Dynamic responses of an inclined fgsw beam traveled by a moving mass based on a moving mass element theory

Nội dung text: Dynamic responses of an inclined fgsw beam traveled by a moving mass based on a moving mass element theory

1. Vietnam Journal of Mechanics, VAST, Vol.41, No. 4 (2019), pp. 319 – 336 DOI: DYNAMIC RESPONSES OF AN INCLINED FGSW BEAM TRAVELED BY A MOVING MASS BASED ON A MOVING MASS ELEMENT THEORY Tran Thi Thom1,2,∗, Nguyen Dinh Kien1,2, Le Thi Ngoc Anh3 1Institute of Mechanics, VAST, 18 Hoang Quoc Viet, Hanoi, Vietnam 2Graduate University of Science and Technology, VAST, 18 Hoang Quoc Viet, Hanoi, Vietnam 3Institute of Applied Information and Mechanics, Ho Chi Minh City, Vietnam ∗E-mail: thomtt0101@gmail.com Received: 02 August 2019 / Published online: 14 November 2019 Abstract. Dynamic analysis of an inclined functionally graded sandwich (FGSW) beam traveled by a moving mass is studied. The beam is composed of a fully ceramic core and two skin layers of functionally graded material (FGM). The material properties of the FGM layers are assumed to vary in the thickness direction by a power-law function, and they are estimated by Mori–Tanaka scheme. Based on the first-order shear deformation theory, a moving mass element, taking into account the effect of inertial, Coriolis and centrifugal forces, is derived and used in combination with Newmark method to compute dynamic responses of the beam. The element using hierarchical functions to interpolate the dis- placements and rotation is efficient, and it is capable to give accurate dynamic responses by small number of the elements. The effects of the moving mass parameters, material dis- tribution, layer thickness ratio and inclined angle on the dynamic behavior of the FGSW beam are examined and highlighted. Keywords: inclined FGSW beam; hierarchical functions; moving mass element; Mori– Tanaka scheme; dynamic responses. 1. INTRODUCTION Sandwich beams are widely used in the aerospace industry as well as in other indus- tries due to their high stiffness to weight ratio. Functionally graded materials (FGMs), initiated by Japanese scientists in 1984, are employed to fabricate functionally graded sandwich (FGSW) beams to improve their performance in severe conditions. Investiga- tions on mechanical behavior of the FGSW beams have been recently reported by several researchers. Bhangale and Ganesan [1] studied thermo-elastic buckling and vibration behavior of a FGSW beam having constrained viscoelastic core using a finite element for- mulation. Amirani et al. [2] analyzed free vibration of sandwich beam with FGM core c 2019 Vietnam Academy of Science and Technology
2. 320 Tran Thi Thom, Nguyen Dinh Kien, Le Thi Ngoc Anh by a mesh-less method. Bui et al. [3] proposed a novel truly mesh-free radial point inter- polation method to investigate transient responses and natural frequencies of sandwich beams with FGM core. Using a mesh-free boundary-domain integral equation method, Yang et al. [4] studied free vibration of the FGSW beams. Based on a refined shear defor- mation theory and a quasi-3D theory, Vo et al. [5,6] derived finite element formulations for free vibration and buckling analyses of FGSW beams. Nguyen et al. [7] obtained an analytical solution for buckling and vibration analysis of FGSW beams using a quasi-3D shear deformation theory. Again, a quasi-3D theory is used by Vo et al. [8] to study static behavior of FGSW beams. Finite element model and Navier solutions are developed by the authors to determine the displacements and stresses of FGSW beams with various boundary conditions. Su et al. [9] considered free vibration of FGSW beams resting on a Pasternak elastic foundation. The effective material properties of FGM are estimated by both Voigt model and Mori–Tanaka scheme, and the governing equations are solved using the modified Fourier series method. Based on Timoshenko beam theory, S¸ims¸ek and Al-shujairi [10] examined static, free and forced vibration of FGSW beams under the action of two moving harmonic loads. The equations of the motion are obtained by the authors using Lagrange’s equations, and they are solved by the implicit Newmark-β method. The problem of beams traveled by a moving mass has drawn much attention from scientists [11–15]. The inertial effects of the moving mass including Coriolis, inertia and centrifugal forces are taken into consideration by the authors. Most of the works, how- ever considered the horizontal beams. When the beams are inclined, then the approaches presented in the foregoing researches cannot be directly applied to solve the problem. For this reason, Wu [16] used the theory of moving mass element to determine the dynamic response of an inclined homogeneous Euler-Bernoulli beam due to a moving mass. The property matrices of the moving mass element are derived by taking into account of the effects of inertial force, Coriolis force and centrifugal force induced by a moving mass. Mamandi and Kargarnovin [17] studied dynamic behavior of inclined pinned-pinned Timoshenko beams made of linear, homogenous and isotropic material subjected to a traveling mass/force. The inertial force due to the motion of the traveling mass on the deformed shape of the beam is considered. Bahmyari et al. [18] presented the finite el- ement dynamic analysis of inclined composite laminated beams under a moving dis- tributed mass with constant speed. The algorithm developed accounts for inertial, Cori- olis, and centrifugal forces due to the moving distributed mass and friction force between the beam and the moving distributed mass. According to authors’ best knowledge, there have not been any studies on dynamic analysis of inclined FGSW beams subjected to moving mass reported in the literature so far. In this paper, dynamic analysis of an inclined FGSW beam subjected to traveling mass is studied using a moving mass element. The beam is composed of a fully ceramic core and two skin layers of FGM. The material properties of the FGM skin layers are assumed to vary continuously through the thickness of the beam according to a power- law. Mori–Tanaka scheme is employed to evaluate the effective properties. The effects of interaction forces due to the action of the traveling mass including the inertia force, Coriolis force and centrifugal force are considered. The overall matrices are received by
