This paper investigates the effect of considering soilstructure interaction (SSI) in seismic responses of reinforced concrete (RC) chimneys installed by distributed tuned vibration absorbers vertically (dMTVAs). A multimode control approach is used to design the dMTVAs. Twodimensional (2D) RC chimney is the assembly of beam elements. Frequencyindependent constants for the springs and dashpots are used for modeling the raft and the surrounding soil. The equations of motion for nonclassically damped systems are derived and solved using Newmark’s method. The effectiveness of the dMTVAs is weighed against the case of single tuned vibration absorber (STVA), dMTVAs suppressing the first modal responses (dMTVAs1), and randomly distributed MTVAs (adMTVAs). Additionally, parametric studies are conducted for varying mass and damping ratios in the STVA, dMTVAs1, adMTVAs, and dMTVAs. In order to show the efficiency in the STVA, dMTVAs1, adMTVAs, and dMTVAs cases, responses (displacement and acceleration) at top of the RC chimney while subjected to different real earthquake excitations are computed. It is concluded that the STVA, dMTVAs1, adMTVAs, and dMTVAs are effective in response mitigation of the RC chimney; however, dMTVAs are more efficient while considering equal total mass of the TVA(s). Moreover, the soil type significantly influenced the design parameters of the STVA/dMTVAs1/adMTVAs/dMTVAs and seismic response of the RC chimney.
Industries generally used reinforced concrete (RC) chimneys with varied geometries. Earthquake forces caused damages or collapses to several chimneys. Kocaeli earthquake in 1999 and Chile earthquake in 2010 are the examples which caused collapses to the RC chimney. The design of chimneys is a wellestablished procedure for working engineers and researchers. Many of the researchers believe that if the chimney is considered to be located in a location with medium to soft soil, the modeling will depend on the type of foundation. In such a site, the structure will be supported on very deep foundation if the rock is too deep or it will be supported on a combination of mat foundation and deep foundation (where part of the site is reinforced using piles) or it will be supported on rocksocketed piles or drilled shafts. The researchers believe that if the chimney is modeled with these foundations, then the behavior of such foundations will be mobilized because of dynamic loads; i.e., there will not be much soilstructure interaction (SSI) under dynamic loads. Hence, many of them ignored the effect of SSI in their studies [
The beam elements are assembled to model the chimney with sway degrees of freedom (DOFs). The DOFs are considered to be the dynamic degrees of freedom of the chimney with consideration of the soilstructure interaction (SSI). The hypothetical modeling is based on the hypothesis that the crosssectional dimension in the element residue is the same. More hypotheses prepared for the systematic formulation are as follows: (i) the chimney is measured to stay in the elastic boundary under earthquake excitations and (ii) each scheme is considered to be under a single horizontal (unidirectional) component of the earthquake ground motion. Figures
(a) Details of the chimney with no control, i.e., uncontrolled (NC); (b) schematic diagram of TVA and section AA; (c) lumped mass idealization for the chimney including SSI and installed with (d) STVA, (e) adMTVAs, and (f) dMTVAs.
Many of the standards such as the Indian standard or the Chilean code supervision indicate that 90% or above of the modal mass has to be taken into consideration for dynamic analysis. Therefore, for the present study, the author decided that the controlled modal responses should have modal mass greater than or equal to 90%. The first three modal responses of the chimney are controlled by installation of STVA, dMTVAs, adMTVAs, and dMTVAs. The mass participation factors for the first mode, for the second mode, and for the third mode are 0.615, 0.190, and 0.100, respectively. Thus, it is decided to have more number of TVAs to control the first modal responses as compared to the number of TVAs for second and third modal responses. The modal frequencies and mode shapes of the chimney with placement of the 9dMTVAs are shown in Figure
Mode shapes and frequencies of the uncontrolled chimney.
The mass (
The damping (
Tuning frequency ratio (
In this study, the RC chimney properties are taken from the model investigated by Datta and Jain [
The soil is represented in a single layer under the footings, which consist of annular raft footing with the inner and outer diameter of 15 m and 40 m, respectively, and with the height of 2.5 m. The raft footing and the neighboring soil are modeled taking into account the springs and related dashpots as shown in Figure
Time variation of displacement and acceleration at the topmost node of the chimney under Llolleo (1985); TVAs are with the mass ratio (
Time variation of displacement and acceleration at the topmost node of the chimney under Nahanni (1985); TVAs are with the mass ratio (
Variations of damping ratios (
Variations of damping ratios (
Variations of damping ratios (
Variations of damping ratios (
Variations of damping ratios (
The assessments between the efficiency of the four TVA schemes for seismic response mitigation of the RC chimney are presented in this section. These TVA schemes are used to control the response of the fixedbase chimney, chimney including the SSI effect. The design parameters for the TVA schemes installed on the RC chimney are provided in Table
Design parameters for the TVA schemes installed on the RC chimney.
Schemes  TVAs  Frequency, 
Mass, 
Stiffness, 
Damping, 

STVA  TVA1  2.05  106.84 (2% of 
447.71  3.22 


9dMTVAs1  TVA1  1.77  11.87  37.03  0.03 
TVA2  1.85  11.87  40.45  0.04  
TVA3  1.93  11.87  44.02  0.04  
TVA4  2.00  11.87  47.27  0.04  
TVA5  2.08  11.87  51.53  0.04  
TVA6  2.16  11.87  55.45  0.04  
TVA7  2.24  11.87  59.30  0.04  
TVA8  2.31  11.87  63.61  0.05  
TVA9  2.39  11.87  68.07  0.05  


9adMTVAs  TVA1  1.77  11.87  37.03  0.03 
TVA2  1.93  11.87  44.02  0.04  
TVA3  2.08  11.87  51.53  0.04  
TVA4  2.25  11.87  59.83  0.04  
TVA5  2.39  11.87  68.07  0.05  
TVA6  6.73  11.87  536.89  0.13  
TVA7  7.91  11.87  743.21  0.16  
TVA8  9.10  11.87  983.00  0.18  
TVA9  18.76  11.87  4177.14  0.37  


9dMTVAs  TVA1  1.77  11.87  37.03  0.03 
TVA2  1.93  11.87  44.02  0.04  
TVA3  2.08  11.87  51.53  0.04  
TVA4  2.25  11.87  59.83  0.04  
TVA5  2.39  11.87  68.07  0.05  
TVA6  6.73  11.87  536.89  0.13  
TVA7  7.91  11.87  743.21  0.16  
TVA8  9.10  11.87  983.00  0.18  
TVA9  18.76  11.87  4177.14  0.37 
Figures
In addition, the figures show the peak displacement relative to ground and peak absolute acceleration at top of the chimney for uncontrolled and controlled cases of using different configurations of the TVAs. The uncontrolled peak displacement responses of chimney with fixed base, dense soil, medium soil, and soft soil, respectively, are 0.630 m, 0.633 m, 0.640 m, and 0.660 m under Llolleo (1985) earthquake excitations. The values of the peak displacement response for different uncontrolled cases are 0.555 m, 0.553 m, 0.553 m, and 0.512 m under the Nahanni (1985) earthquake excitations. Similarly, the peak acceleration responses for the cases of fixed base, dense soil, medium soil, and soft soil, respectively, are 2.847
It is observed there are up to 10% variations in peak displacement response under consideration of different soil types. Furthermore, it is seen that there are up to 20% variations in peak acceleration response. Moreover, it is observed that the TVAs are effective in controlling the displacement response of the chimney in all the configurations considered herein except the STVA case. The responses of the uncontrolled chimney with different soil types are amplified by installing the STVA. Generally, from the figures, it is observed that the postpeak response (topmost node displacement) diminishes significantly when the MTVAs are added as compared to the NC and STVA cases. Similarly, it is seen that the acceleration at top of the chimney is reduced by installing different TVA schemes. The 9dMTVAs are generally observed to have maximum reduction of top node acceleration of the chimney as compared to the STVA, 9dMTVAs1, and 9adMTVAs. Hence, it is concluded that the dMTVAs controlling multimodal response are more consistent in efficiently mitigating the displacement and the acceleration responses.
In this section, the effect of the change in mass ratio (
It is generally observed from the figure that the pattern of variation of the reduction in responses is uniform for different types of responses and excitations for MTVAs schemes. However, it varies for the STVA scheme. Furthermore, it is seen that, in case of STVA, by increasing the mass ratio, there is significant reduction in performance of the STVA. It is due to mistuning effect of the STVA. It is seen that, for the case of STVA, optimum damping exists which could be between 5 and 8%. The better response reduction is observed by installing different MTVAs schemes. The mass ratio increased, the performance of the MTVAs schemes improved. Also, it is observed that the optimum damping ratio exists for the MTVAs schemes which is smaller as compared to the STVA scheme. Besides, it is also noticed that highest response diminution is achieved with equipment of the 9dMTVAs as compared to STVA, 9dMTVAs1, and 9adMTVAs. Therefore, it is concluded that, by increasing the mass ratio of MTVAs schemes, the response diminution is increased as it is not the same for the STVA scheme. Figures
Four different types of soil are considered in order to compare the performance of the different TVA schemes. It is noticed that soil properties significantly condensed the efficiency of the STVA. Conversely, in MTVAs schemes, it is found that they are more robust as compared to the STVA scheme. In addition, it is noticed that the increase in damping ratios (
Therefore, it is concluded that the optimum damping exists for the fixedbase chimney installed with TVAs. However, damping ratios (
It is observed in Figures
It is also observed that the 9dMTVAs performance is unchanged under different soil types considered herein, which means that they are more robust. Therefore, it is concluded that the increase in the mass ratio (
Multimode control of chimneys including soilstructure interaction (SSI) under earthquake ground motions is presented. Distributed multiple tuned vibration absorbers (dMTVAs) are used for multimode control of the chimney including SSI. Assessment of seismic responses is made for the chimney equipped with a single tuned vibration absorber (STVA), the dMTVAs all suppressing the primary modal responses (dMTVAs1), arbitrarily placed dMTVAs (adMTVAs), and dMTVAs under different real earthquake excitations. The following conclusions are drawn from the results of the numerical study presented here:
The dMTVAs controlling multimodal response are more consistent in efficiently mitigating the displacement and the acceleration responses.
The optimum damping exists for the fixedbase chimney installed with TVAs. However, damping ratios (
The increase in the mass ratio (
The soil type significantly influenced the design parameters of the STVA/dMTVAs1/adMTVAs/dMTVAs schemes and seismic responses of the chimney with flexible foundation. Moreover, the dMTVAs are more robust as compared to the STVA, dMTVAs1, and adMTVAs.
The data used to support the findings of this study are available from the corresponding author upon request.
The author declares that there are no conflicts of interest.