Torsional balance of seismically isolated asymmetric structures

Engineering Structures 46 (2013) 703–717

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Torsional balance of seismically isolated asymmetric structures Carlos E. Seguin, Jose L. Almazán ⇑, Juan C. De la Llera Department of Structural and Geotechnical Engineering, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago, Chile

a r t i c l e

i n f o

Article history: Received 27 June 2011 Revised 25 November 2011 Accepted 26 August 2012 Available online 6 October 2012 Keywords: Seismic isolation Torsional balance Structural optimization Superstructure protection

a b s t r a c t Seismic isolation systems reduce the destructive effects of earthquakes through introducing a flexible interface between the superstructure and the foundation. Numerous studies, both analytical and experimental, have shown that such devices can control the translational response of the superstructure. However, no generally consistent design recommendations to control the torsional response are available. The objective of this paper is to present a methodology that lead to achieve optimal torsional control. The assumption behind is that the isolation–superstructure system can be split into an isolated base and superstructure subjected to the filtered ground motion excitation produced by the base response. The superstructure acts as a rigid body subjected this filtered acceleration responses which are applied as quasi-static inputs into the superstructure. Under rather weak assumptions, this simplification provides excellent results in estimating the response of the system. Such response has been used in this study to choose the optimal eccentricity and torsional stiffness parameters of the isolation system that minimize the lateral-torsional response of the superstructure. Results are obtained using probabilistic techniques and show that the response of the superstructure may be substantially improved if the isolation system is torsionally flexible, and if the center of stiffness of the isolated base should lie in the vicinity of the (average) center of stiffness of the superstructure. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Passive seismic isolation systems are devised to mitigate the destructive effects of earthquakes in buildings and their contents by controlling the seismic input. Although a great number of studies performed during the last 25 years have increasingly advanced to this goal, no systematic methodology has been developed to deal with asymmetric superstructures. It is customary to design the isolation system by simply ignoring torsion in the superstructure, or perhaps by considering it only as a secondary effect. For instance, some studies (i.e., [1–4]) assume that the best solution is the one that minimizes torsion in the isolation system, whereas a more sound strategy should always consider the design of an isolation system to protect the superstructure. Currently, the designer has no design criteria to provide effective control of torsion in the superstructure. The effect of seismic isolation is traditionally considered as a shift of the fundamental frequency of the structure to a range where the spectral pseudo-accelerations are substantially lower. However, as recently shown [5] the behavior of isolated structures may be considered as a system in cascade, in which the first system corresponds to an isolated rigid superstructure, and the second, the ⇑ Corresponding author. Tel.: +56 2 354 4204; fax: +56 2 354 4243. E-mail address: [email protected] (J.L. Almazán). 0141-0296/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2012.08.025

actual superstructure subjected to quasi-static input resulting from the total accelerations obtained as response of the first system. In other words, it is possible to model the isolation system as a second order band-pass filter. This important property allows the design of the isolation system in such a way that the translational and rotational base accelerations introduced into the superstructure combine in such a way as to balance its torsional response. More recently, Kilar and Koren [6] studied the behavior of base isolated asymmetric structures with different plan distribution of the isolators, two of them symmetric and four asymmetric. The results indicate that all six considered distributions of isolators, however differently, substantially reduce the torsional effects. They observed also that symmetric distribution (called CI = CM), favoured by typical building codes, is the best solution only for the base isolation system. By the contrary, such distribution might cause more damage in the flexible side frames of the superstructure. These important topics will be also analyzed in this research. Applying the traditional philosophy of seismic design, we can say that the general behavior of seismic structures in Chile during the Mw8.8 February 27th 2010 Earthquake (Maule’s Earthquake) has been very good. However, many new earthquake-resistant buildings collapsed or suffered major damage, even irreparable. Due to plan and/or height irregularities, in most of these buildings the damage was concentrated on lower floors, whereas the remainder with virtually no damage.

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At the time of the earthquake, there were a total of 11 base isolated structures in Chile [7]: one residential building, three office buildings, one educational building, two hospitals, two viaducts, one container dock, and one storage LNG tank. All seismic isolation devices used in this structures were activated during the earthquake. The behavior of both structures and their contents was excellent in all cases. As a result, there has been a large increase in demand for seismic isolation devices in Chile. In many cases the projects include irregular plans. For a better understanding of the nature of the problem, a simplified two-story model will be first considered; the lower story represents the isolated base, and the upper story represents the superstructure. Application of probabilistic techniques to this model will produce the conditions required for the isolation system to optimize the torsional response of the superstructure for a white-noise input. These results are then extended to modal response spectral analysis. Finally, the results are generalized to multistory buildings.

2. Model and equations of motion The final verification of the design of isolated structures requires the use of analytical and numerical tools capable of handling inelastic behavior of the materials. However, a small variation in the fundamental vibration period of the buildings is observed for displacements within a rather wide range [5,8]. Therefore, an equivalent linear behavior can be considered as a reasonable approach to the initial design of the isolation interphase. Thus, this study begins with the model of an isolated two-story linear structure, schematically shown in Fig. 1. Both plans are rectangular with the same side dimensions b and a in the X- and Y-directions, respectively. The structure is subjected to a single €g . horizontal component of ground motion in the Y-direction, y Two degrees of freedom (DOFs) are defined at the center of mass (CM) of each floor and are denoted by u = [uy, uh]T, where uy is

Isolation

the Y-direction displacement of the CM of the superstructure plan relative to the isolated base, uh = qh(s), where h(s) is the rotation of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 the superstructure plan, and q ¼ ða2 þ b Þ=12 is the radius of gyration of the plan. The vector of degrees of freedom of the isolated base is q = [qy,qh]T, where qy is the Y-direction displacement of the base relative to the ground, and qh = qh(b), where h(b) is the rotation of the base plan, also relative to the ground. The centers of mass of the base and superstructure are assumed vertically aligned, while the position of the centers of stiffness (CS) of each level is arbitrary. The equations of motion can be written as (e.g., [5,8]),

"

#  " € u cðsÞ þ € 0 mðsÞ mðtÞ q " # ðsÞ ðsÞ m r €g ¼ rðbÞ y mðtÞ mðsÞ

b/2

" k

ðsÞ

¼k

ðsÞ

1 ^eðsÞ

CSb

a/2

qx

X

CMb e(b)

a/2

isolators

Y Superstructure

uθ CMs

ð1Þ

# ^eðsÞ ; XoðsÞ2 þ ^eðsÞ2

" k

ðbÞ

¼k

ðbÞ

1 ^eðbÞ

^eðbÞ

#

XoðbÞ2 þ ^eðbÞ2

ðbÞ ðsÞ2 ðbÞ2 ¼ y mðsÞ and k ¼ y mðtÞ are ðsÞ ðbÞ and are the translational y y ðbÞ ^ðbÞ

where k x x the translational stiffnesses; x x nominal frequencies; ^eðsÞ ¼ eðsÞ =q and e ¼ e =q are the normalized static eccentricities; and XoðsÞ and XoðbÞ are the ratios between the uncoupled torsional and translational frequencies:

xðsÞ ,ðsÞ h ¼ ; ðsÞ q xy ðsÞ

qy

#  u ðbÞ q k 0

ð2Þ ðsÞ

b/2

qθ

#  " ðsÞ u_ k þ cðbÞ q_ 0 0

where the superindices (s) and (b) refer to the superstructure and base, respectively; m(s) = m(s)I and m(b) = m(b)I are the mass matrices, with I the order 2 identity matrix; m(s) and m(b) are the translational masses of the superstructure and isolated base, respectively (m(s) = m(b) = 1 in this research); m (t) = m(s) + m(b) = m(t)I is the total mass matrix of the system considering the superstructure as a rigid body, so that m(t) = m(s) + m(b); and r(s) = I and r(b) = [1, 0]T are the input incidence matrices. The normalized stiffness matrices of the superstructure with fixed base and of the isolated base are:

XðsÞ o ¼

Y

mðsÞ

XðbÞ o ¼

xðbÞ ,ðbÞ h ¼ ðbÞ q xy

ð3Þ

ðbÞ

where xh and xh are the uncoupled rotational frequencies; ,ðsÞ and ,ðbÞ are the stiffness radii of gyration of the fixed base superstructure and of the isolation system with respect to their respective CS. It should be pointed out that the stiffness matrices of Eq. (2) make explicit the offset between the CM and CS. While there are other ways in which the eccentricity in the structure can be introduced, each leading to different values of the term associated with the rotational DOF [9], this is a purely formal question which does not affect the final results. Moreover, the damping matrices of the superstructure and isolation system are:

cðsÞ ¼ mðsÞ /ðsÞ ^cðsÞ /ðsÞT mðsÞ

ð4:aÞ

cðbÞ ¼ mðtÞ /ðbÞ ^cðbÞ /ðbÞT mðtÞ

ð4:bÞ

where the diagonal damping matrices ^ cðsÞ and ^ cðbÞ have the common h i ðs;bÞ ðs;bÞ ðs;bÞ ðs;bÞ ðs;bÞ ðs;bÞ ðs;bÞ structure ^ c ¼ diag 2f1 x1 =m1 ; 2f2 x2 =m2 ; / (s,b),

uy

ux

CSs

X

(s)

e

ðs;bÞ

ðs;bÞ

ðs;bÞ

fi , xi , and mi (i = 1, 2) are the modes, damping ratios, coupled frequencies, and modal masses of the superstructure with fixed base and of the isolated base with rigid superstructure. A constant damping ratio f(s) = 5% will be assumed for the superstructure, while for the isolation system two cases are considered: (i) constant ðbÞ

Fig. 1. Schematic plan view of the base isolated two-story model considered and definition of degrees of freedom.

ðbÞ

modal damping, i.e., f1 ¼ f2 ; and (ii) stiffness proportional damping, i.e.,

ðbÞ ðbÞ f1 =f2

¼x

ðbÞ 1 =

x

ðbÞ 2 .

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C.E. Seguin et al. / Engineering Structures 46 (2013) 703–717

3. The quasi-static model (QSM) The system of coupled Eq. (1) can be approximated through the following system of equation of motions [5]: ðbÞ

ð5Þ

€o k u  mðsÞ rðsÞ q

ð6Þ

€ þ cðbÞ q_ þ k q  mðtÞ rðbÞ y €g mðtÞ q ðsÞ

where

€ þ r ðbÞ y €o ¼ q €g ¼ mðtÞ q

1

  ðbÞ k q þ cðbÞ q_

ð7Þ

is the total acceleration vector of the base with rigid superstructure. For this system the undamped eigenvalue problem can be expressed as [8]: ðbÞ

ðbÞ

ðbÞ2

k /j ¼ xj ðbÞ r

x

ðbÞ t

x

ðbÞ

mðtÞ /j

j ¼ r; t

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 s  signð1  XðbÞ ¼x o Þd ; 2 ðbÞ y

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 s þ signð1  XðbÞ ¼x o Þd ; 2 ðbÞ y

ð8Þ /ðbÞ r

ðbÞ /t

" # ^eðbÞ 1 ¼ d x ^ rðbÞ2  1

ð9:aÞ

" # ^ rðbÞ2  1 1 x ¼ d ^eðbÞ

ð9:bÞ

ðbÞ

where fxr ; /rðbÞ g are the natural frequencies and modes of the pren o ðbÞ ðbÞ are dominantly rotational vibration mode (subscript r); xt ; /t the natural frequencies and modes of the predominantly translaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r 2 ^ rðbÞ2  1 þ ^eðbÞ2 ; tional vibration mode (subscript t); d ¼ x ðbÞ

^ rðbÞ ¼ xrðbÞ is the normalized coupled rotational frequency; x x y

s ¼ 1 þ XoðbÞ2 þ ^eðbÞ2 ; and d ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2  4XoðbÞ2 . If XoðbÞ ¼ 1, an indetermi-

nacy appears in Eqs. (9.a) and (9.b) which is solved by assigning sign(0) = 1. The modal transformation can be written as ðbÞ

q ¼ /ðbÞ g ¼ /ðbÞ r gr þ /t gt

ð10Þ

T

where g ¼ ½ gr gt  is the modal coordinate vector, so that by substituting from (10) into (5), the following pair of uncoupled equations result 2

ðbÞ ðbÞ _ € j ¼ r; t g€ j þ 2fðbÞ gj ¼ CðbÞ j xj gj þ xj j yg ðbÞ r

where C

CðbÞ r ¼  ¼

and C

ðbÞ t

ð11Þ

are the modal participation factors,

/rðbÞT mðtÞ rðbÞ /rðbÞT mðtÞ /rðbÞ

^e ¼ ; d

ðbÞT

CtðbÞ ¼ 

/t

mðtÞ rðbÞ

ðbÞT ðbÞ /t mðtÞ /t

^ rðbÞ2  1 x

ð12Þ

d

€ o as defined by On the other hand, the total acceleration vector q Eq. (7), can be expressed as [5,8],

€o

ðbÞ € o

q ¼/ g ¼

€ or /ðbÞ r

g þ

ðbÞ /t € ot

g

ð13Þ

where F(s) is the pseudo-static force vector applied to the superðsÞ structure DOFs, u, with F yðsÞ being the force in Y direction, and F h the torque divided by the radius of gyration q; the expression involves the definition: G(s) = m(s)r(s). In summary, the computation of the approximate response of the superstructure will require the following steps: (1) solution of the undamped eigenvalue problem of the isolated base with rigid superstructure (Eq. (8)); (2) integration of the modal differential equations (Eq. (11)); (3) evaluation of the modal acceleration € o (Eq. (14)); (4) evaluation of the pseudo-static force vector vector g F(s) (Eq. (15.b)); and (5) solution of the linear algebraic Eq. (15.a). Despite the ability of the model to supply excellent results [5], its greater use is of conceptual nature, because it allows understanding of the essential features of the torsional response of seismically isolated structures in an intuitive form. In this respect, the analysis of the response of an asymmetric structure with nominally symmetric isolation, i.e., isolation in which the CS of the isolated base coincides with the vertical projection of the CM of the superstructure taken as a whole (i.e., the case of structures with FPS isolation [3]) is instructive. Indeed, in this case, with no calculation needed, the QS model predicts that: (i) the isolation system will describe a translational motion, since the asymmetry of the superstructure is irrelevant; and (ii) the superstructure will perceive a quasi-static system of lateral forces applied at the CM in each story, inevitably concentrating the deformation at the flexible edge [5,6]. Shown in Fig. 2a is the response of the asymmetric structure subjected to the Newhall (Northridge, 1994) record, with parame(b) ters b/a = 2, T(s) = 0.25 s, f(s) = 0.05, ^ eðsÞ ¼ 0:25, XðsÞ = 2.5 s, o ¼ 1:2, T ðbÞ (b) ðbÞ ^ f = 0.15, e ¼ 0; and Xo ¼ 1. It presents plots of the deformations at the stiff and flexible edges, both for the isolation system (top) and for the superstructure (bottom). All deformations are normalized with respect to the maximum response of the associated symmetric system (^ eðsÞ ¼ 0). These results have been obtained from the exact formulation (Eq. (1)); however, the QS model is equally capable of correctly predicting the fundamental tendencies of the response. The above example shows that suppressing torsion in the isolated base does not lead to a torsionally balanced response of the superstructure. The QS model suggests that it is possible to somehow transfer the lateral forces towards the vicinity of the CS of each story, instead of the CM. For that to happen, the lateral stiffness of the isolation system has to be concentrated in a few isolators placed around the vertical projection of the CS (average) of the superstructure. Fig. 2b illustrates this case, while keeping the parameters of the model of Fig. 2a, if introduces ^ eðbÞ ¼ ^ eðsÞ ¼ 0:25 and XoðbÞ ¼ 0:2. As the QS model suggests, torsion of the superstructure is essentially eliminated, at the expense of introducing torsion in the isolated base, which in this case turns out to be torsionally   very flexible XoðbÞ ¼ 0:2 . It is convenient to define the quasi-static force vector applied to the CS of the superstructure:

