

E1j Ej

d

f yi fki fkN fQi fQmax i fQyi

Q^ j Fy G(x) g Go GT H Id K keq

Ki M m(x) M mi N neq

PGA PGV Q(t) Q(y) Qi qj(t)

coefficient that expresses the base shear force coefficient in a MDOF system of mass M and fundamental frequency o1 as a fraction of the corresponding value in a fictitious SDOF of mass M and frequency o1 damping coefficient damping matrix seismological parameter seismological parameter coefficient associated with a given vibration mode j used to relate E and E1j total input energy coefficient that expresses the input energy in a MDOF system of mass M and fundamental frequency o1 as a fraction of the corresponding value in a fictitious SDOF of mass M and frequency o1 energy input in a fictitious undamped SDOF of mass M and frequency oj energy input in a single mode of vibration j yielding interstory drift of the i-th story of the main structure (without dampers) lateral stiffness of the i-th story of the main structure (without dampers) lateral stiffness of the uppermost N-th story of the main structure (without dampers) restoring shear force at the i-th story of the main structure (without dampers) maximum shear force at the i-th story of the main structure (without dampers) yield shear force at the i-th story of the main structure (without dampers) contribution of the j-th vibration mode to the shear force acting at a given height of a shear strut. yielding force of an equivalent SDOF system shear rigidity per unit length of a shear strut that represents the building acceleration of gravity shear rigidity at the bottom of the building shear rigidity at the top of the building total height of the building seismological parameter ratio between the lateral stiffness of the dampers and the main structure at the i-th story (equal for all stories) stiffness of an equivalent SDOF system of mass and period equal to the total mass and fundamental period of the MDOF system that represents the main structure ratio between the lateral stiffness of the dampers and the lateral stiffness of the main structure at the i-th story mass matrix mass per unit length of a shear strut that represents the building total mass mass of the i-th story number of stories of the building equivalent number of plastic cycles at the maximum value of plastic excursion that an equivalent SDOF system must develop to dissipate the total amount of hysteretic energy input by the earthquake peak ground acceleration peak ground velocity restoring force vector restoring force restoring force at the i-th story of the entire buildingdevice structure time function for the j-th mode of vibration

a

s 1i

a

s i

d

s yi

Sa ski Spv Sv sQi sQyi t T T1 TG

TNH to V1j Va VD VDmax VE Wx We Wk Wp Wpi Ws Wse Wui x x y y_ y€ _ yðtÞ € yðtÞ z€ g

ae

a^ ðxÞ ai a

f max i

ai w1 di

1395

yield shear force at the i-th story of the entire buildingdevice structure strength reduction factor displacement vector resulting from a unit support displacement. shear-force coefficient required on the dampers of the first story so that the maximum interstory drift at the i-th story does not exceed dallow i. shear-force coefficient representing sQyi normalized by the total weight of the upper stories yielding interstory drift of the dampers of the i-th story spectral absolute response acceleration initial lateral stiffness of the dampers of the i-th story spectral relative pseudo-velocity spectral relative velocity restoring force exerted by the dampers on the i-th story yield shear force of the dampers of the i-th story time period of vibration fundamental period of the main structure (without dampers) period corresponding to the change of slope of the VD  T bilinear spectra (predominant period of the ground motion). initial period of medium period region in the Newmark and Hall spectral representation instant when the ground motion fades away equivalent velocity converted from E1j equivalent velocity converted from the absorbed energy equivalent velocity converted from the energy that contributes to damage ( ¼Wp þWe) maximum value of VD in the VD  T design spectrum equivalent velocity converted from the total input energy damping energy elastic vibrational energy kinetic energy irrecoverable plastic strain energy plastic strain energy accumulated at the i-th story absorbed energy recoverable elastic strain energy ultimate energy dissipation capacity of the dampers installed in the i-th story height with respect to the ground at an arbitrary point of a shear strut that represents a building coefficient that represents the position of the story relative horizontal displacement relative horizontal velocity relative horizontal acceleration relative velocity vector relative acceleration vector ground acceleration base shear-force coefficient required on the main structure so that it can absorb by itself (i.e. without the dampers) the amount of energy that contributes to damage input by the earthquake shear-force coefficient representing the maximum total shear force at a given height x normalized by the weight over the height x shear-force coefficient representing Qyi normalized by the total weight of the upper stories shear-force coefficient representing fQmax i normalized by the total weight of the upper stories lateral strength distribution of the entire buildingdevice structure expressed as ai/a1 stiffness ratio representing fk1 normalized by keq interstory displacement of the story i

1396

dallow i dmax i fj(x)

g1 Z Zi Zui m mi o oj

A. Benavent-Climent / Soil Dynamics and Earthquake Engineering 31 (2011) 1385–1396

maximum allowed interstory drift at the i-th story maximum interstory drift of the entire building-device structure at the i-th story mode function of the j-th mode of vibration ratio between Wp and Wp1 cumulated plastic strain energy (equal for all stories) normalized plastic strain energy accumulated at the i-th story normalized ultimate energy dissipation capacity of the dampers installed at the i-th story plastic deformation ratio (equal for all stories) ratio representing the normalized plastic deformation of the dampers of the i-th story circular frequency of a SDOF system circular frequency of the j-th mode of vibration

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