Metaheuristic based optimization of tuned mass dampers under earthquake excitation by considering soil-structure interaction

Soil Dynamics and Earthquake Engineering 92 (2017) 443–461

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Metaheuristic based optimization of tuned mass dampers under earthquake excitation by considering soil-structure interaction ⁎

Gebrail Bekdaş, Sinan Melih Nigdeli

Department of Civil Engineering, Istanbul University, 34320 Avcılar, Istanbul, Turkey

A R T I C L E I N F O

A BS T RAC T

Keywords: Soil-structure interaction Structural control Tuned mass damper Optimization Metaheuristic algorithms Seismic structures

The aim of the study is to propose an optimization approach for optimum design of tuned mass dampers (TMDs) implemented to seismic structures considering soil-structure interaction (SSI). In the methodology, two metaheuristic algorithms such as harmony search algorithm and bat algorithm were employed. The present approaches evaluate time domain analyses of structure and consider the responses under several earthquake records. The optimum design variables defined as mass, period and damping ratio of TMD were searched for the optimization objective (minimization of the maximum displacement of structure) and the design constraint (limitation of the scaled stroke of TMD). The proposed methods were presented by using single degree of freedom structures for different soil characteristics, main structure periods and damping ratios. Also, a 40-story high-rise structure was investigated. For the 40-story structure, the optimally tuned TMDs are effective to reduce the critical displacement up to 25%. The proposed methodologies are both effective and feasible, but bat algorithm has advantages on the minimization of the optimization objective and finding a precise optimum value.

1. Introduction In civil engineering structures, the usage of structural control systems is an increasing trend. The main reason of using control systems is to reduce structural vibrations in order to provide a safe and comfortable life to people. Several undesired vibrations may occur by natural happenings such as wind and earthquakes. For these types of external excitations, both active and passive structural control systems can be employed. Since active control systems require high energy and development cost, passive control systems are preferred in practice of ordinary or special structures. Some of the examples of passive control techniques are the use of fiber-reinforced plastics (FRP), reinforced concrete walls, yielding steel absorbing devices, base isolations systems, friction or viscoelastic dampers and tuned mass dampers (TMDs). Most of these techniques are designed according to wellknown regulation standards, but the optimization and design of several systems need further investigation. Because of the complex and undetermined aspects in the structural dynamics, the optimizations of dynamic mechanical systems are challenging problems like tuning of TMD parameters according to frequency and damping of a controlled structures. The idea of reduction of mechanical vibrations by using an additional mass was first considered by using the vibration absorber

⁎

device of Frahm [1]. This device was lack of damping and the control of mechanical systems subjected to an excitation with changing frequency was not applicable. For that reason, Ormondroyd and Den Hartog [2] proposed to add damping devices to the vibration absorber device. After that, several approaches have been proposed for the tuning of mechanical components of TMDs by using closed form expressions of frequency and damping ratio [3,4]. The inherent damping of controlled systems is an obstacle in the development of the closed form expressions. The optimization of TMDs can be performed by using numerical iterations [5] and the required expressions can be defined by using curve fitting [6]. These expressions are for single degree of freedom (SDOF) systems and can be adopted to multiple degrees of freedom (MDOF) systems by considering a dominant vibration mode, but all assumptions effect the performance of TMD. For general and global optimization of TMDs, metaheuristic algorithms have been employed in the methodologies of optimum parameter estimation of TMDs. Most of these metaheuristic algorithms are nature inspired and different imitations or processes have been used in development of different and effective methods. Natural evolution inspired Genetic Algorithm (GA) is a well-known method [7,8]. For example, the other inspirations are the movement of swarm members in Particle Swarm Optimization (PSO) [9], the behavior of ants seeking a path between colony and food in Ant Colony Optimization (ACO) [10]

Corresponding author. E-mail addresses: [email protected] (G. Bekdaş), [email protected] (S.M. Nigdeli).

http://dx.doi.org/10.1016/j.soildyn.2016.10.019 Received 12 August 2014; Received in revised form 10 August 2016; Accepted 16 October 2016 0267-7261/ © 2016 Elsevier Ltd. All rights reserved.

Soil Dynamics and Earthquake Engineering 92 (2017) 443–461

G. Bekdaş, S.M. Nigdeli

Fig. 1. The TMD implemented structure including SSI effects.

on time-domain analyses for high-rise structures with SSI and TMD [29]. In several studies, active tuned mass dampers have been investigated for asymmetric structures considering soil-structure interaction effects [30–32]. The optimum TMD parameter must be also practical. For that reason, solution ranges are generally defined for all design parameters. Also, design constraints may be considered in the optimum design of TMDs. A design constraints of optimum TMD design is related with the stroke capacity. When the stroke capacity of TMDs are limited, a damping coefficient larger than the optimum value is needed in order to reduce the maximum displacement of the TMD spring [33]. By using numerical optimization techniques, it is possible to find a balance between the performance and the stroke capacity of TMD. Miranda developed an energy-based theoretical model for reducing the seismic responses by using TMDs [34]. A damping maximized TMD was proposed for seismic applications by Miranda [35]. Additionally, Tigli developed closed form expressions for TMDs used for reduction of vibrations resulting from random loads [36]. Salvi and Rizzi proposed minimax optimization for TMDs of frame structures under earthquake effects [37]. In this paper, two metaheuristic based optimization methods for parameter tuning of TMDs implemented on high-rise structures considering SSI effects are presented. In the proposed method, harmony search algorithm and bat algorithm (BA) were employed in the search of the design variables such as mass, period and damping ratio of TMD. Time domain analyses of earthquake excited structures were considered in the proposed optimization methodologies. During the optimization process, the structure was analyzed for different earthquake records and the responses of the critical excitation were considered. Additionally, maximum stroke capacity was also considered in an optimization case using a set of 44 earthquake records. Also, SDOF main structures interacting with soil were also investigated for

and the effort of a musician in Harmony Search (HS) algorithm [11]. In optimum design of TMDs, GA based methodology of Hadi and Arfiadi proposed optimum stiffness and damping properties of TMDs for MDOF shear buildings subjected to earthquake excitations [12]. Marano et al. also optimized the mass of TMD by employing GA [13]. Additionally, GA has been employed in the other studies involving asymmetric buildings [14,15] and active TMDs [16]. PSO was also used for optimization of TMDs and closed form expressions were proposed according to the optimum results [17,18]. Bionic algorithm is another metaheuristic algorithm employed by Steinbuch in order to estimate optimum TMD parameters for earthquake resistance of structures [19]. Bekdas and Nigdeli developed a HS-based method for the optimum design of TMDs [20]. Also, the effect of mass ratio was investigated for TMDs. A methodology employing HS using recorded ground motions during the optimization process was also developed [21]. In order to estimate TMD parameters to prevent brittle fracture of reinforced concrete (RC) structures, the HS-based methodology was also investigated by limiting the mass of TMD with respect to the maximum axial force capacity of RC structures [22]. Ant colony optimization (ACO) [23], Artificial Bee Colony (ABC) [24] and shuffled complex evolution [25] were employed for optimization of TMDs implemented on structures considering SSI effects by Farshidianfar and Soheili. Since the vibrational energy of the structure is directed to the soil through the foundation, dynamic response of structures may be very different than the fixed based structures when SSI effects are taken into consideration. Xu and Kwok proved the influence of SSI effect on the effectiveness of TMDs by investigating the wind-induced motion of tall and slender structures [26]. Wu et al. investigated SSI effect for seismic performance of TMDs and found that TMD becomes less effective when the soil shear wave velocity decreases [27]. Wang and Lin studied the application of TMDs for torsionally coupled irregular buildings including SSI effects [28]. Lui et al. developed a mathematical model based

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different main structural periods and damping ratios.

2. Mathematic representation of a structure with SSI The equations of motion of the TMD implemented structure with SSI effects are presented in this section. The physical model of the shear building with SSI is shown in Fig. 1. In this figure, x0 and θ0 represent the displacement and the rotation of foundation, respectively. The ith story displacement of N-story structure is represented by xi (for i=1 to N). The structural parameters such as the stiffness coefficient, the damping coefficient, the mass and the moment of inertia of each story are represented by ki, ci, mi and Ii, respectively. The location of stories with respect to foundation is defined with zi for the ith story. The symbols; m0 and I0 are the mass and the moment of inertia of the foundation, respectively. The swaying damping, the rocking damping, the swaying stiffness and the rocking stiffness of soil are represented with cs, cr, ks and kr, respectively. The parameters of TMD which are optimized, are represented with md, kd and cd for mass, stiffness coefficient and damping coefficient, respectively. The main form of the equations of motion of structure including SSI effects and a TMD are given in Eq. (1) for ẍg ground acceleration.

[M ]x (̈ t ) + [C ]x (̇ t ) + [K ]x (t ) = − [m*]xg̈

Fig. 2. Block diagram of the equation of motion.

Table 1 Properties of the soil and foundation [29].

(1)

In the equation of motion, [M], [C], [K] and [m*] are the system mass, damping, stiffness and acceleration mass matrices. x(t) represents the response matrix which is given as

⎡ x1 ⎤ ⎢ x2 ⎥ ⎢⋮ ⎥ ⎢ ⎥ xN −1 x (t ) = ⎢⎢ x ⎥⎥ . N ⎢ xd ⎥ ⎢ x0 ⎥ ⎢ ⎥ ⎣ θ0 ⎦

cs (Ns/m)

cr (Ns/m)

ks (N/m)

kr (N/m)

Soft soil Medium soil Dense soil

2.19×108 6.90×108 1.32×109

2.26×1010 7.02×1010 1.15×1011

1.91×109 1.80×1010 5.75×1010

7.53×1011 7.02×1012 1.91×1013

Table 2 Properties of the 40-story building. z1-z40 (m) mi (t) Ii (kgm2) k1-k40 (MN/m) c1-c40 (MNs/m) mo (t) Io (kgm2)

(2)

The required matrices of the equations of motion are defined between Eqs. (3)–(8). [Mf], [Kf] and [Cf] are the classical fixed based system matrices of mass, stiffness and damping of a shear building with a TMD positioned on the top.