3. The problem of beams traveled by a moving mass has drawn much attention from scientists [11-15]. The inertial effects of the moving mass including Coriolis, inertia and centrifugal forces are taken into consideration by the authors. Most of the works, however considered the horizontal beams. When the beams are inclined, then the approaches presented in the foregoing researches cannot be directly applied to solve the problem. For this reason, Wu [16] used the theory of moving mass element to determine the dynamic response of an inclined homogeneous Euler-Bernoulli beam due to a moving mass. The property matrices of the moving mass element are derived by taking into account of the effects of inertial force, Coriolis force and centrifugal force induced by a moving mass. Mamandi and Kargarnovin [17] studied dynamic behavior of inclined pinned-pinned Timoshenko beams made of linear, homogenous and isotropic material subjected to a traveling mass/force. The inertial force due to the motion of the traveling mass on the deformed shape of the beam is considered. Bahmyari et al. [18] presented the finite element dynamic analysis of inclined composite laminated beams under a moving distributed mass with constant speed. The algorithm developed accounts for inertial, Coriolis, and centrifugal forces due to the moving distributed mass and friction force between the beam and the moving distributed mass. AccordingDynamic toresponses authors' of an best inclined knowledge, FGSW beam there traveled have by a not moving been mass any based studies on a moving on dynamic mass element ana theorylysis of 321 inclined FGSW beams subjected to moving mass reported in the literature so far. In this paper, dynamic analysis of an inclined FGSW beam subjected to traveling mass is studied using a moving mass element. Theadding beam is the composed contribution of a fully of ceramic the mass, core and damping two skinand layers stiffness of FGM. matricesThe material of propertiesthe moving of the mass FGelement,M skin layers respectively. are assumed to The vary present continuously work through focuses the thickness on the useof the of beam hierarchical according to functions a power- as law.interpolation Mori-Tanaka scheme functions is employed to derive to evaluate a finite the element effective properties formulation. The effects for the of analysis. interaction forces Numeri- duecal to investigationthe action of the istraveling carried mass out including to show the the inertia effects force, of Coriolis the material force and gradientcentrifugal index, force are layer considered.thickness The ratio,overall inclined matrices angle are received as well by add asing the the weight contribution of the of movingthe mass, damping mass and and itsstiffness velocity matriceson dynamic of the movi responsesng mass e oflement, FGSW respectively. beam. The present work focuses on the use of hierarchical functions as interpolation functions to derive a finite element formulation for the analysis. Numerical investigation is carried out to show the effects of the material gradient index, layer thickness ratio, inclined angle as well as the weight of the 2.moving THEORETICAL mass and its velocity FORMULATION on dynamic responses of FGSW beam. An inclined FGSW beam2. THEORETICAL element with lengthFORMULATl, widthIONb and height h, traveled by a moving mass mc as shown in Fig.1 is considered. The beam element is inclined an angle β toAn the inclined horizontal FGSW plane.beam element The localwith length coordinate l, width (bxand, z) heightis chosenh, traveled such by that a moving the x mass-axis m isc on as shown in Fig. 1 is considered. The beam element is inclined an angle  to the horizontal plane. The the mid-plane, and the z-axis is perpendicular to the mid-plane and directs upward. local coordinate (x,z) is chosen such that the x-axis is on the mid-plane, and the z-axis is perpendicular to the mid-plane and directs upward. z h 3 v mc h2 h1 y h0 b xi l Fig. 1. An inclined FGSW beam element traveled by a moving mass mc The beam element is composed of a fully ceramic core and two skin layers of trans- verse FGM. The vertical positions of the2 bottom, top and of the two interfaces between h h ( ) the layers are denoted by h = − , h , h , h = . The volume fraction function V k of 0 2 1 2 3 2 c ceramic at the kth layer is given by [5]   n (1) z − h0  Vc (z) = , z ∈ [h0, h1]  h1 − h0  (2) Vc (z) = 1 , z ∈ [h1, h2] (1)   n  (3) z − h3  Vc (z) = , z ∈ [h2, h3] h2 − h3 where n is a non-negative material grading index. This paper employs Mori–Tanaka scheme to evaluate the effective material proper- (k) ties. According to the Mori–Tanaka scheme, the effective local bulk modulus K f and the
4. 322 Tran Thi Thom, Nguyen Dinh Kien, Le Thi Ngoc Anh (k) th shear modulus Gf of the k layer of the sandwich beams can be given by [9] (k) (k) (k) K f − Km V = c , (k) (k)  (k)  (k) (k)  (k) (k)  (2) Kc − Km 1 + 1 − Vc Kc − Km / Km + 4Gm /3 (k) (k) (k) Gf − Gm V = c , (k) (k)  (k)  (k) (k) n (k) (k)  (k) (k) h  (k) (k)io (3) Gc − Gm 1 + 1 − Vc Gc − Gm / Gm + Gm 9Km + 8Gm / 6 Km + 2Gm where (k) (k) (k) (k) (k) E (k) E (k) E (k) E K = c , G = c , K = m , G = m , c  (k) c  (k) m  (k) m  (k) (4) 3 1 − 2µc 2 1 + µc 3 1 − 2µm 2 1 + µm are the local bulk modulus and the shear modulus of the ceramic and metal at the kth layer, respectively. (k) Noting that the effective mass density ρ f is defined by Voigt model as [9] (k) (k) (k) (k) (k) ρ f = (ρc − ρm )Vc + ρm . (5) (k) (k) The effective Young’s modulus Ef and Poisson’s ratio υ f are computed via effec- tive bulk modulus and shear modulus as (k) (k) (k) (k) ( ) 9K f Gf ( ) 3K f − 2Gf E k = , υ k = . (6) f (k) (k) f (k) (k) 3K f + Gf 6K f + 2Gf Based on the first-order shear deformation beam theory, the displacements in x- and z-directions, u1(x, z, t) and u3(x, z, t), respectively, at any point of the inclined beam ele- ment are given by u1(x, z, t) = u(x, t) − zθ(x, t), u3(x, z, t) = w(x, t), (7) where z is the distance from the mid-plane to the considering point; u(x, t) and w(x, t) are, respectively, the displacements of the point on the mid-plane in x- and z-directions; θ(x, t) is the cross-sectional rotation. The axial strain (εxx) and the shear strain (γxz) resulted from Eq. (7) are of the forms εxx = u,x − zθ,x, γxz = w,x − θ, (8) where a subscript comma is used to indicate the derivative of the variable with respect to the spatial coordinate x, that is (.),x = ∂ (.) /∂x. Based on the Hooke’s law, the constitutive relation for the FGSW beam element is as follows (k) (k) (k) (k) σxx = Ef (z)εxx, τxz = ψGf (z)γxz, (9) (k) (k) th where σxx and τxz are the axial stress and shear stress at the k layer, respectively; ψ is the shear correction factor, equals to 5/6 for the beams with rectangular cross-section considered herein.
