# Đáp ứng động của dầm đa nhịp có cơ tính biến thiên chiều dưới tác dụng của lực di động

Tóm tắt Đáp ứng động của dầm đa nhịp có cơ tính biến thiên chiều dưới tác dụng của lực di động: ... dictate the variation of the constituent materials in the thickness and longitudinal directions, respectively. z x E (x, z) 0 F x Ec Em Em KẾT CẤU - CÔNG NGHỆ XÂY DỰNG Tạp chí KHCN Xây dựng - số 2/2021 29 Based on the third-order shear deformation theory, the axial and t...34 0 0 2 44 , 44 , 66 66 0 0 ; 2 ; ; 2 ; 2 ; ; l l T T T u w u w u w x l l T T w x y w x u l l T T x y w x N N I N N dx N I N N dx N N I N N dx N I N dx N I N N dx N I N dx m m m m m m (17) Are the elemen...y Şimşek and Kocatürk [13], are computed. The obtained results are listed in Table 3. In the table, the dynamic deflection factor fD is defined as fD = max(w(L/2, t)/w0) with w0 is the static deflection of the steel beam under static load F acting at the mid-span. A very good agreement b...

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re are practical circumstances, in which the unidirectional FGMs may not be so appropriate to resist multi-directional variations of thermal and mechanical loadings. On the other hand, a new type of functionally graded material (FGM) with material properties varying in two or three directions is needed to fulfil the technical requirements such as the temperature and stress distributions in two or three directions for aerospace craft and shuttles where the conventional FGMs (or 1D-FGM) with material properties which vary in one direction are not so efficient. Several models for bi-dimensional KẾT CẤU - CÔNG NGHỆ XÂY DỰNG 28 Tạp chí KHCN Xây dựng - số 2/2021 FGM beams and their mechanical behaviour have been considered recently. In this line of works, Simsek [7] considered the material properties vary in both the length and thickness directions, by an exponential function in vibration study of Timoshenko beam. Polynomials were assumed for the displacement field in computing natural frequencies and the dynamic response of the beam. Wang et al. [8] presented an analytical method for free vibration analysis of a 2D-FGM beam. The material properties are also assumed to vary exponentially in the beam thickness and length. The bending of a two-dimensional FGM sandwich (2D- FGSW) beam was investigated by Karamanli [9] using a quasi-3D shear deformation theory and the symmetric smoothed particle hydro-dynamic method. Nguyen et al. [10] proposed a 2D-FGM beam model formed from four different constituent materials with volume fractions to vary by power-law functions in both the thickness and longitudinal directions. Timoshenko beam theory was adopted by the authors in evaluating the dynamic response of the beam to a moving load. In this paper, a finite element procedure for vibration analysis of multi-span 2D- FGM beams subjected to a moving point load is proposed. The material properties of the beams are assumed to vary continuously in the thickness and longitudinal direction by a power-law distribution. The discrete equations of motion of the beams are solved by using the Newmark method. A parametric study is carried out to highlight the influence of the material heterogeneity, number of spans and loading parameters on the dynamic response of the beam. 2. Problem and formulation Figure 1. A multi-span 2D-FGM beam traversed by a moving load Figure 1 shows a multi-span beam with length of L subjected to a harmonic load, F= const, moving at a constant speed v from left to right. The beam cross-section is assumed to be rectangular with width b and height h. The beam material is assumed to be a 2D-FGM composed from two constituent materials, and the effective properties of materials are graded in the thickness and longitudinal direction (x, z -direction) according to a power-law distribution