**9. Current problem of existing high-rise buildings**

High-rise buildings have relative long natural period from the structural form. This charac‐ teristic is considered to be the most effective to avoid structural damages due to earthquake actions. However, when high-rise buildings subject to the action of the earthquake wave in‐ cluded the excellent long period components, a serious problem which the lateral deflection is remarkably large is produced in Japan. This phenomenon is based on resonance between the long period of high-rise buildings and the excellent long period of earthquake wave.

**Figure 24.** Numerical model

Urayasu 2011 NS

**Figure 25.** Time histories of acceleration (a) EL-Centro 1940 NS, (b) JMA Kobe 1995 NS, (c) Shinjuku 2011 NS, and (d)

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The 2011 Tohoku Earthquake (M 9.0) occurred many earthquake waves, which long period components are distinguished, on everywhere in Japan. These earthquake waves occur many physical and mental damages to structures and people living in high-rise buildings. The damage occurs high-rise buildings existing on all parts of Japan which appears long dis‐ tance from the source. People entertain remarkable doubt about the ability to withstand earthquakes of high-rise buildings. This distrust is an urgent problem to people living and working in high-rise buildings. Existing high-rise buildings are necessary to improve ur‐ gently earthquake resistance. This section presents about an urgency problem which many existing high-rise buildings face a technical difficulty.

Let us consider dynamic behavior for one plane-frame of a high-rise building, as shown in Figure 24.

**Figure 24.** Numerical model

other is the continuous condition corresponding to longitudinally adjoining constituents. Thus, Takabatake et al*.* [30] proved the efficiency of the two-dimensional extended rod theo‐

Two-dimensional extended rod theory has been presented for simply analyzing a large or complicated structure with setback in which the stiffness and mass due to the existence of setback vary rapidly in the longitudinal and transverse directions. The effectiveness of this theory has been demonstrated from numerical results for exemplified numerical models. The transverse-wise distribution of longitudinal stress for structures with setbacks has been clarified to behave remarkable nonlinear behavior. Since the structural form of high-rise buildings with setbacks is frequently adapted in the world, the incensement of stress distri‐ bution occurred locally due to setback is very important for structural designers. The present theory may estimate such nonlinear stress behaviors in the preliminary design stages. The further development of the present theory will be necessary to extend to the three-dimensional extended rod theory which is applicable to a complicated building with three dimensional behaviors due to the eccentric station of many earthquake-resistant struc‐

High-rise buildings have relative long natural period from the structural form. This charac‐ teristic is considered to be the most effective to avoid structural damages due to earthquake actions. However, when high-rise buildings subject to the action of the earthquake wave in‐ cluded the excellent long period components, a serious problem which the lateral deflection is remarkably large is produced in Japan. This phenomenon is based on resonance between the long period of high-rise buildings and the excellent long period of earthquake wave.

The 2011 Tohoku Earthquake (M 9.0) occurred many earthquake waves, which long period components are distinguished, on everywhere in Japan. These earthquake waves occur many physical and mental damages to structures and people living in high-rise buildings. The damage occurs high-rise buildings existing on all parts of Japan which appears long dis‐ tance from the source. People entertain remarkable doubt about the ability to withstand earthquakes of high-rise buildings. This distrust is an urgent problem to people living and working in high-rise buildings. Existing high-rise buildings are necessary to improve ur‐ gently earthquake resistance. This section presents about an urgency problem which many

Let us consider dynamic behavior for one plane-frame of a high-rise building, as shown in

ry to the general structures with setbacks.

276 Advances in Vibration Engineering and Structural Dynamics

tural members, such as shear walls with opening.

existing high-rise buildings face a technical difficulty.

Figure 24.

**9. Current problem of existing high-rise buildings**

**Figure 25.** Time histories of acceleration (a) EL-Centro 1940 NS, (b) JMA Kobe 1995 NS, (c) Shinjuku 2011 NS, and (d) Urayasu 2011 NS

