Determination of mode shapes of a multiple cracked beam element and its application for free vibration analysis of a multi-Span continuous beam

Volume 35 Number 4 4 Vietnam Journal of Mechanics, VAST, Vol. 35, No. 4 (2013), pp. 313 – 323 DETERMINATION OF MODE SHAPES OF A MULTIPLE CRACKED BEAM ELEMENT AND ITS APPLICATION FOR FREE VIBRATION ANALYSIS OF A MULTI-SPAN CONTINUOUS BEAM Tran Van Lien∗, Trinh Anh Hao University of Civil Engineering, Hanoi, Vietnam ∗E-mail: lientv@hotmail.com Abstract. This article presents some results on the determination of the vibration shape function of a multiple cracked elastic beam element, which is modeled as an assembly of intact sub-segments connected by massless rotational springs. Algorithms and computer programs to analyze changes of natural mode shapes of multiple cracked beams have been carried out. Numerical analysis of natural mode shapes of cracked simple support beams using the obtained expression shows a good agreement in comparison with the well-known analytical methods. The methodology approach and results presented in this article are new and basic for building an efficient method to identify cracks in beam structures using wavelet analysis of mode shapes. Keywords: Shape function, cracked beam, transfer matrix, natural frequency, mode shape. 1. INTRODUCTION Current researches on the identification of cracks or damages in a structure us- ing non-destructive test method have been developed primarily based on the dynamic characteristics of the structure such as natural frequency, mode shapes, response spec- trum function [1–11]. These dynamic characteristics are normally determined by analyt- ical method, semi-analytical method, finite element method (FEM), and dynamic stiff- ness method (DSM). The analytical and semi-analytical methods are limited in a simple beam [1, 5, 6] and not applicable for a complex structure such as multi-span continuous beam or frame structure. Therefore, identification of dynamic characteristics of a structure is mainly based on the FEM and the DSM: - In the FEM, a multiple-cracked beam element has been modeled as an assembly of intact sub-segments connected by massless rotational springs at the locations of cracks or damages. Sato [7] has developed a combination of the transfer matrix method (TMM) and the FEM for vibration analysis of a beam with abrupt changes of cross-section. Gounaris and Dimarogonas [8] have conjugated the FEM with the idea of a compliance matrix to develop a specific technique for vibration analysis of a cracked beam. Zheng and Kessis- soglou [10] has used the overall additional flexibility matrix instead of the local additional 314 Tran Van Lien, Trinh Anh Hao flexibility matrix to obtain the total flexibility matrix of a cracked beam. FEM is an ap- proximate method in comparison with the analytical method, especially with high natural frequencies and mode shapes [2, 12]. - The DSM is used to overcome the above limitations of the FEM and extend the advantages of the analytical method to a more complex structure other than beams [12]. In the DSM, a multiple-cracked beam element is also divided into intact sub-segments and joined at crack locations [13]. Combining the DSM and the TMM with the references of [2, 14], the authors have built straight 3D multiple cracked bar element under axial compression, tension, bending and twisting. The authors have analyzed the change of natural frequencies of structures depending of quantities, locations and depths of cracks [2, 9]. The authors also determined the quantities, locations and depths of cracks in a multiple-cracked beam based on natural frequencies measured from experiments [2, 15]. One advantage of the method is that the numbers of input parameters may be less than the number of parameters found out on solving extreme value problem. However, mode shapes corresponding to the measured natural frequencies have not been found