Volume 36 Number

Transcription

1 Volume 6 Number

2 Vietnam Journal of Mechanics, VAST, Vol. 6, No. (), pp. 5 5 SPECTRAL ANALYSIS OF MULTIPLE CRACKED BEAM SUBJECTED TO MOVING LOAD N. T. Khiem,, P. T. Hang Institute of Mechanics, VAST, 8 Hoang Quoc Viet, Hanoi, Vietnam Electric Power University, Hanoi, Viet Nam nthiem@imech.ac.vn Received October 5, Abstract. In present paper, the spectral approach is proposed for analysis of multiple craced beam subjected to general moving load that allows us to obtain explicitly dynamic response of the beam in frequency domain. The obtained frequency response is traightforward to calculate time history response by using the FFT algorithm and provides a novel tool to investigate effect of position and depth of multiple cracs on the dynamic response. The analysis is important to develop the spectral method for identification of multiple craced beam by using its response to moving load. The theoretical development is illustrated and validated by numerical case study. Keywords: Multiple craced beam, moving load problem, frequency domain solution, modal method, spectral analysis.. INTRODUCTION The moving load problem has attracted attention of researchers and engineers in the field of structural engineering and it is so far an actual topic in dynamics of structures. The mathematical fundamentals of the problem were formulated in [ ]. The mathematical representation of the problem is strictly associated with the model adopted for moving load and structure subjected to the load. The models adopted for moving load are constant or harmonic force []; moving mass [5, 6] and more complicated vehicle system [7, 8]. The structure taen into this issue is firstly the simple and intact beam lie structures and, recently, more complicated structures [9 ]. Most of the aforementioned studies have investigated the moving load problem in time domain by using either the mode superposition (modal) method or the finite element one. The modal method [5] relies on the eigenvalue problem that is not easily for damaged structures. On the other hand, the FEM [6, 7] requires a time consumable tas to identify position of moving load for computing nodal load. Moreover, both the methods are poorly applicable for evaluating the shear force [8] and high frequency components [9]. Jiang et al have demonstrated in [, ] that the moving load problem can be investigated straightforwardly in the frequency domain. Khiem et al. [] have proposed a spectral approach to the moving load problem that is solved completely in frequency domain.

