Ideal Drift Response Curve for Robust Optimal Damper Design for Elastic-Plastic MDOF Structures under Multi-Level Earthquakes

Hiroki Akehashi; Izuru Takewaki

doi:10.32604/cmes.2021.017204

Department of Architecture and Architectural Engineering, Graduate School of Engineering, Kyoto University, Kyotodaigaku-Katsura, Nishikyo, Kyoto, 615-8540, Japan
*Corresponding Author: Izuru Takewaki. Email: takewaki@archi.kyoto-u.ac.jp
Received: 22 April 2021; Accepted: 18 June 2021

Abstract: A new method of robust damper design is presented for elastic-plastic multi-degree-of-freedom (MDOF) building structures under multi-level ground motions (GMs). This method realizes a design that is effective for various levels of GMs. The robustness of a design is measured by an incremental dynamic analysis (IDA) curve and an ideal drift response curve (IDRC). The IDRC is a plot of the optimized maximum deformation under a constraint on the total damper quantity vs. the design level of the GMs. The total damper quantity corresponds to the total cost of the added dampers. First, a problem of generation of IDRCs is stated. Then, its solution algorithm, which consists of the sensitivity-based algorithm (SBA) and a local search method, is proposed. In the application of the SBA, the passive added dampers are removed sequentially under the specified-level GMs. On the other hand, the proposed local search method can search the optimal solutions for a constant total damper quantity under GMs’ increased levels. In this way, combining these two algorithms enables the comprehensive search of the optimal solutions for various conditions of the status of the GMs and the total damper quantity. The influence of selecting the type of added dampers (oil, hysteretic, and so on) and the selection of the input GMs on the IDRCs are investigated. Finally, a robust optimal design problem is formulated, and a simple local search-based algorithm is proposed. A simple index using the IDRC and the IDA curve of the model is used as the objective function. It is demonstrated that the proposed algorithm works well in spite of its simplicity.

Keywords: Optimal damper placement; robust damper design; multi-level earthquake; ideal drift response curve elastic-plastic MDOF model; viscous damper; hysteretic damper

Building structures are designed under various uncertainties, e.g., modeling errors and incomplete reported data on geometrical and mechanical properties of structural elements and ground supporting the building. Excessive pursuit of the profitability regarding structural design may lead to a design with a low resistance to natural disasters. Experienced designers can balance the profitability and the effect of these uncertainties based on their deep understanding of structural mechanism. On the other hand, the reliability-based design and the robustness-based design are mathematical ways to deal with such uncertainties directly. Many researches on these design methods and the evaluation of robustness of structures have been accumulated so far [1–7]. The performance-based design is also one of the structural design methods which take account of uncertainties. Incremental dynamic analysis (IDA) [8] is often used to make fragility curves of structures for the increasing level of earthquake ground motions.

Many researchers have tackled the problems of optimal passive dampers (for example, see [9 –11]). Zhang et al. [12] and Garcia [13] developed a simple design procedure for viscoelastic dampers. Takewaki [14] introduced an optimality criteria-based algorithm to minimize the sum of the inter-story drifts under a harmonic excitation with the fundamental natural frequency of the structure. This approach was extended to the base shear response for seismic rehabilitation of planar building structures [15]. Trombetti et al. [16] investigated the effectiveness of mass-proportional damping systems using a finite element model. Whittle et al. [17] compared several methods for optimal damper placement in view of reductions in peak responses, the usability of the methods and the computational effort. Martínez et al. [18] formulated a problem of optimal hysteretic damper placement in the frequency domain using the stochastic equivalent linearization technique. Aydin et al. [19] tackled a problem of optimal damper placement considering the soil-structure interaction. Metaheuristics have been applied widely to the optimal damper placement [20–23]. Some researchers treated simultaneous optimization problems of main structures and added dampers [24,25].

