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Edited by: Tomaso Trombetti, Università di Bologna, Italy

Reviewed by: Emanuele Brunesi, European Centre for Training and Research in Earthquake Engineering, Italy; Michele Palermo, Università di Bologna, Italy

Specialty section: 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) or licensor 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.

A new structural control system using damper-installed shear walls in lower stories with reduced stiffness is proposed for vibration control of high-rise RC buildings. That system has some design variables, i.e., height of shear wall, degree of stiffness reduction at lower stories, and quantity of dampers. In this paper, some parametric studies on the shear-beam model with a stiff beam against two kinds of ground motion, a pulse-type sinusoidal wave and a resonant sinusoidal wave, are conducted to clarify the vibration characteristics of the proposed structural control system. It is shown that the optimal combination of design parameters depends on the input ground motion. It is also shown that it is possible to prevent from increasing the response under the one-cycle sinusoidal input resonant to the lowest mode and reduce the steady-state response under the harmonic input with the resonant fundamental period by reducing the stiffness in the lower structure and increasing the damper deformation.

After the occurrence of unexpected earthquake damage, many structural engineers are striving for resilient building structures tough for extreme earthquake inputs and recoverable fast to an acceptable level (Bruneau and Reinhorn,

In response to these circumstances, various kinds of vibration-controlled systems have been developed in the last three decades (Housner et al.,

Base-isolation systems have been employed mainly in Japan, New Zealand, China, and US. Various types of base-isolation systems have been introduced for pulse-type ground motions (Jangid and Datta,

Base isolation has been applied even to high-rise buildings. One is a base-isolated high-rise building without connection and the other is a base-isolated building connected to another structure with dampers (Murase et al.,

Historically tuned mass dampers (TMDs) have often been used for reducing building responses to wind loading and have been actually installed in many high-rise buildings (Soong and Dargush,

Nevertheless, large mass ratio TMDs have been explored mainly for earthquake inputs (Chowdhury et al.,

Recently, large mass ratio TMDs were tackled for base-isolated buildings (Villaverde,

The use of a braced or mega-braced core is another effective method for improvement of structural responses under seismic loading (Brunesi et al.,

It is also important to develop methods for estimation of displacement and velocity profiles for framed structures with added dampers. The methods proposed by Palermo et al. (

The present authors proposed a new vibration-controlled system in which large deformation in lower stories is induced and oil dampers are installed effectively (Tani et al.,

Proposed vibration-controlled structure and its modeling into multi-degree-of-freedom model and reduced model:

In this paper, parametric study on shear-beam model with stiff beam against two kinds of ground motion, pulsed sinusoidal wave and resonant sinusoidal wave, is conducted to clarify the vibration characteristics. It is shown that the optimal combination of variables depend on input ground motion, however, considering damper deformation growth by shear-wall, slight stiffness reduction at lower stories can achieve smaller story drift than proportional damping.

Consider a 50-story shear building model, as shown in Figure

Since the analysis of the original model shown in Figure _{w}_{e}_{e}_{θ} denote the equivalent mass of the lower structure, the equivalent height of the lower structure, and the rotational stiffness of the spring at the base. When the fundamental natural circular frequency of the lower structure is denoted by _{i}_{i}_{e}_{e}

By requiring the equivalence _{1}_{e}_{1} = _{e} in Eq. _{e}_{e}

As _{e}_{w}_{e}

The model shown in Figure

Change of natural period and participation vector to wall height:

Figure _{w}_{w}

The model shown in Figure _{w}_{w}

Transfer function:

When the building with the proposed system is subjected to pulse-type ground motions, the energy dissipation by repeated vibration cannot be expected. In this case, the strict check of strength is important in the lower parts. In this section, the one-cycle sinusoidal waves resonant to the fundamental and second natural periods are input. The maximum response is evaluated by the mean of “the absolute sum of the maximum fundamental and second vibration components” and “the SRSS value.” The accuracy of this evaluation method will be investigated later.