5. Dynamic responses of an inclined FGSW beam traveled by a moving mass based on a moving mass element theory 323 The strain energy of the beam element (Ue) resulted from Eq. (8) and Eq. (9) is l l Z Z Z 1 (k) (k) 1 h 2 2 2i Ue = (σ εxx + τ γxz)dAdx = A u − 2A u,xθ,x + A θ + ψA (w,x − θ) dx. (10) 2 xx xz 2 11 ,x 12 22 ,x 33 0 A 0 The kinetic energy resulted from Eq. (7) is of the form l l 1 Z Z ( ) 1 Z T = ρ k (z) u˙2 + u˙2
   dAdx = I u˙2 + I w˙ 2 − 2I u˙θ˙ + I θ˙2dx, (11) e 2 f 1 3 2 11 11 12 22 0 A 0 where the overhead dot (.) indicates derivative with respect to time t. In Eqs. (10) and (11), A is the cross-sectional area; A11, A12, A22 and A33 are, respectively, the extensional, extensional-bending coupling, bending rigidities and the shear rigidity, which are de- fined as h h 3 Z k 3 Z k = (k) 2
   = (k) (A11, A12, A22) b ∑ Ef (z) 1, z, z dz, A33 b ∑ Gf (z)dz, (12) k=1 k=1 hk−1 hk−1 and I11, I12, I22 are the mass moments, defined as h 3 Z k = (k) 2
   (I11, I12, I22) b ∑ ρ f (z) 1, z, z dz. (13) k=1 hk−1 3. FINITE ELEMENT FORMULATION The finite element formulation for dynamic analysis of the beam is derived in this section by using hierarchical functions to interpolate the kinematic variables. These shape functions are of the forms [19] 1 1 N = (1 − ξ) , N = (1 + ξ) , N = 1 − ξ2
   , N = ξ 1 − ξ2
   , (14) 1 2 2 2 3 4 x with ξ = 2 − 1 being the natural coordinate. l The beam element based on the hierarchical functions needs middle values of the variables, and this increases the number of degrees of freedom of the element. In order to improve the efficiency of the element, the shear strain is constrained to be constant [20] for reducing the number of degrees of freedom. Using this procedure, the vector of nodal displacements for a generic element (d) has seven components as T d = {u1 u2 w1 θ1 θ3 w2 θ2} . (15)
6. 324 Tran Thi Thom, Nguyen Dinh Kien, Le Thi Ngoc Anh In the above equation and hereafter, the superscript ‘T’ is used to denote the trans- pose of a vector or a matrix. By constraining the shear strain to constant, the displace- ments and rotation are interpolated as [21] 1 1 u = (1 − ξ) u + (1 + ξ) u , 2 1 2 2 1 1 θ = (1 − ξ) θ + (1 + ξ) θ + 1 − ξ2
   θ , (16) 2 1 2 2 3 1 1 l l w = (1 − ξ) w + (1 + ξ) w + 1 − ξ2
   (θ − θ ) + ξ 1 − ξ2
   θ . 2 1 2 2 8 1 2 6 3 In matrix forms, we can write Eq. (16) in the forms u = Nud, w = Nwd, θ = Nθd. (17) where T Nu = {N1 N2 0 0 0 0 0} , N = {0 0 0 N N 0 N }T , θ 1 3 2 (18)  l l l T N = 0 0 N N N N − N , w 1 8 3 6 4 2 8 3 with N1, N2, N3, N4 are defined by Eq. (14). From the displacement field in Eq. (17), one can rewrite the strain energy (10) in the form 1 U = dTk d, with k = k + k + k + k , (19) e 2 uu uθ θθ s where k is the element stiffness matrix; kuu, kuθ, kθθ and ks are, respectively, the stiffness matrices stemming from the axial stretching, axial stretching-bending coupling, bending l l2 2 and shear deformation. Using (.) = (.) ; (.) = (.) ; dξ = dx, these matrices ,ξ 2 ,x ,ξξ 4 ,xx l have the following forms l l Z Z T T kuu = Nu,x A11 Nu,xdx, kuθ = − Nu,x A12 Nθ,xdx, 0 0 (20) l l Z Z T  T T kθθ = Nθ,x A22 Nθ,xdx, ks = ψ Nw,x − Nθ A33 (Nw,x − Nθ) dx. 0 0 Similarly, the kinetic energy (11) can also be written in the form 1 T = d˙ Tm d˙ with m = m + m + m + m , (21) e 2 uu uθ θθ ww
7. Dynamic responses of an inclined FGSW beam traveled by a moving mass based on a moving mass element theory 325 where m denotes the element mass matrix, and l l Z Z T T muu = Nu I11 Nudx, mww = Nw I11 Nwdx, 0 0 (22) l l Z Z T T muθ = − Nu I12 Nθdx, mθθ = Nθ I22 Nθdx, 0 0 are, respectively, the element mass matrices resulted from the axial and transverse trans- lations, axial translation-rotation coupling, cross-sectional rotation. When beam is inclined an angle β to the horizontal plane as in Fig.1, the displace- ment components of an arbitrary point on the inclined beam in the local x and z direc- tions, u and w are related to those in the global x¯ and z¯ directions, u¯ and w¯ u¯ = u cos β − w sin β; w¯ = u sin β + w cos β. (23) Because the local rotations and the global ones are identical, the vector of local degrees of  T freedom d is related to the global one d¯ by d = Td¯ where d¯ = u¯1 u¯2 w¯ 1 θ¯1 θ¯3 w¯ 2 θ¯2 and   cos β 0 sin β 0 0 0 0  0 cos β 0 0 0 sin β 0     − sin β 0 cos β 0 0 0 0    T =  0 0 0 1 0 0 0  , (24)    0 0 0 0 1 0 0     0 − sin β 0 0 0 cos β 0  0 0 0 0 0 0 1 is the transformation matrix between the local coordinate and the global one. The global element stiffness and mass matrices are finally computed as T k¯ = T kT and ¯m = TTmT, (25) with k and m are given in Eqs. (19) and (21). The structural mass matrix M¯ b and stiffness matrix K¯ b of the inclined FGSW beam are obtained by assembling the corresponding element matrices over the total elements. Assumption that the moving mass mc is located at point i of the beam element. The interaction forces in the x- and z-directions due to the action of the traveling mass are respectively given by [16] 2
   Fx = mcu¨c, Fz = mc w¨ c + 2vw˙ c,x + v wc,xx , (26) where v is the velocity of the moving mass; uc, wc represent the displacement compo- nents of the contact point i in the local x and z directions of the beam element, respec- 2 tively; mcu¨c, mcw¨ c represent the inertia forces; and 2mcvw˙ c,x, mcv wc,xx represent the Cori- olis force and centrifugal force, respectively. The equivalent nodal forces of the beam element induced by the two forces given by Eq. (26) are [16] fk = Nuk Fx (k = 1, 2), fk = Nwk Fz (k = 3, 4, 5, 6, 7), (27)
8. 326 Tran Thi Thom, Nguyen Dinh Kien, Le Thi Ngoc Anh where Nuk, Nwk are the hierarchical functions defined in Eq. (18). The displacement com- ponents of the contact point i can be also interpolated from the nodal displacements as uc = Nu1u1 + Nu2u2, wc = Nw3w1 + Nw4θ1 + Nw5θ3 + Nw6w2 + Nw7θ2. (28) From Eq. (28), one can receive the time derivatives of displacement components, then substituting into Eqs. (26), (27), and writing the resulting expressions in matrix form yield fc = mcd¨ + ccd˙ + kcd, (29) with d is given in Eq. (15). In Eq. (29),  2  N1 N1 N2 0 0 0 0 0  2   N1 N2 N2 0 0 0 0 0     2 l l l   0 0 N1 N1 N3 N1 N4 N1 N2 − N1 N3   8 6 8   2 2 2   l l 2 l l l 2   0 0 N N3 N N3 N N2 N3 − N   8 1 64 3 48 4 8 64 3  mc = mc   , (30a)  l l2 l2 l l2   0 0 N N N N N2 N N − N N   1 4 3 4 4 2 4 3 4   6 48 36 6 48   l l l   0 0 N N N N N N N2 − N N   1 2 2 3 2 4 2 2 3   8 6 8   l l2 l2 l l2  0 0 − N N − N2 − N N − N N N2 8 1 3 64 3 48 3 4 8 2 3 64 3   0 0 0 0 0 0 0    0 0 0 0 0 0 0     l l l   0 0 N1 N1,x N1 N3,x N1 N4,x N1 N2,x − N1 N3,x   8 6 8   2 2 2   l l l l l   0 0 N x N3 N3 N3,x N3 N x N2,x N3 − N3 N3,x   8 1, 64 48 4, 8 64  cc = 2mcv   , (30b)  l l2 l2 l l2   0 0 N N N N N N N N − N N   1,x 4 3,x 4 4 4,x 2,x 4 3,x 4   6 48 36 6 48   l l l   0 0 N N N N N N N N − N N   1,x 2 2 3,x 2 4,x 2 2,x 2 3,x   8 6 8   l l2 l2 l l2  0 0 − N N − N N − N N − N N N N 8 1,x 3 64 3 3,x 48 3 4,x 8 2,x 3 64 3 3,x   0 0 0 0 0 0 0    0 0 0 0 0 0 0     l l l   0 0 N1 N1,xx N1 N3,xx N1 N4,xx N1 N2,xx − N1 N3,xx   8 6 8   2 2 2   l l l l l   0 0 N xx N3 N3 N3,xx N3 N xx N2,xx N3 − N3 N3,xx  2  8 1, 64 48 4, 8 64  kc = mcv   , (30c)  l l2 l2 l l2   0 0 N N N N N N N N − N N   1,xx 4 3,xx 4 4 4,xx 2,xx 4 3,xx 4   6 48 36 6 48   l l l   0 0 N N N N N N N N − N N   1,xx 2 2 3,xx 2 4,xx 2 2,xx 2 3,xx   8 6 8   l l2 l2 l l2  0 0 − N N − N N − N N − N N N N 8 1,xx 3 64 3 3,xx 48 3 4,xx 8 2,xx 3 64 3 3,xx are the mass, damping and stiffness matrices of the moving mass element written in the local coordinate system. It can be seen from Eqs. (30b), (30c) that the damping and