3 6 N. T. Khiem, P. T. Hang The present paper aims to use the spectral approach for analysis of frequency response of beam with arbitrary number of cracs. The equivalent spring model [] is adopted to represent open cracs in a beam element. The conventional time history response can be easily calculated from the frequency response in arbitrary frequency range. The theoretical development is illustrated and validated by numerical examples.. THE GOVERNING EQUATIONS OF DYNAMIC SYSTEM Let s consider a dynamic system that comprises a simply supported Euler-Bernoulli beam and a vehicle moving on the beam, see Fig.. Suppose that E, ρ, A, I, l are parameters of the beam and m, c, are respectively the mass, damping coefficient and stiffness of vehicle. Moreover, the beam is assumed to be craced at the position e,..., e n with the depth a,..., a n respectively. m x v ( y c E, I,, A x x w w(x, Figure. Dynamic model of beam subjected to moving vehicle By introducing Fig.. the Dynamic notations w model ( x, t ), z ( t ), of x ( beam t ) respectively subjected for transverse to moving deflection vehicle of the beam at section x, vertical displacement of the vehicle and distance of the vehicle from the left end (x = ) of beam, the governing equations for the system can be derived as follow By introducing the notations w(x,, z(, x ( respectively for transverse deflection of the beam at section w( x, vertical w( xdisplacement, w( x, of the vehicle and distance of the EI A A P( [ x x ( ] ; () vehicle from the left end (x = x ) of beam, t the governing t equations for the system can be derived as follow P( mg cy ( y( m[ g z ( ] ; () m y ( cy ( y( mw ( ; y( [ z( w ( ]; w ( w[ x(, t] [ x( ]. () EI w(x, w(x, In the latter x equation + ρaη + ρa w(x, function (x) represents t rough surface t = P (δ[x x of the beam on which (], () the vehicle is traveling. Furthermore, P ( solution = mg of equation + cẏ( () + is y( subject = to boundary m[g z(], conditions () w(, w (, w(, w (, () mÿ( + cẏ( + y( = mẅ (, y( = [z( w (], w ( = w[x (, t] + ζ[x (]. () and compatibility conditions at the crac positions In the latter w( eequation, w( function e, ; w ( ζ(x) e, represents w ( e, rough ; w ( e surface, wof ( e the, tbeam ); on which the vehicle is traveling. Furthermore, solution of Eq. () is subject to boundary conditions (5) [ w ( e, w ( e, ] w ( e, ; EI ( a Function (a) in equation () is defined in the theory of craced beam []. Note that moving load () expressed in the form P( P [ a b ( ] (6) represents a number of earlier models of the moving load. Namely, for the case when relative vertical displacement of vehicle and acceleration of beam are negligible one has a, b / g, P mg, ( d [ x ( ]/ dt. The conventional case of constant force moving on w(, = w (, = w(l, = w (l, =, () and compatibility conditions at the crac positions w(e +, = w(e, ; w (e +, = w (e, ; w (e +, = w (e,, [w (e +, w (e, ] = γ w (e,, γ = EIθ(a ). Function θ(a) in Eq. (5) is defined in the theory of craced beam []. smooth surface of beam corresponds to a, b. In the case of a concentrated harmonic load, a, b P, ( sin( e t ) that gives rise P( P sin( et ) and in the moving mass case, a, b / g, P mg, i.e. P( m[ g w ( ]. In this study we investigate the problem with moving load given generally in a discrete form P t ),..., P( t ). ( M II. FREQUENCY RESPONSE OF CRACKED BEAM TO GENERAL MOVING LOAD Supposing that the force P ( is travelling on the beam with constant speed, i. e. x ( vt, the Fourrier transform leads Eq. () to ). (5)

4 Spectral analysis of multiple craced beam subjected to moving load 7 Note that moving load () expressed in the form P ( = P [a + bξ(], (6) represents a number of earlier models of the moving load. Namely, for the case when relative vertical displacement of vehicle and acceleration of beam are negligible one has a =, b = /g, P = mg, ξ( = d ζ[x (]/dt. The conventional case of constant force moving on smooth surface of beam corresponds to a =, b =. In the case of a concentrated harmonic load, a =, b = P, ξ( = sin(ω e t + ϕ) that gives rise P ( = P sin(ω e t + ϕ) and in the moving mass case, a =, b = /g, P = mg, i.e. P ( = m[g ẅ (]. In this study we investigate the problem with moving load given generally in a discrete form {P (t ),..., P (t M )}.. FREQUENCY RESPONSE OF CRACKED BEAM TO GENERAL MOVING LOAD Supposing that the force P ( is travelling on the beam with constant speed, i.e. x ( = vt, the Fourrier transform leads Eq. () to d φ(x, ω) dx λ φ(x, ω) = Q(x, ω), λ = (ω iηω)/a, a = EI/ρA, (7) φ(x, ω) = w(x, e iωt dt, Q(x, ω) = P (x/v)e iωx/v /EIv. (8) It is well nown that general solution of Eq. (7) is φ(x, ω) = φ (x, ω) + with φ (x, ω) being general solution of homogeneous equation and Since h() = h () = h () = function x h(x s)q(s, ω)ds, (9) d φ(x, ω) dx λ φ(x, ω) =, () h(x) = (/λ )[sinh λx sin λx]. () φ (x, ω) = x h(x s)q(s, ω)ds () satisfies the conditions φ (, ω) = φ (, ω) = so that solution (9) will satisfy the boundary conditions at the left end of beam together with function φ (x, ω). It is easily to verify that solution φ (x, ω) of Eq. () satisfying conditions [φ (e + ) φ (e )] = γ φ (e ), () φ(e + ) = φ(e ), φ (e + ) = φ (e ), φ (e + ) = φ (e ),

