Seismic performance of pendulum and translational roof-garden TMDs

ARTICLE IN PRESS Mechanical Systems and Signal Processing 23 (2009) 908–921

Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Seismic performance of pendulum and translational roof-garden TMDs Emiliano Matta , Alessandro De Stefano Department of Structural and Geotechnical Engineering, Politecnico di Torino, 24 Corso Duca degli Abruzzi, Torino 10129, Italy

a r t i c l e i n f o

abstract

Article history: Received 31 December 2007 Received in revised form 16 July 2008 Accepted 17 July 2008 Available online 30 July 2008

In a previous paper, the authors already introduced the concept of the roof-garden TMD, an innovative passive vibration absorber for building structures, meant to combine the dynamic response mitigation capabilities of traditional tuned mass dampers (TMDs) with the environmental advantages of traditional roof gardens. In order to limit the mistuning effect and control loss due to the intrinsic variability of its mass, a roofgarden TMD of the rolling-pendulum type, advantageously characterized by a massindependent natural period although unfortunately inherently non-linear, was then proposed and its performance assessed for increasing values of mass uncertainty and excitation levels. In the present study, the rolling-pendulum type is compared with the well-known translational configuration, the latter expectedly more prone to mistuning and yet insensitive to seismic intensity because of its linear behaviour. The trade-off between the two schemes is first explored for the case of a single-degree-of-freedom (SDOF) structure under a variety of design scenarios and then demonstrated through simulations on a building structure recently completed in a seismic site in Central Italy and designed to host a roof-garden atop for architectural purposes. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Passive structural control Seismic protection TMD Non-linearity Roof garden Sustainable development

1. Introduction Commonly used for controlling wind- and traffic-induced vibrations in flexible structures [1], passive tuned mass dampers (TMDs) are rarely employed for the seismic protection of buildings, their effectiveness in earthquake mitigation still being controversial [2–11] and generally acknowledged to be conditional upon the adoption of large mass ratios [12]. In order to meet the latter condition without recurring to cumbersome metal or concrete devices, the authors proposed in a previous study [13] to turn into TMDs non-structural masses already available atop buildings. In particular, the idea was presented of an innovative roof-garden TMD, capable of combining environmental and structural protection into one tool. The new device was conceived as a traditional tuned absorber but for its mass, consisting of a tank filled with planted soil. Such oscillating roof garden would provide vibrations absorption like a common TMD while retaining all the characteristics which make a green roof an unparalleled ingredient for sustainable development: Climate fluctuation mitigation, rain draining regularization, polluting particles capture, energy dispersion reduction; in short: liveability enhancement. In the city of Tokyo, for instance, where the adoption of planted roofs has been stimulated since 2001 through public incentives as a mean of contrasting the 31 average increase in annual temperatures, the roof-garden TMD  Corresponding author. Tel.: +39 011 5644884; fax: +39 011 5644899.

E-mail addresses: [email protected] (E. Matta), [email protected] (A. De Stefano). 0888-3270/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2008.07.007

ARTICLE IN PRESS E. Matta, A. De Stefano / Mechanical Systems and Signal Processing 23 (2009) 908–921

909

would sound like a promising solution, in that it would turn the seismically detrimental mass of a roof garden into a beneficial instrument of dynamic mitigation, potentially even encouraging using thicker soil layers with more massive (and likely more environmentally effective) arboreal species. Unfortunately, while keeping all the robustness drawbacks of the traditional TMD [14–16], the roof-garden TMD adds a new one, namely the uncertainty in the oscillating mass, prone to vary with the soil water content, the amount of variable load and the natural vegetation growth [17,18]. The resulting mass-uncertain TMD (MUTMD) appears thus susceptible to mistuning and control loss. In an attempt to minimize such adverse effects, robust analysis and synthesis against mass variations were applied in [13] to MUTMDs of the rolling-pendulum type (PTMD), a configuration characterized by mass-independent natural period. Through simulations under harmonic and recorded ground motions, the non-linear behaviour of circular and cycloidal rollingpendulum MUTMDs was there assessed and a possible implementation of roof-garden TMD on a real building structure was eventually described. In the present paper, the same study is extended to roof-garden TMDs of the translational type (TTMD). Equivalent in the absence of mass variation and in the linear response domain, PTMD and TTMD diverge from each other under mass perturbations, in that the TTMD will change its natural frequency while the PTMD will keep it unaltered, and under large excitations, in which case the TTMD will ideally remain linear while the PTMD will possibly undergo non-linear oscillations. The aim of this study is to illustrate, through simulations of roof-garden TMDs on SDOF and MDOF structures under harmonic and recorded ground motions, the trade-off between the non-linear PTMD and the linear TTMD, in order to establish which arrangement is preferable for practical implementations. The circular PTMD alone will be addressed, the cycloidal one having resulted inadequate for seismic purposes [13]. For the sake of simplicity, friction is assumed negligible throughout the whole paper for both the PTMD and the TTMD, for reasons explained in the sequel. The material is presented as follows. Section 2 derives the necessary dynamic equations for roof-garden PTMDs and TTMDs. Section 3 compares PTMDs and TTMDs focusing on assessing and minimizing the drawbacks of mass uncertainty in a worst-case HN linearized framework. Section 4 extends the comparison to the seismic performance in both the linear and the non-linear domains. Section 5 simulates two possible implementations of roof-garden TMDs, respectively, a PTMD and a TTMD, on a multi-storey building structure.

2. The equations of motion In this section the equations of motion are derived for an SDOF structure under ground motion, equipped with a roofgarden TMD of either PTMD or TTMD type (Fig. 1). The SDOF assumption, legitimate anytime the structural target mode is far from adjacent modes, is kept all through this paper, except in the multi-storey application of Section 5. The SDOF structure of stiffness ks, damping cs and mass ms is subjected to the ground acceleration x¨g(t). Both TTMD and PTMD have (uncertain) mass ma and damping ca, but whilst for the TTMD the restoring force is provided by the elastic force in the linear spring of stiffness ka, for the PTMD it derives from the tangential component of the gravity force along the circular trajectory (of radius R) descried by its centre of mass (dotted curve). In this scheme, xs(t) denotes the structural displacement relative to the ground, xa(t) the absorber’s horizontal displacement relative to the structure, za(t) the PTMD’s vertical displacement relative to the structure and y(t) the PTMD’s angular deflection. Not depicted for clarity’s sake, a non-linear (gap type) elastic rotational spring of stiffness ky is modelled for the PTMD so as roughly simulate the effect of a non-dissipative impact against a fail-safe

xa R

ca

 G G0

ma

ka

za

xa G0 G ca ks

ks

ms

ms

xg

cs

xs

xg

cs

Fig. 1. Schematics of SDOF structures equipped with a PTMD (left) or a TTMD (right).

