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MINISTRY OF EDUCATION AND VIETNAM ACADEMY OF TRAINING SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY - TRAN THI THOM FINITE ELEMENT MODELS IN VIBRATION ANALYSIS OF TWO-DIMENSIONAL FUNCTIONALLY GRADED BEAMS Major: Mechanics of Solid code: 9440107 SUMMARY OF DOCTORAL THESIS IN MATERIALS SCIENCE Hanoi – 2019 The thesis has been completed at: Graduate University Science and Technology – Vietnam Academy of Science and Technology Supervisors: Assoc Prof Dr Nguyen Dinh Kien Assoc Prof Dr Nguyen Xuan Thanh Reviewer 1: Prof Dr Hoang Xuan Luong Reviewer 2: Prof Dr Pham Chi Vinh Reviewer 3: Assoc Prof Dr Phan Bui Khoi Thesis is defended at Graduate University Science and TechnologyVietnam Academy of Science and Technology at … , on … Hardcopy of the thesis be found at : - Library of Graduate University Science and Technology - Vietnam national library PREFACE The necessity of the thesis Publications on vibration of the beams are most relevant to FGM beams with material properties varying in one spatial direction only, such as the thickness or longitudinal direction There are practical circumstances, in which the unidirectional FGMs may not be so appropriate to resist multi-directional variations of thermal and mechanical loadings Optimizing durability and structural weight by changing the volume fraction of FGM’s component materials in many different spatial directions is a matter of practical significance, being scientifically recognized by the world’s scientists, especially Japanese researchers in recent years Thus, structural analysis with effective material properties varying in many different directions in general and the vibration of FGM beams with effective material properties varying in both the thickness and longitudinal directions of beams (2D-FGM beams) in particular, has scientific significance, derived from the actual needs It should be noted that when the material properties of the 2D-FGM beam vary in longitudinal direction, the coefficients in the differential equation of beam motion are functions of spatial coordinates along the beam axis Therefore analytical methods are getting difficult to analyze vibration of the 2D-FGM beam Finite element method (FEM), with many strengths in structural analysis, is the first choice to replace traditional analytical methods in studying this problem Developing the finite element models, that means setting up the stiffness and mass matrices, used in the analysis of vibrations of the 2D-FGM beam is a matter of scientific significance, contributing to promoting the application of FGM materials into practice From the above analysis, author has selected the topic: Finite element models in vibration analysis of two-dimensional functionally graded beams as the research topic for this thesis Thesis objective This thesis aims to develop finite element models for studying vibration of the 2D-FGM beam These models require high reliability, good convergence speed and be able to evaluate the influence of material parameters, geometric parameters as well as being able to simulate the effect of shear deformation on vibration characteristics and dynamic responses of the 2D-FGM beam Content of the thesis Four main research contents are presented in four chapters of the thesis Specifically, Chapter presents an overview of domestic and foreign studies on the 1D and 2D-FGM beam structures Chapter proposes mathematical model and mechanical characteristics for the 2DFGM beam The equations for mathematical modeling are obtained based on two kinds of shear deformation theories, namely the first shear deformation theory and the improved third-order shear deformation theory Chapter presents the construction of FEM models based on different beam theories and interpolation functions Chapter illustrates the numerical results obtained from the analysis of specific problems Chapter OVERVIEW This chapter presents an overview of domestic and foreign regime of researches on the analysis of FGM beams The analytical results are discussed on the basis of two research methods: analytic method and numerical method The analysis of the overview shows that the numerical method in which FEM method is necessary is to replace traditional analytical methods in analyzing 2D-FGM structure in general and vibration of the 2D-FGM beam in particular Based on the overall evaluation, the thesis