5 8 N. T. Khiem, P. T. Hang can be expressed in the form φ (x, λ) = L (x, λ) + n µ K(x e ), () where L (x, λ) is a particular continuous solution of Eq. () satisfying the condition L (, λ) = L (, λ) = and { { for x for x K(x) = S(x) for x, K (x) = S (x) for x, = S(x) = (sinh λx + sin λx)/λ, S (x) = λ(sinh λx sin λx)/, (5) Representing the solution L (x, λ) as j µ j = γ j [L (e j, λ) + µ S (e j e ) ]. (6) = L (x) = CL (x, λ) + DL (x, λ), (7) and substituting it together with expression () into Eq. (9) one obtains n φ(x, ω) = CL (x, λ) + DL (x, λ) + µ K(x e ) + φ (x, ω). (8) Obviously, the latter function (8) satisfies boundary conditions at the left end of beam and the unnown constants C, D can be determined from the boundary conditions that is rewritten in more detail as = φ(l, ω) = φ (l, ω) =, CL (l, λ) + DL (l, λ) = n µ S( e ) φ (l, ω), = CL (l, λ) + DL (l, λ) = n µ S ( e ) φ (l, ω). = Solution of the latter equations is easily obtained in the form n n C = C + C µ, D = D + D µ, (9) = where C = [L (l, λ)φ (l, ω) L (l, λ)φ (l, ω)], D = [L (l, λ)φ (l, ω) L (l, λ)φ (l, ω)], d (λ) d (λ) () C = [L (l, λ)s (l e ) L (l, λ)s(l e )], D = [L (l, λ)s(l e ) L (l, λ)s (l e )], d (λ) d (λ) d (λ) = L (l, λ)l (l, λ) L (l, λ)l (l, λ). () Now substituting expression (7) with coefficient (9) into (6) yields = [I Γ(γ)B(λ, e)]µ = Γ(γ)b(λ, e), ()

6 Spectral analysis of multiple craced beam subjected to moving load 9 where the following matrices and vectors are introduced B(λ, e) = [b j, j, =,..., n], b j = C L (e j, λ) + D L (e j, λ) + K (e j e ), Γ(γ) = diag{γ,..., γ n }, µ = (µ,..., µ n ) T, e = (e,..., e n ) T, γ = (γ,..., γ n ) T, () b = (b,..., b n ) T, b j = C L (e j, λ) + D L (e j, λ), j =,..., n. Eq. () can be solved with respect to µ as where µ = [I - Γ(γ)B(λ, e)] Γ(γ)b(λ, e). () Therefore, frequency response of multiple craced beam can be represented as n φ(x, ω) = α (x, ω) + µ α (x, e, γ, ω), (5) = α (x, ω) = C L (x, λ) + D L (x, λ) + φ (x, ω), (6) α (x, ω) = C L (x, λ) + D L (x, λ) + K(x e ), =,..., n. Since the static response is defined as the frequency response at ω =, it can be conducted by solving the equation d φ(x, )/dx = Q(x, ). So that the static solution φ(x, ) satisfying the given boundary conditions is φ(x, ) = Q (x) - Q (l)x /6l + [Q (l)l / 6 - Q (l) / l]x, (7) Q (x) = x ds s ds s ds s Q(s, )ds. (8) If the moving load P ( has been given at the time mesh (t,..., t M ) the function Q(x, ω) would be defined in the form Q(x j, ω) = P (t j )e iωt j /EIv, x j = vt j j =,..., M. (9) Hence, the function defined in () can be calculated φ (x, ω) = φ (x, ω) = x x H(x) = h(x s)q(s, ω)ds = (/EI) h (x s)q(s, ω)ds = (/EI) {, x h(x), x, H (x) = M H(x vt r )P (t r ))e iωtr t r, () r= M H (x vt r )P (t r )e iωtr t r, () r= {, x h (x), x, t r = t r t r, () that allow the coefficients C, D to be completely calculated with expressions (). Thus, the frequency response (5), (6) is fully determined for the given discretely moving load. Once the frequency response φ(x, ω) has been nown the time history response w(x, = (/π) φ(x, ω)e iωt dω, ()