h e F ðsÞ ðtÞ ¼ F ðsÞ y ðtÞ

g ¼ x

ðbÞ2 r;t

gr;t 

ðbÞ 2fr;t

ðbÞ r;t

x g_ r;t

ð14Þ

€o ¼ ½ g € ot T is called the total modal acceleration vector. € or g where g Finally, Eq. (6) can be rewritten in modal terms as: ðsÞ

k u ¼ FðsÞ h FðsÞ ¼ F ðsÞ y

ð15:aÞ ðsÞ

Fh

iT

€o € o ¼ GðsÞ /ðbÞ g ¼ mðsÞ rðsÞ q

ð15:bÞ

iT

 T ¼ 1 ~fðtÞ F yðsÞ ðtÞ

ð16Þ

where e F h ðtÞ ¼ F h ðtÞ  ^eðsÞ F yðsÞ ðtÞ is the torque computed at the CS of ðsÞ the superstructure; and ~fðtÞ ¼ e F h ðtÞ=F yðsÞ ðtÞ is the normalized dynamic eccentricity, i.e., the distance measured from the CS to the point of instantaneous application of the lateral force, divided by the radius of gyration. For the two structures of Fig. 2, the left plots of Fig. 3 show the ðsÞ combinations of shear F ðsÞ y versus torque F h (S vs T) at the CM as ðsÞ ðsÞ e well as the shear F y versus torque F h at the CS. The right side plots shows the normalized dynamic eccentricity histograms of ~fðtÞ. For the symmetric isolation case (^eðbÞ ¼ 0 and XðbÞ ¼ 1) the o ðsÞ

€ or;t

ðsÞ e F h ðtÞ ðsÞ

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C.E. Seguin et al. / Engineering Structures 46 (2013) 703–717

Normalized roof−to−base displacement

Normalized base displacement

1

0 x=−b/2 (flexible edge) x=0 x=b/2 (stiff edge)

−1 2 1 0 −1 −2

0

2

4

6

8

10

12

14

16

18

20

Time (s)

Normalized roof−to−base displacement

Normalized base displacement

2 1 0 x=−b/2 (flexible edge) x=0 x=b/2 (stiff edge)

−1 −2 1 0.5 0 −0.5 −1

0

2

4

6

8

10

12

14

16

18

20

Time (s)

ðsÞ ^ðsÞ ¼ 0:25, T(b) = 2.5 s, n(b) = 0.15) subjected to Newhall record: (a) symmetric Fig. 2. Edge deformations for asymmetric models ðTðsÞ ¼ 0:25 s; X0 ¼ 1:2; nðsÞ ¼ 0:05; e isolation system; and (b) asymmetric and torsionally flexible isolation system.

S vs T hardly present any correlation (m = 0.15) at the CM, while at ~ ¼ 1Þ. The opposite occurs for the CS, correlation is very high ðm the asymmetric structure (^eðbÞ ¼ ^eðsÞ and XoðbÞ ¼ 0:2). This is because in the first structure the lateral force F ðsÞ y tends to pass through the CM, while in the second it tends to pass through the CS, as can be observed in the histograms of ~fðtÞ. It should be noted that these histograms show very good agreement with the theoretical Cauchy distribution corresponding to the quotient between two Gaussian variables [8].

4. Spectral Modal Analysis In this section the QS model is used to derive explicit expressions for the response of the 4 DOF structure considered in this study. Though the accuracy of the QS model is excellent [5], these solutions are only intended to estimate the parameters of the isolation system that allow an optimal control of the response of the

superstructure. It will be assumed initially that the excitation is a stationary white noise random process. Afterward the solutions obtained will be extended to spectral modal analysis through the well known relationship [10]:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : jaðtÞjo ¼Efmax jaðtÞjg ¼ pa ðkn;aðtÞ ; sÞ EfaðtÞ2 g

ð17Þ

where a(t) is a stationary random process; the symbol E{:} stands for the expectation of a(t); and pa is the peak factor [8], function of the moments kn,a(t) (n = 0, 1, 2) and of the effective duration of the process, s. This white noise assumption has been used extensively in the past, and it is the essence of the well-known CQC rule [10] which works well for any kind of excitation. In fact, this assumption implies that the power spectral density of the ground motion (considered as a general random process) does not vary significantly around the fundamental frequency of the considered mode. This hypothesis also holds for isolated linear systems [8,16].

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C.E. Seguin et al. / Engineering Structures 46 (2013) 703–717

Histrogram Cauchy distribution

Histrogram Cauchy distribution

ðsÞ Fig. 3. Response histories of Shear and Torque (S vs T) for asymmetric models ðT ðsÞ ¼ 0:25 s; X0 ¼ 1:2; nðsÞ ¼ 0:05; ^ eðsÞ ¼ 0:25, T(b) = 2.5 s, n(b) = 0.15) subjected to Newhall record: (a) symmetric isolation system; and (b) asymmetric and torsionally flexible isolation system.

The procedure begins by calculating the covariance matrix of the pseudo-static force vector F(s) (Eq. (15.b)) for the white noise type excitation:

o oT ðbÞT ðsÞT T € / G € g EfFðsÞ FðsÞ g ¼ GðsÞ /ðbÞ E g 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o n o n o3 m E F yðsÞ2 E F hðsÞ2 7 E F yðsÞ2 6 7 6 7 ¼6 7 6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o n o n o 5 4 ðsÞ2 ðsÞ2 ðsÞ2 E Fh m E Fy E Fh

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o n o

ðsÞ2 ¼ mðsÞ m E F ðsÞ2 E Fh y

2

 2 ^r 1 ^eðbÞ x 4

d

ð18Þ

ð19:bÞ

ð19:dÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi



8 fr ft xr xt 4fr ft xr xt ðft xr þ fr xt Þ þ ft x3r þ fr x3t ffi m ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ð1 þ 4 f2r Þð1 þ 4 f2t Þ ðx2r  x2t Þ þ 4xr xt ðfr xr þ ft xt Þðft xr þ fr xt Þ

where m is the correlation factor between the lateral force F yðsÞ and ðsÞ the torque F h measured at the CM(s);  k4r and  k4t are the variances of the absolute acceleration of the SDOF oscillators with frequencies xr and xt and damping ratios fr and ft, subjected to white noise  is the correlation factor between excitation of intensity W; and m the total modal accelerations [8,10]. The covariance matrix of the superstructure‘s deformation vecðsÞ1 ðsÞ tor u ¼ k F , results in:

Efu uT g ¼ k

ðsÞ1

2

...

   qffiffiffiffiffiffiffiffiffiffiffiffi  2 2 ^ 2r  1 k4t ^ 2r  1  ^eðbÞ2 m  k4r k4t  x ^eðbÞ2 k4r þ x

k4t ¼ ð1 þ 4f2 Þ xt W t 4ft

ð19:eÞ

 n 2 o mðsÞ2  2 2 qffiffiffiffiffiffiffiffiffiffiffiffi  2 4 ^ r  1 k4t ^r 1 m  k4r k4t þ x E F ðsÞ ¼ 4 ^eðbÞ4 k4r þ 2^eðbÞ2 x y d ð19:aÞ  2  qffiffiffiffiffiffiffiffiffiffiffiffi n o ^r 1 2 eðbÞ2 x 2 ^ ðsÞ2 k4  2m k4 k4 þ k4  ¼ mðsÞ E Fh r r t t 4 d

k4r ¼ ð1 þ 4f2 Þ xr W r 4fr

ð19:cÞ

T

ðsÞ1

EfFðsÞ FðsÞ gk

n o E u2y 6 6 6 ¼6 6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o

4 l E u2y E u2h

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o 3

l E u2y E u2h 7 7

E u2h

7 7 7 5

ð20Þ

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C.E. Seguin et al. / Engineering Structures 46 (2013) 703–717

  n o n o 2 2 2 2 ðsÞ2 n o XðsÞ þ ^eðsÞ E F yðsÞ2 þ ^eðsÞ E F h o 2 E uy ¼  4 2 ðsÞ mðsÞ xy XðsÞ o   rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o n o 2 ðsÞ2 ^ 2^eðsÞ XðsÞ þ e m E F yðsÞ2 E F hðsÞ2 o   4 2 ðsÞ mðsÞ xy XðsÞ o



E u2h ¼

n o 2 E F yðsÞ2 ^eðsÞ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o n o n o ðsÞ2 ðsÞ2 ðsÞ ^  2e m E F ðsÞ2 E Fh þ E Fh y  4 2 ðsÞ mðsÞ xy XðsÞ o

  n o n o ðsÞ2  ^eðsÞ3 þ ^eðsÞ XoðsÞ2 E F yðsÞ2  ^eðsÞ E F h rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l¼ ... n o

 4 2 ðsÞ E u2y E u2h mðsÞ xy XoðsÞ   rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o n o 2^eðsÞ2 þ XðsÞ2 m E F yðsÞ2 E F hðsÞ2 o þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o

 4 2 ðsÞ E u2y E u2h mðsÞ xy XoðsÞ

Ar,t (Eq. (22)) may consider the simultaneous effect of the 2 (or 3) components of seismic motion. This can be done using any of the formulas proposed by many researchers (i.e. [11]).

ð21:aÞ

ð21:bÞ

ð21:cÞ

5. Torsional balance of the superstructure The results displayed in Fig. 2b show that the ideal conditions to control torsion of the superstructure are: (i) that the CS of the isolated base should lie in the vicinity of the (average) CS of the superstructure (CS(b)  CS(s)); and (ii) that the torsional  stiffness  of the isolation system should be as low as possible XoðbÞ ! 0 . Under such conditions the seismic isolation system will be able to funnel the lateral seismic forces towards the CS of the superstructure. However, either because of practical limitations or because of the need to keep torsion of the isolated base within certain bounds, an excessive reduction of torsional stiffness is not possible. Therefore, in what follows both the parameters of the superstructure and the torsional stiffness of the isolated base will be assumed to be fixed, and the objective will be to find the optimized eccentricity of the isolated base. 5.1. Force-based criterion

where l is the correlation factor between uy(t) and uh(t). Finally, substituting from Eq. (17) into Eqs. (18) and (20), and assuming that the peak factors corresponding to the participating random processes are approximately equal [10], the following estimators are obtained: ðsÞ 1 jF ðsÞ y jo  m 2 d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi ^ 2r  1 2 m ^ 2r  1 4 S2t  Sr St þ x  ^eðbÞ4 S2r þ 2^eðbÞ2 x

ðsÞ jF h jo

 2 ^ r  1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^eðbÞ x  Sr St þ S2t  mðsÞ S2r  2 m 2 d

k

St ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ 4f2t ÞAt

ð22:bÞ

ð22:dÞ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2   2 2 ðsÞ 2 2 ^ðsÞ XðsÞ ^ðsÞ2 jF ðsÞ XðsÞ þ ^eðsÞ2 jF ðsÞ þ ^eðsÞ2 mjF ðsÞ h jo y jo  2 e y jo jF h jo þ e o o juy jo   2 ðsÞ mðsÞ xðsÞ y Xo ð23:aÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðsÞ 2 ^eðsÞ2 jF yðsÞ j2o  2^eðsÞ mjF yðsÞ jo jF ðsÞ h jo þ jF h jo juh jo   2 ðsÞ mðsÞ xy XðsÞ o



1

0

0 XoðsÞ2



~y u uh

 ¼

F yðsÞ

!

ðsÞ e Fh

ð24Þ

being

ð22:cÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ 4f2r ÞAr ;

ðsÞ

ð22:aÞ

   2  Sr St  x ^ 2r  1 2  ^eðbÞ2 m ^ r  1 2 S2t ^eðbÞ2 S2r þ x m  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi ^ 2r  1 2 m  Sr St þ x ^ 2r  1 4 S2t S2r  2 m  Sr St þ St ^eðbÞ4 S2r þ 2^eðbÞ2 x

Sr ¼

When isolated base has some eccentricity, the superstructure will be subjected to the simultaneous action of lateral forces and torsional moments. It proves convenient to now rewrite the equation for the pseudo-static response of the superstructure (Eq. (15)) considering equilibrium at the CS:

ð23:bÞ

    ðsÞ 2 ^ðsÞ2 þ XðsÞ2 ^ðsÞ ðsÞ 2 jF ðsÞ  ^eðsÞ3 þ ^eðsÞ XðsÞ2 mjF ðsÞ o y jo þ 2e o y jo jF h jo  e jF h jo ljuy jo juh jo   4 ðsÞ mðsÞ2 xy XðsÞ o

~ y ¼ uy þ uh ^eðsÞ u ðsÞ ðsÞ Fe ¼ F  ^eðsÞ F ðsÞ h

h

y

ð25:aÞ ð25:bÞ

ðsÞ ~ y is the lateral displacement and e where u F h is the torque about the CS of the superstructure (CS(s)). Under the quasi-static approximation it becomes obvious that ðsÞ the rotation uh(t) will be zero if b F h is too. This situation occurs only if the isolating system is reduced to a single isolator placed bellow the CS(s) (i.e., ^ eðbÞ ¼ ^ eðsÞ , XðbÞ o ¼ 0). Otherwise, torsional coupling of the superstructure is unavoidable. However it is possible to move the CS(b) so as to make the stae ðsÞ ~ between F ðsÞ tistical correlation m y ðtÞ and F h ðtÞ vanish. This condition, that will be called force-based weak torsional balance (FBWTB) can be expressed as:

n o ðsÞ E e F h F ðsÞ y m~ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o n offi ¼ 0 ðsÞ2 ðsÞ2 E e Fh E Fh

