Passive control of bilinear hysteretic structures by tuned mass damper for narrow band seismic motions

Engineering Structures 54 (2013) 103–111

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Passive control of bilinear hysteretic structures by tuned mass damper for narrow band seismic motions Z. Zhang, T. Balendra ⇑ Department of Civil and Environmental Engineering, National University of Singapore, 117576 Singapore, Singapore

a r t i c l e

i n f o

Article history: Received 2 March 2011 Revised 4 March 2013 Accepted 31 March 2013 Available online 11 May 2013 Keywords: Tuned mass damper Nonlinear system Vibration control

a b s t r a c t Tuned mass damper (TMD) has been extensively used in vibration control of engineering structures. Numerous available results show that TMD can greatly reduce the response of elastic structures under wind, water waves, and earthquakes. The efficiency of TMDs, however, decreases when primary system experiences nonlinear behaviors. In order to make use of TMDs in such situations, this paper investigates the feasibility of adopting TMD in controlling of inelastic structures subjected to seismic motions. The focus is to explore the performance of TMD controlled buildings in areas subjected to long distance earthquakes, where structures usually have low nonlinearity. An optimization criterion is proposed to minimize the maximum nonlinear response within the concerned frequency band of excitations, instead of conventional approach whereby the linear response of the primary system is minimized. The limit on stroke is taken into consideration for practicality. Through numerical studies, it is shown that the primary structures achieve much better damage reduction by adopting the optimal TMD parameters based on the proposed criterion. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Tuned mass damper, as one of the passive control devices, has been widely used for vibration control of structures such as machineries [1], bridges [2], and offshore platforms [3,4], as well as vibration suppression under wind loading [5], earthquakes [6], human induced vibration [7,8]. Good performance can be achieved by obtaining the TMD parameters from optimization procedures. The optimal TMD damping and stiffness parameters vary for certain mass ratios. For instance, Den Hartog [1] proposed optimal parameters for undamped primary systems through the fixed/ invariant-points theory, i.e. by requiring equal frequency response peaks for system with TMD. Using Den Hartog’s strategy, Warburton [9] formulated the problem into minimization of various response parameters and derived formulas for undamped one degree-of-freedom primary systems for harmonic and white noise random excitations. To facilitate the use of TMD in seismic control of buildings, Tsai and Lin [10] proposed curve-fitted formulas based on the findings from Den Hartog. Optimal TMD parameters from these formulas were recommended for support excited and damped systems. A common feature of these traditional methods is that the optimal TMD parameters are obtained based on the concerned mode (i.e. fundamental period) of an elastic structure. For the ease of practical design purpose, researchers studied on using formulas based on elastic response to control nonlinear ⇑ Corresponding author. Tel.: +65 6516 2159; fax: +65 6779 1635. E-mail address: [email protected] (T. Balendra). 0141-0296/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2013.03.044

structures. Pinkaew et al. [11] adopted Warburton’s formulas for designing TMDs for damage reduction of seismically excited nonlinear structures. The formula was used on an equivalent inelastic single degree-of-freedom (DOF) system of a multi-DOF nonlinear system. Ghosh and Basu [12] used Den Hartog’s formulas for designing liquid column dampers for nonlinear structures. The formula was used on the equivalent linear system of a nonlinear system. These optimal TMD parameters are obtained on the assumption that the primary structures undergo linear behavior. As the structures go into nonlinear stage, structural stiffness will deteriorate such that the fundamental frequency of the primary systems changes accordingly. The formulas or results based on optimization of elastic structures might not consistently result in optimal performance as structures step into nonlinear stage. Recent study [13] showed that TMD can be introduced in nonlinear structures for damage reduction, the detailed design of TMD, however, was less reported. In order to obtain optimal TMD design for nonlinear structures, numerical optimization methods are often adopted. Sgobba and Marano [14] presented to control nonlinear single DOF structure by a linear TMD. The optimization is studied on three objective functions, the maximum of the peak structural displacement standard deviation, the average hysteretic dissipated energy of a protected building with reference to an unprotected one, and a functional damage that considers the two indexes previously described. Besides the considerations in structural nonlinearity, the design of TMD should also consider the frequency range of excitations. This might be useful for areas subjected to narrow band

