Multi-objective stochastic optimization of tuned mass dampers under earthquake excitation considering soil–structure interaction

ABSTRACT

Tuned mass dampers (TMDs) are attractive vibration control devices, and their seismic attenuation depends on their frequency and damping coefficient. Structural characteristics are changed by the soil-structure interaction (SSI); therefore, SSI should be considered in the optimization of TMD. A high-rise building with 40 stories, including four different soil conditions, is proposed as a case study. For the objective function, considering the randomness of earthquakes, a stationary white noise stochastic process based on the Kanai-Tajimi spectrum was used as the input. To improve the performance of a TMD in controlling multimodal and overall dynamic responses, three objectives were considered in the objective function: the maximum and root mean square (RMS) displacements of the top story and the maximum displacement of all stories, respectively. The weight coefficient of each objective function was obtained according to the modal mass participation coefficient. To clearly observe the damping effect of the TMD with different parameter combinations, an ergodic parameter calculation is used. The TMD optimization parameters for each benchmark model with different soil conditions were proposed. Far-field ground motions were considered, and four optimized TMDs from reference were compared. The results show that the proposed TMDs can significantly mitigate earthquake oscillations in a great degree. They performed better than contrastively optimized TMDs under several earthquakes.

KEYWORDS:

• Tuned mass damper
• earthquake mitigation
• stochastic excitation
• soil-structure interaction
• multi-objective optimization

1. Introduction

Because of their slender architectural form, high-rise buildings are vulnerable to vibrations. Earthquakes are one of the most dangerous natural disasters affecting tall buildings (Sheng et al. 2020, 2022; Wang, Shi, and Zhou 2022; Wang et al. 2022, 2023; Zhang, Wang, and Shi 2023). Structural control is an effective and economic method for structural earthquake protection (Choi et al. 2023; Shi et al. 2018), whereas tuned mass dampers (TMD) are attractive controllers (Wang, Zhou, and Shi 2023; Wang et al. 2023, 2020). A TMD can be regarded as a single-degree-of-freedom (DOF) vibration absorber. The basic idea of a TMD is to transfer vibrational energy from the primary structure to itself and dissipate it through a damping element (Shi et al. 2019; Wang et al. 2021; Wang, Zhou, and Shi 0000).

Because of the complexity and randomness of earthquake (Wang et al. 2020, 2023; Wang, Shi, and Zhou 2019), to achieve a good structural control effect, the parameters of a TMD should be optimized. Greco et al. (Greco, Marano, and Fiore 2016) optimized a TMD based on performance cost under low-to-moderate earthquakes. Kamgar et al. (Kamgar, Samea, and Khatibinia 2018) optimized the parameters of the TMD under critical earthquakes. Lyu et al. (Lyu et al. 2020) proposed a shared TMD optimization for a twin-tower structure. Bakre and Jangid (Bakre and Jangid 2007) optimized the parameters of the TMD for a damped primary structure. Pozos-Estrada and Gómez (Pozos-Estrada and Gómez 2019) presented an optimized TMD for slender monuments under multiple hazards. Leung et al. (Leung and Zhang 2009; Leung et al. 2008) proposed a particle swarm optimization of a TMD under different excitations. Lavan (Lavan 2017) presented a multiobjective optimal TMD design. Matta (Matta 2018) proposed lifecycle cost optimization of TMDs for inelastic structures. Keshtegar and Etedali (Keshtegar and Etedali 2018) applied an adaptive dynamic harmony search algorithm to optimize the TMDs in nonlinear mathematical models. Bekdaş et al. optimized TMDs using a novel bat algorithm (Bekdaş, Nigdeli, and Yang 2018) and harmony search (Bekdaş and Nigdeli 2011). Greco and Marano (Greco and Marano 2013) optimized the TMDs according to the displacement response and energy dissipation. The application of TMD for structural earthquake mitigation can be found in (Lee et al. 2022; Ozturk, Cetin, and Aydin 2022; Roozbahan and Jahani 2022) as well.