xs

ma

ARTICLE IN PRESS 910

E. Matta, A. De Stefano / Mechanical Systems and Signal Processing 23 (2009) 908–921

retaining system deployed at the end of the allowable stroke. Such spring gets activated whenever |y(t)| exceeds an assigned limit ymax40, in which circumstance it applies the torque My ¼ ky(|y|ymax)sgn y to the PTMD pushing it back towards its at rest position. Choosing xs and y as the two generalized coordinates and applying basic analytical dynamics, it is easy to show [13] that the equations of motion for the non-linear PTMD are 2

ms ðx€ g þ x€ s Þ þ cs x_ s þ ks xs þ ma ðx€ g þ x€ s þ R cos yy€  R sin yy_ Þ ¼ 0

(1)

ma bðx€ g þ x€ s Þ cos y þ Ry€ c þ ca Rcos2 yy_ þ ma g sin y þ wðyÞ ¼ 0

(2)

where the term w(y) ¼ kmag(|y|ymax)sgn y step(|y|ymax) accounts for the said fail-safe constraint whenever |y|4ymax. The dimensionless parameter k ¼ (ky/R)/(mag), representing the ratio between the constraint restoring stiffness and the gravity restoring stiffness at small angular deflections, is herein equalled to 10,000 in order to sufficiently approximate a rigid boundary. Finally, ymax is assumed to equal 901. On the other hand, Newton’s Law provides the equations for the linear TTMD as ms ðx€ g þ x€ s Þ þ cs x_ s þ ks xs ¼ ca x_ a þ ka xa

(3)

ma ðx€ g þ x€ s þ x€ a Þ þ ca x_ a þ ka xa ¼ 0

(4)

At last, linearizing Eqs. (1) and (2), dropping the fail-safe constraint and recalling that xa ¼ Ry, the equations of motion for the linearized PTMD are obtained as ms ðx€ g þ x€ s Þ þ cs x_ s þ ks xs ¼ ca x_ a þ ðma g=RÞxa

(5)

ma ðx€ g þ x€ s þ x€ a Þ þ ca x_ a þ ðma g=RÞxa ¼ 0

(6)

3. Worst-case HN performance of roof-garden TMDs in the linear domain This section presents a procedure for evaluating and minimizing the performance reduction caused in a roof-garden TMD by mass variations. At this stage, assuming a linearized model for the PTMD as well, performance is measured in terms of the worst-case steady-state structural response to harmonic ground motion (worst-case HN approach). Realistic seismic inputs (spectrum-compatible accelerograms) will be introduced only in Section 4. 3.1. Introduction to the robust synthesis and analysis procedure As long as mass uncertainty is neglected, the linearized PTMD and the TTMD are indistinguishable. Introducing the equivalent stiffness as keq ¼ mag/R for the PTMD and as keq ¼ ka for the TTMD, PTMD and TTMD are jointly described by the equations ms ðx€ g þ x€ s Þ þ cs x_ s þ ks xs ¼ ca x_ a þ keq xa

(7)

ma ðx€ g þ x€ s þ x€ a Þ þ ca x_ a þ keq xa ¼ 0

(8)

and by the following normalized transfer function from x¨g(t) to xs(t):   keq þ ica o ms þ ma qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k þ ica o  ma o 2  eq  (9) Hxs x€ g ¼ ð2zs 1  zs o2s Þ  keq þ ica o 2 ks þ ics o  o ms þ ma keq þ ica o  ma o2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 where the normalizing factor 2zs 1  zs os , which exclusively depends on structural parameters and thus keeps constant during worst-case analysis for a given structure, is included to make the peak of the transfer function equal to 1 for the uncontrolled structure. As mass perturbations are admitted, however, PTMD and TTMD will differ even in the linear domain: For the PTMD, the natural frequency remains anchored to its nominal value (the equivalent stiffness changes); for the TTMD, the equivalent stiffness is unaltered (the natural frequency changes). Stemming from this evidence and recalling that TMDs mainly rely on frequency tuning to be effective, the authors proposed in [13] a roof-garden TMD of the pendulum type since expectedly more promising for ensuring robustness against mass fluctuations. Embracing the translational type as well, the present study will partially revise such expectation and depict a more articulated trade-off between PTMDs and TTMDs. To this aim, mass variations are herein described using a worst-case approach, where no probabilistic description is needed and fluctuations are merely assigned lower and upper bounds. So, the absorber’s mass varies around its nominal (mean) value ma0 according to ma ¼ ma0 ð1 þ dÞ;

8d 2 <=jdjpd

(10)

ARTICLE IN PRESS E. Matta, A. De Stefano / Mechanical Systems and Signal Processing 23 (2009) 908–921

911

where d is the admitted percentage bound, and the equivalent stiffness around its nominal value keq0 according to keq ¼ ma g=R ¼ ð1 þ dÞma0 g=R ¼ ð1 þ dÞkeq0 keq ¼ ka ¼ keq0

(11)

ðPTMDÞ

(12)

ðTTMDÞ 2

2

Using Eqs. (10)–(12) and further posing os ¼ ks/ms, zs ¼ cs/2msos, oa0 ¼ keq0/ma0, za0 ¼ ca/2ma0oa0, ma0 ¼ ma0/ms, ra0 ¼ oa0/os, r ¼ o/os, Eq. (9) is specialized as " # m0 ð1 þ dÞðð1 þ dÞr 2a0 þ i2za0 ra0 rÞ 1 þ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ dÞr 2a0 þ i2za0 r a0 r  ð1 þ dÞr 2 2 " # (13) HPxs x€ g ¼  2zs 1  zs m0 ð1 þ dÞðð1 þ dÞr2a0 þ i2za0 ra0 rÞ 2 1 þ i2zs r  r 1 þ ð1 þ dÞr 2a0 þ i2za0 r a0 r  ð1 þ dÞr 2 "  qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 HTxs x€ g ¼  2zs 1  zs

1þ

m0 ð1 þ dÞðr2a0 þ i2za0 ra0 rÞ

#

r 2a0 þ i2za0 r a0 r  ð1 þ dÞr 2 " # m ð1 þ dÞðr 2a0 þ i2za0 r a0 rÞ 1 þ i2zs r  r 2 1 þ 20 r a0 þ i2za0 r a0 r  ð1 þ dÞr 2

(14)

where os and zs are the circular frequency and the damping ratio of the (uncontrolled) structure; oa0, za0, m0 and ra0 are the nominal circular frequency, the nominal damping ratio, the nominal mass ratio and the nominal frequency ratio of the (fixed-base) MUTMD; and r is the input frequency ratio. Superscripts ‘‘P’’ and ‘‘T’’ will denote hereafter PTMDs and TTMDs, respectively. The aim of the chosen worst-case HN approach is to select the control parameters ra0 and za0 so that the worst-case HN norms RP ¼ max kHPxs x€ g k1 ¼ max max jHPxs x€ g j

ðPTMDÞ

(15)