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earthquake motions, since it is suspicious that response could be amplified in some frequencies that it is not accounted for in the conventional TMD design. Limited results are available on design the TMD for nonlinear structures in the areas subjected to narrow band earthquakes. This is important as structures in these areas are usually designed with low ductility ratios. Therefore, another focus of this study is on the application of TMDs for nonlinear structures subjected to ground motions amplified by soft soil sites due to far field earthquakes. Unlike near field earthquakes that are dominant in broad band high-frequency excitations of short duration, the characteristics of long-distance earthquake excitations are narrow band with long durations. Under this long-distance earthquake excitation, the structures are prone to produce resonant responses. Therefore the base excitation could be considered as harmonic excitations. Designing TMD for inelastic structures in long distance base motion areas has already attracted attention by few researchers. Lukkunaprasit and Wanitkorkul [15] assumed elastic perfectly plasticity for main structure and considered to reduce the accumulated hysteretic energy dissipated by main structure and the peak displacement. The TMD would be effective in reducing the hysteretic energy absorption demand in the critical storeys for buildings in the 1.8–2.8 s range. Nevertheless, the above-mentioned results for optimal TMD did not consider the effects of limited stroke. TMD tends to fail to achieve its optimal performance when neglecting the stroke limit. Recent studies highlighted that design of TMD should account for the stroke limit. Marano et al. [16] proposed to consider the failure probability of TMD due to excessive displacement for linear structures. The TMD was devised for a single DOF system that extracted from a multi-DOF structure based on the first mode. The objective was to minimize the maximum dimensionless peak displacement of controlled structure with respect to the uncontrolled one. As the stroke limit of TMD is considered, designing of optimal TMD is usually treated as constrained optimization [17,18]. It is found that the reduction of TMD stroke and structural response has been identified as a trade-off problem [17]. With these concerns on structural nonlinearity, excitations and TMD stroke, this paper investigates the use of TMD for control of bilinear structures, accounting for excitation frequency range and stroke limit. The problem is cast into a nonlinearly constrained mini-max procedure. The purpose is to explore the effectiveness of TMD with prescribed stroke limit, as structures experience nonlinearity under narrow band ground excitations. 2. Equation of motion for dynamic systems For an n degree-of-freedom (DOF) structural system attached with a TMD, the equations of motion when subjected to base excitation can be written as

€ þ Cx_ þ Kx ¼ Mr€xg Mx

ð1Þ

where M, C, and K are the mass, damping and stiffness matrices of €; x, _ and x are relative (n + 1)  (n + 1) dimension. The quantities x acceleration, velocity and displacement vectors of (n + 1) dimension. On the right hand side of Eq. (1), the quantity r is an (n + 1)dimensional unit vector and €xg is the base accelerations. As illustrated in Fig. 1, Eq. (1) can be applied for a single-DOF primary system as





ms md

€þ x



cs þ cd cd cd

cd



x_ þ



ks þ kd kd kd

kd



 ms x¼

md

 r€xg

(a)

(b)

Fig. 1. (a) Original primary bilinear system; (b) bilinear system with TMD.

stiffness ks and damping coefficient cs could be response dependent quantities. When the base excitation is in the form of harmonic forces, i.e. FðtÞ ¼ Mr€ xg ðtÞ ¼ MrRe½Aeixt  with the amplitude A, then the ixt steady-state response takespthe ffiffiffiffiffiffiffi form x(t) = Re[Re ]. The complex quantity i is defined as i ¼ 1. The quantity x is the forcing frequency. From Eq. (1), the amplitude of response R, i.e. R = [R1, . . . Rj, . . . , Rn+1]T, is given as

R ¼ ðMx2 þ ixC þ KÞ1 ðMrAÞ

ð3Þ

3. Modeling the bilinear hysteresis In this study, structural element undergoing nonlinear behavior is modeled by a bilinear representation. The bilinear model is based on a linearization strategy for the hysteretic systems [19,20]. The linearization operated directly on the equations of motion. Equivalence is maintained for an actual nonlinear system and a linearized system in a statistical sense that the mean-square error of the response between the actual nonlinear and linearized systems is minimized. This method has been recognized as one of the recommended linearization method for solving nonlinear structural responses [21,22]. This commonly used approach has proved useful in most applications [23–25]. If the pre- and post-yielding stiffnesses are denoted by k0 and k1, then their relationship can be illustrated in Fig. 2 as:

k1 ¼ gk0

ð4Þ

The structural member has the yielding displacement x0 and initial damping coefficient c0. Adopting linearization method given by Caughey [19,20], the stiffness ke and damping coefficient ce of an equivalent linear system of bilinear hysteresis under harmonic loading are

ke ðRÞ ¼

PðRÞ ; R

ce ðRÞ ¼ c0 

QðRÞ ; xe R

ð5Þ

where xe is the p frequency corresponding to the equivalent linear ffiffiffiffiffiffiffiffiffiffiffi system, i.e. xe ¼ ke =m. The quantity m is the mass of the system.