From the references reviewed in the second paragraph, it can be found that first, an optimized TMD can have good earthquake mitigation, and second, for the optimization of TMD parameters, at least three issues should be considered: objective structure, objective function, and optimization algorithm. It is well known that a passive TMD is sensitive to frequency deviation (Wang, Wang, and Shi 2021; Shi, Wang, and Lu 2018; Wang et al. 2019), and a mistuned TMD can even perform negatively in vibration control (Fu et al. 2023; Islam, Do, and Kim 2018; Wang et al. 2020). The soil – structure interaction (SSI) can change the structural characteristics, that is, structural modes, structural frequencies, and damping ratio (Chuai 2022; Ganjavi, Gholamrezatabar, and Hajirasouliha 2019; Todorovska et al. 2020; Yang et al. 2023). It was highlighted in (Jia and Jianwen 2019) that neglecting SSI would cause performance degradation of TMDs. Therefore, to obtain a robust and effective TMD, SSI should be considered in the objective structure.

A 40-story model proposed in (Liu et al. 2008) with four different soil conditions presents a benchmark test bed for the earthquake mitigation optimization of TMDs, while base fixed, dense soil, medium soil, and soft soil models are all considered. In (Bekdaş and Nigdeli 2017), proposed by Bekdaş and Nigdeli, the objective function was the minimization of the maximum displacement of the structure and the limitation of the scaled stroke of TMD under several earthquake records, and two metaheuristic algorithms (harmony search and bat algorithms) were used for parameter search. In (Bekdaş et al. 2019), authored by Bekdaş et al., the objective function was the minimization of the acceleration transfer function, which did not depend on a specific earthquake and could show great randomization, and several metaheuristic algorithms were used and compared. In (Salvi, Pioldi, and Rizzi 2018) presented by Salvi et al., the objective function was the minimization of the root mean square (RMS) displacement of the top story under a special earthquake input, and a classical nonlinear gradient-based optimization algorithm was used to calculate it. In (Farshidianfar and Soheili 2013), proposed by Farshidianfar and Soheili, the objective function was to reduce both the maximum displacement and acceleration of stories under the Tabas and Kobe earthquakes, and the ant colony optimization method was utilized. In (Etedali, Akbari, and Seifi 2019) authored by Etedali et al., the objective function was focused on the maximum displacement, acceleration, and drift of floors under special near-field earthquakes, and multi-objective cuckoo search algorithm was chosen in the optimization process.

To include SSI in the optimization of TMDs, a 40-story benchmark tall building (Bekdaş and Nigdeli 2017; Bekdaş et al. 2019; Etedali, Akbari, and Seifi 2019; Farshidianfar and Soheili 2013; Liu et al. 2008; Salvi, Pioldi, and Rizzi 2018) is proposed as a case study in this paper. In (Bekdaş et al. 2019), it was also shown that different metaheuristic algorithms could obtain similar optimum parameters for TMDs. Therefore, special attention should be paid to this objective function. Considering the randomness of earthquakes, it would be better to use a stochastic earthquake as an input in the optimum process, but not a special earthquake. Additionally, to improve the overall dynamic response mitigation effect of a TMD, both the maximum and RMS responses should be considered. Furthermore, tall buildings may exhibit several modal responses under earthquake excitation. Regard all of these, in this study, a stationary white noise stochastic process based on the Kanai-Tajimi spectrum (Shan et al. 2018) is used as the input. To improve the performance of a TMD in controlling multimodal and overall dynamic responses, three objectives were considered in the objective function: the maximum and RMS displacements of the top story and the maximum displacement of all stories, respectively. The weight coefficient of each objective function was obtained according to the modal mass participation coefficient. For a TMD with a constant mass, only its stiffness and damping coefficient must be optimized (Jin et al. 2018; Shi et al. 2018). To clearly observe the damping effect of the TMD with different parameter combinations, an ergodic parameter calculation is used.

In this study, a multi-objective stochastic optimization of TMDs for high-rise buildings was proposed, which included four different soil conditions. The multi-objective stochastic optimization method is introduced in Section 2. The introduction of the 40-story model and the optimized results are presented in Section 3. In Section 4, 44 far-field ground motions are considered in the numerical simulation and four optimized TMDs from (Bekdaş and Nigdeli 2017) are used for comparison. Finally, conclusions and further research are presented in Section 5.