RT ¼ max kHTxs x€ g k1 ¼ max max jHTxs x€ g j

ðTTMDÞ

(16)

d

r

d

d

r

d

are minimum for any given design scenario, i.e. for any given pair (m0, d). In other words, the optimization problem is formalized as the ‘‘robust synthesis’’ problem: RPopt ¼ RPopt ðm0 ; dÞ ¼ min RP ¼ min max max jHPxs x€ g j;

jdjpd ðPTMDÞ

(17)

RTopt ¼ RTopt ðm0 ; dÞ ¼ min RT ¼ min max max jHTxs x€ g j;

jdjpd ðTTMDÞ

(18)

r a0 ;za0

ra0 ;za0

r a0 ;za0

ra0 ;za0

d

d

r

r

and the parameters solving the two min max problems above are, respectively, denoted as r Pa0opt ¼ r Pa0opt ðm0 ; dÞ;

zPa0 opt ¼ zPa0 opt ðm0 ; dÞ ðPTMDÞ

(19)

r Ta0opt ¼ r Ta0opt ðm0 ; dÞ;

zTa0 opt ¼ zTa0 opt ðm0 ; dÞ ðTTMDÞ

(20)

If the control parameters are selected obeying Eqs. (19) and (20) depending on the expected uncertainty d ¼ ds (subscript ‘‘s’’ standing for ‘‘synthesis’’), the worst-case HN response in the presence of the actual uncertainty d ¼ da (subscript ‘‘a’’ standing for ‘‘analysis’’), where in general da6¼ds, is given by the following ‘‘robust analysis’’ equations: RPda ds ¼ RPda ds ðm0 ; ds ; da Þ ¼ max max jHPxs x€ g jrT

;zTa0 opt

RTda ds ¼ RTda ds ðm0 ; ds ; da Þ ¼ max max jHTxs x€ g jrT

;za0opt

d

d

r

r

c0 opt

a0opt

T

;

;

jdjpda ðPTMDÞ

(21)

jdjpda ðTTMDÞ

(22)

3.2. Application of the robust synthesis and analysis procedure The optimization problem defined by Eqs. (17) and (18) is herein solved through a branch & bound search, the same presented for multiple TMDs by the authors in [18]. A variety of design scenarios (m0, d) are considered, with m0 equalling in turn 1%, 2%, 5%, 10%, 20% and d varying between 0% and 50%. In all cases, structural damping is fixed at zs ¼ 0.02. Once the optimal parameters in Eqs. (19) and (20) are determined, the robust analysis is finally performed according to Eqs. (21) and (22) through letting da vary. The results of the robust synthesis are given in Fig. 2, where the optimal parameters and the corresponding optimal response are plotted for the PTMD (left) and the TTMD (right) as a function of the perturbation bound d which, being this a synthesis problem, stands for both ds and da. For d ¼ 0 (nominal design), the PTMD and the TTMD coincide with the

ARTICLE IN PRESS E. Matta, A. De Stefano / Mechanical Systems and Signal Processing 23 (2009) 908–921

1

1

0.95

0.95

0.9

0.9

0.85

0.85

r Ta0opt

r Pa0opt

912

0.8 0 = 1% 0 = 2% 0 = 5% 0 = 10% 0 = 20%

0.75 0.7

0 = 1% 0 = 2% 0 = 5% 0 = 10% 0 = 20%

0.8 0.75 0.7

0.65

0.65 0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

d 0.3

0.4

0.5

0.3

0 = 1% 0 = 2% 0 = 5% 0 = 10% 0 = 20%

0.25

0.25

0.2

0.2 Ta0opt

Pa0opt

0.3 d

0.15

0.15

0.1

0.1

0.05

0.05

0

0 = 1% 0 = 2% 0 = 5% 0 = 10% 0 = 20%

0 0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

0.3

0.4

0.5

d

d 1

1

0 = 1% 0 = 2% 0 = 5% 0 = 10% 0 = 20%

0.8

0 = 1% 0 = 2% 0 = 5% 0 = 10% 0 = 20%

0.8

0.6 RPopt

R Topt

0.6

0.4

0.4

0.2

0.2

0

0 0

0.1

0.2

0.3 d

0.4

0.5

0

0.1

0.2 d

Fig. 2. Robust synthesis—optimal HN parameters (tuning frequency and damping ratio) and corresponding worst-case HN response for mass-uncertain PTMD (left) and TTMD (right) as functions of the design uncertainty d, for varying nominal mass ratios m0.

traditional constant-mass TMD. However, as uncertainty is introduced (d40), PTMD and TTMD perform differently. As to the TTMD, parameters and response increasingly diverge from their nominal values as d increases and/or m0 decreases, with a significant decrease in ra0 opt and an increase both in za0 opt (except for m0 ¼ 20%) and in Ropt. In a way, the ratio d/m0 can be regarded as an inverse measure of the TTMD’s ‘‘capacity for robustness’’. For the PTMD, instead, the deviations from the nominal case are softer and mostly independent on m0, with a nearly constant ra0 opt, a slightly decreasing za0 opt and a limited increase in Ropt. These fundamental differences can be explained as follows. For a TTMD, the main drawback of mass

ARTICLE IN PRESS E. Matta, A. De Stefano / Mechanical Systems and Signal Processing 23 (2009) 908–921

