Continuous Wavelet and Fourier Transform Methods for the Evaluation of the Properties of Critical Excitation

Document Type : Research Article


1 Office No. 57, Department of Civil Engineering, Shahrekord University, Shahrekord, Iran.

2 Shahrekord University, Shahrekord, Iran

3 Associate Professor, Department of Civil Engineering, Shahrekord University, Shahrekord, Iran


A designer needs to design a structure with the aim of obtaining the maximum possible load expected for the structure during its lifetime. In this paper, considering the information obtained from the earthquakes, the critical earthquakes were computed for a shear frame building equipped with a belt truss system and subjected to two constraint scenarios. For this purpose, a nonlinear optimization problem has been solved in which the objective function was the maximization of the roof displacement. In the first constraint scenario, the computed critical earthquake was known as the first state critical earthquake. In addition, for the second constraint scenario, the earthquake was named as the second state critical earthquake. In the first scenario, the energy, the duration of strong ground motion, and peak ground acceleration were considered as the constraints, while in the second scenario, the upper bound Fourier spectrum was added to these constraints. Finally, the properties of the initial and critical earthquakes were investigated using the Fourier analysis method and continuous wavelet transform. The numerical results showed that the Fourier spectrum of the first critical earthquake was 6.86 times higher than the maximum values for the same parameter in case of other earthquakes at a frequency near the first natural frequency of the structure. Also, using the time-frequency curve, it was shown that duration of the strong ground motion of all earthquake places within the dominant duration of the frequencies of the same earthquake was more than 10 sec.


