HBRC Journal (2015) xxx, xxx-xxx

Housing and Building National Research Center HBRC Journal

http://ees.elsevier.com/hbrcj

FULL LENGTH ARTICLE

Vibration control of buildings by using partial floor loads as multiple tuned mass dampers

Tharwat A. Sakr

Department of Structural Engineering, Zagazig University, Zagazig 44519, Egypt

Received 5 February 2015; revised 1 April 2015; accepted 17 April 2015

KEYWORDS

Structures; Buildings;

Tuned mass damper;

Earthquake;

Abstract Tuned mass dampers (TMDs) are considered as the most common control devices used for protecting high-rise buildings from vibrations. Because of their simplicity and efficiency, they have found wide practical applications in high-rise buildings around the world. This paper proposes an innovative technique for using partial floor loads as multiple TMDs at limited number of floors. This technique eliminates complications resulting from the addition of huge masses required for response control and maintains the mass of the original structure without any added loads. The effects of using partial loads of limited floors starting from the top as TMDs on the vibration response of buildings to wind and earthquakes are investigated. The effects of applying the proposed technique to buildings with different heights and characteristics are also investigated. A parametric study is carried out to illustrate how the behavior of a building is affected by the number of stories and the portion of the floor utilized as TMDs. Results indicate the effectiveness of the proposed control technique in enhancing the drift, acceleration, and force response of buildings to wind and earthquakes. The response of buildings to wind and earthquakes was observed to be more enhanced by increasing the story-mass ratios and the number of floor utilized as TMDs. © 2015 The Author. Production and hosting by Elsevier B.V. on behalf of Housing and Building National Research Center. This is an open access article under the CC BY-NC-ND license (http://creativecom-

mons.org/licenses/by-nc-nd/4.0/).

Introduction

Tuned mass dampers (TMDs) are considered as the most commonly used devices for controlling the dynamic response of

structures because of their effectiveness, robustness, and relative ease of installation [1,2]. Because of the efficiency of TMD systems, they have been used in many structures around the world, such as buildings and bridges [1-3]. Although TMDs have been installed in many buildings around the world, such as the CN tower at Toronto, 1975 and Shanghai World Finance Center at Shanghai, 2008, the 660-ton TMD installed at the top of the Taipei Tower at Taiwan, 2004 is considered as the largest and most known TMD [2]. The use of TMDs was studied as a control technique, focusing on the directions of research in the US in structural control [1]. Many investigations have been carried out regarding the mathematical formulations, numerical applications, and

http://dx.doi.org/10.1016/j.hbrcj.2015.04.004

1687-4048 © 2015 The Author. Production and hosting by Elsevier B.V. on behalf of Housing and Building National Research Center. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

* Tel.: +20 11 1191 4504. E-mail address: thsakr@gmail.com.

Peer review under responsibility of Housing and Building National Research Center.

response of TMD-controlled systems [4,5]. TMDs are used in buildings not only to control the dynamic response under lateral loads but also to mitigate the torsional behavior of extremely torsionally coupled buildings [6,7]. The seismic response of severe torsionally coupled buildings was investigated by conducting a large-scale parametric study to obtain the optimum values for the parameters of a TMD system, such as the location of the added mass damper, tuning frequency ratio, tuning mass ratio, and tuned damping ratio [6]. A parametric study was conducted to investigate the effectiveness of multiple TMDs (MTMDs) in reducing the response of a torsionally coupled system [7]; the study concluded that systems with MTMDs were more effective than single TMD systems, even for torsion-ally coupled systems, for a wide range of design parameters. However, the relative advantage of MTMDs over a single TMD decreases with an increase in the eccentricity ratio.

In addition to passive TMDs, other types of TMDs have been investigated, such as a hybrid mass damper (HMD) system driven by a fuzzy logic controller (FLC) [8], semi-active variable stiffness TMD (SAIVS-TMD) [9], and bidirectional and homogeneous TMD (BH-TMD) [10]. The SAIVS-TMD employed a single mass with a variable stiffness spring [9] to control the response of a wind-excited benchmark 76-story concrete building; the results indicated that the top-floor displacement and acceleration response reduced to 32% and 53%, respectively, relative to the corresponding response of an uncontrolled building. This effect is similar to that of an active TMD, although with less power consumption [9]. The BH-TMD, which allows vibration control in both the principal directions, was reported to reduce the displacement response to earthquakes by 60% [10]. Because TMDs were successful in controlling the dynamic response of buildings, the concept of a roof-garden TMD was proposed and investigated [11,12]. As the mass ratio of such a system can be altered, mass-uncertain TMDs (MUTMDs) have been studied by subjecting them to harmonic and earthquake loading [11]. If properly designed, MUTMDs represent a viable alternative to traditional TMDs, compensating for some reduction in effectiveness with their advantages of flexibility and multitasking [11]. From this perspective, a roof-garden MUTMD was shown to be a promising tool for developing a single device that could combine the two functions of structural and environmental protection [11]. Translational and pendulum roof-garden TMDs were compared [12].

