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1

Inspection and characterization of random physical property defects by stochastic finite element method

Berkache Azouaou, Oudni Zehor, Mehaddene Hamid, Mohellebi Hassane, Lee Jinyi

Przegląd Elektrotechniczny

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2019

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R. 95, nr 8

96--101

EN

A stochastic finite element method was applied in the case of 2D magneto dynamic equation to study the detection and evaluation of harmful defect in a conductive structure by InSb Hall sensor and Eddy Currents Non Destructive Testing Technique. The defect is considered as a degradation of physical properties of the material. The electrical conductivity is considered as random variable inside the defect area which is generated using Monte-Carlo method. A statistical treatment was used to study the evaluation of inverse problem. Several global parameters are calculated and discussed.

PL

Przedstawiono przetwornik hallotronowy do badań wiroprądowego nieniszczących. Do projektowania przetwornika wykorzystano stochastyczną metode elementów skończonych. Do rozwiązania odwrotnego problemu wykorzystano metode Monte Carlo. Przetwornik bada zmianę konduktywności materiału spowodowaną istnieniem defektu.

2

Probabilistic mechanical properties and reliability of carbon nanotubes

Esbati A. H., Irani S.

Archives of Civil and Mechanical Engineering

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2018

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Vol. 18, no. 2

532--545

EN

Carbon nanotubes (CNTs) and their products such as polymer nanocomposite (PNC) are an undeniable part of future materials. To use such future materials, it is necessary to have an accurate evaluation of their properties. Several uncertainties such as structural defects and their distributions cause change in the properties of CNTs that could be considered probabilistic variables. A novel procedure is presented for evaluating CNTs’ probabilistic fracture properties and structural reliability using stochastic finite element methods. By employing two dimensionless parameters, both types of Stone–Wales 5-7-7-5 defects are randomly applied to CNTs. Section defect density and critical section defect density are defined and used to manage the distribution and geometrical configuration of CNTs’ structural defects. A probabilistic method is used to evaluate the effect of defects’ distribution on Young's modulus, ultimate strain, and ultimate stress. It has been observed that normal and Weibull distribution functions are suitable for describing Young's modulus distribution and ultimate stress distribution, respectively. Defect density ratio is defined and, using this parameter, the effect of aggregated defects on mechanical properties is evaluated. It is demonstrated that the defects out of critical section have an unavoidable effect on Young's modulus and ultimate strain; but they have an insignificant effect on ultimate stress. A reliability analysis is performed on armchair (15,15) CNTs and it is investigated that the reliability of CNTs depends on critical defect density significantly. In addition, the reliability is equal to one for the stress of less than 50 GPa and this value is equal to zero for the stress of higher than 100 GPa, independent from the changes of critical defect density. Eventually, a procedure is described to estimate the reliability of armchair CNTs using critical defect density and the results’ accuracy is discussed and evaluated.

3

Bridges for pedestrians with random parameters using the stochastic finite elements analysis

Szafran J., Kamiński M.

International Journal of Applied Mechanics and Engineering

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2017

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Vol. 22, no. 1

175--197

EN

The main aim of this paper is to present a Stochastic Finite Element Method analysis with reference to principal design parameters of bridges for pedestrians: eigenfrequency and deflection of bridge span. They are considered with respect to random thickness of plates in boxed-section bridge platform, Young modulus of structural steel and static load resulting from crowd of pedestrians. The influence of the quality of the numerical model in the context of traditional FEM is shown also on the example of a simple steel shield. Steel structures with random parameters are discretized in exactly the same way as for the needs of traditional Finite Element Method. Its probabilistic version is provided thanks to the Response Function Method, where several numerical tests with random parameter values varying around its mean value enable the determination of the structural response and, thanks to the Least Squares Method, its final probabilistic moments.

4

Dual probabilistic analysis of the transient heat transfer by the stochastic finite element method with optimized polynomial basis

Rabenda M., Kamiński M.