⎡ [Mf ] ⎤ [Mv ] [MZ ] ⎢ ⎥ N N T ⎥ (∑i =1 mizi ) + md zN [M ] = ⎢ [MV ] m 0 + (∑i =1 mi ) + md ⎢ ⎥ N N N T 2 2 ⎢⎣[MZ ] (∑i =1 mizi ) + md zN (∑i =1 mizi ) + md zN + I0 + ∑i =1 Ii ⎥⎦ (3)

⎡ m1 ⎤ ⎢ m2 ⎥ ⎢⋮ ⎥ [MV ] = ⎢ m ⎥ ⎢ N −1⎥ ⎢ mN ⎥ ⎢⎣ md ⎥⎦

Soil type

(4)

⎡ m1z1 ⎤ ⎢ m 2z2 ⎥ ⎢⋮ ⎥ [MZ ] = ⎢ m z ⎥ ⎢ N −1 N −1⎥ ⎢ mN zN ⎥ ⎢⎣ md zN ⎥⎦

(5)

⎡ [K f ] 0 0 ⎤ ⎢ ⎥ [K ] = ⎢ 0 … 0 ks 0 ⎥ ⎢⎣ 0 … 0 0 k ⎥⎦ r

(6)

4–160 980 1.31×108 2130–998 42.6–20 1960 1.96×108

⎡ [Cf ] 0 0 ⎤ ⎢ ⎥ [C ] = ⎢ 0 … 0 cs 0 ⎥ ⎢⎣ 0 … 0 0 c ⎥⎦ r

(7)

⎡ m1 ⎤ ⎢ m2 ⎥ ⎢⋮ ⎥ ⎢ ⎥ m ⎢ N −1 ⎥ ⎥ [m*] = ⎢ mN ⎢ md ⎥ ⎢ ⎥ N + (∑ ) + m m m ⎢ 0 d⎥ i =1 i ⎢ N ⎥ ⎣ (∑i =1 mizi ) + md zN ⎦

(8)

In the optimum design, the equations of motion were modeled with Matlab Simulink [38]. The developed block diagram is given in Fig. 2. Matrix operations and integrations are directly calculated by using a loop of the acceleration of the structure. As a solver, the fourth order Runge-Kutta method was chosen in Matlab Simulink and the time step was taken as 0.001 s in the analyses. The methodology of the optimization is presented in the Section 3 for both metaheuristic methods.

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several optimum designs. In this section, two different design methodologies are presented. These methodologies are based on harmony search algorithm and bat algorithm.

Table 3 FEMA P-695 far-field ground motion records [54]. Earthquake no.

Date

Name

Component 1

Component 2

1

1994

Northridge

2

1994

Northridge

3

1999

Duzce, Turkey

4

1999

Hector Mine

5

1979

Imperial Valley

6

1979

Imperial Valley

7 8 9

1995 1995 1999

Kobe, Japan Kobe, Japan Kocaeli, Turkey

10

1999

Kocaeli, Turkey

11

1992

Landers

12

1992

Landers

13

1989

Loma Prieta

14

1989

Loma Prieta

15

1990

Manjil, Iran

16

1987

Superstition Hills

17

1987

Superstition Hills

18

1992

Cape Mendocino

19

1999

Chi-Chi, Taiwan

20

1999

Chi-Chi, Taiwan

21 22

1971 1976

San Fernando Friuli, Italy

NORTHR/ MUL009 NORTHR/ LOS000 DUZCE/ BOL000 HECTOR/ HEC000 IMPVALL/HDLT262 IMPVALL/HE11140 KOBE/NIS000 KOBE/SHI000 KOCAELI/ DZC180 KOCAELI/ ARC000 LANDERS/ YER270 LANDERS/ CLW-LN LOMAP/ CAP000 LOMAP/ G03000 MANJIL/ ABBAR–L SUPERST/BICC000 SUPERST/BPOE270 CAPEMEND/ RIO270 CHICHI/ CHY101-E CHICHI/ TCU045-E SFERN/PEL090 FRIULI/ATMZ000

NORTHR/ MUL279 NORTHR/ LOS270 DUZCE/ BOL090 HECTOR/ HEC090 IMPVALL/HDLT352 IMPVALL/HE11230 KOBE/NIS090 KOBE/SHI090 KOCAELI/ DZC270 KOCAELI/ ARC090 LANDERS/ YER360 LANDERS/ CLW-TR LOMAP/ CAP090 LOMAP/ G03090 MANJIL/ ABBAR–T SUPERST/BICC090 SUPERST/BPOE360 CAPEMEND/ RIO360 CHICHI/ CHY101-N CHICHI/ TCU045-N SFERN/PEL180 FRIULI/ATMZ270

3.1. Harmony search algorithm based methodology Like a designer searching for the best solution, a music performer tries to find the best harmony in order to gain the attention and admiration of listeners. Harmony search is inspired by this phenomena. Because of efficiency of HS in optimization of structural engineering problems, HS has been employed in optimization of structures [39], steel frames [40], trusses [41], cellular beams [42], TMDs [20–22], base isolation systems [43], reinforced concrete members [44], cylindrical walls [45,46] and design of trusses [47]. Three unique options of musicians are used in the HS algorithm [48]. These options are listed as seen below. Option 1: The first option is to play a part of music from the memory of the performer. In HS algorithm, this option is used as a generation of a suitable harmony memory matrix and possible solutions of design variables are stored in it. Option 2: The performer changes the known part of the music in order to gain attention of audience. Like this searching process of musicians, a new harmony vector is generated by considering an existing values in harmony memory matrix as a reference and modifications are done in the harmony memory matrix according to the optimization objective. Option 3: The last option of the performer is to try his luck by using random notes, because the best harmony may be very different from the known parts of music. Thus, new harmony vectors can be randomly generated by using the all solution domain. A possibility called Harmony Memory Considering Rate (HMCR) is used for using an existing vector and the solution range is modified with a parameters called Pitch Adjacent Rate (PAR). The HS based methodology of optimization can be explained in five steps. i. In this step, the structural properties are defined. The parameters of the soil must be also entered. In order to search the optimum TMD parameters, the bounds of the solution ranges are also optimization constants. Differently from the problem constants, the parameters of HS algorithm such as Harmony Memory Size (HMS), HMCR and PAR are defined. Also, earthquake records are loaded. ii. The preliminary analyses are done in the second step. The structure without TMD is analyzed, because the response of uncontrolled structure is contained in the design constraints considering the stroke capacity of the TMD. iii. Then, the generation of the design variables is done by using randomly generated variables from the solution range. By merging

3. The optimization methodology The nature inspired metaheuristic algorithms formulize a happening in life. By the similarity of reaching a final objective of a process and optimization of engineering problems, metaheuristics are employed in

Table 4 The optimum results of SDOF structure on soft soil (HS approach). T(s) ξ md (t) μ (%) Td(s) kd (kN/m) ξ d (%) cd (kNs/m) max(x) (m) without TMD max (x) (m) with TMD max (x ̈ + xg̈ ) (m/s2) without TMD

0.5 0.02 48.280 4.927 0.545 3.497 18.100 201.493 0.173 0.116 26.188

0.5 0.05 48.430 4.942 0.552 3.464 21.000 231.529 0.133 0.098 19.978

1.0 0.02 48.781 4.978 0.889 2.166 7.500 51.715 0.339 0.279 13.221

max (x ̈ + xg̈ ) (m/s2) with TMD

16.820

14.279

10.125

8.882

5.672

5.088

4.688

4.177

Earthquake no. Component

1 2

1 2

3 2

3 2

9 2

9 2

19 2

19 2

446

1.0 0.05 48.165 4.915 0.893 2.129 9.500 64.389 0.290 0.243 11.355

1.5 0.02 48.658 4.965 1.601 1.200 13.400 51.177 0.385 0.327 6.726

1.5 0.05 48.885 4.988 1.698 1.137 19.600 70.909 0.332 0.291 5.840

2.0 0.02 48.835 4.983 2.419 0.797 21.100 53.529 0.595 0.493 5.839

2.0 0.05 48.909 4.991 2.428 0.795 24.200 61.258 0.493 0.441 4.851

Soil Dynamics and Earthquake Engineering 92 (2017) 443–461

G. Bekdaş, S.M. Nigdeli

Table 5 The optimum results of SDOF structure on soft soil (BA approach). T(s) ξ md (t) μ (%) Td(s) kd (kN/m) ξ d (%) cd (kNs/m) max(x) (m) without TMD max (x) (m) with TMD max (x ̈ + xg̈ ) (m/s2) without TMD

0.5 0.02 49.000 5.000 0.569 3.400 18.100 195.872 0.173 0.114 26.188

0.5 0.05 49.000 5.000 0.598 3.235 20.600 212.115 0.133 0.097 19.978

1.0 0.02 49.000 5.000 0.906 2.135 8.500 57.769 0.339 0.278 13.221

1.0 0.05 49.000 5.000 1.009 1.917 14.100 86.047 0.290 0.240 11.355

1.5 0.02 49.000 5.000 1.572 1.231 13.000 50.921 0.385 0.325 6.726

1.5 0.05 49.000 5.000 1.665 1.162 17.400 64.349 0.332 0.288 5.840

2.0 0.02 49.000 5.000 2.405 0.804 20.300 51.974 0.595 0.489 5.839

2.0 0.05 48.979 4.998 2.440 0.792 23.900 60.288 0.493 0.439 4.851

max (x ̈ + xg̈ ) (m/s2) with TMD

16.896

14.442

10.189

9.228

5.626

5.053

4.657

4.171

Earthquake no. Component

1 2

1 2

3 2

3 2

9 2

9 2

19 2

19 2

1.0 0.05 47.568 4.854 0.913 2.057 9.900 64.817 0.288 0.240 11.431

1.5 0.02 48.906 4.990 1.591 1.214 13.600 52.534 0.383 0.325 6.720

1.5 0.05 48.589 4.958 1.717 1.117 17.300 61.521 0.329 0.288 5.824

2.0 0.02 48.524 4.951 2.599 0.737 22.900 53.727 0.585 0.489 5.771

2.0 0.05 46.929 4.789 2.396 0.773 23.800 58.579 0.485 0.436 4.802

Table 6 The optimum results of SDOF structure on medium soil (HS approach). T(s) ξ md (t) μ (%) Td(s) kd (kN/m) ξ d (%) cd (kNs/m) max(x) (m) without TMD max (x) (m) with TMD max (x ̈ + xg̈ ) (m/s2) without TMD