ð26:aÞ

or alternatively, in the equivalent zero covariance form:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o n o n o n o ðsÞ ðsÞ2 E e F h F yðsÞ ¼ E F yðsÞ2 ^eðsÞ  m E F yðsÞ2 E F h ¼0

ð26:bÞ

ð23:cÞ

n o € or;t j are the expected maximum values of the where Sr;t ¼ E max jg absolute modal acceleration; and Ar,t are the spectral pseudoaccelerations. It is noteworthy that, just to simplify the presentation (and interpretation) of the problem, it is considered unidirectional seismic motion. However, the conceptual framework proposed here is quite general. The calculation of the pseudo-spectral accelerations

Three corollaries can be derived from this equation: Corollary 1. the FB-WTB implies minimization of the mean square of the superstructure rotation with respect to the eccentricity of the superstructure:



@E u2h ¼0 @ ^eðsÞ

ð27Þ

709

1 probability f~fðtÞ > 0g ¼ probabilityf~fðtÞ < 0g ¼ 2

NCH code UBC code

2

d

Pseudo−displacement, S (cm)

C.E. Seguin et al. / Engineering Structures 46 (2013) 703–717

10

ð29Þ

e Proof 3. Assuming that F ðsÞ y and F h are gaussian random processes, and given that (by hypothesis) their correlation is zero, ~fðtÞ has a Cauchy probability distribution that is symmetric about the CS(s) (see Fig. 3). Accordingly, the probability of ~fðtÞ is greater than zero is equal to the probability is less than zero. h It is important to point out that all the above expressions are true not only for white noise, but also for limited bandwidth processes. On the other hand, if the process is assumed to be stationary, they are all time independent. From a practical point of view it is useful to have an expression for the torsional balance criterion using a design spectrum. For this purpose, substituting from Eq. (17) into Eq. (26.b), and assuming equal to 1 the ratio between the peak factors associated to the ðsÞ F h ðtÞ and F ðsÞ y ðtÞ processes, the following equation is obtained: ðsÞ

0

10

−2

10

−2

−1

10

0

10

10

1

2

10

10

Period, T (sec) Fig. 4. Design spectrum used in this study.

Proof 1. Using Eq. (21.b),

ðsÞ

¼0 by Hip:

^eðsÞ ¼ m

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl  n o ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl n offl{ 2

ðsÞ ðsÞ ðsÞ2 ðsÞ ^ 2 E F  E F F e y y h @E uh ¼ ¼0  4 2 @ ^eðsÞ ðsÞ ðsÞ mðsÞ xy Xo

jF h jo

ð30Þ

jF ðsÞ y jo ðsÞ

Corollary 2. the FB-WTB implies suppressing correlation between the rotation and the displacement measured at the CS(s):

where jF yðsÞ jo , jF h jo , and m are computed from Eqs. 22.a, 22.b, 22.c, 22.d. It should be noted that the expressions from Eqs. (26) to (30) are independent of the torsional stiffness of the superstructure, making their practical application much easier.



~ y uh ¼ 0 E u

5.2. Displacement-based criterion

Proof

2. Using

¼0by Hip:

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ n o ðsÞ E e F h F ðsÞ y ðkðsÞ XðsÞ o Þ

2

ð28Þ Eq.

(24),

 ðsÞ  ðsÞ  eF h Fy ~ y uh g ¼ E Efu ¼ ðsÞ ðsÞ2 kðsÞ k

Xo

¼0

Within the context of torsional control of asymmetric structures with energy dampers, it has been proposed [12,13] a criterion consisting in reducing to zero the correlation coefficient between the rotation and the displacement at the geometrical center (GC) of the plan, called the deformation-based weak torsional balance (DB-WTB). The condition is:

response Eq. (28) implies that the CS(s) is the point n oof minimal n o ~ 2y < E ðuy þ uh xÞ2 ; 8x. (center of torsional balance, [12,13]), i.e., E u

 y uh g ¼ 0 Efu

Corollary 3. for a given instant, the FB-WTB implies that the (s) probability that F ðsÞ is the same, y be acting at either side of the CS that is:

For the 4 DOF structure being analyzed, in which the GC coincides with the CM, the torsional balance condition is satisfied when the numerator of Eq. (21.c) becomes zero,

Fig. 5. Total pseudo-displacement response spectra for seven records of the Mw8.8 Maule’s Earthquake (Chile, 2010) for a damping ratio n = 0.15.

ð31Þ

710

C.E. Seguin et al. / Engineering Structures 46 (2013) 703–717

UBC code

1.8

1.8 0.1

1.6

0

(b) o

1

1.2

0.8

0.8

0.9 0.95

0.6 0.4

0 0.1 0.2 0.3

1

Ω

(b) o

1.4

0.4 0.5 0.6 0.7

1.2

Ω

1.6

0.2 0.3

−0.1

1.4

.1

−0

−0

2 .2

NCh code 2

0.8

0.4 0.5 0.6 0.7

0.6

0.8 0.9

0.4

0.95

0.2 0

0.2

0

0.2

0.4

0.6

0.8

0

1

0

0.2

0.4

(b)

0.6

0.8

1

0.75

1

(b)

e /ρ

e /ρ Fig. 6. Force correlation factor m (Eq. 22.c).

NCh code

UBC code

1 1.4 1.4

1.0 1.2 1.2

0.75

0.8

1.0 0.5 0.8

0.25

0 −0.25

0

0.25

0.5

0.75

1 −0.25

0

0.25

0.5

ðbÞ Fig. 7. Optimized base eccentricities ^ eop obtained by means of FB-WTB criteria.

  n o   rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o n o n o 2 ðsÞ2 ðsÞ2 E F ðsÞ2 þ 2^eðsÞ þ XðsÞ2 E Fh ¼0  ^eðsÞ3 þ ^eðsÞ XðsÞ2 m E F ðsÞ2  ^eðsÞ E F h o y o y

ð32Þ

In the same way, if a design spectrum is being used, the torsional balance condition is satisfied when the numerator of Eq. (23.c) becomes zero,

    2 2  ^eðsÞ3 þ ^eðsÞ XoðsÞ2 jF yðsÞ j2o þ 2^eðsÞ þ XðsÞ2 mjF yðsÞ jo jF hðsÞ jo  ^eðsÞ jF ðsÞ o h jo ¼ 0 ð33Þ ðsÞ

where jF yðsÞ jo , jF h jo , and m are computed from Eqs. 22.a, 22.b, 22.c, 22.d. It should be noted that in this case the expressions from Eqs. (32) and (33) do depend on the torsional stiffness of the superstructure.

6. Results obtained In this section the results obtained by applying the above defined criteria of torsional balance are presented. The nominal parameters used are: b/a = 2, T(s) = 0.25 s, f(s) = 0.05, T(b) = 2.5 s, and f(b) = 0.15. Fig. 4 shows the two pseudo-displacement spectra used to characterize the seismic excitation. The first one is that of the Chilean seismic isolation code [14], and the second one that of American code [15], both for firm soil and a 5% damping ratio. In what follows these spectra will be referred to as the NCh and the UBC, respectively. Fig. 5 shows the total pseudo-displacement spectra, estimated  1=2 2 2 as Sd ¼ Sd1 þ 0:3Sd2 for seven records [16] of the Maule’s Earthquake, for a damping ratio n = 0.15. It can be observed that for a period To = 2.5 s the displacements range between 10 cm

C.E. Seguin et al. / Engineering Structures 46 (2013) 703–717

711

ðbÞ

Fig. 8. Optimized base eccentricity ^ eop obtained my means of FB-WTB (dased-point) and DB-WTB (dashed) criteria. Shaded areas correspond to 10% of torsional control (Eq. (34)).

and 25 cm. There have also plotted the NCh spectra [14] for seismic zones 2 and 3, using the damping response modification factors given by this code. Considering only these seven records, it is possible to infer that this earthquake represented approximately the expected design ground motion.