(

ð2Þ where ms, cs, ks are the mass, damping coefficient, stiffness properties of the primary system and md, cd, and kd are their counterparts for the TMD system. For systems with nonlinear behavior, the

PðRÞ ¼

k0 R

h

p

k0 R and

ð1  gÞh þ gp  ð12 gÞ sinð2hÞ

i

when R > x0 when R  x0

ð6Þ

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Z. Zhang, T. Balendra / Engineering Structures 54 (2013) 103–111

The quantities xe and ne indicate equivalent linear frequency and damping ratio of the bilinear element, respectively. The transfer function for TMD is H2 ðx;RðxÞÞ ¼

ms ðx2e ðRÞ þ 2i  ne ðRÞ  xe ðRÞ  xÞ=ðx2d  x2 þ 2ine ðRÞ  xe ðRÞ  xÞ ms ðx2e ðRÞ  x2 þ 2i  ne ðRÞ  xe ðRÞ  xÞ  x2 ZðxÞ ð13Þ

For nonlinear systems, the equivalent transfer functions H1 and H2, the equivalent frequency xe and damping ratio ne are found to depend on the response amplitudes. It has to be noted that this lumped linearization method maybe not applicable for areas under strong earthquake motion instead of long distance earthquake motions such that the structure would experience high nonlinearity such that the structural ductility ratio in terms of elemental nonlinear displacement over yielding displacement may reach about 4–6. In such a case, a nonlinear time history analysis is more appropriate. 4. Problem definition

Fig. 2. Bilinear hysteresis.

( QðRÞ ¼

2

 kp0 R sin h when R > x0 0

ð7Þ

when R  x0

and

  2x0 h ¼ cos1 1  R

ð8Þ

It should be noted that this linearization is based on the condition that the amplitude F of driving force F(t) = MrRe[Aeixt] has to be bounded according to Eq. (9), otherwise the system exhibits unbounded resonance [19]. It is thus noted that Eq. (9) stipulates the requirement for systems with weak damping system.

4g F < k0 x0

ð9Þ

p

For a multiple storey structure, this linearization is applied to every nonlinear member. Therefore each member within a MDOF system has a bilinear representation. The bilinear hysteresis in the local element will be activated when relative nodal responses of this element go beyond the yielding response, i.e. yielding displacement in the present study. These elements are assembled in the same way as a linear MDOF lumped mass system. Since each bilinear element has an equivalent stiffness and damping property with respect to the relative nodal responses, the equivalent global stiffness matrix Keq and damping matrix Ceq of a MDOF system can be then formulated through these members. The response is then written as

R ¼ ðMx2 þ ixCeq þ Keq Þ1 ðMrAÞ

ð10Þ

Since matrices Keq and Ceq are response amplitude and frequency dependent, Eq. (10) is a set of nonlinear equations and are solved by Newton–Raphson method. If the primary system is modeled by a single DOF system, i.e. as in Fig. 1, transfer function of the structure with TMD, i.e. for unit input, can be derived as follows

H1 ðx; RðxÞÞ ¼

ms þ ZðxÞ ms ðx2e ðRÞ  x2 þ 2i  ne ðRÞ  xe ðRÞ  xÞ  x2 ZðxÞ ð11Þ

where

ZðxÞ ¼

md ðx2d þ 2ind xd xÞ x2d  x2 þ 2ind xd x

ð12Þ

In this study, designing a tuned mass damper is formulated into a nonlinearly constrained mini-max optimization problem [26]. The purpose is to minimize the top displacement of primary structure over a band of concerned functional frequencies, i.e. [x1, x2], with given mass ratio l and excitation amplitude A. It could also be interpreted as to keep TMD absorbing as much energy as possible when structure undergoes nonlinearity. In that case, primary structure is expected to suffer less damage or even possibly to function elastically with the aid of optimal TMD. The problem can therefore be stated as follows:

min max Rn ðcd ; nd ; l; A; xÞ;

Rn ðcd ;nd Þ

x

x 2 ½x1 ; x2 

subjected to : Rnþ1 ðcd ; nd ; l; A; xÞ 6 s

ð14Þ

where cd and nd are the TMD frequency ratio and damping ratio, respectively. The frequency ratio is given by cd = xd/xs, where xs is the fundamental frequency of the original n-DOF elastic structure while xd is the TMD frequency to be determined. This frequency xs is unaltered as compared to the instantaneous frequency associated with the structure experiencing nonlinear behavior. The quantity Rn is the top displacement and Rn+1 the displacement of TMD relative to top displacement, while s is the stroke limit. It is clear that, given mass ratio and stroke limit, the optimal parameters cd and nd are functions of the frequency band and forcing amplitude. As cast in the form of constrained mini-max optimization problem, Eq. (14) is solved by sequential quadratic programming (SQP) method [26]. The SQP method is an iterative method which solves at each iteration a quadratic programming (QP) problem. In the QP procedure, inequality constraints are linearized and formulated into a subproblem, which is then solved by active-set method [26]. As the iteration of the SQP method converge to the solution, solving the quadratic subproblem becomes very economical since the information from the previous iteration can be used to make a good guess of the optimal solution of the current subproblem. Different from traditional optimal design of TMDs, the optimization criteria presented herein is to minimize the maximum top nonlinear (linear equivalent) displacement over a concerned frequency range. In addition, the limitation of TMD stroke has been taken into consideration, which is due to the possible physical space constraints. It is clear that TMD with longer stoke will absorb more energy and hence would indirectly minimize the energy dissipated by the primary structure, resulting in less damage. As defined in Eq. (14), the objective function quantifies the TMD efficiency in the case of nonlinear primary structures. Obviously, the proposed optimization criteria falls within those method based

Z. Zhang, T. Balendra / Engineering Structures 54 (2013) 103–111

on elastic responses, i.e. Refs. [1,9,10], when the structure is in the elastic stage. A significant feature of the proposed strategy is that the optimization is carried out while enforcing the limited physical space available for stroke movement. To quantify the damage of the nonlinear primary system, the energy formulation has to be involved. As suggested by Uang and Bertero [27], the absolute energy representation is used in this study to quantify the input energy, kinetic energy, damping energy, and strain energy of systems. Reformulating Eq. (1) as

€t þ Ceq x_ þ Keq x ¼ 0 Mx

ð15Þ

€t ¼ x € þ r€xg . The rewhere absolute nodal acceleration vector x € , x, _ and x are relative with respect to the ground. For sponses x the ease of quantifying the dissipated energy, the duration of time t for energy calculation is taken from one cycle of harmonic excitation. Therefore the damping energy, which represents total energy dissipated by the linearized system, is as follows

Ed ðtÞ ¼

Z

4.0 Uncontrolled Tsai and Lin (1993) Minimax

3.5 3.0

Ductility x/x0

106

2.5 2.0 0.30 g 1.5 0.20 g 0.10 g

1.0

0.01 g 0.5 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

ω/ωs

2p

x

_ x_ T Ceq xdt

ð16Þ

0

Fig. 3. Uncontrolled and controlled systems under excitations with increasing amplitude and varying frequencies.

5. Numerical examples To demonstrate the proposed passive control method, results are presented on the numerical studies in this section. Two seismically excited two dimensional structures are considered: (1) a single-degree-of-freedom (SDOF) bilinear system and (2) a multiple degree-of-freedom (MDOF) nonlinear building. Designing the TMD using the proposed minimax optimization method will be implemented for the above mentioned SDOF and MDOF structures. Verifications of the designed TMD structures under long distant earthquake excitation will be presented through a fully nonlinear time history analysis on a SDOF nonlinear structure. 5.1. Single degree of freedom system The system is an inelastic structure with bilinear hysteresis. The mass is 1.8  104 tons and pre-yielding stiffness is 2.85  106 kN/ m, resulting in a fundamental frequency of 12.6 rad/s. The damping ratio corresponding to the fundamental frequency is 1.5%. The post-yielding stiffness is 0.3 times of the pre-yielding stiffness. Yielding displacement is 0.02 m. The mass ratio of the designed TMD is 2% of the total system mass. The harmonic base input considers four magnitudes: 0.01 g, 0.1 g, 0.2 g and 0.3 g. To understand the performance of proposed method as compared to the idea of using design formulas based on the linearized system, the study is carried out on different excitation amplitude and a predefined frequency band of 0.5–1.5 times of the elastic fundamental frequency. Fig. 3 shows, when there is no limit on stroke, the improvement of minimax optimization based TMD as compared to the original uncontrolled system. The results are also compared when TMD parameters are selected using Tsai and Lin’s formula, which is based on equivalent linear system of the original nonlinear structure. In Fig. 3, x0 is the yielding displacement of the primary system, x is the top displacement response of the primary system. Thus the ratio x/x0 gives the ductility of the SDOF system. In addition, xs is the fundamental frequency of structure at elastic state while x is the forcing frequency. When subjected to harmonic loadings, the structural nonlinearity depends on the forcing amplitude as well as the forcing frequency. Therefore Tsai and Lin’s formula is able to design TMD system with respect to one fixed forcing amplitude and frequency. This is different from the minimax optimization method proposed in this study, which considers a forcing frequency band (as in Eq. (14)) of one fixed forcing amplitude. In order to make a fair comparison with the proposed

