Probabilistic finite elements for nonlinear structural dynamics

Liu, W K; Belytschko, Ted; Mani, Adam

doi:10.1016/0045-7825(86)90136-2

Cited by 350 publications

(98 citation statements)

References 4 publications

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“…SFEM has been successfully applied in a wide variety of problems (e.g. solid, structural and fluid mechanics, acoustics, heat transfer) [ There are two main variants of SFEM in the literature: i) the perturbation approach [96,106,107], which is based on a Taylor series expansion of the response vector and, ii) the spectral stochastic finite element method -SSFEM [67] where each response quantity is represented using a series of random Hermite polynomials. Monte Carlo simulation -MCS [138] can also be added to these two variants.…”

Section: The Stochastic Finite Element Methods (Sfem)mentioning

confidence: 99%

“…[18,96,106,107,183]. In this approach, the stochastic field f(x) representing an uncertain system property is discretized into N zero-mean random variables fa i g N i¼1 .…”

Section: The Perturbation Methods -Taylor Series Expansion Of the Stocmentioning

confidence: 99%

“…The main representatives of the first category are the midpoint, nodal point, integration point and interpolation methods [37,106,107], while the local average and weighted integral methods [182,183,31,32] are the main representatives of the second group. These methods have been extensively used by several researchers in the framework of SFEM leading to results of different levels of accuracy e.g.…”

Section: Discretization Of Stochastic Processes and Fieldsmentioning

confidence: 99%

“…A second-order perturbation approach has been used in [106] leading to accurate results for a two d.o.f. dynamic problem.…”

Section: First and Second-order Approximation Of The Response Variabimentioning

confidence: 99%

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The stochastic finite element method: Past, present and future

Stefanou

2009

Computer Methods in Applied Mechanics and Engineering

862

407

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a b s t r a c tA powerful tool in computational stochastic mechanics is the stochastic finite element method (SFEM). SFEM is an extension of the classical deterministic FE approach to the stochastic framework i.e. to the solution of static and dynamic problems with stochastic mechanical, geometric and/or loading properties. The considerable attention that SFEM received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large-scale problems. This article aims at providing a state-of-the-art review of past and recent developments in the SFEM area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future.

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Section: The Stochastic Finite Element Methods (Sfem)mentioning

confidence: 99%

“…[18,96,106,107,183]. In this approach, the stochastic field f(x) representing an uncertain system property is discretized into N zero-mean random variables fa i g N i¼1 .…”

Section: The Perturbation Methods -Taylor Series Expansion Of the Stocmentioning

confidence: 99%

Section: Discretization Of Stochastic Processes and Fieldsmentioning

confidence: 99%

“…A second-order perturbation approach has been used in [106] leading to accurate results for a two d.o.f. dynamic problem.…”

Section: First and Second-order Approximation Of The Response Variabimentioning

confidence: 99%

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The stochastic finite element method: Past, present and future

Stefanou

2009

Computer Methods in Applied Mechanics and Engineering

862

407

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“…An analysis method to account for material uncertainties to solve this class of mechanical problems was proposed by Facchini et al (2005) where, based on a Galerkin approach, the material properties were assumed as a stochastic field. Other possible approaches to account for the randomness that affects the material properties are perturbation methods, originally proposed by Liu et al (1986).…”

Section: Introductionmentioning

confidence: 99%

Simplified Seismic Analysis of Disordered Masonry Towers

Facchini

Betti

2016

ASCE-ASME J. Risk Uncertainty Eng. Syst., Part A: Civ. Eng.

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The structural assessment of existing masonry towers under exceptional loads, such as earthquake loads, requires reliable and expedite methods of analysis. These approaches should take into account both the specific nonlinear behavior of the material (for instance, the small tensile strength) and the randomness that affect the masonry material's properties (in some cases the distribution of the elastic parameters too). Considering the need of simplified but effective methods to assess the seismic response of such a class of structures, the paper proposes an expeditious approach based on an equivalent Bouc and Wen model. As a prototype of masonry towers, a cantilever masonry beam is analyzed assuming that the first mode shape governs the dynamic behavior. With this hypothesis, the nonlinear Bouc and Wen model is employed to reproduce the system hysteretic response. Subsequently, assuming the material properties as random variables, stochastic linearization and the perturbative approach are employed to evaluate the bounds of the seismic response. The results of the simplified approach are compared with the results of finite-element models to show the effectiveness of the method.

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Stochastic finite element methods for the seismic response of soils

Yeh

Rahman

1998

Int. J. Numer. Anal. Meth. Geomech.

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SUMMARYSome of the available stochastic finite element methods are adapted and evaluated for the analyses of response of soils with uncertain properties subjected to earthquake induced random ground motion. In this study, the dynamic response of a soil mass, with finite element discretization, is formulated in the frequency domain. The spectral density function of the response variables are obtained from which the evaluation of the root-mean-squared and the most probable extreme values of the response are made. The material non-linearities are incorporated by using strain compatible moduli and damping of soils using an equivalent linear model for stress-strain behaviour of soils and an iterative solution of the response. The spatial variability of the shear modulus is described through a random field model and the earthquake included motion is treated as a stochastic process. The available formulations of direct Monte-Carlo simulation, first-order perturbation method, a spectral decomposition method with Neumann expansion and a spectral decomposition method with Polynomial Chaos are used to develop stochastic finite element analyses of the seismic response of soils. The numerical results from these approaches are compared with respect to their accuracy and computational efficiency.1998 John Wiley & Sons, Ltd.

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Probabilistic finite elements for nonlinear structural dynamics

Cited by 350 publications

References 4 publications

The stochastic finite element method: Past, present and future

The stochastic finite element method: Past, present and future

Simplified Seismic Analysis of Disordered Masonry Towers

Stochastic finite element methods for the seismic response of soils

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