913

variation is frequency mistuning, especially critical when m0 is small, i.e. when the reduction of the transfer function is circumscribed to a narrow frequency band around the structural frequency. For a PTMD, instead, mass variation does not affect the absorber’s natural frequency, hence its only drawback is mass reduction itself and the consequent control impairment. This also explains why the robust synthesis increases za0 opt for the TTMD and decreases it for the PTMD: In a TTMD, enlarging za0 opt helps keeping flat the transfer function when tuning is lost; in a PTMD, recalling that a smaller mass ratio requires a smaller optimal damping [1], reducing za0 opt is a means for maximizing the control effectiveness for the worst possible scenario, in fact corresponding to the minimum mass ratio. On the other hand, as m0 gets larger, the TTMD too gets more and more insensitive to frequency mistuning (its transfer function gets increasingly flatter) until it too, as the PTMD, is more troubled by mass ratio reduction than by frequency mistuning. Evidently this transition for the TTMD has already occurred for m0 ¼ 20%, as demonstrated by za0 opt becoming a decreasing function of d for this particular mass ratio. But above all, and most surprisingly, this transition reveals the entry into a domain of large m0 where the optimal response curves are in fact lower for the TTMD than for the PTMD. Thus, against all expectations (linearized) PTMDs prove more robust to mass variations only for sufficiently small values of m0; for larger values (for instance m0420%), a roof-garden TMD is more conveniently implemented in the shape of a TTMD. To exemplify the effects of mass variations and the advantages of a robust design, Fig. 3 applies the robust analysis of Eqs. (21) and (22) to the case when m0 ¼ 5%. The two alternatives of nominal (ds ¼ 0%) and robust (ds ¼ 50%) design are examined, and for each alternative the worst-case transfer function is computed twice, first under no uncertainty (da ¼ 0%) and then under the maximum uncertainty (da ¼ 50%). Four curves are thus plotted for both the PTMD (left) and the TTMD (right), from the lowest to the highest representing: (1) the response due to a nominally optimized constant-mass TMD (ds ¼ da ¼ 0%); (2) the worst-case response due to the robustly optimized MUTMD (ds ¼ da ¼ 50%); (3) the response due to the robustly optimized MUTMD when the expected perturbation does not occur (ds ¼ 50%, da ¼ 0%); (4) the worst possible scenario at all, happening when mass variations, erroneously neglected at the design stage, actually occur (ds ¼ 0%, da ¼ 50%). According to Eqs. (21) and (22), the peaks of the four curves in ascending order are denoted as RP0%0%oRP0%50%o RP50%50%oRP50%0% for the PTMD and as RT0%0%oRT0%50%oRT50%50%oRT50%0% for the TTMD. The four curves reveal a trade-off between nominal and robust syntheses: Admitting that mass variation occurs within a given bound, the robust synthesis provides the best possible performance if the expected perturbation occurs (RT50%50%oRT50%0%, RP50%50%oRP50%0%), but on the other hand worsens the performance with respect to the nominal design if no perturbation occurs (RT0%50%4RT0%0%, RP0%50%4RP0%0%). In order to quantify such trade-off, the robust analysis in Fig. 3 is extended in Fig. 4 to various combinations of m0, ds and da. In each diagram, having fixed m0 at, respectively, 1%, 5% and 20% from the top to the bottom, several robust designs are first accomplished, each based on a different expected uncertainty ds (in turn equal to 0%, 10%, 20%, 30%, 40%); then, for each robust design, a robust analysis curve is plotted providing the worst-case response under an increasing actual uncertainty da (ranging from 0% to 50%). The mentioned trade-off reflects in any two curves intersecting one another [13]. The minimum envelope, for a given m0, of all possible robust analysis curves (the thickest line in each picture) is the locus of the optimal robust syntheses, corresponding to the case when the actual perturbation equals the expected one. Such minimum envelope, already reported in Fig. 2, is generally lower for the PTMD than for the TTMD, meaning that PTMD are generally superior whenever ds ¼ da. The new result in Fig. 4 is that the total envelope, for a given m0, of all possible curves is thinner for the PTMD than for the TTMD. This means that PTMDs’ superiority is even larger if ds6¼da or, in other words that PTMDs are inherently more robust than TTMDs, independently on the adoption of any robust design.

0.8

0.8 ds = da = 0%

ds = da = 0%

0.7

ds = 50% da = 0%

0.7

ds = 50% da = 0%

ds = da = 50%

ds = da = 50%

0.6

0.6

ds = 0% da = 50%

0.5 HPxsxg

HPx x

s g

0.5

ds = 0% da = 50%

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0 0.6

0.7

0.8

0.9

1 r

1.1

1.2

1.3

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

r

Fig. 3. Worst-case acceleration-to-displacement transfer functions—trade-off between nominal and robust design for mass-uncertain PTMD (left) and TTMD (right)—nominal mass ratio m0 taken as 5%. Both design (ds) and actual (da) uncertainties taken as either 0% or 50%.

ARTICLE IN PRESS 914

E. Matta, A. De Stefano / Mechanical Systems and Signal Processing 23 (2009) 908–921

1

1 ds = da ds = 0.0 ds = 0.1 ds = 0.2 ds = 0.3 ds = 0.4

a s

0.6

0.8

R Td d

R Pdads

0.8

0.4 0.2

0.6 d s = da ds = 0.0 ds = 0.1 ds = 0.2 ds = 0.3 ds = 0.4

0.4 0.2

0

0 0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

da 1

0.5

a s

0.6

0.3

0.4

0.5

ds = da ds = 0.0 ds = 0.1 ds = 0.2 ds = 0.3 ds = 0.4

0.8

R Td d

R Pdads

0.4

1 ds = da ds = 0.0 ds = 0.1 ds = 0.2 ds = 0.3 ds = 0.4

0.8

0.4 0.2

0.6 0.4 0.2

0

0 0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

da 1

da 1

ds = da ds = 0.0 ds = 0.1 ds = 0.2 ds = 0.3 ds = 0.4

a s

0.6

ds = da ds = 0.0 ds = 0.1 ds = 0.2 ds = 0.3 ds = 0.4

0.8

R Td d

0.8

R Pdads

0.3 da

0.4 0.2

0.6 0.4 0.2

0

0 0

0.1

0.2

0.3 da

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

da

Fig. 4. Robust analysis—worst-case HN response for mass-uncertain PTMD (left) and TTMD (right) as a function of the actual uncertainty da, under varying design uncertainty ds and for different mass ratios m0—top pictures: m0 ¼ 1%. Middle pictures: m0 ¼ 5%. Bottom pictures: m0 ¼ 20%.

In conclusions, as long as steady-state harmonic oscillations are considered, it is shown that, although mass variations inevitably worsen performance, a proper robust synthesis always preserves enough of the nominal effectiveness for (linearized) PTMDs, and for TTMDs as well if the ratio d/m0 is sufficiently small. In particular, if m0 is larger than approximately 20%, the best solution does unexpectedly correspond to using TTMDs.

4. Seismic performance of roof-garden TMDs In this section, the seismic performance of roof-garden PTMDs and TTMDs is compared using horizontal ground acceleration time histories as the input, respectively, in the linear and non-linear domain assumptions. Spectrumcompatible recorded accelerograms are chosen as prescribed by the Eurocode 8 [20] and by the Italian Code [21]. A set of seven accelerograms is employed so that their average response can be used for assessment instead of the most severe response. The set, valid for soil type C, is the one detailed in [13]. The seven accelerograms are scaled so as to have the same peak ground acceleration (pga), evaluated as the product of the site amplification factor S (1.25 for soil type C) times the reference pga on soil type A, indicated as ag and classified by the Italian Code in four hazard levels or zones (from zone 4 to 1:ag ¼ 0.05, 0.15, 0.25, 0.35 g).