Main Subjects

[1] I. Takewaki, Seismic critical excitation method for robust design: A review, Journal of Structural Engineering, 128(5) (2002) 665-672.
[2] R.S. Stein, Earthquake Conversations, Scientific American, 288(1) (2003) 72-79.
[3] R. Kamgar, R. Rahgozar, Critical excitation method for determining the best location of belt truss system in tall buildings, Iranian Journal of Structural Engineering, 4(2) (2018) 76-88.
[4] I. Takewaki, A new method for non‐stationary random critical excitation, Earthquake Engineering and Structural Dynamics, 30(4) (2001) 519-535.
[5] I. Takewaki, Nonstationary random critical excitation for acceleration response, Journal of Engineering Mechanics, 127(6) (2001) 544-556.
[6] B. Westermo, The critical excitation and response of simple dynamic systems, Journal of Sound and Vibration, 100(2) (1985) 233-242.
[7] A. Moustafa, Critical earthquake load inputs for multidegree-of-freedom inelastic structures, Journal of Sound and Vibration, 325(3) (2009) 532-544.
[8] A. Moustafa, Damage-based design earthquake loads for single-degree-of-freedom inelastic structures, Journal of Structural Engineering, 137(3) (2011) 456467.
[9] A.M. Abbas, Critical seismic load inputs for simple inelastic structures, Journal of Sound and Vibration, 296(4-5) (2006) 949-967.
[10] R. Kamgar, R. Rahgozar, Determination of critical excitation in seismic analysis of structures, Earthquakes and Structures, 9(4) (2015) 875-891.
[11] R. Kamgar, S. Shojaee, R. Rahgozar, Rehabilitation of tall buildings by active control system subjected to critical seismic excitation, Asian Journal of Civil Engineering, 16(6) (2015) 819-833.
[12] R. Kamgar, P. Samea, M. Khatibinia, Optimizing parameters of tuned mass damper subjected to critical earthquake, The Structural Design of Tall and Special Buildings, 27(7) (2018) e1460.
[13] M. Khatibinia, H. Gholami, R. Kamgar, Optimal design of tuned mass dampers subjected to continuous stationary critical excitation, International Journal of Dynamics and Control, 6(3) (2018) 1094-1104.
[14] A.       Haar,      Zur         Theorie der          orthogonalen Funktionssysteme, Inaugural-Dissertation. Von Alfred Haar, Druck von Dieterich, 1909.
[15] M. Misiti, Y. Misiti, G. Oppenheim, J. Poggi, Wavelet Toolbox: Computation, Visualization, Programming User’s Guide, Ver, 1.
[16] A. Heidari, J. Raeisi, R. Kamgar, Application of wavelet theory in determining of strong ground motion parameters, International Journal of Optimization in Civil Engineering, 8 (2018) 103-115.
[17] A. Heidari, J. Raeisi, R. Kamgar, The application of wavelet theory with denoising to estimate the parameters of earthquake, Scientia Iranica, International Journal of Science & Technology,  (2019).
[18] A. Kaveh, V. Mahdavi, Modification of ground motions using wavelet transform and VPS algorithm, Earthquakes and Structures, 12(4) (2017) 389-395.
[19] A. Kaveh, V. Mahdavi, Generation of endurance time acceleration functions using the wavelet transform, International Journal of Optimization in Civil Engineering, 2(2) (2012) 203-219.
[20] A. Kaveh, V. Mahdavi, A new method for modification of ground motions using wavelet transform and enhanced colliding bodies optimization, Applied Soft Computing, 47 (2016) 357-369.
[21] M. Najafzadeh, M. Zeinolabedini, Derivation of optimal equations for prediction of sewage sludge quantity using wavelet conjunction models: an environmental assessment, Environmental Science and Pollution Research, 25(23) (2018) 22931-22943.
[22] M. Zeinolabedini, M. Najafzadeh, Comparative study of different wavelet-based neural network models to predict sewage sludge quantity in wastewater treatment plant, Environmental Monitoring and Assessment, .52-1 )9102( )3(191
[23] S. Gholizadeh, E. Salajegheh, P. Torkzadeh, Structural optimization with frequency constraints by genetic algorithm using wavelet radial basis function neural network, Journal of Sound and Vibration, 312(1-2) (2008) 316-331.
[24] S. Gholizadeh, O. Samavati, Structural optimization by wavelet transforms and neural networks, Applied Mathematical Modelling, 35(2) (2011) 915-929.
[25] E. Salajegheh, S. Gholizadeh, P. Torkzadeh, Optimal desigin of structures with frequency constraints using wavelet back propagation neural, Asian Journal of Civil Engineering, 8(1) (2007) 97-111.
[26] S. Seyedpoor, J. Salajegheh, E. Salajegheh, S. Gholizadeh, Optimum shape design of arch dams for earthquake loading using a fuzzy inference system and wavelet neural networks, Engineering Optimization, 41(5) (2009) 473-493.
[27] M. Shinozuka, Y. Sata, Simulation of nonstationary random process, Journal of the Engineering Mechanics Division, 93(1) (1967) 11-40.
[28] A. Arias, Measure of earthquake intensity: seismic design for nuclear power plants, Cambridge, MA, 1970.
[29] A. Abbas, C. Manohar, Critical spatially-varying earthquake load models for extended structures, Journal of Structural Engineering, 29(1) (2002) 39-52.
[30] S.N. 2800, Iranian Code of Practice for Seismic Resistant Design of Buildings, 4rd edition, in, Building and Housing Research Center Tehran, Iran, 2014.
[31] J.S. Arora, Introduction to Optimum Design, Elsevier, Academic Press, USA, 2012.
[32] T. Coleman, M.A. Branch, A. Grace, Optimization Toolbox For Use with MATLAB, The MathWorks, Inc., USA, 1999.
[33] I. Takewaki, A. Moustafa, K. Fujita, Improving the Earthquake Resilience of Buildings: The Worst Case Approach, Springer London, 2013.
[34] S. Addison Paul, The illustrated wavelet transform handbook: introductory theory and applications in science, engineering, medicine and finance, Institute of Physics Publishing, 2002.
[35] R. Polikar, The Wavelet Tutorial-http:\users. rowan. edu/ polikar, WAVELETS/WTpart1. html,  (1999).
[36] M. Schneiders, v.d. Molengraft, M. Steinbuch, Wavelets in control engineering, Technische Universiteit Eindhoven, 2001.
[37] O. Rioul, P. Duhamel, Fast algorithms for discrete and continuous wavelet transforms, IEEE Transactions on Information Theory, 38(2) (1992) 569-586.
[38] COSMOS, Consortium Organizations for StrongMotion Observation Systems, in, 2009.
[39] M.D. Trifunac, A.G.J.B.o.t.S.S.o.A. Brady, A study on the duration of strong earthquake ground motion, Bulletin of the Seismological Society of America, 65(3) (1975) 581-626.