7 5 N. T. Khiem, P. T. Hang could be usually evaluated at the discrete time mesh t r = rt/n, r =,..., N in a finite interval of time [, T ] as N w(x, t r ) = (/T ) φ(x, ω )e iπr/n), r =,..., N, () = where ω = ω = (π/t ) and N is chosen accordingly to the frequency range of interest. For instance, if Ω is Nyquist frequency of the response, then N = Ω/ ω = ΩT/π. (5). RESULTS AND DISCUSSION An example of the beam with E =., ρ = 786 g/m, l = 5 m, h =. m, b =.5 m subjected to moving constant force is examined by using the proposed spectral method. Deflection, slope, bending moment and shear force distributed along the beam length are computed with different speeds of moving load and various scenarios of multiple cracs. Namely, the quantities are computed at the frequencies f = f ;.5f ; f ; f, where f is the fundamental frequency of uncraced beam with speed equal to a half of critical speed v =.5v c. Results of computation are given in Fig.. Fig. presents the deflection, slope, moment and shear response at fundamental frequency for various speed ratios, v/v c =... The frequency response for beam with different scenarios of crac position and depth is presented in Figs., where crac position is roving from 5 m to 5 m and crac depth is varying from % to 5%. Fig. 5 shows the response computed for different numbers of cracs appeared in the beam. In all the figures the deflection, slope, bending, moment and shear are plotted along the beam length, i.e. versus x (, l). It can be noted from Fig. that waveform of defection, slope, moment and shear response vary strongly with frequency and is much dissimilar to the vibration mode shape. The response at lower frequency may appear as higher frequency mode shape that is perhap caused by multi-resonance phenomenon for forced vibration under moving load. Vibration amplitude increases with the speed growing up to the critical one except the speed v =.5v c that shows to be intiresonant speed (Fig. ). Further increase of speed from the critical one leads to reduced vibration amplitude so that maximum effect is observed at critical speed. Furthermore, any crac inside beam maes uniformly distributed change in frequency response so that crac position cannot be visible from the frequency response plotted along the beam length. However, the largest change is observed when crac occurred at position m from the left end. It can be seen from Fig. that symmetric (about the beam middle) cracs lead to not equal change in frequency response that is important to solve the nonunique solution problem in crac detection for beam with symmetric boundary conditions. Figs. 5 and 6 show that while the frequency response monotony increases with crac depth, multiple cracs occurred additionally to the right of beam middle mae reduction of the response. This implies that frequency response is monotony increasing with amount of cracs located on the left of beam midpoint and decreasing with growing number of cracs on the right of the midpoint.

8 Resonant deflection Resonant slope Bending moment at resonance Shear force at fundamental resonance Resonant deflection Slope at fundamental resonance Shear force Bending moment Slope Deflection Vbration amplitude increases with the speed growing up to the critical one except the speed v. 5v c that shows to be intiresonant speed (Fig. ). Further increase of speed from the critical one Spectral analysis of multiple craced beam subjected to moving load 5 leads to reduced vibration amplitude so that maximum effect is observed at critical speed. 6 x x -5 f = *f f =.5*f 8 f =.5*f 6 f = *f f = *f f = f - f = *f - f = f x x -6 f =.5*f.5 f =.5*f f = *f f = f.5.5 f = *f f = *f f = f f = *f Fig.. Frequency response for deflection, slope, bending moment and shear of uncraced beam at natural frequencies f [.;.5;.;.] f, speed v. 5v. c Fig.. Frequency response for deflection, slope, bending moment and shear of uncraced beam at natural frequencies f = [.;.5;.;.] f, speed v =.5v c Futhermore, any crac inside beam maes uniformly distributed change in frequency response. 5 so that crac position cannot be visible from the frequency response plotted along the beam length. x - However, v =.*Vc the largest change v =.5*Vc is observed when v =.*Vc crac occurred at position m from the left end. It can be seen from Figure that symmetric (about the beam middle) cracs lead to not equal change -. v =.*Vc in frequency response that is important to solve the nonunique solution problem in crac detection -. for beam with symmetric boundary conditions. v =.8*Vc -. Figures 5 and 6 show that while the frequency response monotony increases with crac depth, v = *Vc v =.5*Vc v =.*Vc v =.*Vc multiple cracs occurred additionally to the right of beam middle mae reduction of the response. v =.6*Vc -. v =.*Vc - This implies that frequency response is monotony increasing with amount of cracs located on the v = Vc v =.5*Vc -.5 v = *Vc left of beam midpoint and decreasing with growing number - of cracs on the right of the midpoint. -.6 v =.8*Vc v =.*Vc v =Vc - v =.6*Vc v =.5*Vc -.7 v =.*Vc x x -5 v = Vc v =.8*Vc.5 v =.*Vc v =.*Vc v =.5*Vc.5 v = Vc v =.8*Vc v =.6*Vc v = *Vc v =.5*Vc.5 v =.5*Vc v =.*Vc v =.6*Vc.5 v =.*Vc v = *Vc v =.*Vc v =.*Vc v =.5*Vc v =.*Vc x Fig.. Frequency response for deflection, slope, bending moment and shear at fundamental frequency in different speed ratios (.;.;.;.5;.6;.8;.;.;.5;.). Fig.. Frequency response for deflection, slope, bending moment and shear at fundamental x e = e= frequency in different speed ratios (.;.;.;.5;.6;.8;.;.;.5;.) e = 5 e = 5.5 e= 5 e= 5.5 e = e =.5 e= e= 5 e= e= e= 5 e= 5 e = e= and 5 or no crac.5 e = 5 - e = e = and 5 or no crac e =

