# Three models for numerical examples

Consider three shear building models of 12 stories with different story stiffness distributions. Model 1 has a uniform distribution of story stiffnesses. Model 2 has a straight-line lowest eigenmode (the story stiffnesses are obtained by the inverse-mode formulation). Model 3 has a stepped distribution of story stiffnesses (upper four stories, middle four stories, and lower four stories have uniform stiffness distributions with different values; the ratios among them are 1: 1.5: 2). The undamped fundamental natural period of these three 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 interstory drift ratio is 1/150. The story shear-interstory drift relation obeys the elastic-perfectly plastic (EPP) rule.

Figure 11.5 shows the eigenmodes multiplied by the participation factor (participation vectors) and the natural periods for three models.

*Figure 11.5* Eigenmodes multiplied by participation factors (participation vectors) and natural periods for three models: (a) Model I, (b) Model 2, (c) Model 3 (Akehashi and Takewaki 2019).

# Dynamic pushover analysis for increasing critical double impulse (DIP: Double Impulse Pushover)

The Kumamoto earthquake (Japan) in 2016 strongly influenced the definition of design earthquake ground motions. The maximum ground velocity over 2 m/s is far from common sense in the earthquake-resistant design of structures. To determine the input velocity level of the critical double impulse, an idea similar to the incremental dynamic analysis (IDA) procedure (Vamvarsikos and Cornell 2001) is applied to the critical double impulse. It should be reminded that only the critical set (amplitude and time interval of the double impulse) is treated here. In other words, the interval of two impulses of the double impulse is varied depending on the input velocity level (also depending on the maximum interstory drift). Akehashi and Takewaki (2019) called this procedure “Double impulse pushover (DIP).” DIP provides the relation between the maximum interstory drift and the input velocity level of the critical double impulse. While the conventional IDA includes multiple recorded ground motions for taking into account the uncertainty in predominant periods of ground motions, DIP adopts the critical input and enables an efficient analysis of the relation between the maximum response and the input level.

Figure 11.6 shows the maximum interstory drift distributions by DIP. The velocity level is increased from V = 0.2 m/s to V = 1.6 m/s by 0.2 m/s.

Since the input velocity level of the critical double impulse influences the maximum interstory drift and the optimal damper placement greatly, its determination appears very important. The determination process of the input velocity level of the critical double impulse is explained in the next.

- 1. Specify the maximum interstory drift of the initial design model (bare model without dampers).
- 2. Conduct DIP for the initial design model. Find the velocity level V for which the maximum interstory drift of the initial design model exceeds the specified value for the first time. Conduct DIP also for larger values of the velocity level V.
- 3. Draw the maximum interstory drift distributions by DIP as shown in Figure 11.6 with respect to the input velocity level. Realize how easily the plastic deformation is concentrated to a special location. Based on these results, determine the input velocity level of the critical double impulse so that the maximum interstory drift exceeds the specified value.

Model 3 is treated as an example of determining the input level. The double of the yield interstory drift is taken as the specified target value of the maximum interstory drift. Then, it was found that over 0.6 m/s is necessary.

*Figure 11.6* Maximum interstory drift by DIP: (a) Model I, (b) Model 2, (c) Model 3 (Akehashi andTakewaki 2019).

Although the input velocity level should be chosen for each structural model, V = 0.84 m/s and 1.23 m/s are employed in the following section.

# Numerical examples

## Examples for Problem 1 using Algorithm 1

Consider first some examples for Problem 1. The amplitudes of the critical double impulses were determined from the results for the DIP analysis explained in Section 11.5.

Figure 11.7 shows the distribution of damping coefficients of added dampers, ma*x(d _{max}Jd_{y})* with respect to step number and the distribution of

*dmaxjdy*under the critical double impulses with V = 0.84 [m/s] and V = 1.23 [m/s] for Model 1. The condition t/

_{targct},, =

*d*(for all

_{y}*i)*is adopted.

Figure 11.8 presents similar figures for Model 2 and Figure 11.9 illustrates those for Model 3.

Figures 11.7,11.8, and 11.9 show that as the input velocity level increases, the ratios of damping coefficients of added dampers along height change

*Figure 11.7* Distribution of added damping coefficients, max(d_{max} ,/d_{y}) with respect to step number and distribution of d_{max} *,ld _{y}* under critical double impulse for Model I (Problem I): (a)

*V =*0.84 [m/s], (b)

*V =*1.23 [m/s] (Akehashi and Takewaki 20 19).

*Figure 11.8* Distribution of added damping coefficients, rnax(d_{max} ,/d_{y}) with respect to step number and distribution of B_{max} */d _{y}* under critical double impulse for Model 2 (Problem I): (a)

*V =*0.84 [m/s], (b)

*V*= 1.23 [m/s] (Akehashi andTakewaki 2019).

*Figure I 1.9* Distribution of added damping coefficients, rnax(d_{max} ,/d_{y}) with respect to step number and distribution of d_{max} *,ld _{y}* under critical double impulse for Model 3 (Problem I): (a)

*V =*0.84 [m/s], (b)

*V*= 1.23 [m/s]: (a)

*V =*0.84 [m/s], (b)

*V =*1.23 [m/s] (Akehashi and Takewaki 201 9).

and their allocation becomes smooth in the wide height range. This is because as the input level becomes larger, the number of stories experiencing plastic deformation increases. Secondly, the maximum interstory drift distribution in the final model is controlled to become almost uniform by Algorithm 1. Furthermore, the increase of ma*(d _{max}Jd_{y})* in the damper allocation process is allowed.