The optimization of TMD parameters and position has been increasingly investigated using different optimization techniques [13,14,8]. TMD parameters were optimized using a genetic algorithm (GA) [8] and a hybrid-coded GA (combination of binary- and real-coded GAs) [15] considering the location of the TMD. An HMD system driven by an FLC was optimized using a two-branch tournament GA [8]. The use of MTMDs has also attracted considerable attention. Different MTMD systems were mathematically formulated and evaluated on single degree of freedom (SDOF) and multiple degree of freedom systems [16]; an active TMD was also mathematically formulated and evaluated on an SDOF system [17]. Numerical studies on the effectiveness of MTMDs concluded that their effectiveness increases with an increase in the mass ratio and that the use of double TMDs is considerably more effective than the use of a single TMD with the same mass ratio for vibration mitigation under earthquake conditions as well as under sinusoidal acceleration [18,19].

Different scenarios for optimizing MTMD parameters are investigated through the study of an SDOF system with an MTMD [20-22].

This paper presents the theoretical bases of an innovative idea for utilizing a portion of the load of multiple floors to act as MTMDs. A part of the weight of floor slab, floor finishes, and architectural partitions can be utilized, especially in case of steel deck floors, if such weight is isolated using bearing devices similar to that used for base isolation. The realization of relevant special detailing that would allow such building behavior will help us bypass the need to install huge TMDs, which add to the structure load and affect the columns and foundation of a building. In addition, complicated TMD installation procedures can be avoided and the space occupied by the TMD equipment can be saved. In this paper, a technique based on the abovementioned idea is presented and its effects on building response are discussed. The effects of different design variables such as the ratio of the floor load utilized, number of floors used, and excitation characteristics are investigated. Wind effects are considered by applying sinusoidal dynamic loads with different frequencies, whereas earthquake effects are considered by applying three known ground motions. The effects are investigated for low-, mid-, and high-rise buildings.

Mathematical model of multiple-story TMDs

Consider the multistory building with multiple-story TMDs shown in Fig. 1. The building is composed of N stories with Nd TMDs located at different floor levels. The dynamic equation of motion of the building modeled as a shear building with lumped masses can be expressed as

MX + CX + Kx = F (1)

where M, C, and K are the mass, damping, and stiffness matrices of the building, respectively, considering the effect of TMDs; these matrices are defined as

M = Ms + Md

Fig. 1 Model of building with MTMD.

C — Cs + Cd

K — Ks + Kd

where Ms, Cs, and Ks are the mass, damping, and stiffness matrices of the structure without TMDs whereas Md, Cd, and Kd are the corresponding corrections resulting from the existence of TMDs. These matrices for a shear building with lumped masses are defined as follows:

"mi 0 0 . 0 0 0 . 0 0

0 m2 0 . 0 0 0 . 0 0

0 0 m3 . . 0 0 0 . 0 0

0 0 0 . mi_i 0 0 . 0 0

0 0 0 . 0 mi 0 . 0 0

0 0 0 . 0 0 mi+i . . 0 0

0 0 0 . 0 0 0 . mN_i 0

0 0 0 . 0 0 0 . 0 mN

at floor f is defined as pmf (Eq. (9.b)), where mf is the mass of floor f and pi is the story-TMD mass ratio for story-TMD number i:

mdi mf

F defined in Eq. (1) is the applied dynamic load vector, which is defined herein for wind (Fw) as a sinusoidal dynamic load and for an earthquake (Fq) using the ground acceleration record, as shown in Eqs. (12) and (13), respectively. The structure equation of motion is then solved using the Newmark-b procedure [23], which gives the nodal displacement, velocity, and acceleration vectors at each time step.

Fw — Asin(rat)/

Fq — -MKg

(12) (13)

where A is an arbitrary sinusoidal load amplitude, m is the frequency of wind excitation, xg is the earthquake ground acceleration, and I is the unit direction influence vector defined here for both earthquake and wind loads as a unit vector of size N + Nd, where Nd is the number of stories used as TMDs.

"ki + k2 _k2 0. 0 0 0 0 0

_k2 k2 + k3 _k3 . 0 0 0 0 0

0 _k3 k3 + k4 . 0 0 0 0 0

0 0 0 . ki_1 + ki _ki 0 0 0

Ks —

0 0 0 _ki ki + ki+1 _ ki+1 0 0

0 0 0 0 _ki+1 ki+1 + ki+2 0 0

0 0 0 0 0 0 . kN_1 + kN _ kN

0 0 0 0 0 0 _kN kN

Cs = /Ms + /Ks (7)

where mi and ki are the mass and stiffness of story I, respectively, and a1 and a2 are constants derived using the damping ratio of the first two fundamental structural periods. For each TMD number (i) installed on a floor (f), the property matrices that account for such a TMD can be formed as follows:

Kd(N+i,N+i) — Kdi (8.a)

Kd(N+i,f) — _Kdi (8.b)

Kd(f,N+i) — _Kdi (8.c)

Kd(f,f) — Kdi (8.d)

Md(f,f) — -PiWf (9.a)

Md(N+i,N+i) — Pimf (9.b)

Cd(N+i,N+i) — 2^dimdiMdi (10)

where Kdi, ndi, and mdi are the stiffness, damping ratio, and frequency, respectively, of TMD number i. The mass of the TMD

Verification of numerical analysis

To verify the numerical analysis and the developed MATLAB code, the solution for a 10-story shear building previously obtained by Arfiadi and Hadi [15] (reference case) is used. The same building properties shown in Table 1 are considered, and the same TMD properties are applied (Cd = 175.033 kN-s/m, Kd = 4540.369 kN/m, and a 115-ton TMD is located on the 10th floor). First, the fundamental mode shape obtained from the MATLAB code developed using the previously defined equations is compared with the results of Arfiadi and Hadi [15], as listed in Table 2. As can be seen from the table, the mode shape results are identical to the results obtained in the reference case. To verify the numerical integration procedure and the TMD effect, the time history of the top-floor lateral displacement of uncontrolled and controlled 10-story buildings (verification example) subjected to the El-Centro earthquake is obtained (Fig. 2); in the figure, the top-floor lateral displacement is plotted against time. This time history is similar to that for the reference case, with a response peak of 266.8 mm after 4.78 s for the uncontrolled building and a peak of —163.2 mm after 5.88 s for the controlled building. The

Table 1 Properties of reference case (10 story building) [15].

Stories 1-2 3 4-6 7 8-9 10

Story stiffness (kN/m) 1410587.5 1410587.5 1048724.2 1048724.2 367187.5 367187.5

Story mass (ton) 572.92 567.62 562.32 548.82 535.32 489.32

Table 2 Comparison between fundamental mode shapes of present study and reference case.

Source Fundamental mode shape

This study Ref. [15] 0 0.0914 0.18182 0.2686 0.37884 0.48 0.56967 0 0.0917 0.1818 0.2686 0.3788 0.48 0.5697 0.64571 0.6457 0.81979 0.8198 0.9405 0.9405 1.0 1.0

Time [s]

Fig. 2 Time history of top-floor lateral displacement (verification example).

distribution of the maximum story drift along the building height is plotted in Fig. 3; again, the distribution shows the same behavior as that for the reference case with acceptable differences. For the uncontrolled building, a peak is observed at

10 9 8 7 6

ä 5 4 3 2 1

—■— TMD-Controlled -Uncontrolled

0 50 100 150 200 250 300

Max. Drift [mm]

Fig. 3 Distribution of maximum story displacement (verification example).

266.8 mm, which is approximately 6.7% more than the peak observed for the reference case, and for the controlled building, a peak is observed at 163.2 mm, which is approximately 8.8% more than the peak observed for the reference case [15]. The above discussion illustrates the validity of the numerical procedure used and the acceptable accuracy of the obtained results.

Examples and loads

To illustrate the proposed idea and explore its effect on building response, three example structures were considered to represent low-, mid-, and high-rise buildings. The buildings have 5, 25, and 50 stories with uniform properties along the height as shown in Table 3. For each example, first, TMDs are employed in a few uppermost stories as shown in Fig. 1 because previous studies reported that the optimum response of buildings can be obtained if TMDs are located in the upper floors [24]. Each example building is first subjected to sinusoidal loads with different excitation frequency ratios, and the effect of the existence of story-TMDs on the peak dynamic response and resonance frequency is investigated. Using the frequency that generates the peak response of the building, a relation is derived between the story-TMD mass ratio (qi) and the main response parameters when different number of stories are used as TMDs. The building response to earthquakes is investigated by using three major known earthquakes: El-Centro, Parkfield, and Loma Prieta. The average response is studied in terms of pi to show how the seismic response of a building is affected by different TMD arrangements.

Results and discussion

Dynamic response of low-rise building

To investigate the response of the five-story low-rise building to dynamic loads, the TMD is located at the uppermost story

Table 3 Main properties of building examples.

