Beam Model Validation Based on Finite Element Analysis

Transcription

1 Beam Model Validation Based on Finite Element Analysis CARLA PROTOCSIL, PETRU FLORIN MINDA, GILBERT-RAINER GILLICH Department of Mechanical Engineering Eftimie Murgu University of Resita P-ta Traian Vuia 1- ROMANIA ANDREA AMALIA MINDA Department of Mathematics Eftimie Murgu University of Resita P-ta Traian Vuia 1- ROMANIA OVIDIU VASILE Department of Mechanics Politehnica University of Bucharest Splaiul Independentei 313, 6 Bucharest ROMANIA ovidiu_vasile@yahoo.co.uk Abstract: - The paper present the newest researches performed by the authors, destined to find out the proper beam model, in concordance with its geometrical characteristics. This knowledge is used to choose the adequate relations in the difficult process of damage detection. First, we calculated, for beams with various lengths and cross-section areas the first ten natural frequencies of the weak-axis bending vibration modes, basing on two models which describe the beam s dynamic behavior: the Euler-Bernoulli model and the Share model. Next, using FEM simulations we found out these frequencies for the considered beams. Finally, comparing the natural frequencies obtained analytically and by means of FEM analysis, we found out which theoretical models best fit for the various investigated cases. Key-Words: - beam, natural frequency, FEM analysis, Euler-Bernoulli model, Share model 1 Introduction Non-destructive evaluation of engineering structures is nowadays an important tool in risk evaluation, guarantying integrity and reliability of these structures [1], [] and [3]. Pervious researches made in the field of damage detection show that it is possible to detect damages in structures, [] and [5], using a lot of features extracted from the time-history; however, the designed applications fit particular cases nor one cover all possible situations. Researches made by the authors of this paper highlighted a specific phenomenon which appears by bending beams, introducing a new concept upon dynamic behavior of damaged beams [6]. This permitted us to find a relation that correlates the natural frequencies with damage depth and position [7] and [8]. The method is generally available if using the Timoshenko model (T). Because the involvement of this model is more complicated and time-extensive, for beams with specific geometries (less rigid beams) other models, like Euler-Bernoulli (E-B), Rayleigh (R) or Share (Sh) can be used with less effort. In this paper is analyzed the adequacy of the Euler-Bernoulli and Share models for some beams which usually appear in practice, having following geometries: cross-section 11x11 mm and x mm respectively and length 1 mm and mm respectively. Consequently, the dynamic behavior of four beams is investigated, trying to find out the limits for which various models can be used. ISBN:

2 The analytical calculus of natural frequencies A beam is subjected to external and internal loads, in terms of forces and moments, producing stress, deflection/displacements and rotations, like shown in figure 1. Figure 1. Incremental beam element In literature are presented four beam theories, based on common and particular assumptions. The basic assumptions made by all these models [9], are: - one dimension (axial direction) is considerably larger than the other two; - the material is linear elastic; - the Poisson effect is neglected; - the cross-sectional area is symmetric so that the neutral and centroidal axes coincide; - planes perpendicular to the neutral axis remain perpendicular after deformation. - the angle of rotation is small so that the small angle assumption can be used. In particular, each model considers more or less influences of external and internal loads [9]; a summary of these considerations is presented in table 1. Table 1 Particularities of the considered models Beam models Bending Moment Lateral Displacement Shear Deformation Rotary inertia (E-B) x x (R) x (Sh) x (T) From the four theories of the dynamics of transversely vibration of beams we used two of them (Euler-Bernoulli and Shear) to analyze the beams with L 1 = 1 mm and L = mm respectively, having the cross-sections mentioned in the first section..1 Natural frequency calculus using the Euler-Bernoulli model To find the natural frequencies of the beams we started from the governing differential equation of motion, [6] and [9], given below: v( x,t) ρa v x, t + = f x,t EI t (1) where v is the vertical displacement, ρ the mass density, A the cross-section area, x and t are variables in correspondence with displacement and time, and f stay for external loads. The boundary condition to be satisfied by the cantilever beam, are: v - for clamped end : = ; v= - for free end: 3 v v = ; = 3 f x,t v x, t into Taking = and separating two functions such that v( x, t) W( x) T( t) =, the equation of motion (1) can be separated into two ordinary differential equations. d T t dt d W x dx +ω = T t γ = W x () γ is in correspondence with the angular frequency ω by: ρ ω γ = (3) EI A From the second equations in () result the solution W(x) as following: 1 W x = C sinγ x+ C cosγ x+ + C sinhγ x+ C coshγx 3 () where Ci, i = 1,,3, are constant coefficients. Deriving the function from () three times, result a system of equations for which, imposing the boundary conditions for a cantilever, results the relation to determine the natural frequencies as follows (see [3]): where λ j =γ j L. f λ = (5) π ρ AL j EI j ISBN:

3 Table Natural frequencies for beams with crosssection 11x11 calculated using the (E-B) theory Analyzing these beams the results for the natural frequencies shown in Table and Table 3 were obtained, by using eq. (5). Table 3 Natural frequencies for beams with crosssection x calculated using the (E-B) theory Natural frequency calculus using the Shear model This model adds the effect of shear distortion (but not rotary inertia) to the Euler-Bernoulli model. [5] The new variables are α the angle of rotation of the cross-section due to the bending moment, β the angle of distortion due to shear. The total angle of rotation is approximately the first derivate of the deflection. v x,t α ( x,t) +β ( x, t) = x The boundary conditions are given by: (6) α δα = 1 ' v 1 k GA α δ v = (7) in which: v is the displacement, α the angle of α rotation due bending, the moment and k the shear factor. The boundary conditions are: - for the clamped end : α= ; v= α ' v - for the free end : = ; k GA α = For a rectangle cross section, the shear factor is 1(1 +ν) k = (8) 1+ 11ν When ν(x,t) and α (x,t) are synchronized in time, we can write: v( x, t) W( x) = T( t) (9) α( x, t) Ψ( x) and so equation (6), for the homogeneous problem, can be decoupled to obtain d W x ρω d W x ρa + ω W x = dx k G dx EI d Ψ( x) ρω d Ψ( x) ρa + ω Ψ ( x) = k G dx dx EI (1) We solve now the characteristic equation attached to the previous system: ρω ρa r + r ω = (11) k 'G EI The wave numbers are given by: ρω ρω ρaω ri =± ± + k 'G k 'G EI for j = 1,,3,... If we denote: ρω ρω ρaω a = + +, k 'G k 'G EI ρω ρω ρaω b= + + k 'G k 'G EI (1) (13) ISBN:

4 the solutions for eq. (1) are given by the ax ax W( x) = C1 sin + C cos + bx bx + C3sh + Cch ax ax Ψ ( x) = D1 sin + D cos + bx bx + D3sh + Dch (1) where C i and D i are constant coefficients, with i=1.,3,. The relationship between a and b is given by b= a ( +ν) 1 γ k a + 1 (15) 1 where γ=. k Taking the boundary conditions we have that [9]: b a absin a shb+ + b + a cos a chb+ a b = (16) Solving this equation we obtain the values for a i (and thus for b i ). The natural frequencies are given by [9]: Table 5 Natural frequencies for beams with crosssection x calculated using the (Sh) theory FEM analysis The FEM analysis is performed on a steel beam [8] shown in figure, first for the beam of length L = 1 mm and next for the beam having the 1 length L = mm, each with cross-section of 11x11 and x and meshed with mm and 5 mm elements respectively. f = a b E k π ρ +ν i i L 1 (17) for j = 1,,3,... Table Natural frequencies for beams with crosssection 11x11 calculated using the (Sh) theory Figure. Meshed beam in 3D representation The physical and mechanical properties of the analyzed beams are presented in table 6, while the geometry of the cross-section and consequently the geometrical characteristics are presented in table 7. Table 6 Physical and mechanical properties Property U.M. Value Mass density kg/m Youngs modulus N/m, 1 11 Poissons ratio -,3 ISBN:

5 High H [m] Wide B [m] Table 7 Geometrical characteristics Crosssection A [m ] Moment of inertia I [m ],11,11 1,1 1-1,8 1-9,,, 1-1, In tables 8 and 9 we have for different beam type the number of nodes and the number of elements corresponding to the type of mesh. The analysis demonstrated that no significant differences occur between results obtained by meshing with and 5 mm respectively. The natural frequencies obtained by means of FEM analysis, corresponding to the different beam type for the mesh with mm elements, for the first ten modes are given in tables 1 and 11. Mesh mm 5 mm Mesh mm 5 mm Table 8 Number of nodes and elements for the beams meshed with a 5 mm element Beam type 11x11x xx Table 9 Number of nodes and elements for the beams meshed with a mm element Beam type 11x11x1 xx Table 1 Natural frequencies for the beam with cross-section 11x11 determined by FEM analysis Lenght [mm] Table 11 Natural frequencies for the beam with cross-section x determined by FEM analysis Lenght [mm] Figure 5 present the relative deviation for the first ten natural frequencies obtained by FEM analysis and calculated using the (E-B) and (Sh) model respectively for the beam of length mm, whereas figure 5 present the same information for the beam of length 1 mm EB model - beam Sh model - beam EB model - beam Sh model - beam Figure 5. Relative deviation for the first ten natural frequencies obtained by FEM analysis and calculated using the (E-B) and (Sh) model respectively beam of length mm We observe, in figure 5, that for the beam of length mm the (Sh) model is proper for both presented cases (11 11 and ), presenting deviations less than.3% for all 1 weak-axis bending vibration modes. For the beam with length mm and cross-section the (E-B) model is proper too, while for more rigid beams (i.e. ) it provides deviations up to 1.5% in higher vibration modes. ISBN:

6 EB model - beam Sh model - beam EB model - beam 1 Sh model - beam 1 Figure 6. Relative deviation for the first ten natural frequencies obtained by FEM analysis and calculated using the (E-B) and (Sh) model respectively beam of length 1 mm The results for the shorter beam, with length 1 mm, presented in figure 6, reveal that for applications needing accurate results the (Sh) model can be involved for beams with reduced rigidity (i.e ), deviations being less than.%. For more rigid beams deviations are significant even for this model, therefore the use of (T) model is recommended. The (E-B) model cannot be used for short beams at all.. Conclusion The researches presented in this paper confirm that as shorter and rigid the beam, the Euler- Bernoulli model dose not provide enough precise information regarding the natural frequencies and mode shapes and the Share model is necessary to be involved. Moreover, for more severe cases, like the beam of length 1 mm and cross-section mm, a improved model, like the Timoshenko one is needed. Acknowledgement The work has been funded by the Sectoral Operational Programme Human Resources Development 7-13 of the Romanian Ministry of Labour, Family and Social Protection through the Financial Agreement POSDRU/89/1.5/S/6557 and POSDRU/17/1.5/S/ References: [1] A. Morassi, F. Vestroni, Dynamic Methods for Damage Detection in Structures, CISM Courses and Lectures, Vol. 99, Springer Wien New York, 8 [] D. Balageas, C.P. Fritzen, A. Güemes, Structural health monitoring, ISTE Ltd, London, 6 [3] G.-R.Gillich, Z.I. Praisach, D. Moaca-Onchis, About the effectiveness of damage detection methods based on vibration measurements, 3 rd WSEAS International Conference on Engineering Mechanics, Structures, Engineering Geology, Corfu Island, 1 [] S.W. Doebling, C.R. Farrar, M.B. Prime, D.W. Shevitz, Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review, Reprot No. LA 137-MS, Los Alamos National Laboratory, Los Alamos, NM, 1996 [5] O.S. Salawu, Detection of structural damage through changes in frequency: a review, Engineering Structures, 19(9), 1997, pp [6] Z.I. Praisach, G.-R. Gillich, D. Birdeanu, Considerations on Natural Frequency Changes in Damaged Cantilever Beams Using FEM, 3 rd WSEAS International Conference on Engineering Mechanics, Structures, Engineering Geology, Corfu Island, 1 [7] G.R. Gillich, Z.I. Praisach, Robust method to identify damages in beams based on frequency shift analysis, SPIE Smart Structures and Materials & Nondestructive Evaluation and Health Monitoring (Vol. 838), March 1, San Diego, California, USA [8] G.R. Gillich, Z.I. Praisach, C.M. Iavornic, Reliable method to detect and assess damages in beams based on frequency changes, Proceedings of the ASME 1 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, IDETC/CIE 1, August 1-15, 1, Chicago, Illinois, USA [9] Seon M. Han, Haym Bernaroya, Timothy Wei Dynamics of Transversely Vibrating Beams Using Four Engineering Theories, Journal of Sound and Vibration, 1999 ISBN: