### A Mathematical Formulation to Estimate the Fundamental Period of High-Rise Buildings Including Flexural-Shear Behavior and Structural Interaction

Document Type : Research Paper

Authors

1 Structural Engineering Research Center, International Institute of Earthquake Engineering and Seismology, Tehran

2 Multimedia University, Cyberjaya, Malaysia

3 Civil Engineering Department, University of Technology Malaysia, Johor Bahru, Malaysia

Abstract

The objective of the current study is to develop a simple formula to estimate the fundamental vibration period of tall buildings for using in equivalent lateral force analysis specified in building codes. The method based on Sturm-Liouville differential equation is presented here for estimating the fundamental period of natural vibration. The resulting equation, based on the continuum representation of tall buildings with various lateral resisting systems for natural vibration of the buildings, is proved to be the forth-order Sturm-Liouville differential equation, and a quick method for determining the fundamental period of natural vibration of the building is presented. Making use of the coupled wall theory for natural vibration, the method is extended to deal with vibration problem of other buildings braced by frame, walls or/and tube. The proposed formulation will allow a more consistent and accurate use of code formulae for calculating the earthquake-induced maximum base shear in a building. Use of the method is economical with respect to both computer time and equipment and can be used to verify the results of the finite element analyses where the time-consuming procedure of handling all the data can always be a source of errors.

Keywords

[1] Goel R. K., Chopra A. K., 1997, Period formulas for moment resisting frame buildings, Journal of Structural Engineering 123(11):1454-1461.
[2] Goel R. K., Chopra A. K., 1998, Period formulas for concrete shear wall buildings, Journal of Structural Engineering 124(4):426-433.
[3] Heidebrecht A. C., Smith S. S., 1973, Approximate analysis of tall wall-frame structures, Journal of the Structural Division 99(2):199-221.
[4] Smith B. S., Yoon Y. S., 1991, Estimating seismic base shears of tall wall-frame buildings, Journal of Structural Engineering 117(10):3026-3041.
[5] Wallace J.W., 1995, Seismic design of RC structural walls, Part I: New code format, Journal of Structural Engineering 121(1):75-87.
[6] Chaallal O., Gauthier D., Malenfant P., 1996, Classification methodology for coupled shear walls, Journal of Structural Engineering 122(12):1453-1458.
[7] Trifunac M. D., Ivanovic S. S., Todorovska M. I., 2001, Apparent periods of a building, Part II: Time-frequency analysis, Journal of Structural Engineering 127( 5):527-537.
[8] Goel R. K., Chopra A. K., 1997, Vibration Properties of Buildings Determined from Recorded, Earthquake Engineering Research Center , University of California, Berkeley.
[9] Rutenberg A., 1975, Approximate natural frequencies for coupled shear walls, Earthquake Engineering & Structural Dynamics 4(1):95-100.
[10] Wang Y.P., Reinhorn A.M., Soong T.T., 1992, Development of design spectra for actively controlled wall frame buildings , Journal of Engineering Mechanics 118(6):1201-1220.
[11] Smith S. B., Coull A., 1991, Tall Building Structures: Analysis and Design, Wiley, New York.
[12] Iwan W.D., 1997, Drift spectrum measure of demand for earthquake ground motions, Journal of Structural Engineering 124(4):397-404.
[13] Miranda E., Akkar S. D., 2006, Generalised interstorey drift spectrum, Journal of Structural Engineering 132(6):840-852.
[14] Miranda E., Reyes C. J., 2002, Approximate lateral drift demands in multistorey buildings with nonuniform stiffness, Journal of Structural Engineering 128(7):840-849.
[15] Computer and Structures Inc., 2011, SAP2000 Version 15, Structural Analysis Program, University of Berkeley, California.