9. Dynamic responses of an inclined FGSW beam traveled by a moving mass based on a moving mass element theory 327 stiffness matrices of the moving mass element are generated from transverse displace- ment only. Using Eq. (23) one can also get uc = T ¯uc, wc = T ¯wc. (31) Similarly, the nodal forces and the time derivatives of displacement components in local coordinate system can be also transformed into those in global coordinate system. Since, one receives ¨ ˙ f¯c = ¯mcd¯ + ¯ccd¯ + k¯ cd¯ , (32) where T T T ¯mc = T mcT; ¯cc = T ccT; k¯ c = T kcT, (33) are the mass, damping and stiffness matrices of the moving mass element written in global coordinate system, respectively. The finite element equation for the dynamic analysis of the beam can be written in the form M¯ D¯¨ + C¯ D¯˙ + K¯ D¯ = F¯ ex, (34) where M¯ , K¯ are the instantaneous overall mass and stiffness matrices, respectively. They composed of the constant overall mass and stiffness matrices of the entire inclined beam itself and the time-dependent element property matrices of the moving mass element [16]. The instantaneous overall damping matrix C¯ is received by adding the damping matrix of the moving mass element ¯cc to the damping matrix of the inclined beam itself C¯ b. The overall damping matrix C¯ b of the inclined beam is proportional to the instanta- neous overall mass and stiffness matrices by using the theory of Rayleigh damping [16]. The equivalent force vector Fex has the following form  T   ex  l l l  F = 0 0 . . . 0 0 . . . Px N | Px N2| Pz N | Pz N3 Pz N Pz N2| − Pz N3 . . . 0 0 . . . 0 0 ,  1 xi xi 1 xi 8 6 4 xi 8   xi xi xi  | {z } element under moving mass (35) where Px, Pz are the corresponding force components of the equivalent force vector P induced by the mc at any time t. They are given by Px = −mcg sin β, Pz = −mcg cos β, (36) in which g = 9.81 m/s2 is the acceleration of gravity. Noting that the effect of frictional force at the contact point i between the moving mass and the inclined beam is small [16], and it is neglected in this paper. The local equivalent force vector in Eq. (35) must also transform into global coordinate to form the vector F¯ ex. The system of Eq. (34) can be solved by the direct integration Newmark method. The average acceleration method which ensures the unconditional convergence is adopted in the present work.
10. 328 Tran Thi Thom, Nguyen Dinh Kien, Le Thi Ngoc Anh 4. NUMERICAL RESULTS AND DISCUSSION The dynamic responses of a simply inclined supported FGSW beam subjected to a moving mass are numerically examined in this section. In the below, it is assumed that the core of the beam is pure Si3N4 and FGM parts are composed of SUS304 and Si3N4. The properties of these constituent materials are given in room temperature (T = 300 K) as [22]: 3 - SUS304: Em = 207.8 GPa; ρm = 8166 kg/m ; υm = 0.3; 3 - Si3N4: Ec = 322.3GPa; ρc = 2370 kg/m ; υc = 0.3. Otherwise stated, an aspect ratio L/h = 20 is assumed,3 where L is the total length Si3N4: Ec=322.3 GPa; c =2370 kg/m ; c =0.3 of the beam. To facilitate the discussion, the dynamic magnification factor (Dd) is intro- Otherwise stated, an aspect ratio L/h=20 is assumed, where L is the total length of the beam. To facilitate w¯ (L/2, t) 3 duced as Dd = max ; where w¯ st = mcgL /48Em I is the w Lstatic/ 2, t deflection of w¯ st the discussion, the dynamic magnification factor (Dd) is introduced as Dd max ; where wst a full metal beam under mid-span concentrated load of size mcg; I is secondwst moment of 3 area= ofmcg the L /48 cross-section.Em I is the static deflection The weight of a full of themetal moving beam under mass mid- isspan defined concentrated through load of mass size m ratiocg; mr =I is msecondc/ρm momentAL, and of area the of layer the cross thickness-section. ratioThe weight is defined of the moving using mass three is defined number through as (1-0-1), mass (2-1-2),ratio (1-1-1),mr=mc / (2-2-1),m AL, and (1-2-1),the layer thickness (1-8-1), forratio example is defined using (1-1-1) three means number the as ( thickness1-0-1), (2-1-2), ratio (1-1 of-1), the bottom,(2-2-1) core,, (1-2- and1), (1 top-8-1) layers, for example is 1:1:1. (1-1-1) means the thickness ratio of the bottom, core, and top layers is 1:1:1. x 10-3 2 Mamandi and Kargarnovin, =0.25 Present, =0.25 1 Mamandi and Kargarnovin, =0.5 Present, =0.5 0 -1 w* -2 -3 -4 0 0.2 0.4 0.6 0.8 1 vt/L Fig. 2. Time histories for normalized mid-point deflection of homogenous beam Fig. 2. Time histories for normalized mid-point deflection of homogenous beam To confirm the convergence and accuracy of the derived formulation, we have to consider some special cases of this study to be compared with results in the literature. To this end, the time histories for normalizedTo confirm mid the-point convergence deflection of and homogenous accuracy beam of are the compared derived with formulation, that of Mamandi we have and to consider some special cases of this study to be compared* with results in the literature. To Kargarnovin [17] as shown in Fig. 2. In the figure, w w( L / 2, t ) / wst is the dimensionless mid-span thisdeflection; end, the time and historiesthe velocity for ratio normalized is defined mid-point according deflection to in Ref. of [17] homogenous as vv/ beam, with are compared with that of Mamandi and Kargarnovin [17] as shown in Fig.2. In thecr figure, ∗ vcr (/)/ l EI A is the critical velocity of a moving force on a simply supported Eurler-Bernoulli w = w¯ (L/2, t)/w¯ st is the dimensionless mid-span deflection; and the velocity ratio is beam. It can be seen from the figure that the time histories received in this studyp are in good agreement definedwith that according of Ref. [17], to regardless in Ref. [of17 the] as velocityα = ratio.v/v cr , with vcr = (π/l) EI/ρA is the critical Table 1 compares the fundamental frequency parameters of a simply supported FGSW beam of the present paper with that of Ref. [9], where the modified Fourier series method is used. The fundamental L2 frequency parameter is defined as  / E , with  is the fundamental natural frequency. Very h mm good agreement between the results of the present work with that of Ref. [9] is noted from Table 1. It is worth mentioning that convergence of the results obtained in Fig. 2 and Table 1 has been achieved by using twenty elements, and this number of the elements will be used in the below computations. 9