ARTICLE IN PRESS E. Matta, A. De Stefano / Mechanical Systems and Signal Processing 23 (2009) 908–921

915

4.1. Linear domain Time-history simulations are conducted based on Eqs. (7) and (8), with the uncertain mass and the equivalent stiffness given by Eqs. (10)–(12), the structural damping still assumed as zs ¼ 2%, and TMD’s frequency ratio and damping ratio equal to the optimum values derived from the HN robust synthesis. As for the HN analysis, a worst-case approach is here adopted whereby the response to each accelerogram is evaluated as the maximum over the entire range of admissible perturbations. The average spectrum of the peak structural pseudo-accelerations is therefore introduced as Ap ðTÞ ¼

n n 1X 1X A ðTÞ ¼ max max jai;T ðtÞj; t n i¼1 p;i n i¼1 d

(23)

jdjpda

where ai,T(t) ¼ xs,i(t)(2p/T)2 is the pseudo-acceleration of a structure of period T under the i-th accelerogram (out of n ¼ 7 records), equipped with a roof-garden TMD subjected to the mass perturbation d and designed for the expected uncertainty ds. Based on Eq. (23), seismic performance is measured by comparing controlled and uncontrolled spectra for various combinations of m0, ds and da. The nominal case (ds ¼ da ¼ 0) is presented in Fig. 5(a), where the spectra are plotted for different values of m0. Since uncertainty is excluded, results hold for both PTMD and TTMD. A considerable loss of control effectiveness is observed with respect to the harmonic input case so that a large m0 is required to acceptably cut down the peak response. Mass uncertainty is then introduced in Fig. 5(b), where PTMD and TTMD are compared for m0 ¼ 5% under increasing perturbation levels under the assumption that ds ¼ da ¼ d. As d increases from 0% to 10% and up to 50%, the worst-case controlled spectra increasingly approach the uncontrolled spectrum, less rapidly for the PTMD than for the TTMD. A notable feature in Fig. 5 is that TMD’s control effectiveness is approximately constant over the entire range of periods, for any assigned pair (m0, d). This allows for a more concise representation, provided that a normalized pseudo-acceleration index Ap,norm is introduced, obtained for every pair (m0, d) dividing the controlled spectrum by the uncontrolled one (separately for each of the seven accelerograms) and then averaging over the whole range of periods. Fig. 6 plots Ap,norm on the vertical axis as a function of the mass uncertainty, for different mass ratios. Although the differences between TTMDs and PTMDs are somewhat smoothed, the main conclusion drawn in Section 3 for the steady-state response is confirmed for the seismic case, i.e. PTMDs are decisively preferable for small values of m0, TTMDs slightly preferable for large values, the transition occurring at m0 slightly less than 20%. 4.2. Non-linear domain Non-linear equations (1) and (2) are here solved for the mass-uncertain PTMD instead of the linearized equations (7) and (8) used in Section 4.1, under the same set of accelerograms, in order to account non-linearities under large seismic inputs. As shown in [13], the tuning condition for a PTMD results into a geometric constraint on its available stroke, particularly strict for rigid structures and for small mass ratios. PTMD’s effectiveness may be significantly impaired in high seismicity regions and for target periods in the range of maximum spectral amplifications, particularly when small mass ratios are used.

4

4 Uncontrolled 0 = 1% 0 = 2% 0 = 5% 0 = 10% 0 = 20%

3.5

2.5 2

3 Ap / (Sag)

Ap / (Sag)

3

Uncontrolled Ideal TMD - d = 0% MU TTMD - d = 10% MU PTMD - d = 10% MU TTMD - d = 50% MU PTMD - d = 50%

3.5

d=0

2.5 2

1.5

1.5

1

1

0.5

0.5

0

0 = 5%

0 0

0.5

1

1.5

2 T (s)

2.5

3

3.5

4

0

0.5

1

1.5

2 T (s)

2.5

3

3.5

4

Fig. 5. Average response to a set of 7 spectrum-compatible recorded accelerograms—linear pseudo-acceleration spectra in case of: (a) traditional constant-mass TMDs (d ¼ 0) for different mass ratios m0; (b) PTMDs and TTMDs in case of m0 ¼ 5% for different uncertainties d.

ARTICLE IN PRESS 916

E. Matta, A. De Stefano / Mechanical Systems and Signal Processing 23 (2009) 908–921

1

Ap,norm

0.9 0.8 0.7 PTMD

0.6 TTMD

0.5 0

0.1

0.2

0.3

0.4

0.5

d Fig. 6. Average response to a set of 7 spectrum-compatible recorded accelerograms—normalized linear pseudo-acceleration indices for PTMD and TTMD as a function of mass uncertainty d for different mass ratios m0—from the thinner to the thicker curve: m0 ¼ 1%, 2%, 5%, 10%, 20%.

4 Uncontrolled ag = 0.05g

3.5

ag = 0.15g

Ap / (Sag)

3

ag = 0.25g ag = 0.35g

2.5

0 = 5%

2 1.5 1 0.5 0 0

0.5

1

1.5

2 T (s)

2.5

3

3.5

4

Fig. 7. Average response to a set of 7 spectrum-compatible recorded accelerograms—non-linear pseudo-acceleration spectra for traditional constantmass PTMDs (d ¼ 0) in case of m0 ¼ 5%, under increasing seismicity (ag/g ¼ 0.05–0.35).

Still neglecting mass uncertainty, an example is given for m0 ¼ 5% in Fig. 7, where the pseudo-acceleration non-linear spectra are computed for PTMDs under various seismic hazard levels. As anticipated, non-linearity affects rigid structures much more than flexible structures. Under these circumstances, the response index Ap,norm introduced in Section 4.1 must be conveniently modified by restricting the averaging to limited intervals of the spectrum, namely to either the range of periods 0.15–0.5 s (corresponding to ‘‘rigid structures’’ in the spectral plafond) or to the range 1.5–2.5 s (corresponding to ‘‘flexible structures’’ in a lower ordinates spectral region). The response index Ap,norm is presented in Fig. 8. The charts on the top refer to m0 ¼ 2%, those on the bottom to m0 ¼ 10%, those on the left to rigid structures and those on the right to flexible structures. Each chart compares the structural response using alternatively a linear TTMD or a non-linear PTMD, for increasing values of mass uncertainty (on the horizontal axis) and under different seismic hazard levels (from the thinner to the thicker curve: ag ¼ 0.05, 0.15, 0.25, 0.35 g). Unifying mass variability and amplitude-dependent non-linearity, these figures capture the possible trade-offs between alternative mass-uncertain roof-garden TMDs for the seismic case and justify the following conclusions. For the lowest hazard level non-linearity is virtually negligible both for rigid and flexible structures, and under large mass variations the PTMD is preferable, although its advantage is nearly insignificant already for m0 ¼ 10% (for m0X20% the TTMD is superior instead). As the hazard level increases, rigid and flexible structures must be distinguished. For rigid structures, under any circumstance the TTMD becomes the most advantageous option. A particularly large mass uncertainty is therefore little compatible with rigid structures in high seismicity areas as long as small mass ratios are used, since the need for a TTMD, inherently less robust, would impair control effectiveness. For flexible structures, if a small mass ratio is used, the performance reduction due to non-linearity in the PTMD subjected to the highest hazard level is comparable with the performance reduction due to mass uncertainty in the TTMD

ARTICLE IN PRESS E. Matta, A. De Stefano / Mechanical Systems and Signal Processing 23 (2009) 908–921