yet. To determine the mode shapes, it is necessary to find the shape functions for the multiple- cracked beam with arbitrary depths and positions of cracks with an arbitrary coefficient of damping. This issue is quite complex and has not been published. This paper presents some results on the determination of shape functions of a mul- tiple cracked elastic beam subjected to bending, which is modeled as intact sub-segments joined together at the crack positions by rotation spring. Numerical analysis of the shape function of the cracked simple support beams using the obtained expression shows a good agreement in comparison with the well-known analytical methods. A computer program for analyzing changes in mode shapes of a beam structure with multiple-cracked elements is built. The proposed method and the obtained results are new and they are the basis for determination of cracks in the structures using wavelet analysis of mode shapes. 2. SHAPE FUNCTION OF INTACT LATERAL BEAM According to FEM as described in [16], the shape function of intact lateral beam (Fig. 1) is the solution of static equilibrium differential equation without external forces as follows EIz d4w dx4 = 0 (1) P2 u3 P4 u1 u4 P3 x P1 y L u2 Fig. 1. Modeling of intact lateral beam Determination of mode shapes of a multiple cracked beam element and its application for free ... 315 The boundary conditions are w (0) = 1 ; w′ (0) = 0 ; w (L) = 0 ; w′ (L) = 0 (2-a) w (0) = 0 ; w′ (0) = 1 ; w (L) = 0 ; w′ (L) = 0 (2-b) w (0) = 0 ; w′ (0) = 0 ; w (L) = 1 ; w′ (L) = 0 (2-c) w (0) = 0 ; w′ (0) = 0 ; w (L) = 0 ; w′ (L) = 1 (2-d) The solutions of Eq. (1) with boundary conditions (2a-d) are the shape functions N1 −N4, called Hermit functions N1 (x) = 1− 3 (x L )2 + 2 (x L )3 ; N2 (x) = x− 2 x2 L + x3 L2 ; N3 (x) = 3 (x L )2 − 2 (x L )3 ; N4 (x) = − x2 L + x3 L2 (3) Let boundary values be u1; u2; u3; u4 at both end nodes, the transverse displacement of beam at the section x is w (x) = N1 (x)u1 +N2 (x)u2 +N3 (x)u3 +N4 (x)u4 (4) According to DSM developed by Leung [12], shape functions of the intact beam are solutions of the undamping free vibration equation EIz d4Φ(x, ω) dx4 − ρAω2Φ(x, ω) = 0 (5) Solutions of Eq. (5) with boundary conditions (2a-d) are shape functions N1 − N4 as follows  N1 N2 N3 N4   T =   cos λx/L sinλx/L coshλx/L sinhλx/L   T 1/2− F4/2λ2 F2L/2λ2 − F3/2λ2 F1/2λ2 − F6/2λ3 L/2λ+ F4L/2λ3 − F5/2λ3 − F3L/2λ3 1/2 + F4/2λ 2 − F2L/2λ2 F3/2λ2 − F1/2λ2 F6/2λ 3 L/2λ− F4L/2λ3 F5/2λ3 F3L/2λ3   (6) Where λ = 4 √ ω2 ρAL4 EI is the dynamic parameter; ω is the circular frequency (rad/s); Fi(i = 1, ..., 6) are trigonometric functions F1 = −λ(sinhλ− sinλ)/δ ; F2 = −λ(coshλ sinλ− sinhλ cosλ)/δ ; F3 = −λ2(coshλ− cos λ)/δ; F4 = λ2(sinhλ sinλ)/δ ; F5 = λ3(sinhλ+ sinλ)/δ ; F6 = −λ3(coshλ sinλ+ sinhλ cosλ) / δ ; δ = coshλ cosλ− 1 When ω = 0 corresponding to the static problem, we receive the Hermit shape functions (3) from the shape functions (6). In the case of intact beam, the determination of shape functions is the first step to obtain the dynamic stiffness matrix of beam subject to bending. In the case of multiple cracked beam, it will be modeled by rotation spring in the position of the crack, the determination of shape function is more complicated problem. In this case, it is necessary to base on the dynamic stiffness matrix obtained by the TMM. 316 Tran Van Lien, Trinh Anh Hao 3. DYNAMIC STIFFNESS MATRIX OF A MULTIPLE CRACKED BEAM Let consider a beam of length L subjected to bending on surface 0xy, cross-section area A = b×h, moment of inertia I and Young’s modulus E, mass density ρ. Free vibration of the beam is described by the following equation [9] d4Φ(x, ω) dx4 − λ4Φ(x, ω) = 0 (7) Where Φ(x, ω) denotes complex amplitude of vibration; λ = 4 √ ω2 ρA EˆIz ( 1− iµ2 ω ) ; i = √ −1 is dynamic parameter; Eˆ = E (1 + iµ1ω) is complex modulus; µ1, µ2 are the material and viscous damping coefficients, respectively; ω is