Synonyms
Earthquake engineering; Hysteresis; Nonlinearity; Random vibrations
Introduction
The present chapter is devoted to the probabilistic analysis of the random response of nonlinear structural systems exposed to random excitation with special attention to earthquake action. The random system response may be due to random excitation, to random system properties, to random boundary conditions. The nonlinear character of the response is mainly due to the nonlinear materials properties and to the effect of large displacements (the so-called P-Δ effect).
The first pioneering studies on this topic took place in the 1960s and 1970s (VanMarcke et al. 1970; Iwan 1973; Atalik and Utku 1976; Spanos 1976), when equivalent linearization techniques were used, taking advantage of the availability of first digital computers.
The attention is limited herein to systems having deterministic properties, including deterministic boundary conditions, considering those types of nonlinearity that can...
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Atalik T, Utku S (1976) Stochastic linearization of multi-degree of freedom nonlinear systems. Earthquake Eng Struct Dyn 4(4):411–420
Booton R (1953) The analysis of nonlinear control systems with random inputs. Polytechnic Inst, Brooklyn
Bouc R (1967) Forced vibrations of mechanical systems with hysteresis. Proceedings of the Fourth Conference on Nonlinear Oscillation, Prague
Breccolotti M, Materazzi A, Venanzi I (2008) Identification of the nonlinear behavior of a cracked RC beam through the statistical analysis of the dynamic response. Struct Control Health Monitor 15:416–435
Caughey T (1953) Response of nonlinear systems to random excitation. Lecture Notes California Institute of Technology, Pasadena, Calif
Caughey T (1960) Random excitation of a system with bilinear hysteresis. J Appl Mech 27:649–652
Crandall S (1963) Perturbation techniques for random vibration of nonlinear systems. J Acoust Soc Am 35(11):1700–1705
Fokker A (1914) The median energy of rotating electrical dipoles in radiation fields. Annalen Der Physik 43:810–820
Iwan WD (1973) A generalization of the concept of equivalent linearization. Int J Nonlinear Mech 8:279–287
Iwan WD, Lutes LD (1968) Response of the bilinear hysteretic system to stationary random excitation. J Acoust Soc Am 48(3):545–552
Khasminskii R (1966) A limit theorem for the solution of differential equations with random right-hand sides. Theory Probab Appl 11:390–405
Kolmogorov A (1931) Ober analytische Methoden in der Wahrscheinlichkeitsrechnung. Math Ann 104:415–458
Landau P, Stratonovich R (1962) Theory of stochastic transitions of various systems between different states. Moscow University, Vestinik
Lin Y, Cai G (1995) Probabilistic structural dynamics advanced theory and applications. McGraw-Hill, New York
Masing G (1926) Eigenspannungen und Verfestigung beim Messing (in German). In Proceedings of the 2nd international congress of applied mechanics, Zurich, pp 332–335
Naess A, Johnsen J (1993) Response statistics of nonlinear, compliant offshore structures by the path integral solution method. Probab Eng Mech 8(2):91–106
Park R, Kent D, Sampson R (1972) Reinforced concrete members with cyclic loading. ASCE J Struct Div 98(7):1341–1360
Pirrotta A, Santoro R (2011) Probabilistic response of nonlinear systems under combined normal and Poisson white noise via path integral method. Prob Eng Mech 26(1):26–32
Piszczec K, Nizioł J (1986) Random vibration of mechanical system. Ellis Horwood, Chichester
Planck M (1915) Sgr. preuss. Akad. Wiss. p 512
Popov E, Bertero V, Krawinkler H (1972) Cyclic behavior of three r.c. flexural members with high shear. In: EERC report 72-5. Earthquake Engineering Research Center, University of California, Berkeley
Roberts J, Spanos P (1986) Stochastic averaging: an approximate method of solving random vibration problems. Int J Nonlinear Mech 21(2):111–134
Shinozuka M (1972) Monte Carlo solution of structural dynamics. Comput Struct 2(5/6):855–874
Shinozuka M, Deodatis G (1991) Simulation of stochastic processes by spectral representation. Appl Mech Rev 44(4):191–204
Spanos P-TD (1976) Linearization techniques for nonlinear dynamical systems, EERL 76-04. California Institute of Technology, Pasadena, Calif
Stratonovich R (1963) Topics in the theory of random noise. Gordon & Breach, New York
Sun J, Hsu C (1990) The generalized cell mapping method in nonlinear random vibration based upon short-time Gaussian approximation. J Appl Mech 57:1018–1025
Takeda T, Sozen M, Nielson N (1970) Reinforced concrete response to simulated earthquakes. Proc ASCE J Struct Div 96(ST12):2257–2573
VanMarcke E, Yanev P, De Estrada M (1970) Response of simple hysteretic systems to random excitation, s.l., Research report R70-66, Department of Civil Engineering, MIT
Wehner M, Wolfer W (1983) Numerical evaluation of path-integral solutions to Fokker–Planck equations. Phys Rev A 27(5):2663–2670
Wen Y (1976) Method for random vibration of hysteretic systems. ASCE J Eng Mech 120:2299–2325
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this entry
Cite this entry
Materazzi, A.L., Breccolotti, M. (2015). Stochastic Analysis of Nonlinear Systems. In: Beer, M., Kougioumtzoglou, I.A., Patelli, E., Au, SK. (eds) Encyclopedia of Earthquake Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35344-4_334
Download citation
DOI: https://doi.org/10.1007/978-3-642-35344-4_334
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35343-7
Online ISBN: 978-3-642-35344-4
eBook Packages: EngineeringReference Module Computer Science and Engineering