1

917

0.9 Rigid structures 0 = 2%

Flexible structures 0 = 2%

Ap,norm

Ap,norm

0.9 0.8

0.8 PTMD PTMD TTMD

TTMD 0.7

0.7 0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

d

0.3

0.4

0.5

0.4

0.5

d

1

0.9 PTMD

PTMD

TTMD

TTMD

Ap,norm

Ap,norm

0.9 Flexible structures 0 = 10%

0.8

0.8 Rigid structures 0 = 10%

0.7

0.7 0

0.1

0.2

0.3

0.4

0.5

d

0

0.1

0.2

0.3 d

Fig. 8. Average response to a set of 7 spectrum-compatible recorded accelerograms—normalized non-linear PTMD and TTMD pseudo-acceleration indices for ‘‘rigid’’ structures (left) and ‘‘flexible’’ structures (right), as a function of mass uncertainty d—top pictures: m0 ¼ 2%. Bottom pictures: m0 ¼ 10%. From the thinner to the thicker curve: ag/g ¼ 0.05, 0.15, 0.25, 0.35.

subjected to the 50% perturbation. Therefore, based on the expected seismicity of the area and variability of the mass, either the TTMD or the PTMD should be chosen. If a large mass ratio is used, all configurations are substantially equivalent, with the PTMD type slightly superior for m0o20% and the TTMD slightly superior for m0X20%. 5. Design of a roof-garden TMD on a multi-storey building These conclusive pages exemplify the design of a roof-garden TMD of either pendulum or translational type on an MDOF structure. The case study is the roof-garden TMD proposed by the authors to reduce the seismic response of a multi-storey building structure recently constructed in Siena, one of the most beautiful medieval towns in Central Italy. The building is the D Unit of the Portasiena Linear Building complex,1 a polyfunctional commercial centre on which the use of roof gardens was imposed for architectural reasons. A rendering view of the complex and an axonometric projection of the full 3D FEM model for the D Unit are given in Fig. 9. Siena is classified in the seismic hazard zone 2 (moderate seismicity), corresponding to ag ¼ 0.25 g. For an in-situ soil type C category (whence S ¼ 1.25) and an importance factor gI ¼ 1.2, this results in a no-collapse limit state pga of 0.375 g. Roof gardens of the ‘‘intensive’’ kind were chosen, implying a soil mass of about 500 kg/m2, 5.8% of the total mass of the building. Involved in the structural design of the complex, the first author conceived the idea of turning the additional mass of the roof garden into a passive TMD, capable of combining into one tool two functions, the architectural and the structural ones. The same case study has already been proposed in [13] for the PTMD configuration alone. The TTMD option is here added and a comparison is drawn to identify the best solution. A thorough description of the simplified planar 6DOF model representing the D Unit 6-storey building is provided in [13]. In such model, permanent and variable masses are combined into seismic masses through coefficients from the Italian Code (from bottom to top: 1360, 1410, 1600, 1600, 1570, 1870 Mg); a 5% modal damping is assigned to each mode; the three lower modes result to have natural frequencies 2.35, 9.29 and 18.9 Hz and modal mass ratios 57.9%, 25.4% and 9.3%. Although a traditional roof garden was finally preferred for practical 1 Reference in this paper kindly authorized by La Policentro S.p.A. and Policentro Progetti S.r.l., Agrate Brianza (MI), Italy, as the general planner and the owner, and by Si.Me.Te. s.n.c. and Studio Siniscalco, Torino, Italy, as the structural designer.

ARTICLE IN PRESS 918

E. Matta, A. De Stefano / Mechanical Systems and Signal Processing 23 (2009) 908–921

6

7

4

3

6

1

8

5

9

3

4

1

2

5 2

22cm 17cm

10cm

15cm

8cm

a

b

1 2 3 4 5

trapdoor for inspection (behind the column) limited load-bearing capacity r.c. floor prestressed-precast r.c. beams prestressed-precast r.c. panels completion r.c. slab

6 7 8 9

lanted soil (roof-arden) rolling-pendulum bearing commercial viscous dampers HDRB

Fig. 9. Simulation of a roof-garden TMD in Portasiena Linear Building—top left: architectural rendering of the complex. Top right: FEM model of the D Unit. Bottom: two alternative arrangements for the roof-garden TMD (PTMD on the left, TTMD on the right). Detail of the rolling-pendulum bearing: (a) at rest position; (b) maximum expected displacement position.

reasons (e.g. the Italian Code does not explicitly recognize TMDs for seismic protection), this simulation may prove interesting in its attempt to touch on some technological issues useful in practical design. On the other hand, if realized, the proposed device would have been, to the authors’ knowledge, the first purely seismic application of passive TMDs on a building structure. In fact, as mitigation of moderate earthquakes is generally considered a mere by-product of windoriented installations on tall buildings (Chiba Port Tower, MHS Bldg, Fukuoka Tower), the only truly seismic precedent appears the TMD deployed on Sakhalin 1 Exxon Mobil’s offshore platform (not exactly an ordinary building structure) to minimize lateral accelerations at the top of the drilling rig derrick structure during seismic events (Sakhalin Islands, Russia, 2005). Anyway, no reference is known to the authors reporting on the actual performance of such device. In order for the roof garden to work as a TMD, its mass must be evidently disconnected from the top storey. The simplest solution is to lay the planted soil onto a single additional floor resting on bearings coaxial with the columns below. The crucial point rests in the selection and design of the appropriate kind of bearing. A rolling-pendulum bearing would constitute a PTMD system, whilst the linear TTMD system could be roughly obtained through a rubber bearing. In this simulation, both options are explored for the sake of a comparison. For the PTMD system, rolling-pendulum bearings are adopted, consisting of a steel cylinder sandwiched between two identical cylindrical steel cavities of the circular type. Dissipation is granted by horizontal viscous dampers, placed in parallel with the rolling bearings and assumed as linear devices in this study for the sake of simplicity. As to the TTMD system, rubber bearings accomplish both the functions of stiffness and damping. Both the proposed PTMD and TTMD arrangements are so conceived as to keep friction and hysteresis small enough to allow for a linear viscous modelling of the dissipation mechanism in the device. For the PTMD, once the viscous dampers are admitted as linear dashpots, the sandwich-like rolling-pendulum configuration reduces non-linear dissipation to the steel–steel rolling-friction between the ball and the cavities, negligible in operational conditions. For the TTMD, the rubber bearings virtually present no

ARTICLE IN PRESS E. Matta, A. De Stefano / Mechanical Systems and Signal Processing 23 (2009) 908–921