11. Dynamic responses of an inclined FGSW beam traveled by a moving mass based on a moving mass element theory 329 velocity of a moving force on a simply supported Euler-Bernoulli beam. It can be seen from the figure that the time histories received in this study are in good agreement with that of Ref. [17], regardless of the velocity ratio. Tab.1 compares the fundamental frequency parameters of a simply supported FGSW beam of the present paper with that of Ref. [9], where the modified Fourier series method ωL2 p is used. The fundamental frequency parameter is defined as µ = ρ /E , with ω h m m is the fundamental natural frequency. Very good agreement between the results of the present work with that of Ref. [9] is noted from Tab.1. It is worth mentioning that con- vergence of the results obtained in Fig.2 and Tab.1 has been achieved by using twenty elements, and this number of the elements will be used in the below computations. Table 1. Comparison of fundamental frequency parameter of FGSW beam (L/h = 10) n Source (1-1-1) (1-2-1) (1-3-1) (1-4-1) Su et al. [9] 5.3988 5.3988 5.3988 5.3988 0 Present 5.3934 5.3934 5.3934 5.3934 Su et al. [9] 3.7388 4.0246 4.2394 4.4004 0.6 Present 3.7330 4.0187 4.2336 4.3946 Su et al. [9] 3.4480 3.7782 4.0314 4.2220 1 Present 3.4422 3.7723 4.0255 4.2162 Su et al. [9] 2.9387 3.3101 3.6263 3.8709 5 Present 2.9328 3.3040 3.6201 3.8649 Tab.2 lists the dynamic magnification factors of the beam with two values of the aspect ratio, L/h = 5 and 20, for various values of the grading index, the layer thickness ratio and the inclined angle of the beam. The velocity of the moving mass is taken by v = 20 m/s and the mass ratio is mr = 0.5. Consider the case of L/h = 5, it is clear that the factor Dd increases as the grading index n increases. The effect of the grading index on the factor Dd can be explained by the dependence of the rigidities on this index. When the grading index increases, the beam contains more metal, and thus, its rigidities are lower, and this is the reason for the increases in the factor Dd when raising n, no matter what the values of the layer thickness ratio and the inclined angle of the beam would be. On the contrary, the increase in the thickness of the core layer leads to the decrease in the factor Dd. This dependence is explained by the fact that for the present FGSW beam with ceramic hardcore, the rigidities of the beam are higher when the thickness of the core layer increases, and this leads to the factor Dd decreases. In the case of L/h = 20, the effect of the grading index, the layer thickness ratio and the inclined angle of the beam on the factor Dd is similar to the case of L/h = 5. That is, the factor Dd of the FGSW beam increases as the grading index increases while it decreases as the layer thickness ratio and the inclined angle of the beam increase. The value of the factor Dd is also dependent on
12. 330 Tran Thi Thom, Nguyen Dinh Kien, Le Thi Ngoc Anh Table 2. Variations of the dynamic magnification factor with the grading indexes, layer thickness ratio and inclined angle for v = 20 m/s, mr = 0.5 L/h = 5 L/h = 20 β n (1-0-1) (2-1-2) (1-1-1) (2-2-1) (1-2-1) (1-8-1) (1-0-1) (2-1-2) (1-1-1) (2-2-1) (1-2-1) (1-8-1) 0 0.7299 0.7299 0.7299 0.7299 0.7299 0.7299 0.6557 0.6557 0.6557 0.6557 0.6557 0.6557 0.5 0.9137 0.9008 0.8707 0.8571 0.8509 0.7815 0.8326 0.8223 0.8125 0.8011 0.7908 0.7195 0 1 0.9986 0.9795 0.9371 0.9238 0.9096 0.8036 0.9531 0.9092 0.8681 0.8382 0.8327 0.7491 2 1.0802 1.0352 1.0099 0.9869 0.9527 0.8297 1.0306 0.9988 0.9591 0.9187 0.8839 0.7750 5 1.1172 1.1000 1.0599 1.0185 1.0094 0.8560 1.0598 1.0512 1.0236 0.9824 0.9528 0.7968 0 0.7053 0.7053 0.7053 0.7053 0.7053 0.7053 0.6333 0.6333 0.6333 0.6333 0.6333 0.6333 0.5 0.8818 0.8702 0.8416 0.8267 0.8216 0.7551 0.8043 0.7942 0.7848 0.7736 0.7639 0.6950 π 1 0.9642 0.9462 0.9059 0.8910 0.8786 0.7757 0.9207 0.8782 0.8386 0.8093 0.8043 0.7236 12 2 1.0438 1.0002 0.9754 0.9510 0.9210 0.8019 0.9955 0.9648 0.9264 0.8867 0.8538 0.7486 5 1.0784 1.0626 1.0238 0.9817 0.9744 0.8266 1.0236 1.0154 0.9887 0.9481 0.9204 0.7696 0 0.5174 0.5174 0.5174 0.5174 0.5174 0.5174 0.4633 0.4633 0.4633 0.4633 0.4633 0.4633 0.5 0.6442 0.6359 0.6212 0.6002 0.5985 0.5532 0.5892 0.5811 0.5743 0.5656 0.5591 0.5090 π 1 0.7037 0.6909 0.6688 0.6445 0.6427 0.5667 0.6741 0.6432 0.6143 0.5913 0.5884 0.5298 4 2 0.7637 0.7341 0.7112 0.6905 0.6792 0.5891 0.7286 0.7064 0.6784 0.6480 0.6254 0.5480 5 0.7863 0.7772 0.7527 0.7132 0.7104 0.6012 0.7489 0.7431 0.7239 0.6926 0.6740 0.5633 the change of the L/h. In particular, with the velocity value considered in Tab.2, v = 20 m/s, the factor Dd decreases as L/h increases, however the reduction is negligible. In addition, it can be seen from Tab.2 that for any values of the grading index and the layer thickness ratio, the factor Dd decreases as the inclined angle of the beam increases. This phenomenon has been explained as follows. Since the axial stiffness of the beam is much higher than its transverse stiffness, the axial displacement is much smaller than the transverse one. In this case, the global displacement components in Eq. (23) can be approximated as w¯ ≈ w cos β, u¯ ≈ −w sin β. Thus, the value of u¯ increases and the value of w¯ decreases when the inclined angle of the beam increases. This leads to the decrease in the transverse response of the beam. Tab.3 shows the effect of grading indexes, the layer thickness ratio and the inclined angle of the beam on the dynamic magnification factor Dd with a velocity v = 100 m/s. From Tab.3, one can see that the rule of dependence of above dynamic parameters on the factor Dd is similar to the case v = 20 m/s. However, the difference is that a higher value of the L/h, the factor Dd increases more significantly. The dependence of the factor Dd on the aspect ratio L/h with two values of the velocity of the moving mass as seen in Tab.2 and Tab.3 shows the effect of the shear deformation on the dynamic behavior of the beam. The effect of the layer thickness ratio and inclined angle of the beam on the normal- ized mid-span deflection is depicted in Fig.3 for n = 1, v = 30 m/s, mr = 0.5. In the