2. Multi-objective stochastic optimization method

2.1. Stochastic dynamic response

For a multiple DOF linear structure, the dynamic equation under base excitation $\underset{\mathit{g}}{\ddot{u}}$ can be written as:

(1) $\mathit{M}\ddot{u}+\mathit{K}\mathit{u}+\mathit{C}\dot{u}=-\text{M}\underset{\mathit{g}}{\ddot{u}}$(1)

where $\mathbf{M}$, $\mathbf{K}$, and $\mathbf{C}$ denote the mass, stiffness, and damping matrices of the structure, respectively. $\mathbf{u}$ represents the structural relative displacement matrix, and the overdot is the derivative with respect to time.

Considering the randomness of earthquakes, it would be better to use a stochastic earthquake in the optimal design of TMDs, but not a special earthquake. Therefore, the Tajimi spectrum $S_a(\mathit{\omega })$ (Chen et al. 2016) is presented in this study.

(2) $S_a(\mathit{\omega })=\frac{{\omega }_g^4+4{\zeta }_g{\omega }_g^2{\omega }^2}{{({\omega }_g^2-{\omega }^2)}^2+4{\zeta }_g^2{\omega }_g^2{\omega }^2}{S}_0$(2)

where ${\omega }_g$ and ${\zeta }_g$ indicate the dominant frequency and damping ratio of the soil, respectively; $\mathit{\omega }$ is the excitation frequency; and $S_0$ represents a constant intensity.

A stationary white noise stochastic process based on Equation (2)(2) $S_a(\mathit{\omega })=\frac{{\omega }_g^4+4{\zeta }_g{\omega }_g^2{\omega }^2}{{({\omega }_g^2-{\omega }^2)}^2+4{\zeta }_g^2{\omega }_g^2{\omega }^2}{S}_0$(2) was used as the input in the following case study, while ${\omega }_g$ and ${\zeta }_g$ were set to $5\mathit{\pi }$ and 0.6, respectively.

2.2. Multi-objective optimization

Typically, the TMD mass ratio is set to 1%–5%. After determining the mass ratio, this study focused on the optimization of the frequency ratio $\mathit{f}$ and damping ratio $\mathit{\zeta }$.

The optimization results and earthquake mitigation effect mainly depend on the objective function $\mathit{y}(\mathit{f},\mathit{\zeta })$. To improve the overall earthquake mitigation effect of a TMD, a multiobjective function is proposed in this section. Excessive displacement responses endanger the structure under earthquake excitations; therefore, both the maximum and RMS displacement responses are considered. A tall building may exhibit several modal responses under earthquake excitation. To enhance the control effect of a TMD in controlling multimodal and overall dynamic responses, the third objective function was chosen to be the maximum displacement of all stories. The weight coefficient $\mathit{\alpha }$ of each objective function $\mathit{y}(\mathit{f},\mathit{\zeta })$ is obtained according to the modal mass participation coefficient, and the first two modes are usually considered to be sufficient.

The i^th objective function ${\bar{y}}_{\mathit{i}}(\mathit{f},\mathit{\zeta })$ is written in a normalized form (Chen et al. 2016):

(3) ${\bar{y}}_{\mathit{i}}(\mathit{f},\mathit{\zeta })=\frac{{y_i}^{max}(\mathit{f},\mathit{\zeta })-y_i(\mathit{f},\mathit{\zeta })}{{y_i}^{max}(\mathit{f},\mathit{\zeta })-{y_i}^{min}(\mathit{f},\mathit{\zeta })}$(3)

where ${y_i}^{max}(\mathit{f},\mathit{\zeta })$ and ${y_i}^{min}(\mathit{f},\mathit{\zeta })$ are the maximum and minimum values, respectively, of the i^th function. Then, it can be known that the performance is better when ${\bar{y}}_{\mathit{i}}(\mathit{f},\mathit{\zeta })$ is closer than one.