Czasopismo Inżynierii Lądowej, Środowiska i Architektury / Journal of Civil Engineering, Environment and Architecture

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2017

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z. 64, nr 2/II

211--225

EN

The main aim of this work is to contrast three various probabilistic computational techniques, namely analytical, simulation and perturbation-based, in a solution of the transient heat transfer problem in specific axisymmetric problem with Gaussian uncertainty in physical parameters. It is done thanks to a common application of the Finite Element Method program ABAQUS (for the deterministic part) and symbolic algebra system MAPLE, where all probabilistic procedures have been programmed. We determine up to the fourth order probabilistic characteristics of the resulting temperatures, i.e. expectations, coefficients of variation, skewness and kurtosis together with the histograms – all as the functions of the input coefficient of variation of random heat conductivity coefficient. Stochastic perturbation technique is implemented here using the tenth order Taylor series expansion and traditional Least Squares Method released with polynomial basis whose final order is a subject of the separate statistical optimization. Probabilistic results computed show almost perfect agreement of all the probabilistic characteristics under consideration, which means that the traditional simulation method may be replaced due to the time and computer scale savings with the stochastic perturbation method.

PL

Głównym celem niniejszej pracy jest porównanie trzech różnych probabilistycznych metod numerycznych, tj. metody analitycznej, symulacyjnej, a także metody perturbacji, podczas rozwiązywania pewnego zagadnienia osiowo-symetrycznego, w którym współczynnik przewodnictwa ciepła jest Gaussowskim parametrem losowym. Porównanie takie jest przeprowadzone przy użyciu systemu Metody Elementów Skończonych ABAQUS (dla części deterministycznej rozwiązania), a także pakietu algebry komputerowej MAPLE, w którym zaimplementowano wszystkie procedury losowe. W pracy wyznacza się centralne momenty probabilistyczne do rzędu czwartego włącznie, tj. wartości oczekiwane, współczynniki wariancji, skośność i kurtozę, jak również odpowiednie histogramy – wszystkie one są wyznaczone w funkcji wejściowego współczynnika wariancji. Metoda perturbacji stochastycznej jest zaimplementowana przy użyciu rozwinięcia w szereg Taylora rzędu dziesiątego, a także z wykorzystaniem tradycyjnej Metody Najmniejszych Kwadratów. Metoda ta umożliwia wyznaczenie wielomianowej funkcji odpowiedzi, której rząd jest przedmiotem oddzielnej optymalizacji statystycznej. Otrzymane wyniki probabilistyczne pokazują bardzo dobrą zgodność wszystkich wyznaczanych charakterystyk losowych, co oznacza w praktyce, iż tradycyjna metoda symulacji może zostać zastąpiona przez metodę perturbacji stochastycznej w celu wielokrotnego zmniejszenia czasu oraz mocy obliczeniowej.

5

From full-scale testing of steel lattice towers to stochastic reliability analysis

Szafran J., Kamiński M.

Archives of Mechanics

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2017

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Vol. 69, nr 4-5

371--388

EN

Analysis of the steel lattice telecommunication structure exposed to the wind pressure, whose average value is treated as the Gaussian random variable. The Least Squares Method is provided here for symbolic recovery of the polynomial responses of this structure in addition to the given uncertainty source and it serves to the twentieth order perturbation-based approximations for the first four probabilistic moments and coefficients. Static numerical analysis has been carried out by the use of the incremental BFGS (Broyden–Fletcher–Goldfarb–Shanno) procedure necessary for the so-called P-delta effect in steel structures, while the basic statistics of the ultimate limit state have been included into the formulas for the reliability indices of both first and second order. This study shows that the safety margin of such structures is definitely wider than it follows the basic Eurocodes statements, which means that designed durability period for these telecommunication structures is definitely longer.

6

Probabilistic buckling analysis of the beam steel structures subjected to fire by the stochastic finite element metho

Świta P., Kamiński M.