9 Resonant bending moment Resonant bending moment Second resonant deflection Resonant deflection Bending moment at resonance Second resonant bending moment Second resonant deflection Second resonant bending moment v =.*Vc Shear force at second resonance Shear force at second resonance Second resonant slope Second resonant slope resonant shear force resonant shear force Resonant slope Shear force at fundamental resonance v =.5*Vc.5 v = Vc v =.8*Vc v =.6*Vc v = *Vc v =.5*Vc.5 v =.5*Vc v =.*Vc v =.6*Vc.5 v =.*Vc v = *Vc v =.*Vc v =.*Vc v =.5*Vc v =.*Vc N. T. Khiem, P. T. Hang Fig.. Frequency response for deflection, slope, bending moment and shear at fundamental frequency in different speed ratios (.;.;.;.5;.6;.8;.;.;.5;.) x - x e = e= e = 5 e = 5.5 e= 5 e= 5.5 e = e =.5 e= e= 5 e= e= e= 5 e= 5 e = e= and 5 or no crac.5 e = 5 - e = e = and 5 or no crac e = x -6.5 x -6 e = 5 e = and 5 or no crac 6 x -6 e =.5 x -6 e = 5 e = 5 e = e = 5 e = and 5 or no crac.5 e= e= 5 - e = e = 5 e = e = 5 e = 5 e = e= e= 5 e= 5 e= e= e= 5 e= e= e = e 5 = e = e = e= 5 e= e= e= 5 e= e= 5 e= e = e = e= and 5 or no crac e= e= and 5 or no crac Fig.. Effect of crac position (5; ; 5; ; 5; ; 5; ; 5m) on the deflection, slope, bending moment and shear force response at fundamental resonant Fig.. Effect of crac position (5; ; 5; ; 5; ; 5; ; 5m) the frequency deflection, slope, Fig.. Effect of crac position (5; ; 5; ; 5; ; 5; ; 5m) on the deflection, slope, x 5 force at frequency x -5 bending moment and shear force response at fundamental resonant frequency x a = 5% x -5 a = 5% a = 5% a = % a = a % = 5% a = % a = % a = % or no crac a = % a = % a = % a = % a = % a = % a = % a = % a = % or no crac - a = % or no crac a = % a = % - a = % a = % - - a = % or no crac x -6 - x a = 5% 6 x -6 x a = % or no crac a = % a = 5% a = % a = % or no crac.5 a = % or no crac a = % a = % a = % a = % a = % a = % a = % or no crac -.5 a = % - a = % a = % a = % a = % a = % a = 5% a = % a = % a = 5% Fig. 5. Effect of crac depth (; ; ; ; ; 5%) on the deflection, slope, bending moment and shear force response at fundamental resonant frequency Fig. 5. Effect of crac depth (; ; ; ; ; 5%) on the deflection, slope, bending moment and shear force response at fundamental resonant frequency. Fig. 5. Effect of crac depth (; ; ; ; ; 5%) on the deflection, slope, bending moment and shear force response at fundamental resonant frequency