as Karamanli [9]: 1 ( , ) 1 ; ( , ) ( , ) 1; 2 2 1 ( , ) ( ) 1 2 2 ; 0 2 2 nx nz c c m nx nz c m m x z V x z V x z V x z L h x z P x z P P P L h h h z x L (1) where Vc and Vm denote the volume fractions of ceramic and metal material, respectively; P(x,z) represents the effective material properties, including Young's modulus, shear modulus and mass density; the ()c and ()m subscripts respectively denote the ceramic and metal; nz and nx are the grading indexes, which dictate the variation of the constituent materials in the thickness and longitudinal directions, respectively. z x E (x, z) 0 F x Ec Em Em KẾT CẤU - CÔNG NGHỆ XÂY DỰNG Tạp chí KHCN Xây dựng - số 2/2021 29 Based on the third-order shear deformation theory, the axial and transverse displacements at any point of the beam, u(x,z,t) and w(x,z,t), respectively, are given. 3 0 0 0, 0 0 ( , , ) ( , ) ( ) ( , , ) ( , ). xu x z t u x t z w z w x z t w x t (2) where t is the time variable and = 4/3 h 2 , u0(x, t) and w0(x, t) are, respectively, the axial and transverse displacements of the point on the x-axis, γ0 is the transverse shear rotation. The axial strain and shear strain resulted from Eq. (2) are of the forms. 30, 0, 0, 0, 2 0 03 xx x x xx x xz u z w z z (3) Based on the assumption of Hooke’s law, the constitutive relation for 2D- FG beam is as follows. 3 0, 0, 0, 0, 2 0 0 ( , ). ( , )[ ( ) ] ( , ) ( , ) 3 2(1 ) xx xx x x xx x xz xz E x z E x z u z w z E x z G x z z (4) where E(x,z) and G(x,z) are respectively the elastic modulus and shear modulus, which are functions of both the coordinates x, z, xx and xz are the axial stress and shear stress, respectively. The strain energy U of the beam resulted from Eqs. (3) and (4) is of the form. 2 2 11 0, 12 0, 0, 0, 22 0, 0, 34 0, 0, 2 2 2 0 44 0, 0, 0, 66 0, 44 0 2 ( ) ( ) 21 2 2 ( ) L x x x xx x xx x x x x xx x A u A u w A w A u U dx A w A B (5) where A11, A12, A22, A34, A44, A66 and B44 are the beam rigidities, defined as: 2 3 4 6 11 12 22 34 44 66 2 2 4 44 ( , , , , , )( , ) ( , )(1, , , , , ) ( , ) ( , )(1 6 9 ) A A A A A A A A x z E x z z z z z z dA B x z G x z z z dA (6) where E(x,z), G(x,z) are respectively the elastic modulus and shear modulus of the beam. The kinetic energy (T) of the beam is then given by. 2 2 2 2 2 11 0 0 22 0 0, 66 0 12 0 0 0, 34 0 0 0 44 0 0 , ( ) ( ) 2 ( ) 21 2 2 ( ) L x x o x I u w I w I I u w I u T dx I w (7) In Eqs. (7), I11, I12, I22, I34, I44, I66 are the mass moments, defined as 2 3 4 6 11 12 22 34 44 66( , , , , , )( , ) ( , )(1, , , , , ) A I I I I I I x z x z z z z z z dA (8) where, ρ(x,z) is the mass density of the beam. The potential of the moving load is simply given by ( , ) ( )FV Fw x t x vt (9) in which F, v are respectively the amplitude, speed of the moving load and δ (.) is the Kronecker delta. Using the finite element method, the beam is assumed to be divided into numbers of two-node beam elements of length l. The vector of nodal displacements (d) for the element considering the transverse shear rotation 0 as an independent variable contains eight components as. , ,, , , , , , , T i i i x i j j j x ju w w u w w d (10) where , ,, , , , , , ,i i i x i j j j x ju w w u w w are the values of u0, w0, w0,x and γ0 at the node i and at the node j, respectively. In Eq. (10) and hereafter, a superscript ‘T’ is used to denote the transpose of a vector or a matrix. 