International Journal of Applied Mechanics and Engineering

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2016

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Vol. 21, no. 2

485--510

EN

The main purpose is to present the stochastic perturbation-based Finite Element Method analysis of the stability in the issues related to the influence of high temperature resulting from a fire directly connected with the reliability analysis of such structures. The thin-walled beam structures with constant cross-sectional thickness are uploaded with typical constant loads, variable loads and, additionally, a temperature increase and we look for the first critical value equivalent to the global stability loss. Such an analysis is carried out in the probabilistic context to determine as precisely as possible the safety margins according to the civil engineering Eurocode statements. To achieve this goal we employ the additional design-oriented Finite Element Method program and computer algebra system to get the analytical polynomial functions relating the critical pressure (or force) and several random design parameters; all the models are state-dependent as we consider an additional reduction of the strength parameters due to the temperature increase. The first four probabilistic moments of the critical forces are computed assuming that the input random parameters have all Gaussian probability functions truncated to the positive values only. Finally, the reliability index is calculated according to the First Order Reliability Method (FORM) by an application of the limit function as a difference in-between critical pressure and maximum compression stress determined in the given structures to verify their durability according to the demands of EU engineering designing codes related to the fire situation.

7

The least squares stochastic finite element method in structural stability analysis of steel skeletal structures

Kamiński M., Szafran J.

International Journal of Applied Mechanics and Engineering

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2015

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Vol. 20, no. 2

299--318

EN

The main purpose of this work is to verify the influence of the weighting procedure in the Least Squares Method on the probabilistic moments resulting from the stability analysis of steel skeletal structures. We discuss this issue also in the context of the geometrical nonlinearity appearing in the Stochastic Finite Element Metod equations for the stability analysis and preservation of the Gaussian probability density function employed to model the Young modulus of a structural steel in this problem. The weighting procedure itself (with both triangular and Dirac-type) shows rather marginal influence on all probabilistic coefficients under consideration. This hybrid stochastic computational technique consisting of the FEM and computer algebra systems (ROBOT and MAPLE packages) may be used for analogous nonlinear analyses in structural reliability assessment.

8

Some reliability issues of the corrugated i-beam girder

Sokołowski D, Kamiński M.

Engineering Transactions

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2015

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Vol. 63, iss. 3

297--315

EN

The main aim of this paper is reliability analysis of the corrugated-web I-girder carried out to verify its susceptibility to the random corrosion of the web and to make a comparison of the results of the first and of the second order reliability analysis. The methodology implemented in the study is based on the stochastic finite element method related to the generalized stochastic perturbation technique, where a discretization of the entire structure is carried out with four-node quadrilateral shell finite elements. This is numerically implemented using the FEM engineering system ROBOT and computer algebra system MAPLE, where all probabilistic procedures are programmed. The perturbation-based results are compared with these coming from the Monte-Carlo simulation and with an analytical solution obtained via symbolic integration carried out in MAPLE also. The indices of reliability are determined for the maximum deflections of the beam as the function of an input coefficient of variation of the web’s thickness whose meaning is the extent of a corrosion process.

9

Structural stability and reliability of the underground steel tanks with the Stochastic Finite Element Method

Kamiński M., Świta P.

Archives of Civil and Mechanical Engineering

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2015

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Vol. 15, no. 2

593--602

EN

The main aim is to present the Stochastic perturbation-based Finite Element Method analysis of the stability and also reliability of the underground steel vertical cylindrical structure of the waste container. This thin walled structure with constant cross-sectional thickness is loaded with subsoil pressure, snow and surface as well as dead loads and we look for the critical pressure value, when stability loss would be observed; it is done to design in the probabilistic context the safety margins. We employ to achieve this goal the Finite Element Method program ROBOT and computer algebra system MAPLE to get the analytical polynomial functions relating the critical pressure and random design parameters – shell thickness and its Young's modulus as well as to provide all probabilistic calculations. We determine up to the fourth order probabilistic characteristics of the structural response assuming that the input random parameters have Gaussian probability functions truncated to the positive values only. Finally, the reliability index is calculated according to the first order method using a difference in-between critical pressure and maximum tensile stress determined in this structure to verify its durability according to the demands of EU engineering codes.