0.5 0.02 48.644 4.964 0.562 3.417 17.700 192.520 0.192 0.121 30.055

0.5 0.05 48.463 4.945 0.565 3.386 20.300 218.810 0.140 0.100 22.049

1.0 0.02 48.879 4.988 0.907 2.128 8.900 60.272 0.338 0.279 13.350

max (x ̈ + xg̈ ) (m/s2) with TMD

18.676

15.301

10.346

9.013

5.650

5.087

4.742

4.162

Earthquake no. Component

1 2

1 2

3 2

3 2

9 2

9 2

19 2

19 2

1.0 0.05 49.000 5.000 0.951 2.034 11.300 73.165 0.288 0.238 11.431

1.5 0.02 48.911 4.991 1.570 1.230 12.300 48.153 0.383 0.323 6.720

1.5 0.05 49.000 5.000 1.652 1.171 17.100 63.737 0.329 0.286 5.824

2.0 0.02 49.000 5.000 2.370 0.816 20.100 52.222 0.585 0.483 5.771

2.0 0.05 49.000 5.000 2.375 0.815 23.100 59.890 0.485 0.434 4.802

Table 7 The optimum results of SDOF structure on medium soil (BA approach). T(s) ξ md (t) μ (%) Td(s) kd (kN/m) ξ d (%) cd (kNs/m) max(x) (m) without TMD max (x) (m) with TMD max (x ̈ + xg̈ ) (m/s2) without TMD

0.5 0.02 49.000 5.000 0.570 3.394 16.600 179.324 0.192 0.119 30.055

0.5 0.05 49.000 5.000 0.573 3.376 19.400 208.475 0.140 0.098 22.049

1.0 0.02 49.000 5.000 0.909 2.128 8.400 56.901 0.338 0.278 13.350

max (x ̈ + xg̈ ) (m/s2) with TMD

18.423

15.155

10.309

9.070

5.611

5.026

4.612

4.126

Earthquake no. Component

1 2

1 2

3 2

3 2

9 2

9 2

19 2

19 2

1.0 0.05 48.483 4.947 0.941 2.034 11.300 73.162 0.288 0.238 11.418

1.5 0.02 48.797 4.979 1.561 1.234 13.700 53.817 0.396 0.326 6.950

1.5 0.05 48.464 4.945 1.687 1.134 18.200 65.703 0.329 0.287 5.818

2.0 0.02 48.819 4.982 2.508 0.768 21.900 53.569 0.584 0.485 5.763

2.0 0.05 47.638 4.861 2.310 0.814 22.400 58.050 0.485 0.436 4.797

Table 8 The optimum results of SDOF structure on dense soil (HS approach). T(s) ξ md (t) μ (%) Td(s) kd (kN/m) ξ d (%) cd (kNs/m) max(x) (m) without TMD max (x) (m) with TMD max (x ̈ + xg̈ ) (m/s2) without TMD

0.5 0.02 48.557 4.955 0.539 3.557 17.200 194.716 0.190 0.122 29.983

0.5 0.05 48.677 4.967 0.584 3.291 21.100 221.006 0.139 0.099 21.973

1.0 0.02 48.831 4.983 0.989 1.949 13.200 81.900 0.338 0.280 13.340

max (x ̈ + xg̈ ) (m/s2) with TMD

18.395

15.444

10.727

9.067

5.612

5.061

4.682

4.130

Earthquake no. Component

1 2

1 2

3 2

3 2

1 2

9 2

19 2

19 2

447

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Table 9 The optimum results of SDOF structure on dense soil (BA approach). T(s) ξ md (t) μ (%) Td(s) kd (kN/m) ξ d (%) cd (kNs/m) max(x) (m) without TMD max (x) (m) with TMD max (x ̈ + xg̈ ) (m/s2) without TMD

0.5 0.02 49.000 5.000 0.564 3.430 16.500 180.140 0.190 0.118 29.983

0.5 0.05 49.000 5.000 0.576 3.358 19.400 207.389 0.139 0.097 21.973

1.0 0.02 49.000 5.000 0.891 2.171 7.000 48.376 0.338 0.277 13.340

1.0 0.05 49.000 5.000 0.922 2.098 9.600 64.113 0.288 0.237 11.418

1.5 0.02 49.000 5.000 1.573 1.230 12.500 48.931 0.382 0.323 6.714

1.5 0.05 49.000 5.000 1.646 1.175 17.100 63.969 0.329 0.285 5.818

2.0 0.02 49.000 5.000 2.412 0.802 20.700 52.844 0.584 0.483 5.763

2.0 0.05 49.000 5.000 2.473 0.782 24.600 61.252 0.485 0.433 4.797

max (x ̈ + xg̈ ) (m/s2) with TMD

18.262

15.113

10.196

8.935

5.610

5.016

4.629

4.153

Earthquake no. Component

1 2

1 2

3 2

3 2

9 2

9 2

19 2

19 2

1.0 0.05 48.701 4.969 0.906 2.122 9.600 64.847 0.287 0.238 11.409

1.5 0.02 48.070 4.905 1.581 1.200 13.000 49.670 0.382 0.324 6.711

1.5 0.05 48.632 4.962 1.722 1.115 19.100 67.785 0.329 0.288 5.814

2.0 0.02 48.753 4.975 2.372 0.811 20.400 52.690 0.583 0.484 5.759

2.0 0.05 48.898 4.990 2.387 0.809 23.700 61.009 0.484 0.434 4.794

Table 10 The optimum results of fixed based SDOF structure (HS approach). T(s) ξ md (t) μ (%) Td(s) kd (kN/m) ξ d (%) cd (kNs/m) max(x) (m) without TMD max (x) (m) with TMD max (x ̈ + xg̈ ) (m/s2) without TMD

0.5 0.02 47.580 4.855 0.579 3.244 17.800 183.813 0.184 0.121 29.079

0.5 0.05 48.845 4.984 0.572 3.371 19.500 209.251 0.138 0.097 21.859

1.0 0.02 48.879 4.988 0.950 2.031 11.100 71.768 0.337 0.278 13.332

max (x ̈ + xg̈ ) (m/s2) with TMD

19.232

15.040

10.530

8.926

5.641

5.085

4.621

4.134

Earthquake no. Component

7 1

1 2

3 2

3 2

9 2

9 2

19 2

19 2

1.0 0.05 49.000 5.000 0.921 2.100 9.600 64.183 0.287 0.237 11.409

1.5 0.02 49.000 5.000 1.567 1.234 12.300 48.333 0.382 0.322 6.711

1.5 0.05 49.000 5.000 1.645 1.176 17.200 64.383 0.329 0.285 5.814

2.0 0.02 48.888 4.989 2.442 0.790 21.000 52.831 0.583 0.483 5.759

2.0 0.05 49.000 5.000 2.390 0.809 23.300 60.029 0.484 0.432 4.794

Table 11 The optimum results of fixed based SDOF structure (BA approach). T(s) ξ md (t) μ (%) Td(s) kd (kN/m) ξ d (%) cd (kNs/m) max(x) (m) without TMD max (x) (m) with TMD max (x ̈ + xg̈ ) (m/s2) without TMD

0.5 0.02 49.000 5.000 0.562 3.442 16.500 180.781 0.189 0.117 29.809

0.5 0.05 49.000 5.000 0.570 3.394 19.500 210.652 0.138 0.097 21.859

1.0 0.02 49.000 5.000 0.892 2.169 7.100 49.012 0.337 0.277 13.332

max (x ̈ + xg̈ ) (m/s2) with TMD

18.160

15.013

10.199

8.928

5.598

5.012

4.640

4.125

Earthquake no. Component

1 2

1 2

3 2

3 2

9 2

9 2

19 2

19 2

Table 12 Optimum TMD parameters (HS).

md (t) μm (%) Td (s) kd (MN/m) ξd (%) cd (MNs/m)

Table 13 Optimum TMD parameters (BA).

Fixed base

Dense soil

Medium soil

Soft soil

1917 6.23 5.573 2.436 23.224 1.004

1905 6.19 5.704 2.312 25.172 1.057

1909 6.20 6.103 2.023 25.983 1.021

1948 6.33 4.590 3.651 21.097 1.125

md (t) μm (%) Td (s) kd (MN/m) ξd (%) cd (MNs/m)

448

Fixed base

Dense soil

Medium soil

Soft soil

1960 6.37 5.745 2.344 22.54 1.057

1960 6.37 5.888 2.232 23.47 1.048

1960 6.37 6.127 2.061 24.18 0.997

1960 6.37 4.972 3.130 27.96 1.750

Soil Dynamics and Earthquake Engineering 92 (2017) 443–461

G. Bekdaş, S.M. Nigdeli

Table 14 Optimum TMD parameters and maximum results (Den Hartog). Fixed base

Dense soil

Medium soil

Soft soil

md (t) μm (%) fopt Td (s) kd (MN/m) ξd (%) cd (MNs/m) max(x) (m) max (x ̈ + xg̈ ) (m/s2)

1960 6.37 0.94 4.07 4.66 12.24 0.74 1.73 4.36

1960 6.37 0.94 4.18 4.44 12.24 0.72 1.83 4.36

1960 6.37 0.94 4.35 4.10 12.24 0.69 1.99 4.34

1960 6.37 0.94 6.17 2.03 12.24 0.49 2.68 4.45

g(x)

2.27

2.36

2.45

3.9

Fig. 6. Top story acceleration transfer function (TF) plot of 40-story structure (dense soil).

Fig. 3. Total displacement of the top story of the structure (fixed based).

Fig. 7. Total displacement of the top story of the structure (medium soil).

