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For high rise buildings, if the stiffness distribution is unsuitable, whipping phenomenon can occur at the top, and the upper part will be subject to extreme shaking. On account of this, the section of the upper part is limited not from required strength considerations but by appropriate stiffness distribution. In the case of structural planning of a pure frame building, if the story and span numbers are fixed, column / beam sections of lower stories can be established by long term axial force limitation and base shear coefficient. Then the shear stiffness for lower stories is decided. After establishing a suitable shear stiffness distribution, dimensions of members can be decided.

This paper describes the examination of shear and bending deformation for reinforced concrete frame buildings, and the appropriate stiffness distribution to avoid whipping phenomenon. Finally, the method of deciding the member section is shown.

The bending stiffness *EI* of a frame building of height *H* and span
*m* (building width B = *ml*) as shown in figure 1, is given by
equation (1), in which the column section is square and section area is
*Ac*.

„„„„(1)

The story shear stiffness *GA* is given by equation (2) by the Muto's D
method when the story height is *h* and stiffness coefficients obtained from
stiffness ratio of columns to beams are all "*a*".

„„„„(2)

With a model of the bending-shear system fixed at base with height H

,
constant bending stiffness *EI* and shear stiffness *GA*,
the bending deformation and the shear deformation
at the top are obtained by equation (3) where *x* is the distance
from base.

„„„„(3)

The deformation at the top (*x* = *H*) is given by equation (4).

„„„„(4)

*EI* and *GA* obtained from equations (1) and (2) are substituted
into this expression. When the stiffness ratio of column / beam is equal,
the coefficient "*a*" becomes 0.5. Usually it comes within the range
0.2-0.4 because beam span is longer than height and the section of a
column is greater than the section of a beam for a building designed
by the beam yielding method. is known as the column ratio and it is
about 0.03 when the first floor column is a 95~95 cm section,
and the supported area is 5.5~5.5 m. The ratio of deformation of bending
and shear deformation at the top is given by equation (5).

„„„„(5)

On the other hand, for the bending-shear model in which shear stiffness
*GA* varies linearly to zero toward the top, it is given by the
following equation.

„„„„(6)

at the top,

„„„„(7)

It becomes double that of the model with uniform shear stiffness. The ratio of bending and shear deformation is given by equation (8).

„„„„(8)

The ratio is between equation (5) and (8) for a real building, and would be
0.02(*n*/*m*)^2 . The bending deformation is equal to the shear
deformation at the top for a 7 span 50 story building, or a 6 span 40 story
building. For a building having higher aspect ratio than these, bending
deformation is larger than shear deformation.

From equation (1), bending stiffness is proportional to column cross-sectional area, and shear stiffness is in proportional to the square of column cross-sectional area according to equation (2). Therefore, bending stiffness reduction is proportional to the square root of shear stiffness reduction by decreasing cross-sectional area of the column. The effect of stiffness distribution on eigen mode was examined using this relation in the study.

These eigen modes and story drift modes are shown in figure 2. For the first mode, the story drift is large at the lower part when the bending deformation rate is small, and is greatest at the top part when there is considerable bending deformation. The greater the rate of bending deformation, the larger the story drift in the higher part.

When there are few bending deformation, the change of eigen mode with stiffness reduction is considerable. The shape becomes susceptible to whip with stiffness reduction. The change in eigen mode with stiffness reduction with a large bending deformation rate is small. If shear stiffness at the top is more than 0.5 times that at the base and the bending deformation is smaller than the shear deformation at the top, the first mode shape of story drift never become large at the upper part. When the bending deformation is larger than the shear deformation at the top, the story drift is large in the upper part even for the uniform system. The tendency increases with decreasing the shear stiffness.

For the case where the shear stiffness decreases to 0.2 at the top, the story drift mode becomes considerable at the upper part, and will lead to whipping phenomenon by a combination with response spectrum. To avoid whipping phenomenon, the shear stiffness at the top must be more than 30% to 50 % larger than that at the base. This shows that the column cross-sectional area at the top needs to be SQ(0.5)ą0.7 times that at the top, and column dimensions needs to be SQ(0.7)ą0.85 times those at the base when the top shear stiffness is larger than 50% of the bottom stiffness.