minimax optimization method, Tsai and Lin’s formula is used at individual forcing frequencies of the involved forcing frequency band in the proposed method. This comparison basis is used in all the following figures where such comparison is made, except otherwise stated. It is found that the efficiency of the proposed TMD is impressively good in the elastic stage (x/x0 < 1.0) which reaches 75% reduction in displacement response. Under input amplitudes 0.1 g and 0.2 g, the system undergoes slight nonlinearity. The efficiency of TMD reduces and results in about 8% and 10% response reduction. However, the response reduction increases as the system goes to stronger nonlinearity, for instance, the reductions achieves 15% in the case of 0.3 g excitation. It is noted that Tsai and Lin’s formula is obtained through optimization based on elastic response. Therefore it is expected in Fig. 3, both Tsai and Lin’s formula and the proposed minimax optimization method reach competitively good performance in the elastic stage, i.e. (x/x0 < 1.0), as a result of 0.01 g excitation. This actually reflects the fact that the proposed method can reduce to Tsai and Lin’s formula in the elastic stage. Nevertheless, the method by Tsai and Lin loses efficiency rapidly as compared to the proposed method when the force amplitude increases. As demonstrated in Fig. 3 the Tsai and Lin’s formula amplify the response by about 12% while the latter reduces the response by 15% when the excitation is 0.3 g. The investigation includes the energy dissipation to explore the mechanism of the proposed method in reducing seismic damage. Fig. 4 shows that the effectiveness of proposed method in terms of the hysteresis energy dissipated in the equivalent linear system. It is seen that the system with proposed TMD suffers less damage than the original system, since the former dissipates less hysteresis energy than the latter. In order to study the stroke limit on the performance of the proposed method, investigations are implemented on the structural frequency change with different limits on the TMD stroke. Fig. 5 shows the performance of the proposed method considering different force amplitudes and stroke lengths. In Fig. 5a, the stroke limit is set to be 0.1 m. In Fig. 5b, there is no stroke limit. The quantity xs is the fundamental frequency of the elastic structure while xeq is the fundamental frequency of the linearized primary system which is attached with the TMD. Hence, the ratio xeq/xs = 1.0 means the system is exactly at the elastic stage. Three force amplitudes are involved for illustration, with the basic amplitude is chosen as A0 = 0.2g. By definition of Eq. (14), the forcing frequency band is

107

0.80

12000 Uncontrolled Controlled by minimax optimization

(a)

Limited stroke: 0.10 m No stroke limit

0.70

γd

10000

0.60 0.50

8000

0.40 0.00

0.02

0.04

0.06

0.08

0.10

0.02

0.04

0.06

0.08

0.10

6000 0.40

(b)

0.30

4000

ζd

Hysteresis energy of primary system (kN−m)

Z. Zhang, T. Balendra / Engineering Structures 54 (2013) 103–111

0.20

2000

0.10 0.00 0.00

0 0

0.25 Tm

0.5 Tm

Tm

0.75 Tm

Mass ratio

Time (sec) Fig. 6. Optimal TMD parameters: (a) cd and (b) nd. Fig. 4. Energy dissipation in one cycle for excitation frequency xm ¼ 2p=T m that produces the maximum response under 0.3 g.

Inelastic

1.00 0.1A 0 1.5A 0

0.80

(a) 0.60 0.0

A0 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

ωeq/ωs

1.20 Uncontrolled Tsai et al. (1993) Minimax

1.00 0.1A 0

1.5A 0

0.80

(b) 0.60 0.0

A0 0.5

1.0

1.5

2.0

2.5

3.0

3.5

Ductility x/x0 Fig. 5. Optimal systems with force amplitude of 0.1A0, A0, and 1.5A0: (a) stroke limit of 0.1 m; (b) no stroke limit.

considered from 0.5 to 1.5 times of the elastic fundamental frequency xs. It is seen in Fig. 5 that Tsai and Lin’s formula and the proposed method achieve the same efficiency, as both maintain the system in the elastic stage. The stroke limit effect tends to be significant when the system goes into nonlinear. For example, Fig. 5a shows that the proposed method loses efficiency immediately in the forcing amplitude of 1.0A0 and 1.5A0 while Tsai and Lin’s formula still works. The reason is that Tsai and Lin’s formula is based on minimizing the elastic response and does not consider the limit on stroke. In Fig. 5b, the proposed method is found to be better than Tsai and Lin’s formula, since there is no limit on stroke. Further study is on the effect of stroke limit on the optimal TMD parameters by the proposed minimax optimization. Fig. 6 shows that placing a limit on the stroke affects the optimal TMD parameters greatly. It is found to be reasonable that, with increasing mass ratio, the difference in optimal TMD parameters cd and nd for unlimited and limited stroke tend to reduce. This could be understood as increase in mass ratio reduces the stroke required, and beyond 6% mass ratio, the stroke needed is less than the limit placed on the stroke. In order to verify the performance of presented minimax optimization in designing TMD, nonlinear time history analyses are

Input acceleration (g)