10

On the least squares stochastic finite element analysis of the steel skeletal towers exposed to fire

Kamiński M. M., Strąkowski M. T.

Archives of Civil and Mechanical Engineering

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2013

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Vol. 13, no. 2

242--253

EN

The main issue in this paper is an application of the generalized stochastic perturbation technique to thermal stresses and deformations analysis for the spatial steel tower structure exposed to a fire. This approach is based on the given order Taylor expansion of all random parameters and state functions around their mean values as well as on the Least Squares Technique to determine the analytical functions in-between design parameters and structural responses. A distribution of the temperature in the model is considered for a simplicity as the set of constant values for all structural members. The temperature equivalent to the fire exposure in the analyzed structure is taken as the input Gaussian random variable, where basic material parameters are considered as temperature-dependent and this is the basis to calculate up to the fourth probabilistic moments and characteristics of the stresses and deformations. This study is an example of a hybrid usage of the FEM engineering system ROBOT and the computer algebra system MAPLE in stochastic analysis, where thermal stresses and strains may be straightforwardly used in fire reliability analysis of the civil engineering structures with both temperature-independent and dependent material characteristics.

11

Comparison of the aluminium versus steel telecommunication towers in stochastic finite element method eigenvibrations analysis

Solecka M., Kamiński M., Szafran J.

Mechanics and Mechanical Engineering

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2011

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Vol. 15, nr 2

101-116

EN

The main aim of this paper is to make a comparison of the eigenfrequencies of the high telecommunication towers alternatively manufactured using the stainless steel and the aluminium components. It is provided each time assuming that the Young modulus of the applied materialź is the Gaussian input random variable and using the generalized stochastic perturbation method using the global version of the Response Function Method. Up to the fourth order probabilistic moments and characteristics are computed in the three dimensional Finite Element Method model of the tower composed from the continuous linear elastic edge beams spanned by the large number of the linear elastic bars. A computational part of the work is made using the hybrid usage of the computer algebra system MAPLE and the FEM engineering package ROBOT used widely in the civil engineering practice.

12

Reliability modeling in some elastic stability problems via the generalized stochastic finite element method

Kamiński M., Świta P.

Archives of Civil Engineering

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2011

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Vol. 57, nr 3

275-295

EN

The main idea of this work is to demonstrate an application of the generalized perturbation-based Stochastic Finite Element Method for a determination of the reliability indicators concerning elastic stability for a certain spectrum of the civil engineering structures. The reliability indicator is provided after the Eurocode according to the First Order Reliability Method, and computed using the higher order Taylor expansions with random coefficients. Computational implementation provided by the hybrid usage of the FEM system ROBOT and the computer algebra system MAPLE enables for reliability analysis of the critical forces in the most popular civil engineering structures like simple Euler beam, 2 and 3D single and multi-span steel frames, as well as polyethylene underground cylindrical shell. A contrast of the perturbation-based numerical approach with the Monte-Carlo simulation technique for the entire variability of the input random dispersion included into the Euler problem demonstrates the probabilistic efficiency of the perturbation method proposed.

PL

Zasadniczym problemem omawianym w tej pracy jest zastosowanie Uogólnionej Stochastycznej Metody Elementów Skończonych opartych na metodzie perturbacji stochastycznej do wyznaczania wskaźników niezawodności w przypadku stateczności konstrukcji budowlanych pracujących w zakresie sprężystym. Wskaźnik niezawodności jest modelowany zgodnie z definicją podaną w normie Eurokod odpowiednią dla analizy niezawodności pierwszego rzędu, a obliczony dzięki zastosowaniu rozwinięcia wszystkich funkcji stanu w szereg Taylora ze współczynnikami z uwzględnieniem wyrazów wyższego rzędu. Procedura obliczeniowa jest oparta na jednoczesnym zastosowaniu programu Metody Elementów Skończonych ROBOT oraz systemu algebry komputerowej MAPLE i jest wykorzystana do analizy obciążeń krytycznych w popularnych konstrukcjach inżynierskich takich jak pręt Eulera, dwu i trójwymiarowe modele jedno- i wieloprzęsłowej ramy stalowej, jak również polietylenowy zbiornik podziemny o kształcie cylindrycznym. Porównanie metody perturbacji stochastycznej z techniką symulacji Monte-Carlo w całym zakresie losowej zmienności wykorzystywanych parametrów wejściowych na przykładzie zagadnienia Eulera pokazuje efektywność i ograniczenia zastosowanej metody perturbacji.

