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.
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".
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.
The deformation at the top (x = H) is given by equation (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).
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.
at the top,
It becomes double that of the model with uniform shear stiffness. The ratio of bending and shear deformation is given by equation (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.
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=15n (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:
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.
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