6.1. The correlation coefficient m As the results shown in Figs. 2 and 3 suggest, and as can be seen from Eqs. (26) and (30), it is quite obvious that the correlation facðsÞ tor m between the lateral force F ðsÞ y and the torque about the CM F h ,

712

C.E. Seguin et al. / Engineering Structures 46 (2013) 703–717 (b) o

(b) o

Ω =0.5

(b) o

Ω =0.7

Ω =1.0

2 Flexible edge Stiff edge 1.5

Base edge amplification

NCh code 1

0.5 2

1.5

UBC code 1

0.5

0

0.2

0.4

0.6

0.8

1

0

0.2

(b)

0.4

0.6

0.8

1

0

(b)

e

0.2

0.4

0.6

0.8

1

(b)

e

e

Fig. 9. Torsional edge amplification at the isolation system.

plays a fundamental role in torsional balance of the superstructure. Fig. 6 shows the values of m as a function of ^ eðbÞ and XoðbÞ computed from Eq. (22.c) for the NCh and UBC spectra. It can be noted that in both cases m increases as XoðbÞ decreases, showing scarce sensibility with respect to the eccentricity ^eðbÞ . It can also be observed that the values of m for the NCh spectrum are larger due to the pseudo-displacement plateau it presents starting at 2.5 s (Fig. 4).

 sion areas, correspond to the ^eðsÞ ; ^ eðbÞ combinations satisfying the following condition:

krf  rs k

ro

ðbÞ Fig. 7 shows the optimized isolation eccentricity ^eop curves obtained with the FB-WTB criterion (Eq. (30)) for structures subjected to the NCh and UBC spectra. Three aspects can be clearly singled out: (i) when the isolated base is torsionally flexible   ðbÞ XoðbÞ < 1 ^eðsÞ and ^eop have the same sign and their magnitudes

are alike; (ii) when the isolated base is torsionally stiff (XðbÞ o > 1) ðbÞ ^eop is significantly larger than ^eðsÞ and for small values of ^ eðsÞ even of the opposite sign; and (iii) when the isolated base is torsionally   ðbÞ stiff or hybrid XoðbÞ P< 1 ^eop for UBC spectrum is significantly larger (approximately twice) than NCh spectrum. This is because NCh spectrum has a pseudo-displacement plateau, while the UBC spectrum has a pseudo-velocity plateau. This confirms the convenience of using torsionally flexible isolation systems. Since the sole purpose of the FB-WTB is to make zero the correlation between the forces referred to the CS of the superstructure, it is that the results shown in Fig. 7 do not depend on the torsional stiffness of the superstructure. The influence of this important torsional parameter is assessed through the analysis of Fig. 8, in which ðbÞ the optimized isolation eccentricity ^ eop curves obtained with the DB-WTB criterion (Eq. (33)) are shown for structures subjected to the NCh and UBC spectra. Each column is associated to a value of the torsional stiffness of the superstructure (XðsÞ o ¼ 0:7; 1; 1:3) and each row to a value of the torsional stiffness of the isolated base (XðbÞ o ¼ 0:7; 1:0). The shaded areas, called the controlled tor-

ð34Þ

being

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Efuy  uh bg2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rs ¼ Efuy þ uh bg2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ro ¼ Efuy g2

rf ¼ 6.2. Optimized eccentricities

< 0:10

ð35:aÞ ð35:bÞ ð35:cÞ

where rf and rs are the standard deviation of the deformations at the flexible and stiff edges of the superstructure, respectively; and ro is the standard deviation of the deformation of the associated ðbÞ symmetric system ð^eðsÞ ¼ ^eðbÞ ¼ 0Þ. The curves ^eop vs ^eðsÞ derived from Eq. (33) are the bisectors of the controlled torsion areas, since they satisfy the torsional balance condition (i.e., rf  rs = 0). For ðbÞ comparison sake, the force-based criterion ^eop vs ^eðsÞ curves have also been plotted. The trends observed in Fig. 8 are similar to those already observed in Fig. 7, though now the problem can be analyzed in greater depth. In the first place, it can be observed that increasing the torsional stiffness of the superstructure reduces the eccentricities ^eðbÞ op and that the controlled torsion areas grow larger, meaning that for a certain value of ^eðbÞ it becomes possible to successfully control a larger range of superstructures. In the second place, it can be obðbÞ served that the eccentricities ^eop necessary for DB-WTB are larger than those required for FB-WTB, though the difference between the two diminishes as the torsional stiffness of the superstructure increases. However, it has to be pointed out this difference is circumstantial, and it is due to the GC of the considered model being coincident with the CM (Fig. 1). For instance, if the GC were to coincide with the CS (systems with mass eccentricity) both torsional balance criteria will lead to the same result.

713

C.E. Seguin et al. / Engineering Structures 46 (2013) 703–717

(s)

(s)

Ωo =0.8

(s)

Ωo =1.0

Ωo =1.2

1.3 1.2

Ω(b) =1.0 o

1.1 1

Super−structure edge amplification

0.9 DB−WTB (both edges) FB−WTB (stiff edge) FB−WTB (flexible edge)

0.8 0.7 1.3 1.2

(b) Ωo =0.7

1.1 1 0.9 0.8 0.7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 0

0.1

0.2

(s)

0.3

0.4

0.5

0.6

0.7 0

0.1

0.2

(s)

e

0.3

0.4

0.5

0.6

0.7

0.5

0.6

0.7

(s)

e

e

(a) NCh code (s)

(s)

Ωo =0.8

(s)

Ωo =1.0

Ωo =1.2

1.3 1.2

1

o

Ω(b)=1.0

1.1

Super−structure edge amplification

0.9 0.8 0.7 1.3 1.2

1

o

Ω(b)=0.7

1.1

0.9 0.8 0.7

0

0.1

0.2

0.3

0.4

e(s)

0.5

0.6

0.7 0

0.1

0.2

0.3

0.4

0.5

e(s)

0.6

0.7 0

0.1

0.2

0.3

0.4

e(s)

(b) UBC code Fig. 10. Torsional edge amplification at the superstructure, corresponding to torsionally balanced base isolated structures by means of FB-WTB and DB-WTB criteria.

714

C.E. Seguin et al. / Engineering Structures 46 (2013) 703–717

Table 1 Fixed-base vibration modes of the six-story building considered in this study. Mode 1

Mode 2

Mode 3

Period (s) MMRx MMRy

0.70 0.00 0.20 Component

0.50 0.84 0.00 Component

0.48 0.00 0.63 Component

Story

x

y

qh

x

y

qh

x

y

qh

1 2 3 4 5 6

0 0 0 0 0 0

0.18 0.40 0.61 0.79 0.92 1.00

0.32 0.72 1.10 1.41 1.64 1.78

0.19 0.41 0.61 0.78 0.91 1.00

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0.18 0.38 0.57 0.75 0.89 1.00

0.10 0.21 0.32 0.42 0.50 0.55

those already shown as the controlled torsional areas of Fig. 9, meaning that the torsional amplification decreases either as a consequence of an increasing torsional stiffness of the superstructure or of a decreasing torsional stiffness of the isolated base.

6.3. Torsional amplification at the edges Shown below amplifications in edge deformations obtained according to two proposed torsional balance criteria. The torsional edge amplification at the base and superstructure is defined as:

CðbÞ ðb=2Þ ¼ Efmaxðkqy ðtÞ  qh ðtÞb=2kÞg=qo

ð36:aÞ

CðsÞ ðb=2Þ ¼ Efmaxðkuy ðtÞ  uh ðtÞb=2kÞg=uo

ð36:bÞ

where E{:} stands for the expectation; qo and uo are maximum responses of the symmetric system. Fig. 9 shows the curves of torsional edge amplification in the isolated base for asymmetric structures subjected to the NCh and the UBC spectra, computed for the QS (Eq. (5)) model. For the NCh spectrum the amplifications do not exceed 1.5 at either edge,

(a)

while for the UBC spectrum the amplifications at the flexible edge increase indefinitely. As mentioned above, this is because the NCh spectrum has a pseudo-displacement plateau from 2.5 s on, while the UBC spectrum does not (constant pseudo-velocity). As a consequence, the possibility to control torsional effects in the superstructure under the NCh code is much greater than under the UBC code [5,7]. The torsional amplification curves at superstructure level is shown in Fig. 10. In the same way as in Fig. 8 each column is associated to a value of the torsional stiffness of the superstructure   XðsÞ o ¼ 0:8; 1; 1:2 and each row to a value of the torsional stiff  ness of the isolated base XoðbÞ ¼ 0:7; 1:0 . These results agree with

7. Application to multistory structures Even though the methodology of torsional balance presented here was developed for a 4 DOFs single-story model, it turns out easy to extend it to multistory structures. As an example the monosymmetric 6-story model shown in Fig. 11 has been considered. It represents a building with a rectangular plan with dimensions b = 1500 cm and a = 1000 cm, in which the mass has been considered uniformly distributed over plan and height, with a seismic weight w = 1.0 tons/m2. The resisting planes are RC frames with beams and columns symmetrically distributed with respect to the CM of the structure. The static eccentricity of the superstructure originates in the steel diagonals shown in the figure. Table 1 shows

(b)

b = steel braces (A b = 30 cm2) 30 x 50

30 x 50

b 30x 50 b 30x 50

6 @ 300 1800

b 30x 50 b 30x 50 b 30x 50 b 30x 70

b 30x 50 b 30x 50 b 30x 50 b 30x 50 b 30x 50 b 30x 70

Fig. 11. Base isolated mono-symmetric six-story building used to validate the proposed torsional control methods: (a) building plan; and (b) typical braced frame elevation (plane 2).

715

C.E. Seguin et al. / Engineering Structures 46 (2013) 703–717 Table 2 Plan stiffness distribution for asymmetric 6-story isolated systems. Optimized eccentricities have been obtained from FB-WTB criteria. Isolator number

Plan location

 ¼ 12k =P12 k Normalized isolator stiffness k j j j¼1 j NCh code

UBC code

^ðbÞ ¼ 0:26; XðbÞ o ¼ 1:0; e ðeðbÞ =b ¼ 0:09Þ 1 2 3 4 5 6 7 8 9 10 11 12

1-a 2-a 3-a 4-a 1-b 2-b 3-b 4-b 1-c 2-c 3-c 4-c

0.20 0.80 1.63 0.28 0.20 1.89 2.72 1.37 0.20 0.80 1.63 0.28

XoðbÞ ¼ 0:75; ^eðbÞ ¼ 0:16;

^ðbÞ ¼ 0:52; XðbÞ o ¼ 1:0; e

ðeðbÞ =b ¼ 0:055Þ

^ðbÞ ¼ 0:20; XðbÞ o ¼ 0:75; e

ðeðbÞ =b ¼ 0:18Þ

0.20 0.20 0.52 0.20 0.20 3.00 6.35 0.20 0.20 0.20 0.52 0.20

0.20 0.20 1.66 0.70 0.20 1.31 2.97 2.00 0.20 0.20 1.66 0.70

ðeðbÞ =b ¼ 0:07Þ 0.20 0.20 0.62 0.20 0.20 2.50 6.67 0.20 0.20 0.20 0.62 0.20

Table 3 Peak edge deformations for the base isolated six-story building subjected to NCh spectrum. Optimized eccentricities have been obtained from both, FB-WTB and DB-WTB (between brackets) criteria.

Optimized eccentricity ^ eðbÞ X-dir. edge coordinate Isolation deformation (cm) Superstructure drift (cm)

^ðbÞ ¼ 0 Symmetric isolation XðbÞ o ¼ 1:0; e

Asymmetric isolation XoðbÞ ¼ 1:0

Asymmetric isolation XoðbÞ ¼ 0:75

–

0.26

0.16

b/2 21.46

b/2 21.97

1

0.44

0.14

2

0.50

0.13

3

0.44

0.13

4

0.35

0.12

5

0.26

0.11

6

0.16

0.08

Mean

0.36

0.12

(0.51) b/2 25.5 (26.9) 0.38 (0.36) 0.43 (0.42) 0.37 (0.36) 0.29 (0.28) 0.21 (0.20) 0.13 (0.12) 0.30 (0.29)

b/2 23.4 (23.9) 0.33 (0.38) 0.37 (0.42) 0.32 (0.37) 0.27 (0.30) 0.20 (0.23) 0.14 (0.15) 0.27 (0.31)

(0.17) b/2 18.5 (18.6) 0.27 (0.27) (0.30) (0.30) 0.27 (0.27) 0.22 (0.22) 0.17 (0.16) 0.11 (0.10) 0.23 (0.22)

b/2 27.5 (27.7) 0.27 (0.28) (0.29) (0.29) 0.25 (0.26) 0.21 (0.22) 0.17 (0.17) 0.12 (0.13) 0.22 (0.22)

Table 4 Peak edge deformations for the base isolated six-story building subjected to UBC spectrum. Optimized eccentricities have been obtained from FB-WTB criteria.

Optimized eccentricity ^ eðbÞ Y-dir. edge coordinate Isolation deformation (cm) Superstructure drift (cm)

^ðbÞ ¼ 0 Symmetric isolation XðbÞ o ¼ 1:0; e

Asymmetric isolation XðbÞ o ¼ 1:0

Asymmetric isolation XðbÞ o ¼ 0:75

–

0.52

0.20

1 2 3 4 5 6

b/2 18.22 0.37 0.42 0.37 0.30 0.22 0.14

b/2 18.32 0.12 0.11 0.11 0.10 0.09 0.07

b/2 35.27 0.36 0.42 0.36 0.28 0.19 0.11

b/2 16.91 0.31 0.34 0.30 0.25 0.18 0.12

b/2 21.02 0.26 0.29 0.25 0.20 0.15 0.09

b/2 23.07 0.24 0.25 0.22 0.19 0.15 0.11

Mean

0.30

0.10

0.29

0.25

0.21

0.19

the fundamental periods, modal mass ratios, and modal amplitudes for the first three fixed-base modes of vibration of the model. The method proposed for the torsional balance of multistory structures starts by first obtaining an equivalent single-story model. The procedure is described in Appendix A, and for the 6-story model the equivalent torsional parameters obtained were: e(s) = 135.3 cm (e(s)/q = 0.26, e(s)/b = 0.09) and XðsÞ o ¼ 0:78.