13

Zagadnienie stateczności wybranych konstrukcji inżynierskich o parametrach losowych

Kamiński M., Świta P.

Zeszyty Naukowe. Budownictwo / Politechnika Łódzka

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2010

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Z. 62

7-20

PL

Tematem niniejszej pracy jest zastosowanie probabilistycznej analizy numerycznej opartej na uogólnionej metodzie perturbacji stochastycznej do wyznaczenia probabilistycznych momentów obciążenia krytycznego. Obciążenia te wyznaczono za pomocą Stochastycznej Metody Elementów Skończonych dla losowo określonych parametrów geometrycznych przekroju. Wykorzystując metodę funkcji odpowiedzi wyznaczono wartości oczekiwane i odchylenia standardowe sił krytycznych dla płaskiej oraz przestrzennej ramy stalowej, które następnie będzie mozna wykorzystać do wyznaczenia ich wskaźników niezawodności.

EN

The main aim of this paper is the stability analysis of elastic frame systems with random parameters using the Generalized Stochastic Finite Element Method. The Taylor expansion with random coefficients of nth order is used to express all random functions and to determine the basic probabilistic moments of the critical load. The Response Function Method assists to determine higher order partial derivatives of the structural response instead of the Direct Differentiation Method employed widely before and is sufficient to assure high quality of the resulting moments. This approach is examined on the examples of 2 and 3D steel frames with some geometrical parameters defined as the Gaussian variables. Further application of this methodology is of course in the reliability analysis and reliability-based optimization of the steel engineering structures.

14

O komputerowym modelowaniu niezawodności stalowych wież telekomunikacyjnych

Kamiński M., Szafran J.

Zeszyty Naukowe. Budownictwo / Politechnika Łódzka

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2010

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Z. 62

51-66

PL

Tematem niniejszej pracy jest zastosowanie analizy numerycznej opartej na uogólnionej metodzie perturbacji stochastycznej i funkcji odpowiedzi do wyznaczenia wskaźników niezawodności dla stalowych wież telekomunikacyjnych. Wskaźniki te wyznaczono za pomocą algorytmu Stochastycznej Metody Elementów Skończonych dla losowo określonego obciążenia wiatrem. Modelowanie niezawodności obejmuje tutaj funkcję graniczną określoną dla wychylenia i obrotu badanych konstrukcji. W obliczeniach wykonanych programem MES ROBOT oraz system algebry komputerowej MAPLE analizowano wieżę o wysokości 40,5 m. Osiągnięte wyniki potwierdzają hipotezę, iż wartości wskaźników niezawodności są bardzo wrażliwe na losowy rozrzut parametrów obciążenia wiatrem.

EN

The main aim of this elaboration is an application of the numerical analysis based on the generalized stochastic perturbation method and the response function technique for a determination of the reliability indices. The subject of this study is the stainless steel telecommunication tower spatial structure. Those indices are determined thanks to the Stochastic Finite Element Method for the randomized uncertain external loading of the wind. Reliability analysis obeys the limit function defined for the deflection and rotation of the discretized structures. The computational part is completed using the commercial FEM system ROBOT and computer algebra system MAPLE and is focused on the tower having 40,5 meters. The results confirm the general hypothesis that the values of the reliability indices are very sensitive to the random dispersion of the structural and external loadings parameters.

15

Stochastyczna analiza dynamiczna wrażliwości układu belkowo-kratowego

Grzywiński M., Hien T. D.

Pomiary Automatyka Kontrola

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2009

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R. 55, nr 6

330-333

PL

W pracy przedstawiono procedury numeryczne dynamicznej analizy wrażliwości układów konstrukcyjnych. Oprócz zagadnienia różniczkowania bezpośredniego, sformułowano sprzężone zagadnienie końcowe, bazujące na pierwotnym zagadnieniu początkowym. Całość zilustrowano przykładem numerycznym, układu belkowo-kratowego, obliczonym metodą stochastycznych elementów skończonych.

EN

A numerical formulation for stochastic sensitivity in dynamic analysis of structural systems is presented in the paper. The stochastic finite element method, basing on the perturbation methodology, is an advanced computer technique for solving engineering problems with randomness taken into account. It allows one to analyze a wide class of static and dynamic systems with geometrical and /or material random parameters. Besides the well-known direct differentiation setting, an adjoint terminal-value system is formulated in accordance with the prescribed fundamental initial-value system. Once the original initial-value equations (15) and (16) are solved for by a step-by-step forward integration scheme over time from 0 to t and the adjoint terminal-value equations (23) and (24) are integrated backward form t to 0, the sensitivity gradients (25) are summed up. Analysis of a symmetrical portal frame with two strut trusses is given. The dynamic sensitivity of the displacements at two representative points B and C with respect to the change of the cross-sectional areas of the truss and beam members is discussed. Two complete sets of stochastic solutions are obtained. One includes the expectations and cross-covariances for the nodal displacements, while the other describes the first two probabilistic moments of their sensitivity gradients.

16

Eigenvalue analysis for high telecommunication towers with lognormal stiffness by the response function method and SFEM

Kamiński M. M., Szafran J.

Computer Assisted Mechanics and Engineering Sciences

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2009

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Vol. 16, No. 3/4

279-290

EN

The main aim of this paper is to demonstrate the application of the generalized stochastic perturbation techniąue to model the lognormal random variables in structural mechanics. This is done to study probabilistic characteristics of the eigenvibrations for the high telecommunication towers with random stiffness, which are modeled as the linear elastic 3D trusses. The generalized perturbation technique based on the Taylor expansion is implemented using the Stochastic Finite Element Method in its Response Function version. The main difficulty here, in a comparison to this techniąue previous applications, is a necessity of both odd and even order terms inclusion in all the Taylor expansions. The hybrid numerical approach combines the traditional FEM advantages with the symbolic computing and its visualization power and it enables for a verification of probabilistic convergence of the entire computational procedure.

17

Random eigenvibrations of elastic structures by the response function method and the generalized stochastic perturbation technique

Kamiński M. M., Szfran J.

Archives of Civil and Mechanical Engineering

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2009

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Vol. 9, no 4

5-32

EN

This paper addresses the important question in structural analysis how to efficiently model the eigenvibrations of the spatial structures with random physical and/or geometrical parameters. The entire computational methodology is based on the traditional Finite Element Method enriched with the stochastic perturbation technique in its generalized nth order approach, while the computational implementation is performed by the use of the academic FEM software in conjunction with the symbolic algebra computer system MAPLE. Contrary to the previous straightforward solution techniques, now the response function method is applied to compute any order probabilistic moments and coefficients of the structural eigenvalues. The response function is assumed in the polynomial form, the coefficients of which are computed from the several solutions of the deterministic problem around the mean value of the given input random parameter. This method is illustrated with the stochastic eigenvibrations of the simple single degree of freedom system and small steel tower modelled as the 3D truss structure with random mass density and Young modulus. This technique may find its wide application in reliability analysis of the real existing engineering structures using the commercial Finite Element Method packages as well as the other discrete computational techniques like the Finite Difference Method at least.