Example Story stiffness Story mass Fundamental

(kN/m) (ton) period (s)

Five story 1.20 x 106 900 0.60

Twenty-five 1.80 x 106 1000 2.40

story 2.60 x 106

Fifty stories 1200 4.33

and the uppermost two, three, and four stories; in addition, the case of the original building without TMDs is considered. Dynamic sinusoidal loads are applied with different excitation frequency ratios rx, which is defined as the ratio of the applied load frequency m to the natural fundamental period mn0 of the original building:

The story-TMD mass ratio qi defined in Eq. (11) is taken as 25% for all the cases as a guide value, which indicates that one quarter of the story mass of the selected stories is used as TMDs. The response of the five-story building to sinusoidal loads with different excitation frequencies is illustrated in Figs. 4-9. In Fig. 4, the maximum top drift ratio rd is plotted against rx. rd is defined as the top-story maximum drift normalized with respect to the top-story maximum drift of the original building subjected to excitation with rx =1. The case of the original building and those with one-, two-, and four-story TMDs at the uppermost floors are shown in the figure. It is observed that the maximum drift of the original building varies with rx such that the peak response is located at rx = 1, and as this ratio moves away from unity, the drift response tends to decrease. This peak response is too much as compared to the response for rx much more or less than unity such that the peak response exceeds 3.56 times the response for rx is only 10% more or less than unity. The case in which the uppermost floor is used as a TMD shows a different response from the other cases; that is, the peak response is greatly reduced, and its location is slightly moved toward a higher value of rx. The peak response of rd for the case of a one-story TMD is approximately 18.55% of that for the original building, and resulting value of rx equals 1.08. It should be noted that this case corresponds to the TMD mass ratio of 5% of the overall building mass. Adding more TMDs at the uppermost stories of the building gives similar results. The peak drifts of two- and four-story TMDs are recorded to be 13.73% and 12.11% of the reference value, respectively, and are located at rx = 1.16 and 1.24, respectively. At rx = 1, the maximum top drift is reduced to 15.87%, 10.21%, and 7.88% of that of the original building for one-, two-, and four-story TMDs, respectively.

0.9 0.8

I °.7

ü a6 o 0.5

® 0.4

S °.3 2

0.2 0.1 0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Excitation frequency Ratio

Fig. 5 Relation between maximum inter-story drift and rx for five-story building subjected to sinusoidal loads (qi = 25%).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 Excitation frequency Ratio

Fig. 6 Relation between maximum acceleration and rx for five-story building subjected to sinusoidal loads (qi = 25%).

------1 TMD

------- 2 TMD

........... 4 TMD

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Excitation frequency Ratio

Fig. 4 Relation between maximum top drift and rx for five-story building subjected to sinusoidal loads (qi = 25%).

0.9 0.8

■B 0.7

is 0.6 OT 0.5

£ 0.4 J 0.3 0.2 0.1 0

- No TMD -----1 TMD ------- 2 TMD .......... 4 TMD

___________________J_________i_________

0.6 0.8 1 1.2 1.4 Excitation frequency Ratio

Fig. 7 Relation between maximum base shear and rx for five-story building subjected to sinusoidal loads (qi = 25%).

No TMD

0.2 0.4

1.6 1.8

0 5 10 15 20 25 30 35 40

Max Lateral Drift [mm]

Fig. 8 Maximum story drift distribution for five-story building subjected to sinusoidal loads (pi = 25%).

0 2 4 6 8 10 12 14 16 18 20 Time [s]

Fig. 9 Time history of top drift for five-story building subjected to sinusoidal loads (pi = 25%).

for all the cases are plotted against rx. The behaviors of acceleration and base shear are similar to that of drift in terms of the peak values and resonance criteria. The existence of TMDs reduces the peak maximum acceleration by 22.15% at rx = 1.1, 19.07% at rx = 1.18, and 18.78% at rx = 1.24 for the cases of one-, two-, and four-story TMDs, respectively, compared to the original case in which the maximum acceleration is observed at rx = 1. It is clear that the acceleration response improves for the one-, two-, and four story TMDs much better than the original building. On the other hand, the base shear ratio significantly improves because of the existence of TMDs, with the peak excitation frequency ratio shifting toward higher values. The base shear ratio is reduced to 19.48% at rx = 1.08, 14.78% at rx = 1.16, and 12.35% at rx =1.22 for the cases of one-, two-, and four-story TMDs, respectively, compared to the original case in which the maximum acceleration is observed at rx = 1.

An overview of the improvements in the drift response for buildings with multiple-story TMDs is shown in Fig. 8; here, the variation in the maximum lateral drift with the building height is shown for different cases. As discussed before, the existence of TMDs enhances the drift distribution significantly at all heights and reduces the top drift to 18.55%, 13.73%, and 12.11% of that of the original building because of the addition of the one-, two-, and four-story TMDs. Fig. 9 shows the time history of the top-floor lateral displacement of the five-story building for different cases of multiple-story TMDs. In the plot, the lateral drifts at the top of the building are plotted against time for qi = 25% and rx =1. For the original building with no TMD, the resonance response is clear; that is, the lateral drift continues to increase with time until the excitation stops. On the other hand, the existence of TMDs at one or more stories changes the behavior considerably toward stable vibration with relatively small amplitudes. Although one-and three-story TMDs show such improvement and stability, the values of the maximum lateral drift in the case of the three-story TMDs are reduced by 53.2% compared to that in the case of the one-story TMD. It can be concluded that by considering 25% of the top floor as a TMD, the response of the lateral top drift can be reduced to 15.87% of the original building while considering the same for two extra levels (fourth and third stories) improves this reduction to 7.8