This plane frame is composed of uniform structural members. The sizes of columns and beams are □-800 x 800 x 25 (BCP) and H-400 x 300 x 11 x 18 (SN400B), respectively. The iner‐ tia moment of beams takes twice due to take into account of slab stiffness. This plane frame is a part of a three-dimensional frame structure with the span 6 m between adjacent planeframes. The width and height are 36 m and 120 m, respectively. The main data used in nu‐ merical calculations are given in Table 8. Four kinds of earthquake waves are given in Table 9. El-Centro 1940 NS is converted the velocity to 0.5 m/s; JMA Kobe 1995 NS is the original wave with the maximum velocity 0.965 m/s; Shinjuku 2011 NS is the original wave with the maximum velocity 0.253 m/s; and Urayasu 2011 NS is the original wave with the maximum velocity 0.317 m/s. Figures 25(a) to 25(d) indicate time histories of accelerations for the four earthquake waves. In these earthquake waves, Shinjuku 2011 NS and Urayasu 2011 NS are obtained from K-net system measured at the 2011 Tohoku Earthquake. These earthquake waves are considered as earthquake waves included the excellent long periods. The excel‐ lent periods obtained from the Fourier spectrum of Shinjuku 2011 NS and Urayasu 2011 NS earthquake waves are 1.706 s and 1.342 s, respectively. The maximum acceleration and max‐ imum velocity of these earthquake waves are shown in Table 9.

It is very difficult to sort out this problem. If the existing structure stiffens the transverse shear rigidity of overall or selected stories, the dynamic responses produced by the earthquake wave included excellently long period decrease within initial design criteria for dynamic calcula‐ tions. However, inversely the dynamic responses produced by both EL-Centro 1940 NS with the maximum velocity 0.5 m/s and JMA Kobe 1995 NS exceed largely over the initial design criteria. The original design is based on flexibility which is the most characteristic of highrise buildings. This flexibility brings an effect which lowers dynamic responses produced by earthquake actions excluding long period components. Now, changing the structural stiff‐ ness from relatively soft to hard, this effect is lost and the safety of the high-rise building

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becomes dangerous for earthquake waves excluding the long period components.

**Figure 26.** Distribution of dynamic responses (a) dynamic lateral deflection, (b) story shear force, and (c) overturn‐

Author has not in this stage a clear answer to this problem. This problem includes two situa‐ tions. The first point is to find out the appropriate distribution of the transverse stiffness. The variation of the transverse stiffness is considered to stiffen or soften. In general, existing high-rise buildings are easily stiffening then softening. However, there is a strong probabili‐ ty that the stiffening of the transverse shear stiffness exceeds the allowable limit for the lat‐ eral deflection, story shear force, and overturning moment in the dynamic response subjected to earthquake waves used in original structural design. Therefore, the softening of the transverse shear stiffness used column isolation for all columns located on one or more selected story is considered to be effective. It is clarified from author's numerical computa‐ tions that the isolated location is the most effective at the midheight. The second point is to find out an effective seismic retrofitting to existing high-rise buildings without the move‐ ment of people living and working in the high-rise building. These are necessary to propose urgently these measures for seismic retrofitting of existing high-rise buildings subject to earthquake waves included excellently long wave period. This subject will be progress to

ensure comfortable life in high-rise buildings by many researchers.

ing moment


**Table 8.** Main data for numerical model

Figure 26(a) shows the dynamic maximum lateral displacement subjected to the four kinds of earthquake waves. The maximum dynamic lateral displacement subject to Urayasu 2011 NS is remarkable larger than in the other earthquake waves. Figures 26(b) and (c) indicate the maximum shear force and overturning moment of the plane high-rise building subject to these earthquake actions, respectively. Earthquake wave Urayasu 2011 NS which includes long period components influence remarkable dynamic responses on the current high-rise building.


**Table 9.** Maximum acceleration and maximum velocity of each earthquake wave

It is very difficult to sort out this problem. If the existing structure stiffens the transverse shear rigidity of overall or selected stories, the dynamic responses produced by the earthquake wave included excellently long period decrease within initial design criteria for dynamic calcula‐ tions. However, inversely the dynamic responses produced by both EL-Centro 1940 NS with the maximum velocity 0.5 m/s and JMA Kobe 1995 NS exceed largely over the initial design criteria. The original design is based on flexibility which is the most characteristic of highrise buildings. This flexibility brings an effect which lowers dynamic responses produced by earthquake actions excluding long period components. Now, changing the structural stiff‐ ness from relatively soft to hard, this effect is lost and the safety of the high-rise building becomes dangerous for earthquake waves excluding the long period components.