0 0 w 0u N . ; w N . ; N . u d d d (11) KẾT CẤU - CÔNG NGHỆ XÂY DỰNG 30 Tạp chí KHCN Xây dựng - số 2/2021 with Nu, Nw and Nγ denote the matrices of shape functions for u0, w0 and γ0, respectively. In the present work, linear shape functions are used for the axial displacement and the shear rotation, using the above interpolation schemes, one can write the strain energy of the beam defined by Eqs. (5) as. 1 2 ne TU d kd (12) where ne is the total number of the elements, and k is the element stiffness matrix with the following form. 11 12 22 34 44 66 s k k k k k k k k (13) with: 11 , 11 , 12 , 12 , , 0 0 22 , , 22 , , 34 , 34 , 0 0 2 44 , 44 , , 66 , 66 , 0 0 44 0 ; 2 ( ) ; ( ) ( ) ; 2 ; 2 ( ) ; ; l l T T u x u x u x x w xx l l T T x w xx x w xx u x x l l T T x x w xx x x l T s N A N dx N A N N dx N N A N N dx N A N dx N A N N dx N A N dx N B N dx k k k k k k k (14) Similarly, the kinetic energy in Eq. (7) can be rewritten as. 1 2 Tne T t t d d m (15) where: 11 12 22 34 44 66 m m m m m m m (16) is the element consistent mass matrix, in which: 11 11 12 12 , 0 0 22 , 22 , 34 34 0 0 2 44 , 44 , 66 66 0 0 ; 2 ; ; 2 ; 2 ; ; l l T T T u w u w u w x l l T T w x y w x u l l T T x y w x N N I N N dx N I N N dx N N I N N dx N I N dx N I N N dx N I N dx m m m m m m (17) Are the element mass matrices stemming from axial, transverse translations, axial translation– sectional rotation coupling, and cross-sectional rotation, respectively. Finally, the potential of the external moving load can be written in the form. ( ) T F wV FN x vt (18) Having the element stiffness and mass matrices derived, the equations of motion for the free vibration analysis in the context of finite element analysis can be written in the form. 2 2 ex t D M KD F (19) Where D, M, and K are the structural nodal displacement vector, mass and stiffness matrices, obtained by assembling the element displacement vector d, mass matrix m, and stiffness matrix k over the total elements, respectively; F ex is the vector of the nodal external forces. 3. Numerical results and discussion Using the derived finite element formulation, the dynamic response of multi-span 2D-FG beams is computed in this section. It is assumed that the beam is formed from spans of the same length. Otherwise stated, the beam is assumed to be KẾT CẤU - CÔNG NGHỆ XÂY DỰNG Tạp chí KHCN Xây dựng - số 2/2021 31 composed of Steel and Alumina. The Young’s modulus, mass density and Poisson’s ratio of Steel are respectively 210 GPa, 7800 kg/m3, 0.3177, and that of Alumina are 390 MPa, 3960 kg/m3 and 0.3, respectively. The beam with L=20 m, h=1m and b=0.5 m used by Şimşek and his co-worker [12,13] is chosen in the computations reported below. 3.1 Formulation validation Validation of the derived formulation is necessary to confirm the accuracy before computing the dynamic response of the beam. Firstly, the natural frequencies of a multi-span homogeneous beam are computed, and the obtained numerical results are listed in Table 1, where the corresponding results obtained by Ichikawa et al [4] are also given. The dimensionless natural frequency parameter, μi, in Table 1 is defined as. 2 2 0 0 i i s A L E I (20) Where ωi are the natural frequencies; Ls is the length of a span; E0, ρ0 are Young’s modulus and mass density of the homogeneous beam, respectively. It should be noted that since the Bernoulli beam theory is used in Ichikawa et al [4], and in order to enable the numerical results comparable, the frequencies in Table 1 have been computed with an aspect ratio Ls/h=100, which is large enough to omit the effect of the shear deformation. As seen from the Table 1, a good agreement between the frequencies computed in the present work with that of Ichikawa et al [4] is noted. Table 1. Comparison of first five natural frequencies of multi-span homogeneous beams (nx=0, nz=0) Number of spans 1 2 3 4 5 1 Present Ichikawa [4] 3.1414 6.2817 2 9.4202 3 12.5567 4 15.6921 5 2 Present Ichikawa [4] 3.1414 3.9261 3.9266 6.2817 2 7.0661 7.0686 9.4202 3 3 Present Ichikawa [4] 3.1414 3.5560 3.5564 4.2968 4.2975 6.2817 2 6.7056 6.7076 4 Present Ichikawa [4] 3.1414 3.3929 3.3932 3.9261 3.9266 4.4625 4.4633 6.2817 2 Secondly, the fundamental frequency of a one- span FGM beam composed of Aluminum (Al) and Alumina (Al2O3), previously studied in Sina et