10 Resonant deflection Resonant bending moment Resonant slope Resonant shear force no crac n = 8 n = 5 Spectral analysis of multiple craced beam subjected to moving load 5 5 x - x n = 5 n = n = 6 n = 5 n = 6 n = n = n = 7 n = 7 n = n = 9 n = 8 n = 8 n = n = n = 9 n = no crac n = no crac x -6 x -6 no crac n = n = 9 n = 8 n = n = n = n = 7 n = 9 n = n = 6 n = 8 n = n = 5 n = 7 n = no crac n = 6 n = n = Fig.6. Effect of number of cracs (; ; ; ; 5; 6; 7; 8; 9) on the deflection, slope, bending moment and shear force response at fundamental resonant frequency. Fig. 6. Effect of number of cracs (; ; ; ; 5; 6; 7; 8; 9) on the deflection, slope, bending IV. CONCLUSION moment and shear force response at fundamental resonant frequency In present paper the spectral method has been developed for dynamic analysis of multiple craced beams subjected to general moving load in frequency domain. A closed form solution for frequency response to moving load was conducted for beam with arbitrary number of cracs. The obtained solution is straightforward to calculate time history response and provides a novel tool for 5. CONCLUSION dynamic analysis of response at arbitrary frequency. Numerical results have shown that a localized crac maes uniformly distributed change in waveform of the frequency response; due to moving load the Incracs present occurred paper to the symmetric spectral positions method affect hasnot been symmetrically developed on forthe dynamic response; analysis amplitude of of forced vibration is not monotony increasing with growing number of cracs. multiple The craced proposed beams method subjected can be used to general for dynamic moving analysis load in frequency the case of domain. more complicated A closed moving load and crac detection problem by measurement of dynamic response of beam-lie form solution for frequency response to moving load was conducted for beam with arbitrary number of cracs. The obtained solution is straightforward to calculate time history structure subjected to moving load. response and provides a novel tool for dynamic analysis of response at arbitrary frequency. Numerical results have shown that a localized crac maes uniformly distributed change in waveform of the frequency response; due to moving load the cracs occurred to symmetric positions affect not symmetrically on the response; amplitude of forced vibration is not monotony increasing with growing number of cracs. The proposed method can be used for dynamic analysis in the case of more complicated moving load and crac detection problem by measurement of dynamic response of beam-lie structure subjected to moving load. REFERENCES [] M. Olsson. On the fundamental moving load problem. Journal of Sound and Vibration, 5, (), (99), pp [] A. V. Pesterev and L. A. Bergman. Response of elastic continuum carrying moving linear oscillator. Journal of Engineering Mechanics,, (8), (997), pp [] L. Fryba. Vibration of solids and structures under moving loads. Thomas Telford, (999). [] I. G. Raftoyiannis, T. P. Avraam, and G. T. Michaltsos. A new approach for loads moving on infinite beams resting on elastic foundation. Journal of Vibration and Control, 8, (), (), pp