circular frequency (rad/s). When λ = 0 corresponding to ω = 0 it is a static deformation. Suppose that the beam has cracks at positions xj with the depths of aj, j = 1, 2, . . . , n, where x0 = 0 < x1 < x2 < . . . < xn < xn+1 = L (Fig. 2). The cracks are modeled as rotational springs with the stiffness kzj calculated by converting formu- las [17, 18]. k1 x1 k2 x2 kn xn Nodal x=0 u3 P4 P3 x u2 u1 P1 P2 u4 y .... Nodal x=L L Fig. 2. Modeling of a multiple-cracked beam element The general solution of Eq. (7) for the sub-segment j = 1, 2, . . . , n + 1 with x ∈ (xj−1, xj) has the form of Φj(x) = K1(λx)Z + j−1,1 + K2(λx) λ Z+j−1,2 + K4(λx) EIzλ3 Z+j−1,3 − K3(λx) EIzλ2 Z+j−1,4 ; x = x− xj−1 (8) Where Ki(x) are Krylov functions and Z + j−1,i are initial parameters of this sub- segment K1(x) = cosh x+ cos x 2 ; K3(x) = cosh x− cos x 2 ; K2(x) = sinh x+ sinx 2 ; K4(x) = sinh x− sin x 2˘ Z+j−1,1 , Z + j−1,2, Z + j−1,3, Z + j−1,4 ¯T = “ Φ(xj−1 + 0);Φ ′(xj−1 + 0); EˆIzΦ ′′′(xj−1 + 0); −EˆIzΦ ′′(xj−1 + 0) ”T By using the combination of dynamic stiffness and transfer matrix methods, we yield [9] [P1 P2 P3 P4] T = [Ke] . [u1 u2 u3 u4] T (9) The matrix [Ke]4×4 is called dynamic stiffness matrix for the multiple cracked beam. Determination of mode shapes of a multiple cracked beam element and its application for free ... 317 4. DETERMINATION OF SHAPE FUNCTIONS AND MODE SHAPES OF MULTIPLE CRACKED BEAM ELEMENTS 4.1. Shape functions Using the general solution (8) and Eq. (9), we can determine the free vibration shape function of multiple cracked beam elements as following: a) In order to find the shape function N1, we determine the nodal forces P1 and P2 based on Eq. (9) with the boundary condition (2-a)( P1 P2 ) = ( k11 k12 k13 k14 k21 k22 k23 k24 )( 1 0 0 0 )T = ( k11 k21 ) (10) The initial parameters of first sub-segment (j = 1) are Z+0 = { Z+1 (0) = 1; Z + 2 (0) = 0; Z + 3 (0) = k11; Z + 4 (0) = k21 } (11) Therefore, the shape function N1 for the first sub-segment is N (1) 1 = K1(λx) + K4(λx) EˆIzλ3 k11 − K3(λx) EˆIzλ2 k21 (12) By using TMM, we get the shape function N1 for the next sub-segment [9]. In the case of intact beam, the shape function has the formula N1 = K1(λx) + K˜1K˜2 − K˜3K˜4 K˜23 − K˜2K˜4 K4(λx)− K˜22 − K˜1K˜3 K˜23 − K˜2K˜4 K3(λx) where K˜i = Ki(λL). When damping coefficients are zero µ1 = µ2 = 0, one will get the shape function N1 following the expression (6). b) To find the shape function N2, we determine the nodal forces P1 and P2 based on Eq. (9) with the boundary condition (2-b)( P1 P2 ) = ( k11 k12 k13 k14 k21 k22 k23 k24 )( 0 1 0 0 )T = ( k12 k22 ) (13) Also, the initial parameters of the first sub-segment (j = 1) are Z+0 = { Z+1 (0) = 0; Z + 2 (0) = 1; Z + 3 (0) = k12; Z + 4 (0) = k22 } (14) Then the shape function N2 for the first sub-segment is N (1) 2 = K2(λx) λ + K4(λx) EˆIzλ3 k12 − K3(λx) EˆIzλ2 k22 (15) Similarly, we get the shape function N2 for the next sub-segment. In the case of intact beam, the shape function has the formula N2 = K2(λx) λ + K˜22 − K˜1K˜3 λ(K˜23 − K˜2K˜4) K4(λx)− K˜2K˜3 − K˜1K˜4 λ(K˜23 − K˜2K˜4) K3(λx) When damping coefficients are zero µ1 = µ2 = 0, one will get the shape function N2 following the expression (6). 318 Tran Van Lien, Trinh Anh Hao c) To find the shape function N3, we determine the nodal forces P1 and P2 based on Eq. (9) with the boundary condition (2-c)( P1 P2 ) = ( k11 k12 k13 k14 k21 k22 k23 k24 )( 0 0 1 0 )T = ( k13 k23 ) (16) The initial parameters of the first sub-segment (j = 1) are Z+0 = { Z+1 (0) = 0; Z + 2 (0) = 0; Z + 3 (0) = k13; Z + 4 (0) = k23 } (17) So, the shape function N3 for the first sub-segment is N (1) 3 = K4(λx) EˆIzλ3 k13 − K3(λx) EˆIzλ2 k23 (18) Similarly, we get the shape function N3 for the next sub-segment. In case of the intact beam, the shape function has the formula N3 = − K˜2 K˜23 − K˜2K˜4 K4(λx) + K˜3 K˜23 − K˜2K˜4 K3(λx) When damping coefficients are zero µ1 = µ2 = 0, one will get the