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1

Euler–Bernoulli type beam theory for elastic bodies with nonlinear response in the small strain range

Janečka A., Průša V., Rajagopal K. R.

Archives of Mechanics

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2016

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Vol. 68, nr 1

3--25

EN

The response of many new metallic alloys as well as ordinary materials such as concrete is elastic and nonlinear even in the small strain range. Thus, using the classical linearized theory to determine the response of bodies could lead to a miscalculation of the stresses corresponding to the given strains, even in the small strain regime. As stresses can determine the failure of structural members, such miscalculation could be critical. We investigate the quantitative impact of the material nonlinearity in the Euler–Bernoulli type beam theory. The governing equations for the deflection are found to be nonlinear integro-differential equations, and the equations are solved numerically using a variant of the spectral collocation method. The deflection and the spatial stress distribution in the beam have been computed for two sets of models, namely the standard linearized model and some recent nonlinear models used in the literature to fit experimental data. The predictions concerning the deflection and the spatial stress distribution based on the standard linearized model and the nonlinear models are considerably different.

2

A new approach for buckling analysis of axially functionally graded beams

Rychlewska J.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 2

95--102

EN

The object of considerations are axially functionally graded (FG) beams, which are loaded by an axial force varying along the length of the beam. The main idea presented here is to approximate FG beams by an equivalent beam with piecewise exponentially varying material properties, geometrical properties and axial load. Numerical solutions of the buckling analysis are obtained for four various types of boundary conditions associated with pinned and clamped ends. The usefulness of the proposed method is confirmed by comparing numerical results with those available for graded beams of special polynomial non-homogeneity.

3

Free vibration of axially functionally graded Euler-Bernoulli beams

Kukla S., Rychlewska J.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 1

39--44

EN

In this contribution, free vibration of axially functionally graded beams is analysed within the framework of the Euler-Bernoulli beam theory. The beams with uniaxial variation of the elasticity modulus and mass density are approximated by an equivalent beam with piecewise exponentially varying geometrical and material properties. A numerical example for a beam with pinned ends is presented.

4

Finite element formulation for the large-amplitude vibrations of FG beams

Javid M., Hemmatnezhad M.

Archive of Mechanical Engineering

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2014

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Vol. LXI, nr 3

469--482

EN

On the basis of Euler-Bernoulli beam theory, the large-amplitude free vibration analysis of functionally graded beams is investigated by means of a finite element formulation. The von Kármán type nonlinear strain-displacement relationship is employed where the ends of the beam are constrained to move axially. The material properties are assumed to be graded in the thickness direction according to the powerlaw and sigmoid distributions. The finite element method is employed to discretize the nonlinear governing equations, which are then solved by the direct numerical integration technique in order to obtain the nonlinear vibration frequencies of functionally graded beams with different boundary conditions. The influences of power-law index, vibration amplitude, beam geometrical parameters and end supports on the free vibration frequencies are studied. The present numerical results compare very well with the results available from the literature where possible.

PL

W oparciu o teorię Eulera-Bernouliego przeprowadzono analizę wielkoamplitudowych drgań belki gradientowej posługując się metodą elementów skończonych. Związek między odkształceniem i przemieszczeniem, typu von Kármána, zastosowano tam, gdzie końce belki są utwierdzone i mogą poruszać się osiowo. Zakłada się, że właściwości materiału zmieniają się w kierunku poprzecznym (grubości) zgodnie z funkcją potęgową lub sigmoidalną. Metoda elementów skończonych jest zastosowana w celu dyskretyzacji nieliniowych równań sterujących, z których po rozwiązywaniu metodą bezpośredniego całkowania numerycznego wyznacza się częstotliwości drgań nieliniowych belki gradientowej dla różnych warunków brzegowych. Badany jest wpływ wykładnika funkcji, amplitudy drgań, geometrycznych parametrów belki i podparcia końców na częstotliwości drgań swobodnych. Wyniki numeryczne, przedstawione w artykule, zgadzają się dobrze z wynikami podawanymi w dostępnej literaturze.

5

Delamination identification using machine learning methods and Haar wavelets

Feklistova L., Hein H.

Computer Assisted Methods in Engineering and Science

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2012

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Vol. 19, no. 4

351-360

EN

The present paper focuses on the identification of delamination size and location in homogeneous and composite laminates. The modal analysis methods are employ ed to calculate the data patterns. An aggregated approach combining Haar wavelets, support vector mac hines (SVMs) and artificial neural networks (ANNs) is used to solve identification problems. The usabili ty and effectiveness of the proposed technique are tested by several numerical experiments. The advantages of the proposed method lie in the ability to make fast and accurate calculations.