The i^th weight coefficient ${\alpha }_i$ of each normalized objective function ${\bar{y}}_{\mathit{i}}(\mathit{f},\mathit{\zeta })$ can be obtained according to the modal mass participation coefficient. For the first two functions, which are the minimum of the maximum and RMS displacement responses of the top story, ${\alpha }_1$ and ${\alpha }_2$ are both half of the ratio of the first-order mass participation coefficient to the previous two order mass participation coefficients. Then, for the third objective, which is the minimum of the maximum displacement of all stories, ${\alpha }_3$ is the ratio of the second-order mass participation coefficient to the previous two order mass participation coefficients.

It should be noted that the first two objective functions are focused on the structural first mode, whereas the third focuses on the second mode. If a structure has only the first modal response, then the third function is virtually the same as the first function.

Therefore, the strategy criterion $\mathit{S}(\mathit{f},\mathit{\zeta })$ proposed in this study can be written as:

(4) $\mathit{S}(\mathit{f},\mathit{\zeta })={\alpha }_1{\bar{y}}_1(\mathit{f},\mathit{\zeta })+{\alpha }_2{\bar{y}}_2(\mathit{f},\mathit{\zeta })+{\alpha }_3{\bar{y}}_3(\mathit{f},\mathit{\zeta })$(4)

A flowchart of the proposed multi-objective stochastic optimization method is illustrated in Figure 1.

Figure 1. Flowchart of the multi-objective stochastic optimization method.

It should be noted that the proposed optimization method is suitable and applicable to buildings in which SSI is ignored or considered, as shown in the following case study.

3. Case study

To present the optimization process and results of the proposed multi-objective stochastic optimization method, a 40-story high-rise building is proposed in this section. Four different soil conditions were considered in this benchmark problem. The first is a base fixed model that ignores the SSI and is a normal shear-type MDOF structure. The other three models considered the SSI with different types of soil.

3.1. Model description

A 40-story high-rise building with SSI installed with a TMD is illustrated in Figure 2.

Figure 2. 40-story high-rise building considering SSI installed with a TMD.

In Figure 2, $m_d$, $k_d$ and $c_d$ are the mass, stiffness and damping coefficients of TMD respectively; $\mathit{N}$ is the total number of stories, which is 40 in this study; $m_i$, $k_i$ and $c_i$ mean the mass, stiffness and damping coefficients of i^th story respectively; $m_0$ and $I_0$ are the mass and mass moment of inertia of the foundation respectively, while $I_i$ is the mass moment of inertia of the i^th story; as for soil parameters, $k_s$ and $c_s$ are the swaying parameters respectively, while $k_r$ and $c_r$ represent the rocking parameters respectively; $x_0$ and $x_i$ are relative displacements of the foundation and i^th story respectively, while ${\theta }_0$ is the rotation of the foundation; $Z_i$ is the height of the i^th story, and $\underset{\mathit{g}}{\ddot{x}}$ is the earthquake input.

Detailed information, parameters, dynamic equations, and matrices for this model can be found in (Bekdaş and Nigdeli 2017; Bekdaş et al. 2019; Jia and Jianwen 2019; Liu et al. 2008; Salvi, Pioldi, and Rizzi 2018; Yang et al. 2023). For brevity, they are not repeated here. The model without TMD, ignoring the SSI, is a normal shear-type 40 DOF structure. As for models including SSI, $k_s$, $c_s$, $k_r$, and $c_r$ are different and can be found in (Bekdaş and Nigdeli 2017; Bekdaş et al. 2019; Jia and Jianwen 2019; Liu et al. 2008; Salvi, Pioldi, and Rizzi 2018; Yang et al. 2023) too. Therefore, when the SSI is considered and a TMD is implemented, it has a 43 DOF structure.

Numerical models were built according to these parameters, dynamic equations, and matrices in MATLAB. The first-order frequencies of the four models are 0.2626, 0.2562, 0.2451, and 0.1735 Hz, respectively, according to the model analysis, similar to the results in (Bekdaş and Nigdeli 2017; Bekdaş et al. 2019; Jia and Jianwen 2019; Liu et al. 2008; Salvi, Pioldi, and Rizzi 2018; Yang et al. 2023), which shows the accuracy of the built mathematical model.