919

friction but rather a constitutive law which in common applications (base isolation) proves to be adequately approximated by an equivalent linear damping model over a wide range around the design operational amplitude. Thus, a possible technological implementation for both types is schematized in the cross-sections at the bottom of Fig. 9. Columns are connected on the top by a cast-in-place, low load-bearing capacity, thin floor, perforated by proper trapdoors to allow periodic inspection and/or substitution of the bearings, which are placed on the vertical axis of the columns. The additional floor actually supporting the planted soil is supposed to be built on pre-cast beams and panels, so as to avoid the need for formworks underneath, and completed with a thin concrete slab. For the PTMD case, the viscous dampers connecting the two floors are housed at both sides of the bearing in the transverse direction. Assuming for the PTMD rolling system the non-linear behaviour described by Eqs. (1) and (2) and for the TTMD rubber bearings the linear behaviour described by Eqs. (3) and (4), the controlled structural model is obtained by simply appending the roof-garden TMD to the top floor of the 6-DOF planar model and reducing the sixth-storey mass to 550,000 kg only. The mass of the TMD is uncertain because of possible variations of the amount of soil, of the soil-moisture content, of the vegetation growth, of any masses temporarily on the garden. This uncertainty can be described by upper and lower bounds, thus bypassing complicated probabilistic formulations. The lower bound is assumed as 50% the 500 kg/m2 nominal value, i.e. 250 kg/m2. The upper bound as 150% the nominal value, i.e. 750 kg/m2, plus 240 kg/m2 for the variable masses possibly present (according to the Italian Code) on the garden during an intense seismic event. Including 650 kg/m2 for the structural mass of the floor, the total mass is bounded by 900 and 1640 kg/m2, so a robust design can be applied based on the uncertainty description of Eq. (14) with m0 ¼ 1270 kg/m2  26 m  42 m ¼ 1,386,840 kg (17.1% of the total mass) and d ¼ 30%. All the ingredients being set, an extension to MDOF structures of the (linear) worst-case HN robust design proposed in Section 3 is applied, in both cases of PTMD and TTMD, searching for the optimum frequency and damping ratios as those minimizing the worst-case peak modulus of the transfer function Hdmax x€ g (normalized so as to have a unit peak for the uncontrolled structure) from the ground acceleration to the maximum interstorey drift ratio, this being the interstorey displacement divided by the interstorey height, here taken as the objective function since critical for both structural and non-structural reasons. Differently from Section 3, for the TTMD a bound is now added on the maximum available damping in order to match commercially available rubber bearings (although a larger damping could be attained by adding viscous dashpots in parallel), so that a suboptimally damped TTMD is eventually obtained. No bound is instead imposed in the case of the PTMD, where damping is absolutely uncoupled from stiffness. The results of such optimization are reported in Fig. 10(a), where the worst-case controlled transfer functions corresponding to the optimum (linearized) PTMD and to the optimum TTMD subjected to 30% mass uncertainty are compared with those for the uncontrolled structure and for the structure controlled with a traditional constant-mass TMD. For the PTMD, the (unconstrained) optimum design corresponds to ra0 ¼ 0.390 (nominal frequency ratio) and za0 ¼ 0.465 (nominal damping ratio), or equivalently to oa0 ¼ 7.64 rad/s (circular frequency) and ca ¼ 9.8522  106 N s/m (total damping coefficient). For the selected rolling-pendulum arrangement of two circular cylinders facing each other, such frequency requires a radius of 13.4 cm for the two steel cavities once the rod section diameter is fixed equal to 10 cm. The subsequent maximum horizontal relative displacement of the PTMD allowed by this configuration is 9.3 cm. A total

1

0.5 Uncontrolled TMD PTMD TTMD

0.8

Uncontrolled TMD PTMD TTMD

0.4 0.3

0.6

dmax (%)

Hd max xg

0.2

0.4

0.1 0 -0.1 -0.2

0.2

-0.3 -0.4

0

-0.5 0

2

4

6 f (Hz)

8

10

12

0

5

10

15

t (s)

Fig. 10. Uncontrolled and variously controlled (traditional constant-mass TMD vs. mass-uncertain PTMD and TTMD) structural responses of the D Unit: (a) worst-case transfer functions from the ground acceleration to the maximum inter-storey drift ratio; (b) time histories for the maximum inter-storey drift ratio under the accelerogram no. 2.

ARTICLE IN PRESS 920

E. Matta, A. De Stefano / Mechanical Systems and Signal Processing 23 (2009) 908–921

Table 1 Peak seismic responses for the uncontrolled and the controlled cases Uncontrolled dmax (%) xs,max (cm) (m/s2) x€^ s;max

Tb (MN) xa (cm)

0.381 5.54 12.5 56.1 –

TMD 0.185 2.81 8.54 35.04 4.34

PTMD 0.201 3.07 9.56 38.4 5.68

TTMD 0.182 2.95 10.6 39.7 9.60

number of 28 such bearings are required to support the upper floor. On the other hand, the required total damping ca is distributed among 28 commercial linear dampers with a damping coefficient cai ¼ 0.3519 kN s/mm. For the TTMD, the (constrained) optimum design corresponds to ra0 ¼ 0.353 and za0 ¼ 0.20, or equivalently to ka ¼ 6.6297  107 N/m (total stiffness) and ca ¼ 3.8355  106 N s/m, resulting in 28 linear spring/dashpot systems having kai ¼ 2.3677 kN/mm and cai ¼ 0.140 kN s/mm. The desired kai and cai can be easily approximated using common commercial high damping rubber bearing (HDRB) isolators of the hard compound type (GE1.4 MPa). Finally, for the traditional constant-mass TMD (ideal TMD) the (unconstrained) optimum design corresponds to ra ¼ ra0 ¼ 0.421 and za ¼ za0 ¼ 0.523. With and without TMDs atop, the MDOF model is evaluated under the set of seven natural recordings used in Section 4, with a pga of 0.375 g, and the worst-case time history of the maximum interstorey drift ratio, occurring at the upper storey under the accelerogram no 2, is reported in Fig. 10(b). The satisfactory performance of the MUTMD is confirmed by Table 1, which compares the interesting structural responses for the uncontrolled and controlled cases, obtained averaging the peak time-history responses over the set of seven accelerograms. The maximum interstorey drift ratio dmax (the selected objective function) drastically drops to 52.8% and 47.8% for the controlled response, respectively, in the case of PTMD and TTMD. Note that for the constant-mass TMD dmax drops to 48.6%, revealing a slightly worse performance than that of the TTMD (the HN minimization does not exactly grant the minimum possible time-history response). Significant reductions are also observed for the maximum relative displacement xs,max, the maximum absolute accelerationx€^ s;max and the base shear force Tb, with the constant-mass TMD generally superior to the roof-garden PTMDs and TTMDs, on their turn comparable in effectiveness (the former slightly better in reducing accelerations and base shear force, the latter in controlling interstorey drifts and displacements). Remarkably, the average peak relative displacement xa of the PTMD is only 5.7 cm, corresponding to an oscillation angle of less than 201, indicating that the large mass ratio used in this application results in limited non-linearities. Furthermore, the maximum peak relative displacement among the seven accelerograms is 8.0 cm, which is still less than the available 9.3 cm stroke, whilst the largest relative velocity is 78.2 cm/s, corresponding to a maximum ultimate limit state force in each damper equal to 275 kN. On the other hand, the average peak relative displacement xa of the TTMD is 9.6 cm, fully compatible with commercially available HDRB isolators. To explain so encouraging results, it should be recalled that the TMD’s effective mass ratio, being the damper tuned to only the first modal mass and placed atop the building where the first modal shape is maximum, is much larger than the ratio between TMD’s mass and total structural mass, computed above as an already remarkable 17.1%. Applying Warburton’s formulas [22], the equivalent ‘‘effective mass ratio’’ results 76.3%, emphasizing that roof -gardens are a very promising source of mass, although uncertain, for control purposes.