In this section, a continuum shear-beam model with uniform mass density is dealt with in order to enable the closed-form mathematical treatment. A massless rigid bar with pin at the base is connected to this shear-beam model in the lower part. The lowest mode of the upper part is assumed based on data on the realistic high-rise building models.

In most high-rise buildings, it is often the case that the inter-story drifts are almost uniform in the middle stories and they decrease toward the top and the bottom. Furthermore, in the proposed model, a rigid bar is installed in the lower part. Based on these information, the fundamental mode is assumed to be expressed by Eq.

In Eq. _{w}_{l}_{t}_{w}_{l}_{t}_{w}_{l}_{t}_{w}_{l}_{t}_{l}_{l}_{t}

The fundamental mode has the property of Eq. _{l}_{t}

The sum of the story stiffnesses in the upper part is the same between the investigation model and the comparison model. The sums of the story stiffnesses in the lower part and their distributions are different depending on the adopted mode. In case of

Since the structural damping is the stiffness-proportional one, the sum of damping coefficients of the model becomes smaller for the model with a smaller sum of the story stiffnesses in the lower part and longer fundamental natural period. The additional damping coefficients are assumed not to change.

The relations of mass, stiffness, and damping between the investigation model and the comparison model are expressed as follows:

Figure

Fundamental mode and its story drift angle:

Realistic models are considered here. Figure _{w}_{t}_{w}_{t}

Investigated region of fundamental-mode shape.

The additional damping quantity

When the fundamental mode is expressed by
_{t}

Substitution of Eqs.

The structural damping coefficient of the investigation model can then be expressed by

Consider the additional damping coefficient of the investigation model. First of all, the stiffness and structural damping coefficient of the comparison model can be obtained by substituting

Similarly, the additional damping coefficient of the comparison model can be expressed by

The sum of the additional damping coefficients of the comparison model can then be obtained as

Since the additional damping coefficient of the investigation model is given only at the lower parts as one proportional to the corresponding stiffness, the additional damping coefficient of the investigation model is expressed as follows:

Equations

The lowest-mode additional damping ratio

It is known that the second mode of a shear-beam with uniform mass density and a straight-line fundamental mode can be expressed by a cubic function in the comparison model. However, some modification is necessary in the investigation model. For this reason, the second mode above the point _{w}

The slope of the second mode can be obtained as

The dynamic equilibrium of the part from _{w}_{t}_{L}_{w}

The second damping ratio of the investigation model is evaluated by using an undamped second natural mode. The additional damping ratios (the additional damping quantity D_{2%}) of the investigation model for various shear wall heights are shown in Figure _{w}_{w}_{w}

Damping ratio with respect to

Consider the maximum story drift angle of the model subjected to the resonant (first mode and second mode) one-cycle sinusoidal wave with constant velocity amplitude of 1 m/s. The total response is evaluated approximately by superposing the first and second mode vibrations. The maximum response is evaluated by the mean of the absolute sum of the first and second mode vibrations and the SRSS value.

The displacement response of the SDOF model under a one-cycle sinusoidal acceleration wave

Equation _{dmax}_{p}_{dmax}_{p}

Since the lower structure exhibits a straight-line displacement due to the existence of the rigid shear wall, the story drift angle shows a constant distribution. In case of _{w}_{l}_{t}

Figures _{2%}. It should be noted that the story drift angles are expressed under the condition of _{t}_{w}_{w}_{w}

Maximum story drift angle for lowest-mode resonance pulse:

Figures _{2%}. It can be observed that, although the response by the second mode becomes larger than that by the fundamental mode around the top, the response by the fundamental mode becomes dominant except around the top. Since the difference of the fundamental and second natural periods becomes large as _{w}_{w}

Maximum story drift angle for second-mode resonance pulse:

Figure

Comparison of approximation with time-history response analysis:

Figure _{w}

Maximum story drift angle under various damping quantities (_{w}

When the stiffness in the lower structure is reduced, the response of the investigation model under the one-cycle sinusoidal wave resonant to the lowest mode becomes larger than that of the comparison model. The proposed model has an advantage that the reduction of the stiffness in the lower structure enhances the performance of the dampers. Figure _{w}_{2%} and D_{5%}. As the damper quantity becomes larger, the necessary damper growth rate becomes smaller. The influence of _{w}

Necessary damper deformation growth rate.