11 5 N. T. Khiem, P. T. Hang [5] A. Garinei. Vibrations of simple beam-lie modelled bridge under harmonic moving loads. International Journal of Engineering Science,, (), (6), pp [6] G. V. Rao. Linear dynamics of an elastic beam under moving loads. Journal of Vibration and Acoustics,, (), (), pp [7] A. V. Pesterev, B. Yang, L. A. Bergman, and C. A. Tan. Response of elastic continuum carrying multiple moving oscillators. Journal of Engineering Mechanics, 7, (), (), pp [8] R. Zarfam and A. R. Khaloo. Vibration control of beams on elastic foundation under a moving vehicle and random lateral excitations. Journal of Sound and Vibration,, (6), (), pp. 7. [9] H. P. Lee and T. Y. Ng. Dynamic response of a craced beam subject to a moving load. Acta Mechanica, 6, (), (99), pp.. [] M. A. Mahmoud and M. A. Abou Zaid. Dynamic response of a beam with a crac subject to a moving mass. Journal of Sound and Vibration, 56, (), (), pp [] C. Bilello and L. A. Bergman. Vibration of damaged beams under a moving mass: theory and experimental validation. Journal of Sound and Vibration, 7, (), (), pp [] H. P. Lin and S. C. Chang. Forced responses of craced cantilever beams subjected to a concentrated moving load. International Journal of Mechanical Sciences, 8, (), (6), pp [] J. Yang, Y. Chen, Y. Xiang, and X. L. Jia. Free and forced vibration of craced inhomogeneous beams under an axial force and a moving load. Journal of Sound and Vibration,, (), (8), pp [] L. Deng and C. S. Cai. Identification of dynamic vehicular axle loads: Theory and simulations. Journal of Vibration and Control, 6, (), (), pp [5] M. Shafiei and N. Khaji. Analytical solutions for free and forced vibrations of a multiple craced Timosheno beam subject to a concentrated moving load. Acta Mechanica,, (- ), (), pp [6] J. J. Wu, A. R. Whittaer, and M. P. Cartmell. The use of finite element techniques for calculating the dynamic response of structures to moving loads. Computers & Structures, 78, (6), (), pp [7] M. Moghaddas, E. Esmailzadeh, R. Sedaghati, and P. Khosravi. Vibration control of Timosheno beam traversed by moving vehicle using optimized tuned mass damper. Journal of Vibration and Control, 8, (6), (), pp [8] A. V. Pesterev, C. A. Tan, and L. A. Bergman. A new method for calculating bending moment and shear force in moving load problems. Journal of Applied Mechanics, 68, (), (), pp [9] N. Azizi, M. Saadatpour, and M. Mahzoon. Using spectral element method for analyzing continuous beams and bridges subjected to a moving load. Applied Mathematical Modelling, 6, (8), (), pp [] J. Q. Jiang. Transient responses of Timosheno beams subject to a moving mass. Journal of Vibration and Control, 7, (), (), pp [] J. Q. Jiang, W. Q. Chen, and Y. H. Pao. Reverberation-ray analysis of continuous Timosheno beams subject to moving loads. Journal of Vibration and Control, 8, (6), (), pp [] N. T. Khiem, T. H. Tran, and N. V. Quang. An approach to the moving load problem for multiple craced beam. In Topics in Modal Analysis, Volume 7, pp Springer, (). [] T. G. Chondros, A. D. Dimarogonas, and J. Yao. A continuous craced beam vibration theory. Journal of Sound and Vibration, 5, (), (998), pp. 7.

12 VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY VIETNAM JOURNAL OF MECHANICS VOLUME 6, N., CONTENTS Pages. N. T. Khiem, P. T. Hang, Spectral analysis of multiple craced beam subjected to moving load. 5. Dao Van Dung, Vu Hoai Nam, An analytical approach to analyze nonlinear dynamic response of eccentrically stiffened functionally graded circular cylindrical shells subjected to time dependent axial compression and external pressure. Part : Numerical results and discussion. 55. Lieu B. Nguyen, Chien H. Thai, Ngon T. Dang, H. Nguyen-Xuan, Transient analysis of laminated composite plates using NURBS-based isogeometric analysis. 67. Tran Xuan Bo, Pham Tat Thang, Do Thanh Cong, Ngo Sy Loc, Experimental investigation of friction behavior in pre-sliding regime for pneumatic cylinder 8 5. Vu Thi Thuy Anh, Dao Huy Bich, Nguyen Dinh Duc, Nonlinear post-bucling of thin FGM annular spherical shells under mechanical loads and resting on elastic foundations N. D. Anh, N. N. Linh, A weighted dual criterion for stochastic equivalent linearization method using piecewise linear functions. 7