6. Conclusions Developing the concept of the roof-garden TMD presented in a previous contribution by the same authors, this paper compares the pendulum and the translational arrangements in order to explore alternative technological implementations of the new device and to discuss their respective control efficacy, keeping into account the performance impairment induced by mass variability as well as by non-linearities related to rolling. Recurring to a worst-case perturbation approach and assuming an HN linear robust synthesis as a legitimate optimization criterion for attaining seismic mitigation, mass-uncertain PTMDs and TTMDs are introduced, and their worstcase performance evaluated under a variety of design scenarios, dutifully turning to non-linear models for the PTMD whenever needed. Based on this analysis, the following conclusions are achieved: (1) For a roof-garden TMD designed for seismic mitigation, preference should be granted to either the translational or the pendulum type depending on a number of factors such as the chosen mass ratio, the structural period, the expected mass variability, the seismic hazard. (2) As long as linear models hold, the mass ratio is a determinant factor: For small mass ratios, PTMDs may prove largely superior in reducing the steady-state response but only slightly preferable in reducing the seismic response; for large mass ratios (approximately above 10–20%), quite unexpectedly it is TTMDs which work better. (3) As PTMD’s non-linearity is accounted

ARTICLE IN PRESS E. Matta, A. De Stefano / Mechanical Systems and Signal Processing 23 (2009) 908–921

921

for, a crucial issue becomes the seismic stroke demand, which may exceed the stroke capacity of the device, depending on the period of the target structural mode and on the seismic hazard level. Although the trade-off between the two configurations is thus made even more articulated, it basically appears that for seismic applications, i.e. when small mass ratios have little effect, the advantage of PTMD is negligible and circumscribed to low seismic hazards or to flexible structures, so that in general a TTMD is preferable, particularly if there is little confidence in the expected excitation intensity. (4) Roof-garden TMDs such as the one proposed in the conclusive simulation may entail very large ‘‘effective mass ratios’’ for ordinary building structures (easily beyond 50%), granting a large nominal performance as well as an improved robustness against both mass fluctuations and non-linearities. In these circumstances, mass uncertainty proves of little disturbance and a bunch of possible arrangements (rolling-pendulum bearings plus oil dampers, for instance, or HDRB isolators) is available to assure the innovative roof-garden TMD most of the performance of a traditional constantmass device.

Acknowledgement This work is part of a research project financed by ReLUIS (the Italian National Network of University Laboratories of Earthquake Engineering), which is gratefully acknowledged. References [1] T.T. Soong, G.F. Dargush, Passive Energy Dissipation Systems in Structural Engineering, Wiley, New York, USA, 1997. [2] Y.P. Gupta, A.R. Chandrasekaran, Absorber system for earthquake excitation, in: Proceedings of the Eight World Conference on Earthquake Engineering, Santiago, Chile, 1969. [3] A.M. Kaynia, D. Veneziano, Seismic effectiveness of tuned mass dampers, Journal of the Structural Division ASCE 107 (1981) 1465–1484. [4] J.R. Sladek, R.E. Klinger, Effect of tuned-mass dampers on seismic response, Journal of the Structural Division ASCE 109 (1983) 2004–2009. [5] A.H. Chowdhury, M.D. Iwuchukwu, The past and future of seismic effectiveness of tuned mass dampers, in: Proceedings of the Second International Symposium on Structural Control, The Hague, The Netherlands, 1987. [6] T. Miyama, Seismic response of multi-storey frames equipped with energy absorbing storey on its top, in: Proceedings of the Tenth World Conference on Earthquake Engineering, Madrid, Spain, July 1992. [7] R. Soto-Brito, S.E. Ruiz, Influence of ground motion intensity on the effectiveness of tuned mass dampers, Earthquake Engineering and Structural Dynamics 28 (1999) 1255–1271. [8] P. Lukkunaprasit, A. Wanitkorkul, Inelastic buildings with tuned mass dampers under moderate ground motions from distant earthquakes, Earthquake Engineering and Structural Dynamics 30 (2001) 537–551. [9] G. Chen, J. Wu, Optimal placement of multiple tuned mass dampers for seismic structures, Journal of Structural Engineering 127/9 (2001) 1054–1062. [10] T. Pinkaew, P. Lukkunaprasit, P. Chatupote, Seismic effectiveness of tuned mass dampers for damage reduction of structures, Engineering Structures 25 (2003) 39–46. [11] A. Ghosh, B. Basu, Effect of soil interaction on the performance of tuned mass dampers for seismic applications, Journal of Sound and Vibration 274 (2004) 1079–1090. [12] N. Hoang, Y. Fujino, P. Warnitchai, Optimal tuned mass damper for seismic applications and practical design formulas, Engineering Structures 30 (3) (2008) 707–715. [13] E. Matta, A. De Stefano, Robust design of mass-uncertain rolling-pendulum TMDs for the seismic protection of buildings, Mechanical Systems and Signal Processing, 2007, in press, doi:10.1016/j.ymssp.2007.08.012. [14] J. Park, D. Reed, Analysis of uniformly and linearly distributed mass dampers under harmonic and earthquake excitation, Engineering Structures 23 (2001) 802–814. [15] S.D. Kwon, K.S. Park, Suppression of bridge flutter using tuned mass dampers based on robust performance design, Journal of Wind Engineering and Industrial Aerodynamics 92 (2004) 919–934. [16] N. Hoang, P. Warnitchai, Design of multiple tuned mass dampers by using a numerical optimizer, Earthquake Engineering and Structural Dynamics 34 (2005) 125–144. [17] E. Matta, Mass-uncertain tuned mass dampers for the dynamic protection of buildings, Ph.D. Thesis, DISTR, Turin Polytechnic Institute, Turin, Italy, December 2005. [18] A. De Stefano, E. Matta, Robust design of mass-uncertain TMD on building structures, in: Proceedings of the Eighth International Conference on Computational Structures Technology (CST 2006), Las Palmas de Gran Canaria, Spain, September 2006. [20] Eurocode 8, Design for structures for earthquakes resistance—part 1: General rules, seismic actions and rules for buildings—EN 1998-1, 2004. [21] Ordinanza del Presidente del Consiglio dei Ministri n. 3274 del 20 marzo 2003 e successive modifiche ed integrazioni, Norme tecniche per il progetto, la valutazione e l’adeguamento sismico degli edifici (Italian Seismic Code Ord. 3274), 2003. [22] G.B. Warburton, E.O. Ayorinde, Optimum absorber parameters for simple systems, Earthquake Engineering and Structural Dynamics 8 (1980) 197–217.