• Magazine of the Korean Society of Agricultural Engineers
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• v.36 no.1
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• pp.54-62
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• 1994

The expanding technique of the Stochastic Finite Element Method(SFEM) is proposed in this paper for adapting direct integration methods in stochastical dynamic analysis of structures. Grafting the direct integration methods and the SFEM together, one can deal with nonlinear structures and nonstationary process problems without any restriction. The stochastical central diffrence and stochastic Houbolt methods are introduced to show the expanding technique, and their adaptabilities are discussed. Results computed by the proposed method (the Stochastic Finite Element Method in Dynamics: SFEMD) for two degree-of-free- dom system are compared with those obtained by Monte Carlo Simulation.

• Steel and Composite Structures
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• v.8 no.2
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• pp.129-144
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• 2008

This paper demonstrates an application of the perturbation based stochastic finite element method (SFEM) for predicting the performance of structural systems made of composite sections with random material properties. The composite member consists of materials in contact each of which can surround a finite number of inclusions. The perturbation based stochastic finite element analysis can provide probabilistic behavior of a structure, only the first two moments of random variables need to be known, and should therefore be suitable as an alternative to Monte Carlo simulation (MCS) for realizing structural analysis. A summary of stiffness matrix formulation of composite systems and perturbation based stochastic finite element dynamic analysis formulation of structural systems made of composite sections is given. Two numerical examples are presented to illustrate the method. During stochastic analysis, displacements and sectional forces of composite systems are obtained from perturbation and Monte Carlo methods by changing elastic modulus as random variable. The results imply that perturbation based SFEM method gives close results to MCS method and it can be used instead of MCS method, especially, if computational cost is taken into consideration.

• Structural Engineering and Mechanics
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• v.11 no.4
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• pp.373-392
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• 2001

The main idea behind the paper is to present two alternative methods of homogenization of the heat conduction problem in composite materials, where the heat conductivity coefficients are assumed to be random variables. These two methods are the Monte-Carlo simulation (MCS) technique and the second order perturbation second probabilistic moment method, with its computational implementation known as the Stochastic Finite Element Method (SFEM). From the mathematical point of view, the deterministic homogenization method, being extended to probabilistic spaces, is based on the effective modules approach. Numerical results obtained in the paper allow to compare MCS against the SFEM and, on the other hand, to verify the sensitivity of effective heat conductivity probabilistic moments to the reinforcement ratio. These computational studies are provided in the range of up to fourth order probabilistic moments of effective conductivity coefficient and compared with probabilistic characteristics of the Voigt-Reuss bounds.

• Structural Engineering and Mechanics
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• v.82 no.1
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• pp.31-40
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• 2022

In structural engineering, the material properties of the structures such as elastic modulus, shear modulus, density, and size may not be deterministic and may vary at different locations. The dynamic response analysis of such structures may need to consider these properties as stochastic. This paper introduces a stochastic finite element method (SFEM) approach to analyze moving loads problems. Firstly, Karhunen-Loéve expansion (KLE) is applied for expressing the stochastic field of material properties. Then the mathematical expression of the random field is substituted into the finite element model to formulate the corresponding random matrix. Finally, the statistical moment of the dynamic response is calculated by the point estimation method (PEM). The accuracy and efficiency of the dynamic response obtained from the KLE-PEM are demonstrated by the example of a moving load passing through a simply supported Euler-Bernoulli beam, in which the material properties (including elastic modulus and density) are considered as random fields. The results from the KLE-PEM are compared with those from the Monte Carlo simulation. The results demonstrate that the proposed method of KLE-PEM has high accuracy and efficiency. By using the proposed SFEM, the random vertical deflection of a high-speed railway (HSR) bridge is analyzed by considering the random fields of material properties under the moving load of a train.

• Structural Engineering and Mechanics
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• v.28 no.2
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• pp.129-152
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• 2008

This paper considers the solution of the stochastic differential equations (SDEs) with random operator and/or random excitation using the spectral SFEM. The random system parameters (involved in the operator) and the random excitations are modeled as second order stochastic processes defined only by their means and covariance functions. All random fields dealt with in this paper are continuous and do not have known explicit forms dependent on the spatial dimension. This fact makes the usage of the finite element (FE) analysis be difficult. Relying on the spectral properties of the covariance function, the Karhunen-Loeve expansion is used to represent these processes to overcome this difficulty. Then, a spectral approximation for the stochastic response (solution) of the SDE is obtained based on the implementation of the concept of generalized inverse defined by the Neumann expansion. This leads to an explicit expression for the solution process as a multivariate polynomial functional of a set of uncorrelated random variables that enables us to compute the statistical moments of the solution vector. To check the validity of this method, two applications are introduced which are, randomly loaded simply supported reinforced concrete beam and reinforced concrete cantilever beam with random bending rigidity. Finally, a more general application, randomly loaded simply supported reinforced concrete beam with random bending rigidity, is presented to illustrate the method.

