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* Contribuio tcnica ao 69 Congresso Anual da ABM Internacional e ao 14 ENEMET - EncontroNacional de Estudantes de Engenharia Metalrgica, de Materiais e de Minas,21 a 25 de julho de2014, So Paulo, SP, Brasil.

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1 INTRODUO

One factor indispensable for the development of good models for the analysis ofstructures consists in choosing the most adequate theory.

Advances in structural engineering enable its integration in various fields ofproductive activity where maximization of safety and performance is demanded, such

as civil construction, automotive and aerospace engineering, and oilfield services,among others. But structures are frequently subject to static and dynamicperturbation that affect significantly the mechanical and response characteristics(static responses may be mechanical displacements, strain and stress distributions,and dynamic responses may be frequency response functions and vibration modes)of these structures (Diacenco, 2010). Besides that, structural integrity may beaffected, leading to structural collapse. An important aspect to be considered is theutilization of reliable numerical models, able to give realistic qualitative andquantitative prediction of the structural behavior. In this context, the objective of thiswork is the numerical and computational implementation of the finite element methodto the static and dynamic analysis of the structures of kind beam which will be usedto Beam Theory Euler Bernoulli of Euler-Bernoulli beams (Souza, 2004).

The present work consists of the development a finite element model based onthe theory of beams Euler Bernoulli applied to the static and dynamic analysis ofmetallic beams. Case studies of different conditions are presented. Matlab was thecalculus software tool adopted, in consonance with its great academic relevance.

1.1. Finite Element Formulation Beams

The Euler Bernoulli beam theory assumes that undeformed plane sections remainplane under deformation. The displacement, at a distance yof the beam middle axisis given by (Azevedo, 2003):

dwu y

dx (1)

where u denote the displacements in direction x,and wdenotes the transverse

displacement.

The figure shows the hypothesis of the Euler-Bernoulli.

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Figure 1. Euler-Bernoulli beam adapted f rom L ima (1999).

The deformations are defined as:

2

2x

u wy

x x

(2)

0xyu w

y x

(3)

Equation (2) can be rewritten:

D (4)

Where: / T

v w dw dx . And the D is matrix formed by differential operators

appearing in the straindisplacement relation.

The Bernoulli beam element is shown in the Figure 2.

Figure 2. Bernoul li beam element with two nodes

The transverse displacement is interpolated by functions, defined in Equations(5):

2

1

1( ) (2 3 )

4N

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4

2 3

2

1( ) (1 )

4N (5)

The vector of element shape functions is given by Equation (6):

1 2( ) ( )TN N N (6)

where: N are the element shape interpolation functions formulated in local

coordinates , 11 .

Discretization of the displacement variables is made by using Hermitianinterpolation functions, as shown in Equation (7)

,v t v t N (7)

where:

N is the 2x2 matrix formed by the beam element shape interpolation functions

formulated in local coordinates , 11 .

The displacement and strain fields are expressed in terms of the nodal values,shown in Equations (8) and (9):

( )U AN (8)

( ) ( )DN B (9)

Where: 1 d

A ydx

Based on the stress-strain relations, the strain and kinetic energies of thecomposite beams can be formulated in terms of the natural variables of strain fieldand the mechanical material properties. After, Lagrange equations are used,considering the nodal displacements and rotations as generalized coordinates, inorder to obtain the following elementary mass and stiffness matrices, respectivelyshown in Equations (10) and (11) (Bathe,1982):

1

1( ) ( ) ( )( )TM I N N J d

(10)

1

1-1

( )n

T

b b

k

K B EI B J d

(11)

where Iand Eare, respectively, the inertia moment and the elastic modulus.

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In Equations. (10) and (11), Jdet indicates the determinant of the Jacobian ofthe transformation from the in-plane physical variables yx, to the natural variables

.

The global equations of motion are constructed from the elementary matrices

computed for each element of the finite element mesh, accounting for the nodeconnectivity, using standard finite element assembling procedures, Hughes et al(1987). After assembling, the global equations of motion in the time domain can bewritten as shown in Equation (12):

ttt fKqqM (12)

where nelem

e

e

1

and nelem

e

e

1

are the global finite element mass and stiffness

matrices. Symbol indicates matrix assembling and tq is the vector of globaldegrees of freedom. tf is the vector of generalized external loads.

1.2. Numerical Results

The beam analyzes has an elastic behavior which neglects the transverse sheardeformations.

1.2.1. Static Analysis

The static analysis is done considering one beam, simply-supported beam, and auniform load discretized in 20 elements. The Geometric characteristics andproperties of the beam are shown in table 1. The data in the table were obtainedOliveira (2008).

Table1. Geometric characteristics and properties o f the beamData Value

Length 1.5 mthickness

inertia momentdensity

Poisson ratioelastic modulus

0.075m2.6367.10-6m4

7850 Kg/m3

0.30210 GPa

The Figure 3 shows a Matlab plot of the transverse displacement of the beam.

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Figure 3. Transverse d isplacement

1.2.2. Natural Frequencies

The natural frequencies calculations are done considering the beam with thesame characteristics presented in the previous application, discretized in 20elements.The Geometric characteristics and properties of the beam are shown in table 1.

The first five natural frequencies obtained by the theoretical formulationimplemented in MATLAB are shown in Table 2.

Table 2. First five natural frequencies

Frequencies Value (Hz)

1 221.42345

349.2353.0360.3

371.1

To show the efficiency of the implemented model were quantified naturalfrequencies of a beam of steel whose properties are shown in Table 3

Table 3. Geometric characteristics and properties o f the beam

Data ValueLength 10 m

thicknessinertia moment

densityPoisson ratio

elastic modulus

5m0.025

7850 Kg/m3

0.30210 GPa

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[4]Hughes T.J., 1987, The finite element method: linear static and dynamic finiteelement analysis. Prentice Hall.[5] Souza, R.M., 2004, Elemento Finito de Barra para Anlise Geomtrica No-

Linear e Dinmica atravs da formulao co-rotacional, XXXI Jornadas SudAmericanas de Ingenieria Estructural.

[6] Lima Jr.J.J., 1999, Modelagem de Sensores e Atuadores Piezoeltricos com

Aplicaes em Controle Ativo de Estruturas, Campinas, Brazil, tese de doutorado.[7] Oliveira, 2008, Estudo do Posicionamento de Atuadores Piezoeltricos em

Estruturas Inteligentes, Itajub, Brazil, tese de doutorado.[8] Callister, Jr., W.D., 2002. Cincia e Engenharia de Materais: Uma introduo.

Editora LTC, 5 Ed., Rio de Janeiro, RJ, ISBN: 85216-1288-5.

RESPONSIBILITY NOTICE

The author(s) is (are) the only responsible for the printed material included in thispaper.

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