al [11] and Şimşek [12], is computed. The Young’s modulus, mass density and Poisson’s ratio of Alumina are 70 GPa, 2707 kg/m3 and 0.23, respectively [12]. The computed fundamental frequency parameters of the present work are listed in Table 2 for various values of the aspect ratio, L/h. The corresponding values obtained by using an analytical method by Sina et al [11] and a numerical method by Şimşek [12] are also given in the table. The non-dimensional fundamental frequencies, μ, in Table 2 have been defined according to Sina et al [11] as. 2 11 2 0 ( ) L I L h E z dz (21) Where ω is the fundamental frequency of the FG beam. As seen from the Table 2, the fundamental frequencies computed in the present work are in good agreement with that of Sina et al [11] and Şimşek [12], regardless of the aspect ratios. Thirdly, the maximum dynamic deflection factor at the mid-span and the corresponding speed of one-span FGM beam composed of Steel and Alumina with L = 20 m, h = 0.9 m and b = 0.4 m, previously studied by Şimşek and Kocatürk [13], are computed. The obtained results are listed in Table 3. In the table, the dynamic deflection factor fD is defined as fD = max(w(L/2, t)/w0) with w0 is the static deflection of the steel beam under static load F acting at the mid-span. A very good agreement between the numerical results of the present work KẾT CẤU - CÔNG NGHỆ XÂY DỰNG 32 Tạp chí KHCN Xây dựng - số 2/2021 with that of Şimşek and Kocatürk [13] is seen from the table. Table 2. Comparison of non-dimensional fundamental frequency of one-span FGM beam(nx=0) n L/h=10 L/h=30 L/h=100 0.3 Present 2.7017 2.7382 2.7425 0.3 Sina et al [11] 2.695 2.737 2.742 0.3 Şimşek [12] 2.701 2.738 2.742 Table 3. Maximum deflection factor and corresponding speed of one-span FGM beam under a moving load (nx=0) n fD- [13] fD-Present v(m/s)-[13] v(m/s)- present 0.2 1.0344 1.0377 222 222 0.5 1.1444 1.1476 198 197 1 1.2503 1.2537 179 178 2 1.3376 1.3416 164 163 Pure Alumina 0.9328 0.9379 252 251 Pure Steel 1.7324 1.7418 132 131 The numerical results listed in Tables 1-3 have been computed by using 14 elements for each span. More than 14 elements have been employed, but no improvement in the numerical results have been seen, and in this regard, 14 elements are used to discrete each span in the computations reported below. 3.2 Dynamic deflection The normalized deflection at the midpoint of the first and second spans of a four-span 2D-FGM beam are shown in Fig. 2 for various values of the index nx, nz, speed parameter fv. In the figures, w(Ls/2, t) denotes the dynamic deflection at the midpoint of the i th span, and 3 0 / 48si mw FL E I is the static deflection of a simply supported beam with length of Ls under a static load F at the midpoint. The speed parameter fv is defined in accordance with Ichikawa et al [4], /v s m mf vL A E I , and thus for the given data of the beam and for fv = 1.2, the equivalent speed of the moving load is 90 m/s for the beam with a span length of 20 m. As seen from the figures, the material heterogeneity which governed by the index nx, nz clearly affects the dynamic deflection of the beam. The maximum normalized deflection of the beam associated with a higher index nx, nz is higher than that of the beam with lower index nx, nz. KẾT CẤU - CÔNG NGHỆ XÂY DỰNG Tạp chí KHCN Xây dựng - số 2/2021 33 Figure 2. Normalized deflection at midpoint of the first span (four-span beam) Figure 3. Normalized deflection at the midpoint of the first and second spans for 2D-FG beam with different numbers of spans (nx=1, nz=1) The normalized dynamic deflections at the first and second spans of the 2D-FGM beam with different numbers of spans computed with various values of the speed parameters are shown in Fig. 3 for nx=1, nz=1 and fv=1.2. As seen from the figure, the maximum deflection at the midpoint of