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1

Various response functions in lattice domes reliability via analytical integration and finite element method

Pokusiński B., Kamiński M.

International Journal of Applied Mechanics and Engineering

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2018

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Vol. 23, no. 2

445--469

EN

The main aim of this work is to verify an influence of the response function type in direct symbolic derivation of the probabilistic moments and coefficients of the structural state variables of axisymmetric spherical steel dome structures. The second purpose is to compare four various types of domes (ribbed, Schwedler, geodesic as well as diamatic) in the context of time-independent reliability assessment in the presence of an uncertainty in the structural steel Young modulus. We have considered various analytical response functions to approximate fundamental eigenfrequencies, critical load multiplier, global extreme vertical and horizontal displacements as well as local deformations. Particular values of the reliability indices calculated here can be of further assistance in the reliability assessment by comparing the minimal one with its counterpart given in the Eurocode depending upon the durability class, reference period and the given limit state type.

2

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.

3

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.

4

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.

5

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.

6

The stochastic perturbation - based finite volume method for 1 & 2D fluid and heat flow problems

Kamiński M., Ossowski R. L.

Mechanics and Mechanical Engineering

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2010

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Vol. 14, nr 1

151-173

EN

This article is devoted to the mathematical formulation and computational implementation of the Stochastic Finite Volume Method for 1 and 2D fluid and heat flow problems. It is based on the stochastic generalized perturbation technique, which allows for a determination of the probabilistic moments of the state variables or functions for the general stationary transport equations with random parameters. Both numerical case studies contain a comparison of the stochastic perturbation approach of different orders, their relations to the Monte-Carlo simulation results as well as the effect of the perturbation parameter and input coefficient of variation on the output state functions.

7

A generalized version of the perturbation-based stochastic finite difference method for elastic beams

Kamiński M.

Journal of Theoretical and Applied Mechanics

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2009

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Vol. 47 nr 4

957-975

EN

The main idea of this paper is to demonstrate a stochastic computational technique consisting of the generalized stochastic perturbation method using the Taylor expansions of random variables and the classical Finite Difference Method based on regular grids. As it is documented by computational illustrations, it is possible to determine, using this approach, also higher probabilistic moments for any random dispersion of input variables unlike in the second order second moment technique worked out before. A numerical algorithm is implemented here using straightforward partial differentiation of hierarchical equations with respect to the random input quantity and further symbolic computations of probabilistic moments and characteristics by the system MAPLE.

PL

Głównym celem niniejszej pracy jest zastosowanie probabilistycznej analizy numerycznej składającej się z uogólnionej metody perturbacji opartej na szeregu Taylora ze współczynnikami losowymi oraz z Metody Różnic Skończonych dla siatek regularnych. Jak wykazano w przykładach numerycznych, używając tego podejścia można również wyznaczać momenty probabilistyczne wyższych rzędów dla dowolnych funkcji losowych, co było niemożliwe dla zastosowań metod drugiego rzędu znanych wcześniej. Zastosowany tutaj algorytm numeryczny jest oparty na różniczkowaniu bezpośrednim hierarchicznych równań równowagi względem przyjętych zmiennych i na symbolicznym wyznaczaniu momentów i charakterystyk losowych przy pomocy programu MAPLE.

8

The stochastic perturbation - based finite volume method for the flow problems

Kamiński M., Ossowski R. L.

Journal of Technical Physics

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2009

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Vol. 50, no 4

297-315

EN

This article is devoted to the mathematical formulation and computational implementation of the Stochastic Finite Volume Method for heat and fluid flow problems. It is based on the stochastic generalized perturbation technique, which allows for determination of the probabilistic moments of the state variables or functions of the general stationary transport equations with random parameters. Both numerical case studies contain a comparison of the stochastic perturbation approach of different orders, their relations to the Monte-Carlo simulation results as well as the effect of the perturbation parameter and input coefficient of variation on the output state functions.

9

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.

10

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.