^{1}

^{2}

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^{2}

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Edited by: Dario De Domenico, University of Messina, Italy

Reviewed by: Diego Lopez-Garcia, Pontifical Catholic University of Chile, Chile; Michele Palermo, University of Bologna, Italy

This article was submitted to Earthquake Engineering, a section of the journal Frontiers in Built Environment

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Viscous dampers (VDs) are effective and widely used passive devices for the protection of civil structures, provided that appropriate design is carried out. For this purpose, optimal design and optimum distribution of VDs method are presented for a shear building under the critical excitation by using random vibration theory in the frequency domain. In the optimization, by using Differential Evolution (DE) algorithm and the top floor displacement are evaluated as objective functions taking into consideration upper and lower limits of VDs damping coefficients, so that optimal damper placement and properties of the shear building can be determined. In this design, the VDs-shear building system is tested under the three different ground motions being compared to some methods in the literature and uniformly distributed VDs placed at each story. It is shown that the results of the study are both compatible and very successful in reducing the response of the structure under the different ground motions.

In traditionally designed building structures, reduction of enormous vibration energy is inadequate because of their very limited energy absorption capacity. Therefore, usage of passive, semi-active or active energy dissipation systems has more and more come into prominence in civil engineering structures. Even if limited damping about 40% capacity of damping quantity is applied, owing to the physical and manufactural difficulties, optimally designed and placed viscous dampers (VDs) significantly decrease the response of building structures. Two endpoints of a VD are attached to two subsequent floors. Because of the relative velocity between these two floors, VDs produce a damping force which is proportional to damping coefficient and relative velocities.

The optimal design concept of VD has been widely studied in the literature (Constantinou and Tadjbakhsh,

In practice, linear VDs, which develop forces that are a linear function of (i.e., proportional to) the relative velocity between their ends can be manufactured and applied to structure in practice. However, in order to attain flexibility in design, non-linear type VDs widely manufactured and used in many application (Hart and Wong,

Even though there has been a continuous development of science and technology, prediction and behavior of ground motion is still a difficult issue. Owing to this random excitation, the response of building structure is also random. For this reason, this randomness should be considered during structural design or structural-control system design. In this study, therefore, stationary random process and probabilistic critical excitation method (Takewaki,

Consider N degree of freedom shear frame model with VDs. _{adi} which is shown ^{th} story while stationary random seismic ground acceleration having zero mean is represented by ẍ_{g}. The equation of motion of structure with added VDs system could be expressed as bellow

in which _{s} represent _{ad} is the

Structural damping determination in the structure includes many factors, accordingly, it cannot be easily defined. For simplicity, structural damping matrix _{s} can be evaluated as mass proportional which is _{s} = α_{s}, stiffness proportional which is _{s} = β_{s} or a linear combination of mass and stiffness (Rayleigh damping) which is _{s} = α_{s}+β_{s}. In here α and β are calculated in terms of the first and second normal mode of vibration. α and β coefficients can be calculated with the following equation.

Where ω_{s1}, ω_{s2} first and second mode natural frequencies of the shear building, ξ_{s1} and ξ_{s2} are damping ratios with respect to first and second mode.

Building model with added VDs.

Fourier Transformation of Equation (1) is written as

Where

in which

_{D}(ω) follow as

The matrix

If _{δ}(ω) is defined as interstorey drift transfer function, _{δ}(ω) as follows

where _{δ}(ω) denotes the transfer function of the interstorey drifts. _{δ}(ω) is given as

As similar to the previous Equation (10), absolute acceleration in the frequency domain can be expressed as

where, _{AA}(ω) is the absolute acceleration transfer function. It can be expressed as:

By using the random vibration theory, the mean square of displacement i^{th} floor can be defined as

where, _{g}(ω) is Power Spectral Density (PSD) function of seismic input, |_{Di}(ω)| and |_{AAi}(ω)| are the transfer function amplitude of i^{th} displacement and absolute acceleration, respectively. In these equations, ()^{*} represents the complex conjugate. The above formulas have also been used by Takewaki (

The general concept of probabilistic critical excitation method is explained herein (Takewaki,

In the inequality equation, PSD function _{g}(ω) can be explained as input variance and _{g}(ω) is defined as Takewaki (

where _{g}(ω). Taking into consideration of ground motion records, _{g}(ω) can be evaluated as Dirac delta function. It is well-known that Dirac delta function takes extremely high values in very small intervals. When _{g}(ω) is evaluated as a band-limited white-noise. The frequency interval Ω can be defined according to _{u} and ω_{L} can be explained with respect to this rate (Takewaki,

Using the random vibration theory, in order to obtain objective functions

The passive and active constraints are defined with respect to upper and lower limits and total damping as follow

in which, ^{th} added VD and total limit of added VDs, respectively.

Although direct search methods spend relatively more computation time, their tolerance with respect to noise is more robust (Champion and Strzebonski,

In the case of ^{th} generation consideration, i^{th} target vector

In this equation, NP means the number of populations and the dimension of the problem is symbolized by

In the population matrices, initialization at the first generation is set equal to zero (g = 0). The population matrix in zero generation is defined with respect to the j^{th} component of i^{th} vector as following

In this equation, _{i, j}(0, 1) is randomly chosen from uniformly distributed numbers which are between 0 and 1. In the equation above, _{j, Low} and _{j, up} are the lower and upper limits of j^{th} component which are mentioned as above for DE. Each step can be determined as follows

Random target vectors

Crossover is exposed to donor vector in order to obtain the test vector

In this equation, _{rand} is chosen as random integer number which is between [1-_{r} is the crossover rate which is in between [0,1].

Either target vector

If the optimum solution is obtained, the operation is ended. Otherwise, it is returned to the mutation step to compose the new generation. The steps explained above are shown as a flow chart in _{1},…,x_{n}, represented as vectors of real numbers (“genes”). Every iteration, each x_{i} chooses random integers a, b, and c and constructs the mate y_{i} = x_{i}+γ(x_{a}+(x_{b}−x_{c})), where γ is the value of Scaling Factor. Then x_{i} is mated with y_{i} according to the value of Cross Probability, giving us the child z_{i}. At this point, x_{i} competes against z_{i} for the position of x_{i} in the population. Search Points is Min [10^{*}d, 50], where d is the number of variables. Differential Evolution is quite robust but generally slower than other methods due to the relatively large set of points it maintains.

Differential Evaluation Flowchart.

The optimum design and optimal distribution of VDs is executed for a sample 6 story shear building model which is shown in

Six story shear building model.

In the 6-story sample model, structural damping of the structure is supposed to be Rayleigh damping. First and second mode damping ratios ξ_{1} and ξ_{2} are set equal to 0.02, each floor mass and each story stiffness are taken as _{si} = {3.48, 10.24, 16.40, 21.61, 25.56, 28.03}rad/s. Structural damping is evaluated as Rayleigh damping. It is given as

The upper limit of each VD is taken as

In the example, in order to attain the critical excitation, power limit of power spectral density and amplitude limit are specified as _{L1} = 1.38rad/s and ω_{u1} = 5.58rad/s, and for the second the mode as ω_{L2} = 8.13 rad/s, and ω_{u2} = 12.33 rad/s. First and second modes are considered for design. The PSD is taken as

The variation of objective function _{1}) during the optimization via differential evolution (DE) method under the specified constraints are depicted according to design step numbers in _{1} minimization, allocations focus on the first four floors on the basis of the _{2} minimization. If the proposed method by Aydin (

Variation of objective function _{1} with respect to design step numbers.

Properties of VDs parameters and their locations to the floors.

^{6} Ns/m) |
_{1}) |
|||
---|---|---|---|---|

_{ad6} |
0 | 0 | 0 | 1.20823 |

_{ad5} |
0 | 0 | 0 | 1.20823 |

_{ad4} |
0 | 0 | 0 | 1.20823 |

_{ad3} |
0.08657 | 0 | 0 | 1.20823 |

_{ad2} |
3.004 | 3.3004 | 1.2494 | 1.20823 |

_{ad1} |
4.1588 | 3.949 | 6.000 | 1.20823 |

Total damper | 7.2494 | 7.2494 | 7.2494 | 7.2494 |

While the minimization of acceleration may be important for the elastic structures, the minimization of deformation may be important for the inelastic structures. Therefore, the objective functions which are needed are chosen by engineers considering objectives. In this study, only one objective function is evaluated. In addition to this, three different ground motions are considered for the time history analysis. These ground motions are El Centro (NS), Cape Mendocino (Petrolia NS), and Kobe (NS) ground motions, respectively. Taking into consideration of damage in the structure, whereas the response of the inelastic structure is usually important for the displacements (or deformations), the acceleration or stress response are more important for the non-structural elements and non-structural components in the structures (Viti et al.,

The variation of the transfer function amplitude of top displacement |H_{D6}(ω)|, the transfer function amplitude of top absolute acceleration |H_{AA6}(ω)| and the transfer function amplitude of first interstorey drift |H_{δ1}(ω)| are shown in

Frequency responses of 6th floor with respect to the transfer function amplitude of top displacement |_{D6}| _{AA6}| _{δ1}| _{1}, the methods of Aydin (

Comparison of the results of proposed methods with the other methods for different ground motions.

_{1}) |
|||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Peak displacement (m) | 6 | 0.2214 | 0.1113 | 49.738 | 0.1181 | 46.676 | 0.1111 | 49.819 | 0.1233 | 44.309 | |

Peak absolute acceleration (m/s^{2}) |
6 | 5.2790 | 5.1330 | 2.766 | 5.1099 | 3.202 | 5.1439 | 2.560 | 4.2687 | 19.138 | |

Peak IDR | 1 | 0.0200 | 0.00875 | 56.250 | 0.00796 | 60.200 | 0.0089 | 32.759 | 0.0102 | 48.855 | |

RMS of displacement (m) | 6 | 0.0929 | 0.0332 | 64.263 | 0.0349 | 62.443 | 0.0330 | 64.478 | 0.0399 | 57.029 | |

RMS of acceleration (m/s^{2}) |
6 | 1.5989 | 0.9174 | 42.625 | 0.9406 | 41.171 | 0.923 | 42.271 | 0.8465 | 47.056 | |

Peak displacement (m) | 6 | 0.4864 | 0.3709 | 23.756 | 0.3768 | 22.551 | 0.3713 | 23.673 | 0.3795 | 21.987 | |

Peak absolute acceleration (m/s^{2}) |
6 | 14.2822 | 12.5799 | 11.919 | 12.7331 | 10.846 | 12.600 | 11.776 | 11.979 | 16.126 | |

Peak IDR | 1 | 0.0464 | 0.02233 | 51.910 | 0.0193 | 58.436 | 0.0227 | 51.113 | 0.0301 | 35.177 | |

RMS of displacement (m) | 6 | 0.1223 | 0.0430 | 64.849 | 0.0429 | 64.920 | 0.0429 | 64.920 | 0.0539 | 55.958 | |

RMS of acceleration (m/s^{2}) |
6 | 2.2286 | 1.1291 | 49.336 | 1.1350 | 49.073 | 1.1350 | 49.073 | 1.0277 | 53.887 | |

Peak displacement (m) | 6 | 0.6556 | 0.3964 | 39.5366 | 0.4275 | 34.793 | 0.3966 | 39.506 | 0.4379 | 33.206 | |

Peak absolute acceleration (m/s^{2}) |
6 | 21.4828 | 14.538 | 32.325 | 14.1026 | 34.354 | 14.7575 | 31.306 | 12.503 | 41.800 | |

Peak IDR | 1 | 0.0488 | 0.0250 | 48.770 | 0.0233 | 52.254 | 0.0256 | 47.623 | 0.0336 | 31.148 | |

RMS of displacement (m) | 6 | 0.1129 | 0.0375 | 66.785 | 0.0405 | 64.128 | 0.03755 | 66.740 | 0.0447 | 60.407 | |

RMS of acceleration (m/s^{2}) |
6 | 1.8909 | 1.0438 | 44.799 | 1.0728 | 43.265 | 1.0519 | 44.367 | 0.979 | 48.226 |

Peak displacements _{1} and the methods of Aydin (

Three different ground motions are selected for time history analysis.

Percentage of Arias intensity in different frequency ranges to the total intensity for near-field ground motion (Moustafa and Takewaki,

Cape mendocino (Petrolia NS) | 86.75 | 8.30 | 1.98 | 1.03 | 0.60 |

El centro (NS) | 82.03 | 13.27 | 2.70 | 1.22 | 0.49 |

Kobe (NS) | 91.43 | 6.02 | 1.40 | 0.58 | 0.21 |

For the purpose of plotting the 3D graphics are plotted to observe the variations of the absolute value of transfer functions with respect to excitation frequencies and added viscous dampers (c_{adi}). If _{1} minimization is considered, three numbers of VDs are needed and they are placed to first three floors. As seen in _{ad1}, c_{ad2}, c_{ad3}, and c_{ad4} successively. After the decrease of the transfer function amplitude reaches the optimum point in 3D graphics, this value of transfer function continues almost as stable value, even if c_{adi} value is raised to higher values in these plots.

3D plots of the absolute value of transfer function |_{D6}| based on _{1} minimization with respect to excitation frequency and added damping parameters.

3D plots of the absolute value of transfer function |H_{AA6}| based on f_{1} minimization with respect to excitation frequency and added damping parameters.

3D plots of the absolute value of transfer function |_{δ6}| based on _{1}, minimization with respect to excitation frequency and added damping parameters.

Even if f_{1} function could not decrease acceleration responses as seen in _{1} function. In this 10-story shear frame, each floor mass and stiffness are taken as _{1} function are shown in _{1} function and uniformly distributed viscous damper is added to the manuscript as seen in _{1} function is more successful than the uniformly distributed viscous damper design with respect to displacement minimization and reduction of accelerations responses between these designs is almost close the each other. The other advantage of f_{1} function beside of succession of displacement reduction. While the design with uniformly distributed VDs occupies ten stories, VDs which is designed according to f_{1} function needs only four stories locations. This means design based on f_{1}function needs less workmanship cost than the uniformly distributed VDs design.

Properties of VDs parameters and their locations to the floors for 10 story shear frame.

^{6} Ns/m) |
_{1}) |
|||
---|---|---|---|---|

c_{ad10} |
0 | 0 | 0 | 2.30897 |

c_{ad9} |
0 | 2.30897 | ||

c_{ad8} |
0 | 0 | 2.30897 | |

c_{ad7} |
0 | 2.30897 | ||

c_{ad6} |
0 | 2.30897 | ||

c_{ad5} |
0 | 0 | 0 | 2.30897 |

c_{ad4} |
6 | 5.0897 | 0 | 2.30897 |

c_{ad3} |
6 | 6.0000 | 7.1088 | 2.30897 |

c_{ad2} |
5.0897 | 6.0000 | 7.7456 | 2.30897 |

c_{ad1} |
6 | 6.0000 | 8.2353 | 2.30897 |

Total damper | 23.0897 | 23.0897 | 23.0897 | 23.0897 |

Comparison of the results of proposed methods with the other methods for the 10-story shear frame with respect to El Centro ground motion.

_{1} |
|||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Peak displacement (m) | 6 | 0,6847 | 0,2446 | 64,276 | 0.2438 | 64,393 | 0,24029 | 64,906 | 0,282 | 58,814 | |

Peak absolute acceleration (m/s^{2}) |
6 | 5,0551 | 2,1405 | 57,657 | 1,956 | 57,022 | 2,84576 | 43,705 | 1,7819 | 64,750 | |

Peak IDR | 1 | 0,03436 | 0,0102 | 70,314 | 0,0102 | 70,402 | 0,0093 | 72,934 | 0,014 | 59,255 | |

RMS of displacement (m) | 6 | 0,394 | 0,0724 | 81,624 | 0,0784 | 81,754 | 0,066 | 83,249 | 0,0999 | 74,645 | |

RMS of acceleration (m/s^{2}) |
6 | 2,0796 | 0,5190 | 75,043 | 0,4537 | 74,827 | 0,5899 | 71,634 | 0,4956 | 76,168 |

Time history of 10

A damper optimization method is proposed for building structures using random vibration theory in the frequency domain. Differential Evolution (DE) algorithm is used in order to minimize the objective function which is the top floor displacement function considering design constraints. After the optimal design is found, the three different ground motions are conducted to test the seismic response of model building structure. Additionally, the proposed optimal design method is compared with the other methods in the literature which were proposed by Aydin (

It is observed that optimum designed and placed VDs can decrease the transfer function amplitudes effectively.

The proposed method is very effective in reducing seismic response and is compatible with the other methods in the literature.

The method used for each objective function minimizes its purpose better. The different purpose functions can be important for different types of structures. In this study, the application objective function is shown on a single type of structure.

It is concluded that differential evolution (DE) algorithm can be used to solve the optimal damper problem based on transfer functions by considering critical excitation.

The proposed method is very effective for more than one mode of control in the frequency domain.

The objective function for the proposed method is more successful than uniformly distributed VDs design with respect to displacement minimization. Additionally, design based on f_{1} function needs less workmanship cost than the uniformly distributed VDs design.

The datasets generated for this study are available on request to the corresponding author.

EA, HC, and BO contributed to the concept and idea of the paper. The analyses were carried out by EA and HC. All three authors contributed to organize, write, review, and approve the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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