These cases correspond to TMD mass ratios of 5%, 10%, and 20%. The relation between the excitation frequency ratio and the maximum inter-story drift ratio is plotted in Fig. 5. The maximum inter-story drift ratio ri is defined here as the maximum inter-story drift normalized with respect to that of the original building excited by a sinusoidal load with rx = 1. The inter-story drift is selected because it is a major indicator of story shear identified by design and limited by codes. The peak of the maximum inter-story drift for the original building is observed at rx = 1; the use of TMDs reduces such peaks considerably and shifts their locations to higher values of rx. The peaks for one-, two-, and four-story TMDs are observed to be 19.48, 14.78, and 12.35% of the reference value, respectively, and are located at rx = 1.08, 1.16, and 1.22, respectively.

The acceleration and base shear response of buildings without and with TMDs are shown in Figs. 6 and 7; in these figures, the maximum acceleration ratios and base shear ratios

- 1 TMD ------2 TMDs

3 TMDs 4 TMDs

0 10 20 30 40 50 60 70

Story-TMD Mass Percentage [pi]

Fig. 10 Variation of maximum top drift with for five-story building subjected to sinusoidal loads (rx = 1).

™ 0.4

- 1 TMD ------2 TMDs -------3 TMDs " ........... 4 TMDs

i\ \ "'"•iÄJJ," «•»"rA-N KBMJM«« ■"t'-.t'r-™.*- Wwwwr'sii

0 10 20 30 40 50 60 70

Story-TMD Mass Percentage [pi]

Fig. 11 Variation of maximum top acceleration with pi for five-story building subjected to sinusoidal loads (rx = 1).

£ 0.9 ro

o I—

0 10 20 30 40 50 60 70

stories TMD mass Percentage [pi]

Fig. 12 Variation of maximum top drift with for five-story building subjected to earthquakes.

Owing to the importance of the share of the floor load reserved as TMDs, the effect of pi on the maximum top lateral drift and maximum top acceleration is examined for the case of rx =1 (Figs. 10 and 11). Fig. 10 shows that as pi increases, the lateral top drift of the building decreases for any number of stories used as TMDs. This increase can be simply attributed to the increase in the overall TMD mass ratio, which produces more response enhancements [3,11]. The rate of reduction in the maximum top drift decreases with increasing pi such that in the case of a one-story TMD, the first 30% of pi reduces the top drift by approximately 86.1% whereas the latter 70% of pi increases the reduced value only by 6.1%. The same effect is observed for the cases of two-, three-, and four-story TMDs with greater reduction in the response. The greatly enhanced value of the maximum inter-story drift was only 7.8%, 5.2%, 4.3%, and 3.9% of that of the original building for the cases of the one-, two-, three-, and four-story TMDs, respectively, at pi = 70%. The acceleration response of the five-story building to sinusoidal loads is shown in Fig. 11; in the figure, the top acceleration ratio is plotted against pi. The top acceleration is observed to decrease sharply with increasing pi for small values of pi and continues to decrease at a lower rate for higher values of pi. in all the cases, the acceleration response of the building with TMDs is much lower than that of the original building. The top acceleration reaches its least values of 8.4%, 6.1%, 5.1%, and 4.7% of that of the original building for the cases of the one-, two-, three-, and four-story TMDs, respectively, at the maximum pi value. Approximately 94% of the acceleration enhancements are observed for the first 30% of pi; that is, 85.9% reduction in the top acceleration is observed for pi = 30%, whereas 91.5% reduction in the top acceleration is observed for pi = 70%.

Earthquake response of low-rise building

As discussed earlier, for the low-rise building subjected to sinusoidal loads and having multiple-story TMDs located at the uppermost story and uppermost two, three, and four stories for any value of pi, considerable enhancement in its displacement and force behavior was observed. To investigate the response of the building to earthquakes, time history analysis