ωeq/ωs

Elastic

0.02 0.01 0 -0.01 -0.02 0

50

100

150

200

250

300

Time (sec) x 10

-3

1

Amplitude

1.20

done for uncontrolled and TMD controlled systems. The base excitation, as illustrated in Fig. 7, is taken from the Katong site of Singapore to represent a long distance earthquake. The maximum acceleration of this excitation is 0.021 g and the frequency content furnishes a critical period of 1.92 s. This excitation is stochastically simulated based on the borehole data from Katong site and the rock motions of critical Mw–R combinations in this region (for instance, Mw = 9.3 Aceh earthquake in December 26, 2004 and Mw = 8.7 Nias earthquake in March 28, 2005). The structure is to take a total mass of 1.8  104 ton and damping ratio of 1.5%. The fundamental frequency is 3.276 rad/s. The yielding displacement dy is 0.02 m. The attached TMD is designed to have a 2% of total structural mass. According to Tsai and Lin’s formula, the TMD gives a stiffness of 3130.86 kN/m and damping of 192.16 kN/m/s. Consider the design amplitude of 0.021 g and frequency range of 0.9–1.2 times of structural fundamental frequency, the stiffness is 4262.08 kN/m and damping of 311.16 kN/m/s based on presented minimax optimization. Under the Katong site excitation, the structural response is calculated via Newmark method with a time step of 0.02 s and the result is shown in Fig. 8. It is seen that the maximum displacement yields 0.055 m for the uncontrolled system. The number of

0.5

0 0

2

4

6

8

10

Period (sec) Fig. 7. Time history and period content of a long distance earthquake (Katong site, Singapore).

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Z. Zhang, T. Balendra / Engineering Structures 54 (2013) 103–111

crossing times from elastic to inelastic states (i.e. |d| > dy) is 217 for the uncontrolled system. This number of crossing times can be used as a damage reduction measure for the efficiency of TMD controlled systems. With Tsai and Lin’s design, the maximum displacement gives a 0.059 m and a crossing times of 186. Based on the presented method, the maximum displacement is 0.048 m and a

crossing times of 176. It is found that TMD designed by Tsai and Lin’s method will produce a damage reduction of 16% while TMD by the presented method furnishes a reduction of 19%. Although both methods are constructive in reducing structural damage, the presented method is successfully reduces the peak displacement by 13% and this is not achieved by Tsai and Lin’s design. The better performance of the presented method can be attributed to the consideration of narrow band frequency range in designing TMD. Hysteresis loops of the original structure corresponding to Fig. 8 are

8000.00 6000.00

(a) Uncontrolled

Base shear (kN)

4000.00 2000.00 0.00 −2000.00 −4000.00 −6000.00 −8000.00 −0.060

−0.040

−0.020

0.000

0.020

0.040

0.060

0.040

0.060

0.040

0.060

Displacement (m) 8000.00 6000.00

(b) Tsai and Lin (1993)

Base shear (kN)

4000.00 2000.00 0.00 −2000.00 −4000.00 −6000.00 −8000.00 −0.060

−0.040

−0.020

0.000

0.020

Displacement (m) 8000.00 6000.00

(c) Minimax

Base shear (kN)

4000.00 2000.00 0.00 −2000.00 −4000.00 −6000.00 −8000.00 −0.060

−0.040

−0.020

0.000

0.020

Displacement (m) Fig. 8. Response time history of a bilinear SDOF system (T0 = 1.92 s) to a long distance earthquake (Katong site, Singapore).

Fig. 9. Hysteresis loops for a bilinear SDOF system (T0 = 1.92 s) to a long distance earthquake (Katong site, Singapore).

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Z. Zhang, T. Balendra / Engineering Structures 54 (2013) 103–111