PL

Artykuł ukazuje metody analizy konstrukcji pozwalające efektywnie modelować drgania własne konstrukcji przestrzennych z losowym parametrem fizycznym bądź geometrycznym. Całkowita metodologia komputerowa jest oparta na tradycyjnej Metodzie Elementów Skończonych, wzbogaconej metodą perturbacji stochastycznej i jej podejściem n-tego rzędu. Komputerowa implementacja została wykonana w programie Metody Elementów Skończonych w powiązaniu z systemem komputerowym algebry symbolicznej MAPLE. W przeciwieństwie do poprzednich rozwiązań bezpośrednich, metoda funkcji odpowiedzi jest zastosowana do obliczeń probabilistycznych momentów dowolnego rzędu i współczynników wartości własnych konstrukcji. Funkcja odpowiedzi jest przyjęta w formie wielomianowej, a współczynniki zostały wyznaczone na podstawie kilku rozwiązań zagadnienia deterministycznego w otoczeniu wartości średniej odpowiedniego parametru losowego. Metoda ta jest zilustrowana na przykładzie stochastycznych drgań własnych prostego układu z jednym stopniem swobody i małej wieży stalowej modelowanej, jako kratowa konstrukcja 3D z losową gęstością masy, a także losowym modułem Younga. Metoda może zostać szeroko zastosowana w analizach niezawodności istniejących konstrukcji inżynierskich przy użyciu komercyjnych programów MES, jak również innych dyskretnych metod obliczeniowych, np. Metody Różnic Skończonych, czy Metody Elementów Brzegowych.

18

Stochastic finite element method as a tool for modelling of deterioration of concrete structures

Kerźner Z., Novák D., Rusina R., Teplý B.

Zeszyty Naukowe. Budownictwo / Politechnika Opolska

|

1998

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z. 42

27-36

EN

The carbonation of concrete and the corrosion of reinforcement are significant service life factors. The modelling of these phenomena via random fields and subsequent stochastic finite element analysis (SFEM) is suggested in this paper. The statistical variation of carbonation of concrete and the corrosion of reinforcement are taken into account. A constitutive model of deterioration based on the change of the structural stiffness is applied in order to make the model tractable for the use in subsequent structural analysis. For that reasion deterioration constitutive model is implemented into the stochastic finite element method. The two-dimensional plane elastic stress/strain stochastic isoparametric quadrilateral finite elements are used. A Monte Carlo type simulation of random fields is combined with Neumann expansion of the inverse global stiffness matrix. Numerical results are presented for a reinforced concrete beam under given environmental conditions. It is shown that the proposed method of analysis can be used to estimate the statistical variation of displacements and the stress of general deterioration - sensitive concrete structure. The proposed approach can contribute to the efficient prediction of random behaviour of RC structures under deterioration by carbonation of concrete and the corrosion of reiforcement.

PL

Karbonizacja betonu i korozja zbrojenia żelbetu sa istotnymi faktami wpływającymi na jego żywotność. W pracy zaproponowane zostało podejście do modelowania tych zjawisk z wykorzystaniem podporządkowanych elementów skończonych. Przy czym uwzględnia się statystyczne wariacje karbonizacji betonu i korozji zbrojenia. Konstytutywny model degradacji oparty jest na zmianie strukturalnej sztywności i zastosowany został do zbudowania modelu przydatnego do zastosowania podporządkowanej analizy strukturalnej. W ten sposób konstytutywny model degradacji realizowany jest stochastyczną metodą elementów skończonych, przy wykorzystaniu dwuwymiarowych płaskich, sprężyście odkształcalnych, izoparametrycznych elementów czworokątnych. Symulacja typu Monte Carlo przypadkowych pól jest połączona z ekstrapolacją Neumana odwróconej globalnej macierzy sztywności. Wyniki obliczeń numerycznych przedstawione są dla zbrojenia betonu przy zadanych warunkach zewnętrznych. Wykazano, że zaproponowana metoda może być wykorzystana do oceny statystycznej wariacji przemieszczeń i naprężeń degradującej struktury betonowej. Proponowane podejście może być również zastosowane do efektywnego prognozowania przypadkowego zachowania się żelbetu na skutek pogorszenia się właściwości betonu w efekcie karbonizacji i korozji zbrojenia.