Since the long-period, long-duration ground motions can be represented approximately by long-duration sinusoidal waves and steady-state vibrations are predominant in such vibration, the resonant input to the fundamental natural period is considered here.

It is well known that, since the steady-state response to the resonant harmonic input can be influenced greatly by the magnitude and distribution of damping in the structure, it may be difficult to use the undamped vibration mode for response evaluation. For this reason, the shear beam treated in the previous section is reduced to the 2DOF model in which one mass is located at _{w}_{1} and the other mass is located at _{2}. The accuracy of this model will be investigated later. The height _{2} is determined from the equivalence of the participation vectors of these two models. In addition, the fundamental natural frequencies, the participation vectors at _{w}_{w}_{w}

Correspondence of participation vectors and story shear forces between shear-beam model and 2DOF model.

Let _{w}

The quantities with that indicate the quantities of the 2DOF model. The additional damping coefficient in the lower part can be obtained as

The story shear _{w}

The height _{2} at which the equivalence of the participation vectors of the two models, i.e.,

The amplitude of the steady-state displacement of the 2DOF model under the harmonic input with constant velocity amplitude is computed for various shear wall heights. These values are drawn with respect to the parameter

Figure _{2%} and Figure _{5%}.

Maximum story drift angle ratio (variable: _{w}_{2%}), _{2%}), _{5%}), and _{5%}).

Figure _{w}_{t}_{w}_{t}

Maximum story drift angle ratio (variable: _{w}_{w}_{w}_{w}

It can be observed that, as the stiffness in the lower structure decreases, the story drift angle in the upper structure also decreases. While the story drift angle in the lower structure also decreases according to the decrease of the stiffness in the lower structure in the case of large additional damping in the lower structure, it increases in the case of small additional damping in the lower structure. This means that the effect of response reduction due to the damping in the steady-state vibration is larger than the effect of response amplification due to the stiffness reduction in the case where a certain level of additional damping is introduced. Furthermore, the influence of _{w}

Figure

Comparison of time–history responses between 2DOF and MDOF: _{w}_{5%} and _{w}_{5%}.

A new structural control system using damper-installed shear walls with a pin connection at the bottom has been proposed for vibration control of high-rise RC buildings. The response analysis for one-cycle sinusoidal ground motions resonant to the lowest and second natural modes and for the harmonic ground motion has been performed. The obtained results are summarized as follows.

The lowest-mode component is predominant in the response resonant to the lowest mode and the maximum response becomes the smallest in the model with a straight-line lowest mode. It is more effective to control the stiffness and the mode shape than concentrating the additional dampers into the location with larger story drift angles (see Figure

The lowest-mode component is predominant in the response of the lower structure resonant even to the second mode. Since the difference of the fundamental and second natural periods becomes large as _{w}

The steady-state vibration resonant to the lowest mode becomes smaller by reducing the stiffness in the lower structure (see Figure

Since the rigid shear wall makes the story drift angle distribution uniform and reduces the maximum response, the response amplification by the second mode in the lower structure is prevented as _{w}

It may be possible to prevent from increasing the response under the one-cycle sinusoidal input resonant to the lowest mode and reduce the steady-state response under the harmonic input with the resonant fundamental period by reducing the stiffness in the lower structure and increasing the damper deformation (see Figure

The maximum response evaluation under the resonant one-cycle sinusoidal input using the undamped natural modes is accurate and reliable. However, as the higher-mode effect becomes larger, the accuracy may deteriorate slightly (see Figure

The response amplitude evaluation under the harmonic input can be made within a reliable accuracy by using the 2DOF model which is reduced from the shear-beam model (see Figure

TT carried out the theoretical and numerical analysis. RM carried out the theoretical investigation. IT supervised the theoretical analysis.

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. The reviewer MP and handling editor declared their shared affiliation.