In reinforced concrete column members, limit deformation and relation of axial force ratio are suggested from meaning of ductility considerations (Inai et al., 1997). The following equation is given for a column with constant axial force.

„„„„(9)

Here,*Å* is the axial force ratio for core cross-sectional area and
*R* is the drift limitation. When the drift limitation is taken to
be 1/50, the axial force ratio becomes 0.36. Because core cross-sectional
area is assumed to be 0.75 times that of the total section in the
literature, it becomes around 0.27 at the axial force ratio limitation
for total sections.

As the horizontal load is equivalent to weight of 4 story
(Shimazaki et al., 1994), regardless of building height, column section
is fixed only by the axial force limit for high rise buildings. Based on
the assumption of a 0.25Fc axial force limit, 0.03 column rate and
1.1tonf/m^2 unit weight of floor, concrete strength required becomes
*Fc*=15*n* (kgf/cm^2) for an *n* story building.

First, the sections of the lower story are set. Column sections are established from the long term axial force limit of 0.25Fc. Beam moment is calculated from base shear required, and section / reinforcing arrangement of the beam is established. Column strength at the base is made double the beam strength for yielding column at the base after beam yielding. If reinforcing bar can not be arranged in the column section, the section is changed. The calculation procedure and the section calculated are shown in table 1.

Next, section setting for each story is performed as follows:

- Shear stiffness distribution is set with a linear distribution of 30-50 % at the top.
- On the assumption that the stiffness coefficient "
*a*" is constant, column section and beam section are decided and total bending stiffness and story shear stiffness are calculated from the set member section. If the assumed value of "*a*" is extremely different from that calculated, the section must be revised. - The drift for inverse triangle distribution horizontal load is calculated, and section revised if extremely uniform.
- The eigen value problem is solved using the elastic stiffness calculated in 2). Using this result and design displacement response spectrum, the value of drift is estimated by SRSS (Square Root of Sum of Squares) of the 1st-5th mode drift. Here, the design displacement response spectrum is defined as pseudo-displacement response spectrum converted from the velocity response spectrum which is bilinear with maximum response velocity of 150cm/s. The design criteria are confirmed using this estimated value.
- The required
*CB*is determined from the first eigen period. - The required strength distribution and moment at nodal point are calculated using eigen mode and design response spectrum.
- The beam moment at face position is calculated and necessary beam main reinforcement section is fixed.
- The yield moment of beam at face is converted into nodal point moment, and nodal point moment divided by column's moment. The column shear force and story shear strength are determined, and confirmed to be approximately 1 to 1.2 times the required story strength.
- With a moment strength magnification factor for the column of 1.5, the required column main reinforcement section is found.

Comparison with the estimated reinforcing arrangement of the lower story shown in table 1, the reinforcing arrangement amount becomes large for the 25 story building, because the first period is short and weight increases. For the 60 story building, because the section is less than that of general structures due to the use of high strength materials, and the period is longer than the abbreviated expression, the required shear force becomes small, and the reinforcing arrangement also becomes small.

The eigen periods agree well. There is almost no difference in both total eigen mode and drift mode. It can therefore be said that the eigen period and eigen mode shape obtained by this abbreviated algorithm have sufficient precision.

Calculated results are shown in figure 7. The response values by frame analysis are less than the estimated values and the distribution of drift for every ground motion is similar. Thus it can be said that if the section for a building is set by the proposed method, the building has the earthquake resistance intended by a design method such as drift limitation.

- Shear stiffness at the top need to be 0.3-0.5 times the base stiffness so as not to produce whipping phenomenon.
- Primary setting of the member can be achieve from base shear in the lower story, axial force limitation and stiffness distribution.
- The building designed by the proposed method has the earthquake resistance intended by a design method such as drift limitation.

Shimazaki, K. (1992). Seismic coefficient distribution of high-rise reinforced concrete buildings. Proc. 10th WCEE, 4281-4286, Madrid, Spain

Inai, E. and H. Hiraishi (1997). Structural design charts and equations of deformation capacity of reinforced concrete columns after flexural yielding. Proc. 11th WCEE (will be published), Mexico

Shimazaki, K. and A. Wada (1994). Design base shear coefficients for high-rise reinforced concrete buildings. Journal of structural and construction engineering, AIJ, No. 458, 99-108, Tokyo, Japan