shape function N3 following the expression (6). d) To find the shape function N4, we determine the nodal forces P1 and P2 based on Eq. (14) with the boundary condition (2-d)( P1 P2 ) = ( k11 k12 k13 k14 k21 k22 k23 k24 )( 0 0 0 1 )T = ( k14 k24 ) (19) The initial parameters of the first sub-segment (j = 1) are Z+0 = { Z+1 (0) = 0; Z + 2 (0) = 0; Z + 3 (0) = k14; Z + 4 (0) = k24 } (20) The shape function N4 for the first sub-segment is N (1) 4 = K4(λx) EˆIzλ3 k14 − K3(λx) EˆIzλ2 k24 (21) Similarly, we get the shape function N4 for the next sub-segment. In case of the intact beam, the shape function has the formula N4 = K˜3 λ(K˜23 − K˜2K˜4) K4(λx)− K˜4 λ(K˜23 − K˜2K˜4) K3(λx) When damping coefficients are zero µ1 = µ2 = 0, one will get the shape function N4 following expression (6). 4.2. Mode shapes The dynamic stiffness matrices of structure Kˆ (ω) are assembled similarly as FEM from the dynamic stiffness matrices [Ke] of each beam elements. Thus, the problem of free vibration of structures with the multiple-cracked bar elements leads to the determination of natural frequencies and mode shapes from the following equation Kˆ(ω)U = 0 (22) Determination of mode shapes of a multiple cracked beam element and its application for free ... 319 Where the natural frequencies ωj are determined from the equation: det Kˆ (ω) = 0 (23) The nodal displacements U corresponding to the natural frequencies ωj will be determined by equations (22). Having obtained the nodal displacements u1; u2; u3; u4, we receive the mode shapes of the structure with the multiple-cracked beam elements from the expression (4). 5. NUMERICAL RESULTS AND DISCUSSION 5.1. Simple support beam Let us consider a simple support beam with the following parameters: beam length L = 0.235 m; cross-section area A = b × h = 0.006× 0.0254 m2; Young’s modulus E = 2.06 × 1011 N/m2, the Poisson coefficient ν = 0.35 and material mass density ρ = 7800 kg/m3 [19]. 0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 point measure A m p lit u d e theoretical existing result 0 10 20 30 40 50 60 70 80 90 100 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 A m p lit u d e theoretical existing result Fig. 3. The comparison of the first two mode shapes of the simple support beam with three cracks at positions x = 0.2L, 0.5L, 0.7L from the left node Fig. 3a-b show two first mode shapes of the simple support beam with three cracks at positions: x = 0.2L, 0.5L, 0.7L from the left node end of the beam with the same relative crack depth of 50% calculated by the analytical method (line - - -, [19]) and the proposed method (line —). These figures present a good agreement between the numerical results obtained by the proposed method and the well-known analytical methods. 5.2. Multiple-span continuous beam Let consider the multiple-span continuous beam with the following parameters: L1 = 0.8 m, L2 = 1.1 m, L3 = 0.6 m, cross-section area b×h = 0.04×0.02m2, Young’s modulus E = 2.1×1011 (N/m2), the Poisson coefficient ν = 0.3 and material mass density ρ = 7800 kg/m3 in Fig. 4. Figs. 5, 6, 7 show the changes in the first three mode shapes (the differences between the mode shapes obtained from cracked and uncracked beams) of the structure with one 320 Tran Van Lien, Trinh Anh Hao b h L1=0.8m L2=1.1m L3=0.6m Fig. 4. Multiple-span continuous cracked beam crack, the depth of crack ranges from 10% to 60% and the positions of cracks are as followings: - At 0.2 m (Figs. 5a, 6a, 7a), 0.4 m (Figs. 5b, 6b, 7b), 0.6 m (Figs. 5c, 6c, 7c) from the left node of the first span; - At 0.2 m (Figs. 5d, 6d, 7d), 0.5 m (Figs. 5e, 6e, 7e), 0.8 m (Figs. 5f, 6f, 7f) from the left node of the second span; - At 0.1 m (Figs. 5g, 6g, 7g), 0.3 m (Figs. 5h, 6h, 7h), 0.5 m (Figs. 5i, 6i, 7i) from the left node of the third span. a) b) c) 0 0.5 1 1.5 2 2.5 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 Comparision of the eigenmodes: 1 Three-Span(m) A m p li tu