was carried out on the building using three known earthquake records. El-Centro, Parkfield, and Loma Prieta were selected as known earthquakes with different characteristics. The average response was compared to indicate how the response of a building is affected by earthquakes in general. Fig. 12 shows the seismic average maximum top drift response of the five-story building to the selected earthquakes. The figure plots the relation between the average top drift ratio and pi for buildings with one-, two-, three-, and four-story TMDs subjected to the ground acceleration of the selected earthquakes. it is clear from the plot that the top drift response of the building is enhanced (decreased) when pi is increased, for any number of stories used as TMDs. The decrease in the top drift with increasing pi continues in the case of a one-story TMD, whereas for more TMDs, this decrease continues up to a specific value of pi. it is also observed that the for two-, three-, and four-story TMDs, no response enhancement is gained after pi = 60%, 45% and 45%, respectively which can be attributed to the simultaneous increase of the number of TMDs and the mass ratio leading to the least possible reduced response. in all the cases, the drift of the low-rise building with multiple-story TMDs is less than that of the original building. The optimum values of the top drift reach 53% at pi = 70% for the one-story TMD, 46% at pi = 60% for the two-story TMDs, 45% at pi = 70% for the three-story TMDs, and 44% at pi = 70% for the four-story TMDs. A similar response can be observed in Fig. 13 for the maximum inter-story drift, which is plotted against pi, for buildings with one-, two-, three-, and four-story TMDs subjected to the ground acceleration of the selected earthquakes. The average value of the maximum inter-story drift is also observed to be enhanced when more number of stories are used as TMDs and for higher values of p. The rate of enhancement is observed to be more for lower values of pi. For a one-story TMD, the decrease in the top drift with increasing pi continues, whereas for more TMDs, this decrease continues up to a specific value of pi. The response tends to increase after this pi value, which is 60%, 45%, and 40% for the two-, three-, and four-story TMDs, respectively. This trend can be attributed to the large increase in the overall mass ratio, which reaches 48%, 36%, and 32% for the above-mentioned cases beyond them no

0 10 20 30 40 50 60 70

stories TMD mass Percentage [pi]

Fig. 13 Variation of maximum inter-story drift with for five-story building subjected to earthquakes.

- 1 TMD ------2 TMD

V - 3 TMD 4 TMD

V \\ \s

Vn V. V N. \ V \

0 10 20 30 40 50 60 70

stories TMD mass Percentage [pi]

Fig. 15 Variation of maximum base shear with for five-story building subjected to earthquakes.

enhancement can take place. In all the cases, the maximum inter-story drift of the low-rise building with multiple-story TMDs is less than that of the original building. The optimum values of the maximum inter-story drift reach 56% at pi = 70% for the one-story TMD, 50% at pi = 60% for the two-story TMDs, 51% at pi = 45% for the three-story TMDs, and 48% at pi = 55% for the four-story TMDs.

The top-story acceleration response of the five-story building to earthquakes is shown in Fig. 14; in the figure, the maximum top-story acceleration ratio is plotted against pi. It can be clearly observed from the plot that the acceleration response of the building is significantly enhanced when TMDs are present and when the values of pi are increased. The rate of enhancement is initially sharp at low values of pi and then decreases as pi increases. The values of the top acceleration reach their minimum values of 74%, 69%, 66%, and 66% of the original building response for the one-, two-, three-, and four-story TMDs with pi = 70%. The base shear response to earthquakes of the low-rise building is shown in Fig. 15. For one- and two-story TMDs, the maximum base shear ratios

1.05 1

0.95 0.9 0.85 0.8 0.75 0.7

0 10 20 30 40 50 60 70

stories TMD mass Percentage [pi]

Fig. 14 Variation of maximum top acceleration with for five-story building subjected to earthquakes.

tend to decrease with increasing pi, whereas for three- and four-story TMDs, the base shear ratios decrease up to a certain value of pi, after which the base shear ratios increase with further increase in pi. Thus, for one- and two-story TMDs, the minimum response occurs at pi = 70% and is 55% and 47%, respectively, of the base shear of the original building. The minimum values of the base shear for three- and four-story TMDs are observed to be 47% and 48%, respectively, of the base shear of the original building at values of pi = 45% and 40%, which are the inflection points after which the base shear increases. It is also noted from the above values that the two- or three-story TMDs have the same values of minimum base shear but at different values of pi and these base shear values are more optimized compared to the cases of one-or four-story TMDs.

Dynamic response of mid- and high-rise buildings

As discussed earlier, the use of a portion of the floor load of limited stories as TMDs enhances the behavior of the low-rise building subjected to sinusoidal and earthquake loads. In this section, sample results for mid- and high-rise buildings are presented. Fig. 16 shows the relation between the maximum lateral top drift and the excitation frequency ratio rx for the selected 25-story building by applying multiple-story TMDs at limited floors with pi = 25% for all the cases. As shown in the figure, the peak drift value of the original building is observed to be at rx = 1, and the value decreases as rx becomes more or less than unity. If the peak response frequency for the original building is changed by only 20%, the drift response for the 25-story building may reduce by approximately 47.3%. As more TMDs are used, the peak response is significantly reduced and the peak response frequency ratio shifts toward higher frequency ratios. The existence of four-, eight-, and twelve-story TMDs resulted in a reduction in the peak drift to 53.3%, 42.9%, and 39.5% respectively, at frequency ratios that are 1.08, 1.16, and 1.22 times the natural frequency of the original building. The response of high-rise building (50 stories) to sinusoidal loads is shown in Fig. 17. It can be observed that the drift behavior of the 50-story

0.9 0.8

■в °.у

ОС 0.6

О 0.5 о

н 0.4

2 0.3 0.2 0.1 0

- No TMD

-----4 TMD

------- 8 TMD

.......... 12 TMD

0.6 0.8 1 1.2 1.4 Excitation frequency Ratio

Fig. 16 Relation between maximum top drift and rx for 25-story building subjected to sinusoidal loads (pi = 25%).

o 0.8 fo

a: 0.7

I- 0.5 J 0.4 0.3 0.2 0.1

0 10 20 30 40 50 60 70 Story-TMD Mass Percentage [pi]

Fig. 18 Variation of maximum top drift with pi. for 25-story building subjected to sinusoidal loads (rx =1).

0.9 0.8

о 0.7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Excitation frequency Ratio

Fig. 17 Relation between maximum top drift and rx for 50-story building subjected to sinusoidal loads (pj = 25%).

1.2 1.1 1

■J 0.8

© 0.7

1 0 6 о

к 05 го

0.4 0.3 0.2

- 1 TMD

f 4 TMDs

-------8 TMDs

.......... 12 TMDs

_._._I_I_I_

20 30 40 50 60 70

Story-TMD Mass Percentage [pi]

Fig. 19 Variation of maximum top acceleration with pi for 25-story building subjected to sinusoidal loads (rx =1).

0.2 0.4

1.6 1.8

building is different from the behavior of the low-rise building or the 25-story building. For the original building, the peak drift response is observed to have shifted from the unity frequency ratio such that peaks are found at rx = 0.84 for the 50-story building. Adding multiple-story TMDs reduces the peak drift response and shifts the peak frequency ratios to lower values. Adding five-story TMDs reduces the peak to 94% of that of the original building at rx = 0.74. Adding 10- or 20-story TMDs results in more reduction in the drift response and leads to the generation of two peaks, with the effective one located at a value of rx that is less than the original building peak response frequency ratio. The peak values of the top drift are 91% and 87.7% for the cases of 10- and 20-story TMDs, respectively, at frequency ratios of 0.68 and 0.62. At the frequency ratio of the peak response of the original building, the 5-, 10-, and 20-story TMDs reduced the drift response to 91.2%, 84.5%, and 77.5%, respectively.

The effects of pi on the maximum drift and acceleration of the 25-story building under sinusoidal dynamic loads are shown in Figs. 18 and 19. The top drift ratio normalized with

respect to the drift of the original building is plotted in Fig. 18 against pi for buildings with 1-, 4-, 8-, and 12-story TMDs. it can be clearly observed that the drift and acceleration response of the 25-story building decreases with an increase in pi and with the use of more stories as TMDs. When only the uppermost story is used as the TMD, although the mass ratio of the TMD to the structure load is still low, the drift and acceleration responses decrease to 59.6% and 59.8% of the original drift and acceleration responses, respectively, at pi = 0.7, which is the maximum ratio examined. As the number of stories and TMDs increases, more enhancement of the drift and acceleration response is observed. For 4-, 8-, and 12-story TMDs, the drift response decreases to 31.7%, 23%, and 18.9%, respectively, of the original response and the acceleration response decreases to 33.3%, 24.9%, and 21.4%, respectively, of the original response. The same relations for the drift and acceleration for the 50-story building are shown in Figs. 20 and 21. Only minor enhancement of the drift and acceleration responses is observed for a one-story TMD. The decrease of only 6.1% and 6.5% for the drift and acceleration,

Story-TMD Mass Percentage [pi]

Fig. 20 Variation of maximum top drift with pL for 50-story building subjected to sinusoidal loads (rx = 1).

stories TMD mass Percentage [pi]

Fig. 22 Variation of maximum top drift with pI for 25-story building subjected to earthquakes.

1.05 1

0.95 0.9 0.85 0.8 0.75

Ï" ^ 0.65

0 10 20 30 40 50 60 70

Story-TMD Mass Percentage [pi]

Fig. 21 Variation of maximum top acceleration with for 50-story building subjected to sinusoidal loads (rx =1).

—— \\SsV

_ - 1 TMD 5 TMDs - 10 TMDs 20 TMDs

pi = 60%, after which the average drift ratio begins to increase. This can be attributed to the increase in the overall TMDs mass ratio in addition to the existence of twelve levels of TMDs, which complicates the behavior of the structure in different ways. In all the cases, the increase in the number of TMDs decreases the drift response of the building to earthquakes. The minimum drift responses recorded for one-, four-, and eight-story TMDs at pi = 70% are 95%, 85%, and 78% of the drift of the original building, respectively. For the 12-story TMDs, the minimum average drift value is 76.8% of the original drift at pi = 60%. As shown in Fig. 23, the relation between the average acceleration ratio and pi exhibits similar behavior but with different values of response ratios. The acceleration is less affected by pi, and the maximum reduction in the maximum top acceleration ratio reaches only 5% for the 12-story TMDs with pi = 70%. For the one- and four-story TMDs, the average acceleration ratio decreases with pi, reaching 98.9 and 96.6%, respectively. For the 8- and 12-story TMDs, the average maximum acceleration ratio decreases to an optimal value at pi = 55% and 40%,

respectively, is attributed to the small overall mass ratio in this case. As the number of stories used as TMDs increases, the enhancement increases and a decrease in the drift of 19.7%, 30.9%, and 37.6% is observed for the 5-, 10-, and 20-story TMDs, respectively. The acceleration response shows similar behavior; for the 5-, 10-, and 20-story TMDs, a reduction in the top acceleration of 22.1%, 32.2%, and 43.7%, respectively, is observed.