This plane frame is composed of uniform structural members. The sizes of columns and beams are □-800 x 800 x 25 (BCP) and H-400 x 300 x 11 x 18 (SN400B), respectively. The iner‐ tia moment of beams takes twice due to take into account of slab stiffness. This plane frame is a part of a three-dimensional frame structure with the span 6 m between adjacent planeframes. The width and height are 36 m and 120 m, respectively. The main data used in nu‐ merical calculations are given in Table 8. Four kinds of earthquake waves are given in Table 9. El-Centro 1940 NS is converted the velocity to 0.5 m/s; JMA Kobe 1995 NS is the original wave with the maximum velocity 0.965 m/s; Shinjuku 2011 NS is the original wave with the maximum velocity 0.253 m/s; and Urayasu 2011 NS is the original wave with the maximum velocity 0.317 m/s. Figures 25(a) to 25(d) indicate time histories of accelerations for the four earthquake waves. In these earthquake waves, Shinjuku 2011 NS and Urayasu 2011 NS are obtained from K-net system measured at the 2011 Tohoku Earthquake. These earthquake waves are considered as earthquake waves included the excellent long periods. The excel‐ lent periods obtained from the Fourier spectrum of Shinjuku 2011 NS and Urayasu 2011 NS earthquake waves are 1.706 s and 1.342 s, respectively. The maximum acceleration and max‐

> **Width:** @6 m x 6 = 36 m **Height:** @4 m x 30 floors = 120 m

Figure 26(a) shows the dynamic maximum lateral displacement subjected to the four kinds of earthquake waves. The maximum dynamic lateral displacement subject to Urayasu 2011 NS is remarkable larger than in the other earthquake waves. Figures 26(b) and (c) indicate the maximum shear force and overturning moment of the plane high-rise building subject to these earthquake actions, respectively. Earthquake wave Urayasu 2011 NS which includes long period components influence remarkable dynamic responses on the current high-rise building.

> **Maximum Acceleration m/s2**

> > 5.11 8.18 1.92 1.25

**Table 9.** Maximum acceleration and maximum velocity of each earthquake wave

**Maximum Velocity m/s**

> 0.500 0.965 0.253 0.317

imum velocity of these earthquake waves are shown in Table 9.

Weight per floor (kN/m2) 12 Young modulus *E* (N/m2) 2.06 x 1011 Shear modulus *G* (N/m2) 7.92 x 1010 Mass density ρ (N/m3) 7850 Damping constant 0.02 Poisson ratio 0.3

**Structure shape**

278 Advances in Vibration Engineering and Structural Dynamics

**Table 8.** Main data for numerical model

**Earthquake Wave**

EL-CENTRO 1940 NS JMA KOBE 1995 NS SHINJUKU 2011 NS URAYASU 2011 NS

**Figure 26.** Distribution of dynamic responses (a) dynamic lateral deflection, (b) story shear force, and (c) overturn‐ ing moment

Author has not in this stage a clear answer to this problem. This problem includes two situa‐ tions. The first point is to find out the appropriate distribution of the transverse stiffness. The variation of the transverse stiffness is considered to stiffen or soften. In general, existing high-rise buildings are easily stiffening then softening. However, there is a strong probabili‐ ty that the stiffening of the transverse shear stiffness exceeds the allowable limit for the lat‐ eral deflection, story shear force, and overturning moment in the dynamic response subjected to earthquake waves used in original structural design. Therefore, the softening of the transverse shear stiffness used column isolation for all columns located on one or more selected story is considered to be effective. It is clarified from author's numerical computa‐ tions that the isolated location is the most effective at the midheight. The second point is to find out an effective seismic retrofitting to existing high-rise buildings without the move‐ ment of people living and working in the high-rise building. These are necessary to propose urgently these measures for seismic retrofitting of existing high-rise buildings subject to earthquake waves included excellently long wave period. This subject will be progress to ensure comfortable life in high-rise buildings by many researchers.

## **10. Conclusions**

A simple but accurate analytical theory for doubly symmetric frame-tube structures has been presented by applying ordinary finite difference method to the governing equations proposed by the one-dimensional extended rod theory. From the numerical results, the present theory has been clarified to be usable in the preliminary design stages of the static and dynamic analyses for a doubly symmetric single or double frame-tube with braces, in practical use. Furthermore, it will be applicable to hyper high-rise buildings, e.g. over 600m in the total height, because the calculation is very simple and very fast. Next the approxi‐ mate method for natural frequencies of high-rise buildings is presented in the closed-form solutions. This method is very simple and effective in the preliminary design stages. Fur‐ thermore, the two-dimensional extended rod theory is introduced as for the expansion of the one-dimensional extended rod theory. Last it is stated to be urgently necessary seismic retrofitting for existing high-rise buildings subject to earthquake wave included relatively long period.

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