This section is to show the proposed TMD design method on a multiple degree of freedom system with weakly nonlinearity. An eight storey building with bilinear nonlinearity for each storey is considered. The properties are listed in Table 1. The post-yielding coefficient is defined as 0.5 for all eight stories. The fundamental frequency of the unyielding fixed base building is T0 = 0.9 s. The given damping coefficients in Table 1 yield a damping ratio of 0.4% for the first mode. The mass ratio is taken as 2.0% for TMD design. A comparison study is conducted on the eight-storey building, with considerations on the effect of stroke limit on the proposed minimax optimal TMD. The design earthquake amplitude is 0.1 g and the forcing frequency band of 0.5–1.5 times xs. Similar to Example 1, Tsai and Lin’s is used in discretized frequency points within the design frequency band such that the linearized system is based on the correspondingly individual forcing frequency. It is noted that when there is no limit on the TMD stroke, the TMD stiffness varies from 10189.2 to 10486.3 kN/m, with damping parameters from 442.2 to 420.9 kN s/m by Tsai and Lin design. The TMD frequencies range from 0.9604 to 0.9743 times of the fundamental frequency of the original structure. This range is due to that the lumped linearization is done at every frequency point within the concerned frequency band while this is avoided in the proposed minimax optimization method as the optimized TMD adopts one set of parameters throughout frequency band. The TMD stiffness by the proposed method is 9986.56 kN/m, and damping is 561.427 kN s/m. The TMD frequency is 0.9508 times of the fundamental frequency of the original structure. Comparing the variation of TMD parameters by Tsai and Lin’s method and the proposed minimax optimization method, it is found that the variation in structural stiffness is small while the damping parameter increases from the former to the latter. This observation of variation on TMD parameters in the present example of eight story building is similar to the previous validation example on a fully nonlinear SDOF model by nonlinear time history analysis. Figure Fig. 10a shows the performance of different optimization methods under 0.1 g earthquake. The vertical axis defines the level number of the structure. The horizontal axis presents the maximum ductility of each floor among the design frequency band. In this example of multi-storey nonlinear system, the ductility is defined as a ratio of inter-storey drift over yielding displacement of the corresponding storey. It is found that the middle stories (from the 3rd to 7th storey) experiences much larger nonlinearity with the structural properties defined in Table 1. The proposed minimax optimization TMD can effectively reduce the seismic damage of the system. For instance, the ductility ratio can be reduced 48% by

(a)

8 Uncontrolled Tsai and Lin (1993) Stroke limit: 0.30 m Stroke limit: 0.75 m No stroke limit

7

6

Level number

5.2. Multiple degree of freedom system

minimax optimization with 0.75 m stroke, while the reduction is 29% by Tsai and Lin’s formula and the corresponding maximum stroke is 0.83 m. This reduction effect by the proposed method is more pronounced with longer stroke, even 0.75 m stroke is sufficient for the case of no limit on stroke. It is seen from Fig. 10b both Tsai and Lin’s method and the proposed method can help to reduce the peak floor acceleration, for instance, the presented method can achieve about 35% peak floor acceleration reduction with 0.75 m stroke whereas Tsai and Lin’s method gives a reduction of 33%. As expected, the efficiency of TMD decreases with smaller stroke. Further investigation involves the frequency response of typical floors, since Fig. 10 only gives the upper bound of performance of each floor, i.e. the maximum ductility among the design frequency

5

4

3

Coinciding cases: 0.75m stroke and no stroke limit

2

1 0

0.5

1

1.5

2

2.5

3

3.5

4

Ductility

(b)

8

7

6

Level number

shown in Fig. 9. It is interesting to find that the proposed TMD design method can reduced more on base shear than that by Tsai and Lin design. Both of the TMD designs will help to mitigate the hysteresis energy dissipated by the original system.

Coinciding cases: 0.75m stroke and no stroke limit

5

4

3 Tsai and Lin (1993) Stroke limit: 0.30 m Stroke limit: 0.75 m No stroke limit

2

1 0

10

20

30

40

50

60

70

Reduction of peak floor acceleration (%) Fig. 10. (a) Peak floor responses under base excitation of 0.1 g, (b) peak floor acceleration reduction, as compared to uncontrolled structure.

Table 1 Structural properties of eight-storey building. Storey

Pre-yielding stiffness for the storey (kN/ m)

Mass for the floor (ton)

Viscous damping coefficient (kN s/ m)

Modal damping ratio (%)

Yielding displacement (cm)

1 2 3 4 5 6 7 8

6.24  105 5.98  105 5.20  105 4.94  105 4.42  105 3.90  105 3.25  105 2.60  105

351.2 351.2 351.2 351.2 351.2 351.2 351.2 351.2

637 611 533 507 455 390 312 260

0.4 1.0 1.5 2.1 2.5 2.9 3.4 3.9

2.8 2.7 2.6 2.5 2.4 2.2 2.0 1.8

110

Z. Zhang, T. Balendra / Engineering Structures 54 (2013) 103–111 2.5

Uncontrolled Tsai and Lin (1993) Stroke limit: 0.30 m Stroke limit: 0.75 m No stroke limit

Ductility

2.0

1.5

1.0

0.5

Coinciding cases: 0.75m stroke and no stroke limit 0.0 0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

ω/ωs Fig. 11. Response of the third floor to varying forcing frequencies under 0.1 g base excitation.