Three isolation systems were considered. The first one is the ref  erence symmetric system XoðbÞ ¼ 1; eb ¼ 0 ; the second one is an   asymmetric and torsionally hybrid system XðbÞ o ¼ 1 ; and the third one is an also asymmetric but torsionally flexible system   XoðbÞ ¼ 0:75 . In all three of them a stiffness-proportional damping

716

C.E. Seguin et al. / Engineering Structures 46 (2013) 703–717

has been considered, with a n(b) = 0.15 value for the lateral uncoupled frequency. Plan stiffness distribution of the asymmetric isolation systems are presented in Table 2. The maximum deformations at the edges of the isolated base and those reflecting the maximum story drifts of the superstructure are presented in Tables 3 and 4, obtained for the NCh and the UBC spectra, respectively. The optimized base eccentricities ^ eðbÞ of the asymmetric models have been calculated using the FB-WTB and DB-WTB approaches, which are based in the quasi-static model (Eqs. (5) and (6)). However, the results shown in these tables have been obtained from the exact dynamic equations using a modal superposition method devised for non-classical damping [17] for obtaining spectral estimates of maximum values. The results obtained show that the proposed design procedure is very effective in securing torsional balance of base isolated asymmetric multistory structures. Additionally, the extension of the tendencies observed in single-story models is confirmed, in particular, the one leading to recommend the use of torsionally flexible isolation systems. 8. Conclusions A design method for controlling the torsional response of seismically isolated asymmetric structures is presented. Based on a monosymmetric isolated two-story model with 4 DOFs, subjected to a one-directional excitation defined through its spectrum and analyzed quasi-statically, an explicit formula for estimating the isolation system parameters that achieve torsional balance of the superstructure is provided. Two methods are proposed: (i) a force-based torsional balance criterion which consists of annihilating the statistical correlation between shear and torque with respect to CS of the superstructure; and (ii) a deformation-based torsional balance criterion which consists in annihilating the statistical correlation between the rotation and the lateral displacement of the geometric center of the superstructure. To extend these methods to multistory structures, a procedure is proposed based in first finding an equivalent single-story model for which it is possible to apply the available explicit formulae. The main conclusions and recommendations that are derived from this study can be summarized as follows: (1) To counter-balance torsional effects in base isolated asymmetric superstructures it is necessary to introduce eccentricity in the isolation system. (2) The torsional balance of the superstructure substantially increases as the torsional stiffness of the isolation system decreases and/or the torsional stiffness of the superstructure increases. If the isolation system is torsionally stiff   XðbÞ it is not possible in general to achieve torsional o > 1 balance of the superstructure. (3) The eccentricity ^eðbÞ required in the isolation system to balance the superstructure increases with the eccentricity ^ eðsÞ of the superstructure and decreases with the torsional stiffness of the isolation system. (4) As the torsional stiffness of the isolation system decreases, the balance eccentricity ^eðbÞ tends to ^eðsÞ ; actually when XðbÞ o tend to zero, the two values coincide, and torsion in the superstructure disappears (strong torsional balance). (5) The balance eccentricity ^eðbÞ computed from the force-based torsional balance criterion is independent of the torsional stiffness of the superstructure. In superstructures with nor^eðsÞ< 0:25 and with rather high tormalized eccentricities  sional stiffness XðsÞ > 1 , the eccentricities obtained are o

quite similar to those given by the deformation-based torsional balance criterion. For this reason the application of this criterion is suitable for most practical cases. (6) For superstructures with normalized  eccentricities  ^ eðsÞ > 0:25 and with low torsional stiffness XðsÞ it is o < 1 preferable to use the deformation-based torsional balance criterion so as to assure that the deformations at the building edges do not turn out to be excessively different from one another. (7) The amplifications at the edges of the isolated base which result from introducing eccentricity into the isolation system depend on the specific spectrum considered. For a spectrum having a plateau, as is the case of the Chilean code for seismic isolation, these amplifications do not exceed 1.5, even for large eccentricities and low torsional stiffnesses. For spectra in which the displacements increase linearly with period, the amplifications at the flexible edge also increase, so that this technique will not be assure torsional balance of the superstructure.

Acknowledgments This investigation was funded by the Chilean National Fund for Science and Technology, Fondecyt through Grant #1020774 and #1090334. The authors are grateful for this support. Furthermore the authors thank Professor Jorge Vasquez for the translation of the article. Appendix A. Procedure to account for equivalent single-story model This appendix describes the algorithm used to calculate the torsional parameters (eccentricities and torsional stiffness ratios) of multi-story buildings. The procedure involves the calculation of the stiffness matrix of an equivalent single-story model. (1) Compute the modes of the multi-story building U, and extract the 3  3 sub-matrix,

e ¼ Uðr; pÞ U

ðA1Þ

where r (1  3) represent the roof degree of freedom (usually the last three); and p (1  3) represents the modes with higher modal mass ratio in the directions X, Y and H. (2) Compute the equivalent single-story mass matrix,

c M ¼ diag½m; m; Io  P P X mk xk mk yk m¼ mk xo ¼ yo ¼ m i m X h Io ¼ mk ðxk  xo Þ2 þ ðyk  yo Þ2 þ q2k

ðA2:aÞ ðA2:bÞ ðA2:cÞ

where mk and qk are the mass and the radius of gyration of the kth story; and (xk, yk) are the center of mass coordinates for the kth story. e to the center of mass of the whole structure, (3) Transform U

^ ¼ Lo U e U

ðA3:aÞ

where

2

1 0 ðyo  yr Þ

6 Lo ¼ 4 0 1 0

0

3

7 ðxo  xr Þ 5 1

ðA3:bÞ

is a kinematic transformation matrix; and (xr, yr) are the coordinates of center of mass of the roof.

C.E. Seguin et al. / Engineering Structures 46 (2013) 703–717

b is the modal matrix of an equivalent single(4) Assume that U story model,

bU b2 b ¼c bK K MU

ðA4Þ

b is the equivalent single-story stiffness matrix; and where K  2 b K ¼ diag x2I ; x2II ; x2III , where xk (k = I, II, III) are fundamental circular frequencies. b using Eq. (A4), (5) Compute K

b ¼c b 2U bK ^ 1 K MU

ðA5:aÞ

^Tc

b ¼ I, Eq. (A5.a) may be written as: If U M U

b ¼c b 2U bK ^Tc K MU M¼c M

! 3 X 2 ^T c ^ Uk x k Uk M

ðA5:bÞ

k¼1

^ij has the following elements, b ¼k (6) The stiffness matrix K

2 b ¼6 K 4

kx

0

kx ey

0

ky

ky ex

kx ey

ky ex

From this equation eccentricities,

^11 kx ¼ k ex ¼

^ k 23 ^ k 22

kh þ

we

kx e2y

can

þ

3 ky e2x

7 5

identify

ðA6:aÞ the

stiffnesses

and

^22 ky ¼ k ^

ey ¼  kk^13

ðA6:bÞ

11

^33  k ^ 2 =k ^11  k ^2 =k ^22 kh ¼ k 13 23 (7) Calculate the torsional stiffness ratios,

rffiffiffiffiffi k xx ¼ x m

Xhx ¼

xh xx

rffiffiffiffiffi k xy ¼ y m

Xhy ¼

xh xy

sffiffiffiffiffi k xh ¼ h Io ðA7Þ

717

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