has selected the research topic and proposed research issues in details Chapter GOVERNING EQUATIONS This chapter presents mathematical model and mechanical characteristics for the 2D-FGM beam The basic equations of beams are set up based on two kinds of shear deformation theories, namely the first shear deformation theory (FSDT) and the improved third-order shear deformation theory (ITSDT) proposed by Shi [40] In particular, according to ITSDT, basic equations are built based on two representations, using the crosssectional rotation θ or the transverse shear rotation γ0 as an independent function The effect of temperature and the change of the cross-section are also considered in the equations 2.1 The 2D-FGM beam model The beam is assumed to be formed from four distinct constituent materials, two ceramics (referred to as ceramic1-C1 and ceramic2-C2) and two metals (referred to as metal1-M1 and metal2-M2) whose volume fraction varies in both the thickness and longitudinal directions as follows: VC1 = VC2 = VM1 = VM2 = z nz x nx + 1− h L nz z x nx + h L z nz x 1− 1− + h L nz n z x x 1− + h L (2.1) nx Fig 2.1 illustrates the 2D-FGM beam in Cartesian coordinate system (Oxyz) Z z C2 C1 h y X M1 M2 L, b, h b Fig 2.1 The 2D-FGM beam model In this thesis, the effective material properties P (such as Youngs modulus, shear modulus, mass density, etc.) for the beam are evaluated by the Voigt model as: P = VC1 PC1 +VC2 PC2 +VM1 PM1 +VM2 PM2 (2.2) When the beam is in thermal environment, the effective properties of beams depend not only on the properties of the component materials but also on the ambient temperature Then, one can write the expression for the effective properties of the beam exactly as follows: x nx z nz + + PM1 (T ) − h L nz z x nx PC2 (T ) − PM2 (T ) + PM2 (T ) + h L (2.4) PC1 (T ) − PM1 (T ) P(x, z, T ) = + For some specific cases, such as nx = or nz = 0, or C1 and C2 are identical, and M1 is the same as M2, the beam model in this thesis reduces to the 1D-FGM beam model Thus, author can verification the FEM model of the thesis by comparing with the results of the 1D-FGM beam analysis when there is no numerical result of the 2D-FGM beam Its important to note that the mass density is considered to be temperatureindependent [41] The properties of constituent materials depend on temperature by a nonlinear function of environment temperature [125]: P = P0 (P−1 T −1 + + P1 T + P2 T + P3 T ) (2.7) This thesis studies the 2D-FGM beam with the width and height are linear changes in beam axis, means tapered beams, with the following three tapered cases [138]: x x , I(x) = I0 − c L L x x − c , I(x) = I0 − c L L x x 1−c , I(x) = I0 − c L L Case A : A(x) = A0 − c Case B : A(x) = A0 Case C : A(x) = A0 (2.9) 2.2 Beam theories Based on the pros and cons of the theories, this thesis will use Timoshenko’s first-order shear deformation theory (FSDT) [127] and the improved third-order shear deformation theory proposed by Shi (ITSDT) [40] to construct FEM models 2.3 Equations based on FSDT Obtaining basic equations and energy expressions based on FSDT and ITSDT theory is similar, so Section 2.4 presents in more detail the process of setting up equations based on ITSDT 2.4 Equations based on ITSDT 2.4.1 Expression equations according to θ From the displacement field, this thesis obtains expressions for strains and stresses of the beam Then, the conventional elastic strain energy, UB is in the form L UB = A11 εm2 + 2A12 εm εb + A22 εb2 − 2A34 εm εhs − 2A44εb εhs (2.27) + A66 εhs + 25 1 B11 − B22 + B44 γ02 dx 16 2h h where A11 , A12 , A22 , A34 , A44 , A66 and B11 , B22 , B44 are rigidities of beam and defined as: (A11 , A12 , A22 , A34 , A44 , A66 )(x, T ) = E(x, z, T )(1, z, z2 , z3 , z4 , z6 )dA A(x) (B11 , B22 , B44 )(x, T ) = G(x, z, T )(1, z2 , z4 )dA A(x) (2.28) The kinetic energy of the beam is as follow: L T = 1 I11 (u˙20 + w˙ 20 ) + I12 u˙0 (w˙ 0,x + 5θ˙ ) + I22 (w˙ 0,x + 5θ˙ )2 16 − 10 25 I34 u˙0 (w˙ 0,x + θ˙ ) − I44 (w˙ + θ˙ )(w˙ + 5θ˙ ) + I66 (w˙ 0,x + θ˙ )2 dx 3h 6h 9h (2.29) in which ρ (x, z) 1, z, z2 , z3 , z4 , z6 dA (2.30) (I11 , I12 , I22 , I34 , I44 , I66 )(x) = A(x) are mass moments The beam rigidities and mass moments of the beam are in the following forms: Ai j = AC1M1 − AC1M1 − AC2M2 ij ij ij Bi j = BC1M1 − ij BC1M1 − BC2M2 ij ij x L x L nx nx (2.31) with AC1M1 , BC1M1 are the rigidities of 1D-FGM beam composed of C1 ij ij C2M2 C2M2 are the rigidities of 1D-FGM beam composed of and M1; Ai j , Bi j C2 and M2 Noting that rigidities of 1D-FGM beam are functions of z only, the explicit expressions for this rigidities can easily be obtained 2.4.2 Expression equations according to γ0 Using a notation for the transverse shear rotation (also known as classic shear rotation), γ0 = w0,x + θ as an independent function, the axial and transverse displacements in (2.13) can be rewritten in the following form u(x, z,t) = u0 (x,t) + z 5γ0 − 4w0,x − z3 γ0 3h (2.35) w(x, z,t) = w0 (x,t) Similar to the construction of basic equations according to θ , the thesis also receives basic equations expressed in γ0 2.5 Initial thermal stress Assuming the beam is free stress at the reference temperature T0 and it is subjected to thermal stress due to the temperature change The initial thermal stress resulted from a temperature ∆T is given by [18, 70]: T σxx = −E(x, z, T )α (x, z, T )∆T (2.41) in which elastic modulus E(x, z, T ) and thermal expansion α (x, z, T ) are obtained from Eq.(2.4) T has the form The strain energy caused by the initial thermal stress σxx [18, 65]: L UT = NT w20,x dx (2.42) T: where NT is the axial force resultant due to the initial thermal stress σxx T dA = − σxx NT = A(x) E(x, z, T )α (x, z, T )∆T dA (2.43) A(x) The total strain energy resulted from conventional elastic strain energy UB , and strain energy due to initial thermal stress UT [70] 2.6 Potential of external load The external load considered in the present thesis is a single moving constant force with uniform velocity The force is assumed to cause bending only for beams The potential of this moving force can be written in the following form V = −Pw0 (x,t)δ x − s(t) (2.44) where δ (.) is delta Dirac function; x is the abscissa measured from the left end of the beam to the position of the load P, t is current time calculated from the time when the load P enters the beam, and s(t) = vt is the distance which the load P can travel 2.7 Equations of motion In this section, author presents the equations of motion based on ITSDT with γ0 being the independent function Motion equations for beams based on FSDT and ITSDT with θ is independent function that can be obtained in the same way Applying Hamiltons principle, one obtained the motion equations system for the 2D-FGM beam placed in the temperature environment under a moving force as follows: I11 uă0 + 5ă0 4wă 0,x I12 I34 ă0 A11 u0,x 3h (2.51) + A12 5γ0,x − 4w0,xx A34 0,x 3h I11 wă + I12 uă0 + 5ă0 4wă 0,x I22 I44 ă0 3h + A22 5γ0,x − 4w0,xx − A44 γ0,x 3h ,x − A12 u0,x ,x = NT w0,x ,xx =0 ,x − Pδ x − s(t) (2.52) 1 I12 uă0 + I22 16 + I66 ă0 9h I34 uă0 I44 ă0 wă 0,x 3h2 3h 1 A12 u0,x + A22 5γ0,x − 4w0,xx − A34 u0,x 16 3h 5ă0 4wă 0,x − 5 A44 γ0,x − w0,xx − A66 γ0,x 3h2 9h +5 ,x 1 B11 − B22 + B44 γ0 = 16 2h h (2.53) Notice that the coefficients in the system of differential equations of motion are the rigidities and mass moments of the beam, which are the functions of the spatial variable according to the length of the beam and the temperature, thus solving this system using analytic method is difficult FEM was selected in this thesis to investigate the vibration characteristics of beams Conclusion of Chapter Chapter has established basic equations for the 2D-FGM beam based on two kinds of shear deformation theories, namely FSDT and ITSDT The effect of temperature and the change of the cross-section is considered in establishing the basic equations Energy expressions are presented in detail for both FSDT and ITSDT in Chapter In particular, with ITSDT, basic equations and energy expressions are established on the cross-sectional rotation θ or the transverse shear rotation γ0 as independent functions The expression for the strain energy due to the temperature rise and the potential energy expression of the moving force are also mentioned in this Chapter Equations of motion for the 2D-FGM beam are also presented using ITSDT with γ0 as independent function These energy expressions are used to obtain the stiffness matrices and mass matrices used in the vibration analysis of the 2D-FGM beam in Chapter Chapter FINITE ELEMENT MODELS This chapter builds finite element (FE) models, means that establish expressions for stiffness matrices and mass matrices for a characteristic element of the 2D-FGM beam The FE model is constructed from the energy expressions received by using the two