Earthquake response of mid- and high-rise buildings

This section discusses the response of the 25- and 50-story buildings to earthquakes by using the three previously mentioned earthquake records. For the 25-story building, the average maximum lateral top drift is plotted in Fig. 22 against pi when different number of stories are used as TMDs. As shown in the figure, the average drift decreases as pi increases for the one-, four-, and eight-story TMDs. For the twelve-story TMDs, the average drift decreases with pi up to

0.97 &

X 0.96

V\ \ \ \ /

\ / / __________.¿i

1 TMD 4 TMD y y y *

------- 8 TMD 12 TMD

0 10 20 30 40 50 60 70

stories TMD mass Percentage [pi]

Fig. 23 Variation of maximum top acceleration with pI for 25-Story Building Subjected to Earthquakes.

- 1 TMD II

-----5 TMD

------- 10 TMD

.......... 20 TMD

0 10 20 30 40 50 60 70

stories TMD mass Percentage [pi]

Fig. 24 Variation of maximum base shear with for 50-story building subjected to earthquakes.

1.004 1.002 1

¿S 0.998 c

<D 0.994

5 0.992 o

H 0.99

5 0.988 0.986 0.984

- 1 TMD

------5 TMD

------- 10 TMD

........... 20 TMD

0 10 20 30 40 50 60 70

stories TMD mass Percentage [pi]

Fig. 25 Variation of maximum top acceleration with for 50-story building subjected to earthquakes.

respectively, and then increases for higher values of pi. This behavior emphasizes the importance of selecting the proper number and properties of TMDs to avoid any adverse effects on the building behavior.

For the 50-story building, the average maximum lateral top drift is plotted in Fig. 24 against pi when different number of stories are used as TMDs. As shown in the figure, the average drift decreases as pi increases for the 1-, 5-, and 10-story TMDs. For the 20-story TMDs, the average maximum drift decreases with pi up to pi = 60%, after which the average drift ratio tends to increase. This trend can be attributed to the abnormal increase in the overall TMD mass ratio, which affects the behavior of the building in different ways. In all the cases, the increase in the number of TMDs decreases the drift response of the building to earthquakes. The minimum drift responses recorded for the 1-, 5-, and 10-story TMDs at pi = 70% are 99.5%, 98%, and 96.6% of the drift of the original building; for the 20-story TMDs, the minimum average drift value is 95% of the original drift at pi = 60%. The relation between the average acceleration ratio and pi is plotted in

Fig. 25, which shows a slight improvement in the response. In all the cases, the average top acceleration ratio is enhanced as pi increases and the number of TMDs increases. The decrease in the acceleration response is as little as 0.1%, 0.4%, 0.8%, and 1.4% of the original acceleration for the 1-, 5-, 10-, and 20-story TMDs, respectively.

Summary and conclusions

In this paper, the basic idea and theoretical bases of using limited stories at the top of a building as multiple TMDs are discussed. After the proposed system was formulated, the response of selected example buildings to wind and earthquakes is investigated. For the investigation, 5-, 25-, and 50-story buildings are selected to represent low-, mid-, and high-rise buildings. Wind loads are considered by applying sinusoidal dynamic loads with different frequencies, whereas earthquake loads are considered by carrying out seismic analysis using the records of El-Centro, Parkfield, and Loma Prieta, which are known major earthquakes. The validity of the proposed idea in improving the response of buildings to wind and earthquakes was verified based on the following observations:

• The existence of multiple-story TMDs significantly reduces the drift, acceleration, and force response of all examined buildings subjected to sinusoidal dynamic loads.

• The peak response of the original buildings without TMDs to sinusoidal loads is observed at rx = 1 for the 5-, and 25-story buildings and slightly below this value for the 50-story buildings. The use of multiple-story TMDs shifts the peak toward higher excitation frequency ratios for the 5- and 25-story buildings and toward lower excitation frequency ratios for the 50-story building.

• An increase in pi and the number of stories utilized as TMDs significantly enhances the response of all types of buildings to sinusoidal loads.

• The response of buildings to earthquakes is also enhanced by the use of more number of stories as TMDs and an increase in pi, especially for low- and mid-rise buildings. For high-rise buildings, this enhancement is not substantial because of the nature of the buildings and the earthquake ground motions selected. Better selection of the building and TMD parameters might provide better results in terms of the response of buildings to earthquakes.

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