The awareness of the significance of treating elastic-plastic structures in the optimal damper placement has been prevailing recently [25–30]. Since the setting of the input ground motions (GMs) level used for the optimization affects the final design, the level should be determined carefully. For example, a model optimized under lower-level GMs may exhibit large plastic deformation concentrations in specific stories for larger-level GMs [25,31]. However, the model optimized under a quite large level of GMs may not reduce the responses effectively for lower-level GMs. Moreover, earthquakes which will hit buildings are unpredictable. Therefore, an optimization method is needed which can properly deal with the uncertainty of the level of GMs.

This paper presents a new method of robust damper design for elastic-plastic multi-degree-of-freedom (MDOF) building structures under multi-level GMs. This method realizes a design that is effective for various levels of GMs. The robustness of such design is measured by an IDA curve [8] and an ideal drift response curve (IDRC) [32]. The IDRC is a plot of the optimized maximum deformation under a constraint on the total damper quantity versus the design level of the GMs. The total damper quantity corresponds to the total cost of the added dampers. A problem of generation of IDRCs is stated and its solution algorithm is proposed. The influence of selecting the type of added dampers (oil, hysteretic,…) and the selection of the input GMs on the IDRCs are investigated. Finally, a robust optimal design problem is formulated, and a simple local search-based algorithm is proposed. A simple index using the IDRC and the IDA curve of the model is used as the objective function.

This section explains the concept of robust damper design based on an ideal drift response curve (IDRC). Then the generalized problem of optimal damper placement under consecutive-level GMs (G-PODPCG) is stated as an extended version of the problem presented by Akehashi et al. [32]. Finally, its solution algorithm, consisting of the sensitivity-based algorithm (SBA) and a local search method, is proposed.

Akehashi et al. [32] proposed an IDRC, which is a new concept on the optimal damper placement for elastic-plastic MDOF structures to visually capture effectiveness of a damper design for various levels of GMs (Fig. 1a). Wc,IMj in the figure represents the optimal distribution vector of the added dampers which minimizes the maximum inter-story drift under a constant sum Wc of the added damping coefficient and IM denotes the level of input GMs. Wc,IMj satisfies the following.

where Wc denote the added damping coefficient vector with W = Wc and the maximum interstory drift of the model with Wc under the GMs with IM = IMj. The red line in the figure represents the IDRC, which is the plot of Wc,IM versus the design IM. The IDA curve for the model with Wc,IMj is tangent to the IDRC only at the point where IM = IMj. Moreover, the IDA curves for any damper designs with W = Wc never cross the IDRC. When the added dampers are not installed much enough, or the type of the added dampers (oil, hysteretic,…) is not effective for the input GMs, the IDRC inclines towards the horizontal direction in the IM-dmax graph.

Fig. 1b shows the difference between the robust damper design based on the IDRC and the non-robust damper designs. The IDA curves of the non-robust damper designs are plotted locally away from the IDRC. This results from the correspondence of the equivalent natural periods of the designs and the predominant periods of the input GMs. On the other hand, the IDA curve of the robust damper design runs near the IDRC. This means that the design is effective for various levels of the input GMs.

Figure 1: Concept of IDRC and robust damper design, (a) IDRC, (b) robust damper design and non-robust damper design

The IDRC of the viscous dampers W = Wc is obtained by solving the following problem.

Akehashi et al. [32] proposed a consecutive design generation method (CDGM) as the solution algorithm for the problem PODPCG. In this paper, an extended version of the problem PODPCG, a generalized PODPCG (G-PODPCG) is treated.

It is noted that the value of IM becomes large when the number of the subscript of IM becomes large, although the value of W becomes small when the number of the subscript of W becomes large.

Step 4 Obtain Wc,iD,IMiL+1 by a local search around Wc,iD,IMiL as the initial design.

Step 4 Evaluate the sensitivity fij=dmaxj+1,i−dmaxj.Then update ci→ci+Δcs and set i→i+1. If i < N, return to Step 3.

It is noted that the local search method adopted in this paper can be described as follows:

Step 2 Find the story i which exhibits the largest value of the interstory drift.

Step 3 Find the story {k;ck>0} which exhibits the minimum value of the interstory drift.