• 한국지구물리탐사학회:학술대회논문집
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• 2003.11a
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• pp.192-197
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• 2003

It can be said that rock mass properties are characterized not by a mean value but by values with variation due to its characteristic uncertainty. This characteristic is one of the most important parts for the design of underground structures, but yet to be fully examined. Stochastic finite element method (SFEM) has been developed in order to take the randomness of structural systems into account. Using SFEM, the response variability of structural system can be obtained and it leads probabilistic stability of structure to be analyzed. In this study, displacements response variability of circular opening with hydrostatic stress field are analyzed in terms of rock mass properties having a certain mean and a standard deviation using the SFEM. The analyzed response variability shows that the necessity of probabilistic stability analysis of underground structures using reliable mean value and standard deviation of deformation modulus.

• Computational Structural Engineering
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• v.5 no.1
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• pp.97-108
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• 1992

This paper is an attempt to account for the uncertainty of the residual strength in the reliability analysis of structural systems. For this purpose the stochastic finite element method(SFEM) is linked to the system reliability analysis procedure. The stochastic finite element is known to be able to a more explicitly consider the effect of uncerainties of material and geometric variables on those of load effects in structural analysis procedure. The method has been applied to system as well as component reliability analysis of a plane structure. Comparison of the results by the present approach is made with the method in which the residual strength of failed component is treated as deterministic variable. Several case studies have been carried to show the effect of uncertainty in residual strength of a member after failure. Is has been conformed that reidual strength very much affect the system reliability level. It can be, hence, concluded that the uncertainties in the post-ultirnate behaviour may have to be taken into account in the system reliability analysis for a better a ssessment of the system reliability especially for a structure of which member behaviour is modelled as asemi-brittle model. And then the stochastic finite element method can efficiently evaluate the system reliability.

• Steel and Composite Structures
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• v.9 no.6
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• pp.499-518
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• 2009

The present paper investigates the stochastic seismic responses of steel structure systems with Partially Restrained (PR) connections by using Perturbation based Stochastic Finite Element (PSFEM) method. A stiffness matrix formulation of steel systems with PR connections and PSFEM and MCS formulations of structural systems are given. Based on the formulations, a computer program in FORTRAN language has been developed, and stochastic seismic analyses of steel frame and bridge systems have been performed for different types of connections. The connection parameters, material and geometrical properties are assumed to be random variables in the analyses. The Kocaeli earthquake occurred in 1999 is considered as a ground motion. The connection parameters, material and geometrical properties are considered to be random variables. The efficiency and accuracy of the proposed SFEM algorithm are validated by comparison with results of Monte Carlo simulation (MCS) method.

• Structural Engineering and Mechanics
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• v.28 no.3
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• pp.281-294
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• 2008

In this paper, linear elastic isotropic structures under the effects of both stochastic operators and stochastic excitations are studied. The analysis utilizes the spectral stochastic finite elements (SSFEM) with its two main expansions namely; Neumann and Homogeneous Chaos expansions. The random excitation and the random operator fields are assumed to be second order stochastic processes. The formulations are obtained for the system solution of the two dimensional problems of plane strain and plate bending structures under stochastic loading and relevant rigidity using the previously mentioned expansions. Two finite element programs were developed to incorporate such formulations. Two illustrative examples are introduced: the first is a reinforced concrete culvert with stochastic rigidity subjected to a stochastic load where the culvert is modeled as plane strain problem. The second example is a simply supported square reinforced concrete slab subjected to out of plane loading in which the slab flexural rigidity and the applied load are considered stochastic. In each of the two examples, the first two statistical moments of displacement are evaluated using both expansions. The probability density function of the structure response of each problem is obtained using Homogeneous Chaos expansion.

• Transactions of the Korean Society of Mechanical Engineers
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• v.18 no.8
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• pp.1920-1929
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• 1994

The stochastic finite element method(SFEM) based structural optimal design is presented. Random system response including uncertainties for the design variable is calculated with first order perturbation method. A method for calculating the sensitivity coefficients is developed using the equilibrium equation and first-order perturbed equation. Numerical results are presented for a truss, frame and plate structures with displacement and stress constraints. The sensitivity calculation proposed here is compared with finite difference method. A nonlinear programming technique is used to solve the problem. The procedure is easily incorporated with existing deterministic structural optimization.