the first and the second spans of the beam slightly reduces for the beam with more spans. Figure 4. Maximum normalized deflection at the midpoint of the first span (four-span beam) KẾT CẤU - CÔNG NGHỆ XÂY DỰNG 34 Tạp chí KHCN Xây dựng - số 2/2021 In Fig. 4, the relation between the moving speed v and the maximum dynamic deflection at the midpoint of the first span is shown for various values of the index nx, nz. The effect of the material heterogeneity and the moving speed is clearly seen from the figure, and the maximum dynamic deflection is higher for the beam associated with a higher index nx, nz, regardless of the moving speed. 4. Conclusion In this paper, a finite element procedure for vibration analysis of multi-span 2D-FGM beams subjected to a moving point load has been presented. The dynamic responses of the beam have been computed with the aid of the Newmark method. The obtained numerical results have shown that the formulated element is capable to give accurate dynamic characteristics of the beams. A parametric study has been carried out to highlight the influence of material heterogeneity, the number of spans and the loading parameters on the dynamic response of the beam. It has been shown that the beam associated with lower index nx, nz endures a smaller dynamic deflection than that of the beam with higher index nx, nz. REFERENCES 1. W.H. Lin and M.W. Trethewey (1990), Finite element analysis of elastic beams subjected to moving dynamic loads. J. Sound and Vibration, 2, 323-342. https://doi.org/10.1016/0022-460X(90)90860-3. 2. L. Fryba (1972), Vibration of solids and structures under moving loads, Academia, Prague Garvan, The Maple book, Chapman & Hall/CRC, Florida. 3. M. Olsson (1991), On the fundamental moving load problem, J. Sound and Vibration, 2, 299-307. https://doi.org/10.1016/0022-460X(91)90593-9. 4. M. Ichikawa, Y. Miyakawa and A. Matsuda (2000), Vibration analysis of the continuous beam subjected to a moving mass, J. Sound and Vibration, 3, 611-628. https://doi.org/10.1006/jsvi.1999.2625. 5. M. Koizumi (1997), FGM activities in Japan, Composites: part B, 1-2, 1-4. https://doi.org/10.1016/S1359- 8368(96)00016-9. 6. V. Birman, and L.W. Byrd (2007), Modeling and analysis of functionally graded materials and structures. Applied Mechanics Reviews, 5, 195-216. https://doi.org/10.1115/1.2777164. 7. M. Simsek (2015). Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions. Composite Structures, 133, 968-978. 8. Z.-H. Wang, X.-H. Wang, G.-D. Xu, S. Cheng, and T. Zeng (2015). Free vibration of twodirectional functionally graded beams. Composite Structures, 135(2016), 191- 198. https://doi.org/10.1016/j.compstruct.09.013. 9. A. Karamanlı (2017). Bending behaviour of two directional functionally graded sandwich beams by using a quasi-3D shear deformation theory. Composite Structures, 174, 70- 86. https://doi.org/10.1016/j.compstruct.2017.04.046. 10. D. K. Nguyen, Q. H. Nguyen, T. T. Tran, and V. T. Bui (2017). Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load. Acta Mechanica, 1, 141-155. DOI 10.1007/s00707-016- 1705-3. 11. S.A. Sina, H.M. Navazi, and H. Haddadpour (2009), An analytical method for free vibration analysis of functionally graded beams, Materials & Design, 3, 741- 747. https://doi.org/10.1016/j.matdes.2008.05.015. 12. M. Şimşek (2009), Vibration analysis of a functionally graded beam under a moving mass by using different beam theories, Composite Structures, 4 (2010), 904- 917. https://doi.org/10.1016/j.compstruct.09.030. 13. M. Şimşek, and T. Kocatürk (2009), Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load, Composite Structures, 4 (2009), 465- 473. https://doi.org/10.1016/j.compstruct.04.024. Ngày nhận bài: 20/5/2021. Ngày nhận bài sửa: 30/6/2021. Ngày chấp nhận đăng: 30/6/2021.

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