Step 4 Update ci→ci+Δcl,ck→ck−Δcl(Δcl>0). The value of Δcl is determined by a uniform random number. If ck < 0, revise the value of Δcl so that the value of ck becomes 0. Then update j→j+1. If j < nLS, return to Step 2. If j = nLS, select the design which exhibits the minimum value of dmax from the nLS designs, and finalize the process.

Fig. 2 shows the overview and the schematic diagram of the proposed solution algorithm. The proposed algorithm consists of the SBA and the local search method. In the application of the SBA, the inactive added dampers are removed sequentially under the specified-level GMs. On the other hand, the proposed local search method can search the optimal solutions with a constant sum of added damping coefficients under the increased levels of the GMs. Therefore, the combination of these two algorithms enables the comprehensive search of the optimal solutions for various conditions of the level of the GMs and the sum of the added damping coefficients.

It is also noted that the proposed algorithm is applicable to the optimal hysteretic damper placement. In such case, added damping coefficient of viscous dampers is replaced to added stiffness of hysteretic dampers.

Figure 2: Overview and schematic diagram of A-CDGM, (a) overview, (b) schematic diagram

This section investigates the influence of the selection of the input GMs and the type of added dampers on the IDRCs. El Centro NS component during the Imperial Valley earthquake in 1940 and Rinaldi station fault-normal component during the Northridge earthquake in 1994 are employed here (Fig. 3). The former is a representative GM of random nature, and the latter is a representative GM of pulse type. The level of the GM is adjusted by the peak ground velocities (PGV). 51 levels of the GM (from PGV = 0.5 [m/s] to PGV = 1.5 [m/s] by the increment 0.02 [m/s]) are used. Linear viscous dampers are treated in Sections 3.2 and 3.3, and hysteretic dampers are treated in Sections 3.4 and 3.5.

Figure 3: Earthquake ground motions adopted for numerical examples, (a) El centro NS component, (b) Rinaldi sta. FN component

Consider two shear building models of 12 stories with different story stiffness distributions. Model 1 has a trapezoidal distribution of story stiffnesses (k1/k12 = 4). Model 2 has a uniform distribution of story stiffnesses. The undamped fundamental natural period of these two models is 1.2 [s] and the structural damping ratio is 0.01 (stiffness proportional type). All the floor masses have the same value. The common story height is 4 [m] and the common yield inter-story drift dy is 4/150 [m]. The story shear-interstory drift relation obeys the elastic perfectly-plastic rule.

The targeted values of the sum of the added damping coefficients are set to Wc,1=30×107[Ns/m],Wc,2=20×107[Ns/m], Wc,3=10×107[Ns/m]. The values of Wc, 1, Wc, 2, Wc, 3 are given so that the final designs has the fundamental-mode damping ratio of about 0.15, 0.10, 0.05. The initial damper design is set to c1,0⋯c12,0. Δcs=(50/600)×107[Ns/m] is used for the SBA, and nLS = 50 and Δcl=Unif[01]×{0.03Wc,iD} are employed for the local search.

Fig. 4 shows the IDRCs. It can be observed that the IDRCs are monotonically increasing with respect to PGV. In the case of Wc,3=10×107[Ns/m], the PGVs corresponding to dmax/dy=1 (elastic limit) are almost the same both under El Centro NS component and Rinaldi Sta. FN component. However in the case of Wc,1=30×107[Ns/m], the PGVs corresponding to dmax/dy=1 under Rinaldi Sta. FN component is smaller than that under El Centro NS component. Moreover, the slope of IDRCs under El Centro NS component is almost constant in the full range of PGV, however the slope of IDRCs under Rinaldi Sta. FN component inclines towards the horizontal direction in the upper range of PGV. This is because viscous dampers are not necessarily effective for pulse type GMs. Figs. 5, 6 show the changes of the added damping coefficient of each story. Figs. 7, 8 illustrate the distributions of the interstory drifts and the distributions of the added damping coefficient of the models designed for PGV = 0.5, 1.0, 1.5 [m/s]. The sum of the transfer functions of interstory velocity is also shown in Figs. 7, 8. It is noted that those values are normalized for Wc,iD,PGV=0.5, where ω1(=2π/1.2) denotes the undamped fundamental natural circular frequency. helps to understand the response control performance of viscous damper placements [31]. It can be pointed out that, as PGV used for the optimization becomes larger, the dampers concentrated to the specified stories spread around those stories to prevent the plastic deformation concentrations of the surrounding stories without dampers for larger levels of GMs [32]. Moreover, as PGV used for the optimization becomes larger, the amplitudes of the transfer functions become small in the higher range of frequency. These correspond to the result [31] that designs whose transfer function amplitudes are well controlled on a broader frequency range can effectively reduce plastic deformations because the higher-mode responses are amplified in the plastic response range. It is also observed that relatively large added damping coefficients are allocated to the lower stories under Rinaldi Sta. FN component compared to the models designed under El Centro NS component. This is due to the difference of the nature of the 2 GMs.

Figure 5: Change of added damping coefficient of each story with respect to PGV (Model 1), (a) El centro NS component, (b) Rinaldi sta. FN component

Figure 6: Change of added damping coefficient of each story with respect to PGV (Model 2), (a) El centro NS component, (b) Rinaldi sta. FN component

Figure 7: Distributions of added damping coefficient, distributions of interstory drifts and sum of transfer functions of interstory velocity (Model 1), (a) El centro NS component, (b) Rinaldi sta. FN component

Figure 8: Distributions of added damping coefficient, distributions of interstory drifts and sum of transfer functions of interstory velocity (Model 2), (a) El centro NS component, (b) Rinaldi sta. FN component

Fig. 9 shows the maximum interstory drift distributions by the IDA analysis for the models designed Wc,2=20×107[Ns/m] under each GM with PGV = 0.5, 1.0, 1.5 [m/s]. PGV in the IDA analysis is increased from 0.5 [m/s] to 1.5 [m/s] by 0.02 [m/s]. The input level PGV = 0.5 corresponds to the elastic design. It can be observed that the models designed for PGV = 0.5, 1.0 [m/s] exhibit large deformation concentrations in specific stories for these two GMs with larger PGV. The IDA curves are away from the IDRCs in the upper range of PGV. On the other hand, the models designed for PGV = 1.5 [m/s] are effective for various levels of the GMs and the IDA curves run near the IDRCs.

Figure 9: IDA analysis for 3 models designed under GMs with PGV = 0.5, 1.0, 1.5 [m/s] (Wc,2=20×107[Ns/m]), (a) Model 1, (b) Model 2

The yield deformations of hysteretic dampers are set to 4/800 [m]. The restoring force-deformation relation obeys the elastic perfectly-plastic rule. The target values of the sum of the added stiffnesses are set to Wk,1=15×109[N/m], Wk,2=10×109[N/m], Wk,3=5×109[N/m]. These values are given so that the equivalent fundamental damping ratios become 0.15, 0.10, 0.05 when the maximum interstory drifts of the models are four times of the yield deformation of the dampers under the harmonic wave with the undamped fundamental natural frequency. The initial damper design is set to k1,0⋯k12,0. Δks=(20/200)×109[N/m] is used for the SBA, and nLS = 100 and Δkl=Unif[01]×{0.001Wk,iD} are employed for the local search.