d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 Comparision of the eigenmodes: 1 Three-Span(m) A m p li tu d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Comparision of the eigenmodes: 1 Three-Span(m) A m p li tu d e 10 % 20% 30% 40% 50% 60% d) e) f) 0 0.5 1 1.5 2 2.5 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Comparision of the eigenmodes: 1 Three-Span(m) A m p li tu d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Comparision of the eigenmodes: 1 Three-Span(m) A m p li tu d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Comparision of the eigenmodes: 1 Three-Span(m) A m p li tu d e 10 % 20% 30% 40% 50% 60% g) h) i) 0 0.5 1 1.5 2 2.5 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Comparision of the eigenmodes: 1 Three-Span(m) A m p li tu d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 Comparision of the eigenmodes: 1 Three-Span(m) A m p li tu d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Comparision of the eigenmodes: 1 Three-Span(m) A m p li tu d e 10 % 20% 30% 40% 50% 60% Fig. 5. The comparison of first mode shapes of the multi-span continuous beam with one crack and variable position Fig. 8 shows the changes in the first three mode shapes of multiple cracked multiple- span continuous beam when the number of cracks increases from 1 to 6 with an equidistance 0.15 m on the second span and with invariable crack depth of 30%. Determination of mode shapes of a multiple cracked beam element and its application for free ... 321 a) b) c) d) e) f) g) h) i) 0 0.5 1 1.5 2 2.5 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Comparision of the eigenmodes: 2 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Comparision of the eigenmodes: 2 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Comparision of the eigenmodes: 2 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Comparision of the eigenmodes: 2 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 Comparision of the eigenmodes: 2 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Comparision of the eigenmodes: 2 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 Comparision of the eigenmodes: 2 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 Comparision of the eigenmodes: 2 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Comparision of the eigenmodes: 2 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% Fig. 6. The comparison of second mode shapes of the multi-span continuous beam with one crack and variable position d) e) f) a) b) c) g) h) i) 0 0.5 1 1.5 2 2.5 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 Comparision of the eigenmodes: 3 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Comparision of the eigenmodes: 3 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Comparision of the eigenmodes: 3 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Comparision of the eigenmodes: 3 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Comparision of the eigenmodes: 3 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 Comparision of the eigenmodes: 3 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Comparision of the eigenmodes: 3 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Comparision of the eigenmodes: 3 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% 0 0.5 1 1.5 2 2.5 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Comparision of the eigenmodes: 3 Three-Span(m) A m p lit u d e 10 % 20% 30% 40% 50% 60% Fig. 7. The comparison of third mode shapes of the multi-span continuous beam with one crack and variable position 322 Tran Van Lien, Trinh Anh Hao a) b) c) 0 0.5 1 1.5 2 2.5 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Comparision of the eigenmodes: 2 Three-Span(m) A m p li tu d e 1 Crack 2 Cracks 3 Cracks 4 Cracks 5 Cracks 6 Cracks 0 0.5 1 1.5 2 2.5 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Comparision of the eigenmodes: 1 Three-Span(m) A m p li tu d e 1 Crack 2 Cracks 3 Cracks 4 Cracks 5 Cracks 6 Cracks 0 0.5 1 1.5 2 2.