Through model analysis of the base fixed model, it can be seen that the first two order mass participation coefficients were 0.7866 and 0.1058, respectively. Therefore, in Equation (4)(4) $\mathit{S}(\mathit{f},\mathit{\zeta })={\alpha }_1{\bar{y}}_1(\mathit{f},\mathit{\zeta })+{\alpha }_2{\bar{y}}_2(\mathit{f},\mathit{\zeta })+{\alpha }_3{\bar{y}}_3(\mathit{f},\mathit{\zeta })$(4) , the weight coefficients are ${\alpha }_1:{\alpha }_2:{\alpha }_3=0.7866/2:0.7866/2:0.1058=0.44\text{break}:0.44:0.12$.

3.2. Optimization results

According to the multi-objective stochastic optimization method proposed in Figure 1, four TMDs were specifically optimized for the four models.

In (Bekdaş et al. 2019), the objective function was the minimization of the acceleration transfer function, which did not depend on a specific earthquake and could show great randomization. Several metaheuristic algorithms were used and compared. It was found in (Bekdaş et al. 2019) that the Jaya (JA) algorithm is an effective and robust method for finding the best parameters. Therefore, the JA-TMDs proposed in (Bekdaş et al. 2019) were used for the earthquake mitigation comparisons in Section 4. Similar to JA-TMDs, the mass of the TMD was set to 784 t in this study. In the parameter search, the TMD frequency ratio range was 0.90 ~ 1.10, and the damping ratio range was 0.0500 ~ 0.1600.

The optimization results and comparative JA-TMDs are proposed and compared in Table 1.

Table 1. Proposed TMDs and comparative TMDs.

Model	Structural first order frequency (Hz)	Proposed TMD			JA-TMD
		Mass (t)	Frequency (Hz)	Damping ratio (%)	Mass (t)	Frequency (Hz)	Damping ratio (%)
Base fixed	0.2626	784	0.2639	7.12	784	0.2532	13.17
Dense soil	0.2562	784	0.2447	8.49	784	0.2502	11.02
Medium soil	0.2451	784	0.2200	5.57	784	0.2370	12.67
Soft soil	0.1735	784	0.1640	6.48	784	0.1658	14.58

To highlight the overall structural control effect of the optimized TMDs under stationary white noise stochastic excitation, comparisons of the displacement responses of every story in the four control cases are presented in Figure 3.

Figure 3. Comparisons of displacement responses of every story. (a) maximum displacement; (b) RMS displacement.

It can be seen in Figure 3 that first, it is clear that the optimized TMD has the best structural control effect, regardless of the maximum or RMS displacements. It is interesting to note that the building has several modal responses, particularly the model with soft soil, as shown in Figure 3(a). In this model, the maximum displacement appeared on the 34^th floor. The optimized TMD can decrease the maximum displacement on the 34^th floor, which is the focus of the third objective function, but it increases the maximum displacement in the top story to a small degree. It can be concluded that the optimized TMD achieves a balance in multimodal response control.

4. Numerical results and discussion

4.1. Responses comparisons of the top story

44 far-field ground motions (two components of 22 stations) (Bekdaş and Nigdeli 2017; Bekdaş et al. 2019) with maximum amplitudes normalized to 1 m/s² were used to verify the earthquake mitigation of the proposed TMDs. These data were downloaded from (Pacific Earthquake Engineering Research Center PEER Ground Motion Database). The displacement responses of the top story of the two TMD cases of the base fixed structure and the model with soft soil are presented in Tables 2 and 3, respectively, while the displacement and acceleration responses of the four models without TMD, with an optimized TMD case, and with a comparative JA-TMD case, respectively, are illustrated in Figures 4 and 5. Meanwhile, the reductions in the proposed TMD are shown in Figures 4 and 5 as well, compared to the other two cases.

Figure 4. Displacements responses reductions of proposed TMDs in the benchmark high-rise building with different soil condition. (a) &; (b) maximum and RMS displacements compared to the case without TMD; (c) &; (d) maximum and RMS displacements compared to the JA-TMD case.