Fig. 10 shows the IDRCs. It can be observed that the IDRCs exhibit a roughly monotonic increase with respect to PGV. Unlike the cases of viscous dampers, the values of dmax under Rinaldi Sta. FN components are greatly improved as the sum of the added stiffnesses increases. This is because hysteretic dampers are more effective for pulse type GMs than viscous dampers [34]. Figs. 11, 12 show the changes of the added stiffness of each story. Figs. 13, 14 illustrate the distributions of the inter-story drifts and the distributions of the added stiffness of the models designed for PGV = 0.5, 1.0, 1.5 [m/s]. It can be pointed out that, unlike the cases of viscous dampers, the added stiffnesses are allocated to the stories which exhibit the large deformations as PGV used for the optimization becomes larger. For Model 1, the added stiffness in the middle and upper stories becomes large as PGV increases. For Model 2, the added stiffness in the lower stories becomes large as PGV increases. The dissipated hysteretic energy of the dampers is small under the lower levels of the GMs. Therefore, the control of the distribution of stiffness mainly contributes to the uniformization of the distribution of the maximum inter-story drifts under the lower levels of GMs. On the other hand, both the hysteretic energy dissipation of the dampers and the control of the distribution of stiffness play an essential role for the uniformization under the larger levels of GMs. It is also said that, like the cases of viscous dampers, relatively large added damping coefficients are allocated to the lower stories under Rinaldi Sta. FN component compared to the models designed under El Centro NS component.

Figure 11: Change of added stiffness of each story with respect to PGV (Model 1), (a) El centro NS component, (b) Rinaldi sta. FN component

Figure 12: Change of added stiffness of each story with respect to PGV (Model 2), (a) El centro NS component, (b) Rinaldi sta. FN component

Figure 13: Distributions of added stiffness and distributions of interstory drifts (Model 1), (a) El centro NS component, (b) Rinaldi sta. FN component

Figure 14: Distributions of added stiffness and distributions of interstory drifts (Model 2), (a) El centro NS component, (b) Rinaldi sta. FN component

Fig. 15 shows the maximum interstory drift distributions by the IDA analysis for the models designed for Wk,2=10×109[N/m] under each GM with PGV = 0.5, 1.0, 1.5 [m/s]. Almost all the tendencies are in common with the case of viscous dampers. Models designed for PGV = 0.5, 1.0 [m/s] exhibit large deformation concentrations in specific stories for these two GMs with larger PGV and the IDA curves are plotted away from the IDRCs in the upper range of PGV. On the other hand, the models designed for PGV = 1.5 [m/s] are effective for various levels of the GMs and the IDA curves run near the IDRCs.

Figure 15: IDA analysis for 3 models designed under GMs with PGV = 0.5, 1.0, 1.5 [m/s] (Wk,2=10×109[N/m]), (a) Model 1, (b) Model 2

This section treats a robust optimal design problem, and a simple local search-based algorithm is proposed. A simple index using the IDRC and the IDA curve of the model is used as the objective function.

Consider the following problem to mathematically obtain damper designs whose IDA curves run near the IDRCs.

The objective function represents the maximum value of the horizontal distances between the IDRC with W = Wc and the IDA curve of the design with at every IM in the IM-dmax graph. It is noted that the value of Wc,IMi is strictly larger than 0 for all i because Wc,IMi is optimally designed to reduce the maximum interstory drift under the GMs with IMi.

Step 3 Find the level of the GMs IMi where the value of Wc,IMi becomes the maximum by conducting the IDA analysis to the model with Wc,IMnL.

Step 4 Find the story j that exhibits the largest value of the interstory drift under the GMs IM = IMi.

Step 5 Find the story {k;ck>0} which exhibits the minimum value of the interstory drift under the GMs with IM = IMi.

Step 6 Update cj→cj+Δcl,2,ck→ck−Δcl,2(Δcl,2>0). The value of Δcl,2 is determined by a uniform random number. If ck < 0, revise the value of Δcl,2 so that the value of ck becomes 0. Then update iLS→iLS+1. If iLS < nLS, 2, return to Step 3. If iLS = nLS, 2, select the design which exhibits the minimum value of dmax from the nLS, 2 designs, and finalize the process.

The procedures of Steps 4–6 in the algorithm is common with the proposed local search method in Section 2.3. Wc,IMnL is used as the initial design because the objective function of models designed under larger levels of GMs tends to exhibit a small value as shown in Sections 3.3 and 3.5. It is also noted that the proposed algorithm is applicable to the optimal hysteretic damper placement.