Fig. 4. Top story acceleration transfer function (TF) plot of 40-story structure (fixed based). Fig. 8. Top story acceleration transfer function (TF) plot of 40-story structure (medium soil).

Fig. 5. Total displacement of the top story of the structure (dense soil). Fig. 9. Total displacement of the top story of the structure (soft soil).

harmony vectors (HV) (Eq. (9)) containing the possible TMD parameters, the initial harmony memory (HM) matrix (Eq. (10)) is constructed. The optimized TMD parameters are mass (md), period (Td) (or stiffness coefficient) and damping ratio (ξd) (or damping coefficient).

HM = [ HV1 HV2 … … HVHMS ]

449

(9)

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Rule 1: By echolocation, bats sense distance in order to know the difference of prey (food) and background barriers. Rule 2: Bats fly with a random velocity with a fixed frequency. The wavelength and loudness may vary. Rule 3: The loudness can change from a large value to a small value. The bat algorithm has been employed in several structural engineering problems [50,51] including optimum design of trusses [52] and reinforced concrete members [53]. The BA based methodology for TMD design can be also given in five steps as the HS based method. The only difference is in step iv. The parameters and the name of the solution vectors (harmony vectors in HS) are different. The solutions are stored in position vectors in BA and the solution vectors construct the solution matrix. The parameters of BA are the limits of the pulse frequency (fmin and fmax), the initial values of the pulse emission rate (ri) and the loudness (Ai). Since BA is a swarm based algorithm, all vectors in the solution matrix are updated in step iv. In the iterative analyses, the position vector (di) is updated according to Eqs. (13)– (15).

Fig. 10. Top story acceleration transfer function (TF) plot of 40-story structure (soft soil).

⎡ mdi ⎤ ⎢ ⎥ HV = ⎢ Tdi ⎥ ⎣ ξdi ⎦

(10)

iv. In this step, the essential optimization starts by generating a new HV in two ways by controlling a possibility called HMCR. A new HV can be generated around an existing solution by using a small range which is PAR times of the area of the initial range. In that case, the convergence of the optimization process is provided, but the algorithm may trap to a local solution. For that reason, the whole solution range is used as the second way of the generation of possible design variables. If the objective function of the algorithm is better in solution by using the set of newly generated design variables comparing to existing worst solution, the worst one is replaced with the new set of variables. The objective function (OF) of the optimization problem is to reduce the maximum displacement of the structure with TMD to a user defined value. It is formulated as shown in Eq. (11). If the objective function can not be reduced to a user defined value for the defined range, the value defined by user can be iteratively increased. In order to minimize the objective function, the user defined value can be entered as zero.

OF = max(x 0 + xN + zN θ0 )

g (X ) =

max [ xN ]without TMD

(13)

vit = vit−1 + (dit − d )fi *

(14)

dit = dit −1 + vit

(15)

In these equations, the adjusting frequency and the updating velocity are shown with fi and vi, respectively. β is a real number between 0 and 1. The superscript represents the current iteration (t). The best solution according the optimization objective is defined with d *. The updated position vectors are accepted or not according to a criterion test. When the pulse rate is smaller than a randomly generated number between 0 and 1, local solutions around the best vector are generated by controlling the user defined solution range. When a randomly generated number is smaller than the loudness and objective for the updated solution is better than the previous solution, the solution generated by using Eqs. (13)–(15) are accepted. Also, the pulse rate (ri) and the loudness (Ai) are updated in that case by using the following equations. Constant values such as α and γ are defined as 0.9.

(11)

Also, a design constraint defined as Eq. (12) is checked in order to consider the stroke capacity of TMD. The maximum relative displacement of TMD scaled by the maximum top story displacement of structure without TMD must not exceed a user defined value called st_max. If the defined constraint violates, the elimination of the possible solution is done with respect to the amount of the violation.

max [ xN +1 − xN ]with TMD

fi = fmin + (fmax − fmin )β

Ait +1 = αAit

(16)

rit +1 = rit[1 − exp( − γt )]

(17)

4. Numerical examples

≤ st max The methods were tested with SDOF main structure with period between 0.5 s and 2 s. Also, the inherent damping ratio of the structure was investigated for two values such as 0.02 and 0.05. The three different soil properties used in the study are shown in Table 1. Also, the optimum values were investigated for MDOF structure. The structural model is a 40-story building including a foundation positioned on soil [29]. The properties of the 40-story structure are shown in Table 2. While, each story has the same mass, height and moment of inertia, the stiffness of the structure linearly decrease when zi distances increase. For SDOF structures, the mass and the moment of inertia properties of the foundation and structure were taken as the same of the foundation and the first story of the 40-story building, respectively. The stiffness and damping coefficient values were calculated with respect to the period and damping ratio of the structure.

(12) v. The last step is about the stopping criterion of the optimization. When the value of the minimum objective function is lower than the user defined value, the optimization process ends. If this situation is not provided, new vectors are generated and the iterative process continues. 3.2. Bat algorithm based methodology Swarm intelligence is also an inspiration used in the metaheuristic algorithms. Bat algorithm developed by Yang [49] formulates the echolocation characteristic of micro bats. Three different rules of nature are used in the algorithm.

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frequencies between fixed based structure and structure on soil, the optimum parameters of TMD are very different. Rayleigh damping is used for the structure and the damping matrix is only calculated as 0.02 times of the stiffness matrix. In that case, the fixed based structure has nearly 2% damping ratio for the first mode [29]. The other modes have different damping ratios. The optimum TMD parameters are presented in Tables 12 and 13 for the HS and BA approaches, respectively. A significant increase of the period of TMD is observed for the BA based method comparing to HS approach. In these results, μm is ratio of mases of TMD and the modal mass for the first mode of the structure. The allowed mass of TMD during the optimization process is the ratio with respect to the total mass of the main structure without the weight of the foundation. A TMD was also designed for 40-story structure by using the classical formulation of Den Hartog [3]. The mass ratio was taken as the same with BA based method. The optimum damping ratio (ξd) were found according to Eqs. (18) and (19), respectively. The first mode was considered in using these equations. The results of TMD parameters and maximum results for the Den Hartog's classical method are presented in Table 14. As seen from the results, the reduction of maximum displacements is low according to the metaheuristic methods. Since the damping ratio of TMD cannot be optimized according to the special conditions of stroke capacity, the g(x) value is higher than the allowed limit. In that case, the reduction of accelerations are better than the metaheuristic methods, but the TMD designed according to Den Hartog's formulations are not practical since it is not possible to control design constraints.

The stroke capacity was considered by taking st_max (defined in Eq. (12)) as 2 for all numerical examples. The maximum relative displacement of TMD scaled by the maximum top story displacement of the structure without TMD may exceed 2 for uncritical earthquakes. The unscaled value is always lower than the occurred under the critical excitation and minimization of the stroke of the TMD is aimed in the study. During the optimization process, 44 recorded ground motions (two components of 22 stations) were separately used. These records are farfield ground motion records defined in FEMA P-695 [54]. The detailed information about these records are given in Table 3 and all earthquake records used in the study were downloaded from the database of Pacific Earthquake Engineering Research Centre (PEER) [55]. For practical applications, the mass of TMD was searched between 1% and 5% of the mass of structure excluding the mass of the foundation. Also, the damping ratio of TMD was optimized for the values between 0.1% and 30%. The range of the period of TMD was taken between 0.5 and 1.5 times of the period of the main structure including SSI. The parameters of HS algorithm such as HMS, HMCR and PAR were taken as 5, 0.5 and 0.1, respectively. Required BA parameters such as the bat population (n), the minimum frequency (fmin), the maximum frequency (fmax), the initial pulse rate (ri) and initial loudness (Ai) were taken as 5, 0, 1, 0.5 and 1, respectively. 4.1. Results of SDOF main structures with different periods and damping ratio The optimum TMD properties were investigated for eight SDOF structures. Four different periods (T=0.5 s, 1 s, 1.5 s and 2 s) and two different damping ratios (ξ=0.02 and 0.05) were taken as structural cases. In Tables 4, 5, the optimum TMD parameters and structural responses of the structure on soft soil are shown for HS and BA approaches, respectively. The tables also contain the critical earthquake record and component. The maximum structural responses are presented for critical excitation in this section. According to the results of the structure on soft soil, BA approach is more effective than HS approach. For the structure with low damping (ξ=0.02), the amount of the reduction of the structural displacements is more than the structure with 5% damping ratio. Also, the advantages of TMD can be clearly seen for the structures with short periods. The results for SDOF structures on medium soil are shown in Tables 6 and 7. Similarly, the advantage of BA approach can be seen from the results. In Tables 8 and 9, the results of SDOF structures for dense soil are presented. For the structure with 1.5 s period and 0.02 damping ratio, the critical excitation is different for the HS approach. BA is more effective than HS to reduce the maximum displacement for a different excitation. In that case, HS algorithm is not effective to find the best optimum solution because the maximum responses under two earthquake records are nearly the same. Finally, the SDOF structure was also analyzed without SSI effects by considering a fixed base (Tables 10 and 11). Similarly, BA is a better approach comparing to HS and HS is also not effective on finding the exact optimum solution for the structure with 0.5 s period and 0.02 damping ratio.

fopt =

ξd =

1 1 + μm

(18)

3μm 8(1 + μm )

(19)

The critical excitation is the second component of 19th record and the data were recorded during Chi-Chi earthquake. This record is the critical one for all soil cases. The maximum displacements, accelerations, scaled relative displacement of TMD (including the maximum stroke) are given in the Appendix A as Tables 15–22. The plots of the frequency domain results and the total displacement of the top story of the structure (xt) under the critical excitation are presented in this section. For the fixed based structure, the maximum displacement (1.93 m) is reduced to 1.54 m and 1.52 m for HS and BA approaches, respectively. The time history plot and the transfer function plot for the fixed based 40-story structure are shown in Figs. 3 and 4, respectively. The optimum TMD is also effective on reduction of the maximum acceleration of the structure up to 21.39%. The time history plot for the dense soil is shown in Fig. 5. The reduction of maximum displacement is 21.52% for BA based method while the reduction is 19.71% for HS based method. Also, a significant reduction of transfer function value for the first resonance peak is observed in Fig. 6. The plots of the medium soil case are shown in Figs. 7 and 8 for the time and the frequency responses, respectively. The BA based method is slightly better than the HS based method as seen from the results (by 0.03 m additional reduction). The structural responses are high for soft soil comparing to other soil types. For this soil type, the optimum TMD is also more effective (25% for critical excitation and BA based method) than the other soil cases. The plots for the soft soil are presented in Figs. 9 and 10.

4.2. Results of 40-story building For the 40-story structure, the first natural frequency of the fixed based structure is 1.65 rad/s. The natural frequencies are 1.61 rad/s, 1.54 rad/s and 1.09 rad/s for dense, medium and soft soils, respectively. Since the frequencies of MDOF structure has great difference of

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G. Bekdaş, S.M. Nigdeli

5. Conclusions Two metaheuristic algorithms (HS and BA) were separately used in the development of the methodology for optimization of TMDs for structures considering soil interaction. The results of the two approaches were compared and several conclusions were done after the numerical analyses of SDOF and MDOF (40-strory building) main structure excited by earthquake excitation. The conclusions can be listed as follows. –

– The proposed methods were tested on SDOF main structures with different periods and damping ratios. For the structures with 2% damping ratio, the optimum TMD is more effective than the structure with 5% damping ratio. Similarly, TMDs are better in reduction of displacements of structures with short period. For example with respect to BA results, the reduction percentage of displacement is 34% for the SDOF structure (T=0.5 s and ξ=0.02) interacting with soft soil, while this value is only 11% for the structure with long period (t=2 s) and high damping (ξ=0.05). – According to the results of the SDOF main structures, a minor advantage of BA can be seen for the reduction of seismic responses, but both methods are feasible for optimization of TMDs. In several cases, HS shows some failures on finding the critical excitation, but it is the reason of using several records and these records may have near maximum effects on the structures. – In the study, the optimum results were found by using several earthquake records, but the final optimum results are only true optimum of a single component of a record. This component is the critical excitation for the example structures. In that case, the aim of the optimization is to reduce the maximum responses. When the results of the SDOF main structures are investigated, it is clearly seen that the critical excitation is different for all structures with different periods. In that case, the critical seismic excitation may change according to the characteristics of the structure. This situation is the reason and shows the importance of using several records during the optimization process. Since TMD have an effect on the modal response of the structure, the randomly assigned TMD parameters may change the characteristic of the structures and the critical excitation may change with respect to assigned parameters. In that case, critical excitation for TMD implemented structure always changes during the optimization process. – The effect of soil-structure interaction cannot be clearly seen for SDOF main structure. For MDOF structures, (especially for high rise structures) the rocking effect of soil can be effective on the structure. Thus, the optimum TMD parameters are very different in value for the soil types. – All maximum results of 40-story building for different records were presented in the Appendix A. As seen from the tables, the optimum TMD is not effective on several records and components, but these excitations are not critical ones and the huge gap of structural responses between these records and critical one can be clearly seen. For example, in Table 16 (fixed based structure and BA approach), the maximum displacement of the structure with TMD is 0.27 m (12.5% increase of the value comparing to structure without TMD) while the value of the critical one (1.52 m) is very high. 21.24% reduction is occurred for TMD implemented structure for this excitation. – The classical method of Den Hartog is also presented in the study. Since the method is not effective on controlling the stroke capacity of the TMD, it is not practical. Numerical algorithms including design constraints are needed in order to design a TMD for practical purposes. Additionally, the reductions of maximum displacements are far from the proposed methods because of the consideration of a single vibration mode. – The transfer function plots of the proposed methods are quite different with the classical method of Den Hartog. A mode splitting

–

–

–

is not observed from the frequency domain results. In the Den Hartog's method, the frequency ratio is 0.94 and the period of TMD is close to the fundamental period of structural system. Whereas, the optimum periods are very different for metaheuristic based methods. In that case, a mode splitting does not occur, but a significant reduction of the amplitude of the first resonance peak is observed. This situation is the results of not using a theoretical optimum value. In the proposed method, the consideration of the stroke capacity and earthquake records with random frequency effect the optimum results in the practical design of TMD. As seen from the time history responses of the 40-story structure, the optimum TMD is effective on obtaining a steady state response. In that case, the structural vibration are quickly damped. Thus, the comfort and safety of residents are provided. It is an important effect in additional to technical issues like reduction of displacements and internal forces. The non-structural components of structures like nuclear devices, huge shelfs and equipment, valuable devices and historical values must be protected because of possible damage of the components and hazards of these components. For the 40-story structure considering soil-structure interaction, the optimum TMDs are effective on reduction of structural displacements. For the dense soil, the structural displacements are reduced up to 40.26% and 38.93 for HS and BA approaches, respectively. But, BA has a little advantage on reduction of critical response (21.52% for the BA and 19.71% for HS). For medium soil, BA based method is also more effective than HS based method by 1.22% additional reduction of the critical response. For soft soil, both methods have nearly the same effects on the critical response with a slight advantage of BA. The reduction percentage of the critical response is nearly 25%. The optimization objective is about the maximum displacements of the structure, but the optimum TMD are also effective on reduction of accelerations. Generally, a significant reduction are not observed for the maximum accelerations occurred under several records, but BA based method is effective to reduce the critical acceleration by 21.52%, 22.34% and 22.20% for dense, medium and soft soils, respectively. The BA based method is slightly better than HS based method on reduction of maximum responses, but the major advantage of the BA based method can be seen from the stroke values. When the stroke values of the BA based method investigated, the limit value is obtained for the critical response. For that reason, BA employs an effective search and uses the allowed limit of stroke effectively. Additionally, the optimum mass is the same as the upper limit of the range of the mass. Thus, BA has also an effective search capacity for the limits.

According to the numerical investigations, the metaheuristic based methods are feasible in finding the optimum values of TMD for structures considering soil-structure interaction. In future studies, complex structural models with a detailed soil-structure interaction may be considered, but the computation time is a constraint for the application. In that case, hybrid methods combining several algorithms or new variants of the algorithms may be developed.

Acknowledgement This work was supported by Scientific Research Projects Coordination Unit of Istanbul University with Project no. 31688.

Appendix A See Tables A1–A8 here 452

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Table A1 Maximum responses under FEMA P-695 far-field ground motion records for fixed based 40-story structure (HS approach). Earthquake no.

Component

max. (x ̈ + xg̈ ) (m/s2)

max. (x) (m) Without TMD

With TMD

Reduction (%)

Without TMD

With TMD

Reduction (%)

g (X)

Stroke (m)

1

1 2

0.29 0.27

0.27 0.25

6.90 7.41

4.86 4.94

4.74 4.80

2.47 2.83

1.27 1.00

0.37 0.27

2

1 2

0.33 0.35

0.31 0.32

6.06 8.57

3.77 4.48

3.77 4.48

0.00 0.00

0.94 0.95

0.31 0.33

3

1 2

0.49 0.29

0.41 0.27

16.33 6.90

6.02 7.28

6.02 7.28

0.00 0.00

1.58 1.36

0.77 0.39

4

1 2

0.24 0.32

0.20 0.30

16.67 6.25

2.34 3.06

2.34 2.92

0.00 4.58

1.93 1.36

0.46 0.44

5

1 2

0.55 0.37

0.39 0.36

29.09 2.70

1.98 2.82

1.98 2.82

0.00 0.00

1.25 1.15

0.69 0.43

6

1 2

0.48 0.35

0.44 0.31

8.33 11.43

3.19 3.41

3.18 3.41

0.31 0.00

1.42 2.11

0.68 0.74

7

1 2

0.25 0.24

0.22 0.19

12.00 20.83

4.74 4.48

4.74 4.47

0.00 0.22

0.89 1.89

0.22 0.45

8

1 2

0.31 0.20

0.28 0.17

9.68 15.00

3.19 1.99

3.03 1.98

5.02 0.50

1.10 1.71

0.34 0.34

9

1 2

1.44 0.53

1.07 0.39

25.69 26.42

4.81 3.96

4.00 3.82

16.84 3.54

1.40 1.57

2.02 0.83

10

1 2

0.17 0.48

0.16 0.42

5.88 12.50

1.78 1.55

1.78 1.42

0.00 8.39

1.93 2.29

0.33 1.10

11

1 2

0.49 0.41

0.50 0.36

−2.04 12.20

3.41 2.14

3.33 2.07

2.35 3.27

2.07 1.88

1.01 0.77

12

1 2

0.24 0.22

0.20 0.22

16.67 0.00

2.59 4.37

2.59 4.15

0.00 5.03

2.12 1.55

0.51 0.34

13

1 2

0.19 0.19

0.19 0.14

0.00 26.32

4.52 3.49

4.52 3.49

0.00 0.00

1.12 1.24

0.21 0.24

14

1 2

0.29 0.40

0.24 0.39

17.24 2.50

4.27 2.99

4.27 3.00

0.00 −0.33

1.33 1.13

0.39 0.45

15

1 2

0.57 0.56

0.47 0.47

17.54 16.07

4.25 4.21

4.24 4.21

0.24 0.00

1.28 1.20

0.73 0.67

16

1 2

0.73 0.46

0.41 0.35

43.84 23.91

3.53 2.35

3.42 2.35

3.12 0.00

1.08 1.41

0.79 0.65

17

1 2

0.37 0.41

0.26 0.31

29.73 24.39

4.05 2.59

4.05 2.59

0.00 0.00

1.03 1.30

0.38 0.53

18

1 2

0.22 0.21

0.22 0.18

0.00 14.29

3.60 5.17

3.60 5.17

0.00 0.00

1.50 1.39

0.33 0.29

19

1 2

1.64 1.93

1.19 1.54

27.44 20.21

5.06 6.17

3.99 4.85

21.15 21.39

1.22 1.98

2.00 3.82

20

1 2

0.24 0.18

0.26 0.14

−8.33 22.22

3.94 4.37

3.94 4.37

0.00 0.00

2.82 1.70

0.68 0.31

21

1 2

0.59 0.24

0.43 0.17

27.12 29.17

2.45 1.35

2.28 1.35

6.94 0.00

1.31 1.35

0.77 0.32

22

1 2

0.10 0.14

0.09 0.12

10.00 14.29

2.79 2.98

2.79 2.98

0.00 0.00

1.43 1.61

0.14 0.23

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Table A2 Maximum responses under FEMA P-695 far-field ground motion records for fixed based 40-story structure (BA approach). Earthquake no.

Component

max. (x ̈ + xg̈ ) (m/s2)

max. (x) (m) Without TMD

With TMD

Reduction (%)

Without TMD

With TMD

Reduction (%)

g (X)

Stroke (m)

1

1 2

0.29 0.27

0.27 0.25

6.90 7.41

4.86 4.94

4.74 4.80

2.47 2.83

1.27 1.00

0.37 0.27

2

1 2

0.33 0.35

0.32 0.32

3.03 8.57

3.77 4.48

3.77 4.48

0.00 0.00

0.94 0.96

0.31 0.34

3

1 2

0.49 0.29

0.41 0.27

16.33 6.90

6.02 7.28

6.02 7.28

0.00 0.00

1.59 1.37

0.78 0.40

4

1 2

0.24 0.32

0.20 0.30

16.67 6.25

2.34 3.06

2.34 2.93

0.00 4.25

2.03 1.38

0.49 0.44

5

1 2

0.55 0.37

0.40 0.36

27.27 2.70

1.98 2.82

1.98 2.82

0.00 0.00

1.27 1.10

0.70 0.41

6

1 2

0.48 0.35

0.44 0.30

8.33 14.29

3.19 3.41

3.18 3.41

0.31 0.00

1.46 2.17

0.70 0.76

7

1 2

0.25 0.24

0.22 0.19

12.00 20.83

4.74 4.48

4.74 4.47

0.00 0.22

0.91 1.95

0.23 0.47

8

1 2

0.31 0.20

0.29 0.16

6.45 20.00

3.19 1.99

3.03 1.98

5.02 0.50

1.11 1.73

0.34 0.35

9

1 2

1.44 0.53

1.08 0.39

25.00 26.42

4.81 3.96

4.04 3.83

16.01 3.28

1.43 1.57

2.06 0.83

10

1 2

0.17 0.48

0.16 0.41

5.88 14.58

1.78 1.55

1.78 1.43

0.00 7.74

2.01 2.36

0.34 1.13

11

1 2

0.49 0.41

0.50 0.36

−2.04 12.20

3.41 2.14

3.33 2.07

2.35 3.27

2.15 1.91

1.05 0.78

12

1 2

0.24 0.22

0.19 0.22

20.83 0.00

2.59 4.37

2.59 4.16

0.00 4.81

2.15 1.59

0.52 0.35

13

1 2

0.19 0.19

0.19 0.14

0.00 26.32

4.52 3.49

4.52 3.49

0.00 0.00

1.14 1.23

0.22 0.23

14

1 2

0.29 0.40

0.24 0.39

17.24 2.50

4.27 2.99

4.27 3.00

0.00 −0.33

1.33 1.14

0.39 0.46

15

1 2

0.57 0.56

0.48 0.47

15.79 16.07

4.25 4.21

4.24 4.21

0.24 0.00

1.28 1.19

0.73 0.67

16

1 2

0.73 0.46

0.42 0.35

42.47 23.91

3.53 2.35

3.42 2.35

3.12 0.00

1.11 1.39

0.81 0.64

17

1 2

0.37 0.41

0.27 0.31

27.03 24.39

4.05 2.59

4.05 2.59

0.00 0.00

1.03 1.29

0.38 0.53

18

1 2

0.22 0.21

0.22 0.18

0.00 14.29

3.60 5.17

3.60 5.17

0.00 0.00

1.52 1.45

0.33 0.30

19

1 2

1.64 1.93

1.20 1.52

26.83 21.24

5.06 6.17

4.05 4.90

19.96 20.58

1.23 2.00

2.02 3.86

20

1 2

0.24 0.18

0.27 0.14

−12.50 22.22

3.94 4.37

3.94 4.37

0.00 0.00

2.99 1.79

0.72 0.32

21

1 2

0.59 0.24

0.43 0.17

27.12 29.17

2.45 1.35

2.28 1.35

6.94 0.00

1.29 1.36

0.76 0.33

22

1 2

0.10 0.14

0.09 0.12

10.00 14.29

2.79 2.98

2.79 2.98

0.00 0.00

1.44 1.58

0.14 0.22

454

Soil Dynamics and Earthquake Engineering 92 (2017) 443–461

G. Bekdaş, S.M. Nigdeli

Table A3 Maximum responses under FEMA P-695 far-field ground motion records for the 40-story structure on the dense soil (HS approach). Earthquake no.

Component

max. (x ̈ + xg̈ ) (m/s2)

max. (x) (m) Without TMD

With TMD

Reduction (%)

Without TMD

With TMD

Reduction (%)

g (X)

Stroke (m)

1

1 2

0.28 0.28

0.28 0.25

0.00 10.71

4.88 4.92

4.75 4.77

2.66 3.05

1.34 0.97

0.36 0.26

2

1 2

0.33 0.35

0.32 0.32

3.03 8.57

3.78 4.50

3.78 4.50

0.00 0.00

0.94 1.01

0.30 0.33

3

1 2

0.49 0.29

0.39 0.26

20.41 10.34

6.06 7.31

6.06 7.31

0.00 0.00

1.71 1.43

0.79 0.39

4

1 2

0.24 0.34

0.21 0.31

12.50 8.82

2.35 3.08

2.36 2.93

−0.43 4.87

2.07 1.38

0.46 0.44

5

1 2

0.43 0.34

0.38 0.34

11.63 0.00

1.98 2.84

1.98 2.84

0.00 0.00

1.60 1.15

0.66 0.77

6

1 2

0.47 0.36

0.42 0.32

10.64 11.11

3.21 3.42

3.21 3.43

0.00 −0.29

1.49 2.18

0.66 0.73

7

1 2

0.26 0.26

0.22 0.20

15.38 23.08

4.76 4.51

4.76 4.50

0.00 0.22

0.86 1.90

0.22 0.47

8

1 2

0.30 0.23

0.27 0.17

10.00 26.09

3.14 1.98

2.98 1.99

5.10 −0.51

1.16 1.55

0.33 0.34

9

1 2

1.39 0.55

1.09 0.40

21.58 27.27

4.57 3.95

3.90 3.81

14.66 3.54

1.49 1.59

1.96 0.83

10

1 2

0.16 0.53

0.15 0.47

6.25 11.32

1.79 1.55

1.80 1.43

−0.56 7.74

2.10 2.27

0.31 1.14

11

1 2

0.54 0.45

0.52 0.38

3.70 15.56

3.46 2.19

3.36 2.07

2.89 5.48

1.96 1.84

1.00 0.78

12

1 2

0.27 0.24

0.21 0.24

22.22 0.00

2.60 4.37

2.60 4.14

0.00 5.26

2.06 1.53

0.52 0.35

13

1 2

0.19 0.21

0.18 0.15

5.26 28.57

4.55 3.50

4.55 3.50

0.00 0.00

1.18 1.19

0.21 0.24

14

1 2

0.28 0.40

0.25 0.39

10.71 2.50

4.29 3.01

4.29 3.02

0.00 −0.33

1.39 1.14

0.36 0.44

15

1 2

0.58 0.60

0.49 0.47

15.52 21.67

4.28 4.23

4.28 4.23

0.00 0.00

1.26 1.17

0.70 0.67

16

1 2

0.77 0.49

0.46 0.36

40.26 26.53

3.50 2.36

3.39 2.36

3.14 0.00

1.11 1.35

0.80 0.62

17

1 2

0.35 0.40

0.25 0.31

28.57 22.50

4.06 2.61

4.06 2.60

0.00 0.38

1.11 1.36

0.37 0.52

18

1 2

0.22 0.22

0.22 0.18

0.00 18.18

3.61 5.18

3.61 5.18

0.00 0.00

1.55 1.47

0.33 0.30

19

1 2

1.58 2.08

1.18 1.67

25.32 19.71

4.88 6.12

3.90 4.75

20.08 22.39

1.29 1.95

1.93 3.83

20

1 2

0.26 0.18

0.28 0.15

−7.69 16.67

3.95 4.38

3.94 4.38

0.25 0.00

2.90 1.82

0.71 0.32

21

1 2

0.62 0.27

0.46 0.19

25.81 29.63

2.44 1.36

2.26 1.36

7.38 0.00

1.26 1.28

0.74 0.32

22

1 2

0.10 0.16

0.09 0.13

10.00 18.75

2.82 2.98

2.82 2.98

0.00 0.00

1.43 1.45

0.14 0.22

455

Soil Dynamics and Earthquake Engineering 92 (2017) 443–461

G. Bekdaş, S.M. Nigdeli

Table A4 Maximum responses under FEMA P-695 far-field ground motion records for the 40-story structure on the dense soil (BA approach). Earthquake no.

Component

max. (x ̈ + xg̈ ) (m/s2)

max. (x) (m) Without TMD

With TMD

Reduction (%)

Without TMD

With TMD

Reduction (%)

g (X)

Stroke (m)

1

1 2

0.28 0.28

0.28 0.25

2.86 10.19

4.88 4.92

4.75 4.79

2.60 2.73

1.35 0.97

0.37 0.26

2

1 2

0.33 0.35

0.32 0.32

4.15 7.35

3.78 4.50

3.78 4.50

0.01 −0.01

0.94 1.04

0.30 0.34

3

1 2

0.49 0.29

0.40 0.26

18.53 8.32

6.06 7.31

6.06 7.31

−0.01 −0.01

1.74 1.47

0.80 0.40

4

1 2

0.24 0.34

0.22 0.31

8.92 8.76

2.35 3.08

2.36 2.94

−0.09 4.64

2.21 1.42

0.49 0.45

5

1 2

0.52 0.48

0.38 0.34

26.20 28.85

1.98 2.84

1.98 2.84

−0.01 0.04

1.38 1.73

0.68 0.78

6

1 2

0.47 0.36

0.43 0.32

8.81 10.21

3.21 3.42

3.21 3.43

0.21 −0.09

1.55 2.27

0.69 0.76

7

1 2

0.26 0.26

0.22 0.20

16.31 23.99

4.76 4.51

4.76 4.50

0.03 0.07

0.88 1.98

0.22 0.49

8

1 2

0.30 0.23

0.28 0.17

8.22 27.02

3.14 1.98

2.99 1.99

4.68 −0.06

1.18 1.59

0.34 0.34

9

1 2

1.39 0.55

1.10 0.40

20.24 28.13

4.57 3.95

3.95 3.82

13.42 3.28

1.56 1.62

2.04 0.85

10

1 2

0.16 0.53

0.15 0.46

3.36 13.64

1.79 1.55

1.80 1.43

−0.09 7.78

2.22 2.38

0.33 1.20

11

1 2

0.54 0.45

0.52 0.37

3.30 16.44

3.46 2.19

3.36 2.08

2.79 5.11

2.07 1.92

1.05 0.81

12

1 2

0.27 0.24

0.20 0.23

22.96 2.08

2.60 4.37

2.60 4.15

−0.09 4.94

2.19 1.59

0.55 0.36

13

1 2

0.19 0.21

0.18 0.15

1.16 27.15

4.55 3.50

4.55 3.50

0.01 0.05

1.21 1.20

0.22 0.24

14

1 2

0.28 0.40

0.25 0.39

9.70 2.86

4.29 3.01

4.29 3.02

0.00 −0.29

1.41 1.16

0.37 0.45

15

1 2

0.58 0.60

0.49 0.48

16.22 20.50

4.28 4.23

4.28 4.23

0.07 0.10

1.29 1.17

0.71 0.67

16

1 2

0.77 0.49

0.47 0.35

38.93 27.23

3.50 2.36

3.40 2.36

3.06 0.18

1.16 1.34

0.84 0.62

17

1 2

0.35 0.40

0.25 0.31

27.22 22.81

4.06 2.61

4.06 2.60

0.00 0.29

1.13 1.38

0.37 0.53

18

1 2

0.22 0.22

0.22 0.18

0.59 14.89

3.61 5.18

3.61 5.18

0.02 0.01

1.61 1.55

0.34 0.32

19

1 2

1.58 2.08

1.20 1.63

24.16 21.52

4.88 6.12

3.96 4.80

18.85 21.52

1.34 1.99

2.01 3.93

20

1 2

0.26 0.25

0.28 0.15

−8.92 41.92

3.95 4.38

3.95 4.38

0.09 −0.02

3.14 1.42

0.77 0.34

21

1 2

0.62 0.27

0.46 0.19

25.87 29.48

2.44 1.36

2.28 1.36

6.79 −0.02

1.26 1.30

0.74 0.33

22

1 2

0.10 0.16

0.09 0.13

11.48 19.23

2.82 2.98

2.82 2.98

0.00 0.01

1.45 1.44

0.14 0.22

456

Soil Dynamics and Earthquake Engineering 92 (2017) 443–461

G. Bekdaş, S.M. Nigdeli

Table A5 Maximum responses under FEMA P-695 far-field ground motion records for the 40-story structure on the medium soil (HS approach). Earthquake no.

Component

max. ( x ̈ + xg̈ ) (m/s2)

max. (x) (m) Without TMD

With TMD

Without TMD

With TMD

g (X)

Stroke (m)

1

1 2

0.28 0.29

0.28 0.26

0.00 10.34

4.91 4.89

4.77 4.75

2.85 2.86

1.39 0.95

0.35 0.25

2

1 2

0.33 0.34

0.32 0.32

3.03 5.88

3.77 4.49

3.77 4.49

0.00 0.00

0.94 1.19

0.27 0.36

3

1 2

0.63 0.32

0.44 0.27

30.16 15.63

6.01 7.28

6.01 7.28

0.00 0.00

1.55 1.44

0.84 0.40

4

1 2

0.23 0.35

0.23 0.32

0.00 8.57

2.34 3.11

2.34 2.95

0.00 5.14

2.46 1.49

0.50 0.45

5

1 2

0.38 0.39

0.33 0.32

13.16 17.95

1.98 2.82

1.98 2.82

0.00 0.00

1.79 1.01

0.60 0.89

6

1 2

0.48 0.38

0.4 0.34

16.67 10.53

3.2 3.41

3.2 3.41

0.00 0.00

1.57 2.31

0.65 0.76

7

1 2

0.26 0.33

0.22 0.24

15.38 27.27

4.75 4.49

4.75 4.49

0.00 0.00

1.02 1.78

0.23 0.51

8

1 2

0.3 0.3

0.26 0.2

13.33 33.33

3.05 1.98

2.92 1.98

4.26 0.00

1.21 1.31

0.32 0.34

9

1 2

1.33 0.58

1.1 0.41

17.29 29.31

3.97 3.93

3.63 3.8

8.56 3.31

1.70 1.74

1.94 0.88

10

1 2

0.17 0.6

0.15 0.55

11.76 8.33

1.78 1.56

1.78 1.43

0.00 8.33

2.24 2.40

0.33 1.25

11

1 2

0.61 0.52

0.55 0.41

9.84 21.15

3.55 2.28

3.42 2.11

3.66 7.46

1.95 1.87

1.04 0.85

12

1 2

0.3 0.27

0.23 0.26

23.33 3.70

2.59 4.35

2.6 4.13

−0.39 5.06

2.23 1.63

0.59 0.39

13

1 2

0.19 0.21

0.18 0.17

5.26 19.05

4.52 3.5

4.52 3.5

0.00 0.00

1.22 1.19

0.22 0.23

14

1 2

0.3 0.41

0.27 0.4

10.00 2.44

4.28 2.97

4.28 2.98

0.00 −0.34

1.32 1.17

0.34 0.42

15

1 2

0.59 0.66

0.5 0.49

15.25 25.76

4.24 4.23

4.24 4.22

0.00 0.24

1.35 1.16

0.70 0.67

16

1 2

0.63 0.6

0.47 0.4

25.40 33.33

3.47 2.37

3.35 2.36

3.46 0.42

1.45 1.10

0.79 0.57

17

1 2

0.33 0.38

0.25 0.32

24.24 15.79

4.05 2.61

4.05 2.6

0.00 0.38

1.13 1.54

0.33 0.52

18

1 2

0.23 0.23

0.22 0.19

4.35 17.39

3.61 5.18

3.61 5.18

0.00 0.00

1.67 1.70

0.35 0.34

19

1 2

1.51 2.38

1.16 1.87

23.18 21.43

4.48 5.86

3.81 4.58

14.96 21.84

1.49 1.94

1.96 3.99

20

1 2

0.29 0.18

0.3 0.15

−3.45 16.67

3.95 4.37

3.94 4.37

0.25 0.00

3.25 2.20

0.81 0.35

21

1 2

0.65 0.3

0.49 0.21

24.62 30.00

2.41 1.34

2.25 1.35

6.64 −0.75

1.26 1.34

0.71 0.35

22

1 2

0.1 0.18

0.09 0.15

10.00 16.67

2.8 2.98

2.8 2.98

0.00 0.01

1.61 1.24

0.13 0.20

457

Soil Dynamics and Earthquake Engineering 92 (2017) 443–461

G. Bekdaş, S.M. Nigdeli

Table A6 Maximum responses under FEMA P-695 far-field ground motion records for the 40-story structure on the medium soil (BA approach). Earthquake no.

Component

max. ( x ̈ + xg̈ ) (m/s2)

max. (x) (m) Without TMD

With TMD

Without TMD

With TMD

g (X)

Stroke (m)