Figure 5. Accelerations responses reductions of proposed TMDs in the benchmark high-rise building with different soil condition. (a) &; (b) maximum and RMS accelerations compared to the case without TMD; (c) &; (d) maximum and RMS accelerations compared to the JA-TMD case.

Table 2. Displacement response of the base fixed model.

No.	Component	Maximum displacement (cm)		RMS displacement (cm)
		Optimized TMD	JA-TMD	Optimized TMD	JA-TMD
1	X	7.750	7.751	1.855	1.896
	Y	5.779	5.886	1.570	1.495
2	X	8.057	8.049	2.112	2.231
	Y	7.267	7.242	1.922	1.941
3	X	6.648	6.485	1.732	1.966
	Y	3.887	4.138	0.995	1.038
4	X	8.284	8.185	2.225	2.221
	Y	10.341	10.371	2.268	2.270
5	X	15.212	14.935	5.683	5.967
	Y	10.930	13.718	4.011	4.499
6	X	13.832	13.572	4.373	4.633
	Y	9.262	9.214	2.982	3.118
7	X	4.562	4.610	1.151	1.184
	Y	4.592	4.536	1.687	1.890
8	X	13.219	13.116	3.195	3.490
	Y	9.508	9.597	2.793	3.131
9	X	35.195	32.689	12.961	11.131
	Y	11.458	11.349	4.844	4.792
10	X	8.281	8.068	2.413	2.433
	Y	35.951	36.876	12.865	13.339
11	X	20.052	19.820	6.040	6.352
	Y	27.801	28.521	7.931	8.855
12	X	8.797	9.322	3.330	3.479
	Y	6.095	6.199	1.605	1.639
13	X	3.749	3.810	1.103	1.141
	Y	3.939	3.776	1.074	1.139
14	X	4.679	4.666	1.343	1.359
	Y	11.214	11.234	2.590	2.637
15	X	10.026	9.830	2.065	2.211
	Y	11.285	11.693	3.391	3.474
16	X	11.389	9.479	4.018	3.725
	Y	14.741	14.104	6.079	6.368
17	X	6.645	6.612	2.697	2.482
	Y	13.102	13.069	5.220	5.490
18	X	5.867	5.906	1.598	1.638
	Y	3.6982	3.700	0.800	0.854
19	X	37.901	36.062	8.530	8.043
	Y	46.590	47.174	8.013	9.804
20	X	6.374	6.273	1.524	1.729
	Y	3.150	3.189	0.870	0.881
21	X	24.907	23.896	5.423	5.782
	Y	12.283	12.322	2.472	2.802
22	X	2.824	2.822	0.879	0.896
	Y	4.342	4.292	1.267	1.368

Table 3. Displacement response of the soft soil model.

No.	Component	Maximum displacement (cm)		RMS displacement (cm)
		Optimized TMD	JA-TMD	Optimized TMD	JA-TMD
1	X	3.200	3.192	0.952	0.928
	Y	3.771	3.792	0.849	0.840
2	X	2.822	2.837	1.009	1.013
	Y	3.640	3.654	1.032	1.001
3	X	3.917	3.846	1.110	1.135
	Y	2.087	2.059	0.469	0.472
4	X	7.960	7.867	2.171	2.249
	Y	5.892	5.915	1.685	1.612
5	X	10.906	12.776	3.333	3.600
	Y	7.845	8.446	2.434	2.463
6	X	7.358	7.212	2.763	2.771
	Y	9.176	8.766	3.309	3.333
7	X	2.075	2.057	0.574	0.570
	Y	4.398	4.306	1.384	1.384
8	X	3.884	3.891	1.033	1.066
	Y	5.117	5.091	1.614	1.500
9	X	17.231	16.595	7.759	7.183
	Y	8.805	8.347	4.233	3.756
10	X	5.403	5.454	1.922	1.914
	Y	32.482	31.919	12.451	11.456
11	X	20.438	19.717	6.581	6.190
	Y	21.468	21.320	6.634	6.326
12	X	10.126	9.869	3.230	3.076
	Y	5.413	5.157	2.154	2.044
13	X	2.726	2.758	0.686	0.692
	Y	1.820	1.856	0.539	0.549
14	X	2.097	2.111	0.745	0.711
	Y	5.522	5.531	1.092	1.182
15	X	4.496	3.962	2.030	1.647
	Y	7.660	7.374	2.100	2.261
16	X	10.834	9.735	3.799	3.531
	Y	17.959	15.134	6.771	5.793
17	X	2.032	2.052	0.690	0.696
	Y	5.843	5.350	2.736	2.529
18	X	4.868	4.782	1.635	1.584
	Y	2.663	2.650	0.779	0.840
19	X	19.744	19.299	3.948	4.130
	Y	35.061	34.302	6.226	6.305
20	X	8.437	8.234	1.881	1.965
	Y	3.457	3.207	0.963	0.907
21	X	9.045	9.074	2.002	2.188
	Y	6.436	6.326	1.715	1.906
22	X	1.919	1.920	0.378	0.402
	Y	2.086	2.089	0.646	0.650

From Tables 2, 3, Figures 4 and 5, because of the randomness and complexity of earthquakes, there is a significant difference in the structural displacement and acceleration responses and control effects of TMDs. Compared to the structure without the controller, the optimized TMD has satisfactory earthquake mitigation in displacement and acceleration RMS responses; the reduction in the maximum response is not as obvious as the RMS response, and the reduction in maximum displacement is better than acceleration. Compared with JA-TMDs, under some earthquakes, the proposed TMDs have a better performance, but they are generally comparable. Because comparative JA-TMDs are also specially optimized, the effectiveness of the optimization method proposed in this paper can be verified.

To clearly observe the control effect in the time history, the No. 19 Y-directional earthquake, which is the “CHICHI/CHY101-N” component of “Chi-Chi, Taiwan” motion, as an example, the displacement responses of the top story are presented and compared in Figure 6.

Figure 6. Comparisons of displacement responses of the top story under the elected earthquake. (a) base fixed; (b) dense soil; (c) medium soil; (d) soft soil.

As shown in Figure 6, in the base fixed model, the proposed TMD was not as effective as the JA-TMD. However, in the other three models considering SSI, the proposed optimized TMDs exhibit better performance than JA-TMDs. Especially in medium soil and soft soil models, the proposed TMDs can significantly decrease the displacement responses of the top story in a great degree.

4.2. Structural responses comparisons

To evaluate the overall earthquake mitigation, five earthquake motions that can cause large dynamic responses in Tables 2 and 3 were chosen and are listed in Table 4 (Jia and Jianwen 2019; Liu et al. 2008). The maximum and RMS displacement response comparisons for each story are shown in Figures 7 and 8 respectively.

Figure 7. Comparisons of maximum displacement responses of every story. (a) maximum displacement under No. 9 FN earthquake; (b) maximum displacement under No. 10 FP earthquake; (c) maximum displacement under No. 19 FN earthquake; (d) maximum displacement under No. 19 FP earthquake; (e) maximum displacement under No. 21 FN earthquake.

Figure 7. (Continued)

Figure 8. Comparisons of RMS displacement responses of every story. (a) RMS displacement under No. 9 FN earthquake; (b) RMS displacement under No. 10 FP earthquake; (c) RMS displacement under No. 19 FN earthquake; (d) RMS displacement under No. 19 FP earthquake; (e) RMS displacement under No. 21 FN earthquake.

Figure 8. (Continued)

Table 4. The considered five earthquake motions.