For Model 3, it can be seen that the dampers are not allocated in the fourth and eighth stories for the input level V = 0.84 [m/s], but those are allocated for the input level V = 1.23 [m/s]. However, the maximum interstory drifts in those stories are smaller than the elastic limit in the initial stage. This indicates that Algorithm 1 is applied first so that the dampers are allocated to the first, fifth, and ninth stories experiencing large plastic deformation. This process helps the energy required for inducing deformation distribute to the neighboring stories. As a result, among the stories neighboring to the fifth and ninth stories, the deformations in the sixth and tenth stories with relatively small stiffness becomes larger. Since the fourth and eighth stories with relatively large stiffness go into the plastic range for the input level of V = 1.23 [m/s], the dampers are allocated so as to strengthen the model. A similar observation may be possible also for Model 1 and 2.

It can be summarized that Algorithm 1 is more apt to allocate added dampers to the stories where the plastic deformation develops first then to allocate additional ones to rather weak stories after the strengthening is completed.

## Examples for Problem 2 using Algorithm 2

Consider some examples for Problem 2 in the following. The parameter specification ЮОДс = *c ^{T}add ■* 1 is given here.

Figure 11.10 shows the distribution of damping coefficients of added dampers, £*ma x,M_{y} with respect to step number and the distribution of d_{m3X }j/d_{y} under the critical double impulses with V = 0.84 [m/s] and V= 1.23 [m/s] for Model 1. The distributions for the elastic limit are also shown for reference (ma(d_{mas}Jd_{y}) = 1).*

Figure 11.11 presents similar figures for Model 2 and Figure 11.12 illustrates those for Model 3.

It can be observed that when the elastic limit is employed as the target for determining the input velocity level, the dampers are allocated so that the maximum interstory drifts become almost uniform for all models (Models 1-3). For Model 1, the dampers are allocated so that the maximum interstory drifts become almost uniform regardless of the input velocity level. On the other hand, for Models 2 and 3, the damper distributions are different depending on the input velocity level. Furthermore, for Models 2 and 3, the maximum interstory drifts in specific stories do not change between the initial model and the final model. This means that, since Algorithm 2 is aimed at finding the optimal damper allocation by seeking the steepest

*Figure I 1.10* Distribution of added damping coefficients, *£,d _{maxl}/d_{y}* with respect to step number and distribution of

*d*under critical double impulse for Model I (Problem 2): (a) elastic limit, (b)

_{max}Jd_{y}*V =*0.84 [m/s], (с)

*V =*1.23 [m/s] (Akehashi and Takewaki 2019).

direction (most effective direction for deformation reduction), it does not provide better damper allocation from the viewpoint of uniform reduction of the maximum interstory drifts in all stories.

## Examples for Mixed Problem (Problem 3) of Problem 1 and 2 using Algorithm 3

Examples for Mixed Problem (Problem 3) of Problem 1 and 2 using Algorithm 3 are presented in this section. The parameter specification by ЮОДс = *c ^{T}aii* • 1 (100 step allocation of total dampers) is used for

V = 0.84 [m/s] and the parameter specification by 250Дс = *c ^{T}add ■* 1 (250 step allocation of total dampers) is used for V = 1.23 [m/s].

Figure 11.13 shows the distribution of damping coefficients of added dampers, * _{x}Jd_{y}* with respect to step number and the distribution of

*d*under the critical double impulses with V = 0.84 [m/s] and

_{maxj}/dyV =1.23 [m/s] for Model 1.

*Figure I l.l I* Distribution of added damping coefficients, £d_{max} with respect to step number and distribution of d_{max} *,ld _{y}* under critical double impulse for Model 2 (Problem 2): (a) elastic limit, (b)

*V*= 0.84 [m/s], (с)

*V =*1.23 [m/s] (Akehashi and Takewaki 20 I 9).

Figure 11.14 presents similar figures for Model 2 and Figure 11.15 illustrates those for Model 3.

From the above analysis, we can observe the following facts. The analysis in the elastic range is not easy by Algorithm 1 for Problem 1 owing to the inability to set the total damper quantity and the maximum interstory drift distribution of the final model obtained by Algorithm 2 for Problem 2 is unstable (not uniform) depending on the model. On the other hand, a stable damper allocation is possible by Algorithm 3 for Problem 3 regardless of the input velocity level of the critical double impulse and the final maximum interstory drift distributions are apt to become uniform. From these results, it may be said that Algorithm 3 enables the procedure to guarantee the minimum performance by using a small amount of added dampers and to reduce the structural response in a global sense by using the additional amount of added dampers.

*Figure I 1.12* Distribution of added damping coefficients, £d_{max} ,/d_{x} with respect to step number and distribution of *d _{m3x} Jd* under critical double impulse for Model 3 (Problem 2): (a) elastic limit, (b)

*V =*0.84 [m/s], (с)

*V =*1.23 [m/s] (Akehashi and Takewaki 2019).