Studying of some limitations of modal analysis versus D’Alembert solution in analysis of shear beam under impulsive loads

Document Type : Original Research

Authors
S. R. Naeemi 1
R. Saleh Jalali 2
1 MS Student in Structural Engineering, Department of Civil Engineering, Faculty of Engineering, University of Guilan, P.O.Box 3756, Rasht, Iran
2 Assistant Professor, Department of Civil Engineering, Faculty of Engineering, University of Guilan, P.O.Box 3756, Rasht, Iran
10.22034/22.4.7
Abstract
Using modal analysis is a lot easier and more widespread among structures, but the important question is about the number of modes should be considered in the modal analysis method to reach an answer with an inevitable error but in logical tolerance. In this regard, the ratio of the dominant period of the earthquake to the main period of the structure is used as a criterion for selecting the number of modes in the modal analysis method. On the other hand, although the maximum displacement of the structure occurs above it, but when the period of the pulse is less than the main period of the structure, due to wave motion along the structure, the maximum shear strain can occur not only at the base but also in other places along the structure. In this paper, some limitations of modal analysis versus Dchr('39')Alembert solution have been studied in analysis of shear beam under impulsive loads. For this purpose, the structure is modeled with a shear beam with linear material and zero damping, and it is analyzed by discrete (modal analysis) and continuous (Dchr('39')Alembert solution) methods. The time response of modal analysis has been done by the fourth-order Runge-Kutta method. The shear beam is subjected to short, medium, and long period half-sine pulses, relative to the main period of the structure, as well as two near-field earthquakes with distinct pulse. The envelope of maximum induced displacement and shear strain (drift) along the beam have been selected to compare the two methods. The necessary number of modes in modal analysis are determined in such a way that its difference with the exact method (Dchr('39')Alembert solution) would be in acceptable range. For shear beam with linear material and zero damping, as it is expected, the results indicate that for convergence of shear strain (drift) response to the exact solution more number of modes are needed than convergence of displacement response in the modal analysis. Under short period pulse , when the ratio of the period of the pulse or the predominant period of earthquake to the main period of the beam is less than , if the minimum number of modes in modal analysis would be 20 and 50 modes for displacement and shear strain, respectively, then the percentage of error of envelope of maximum induced displacement and shear strain (drift) in beam, calculated by modal analysis, would be less than 10 percent, respect to Dchr('39')Alembert solution. Under medium period pulse , when the ratio of the period of the pulse or the predominant period of earthquake to the main period of the beam is greater than and less than , for having ten percent difference between two methods of analyses, the necessary number of modes in modal analysis of beam would be and modes for displacement and shear strain, respectively. For the beam under long period pulse , when the ratio of the period of the pulse or the predominant period of earthquake to the main period of the beam is greater than , the necessary number of modes in modal analysis would be 1 and 5 modes for displacement and shear strain, respectively.

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[1] Biot, M. "A Mechanical Analyzer for the Prediction of Earthquake Stresses", Bull. Seismol. Soc. Am, vol. 31, pp. 151-171, (1941).
[2] Housner, G. "Spectrum Analysis of Strong Motion Earthquakes,", Bull. Seismol. Soc. Am, vol. 43, pp. 97-119, (1953).
[3] Algan, B. "Drift and Damage Considerations in Earthquake Resistant Design of Reinforced Concrete Buildings", Ph.D. Thesis,University of Illinois, Urbana Champaign, (1982).
[4] Moehle, J. "Strong Monitor Drift Estimate for R/C Structures", Journal of Structural Engineering, (1984).
[5] Gicev, V. and Trifunac, M. "Amplification of linear strain in a layer excited by a shear-wave Earthquake Pulse," Journal of Soil Dynamics and Earthquake Engineering, vol. 30, pp. 1073-1081, (2010).
[6] Iwan, W. "Drift Spectrum: measure of demand for earthquake ground motions," Structural Engineering, ASCE, vol. 123(4), pp. 397-404, 123(4)(1997).
[7] Miranda, E. "Approximate Seismic Lateral Deformation Demands in Multistory Buildings", Structural Engineering, vol. 125, no. 4, pp. 417-425, (1999).
[8] Chopra, A. K. and Chintanapakdee, C. "Drift Spectrum vs. Modal Analysis of Structural Response to Near-Fault Ground Motion", Earthquake Spectra, vol. 17, pp. 221-234, (2001).
[9] Roberts, M. and Lutes, L. D. "Potential for Structural Failure in the Seismic Near Field", journal of Engineering Mechanics, vol. 129, pp. 927-934, (2003).
[10] Akkar, S. and Yazgan, U. and Gülkan, P. "Drift Estimates in Frame Buildings Subjected to Near-Fault Ground Motions", Structural Engineering, vol. 131, no. 7, pp. 1014-1024, (2005).
[11] E. Miranda and S. D. Akkar, "Generalized Interstory Drift Spectrum", Structural Engineering, vol. 132, no. 6, pp. 840-852, (2006).
[12] Sasani, M. and Makris, N. and Bolt, B. A. "Damping in Shear Beam Structures and Estimation of Drift Response", Engineering Mechanics, vol. 132, no. 8, pp. 851-858, (2006).
[13] Xie, J. and Wen, Z. "A Measure of Drift Demand for Earthquake Ground Motions Based on Timoshenko Beam Model", The 14th Proceeding of World Conference on Earthquake Engineering, Beijing, (2008)
[14] Cichowicz, A. "Near-field Ground Motion Modal Versus Wave Propagation Analysis", Shock and Vibration Journal, vol. 17,pp. 611-617, (2010)
[15] Rupakhty, R. Sigbjornsonn, R. "Can Simple Pulses Adequately Represent Near-Fault Ground Motions?" Journal of earthquake engineering, vol 11, pp. 1260-1272, (2011)
[16] Raoofi, F. R. and Shodja, A. H. "Comparison of the Drift Spectra Generated Using Continuous and Lumped-mass Beam Models", Journal of Scientica Iranica, vol. 20, pp. 1337-1348, (2013)
[17] Ghanbari, B. and Khosravi, S. and Akhaveissy, A. "Effect of Pulse-like Ground Motions Parameters on Inter-story Drift Spectra", International Journal of Earthquake Engineering and Hazard Mitigation, vol. 3, No. 2, (2015)
[18] Liossatou, E. and Fardis, M. N. "Near-Fault Effects on Residual Displacements of RC Structures", Journal of Earthquake Engineering and Soil Dynamics, vol. 45, Issue 9, pp. 1391-1409, (2016)
[19] Gicev, V., Kisomi, H.B., Trifunac, M.D., Jalali, R.S., Flexibility of foundation increases the base shear and horizontal strains during an out–of– plane response to an SH pulse in linear and nonlinear soil. Soil Dyn Earthq Eng. 2019;127: 17.105837. https://doi.org/10.1016/j.soildyn. 2019.105837
[20] Todorovska, M.I., Girmay, E.A., Wang, F., and Rahmani, M. " Wave propagation in a doubly tapered shear beam: Model and application to a pyramid-shaped skyscraper," Journal of Earthquake Engineering and Structural Dynamics, pp. 1-29, (2021)
[21] Todorovska, M.I., Niu, B., Lin, G., Cao, C., Wang, D., Cui, J., Wang, F., Trifunac, M.D., Liang, J., A new full-scale testbed for structural health monitoring and soil–structure interaction studies: Kunming 48-story office building in Yunnan province, China. Struct Control Health Monit. 2020. https://doi.org/10.1002/stc.2545
[22] Todorovska, M.I., and Trifunac, M.D. (2006). Impulse response analysis of the Van Nuys 7-story hotel during 11 earthquakes (1971-1994): one-dimensional wave propagation and inferences on global and local reduction of stiffness due to earthquake damage, Report CE 06-01, Dept. of Civil Eng., University of Southern California, Los Angeles, California.
[23] Gicev, V. and Trifunac, M. " Permanent deformations and strains in a shear building excited by a strong motion pulse," Journal of Soil Dynamics and Earthquake Engineering, vol. 27, pp. 774-792, (2007)
[24] Trifunac, M.D. "POWER DESIGN METHOD",Proceeding of Earthquake Engineering in the 21th Century, Skopje-Ohrid, Macedonia, August 27-September 1, 2005.
[25] Gicev, V., and Trifunac, M.D. (2006). ROTATIONS IN THE TRANSIENT RESPONSE OF NONLINEAR SHEAR BEAM, Report CE 06-02, Dept. of Civil Eng., University of Southern California, Los Angeles, California.
[26] Rathje, E.M, Abrahamson, N.A, Bray, J.D., " SIMPLIFIED FREQUENCY CONTENT ESTIMATES OF EARTHQUAKE GROUND MOTIONS," Journal of Geotechnical and Geoenvironmental Engineering, 1998. 124: 150-159.
Modares Civil Engineering journal
Volume 22, Issue 4
July and August 2022
Pages 95-110