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Comparision of the eigenmodes: 3 Three-Span(m) A m p li tu d e 1 Crack 2 Cracks 3 Cracks 4 Cracks 5 Cracks 6 Cracks Fig. 8. The comparison of the first three mode shapes of multi-span continuous beam with the number of cracks increases from 1 to 6 with equidistance 0.15 m on the second span and with invariable crack depth of 30% We have remarks: a) At crack positions, the differences of the mode shape have bumpy line with the peak on the same crack position, but it is not maximum value (Figs. 5g, 6c, 6d, 6g, 7c, 7e, 7f, 7g, 7i). b) The differences of the mode shapes increase with the increase of the crack depth. c) At the cracked span, the mode shapes change suddenly, but on the other uncracked spans the mode shapes change smoothly, it is also relative to change large scale on adjacent spans. d) There are some positions in which the cracks do not influence the mode shapes, for example: the crack at position of 0.2 m causes the change in the first two mode shapes but does not cause the change in the third mode shape (Fig. 7a). Thus, these positions of the cracks are called the invariable point of mode shape to discriminate between them with the fixed position in which the mode shapes have a zero amplitude value. e) When gradually increasing the number of cracks with equidistance, then the change amplitude of the mode shape also increases but the magnitude are not necessarily the largest (Fig. 8). 6. CONCLUSION The article presents some results on the determination of vibration shape func- tions of a multiple cracked elastic beam element, where the multiple cracked beam ele- ment is modeled as an assembly of intact sub-segments connected by massless rotational springs. Also, algorithms and computer programs are established for analyzing changes in mode shapes of multiple cracked multi-span continuous beams. Numerical analysis of mode shapes of the cracked simple support beams using the proposed method shows a good agreement in comparison with the well-known analytical methods. 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[18] Haisty B.S. and Springer W.T., A general beam element for use in damage assessment of complex structures, Journal of Vibration, Acoustics, Stress and Reliability in Design, 110, (1988), pp. 389–394. [19] Nguyen Tien Khiem, Hai Tran Thanh, A procedure for multiple crack identification in beam-like structures from natural vibration mode, Journal of Sound and Control, DOI: 10.1177/1077546312470478, first published on April 10, (2013), jvc.sagepub.com. Received October 29, 2012 VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY VIETNAM JOURNAL OF MECHANICS VOLUME 35, N. 4, 2013 CONTENTS Pages 1. Nguyen Manh Cuong, Tran Ich Thinh, Ta Thi Hien, Dinh Gia Ninh, Free vibration of thick composite plates on non-homogeneous elastic foundations by dynamic stiffness method. 257 2. Vu Lam Dong, Pham Duc Chinh, Construction of bounds on the effective shear modulus of isotropic multicomponent materials. 275 3. Dao Van Dung, Nguyen Thi Nga, Nonlinear buckling and post-buckling of eccentrically stiffened functionally graded cylindrical shells surrounded by an elastic medium based on the first order shear deformation theory. 285 4. N. T. Khiem, L. K. Toan, N. T. L. Khue, Change in mode shape nodes of multiple cracked bar: II. The numerical analysis. 299 5. Tran Van Lien, Trinh Anh Hao, Determination of mode shapes of a multiple cracked beam element and its application for free vibration analysis of a multi- span continuous beam. 313 6. Phan Anh Tuan, Pham Thi Thanh Huong, Vu Duy Quang, A method of skin frictional resistant reduction by creating small bubbles at bottom of ships. 325 7. Nguyen Thoi Trung, Bui Xuan Thang, Ho Huu Vinh, Lam Phat Thuan, Ngo Thanh Phong, An effective algorithm for reliability-based optimization of stiffened Mindlin plate. 335