No.	Name	Recording station	Year	Magnitude	Component
9	Kocaeli, Turkey	Duzce	1999	7.5	KOCAELI/DZC180
10	Kocaeli, Turkey	Arcelik	1999	7.5	KOCAELI/ARC090
19	Chi-Chi, Taiwan	CHY101	1999	7.6	CHICHI/CHY101-E
19	Chi-Chi, Taiwan	CHY101	1999	7.6	CHICHI/CHY101-N
21	San Fernando	LA – Hollywood Stor	1971	6.6	SFERN/PEL090

Figure 7 and Figure 8 show that under these five earthquake excitations, which can cause relatively large dynamic responses, the first modal response is dominant. Under the No. 9 FN earthquake, the proposed TMD performs better than the JA-TMD in the base fixed and soft soil models; they perform similarly in the dense soil model, and in the medium soil model, the JA-TMD performs better and they are generally similar. Under the No. 10 FP earthquake, except for the base fixed model, the TMDs proposed in this study have a better control effect than the JA-TMDs. Under the No. 19 FN earthquake, as for the maximum displacement, they have a similar performance; for the RMS displacement, JA-TMDs have a better earthquake mitigation in dense soil and medium soil models, and their performances are similar in the base fixed and soft soil models. Under the No. 19 FP earthquake, as for the maximum displacement, the proposed TMD has a better control effect in the medium soil model, and they perform similarly in the other three models. For the RMS displacement, the JA-TMD performs better in the base fixed model, whereas the proposed TMD has a better performance in the medium soil model. Finally, under the No. 21 FN earthquake, they generally had a comparable control effect.

5. Conclusions

Considering the complexity and randomness of earthquakes, to achieve a good earthquake mitigation, the parameters of TMD should be optimized specially and a multi-objective stochastic optimization method is proposed in this paper. SSI can change the structural characteristics and the neglect of SSI may lead to a performance degradation of TMDs. A benchmark 40-story high-rise building with four different soil conditions is proposed as a case study. A stationary white noise stochastic excitation based on the Kanai-Tajimi spectrum is used as the input in the optimal design. Three objectives are considered in the objective function, which are the maximum and RMS displacements of the top story and the maximum displacement of all stories respectively. The weight coefficient of each objective function is obtained according to the modal mass participation coefficient.

The TMD optimization parameters for every model with different soil conditions were proposed. 44 far-field ground motions were considered in the simulation to verify earthquake mitigation, and four JA-TMDs from (Bekdaş et al. 2019) were used for comparison. From this study, it can be concluded that:

1. The proposed multi-objective stochastic optimization method can cover the maximum and RMS displacement responses of the first mode and the maximum displacement response of the second mode and is suitable for buildings that ignore or consider SSI. In addition, it is based on a stationary stochastic earthquake excitation, which contributes to its wide applicability. Therefore, it can be concluded that the proposed method and optimized TMD has a better robust and is suitable for different structures.

2. Compared to the case without TMD, the proposed optimized TMDs can mitigate earthquake oscillations to a great degree, both in terms of displacement and acceleration responses, and maximum and RMS responses. It has a better earthquake mitigation performance and effect.

3. Under several earthquakes, the proposed TMDs perform better than the contrastive JA-TMDs, and both have excellent earthquake mitigation performance. Therefore, the proposed optimized TMD can protect a tall building from different earthquake excitations very effectively.

In future research, different optimization algorithms can be used to optimize the parameters of TMD, different optimization objectives can be considered, nonlinear structures are worth studying, multiple TMD systems are worth exploring, semi-active and active TMD are also worth studying.

Acknowledgements

The authors are grateful for the financial support received from the formation mechanism and countermeasures of “the Belt and Road power investment project safety cost (17BGL010).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Data will be made available on request.

Additional information

Funding

The work was supported by the formation mechanism and Countermeasures of “the belt and road” power investment project safety cost [17BGL010].

Notes on contributors

Yang Wang

Yang Wang, male, associate professor, Doctor of Engineering, graduated from College of Civil Engineering in Tongji University in 2008. Currently, he is the Director of the Department of Engineering Management, Shanghai Electric Power University. His research interests include seismic reduction and health monitoring of engineering structures, intelligent construction and innovation, and building carbon emission management. E-mail: [email protected].

Haotian Ma

Haotian Ma, male, is currently studying for a master’s degree in the School of Economics and Management of Shanghai Electric Power University. He is a member of the digital intelligent construction laboratory. His research activities have centered on energy system resilience, short-term power forecasting, energy system optimization. E-mail: [email protected].