band. The investigation herein then takes the third storey for instance. The performance of different TMD designs is illustrated in Fig. 11. It has to be bear in mind that formulas by Tsai and Lin [10], Den Hartog [1], and Warburton [9] did not consider the limit on stroke in designing the optimal TMD parameters. From their research work, optimal TMD will traditionally introduce two peaks around the controlled mode. This is observed in the proposed method as well, as seen in Fig. 11, when no stroke limit is allowed. Nevertheless, as stroke limit gets smaller as 0.3 m, the dual-peak phenomenon disappears. This is expected since the limitation of damper stroke will limit the energy that can be transferred to the TMD system and hence introduce higher response for the primary system. It is also noted from Fig. 11 that the maximum ductility does not occur at the same excitation frequency for the original system as well as TMD controlled systems. Furthermore, it is seen that using Tsai and Lin’s formula produces higher damage reduction than the proposed method without stroke limit. This is because a longer stroke of 0.83 m in Tsai and Lin’s case, compared

to 0.75 m in the proposed method. It is, however, noted in Fig. 11 that this damage reduction is achieved only around a small frequency span of 0.92xs to 1.08xs. As a comparison, the proposed method gives much robust damage reduction along the whole design frequency band of 0.5xs to 1.5xs. Since the proposed TMD design method is based on a fixed excitation amplitude and a predefined frequency band, it is therefore of great interest to investigate the robustness of the proposed method under changing base input amplitude. The study is carried out by getting the optimal TMD parameters for a predefined design amplitude A0 and frequency band. The system robustness is then investigated by changing the impute design amplitude from 0.8A0 to 1.2A0 but using the optimal TMD parameters obtained for the predefined amplitude of A0 = 0.1 g and frequency band of 0.5xs to 1.5xs. The result presented in Fig. 12b is the basic design for fixed excitation amplitude and predefined frequency band. The vertical axis represents the ductility of the third storey. It is found that the ductility ratio reduced 48% by the proposed minimax optimization method as compared to the uncontrolled system. The structure with proposed TMD now only experiences slightly inelastic behavior at the third storey. Fig. 12a and c separately illustrated the ductility of the third floor in the basic design when excitation amplitudes are at 0.8A0 and 1.2A0. The proposed TMD system can reduce the ductility by 54% in 0.8A0 and the whole structural system performs in an elastic range. The ductility reduction is 41% for 1.2A0 using the optimal parameters designed for targeted excitation amplitude A0. Fig. 13 shows the relationship of mass ratio and stroke limit. It is seen that with sufficient stroke, Fig. 13a demonstrates that increasing mass ratio of the TMD will reduce the requirement on the stroke. From Fig. 13b, it can be seen that the top displacement is consistently larger when stroke limit is set to be 0.3 m than that response in the case of adequate stroke. This is observed until the mass ratio is high enough such that the predefined 0.3 m stroke is sufficient to control the system. In that case, both systems with and without limit on the stroke achieves the same top displacement. In addition, for both limited and sufficient stroke cases, increasing the mass ratio is found to be constructive in reducing the top displacement.

2.5

(a) A/A 0 = 0.8

(b) A/A 0 = 1.0

(c) A/A 0 = 1.2

Uncontrolled 2.0

Minimax

Ductility

1.5

1.0

0.5

0.0 0.5

1.0

ω/ωs

1.5

0.5

1.0

ω/ωs

1.5

0.5

1.0

1.5

ω/ωs

Fig. 12. Responses under excitations with amplitude of 0.8A0 and 1.2A0, while the design is based on minimax optimal of fixed excitation amplitude A0 (0.1 g) and frequency band of 0.5–1.5xs.

Z. Zhang, T. Balendra / Engineering Structures 54 (2013) 103–111

Stroke (m)

2.00 1.50

(a)

1.00 0.50 0.00 0.00

Top displacement (m)

Limited stroke: 0.30 m No stroke limit

0.60

0.02

0.04

0.06

0.08

0.10

0.02

0.04

0.06

0.08

0.10

(b)

0.40

0.20

0.00 0.00

Mass ratio Fig. 13. Change of mass ratio and stroke limit when excited by 0.1 g ground motion.

6. Conclusions This study presented a novel design of TMD for weakly nonlinear structures by minimizing the maximum inelastic response within a concerned frequency range. This method is applicable for weakly nonlinear structures under narrow band excitations of long distance earthquakes. The stroke limit is considered as constraints in order to achieve the designing TMD parameters. It is found that the proposed method can produce identically excellent performance as conventional method for elastic structures. While for weakly inelastic structures under narrow band excitations of long distance earthquakes, the proposed method is able to provide better damage reduction than traditional design formula based on the reduction of elastic response. In the considered frequency range of excitations, the proposed method produces response reduction much evenly while the traditional method tends to amplify the response in certain sub-band of the considered frequency range. Longer stroke is revealed to be constructive in reducing the structural damage and the requirement on the stroke length can be alleviated with an increase in mass ratio of TMD to the total structural mass. References [1] Den Hartog JP. Mechanical vibrations. Dover Publications; 1985. [2] Hoang N, Fujino Y, Warnitchai P. Optimal tuned mass damper for seismic applications and practical design formulas. Eng Struct 2008;30:707–15.

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