13. Dynamic responses of an inclined FGSW beam traveled by a moving mass based on a moving mass element theory 331 Table 3. Variations of the dynamic magnification factor with the grading indexes, layer thickness ratio and inclined angle for v = 100 m/s, mr = 0.5 L/h = 5 L/h = 20 β n (1-0-1) (2-1-2) (1-1-1) (2-2-1) (1-2-1) (1-8-1) (1-0-1) (2-1-2) (1-1-1) (2-2-1) (1-2-1) (1-8-1) 0 0.7144 0.7144 0.7144 0.7144 0.7144 0.7144 0.9282 0.9282 0.9282 0.9282 0.9282 0.9282 0.5 0.9379 0.9275 0.9134 0.8993 0.8850 0.7964 1.3811 1.3209 1.2716 1.2347 1.1988 1.0439 0 1 0.9966 0.9660 0.9604 0.9465 0.9374 0.8322 1.5802 1.5095 1.4428 1.3878 1.3380 1.0991 2 1.1254 1.0556 0.9926 0.9707 0.9696 0.8629 1.7323 1.6637 1.5957 1.5265 1.4705 1.1559 5 1.2109 1.1511 1.0900 1.0292 0.9827 0.8912 1.8392 1.7771 1.7111 1.6346 1.5866 1.2117 0 0.6905 0.6905 0.6905 0.6905 0.6905 0.6905 0.8966 0.8966 0.8966 0.8966 0.8966 0.8966 0.5 0.9051 0.8952 0.8811 0.8669 0.8549 0.7699 1.3340 1.2759 1.2281 1.1919 1.1579 1.0081 π 1 0.9632 0.9324 0.9270 0.9122 0.9051 0.8039 1.5259 1.4579 1.3936 1.3394 1.2923 1.0617 12 2 1.0880 1.0200 0.9597 0.9358 0.9361 0.8341 1.6729 1.6066 1.5410 1.4729 1.4203 1.1164 5 1.1712 1.1124 1.0534 0.9909 0.9479 0.8610 1.7760 1.7161 1.6523 1.5767 1.5323 1.1703 0 0.5134 0.5134 0.5134 0.5134 0.5134 0.5134 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.5 0.6562 0.6501 0.6422 0.6317 0.6241 0.5661 0.9758 0.9337 0.8984 0.8703 0.8467 0.7373 π 1 0.7126 0.6743 0.6701 0.6607 0.6565 0.5894 1.1129 1.0653 1.0194 0.9780 0.9457 0.7770 4 2 0.8024 0.7547 0.7107 0.6763 0.6762 0.6107 1.2209 1.1718 1.1249 1.0734 1.0390 0.8167 5 0.8595 0.8206 0.7784 0.7243 0.7023 0.6279 1.2943 1.2523 1.2056 1.1467 1.1192 0.8558 figures, t∗ = t/∆T with ∆T is the total time necessary for the mass crossing the beam. From the figure one can point out the dynamic deflection of the beam decreases as the layer thickness ratio increases, and this is explained by the increase in stiffness of the beam as mentioned above. Also, it can be observed again from Fig.3 that the increase in the inclined angle of the beam leads to the decrease in the dynamic deflection. Thus, by increasing the inclined angle of the beam and the layer thickness ratio, it can be reduced the dynamic deflection. In Fig.4, the time histories for normalized mid-span deflection of the (1-2-1) beam are depicted for various values of the moving mass speed and mass ratio. The other π parameters are given as: β = , n = 1. From Fig.4, it is clear that the velocity of 5 the moving mass has a significant effect on both the dynamic deflection and the way the beam vibrates. For a given mass ratio, the beam performs more vibration cycles when the velocity is smaller. The values of the normalized mid-span deflection are also strongly influenced by the mass ratio. The dynamic deflection of the beam increases and reaches maximum value at a later time when the mass ratio increases. In Fig.5, the relation between the dynamic magnification factor Dd and the moving mass velocity is illustrated with different mass ratio and inclined angle of the beam. As seen from the figure, the relation between Dd and v is similar to that of isotropic beams under a moving load, that is, the factor Dd both increases and decreases when the velocity
14. dependence of the factor Dd on the aspect ratio L/h with two values of the velocity of the moving mass as seen in Table 2 and Table 3 shows the effect of the shear deformation on the dynamic behavior of the beam. The effect of the layer thickness ratio and inclined angle of the beam on the normalized mid-span * deflection is depicted in Fig. 3 for n=1, v=30 m/s, mr =0.5. In the figures, t t/ T with ΔT is the total time necessary for the mass crossing the beam. From the figure one can point out the dynamic deflection of the beam decreases as the layer thickness ratio increases, and this is explained by the increase in stiffness of the beam as mentioned above. Also, it can be observed again from Fig. 3 that the increase in the inclined 332angle of the beam leads Tran to the Thi decrease Thom, in Nguyen the dynamic Dinh deflection. Kien, Le Thus, Thi Ngocby increasing Anh the inclined angle of the beam and the layer thickness ratio, it can be reduced the dynamic deflection. =0 = /12 = /6 = /4 0 0 -0.2 -0.2 -0.4 -0.4 w* w* -0.6 -0.6 -0.8 -0.8 (a) (1-0-1) (b) (1-1-1) -1 -1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 t* t* 0 0 -0.2 -0.2 -0.4 -0.4 w* w* -0.6 -0.6 -0.8 -0.8 (c) (1-4-1) (d) (1-8-1) -1 -1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 t* t* Fig. 3 byFig. the 3. mass Time ratio. histories The for dynamic normalized deflection mid-span of thedeflection beam increasesof beam with and different reaches layermaximum thickness value ratio at anda later inclined time . Time historieswhen the mass for ratio normalized increases. angle mid-span of the beam: n deflection=1, v=30 m/s, mr =0.5 of beam with different layer thickness ratio and inclined angle of the beam: n = 1, v = 30 m/s, mr = 0.5 In0 Fig. 4, the time histories for normalized mid-span0 deflection of the (1-2-1) beam are depicted for various values of the moving mass speed and mass ratio. The other parameters are given as:  , n=1. 5 From Fig. 4, it is clear that the velocity of the moving mass has a significant effect on both the dynamic deflection-0.5 and the way the beam vibrates. For a given mass-0.5 ratio, the beam performs more vibration cycles when the velocity is smaller. The values of the normalized mid-span deflection are also strongly influenced w* w* -1 12 -1 v=30 m/s v=30 m/s v=60 m/s v=60 m/s v=100 m/s (a) m =0.25 v=100 m/s (b) m =0.5 r r -1.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 t* t* 0 0 -0.5 -0.5 w* w* -1.0 -1 v=30 m/s v=30 m/s v=60 m/s v=60 m/s v=100 m/s (c) m =0.75 v=100 m/s (d) m =1 r r -1.5 -1.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 t* t* Fig. 4. Time historiesFig. 4. Time for histories normalized for normalized mid-span mid-span deflection deflection of (1-2-1) beam of with (1-2-1) different beam mass ratio with and moving different mass ratio mass speed:  , n=1 π and moving mass speed:5 β = , n = 1 5 In Fig. 5, the relation between the dynamic magnification factor Dd and the moving mass velocity is illustrated with different mass ratio and inclined angle of the beam. As seen from the figure, the relation between Dd and v is similar to that of isotropic beams under a moving load, that is, the factor Dd both increases and decreases when the velocity of moving mass is low. When moving mass velocity increases, the factor Dd increases and it reaches a maximum value. This dependency rule is true for any values of the mass ratio and inclined angle of the beam. In addition, the increase in the mass ratio leads to the decrease in the factor Dd and the factor Dd reaches the maximum value at the lower velocity of moving mass. Also, it is seen from this figure that the factor Dd decreases as the inclined angle of the beam increases. 13
15. Dynamic responses of an inclined FGSW beam traveled by a moving mass based on a moving mass element theory 333 of moving mass is low. When moving mass velocity increases, the factor Dd increases and it reaches a maximum value. This dependency rule is true for any values of the mass ratio and inclined angle of the beam. In addition, the increase in the mass ratio leads to the decrease in the factor Dd and the factor Dd reaches the maximum value at the lower velocity of moving mass. Also, it is seen from this figure that the factor Dd decreases as the inclined angle of the beam increases. m =0.25 m =0.5 m =0.75 m =1 r r r r 2.5 2.5 (a) =0 (b) = /12 2 2 d 1.5 d 1.5 D D 1 1 0.5 0.5 50 100 150 200 250 300 0 50 100 150 200 250 300 v (m/s) v (m/s) 2.5 2.5 (c) = /6 (d) = /4 2 2 d d 1.5 1.5 D D 1 1 0.5 0.5 0 50 100 150 200 250 300 0 50 100 150 200 250 300 damping and stiffness matricesv (m/s) of the moving mass element generated by the vinteraction (m/s) forces including the inertia force, Coriolis force and centrifugal force. These matrices must be added to the corresponding onesFig. 5.of Variationthe entire of inclined the dynamic beam magnification itself to receive factor ofthe (1 instantaneous-2-1) beam with overall different mass, mass damping ratio and andinclined stiffness angle: Fig. 5. Variationmatrices.n=1 of The the system dynamic of motion magnification equations is solved with factor the aid of of (1-2-1) Newmark beam method. withThe accuracy different of the mass ratio derived formulation was validated by comparing the numericaln results= obtained in the present paper with the availableIn Fig. data 6 and in theFig. literature. 7, the thicknessand The numerical inclined distribution results angle: of showthe normalized a clear effect1 axial of stressthe gradient at mid index,-span sectionthe layer of thickness(1-1-1) beam ratio, and moving (4-2- 1)mass beam speed, are depicted mass ratio for and various the inclined values ofangle inclined of th eangle beam of on the the beam dynamic with responsev=30 m/s ofand the v =100beam. m/s, respectively. The stress in these figures was computed at the time when the moving mass arrives at the mid-span of the inclined beam, and it was normalized as  /  , where  = PLh/8I, 0.5 0.5 xx 0 0 P=100 kN. At a given value of moving mass velocity, the maximum amplitude of both the compressive =0 =0 and tensile stresses decrease as the inclined angle of the beam increases. Thus, by raising the inclined angle of the beam, we could= /12 decrease not only the dynamic magnification factor,= /12but also the maximum amplitude of 0.25the axial stress.= Specially, /6 it can be observed from these0.25 figures that =in /6the case beam is unsymmetrical (Fig. 6b, 7b), the stress= /4 does not vanish at the mid-span. = /4 5. CONCLUSION 0 0 z/h z/h The dynamic analysis of an inclined FGSW beam subjected to moving mass is studied using the first- order shear deformation theory. The effective material properties of FGSW beam are estimated by Mori– T-0.25anaka’s scheme. The hierarchical functions are used to-0.25 interpolate the displacements at the contact point i between the moving mass and beam element, and these shape functions are also used to interpolate the kinematic variables of the beam. The theory of moving mass element has been used to establish the mass, (a) (1-1-1) (b) (4-2-1) -0.5 -0.5 -10 -5 0 5 10 14 -10 -5 0 5 10 * * Fig. 6. Thickness distribution of normalized axial stress at mid-span section of inclined FGSW beam with different Fig. 6. Thickness distribution of normalizedinclined angle: axial v=30 m/s, stress n=1, mr=0.5 at mid-span section of inclined FGSW beam with different inclined angle: v = 30 m/s, n = 1, mr = 0.5 0.5 0.5 =0 =0 = /12 = /12 0.25 = /6 0.25 = /6 = /4 = /4 0 0 z/h z/h -0.25 -0.25 (a) (1-1-1) (b) (4-2-1) -0.5 -0.5 -10 -5 0 5 10 15 -10 -5 0 5 10 15 * * Fig. 7. Thickness distribution of normalized axial stress at mid-span section of inclined FGSW beam with different inclined angle: v=100 m/s, n=1, mr=0.5 15
16. damping and stiffness matrices of the moving mass element generated by the interaction forces including the inertia force, Coriolis force and centrifugal force. These matrices must be added to the corresponding ones of the entire inclined beam itself to receive the instantaneous overall mass, damping and stiffness matrices. The system of motion equations is solved with the aid of Newmark method. The accuracy of the derived formulation was validated by comparing the numerical results obtained in the present paper with the available data in the literature. The numerical results show a clear effect of the gradient index, the layer thickness ratio, moving mass speed, mass ratio and the inclined angle of the beam on the dynamic response of the beam. 0.5 0.5 =0 =0 = /12 = /12 0.25 = /6 0.25 = /6 = /4 = /4 0 0 z/h z/h -0.25 -0.25 (a) (1-1-1) (b) (4-2-1) -0.5 -0.5 -10 -5 0 5 10 -10 -5 0 5 10 * * 334Fig. 6. Thickness distribution Tran Thiof normalized Thom, Nguyen axial stress Dinh at mid Kien,-span Lesection Thi of Ngoc inclined Anh FGSW beam with different inclined angle: v=30 m/s, n=1, mr=0.5 0.5 0.5 =0 =0 = /12 = /12 0.25 = /6 0.25 = /6 = /4 = /4 0 0 z/h z/h -0.25 -0.25 (a) (1-1-1) (b) (4-2-1) -0.5 -0.5 -10 -5 0 5 10 15 -10 -5 0 5 10 15 * * Fig. 7. Thickness distribution of normalized axial stress at mid-span section of inclined FGSW beam with different Fig. 7. Thickness distribution of normalizedinclined angle: axialv=100 m/s, stress n=1, m atr=0.5 mid-span section of inclined FGSW beam with different inclined angle:15 v = 100 m/s, n = 1, mr = 0.5 In Fig.6 and Fig.7, the thickness distributions of the normalized axial stress at mid-span section of (1-1-1) beam and (4-2-1) beam are depicted for various values of inclined angle of the beam with v = 30 m/s and v = 100 m/s, respectively. The stress in these figures was computed at the time when the moving mass arrives at the mid-span ∗ of the inclined beam, and it was normalized as σ = σxx/σ0, where σ0 = PLh/8I, P = 100 kN. At a given value of moving mass velocity, the maximum amplitude of both the compressive and tensile stresses decrease as the inclined angle of the beam increases. Thus, by raising the inclined angle of the beam, we could decrease not only the dynamic magnification factor, but also the maximum amplitude of the axial stress. Specially, it can be observed from these figures that in the case beam is unsymmetrical (Fig.6(b),7(b)), the stress does not vanish at the mid-span. 5. CONCLUSION The dynamic analysis of an inclined FGSW beam subjected to moving mass is stud- ied using the first-order shear deformation theory. The effective material properties of FGSW beam are estimated by Mori–Tanaka’s scheme. The hierarchical functions are used to interpolate the displacements at the contact point i between the moving mass and beam element, and these shape functions are also used to interpolate the kinematic variables of the beam. The theory of moving mass element has been used to establish the mass, damping and stiffness matrices of the moving mass element generated by the interaction forces including the inertia force, Coriolis force and centrifugal force. These matrices must be added to the corresponding ones of the entire inclined beam itself to re- ceive the instantaneous overall mass, damping and stiffness matrices. The system of mo- tion equations is solved with the aid of Newmark method. The accuracy of the derived formulation was validated by comparing the numerical results obtained in the present paper with the available data in the literature. The numerical results show a clear effect of the gradient index, the layer thickness ratio, moving mass speed, mass ratio and the inclined angle of the beam on the dynamic response of the beam.