1

1 2

0.28 0.29

0.28 0.26

0.02 11.22

4.91 4.89

4.78 4.76

2.77 2.57

1.41 0.97

0.36 0.25

2

1 2

0.33 0.34

0.32 0.32

4.15 6.71

3.77 4.49

3.77 4.49

0.01 −0.01

0.95 1.23

0.28 0.37

3

1 2

0.63 0.32

0.43 0.27

30.76 16.29

6.01 7.28

6.01 7.28

−0.02 −0.01

1.60 1.47

0.87 0.41

4

1 2

0.23 0.35

0.23 0.32

0.53 6.74

2.34 3.11

2.34 2.96

−0.10 4.77

2.55 1.53

0.51 0.46

5

1 2

0.52 0.69

0.34 0.42

35.31 38.83

1.98 2.82

1.98 2.82

−0.01 0.04

1.38 1.51

0.62 0.90

6

1 2

0.48 0.38

0.40 0.34

16.66 11.14

3.20 3.41

3.20 3.41

0.21 −0.09

1.62 2.38

0.68 0.78

7

1 2

0.26 0.33

0.22 0.24

13.66 27.16

4.75 4.49

4.75 4.49

0.03 0.07

1.04 1.84

0.24 0.53

8

1 2

0.30 0.30

0.26 0.20

15.28 34.24

3.05 1.98

2.92 1.98

4.16 −0.05

1.22 1.33

0.32 0.34

9

1 2

1.33 0.58

1.10 0.41

16.98 29.92

3.97 3.93

3.65 3.80

7.99 3.17

1.76 1.80

2.02 0.91

10

1 2

0.17 0.60

0.15 0.54

9.11 10.32

1.78 1.56

1.78 1.43

−0.08 7.93

2.33 2.48

0.34 1.29

11

1 2

0.61 0.52

0.55 0.41

9.72 21.69

3.55 2.28

3.42 2.11

3.59 7.34

2.02 1.94

1.08 0.88

12

1 2

0.30 0.27

0.23 0.26

25.63 6.06

2.59 4.35

2.60 4.14

−0.08 4.72

2.31 1.69

0.61 0.41

13

1 2

0.19 0.21

0.18 0.17

7.07 22.62

4.52 3.50

4.52 3.50

0.00 0.07

1.24 1.20

0.23 0.23

14

1 2

0.30 0.41

0.27 0.40

10.49 3.21

4.28 2.97

4.28 2.98

0.00 −0.30

1.34 1.20

0.35 0.43

15

1 2

0.59 0.66

0.50 0.49

15.73 26.23

4.24 4.23

4.24 4.22

0.09 0.08

1.40 1.17

0.72 0.68

16

1 2

0.63 0.60

0.48 0.40

24.28 33.11

3.47 2.37

3.36 2.36

3.19 0.18

1.51 1.11

0.83 0.58

17

1 2

0.33 0.38

0.25 0.32

25.31 16.64

4.05 2.61

4.05 2.60

0.00 0.38

1.15 1.57

0.33 0.53

18

1 2

0.23 0.23

0.22 0.19

5.82 16.62

3.61 5.18

3.61 5.18

0.02 0.01

1.72 1.75

0.36 0.35

19

1 2

1.51 2.38

1.16 1.84

23.58 22.65

4.48 5.86

3.81 4.55

15.05 22.34

1.53 2.00

2.02 4.10

20

1 2

0.29 0.23

0.31 0.16

−6.68 31.16

3.95 4.37

3.94 4.37

0.07 −0.02

3.39 1.85

0.84 0.36

21

1 2

0.65 0.30

0.49 0.21

24.73 30.11

2.41 1.34

2.26 1.35

6.37 −0.02

1.29 1.37

0.73 0.36

22

1 2

0.10 0.18

0.09 0.14

3.02 19.61

2.80 2.98

2.80 2.98

0.00 0.01

1.64 1.26

0.14 0.20

458

Soil Dynamics and Earthquake Engineering 92 (2017) 443–461

G. Bekdaş, S.M. Nigdeli

Table A7 Maximum responses under FEMA P-695 far-field ground motion records for the 40-story structure on the soft soil (HS approach). Earthquake no.

Component

max. ( x ̈ + xg̈ ) (m/s2)

max. (x) (m) Without TMD

With TMD

Without TMD

With TMD

g (X)

Stroke (m)

1

1 2

0.32 0.29

0.30 0.27

6.25 6.90

4.83 4.73

4.64 4.57

3.93 3.38

1.45 1.38

0.26 0.23

2

1 2

0.28 0.26

0.26 0.23

7.14 11.54

3.46 3.91

3.46 3.91

0.00 0.00

1.41 1.77

0.19 0.22

3

1 2

0.78 0.28

0.69 0.29

11.54 −3.57

5.06 6.61

5.07 6.61

−0.20 0.00

2.19 2.09

0.72 0.30

4

1 2

0.51 0.49

0.51 0.37

0.00 24.49

2.09 2.91

2.09 2.72

0.00 6.53

2.02 1.45

0.44 0.30

5

1 2

0.51 0.46

0.33 0.48

35.29 −4.35

1.82 2.88

1.82 2.59

0.00 10.07

1.59 2.01

0.65 0.68

6

1 2

0.55 0.89

0.44 0.79

20.00 11.24

2.87 3.04

2.86 3.04

0.35 0.00

1.93 1.86

0.43 0.68

7

1 2

0.26 0.52

0.24 0.40

7.69 23.08

4.31 4.04

4.31 4.03

0.00 0.25

1.94 2.05

0.21 0.45

8

1 2

0.20 0.33

0.19 0.25

5.00 24.24

2.56 1.95

2.46 1.91

3.91 2.05

2.64 1.82

0.22 0.25

9

1 2

1.57 1.09

1.48 0.74

5.73 32.11

2.81 3.72

2.80 3.58

0.36 3.76

2.25 1.67

1.45 0.75

10

1 2

0.31 1.24

0.30 1.06

3.23 14.52

1.52 1.71

1.52 1.31

0.00 23.39

2.08 2.04

0.27 1.04

11

1 2

1.34 0.85

1.23 0.67

8.21 21.18

3.83 2.12

3.56 1.90

7.05 10.38

2.09 2.07

1.17 0.75

12

1 2

0.64 0.59

0.56 0.54

12.50 8.47

2.32 3.81

2.32 3.78

0.00 0.79

1.97 1.87

0.54 0.45

13

1 2

0.26 0.20

0.24 0.16

7.69 20.00

4.03 3.30

3.93 3.30

2.48 0.00

1.54 1.42

0.25 0.15

14

1 2

0.31 0.39

0.22 0.36

29.03 7.69

3.84 2.63

3.84 2.64

0.00 −0.38

1.58 1.35

0.21 0.26

15

1 2

0.86 0.89

0.68 0.82

20.93 7.87

3.74 3.94

3.73 3.93

0.27 0.25

1.83 1.66

0.66 0.65

16

1 2

0.41 0.75

0.41 0.73

0.00 2.67

3.06 2.21

2.97 2.21

2.94 0.00

2.50 2.22

0.42 0.68

17

1 2

0.23 0.49

0.22 0.40

4.35 18.37

3.68 2.30

3.68 2.32

0.00 −0.87

1.23 1.62

0.15 0.35

18

1 2

0.43 0.35

0.42 0.32

2.33 8.57

3.43 4.68

3.42 4.68

0.29 0.00

1.99 1.92

0.40 0.30

19

1 2

1.74 3.88

1.40 2.92

19.54 24.74

3.02 5.54

2.83 4.32

6.29 22.02

2.11 1.98

1.52 3.18

20

1 2

0.98 0.53

1.00 0.47

−2.04 11.32

3.55 4.02

3.54 4.01

0.28 0.25

2.09 1.92

0.85 0.42

21

1 2

0.39 0.30

0.33 0.19

15.38 36.67

2.07 1.16

1.94 1.16

6.28 0.00

2.44 1.84

0.42 0.23

22

1 2

0.10 0.14

0.09 0.13

10.00 7.14

2.48 2.78

2.48 2.78

0.00 0.00

1.46 2.02

0.08 0.13

459

Soil Dynamics and Earthquake Engineering 92 (2017) 443–461

G. Bekdaş, S.M. Nigdeli

Table A8 Maximum responses under FEMA P-695 far-field ground motion records for the 40-story structure on the soft soil (BA approach). Earthquake no.

Component

max. ( x ̈ + xg̈ ) (m/s2)

max. (x) (m) Without TMD

With TMD

Without TMD

With TMD

g (X)

Stroke (m)

1

1 2

0.32 0.29

0.30 0.27

6.25 6.90

4.83 4.73

4.61 4.54

4.55 4.02

1.35 1.30

0.24 0.21

2

1 2

0.28 0.26

0.26 0.23

7.14 11.54

3.46 3.91

3.46 3.91

0.00 0.00

1.36 1.69

0.18 0.21

3

1 2

0.78 0.28

0.69 0.28

11.54 0.00

5.06 6.61

5.07 6.61

−0.20 0.00

2.21 1.89

0.72 0.27

4

1 2

0.51 0.49

0.50 0.37

1.96 24.49

2.09 2.91

2.09 2.70

0.00 7.22

2.03 1.57

0.44 0.33

5

1 2

0.58 0.70

0.59 0.63

−1.72 10.00

1.82 2.88

1.82 2.59

0.00 10.07

2.80 2.32

0.66 0.66

6

1 2

0.55 0.89

0.41 0.77

25.45 13.48

2.87 3.04

2.86 3.04

0.35 0.00

2.08 2.02

0.46 0.74

7

1 2

0.26 0.52

0.24 0.37

7.69 28.85

4.31 4.04

4.31 4.03

0.00 0.25

1.87 1.98

0.20 0.43

8

1 2

0.20 0.33

0.19 0.25

5.00 24.24

2.56 1.95

2.44 1.90

4.69 2.56

2.40 1.83

0.20 0.26

9

1 2

1.57 1.09

1.34 0.71

14.65 34.86

2.81 3.72

2.80 3.55

0.36 4.57

2.20 1.68

1.42 0.76

10

1 2

0.31 1.24

0.30 1.06

3.23 14.52

1.52 1.71

1.52 1.33

0.00 22.22

2.17 2.08

0.28 1.06

11

1 2

1.34 0.85

1.18 0.68

11.94 20.00

3.83 2.12

3.54 1.88

7.57 11.32

2.17 2.00

1.21 0.72

12

1 2

0.64 0.59

0.56 0.52

12.50 11.86

2.32 3.81

2.32 3.78

0.00 0.79

1.93 2.07

0.53 0.50

13

1 2

0.26 0.20

0.23 0.17

11.54 15.00

4.03 3.30

3.93 3.30

2.48 0.00

1.51 1.38

0.25 0.14

14

1 2

0.31 0.39

0.20 0.36

35.48 7.69

3.84 2.63

3.84 2.63

0.00 0.00

1.66 1.34

0.22 0.25

15

1 2

0.86 0.89

0.60 0.80

30.23 10.11

3.74 3.94

3.73 3.93

0.27 0.25

1.80 1.72

0.65 0.67

16

1 2

0.41 0.75

0.42 0.71

−2.44 5.33

3.06 2.21

2.93 2.21

4.25 0.00

2.57 2.32

0.43 0.72

17

1 2

0.23 0.49

0.22 0.37

4.35 24.49

3.68 2.30

3.68 2.32

0.00 −0.87

1.18 1.56

0.14 0.34

18

1 2

0.43 0.35

0.41 0.32

4.65 8.57

3.43 4.68

3.42 4.68

0.29 0.00

2.10 1.95

0.42 0.30

19

1 2

1.74 3.88

1.38 2.91

20.69 25.00

3.02 5.54

2.82 4.31

6.62 22.20

1.96 1.98

1.41 3.17

20

1 2

0.98 0.53

0.99 0.45

−1.02 15.09

3.55 4.02

3.54 4.01

0.28 0.25

2.23 1.93

0.90 0.42

21

1 2

0.39 0.30

0.33 0.19

15.38 36.67

2.07 1.16

1.91 1.16

7.73 0.00

2.25 1.81

0.39 0.23

22

1 2

0.10 0.14

0.09 0.13

10.00 7.14

2.48 2.78

2.48 2.78

0.00 0.00

1.35 2.05

0.07 0.13

460

Soil Dynamics and Earthquake Engineering 92 (2017) 443–461

G. Bekdaş, S.M. Nigdeli

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