The targeted value of the sum of the added damping coefficients are set to Wc=20×107[Ns/m]. El Centro NS component is employed here. The parameters of the A-CDGM are the same in Section 3.2. nLS = 200 and Δcl,2=Unif[01]×{0.03Wc} are employed.

Fig. 16 shows the results of the optimization. The IDA curve of the obtained design runs quite near the IDRC for various PGVs. Moreover, the distributions of the interstory drifts are uniformized for various PGVs.

Figure 16: Results of optimization (Wc=20×107[Ns/m]), (a) IDA curve of optimally designed model and IDRC, (b) distribution of maximum interstory drifts, (c) distribution of added damping coefficient

It may be concluded that the proposed algorithm provides robust damper designs for various levels of GMs. The proposed algorithm works well in spite of simplicity.

A new method of robust damper design was presented for elastic-plastic MDOF building structures under multi-level ground motions (GMs). The main conclusions can be summarized as follows.

(1) An ideal drift response curve (IDRC) is a plot of the optimized maximum deformation under a constraint on the total damper quantity versus the design IM (level of input GMs). An effectiveness of the design for various levels of GMs can be captured visually through plotting the incremental dynamic analysis (IDA) curve and the corresponding IDRC.

(2) A problem of generation of IDRCs was stated and the solution algorithm ‘advanced consecutive design generation method (A-CDGM)’ was developed. A-CDGM consists of the sensitivity-based algorithm and a local search method. The combination of these two algorithms enables the comprehensive search of the optimal solutions for various conditions of the level of the GMs and the sum of the added damping coefficient.

(3) The slope of IDRCs of viscous dampers under pulse type GMs inclines towards the horizontal direction in the upper range of IM. This is because viscous dampers are not necessarily effective for pulse type GMs. On the other hand, the values of the maximum interstory drifts under pulse type GMs are greatly improved as the sum of the added stiffnesses of hysteretic dampers increases. This is because hysteretic dampers are more effective for pulse type GMs than viscous dampers. In the case of GMs of random nature, both types of dampers effectively reduce the maximum interstory drifts.

(4) In the case of optimization of viscous dampers, as IM used for the optimization becomes larger, the dampers concentrated to the specified stories spread around those stories to prevent the plastic deformation concentrations at the surrounding stories without dampers for larger levels of GMs. Moreover, as IM used for the optimization becomes larger, the amplitudes of the transfer functions become small in the higher range of frequency.

(5) In the case of optimization of hysteretic dampers, the added stiffness is allocated to the stories which exhibit the large deformations as IM used for the optimization becomes larger. The dissipated hysteretic energy of dampers is small under the lower levels of the GMs. Therefore, the control of the distribution of stiffness mainly contributes to the uniformization of the distribution of the maximum interstory drifts under the lower levels of GMs. On the other hand, both the hysteretic energy dissipation of the dampers and the control of the distribution of stiffness play an important role for the uniformization under the larger levels of GMs.

(6) For both types of dampers, the IDA curves for the models designed under larger levels of GMs run near the corresponding IDRCs for various IMs. On the other hand, the IDA curves for the models designed under lower levels of GMs are away from the IDRCs in the upper range of IM. In other words, such designs may exhibit large plastic deformation concentrations in specific stories for larger-level GMs.

(7) A robust optimal design problem was formulated and a simple local search-based algorithm was proposed. A simple index using an IDRC and an IDA curve was used as the objective function. This method realizes a design which is effective for various levels of GMs and this means the robust optimal design. Moreover, the proposed algorithm is easy to use because of its simplicity.

In this paper, a single type of dampers was applied to the optimization. It is noted that the proposed method is applicable to the simultaneous optimization of various types of dampers. It is also noted that the proposed method is suitable for optimization under plural GMs. In such case, the envelope response under such plural GMs should be treated as the objective function.

Funding Statement: Part of the present work is supported by the Grant-in-Aid for Scientific Research (KAKENHI) of the Japan Society for the Promotion of Science (Nos. 18H01584, JP20J20811). This support is greatly appreciated.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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