17. Dynamic responses of an inclined FGSW beam traveled by a moving mass based on a moving mass element theory 335 ACKNOWLEDGMENTS This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 107.02-2018.23. The authors gratefully thank the Reviewers for their valuable comments and suggestions to improve the quality of the paper. REFERENCES [1] R. K. Bhangale and N. Ganesan. Thermoelastic buckling and vibration behavior of a func- tionally graded sandwich beam with constrained viscoelastic core. Journal of Sound and Vi- bration, 295, (1-2), (2006), pp. 294–316. [2] M. C. Amirani, S. M. R. Khalili, and N. Nemati. Free vibration analysis of sandwich beam with FG core using the element free Galerkin method. Composite Structures, 90, (3), (2009), pp. 373–379. [3] T. Q. Bui, A. Khosravifard, C. Zhang, M. R. Hematiyan, and M. V. Golub. Dy- namic analysis of sandwich beams with functionally graded core using a truly mesh- free radial point interpolation method. Engineering Structures, 47, (2013), pp. 90–104. [4] Y. Yang, C. C. Lam, K. P. Kou, and V. P. Iu. Free vibration analysis of the functionally graded sandwich beams by a meshfree boundary-domain integral equation method. Com- posite Structures, 117, (2014), pp. 32–39. [5] T. P. Vo, H.-T. Thai, T.-K. Nguyen, A. Maheri, and J. Lee. Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Engineering Structures, 64, (2014), pp. 12–22. [6] T. P. Vo, H.-T. Thai, T.-K. Nguyen, F. Inam, and J. Lee. A quasi-3D theory for vibration and buckling of functionally graded sandwich beams. Composite Structures, 119, (2015), pp. 1–12. [7] T.-K. Nguyen, T. P. Vo, B.-D. Nguyen, and J. Lee. An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi-3D shear deformation theory. Composite Structures, 156, (2016), pp. 238–252. [8] T. P. Vo, H.-T. Thai, T.-K. Nguyen, F. Inam, and J. Lee. Static behaviour of functionally graded sandwich beams using a quasi-3D theory. Composites Part B: Engineering, 68, (2015), pp. 59– 74. [9] Z. Su, G. Jin, Y. Wang, and X. Ye. A general Fourier formulation for vibration analysis of func- tionally graded sandwich beams with arbitrary boundary condition and resting on elastic foundations. Acta Mechanica, 227, (5), (2016), pp. 1493–1514. 016-1575-8. [10] M. S¸ims¸ek and M. Al-shujairi. Static, free and forced vibration of functionally graded (FG) sandwich beams excited by two successive moving harmonic loads. Composites Part B: Engi- neering, 108, (2017), pp. 18–34. [11] T. O. Awodola, S. A. Jimoh, and B. B. Awe. Vibration under variable magnitude moving distributed masses of non-uniform Bernoulli-Euler beam resting on Pasternak elastic foun- dation. Vietnam Journal of Mechanics, 41, (1), (2019), pp. 63–78. 7136/12781.
18. 336 Tran Thi Thom, Nguyen Dinh Kien, Le Thi Ngoc Anh [12] A. O. Cifuentes. Dynamic response of a beam excited by a moving mass. Finite Elements in Analysis and Design, 5, (3), (1989), pp. 237–246. 2. [13] E. Esmailzadeh and M. Ghorashi. Vibration analysis of a Timoshenko beam subjected to a travelling mass. Journal of Sound and Vibration, 199, (4), (1997), pp. 615–628. [14] M. S¸ims¸ek. Vibration analysis of a functionally graded beam under a moving mass by using different beam theories. Composite Structures, 92, (4), (2010), pp. 904–917. [15] I. Esen, M. A. Koc, and Y. Cay. Finite element formulation and analysis of a func- tionally graded Timoshenko beam subjected to an accelerating mass including iner- tial effects of the mass. Latin American Journal of Solids and Structures, 15, (10), (2018). [16] J.-J. Wu. Dynamic analysis of an inclined beam due to moving loads. Journal of Sound and Vibration, 288, (1-2), (2005), pp. 107–131. [17] A. Mamandi and M. H. Kargarnovin. Dynamic analysis of an inclined Timoshenko beam traveled by successive moving masses/forces with inclusion of geometric nonlinearities. Acta Mechanica, 218, (1-2), (2010), pp. 9–29. [18] E. Bahmyari, S. R. Mohebpour, and P. Malekzadeh. Vibration analysis of inclined laminated composite beams under moving distributed masses. Shock and Vibration, 2014, (2014), pp. 1– 12. [19] J. E. Akin. Finite elements for analysis and design. Academic Press, London, (1994). [20] A. Tessler and S. B. Dong. On a hierarchy of conforming Timoshenko beam ele- ments. Computers & Structures, 14, (3-4), (1981), pp. 335–344. 7949(81)90017-1. [21] D. K. Nguyen and T. T. Tran. Free vibration of tapered BFGM beams using an efficient shear deformable finite element model. Steel and Composite Structures, 29, (3), (2018), pp. 363–377. [22] A. Fallah and M. M. Aghdam. Nonlinear free vibration and post-buckling analysis of func- tionally graded beams on nonlinear elastic foundation. European Journal of Mechanics - A/- Solids, 30, (4), (2011), pp. 571–583.