beam theories in Chapter Different shape functions are selected appropriately so that beam elements get high reliability and good convergence speed Nodal load vector 11 in which l kSmθ Sθ Bm = T l Sθ A11 Bm dx ; kSbθ BSbθ = T A22 BSbθ dx 0 l kSs θ = 25 T BSs θ 1 B11 − B22 + B44 BSs θ dx 16 2h h l Sθ khs = Sθ Bhs T Sθ dx A66 Bhs l kSc θ BSmθ = T Sθ A12 BSbθ − Bm T T A34 BShsθ − BbSθ A44 BShsθ dx (3.36) One write the kinetic energy in the following form nE ˙ K T ˙ K (d ) m d 2∑ in which the element consistent mass matrix is in the form T = (3.13) 22 34 44 66 11 12 m = m11 uu + muθ + mθ θ + muγ + mθ γ + mγγ + mww (3.37) with l m11 uu NTu I11 Nu dx = l = NTu I12 (Nw,x + 5Nθ )dx l m22 θθ ; m12 uθ = NTu I34 (Nw,x + Nθ )dx 0 mθ44γ = − l T (Nw,x + 5NθT )I22 (Nw,x + 5Nθ )dx ; m34 uγ = − 16 3h2 l 12h2 (NTw,x + 5NTθ )I44 (Nw,x + Nθ )dx m66 γγ l l 25 = 9h (NTw,x + NθT )I66 (Nw,x + Nθ )dx ; m11 ww NTw I11 Nw dx = (3.38) are the element mass matrices components 12 3.2.2 TBSγ model With γ0 is the independent function, the vector of nodal displacements for a generic element, (i, j), has eight components: dSγ = {ui wi wi,x γi u j w j w j,x γ j }T (3.39) The axial displacement, transverse displacement and transverse shear rotation are interpolated from the nodal displacements according to u0 = Nu dSγ , w0 = Nw dSγ , γ0 = Nγ dSγ (3.40) with Nu , Nw and Nγ are the matrices of shape functions for u0 , w0 and γ0 , respectively Herein, linear shape functions are used for the axial displacement u0 (x,t) and the transverse shear rotation γ0 , Hermite shape functions are employed for the transverse displacement w0 (x,t) The construction of element stiffness and mass matrices are completely similar to TBSθ model 3.3 Element stiffness matrix due to initial thermal stress Using the interpolation functions for transverse displacement w0 (x,t), one can write expressions for the strain energy due to the temperature rise (2.42) in the matrix form as follows UT = where nE T d kT d 2∑ (3.44) l BtT NT Bt dx kT = (3.45) is the stiffness due to temperature rise For different beam theories, the element stiffness matrix due to temperature rise has the same form (3.45) The only difference is that the difference of the shape functions Nw is chosen for w0 (x,t) leading to the difference of the strain-displacement matrix Bt = (Nw ),x in (3.45) 3.4 Discretized equations of motion Ignoring damping effect of the beam, the equations of motion for 2DFGM beam can be written in the context of the finite element analysis as ¨ + KD = Fex MD (3.49) 13 ¨ are, respectively, the vectors of structural nodal displacein which D, D ments and accelerations, K, M, Fex are the stiffness matrices due to the beam deformation and temperature rise, the mass matrix and the nodal load vector of the structure, respectively In the free vibration analysis, the right-hand side of (3.49) is set to zero 0: ă + KD = MD (3.52) 3.5 Numerical procedure Solving the equation (3.52) is brought about solving the eigenvalue problem Eq (3.49) can be solved by the direct integration Newmark method The constant average acceleration method which ensures the unconditional stability is employed in this thesis Conclusion of Chapter Chapter builds FE model for a two-node element based on two kinds of shear deformation theories for beams Based on FSDT, FE models are constructed by using two different shape functions, such as the Kosmatka function and hierarchical shape functions Based on ITSDT, FE models are constructed by linear and Hermite shape functions The expression for stiffness and mass matrix for the models based on ITSDT is built on the basis of considering the cross-section rotation or transverse shear rotation as independent functions The expression for the stiffness matrix due to temperature rise and the vector of nodal force is also built into Chapter Chapter NUMERICAL RESULTS AND DISCUSSION The numerical results are presented on the basis of analyzing three problems: (1) Free vibration analysis of the 2D-FGM beam in thermal environment; (2) Free vibration analysis of the tapered 2D-FGM beam; (3) Forced vibration analysis of the 2D-FGM beam excited by a moving force From the numerical results obtained, some conclusions relate to the influence of the material parameter, the taper ratio, aspect ratio and temperature rise on the fundamental frequency and the vibration mode to be extracted Dynamic behaviour of 2D-FDM beams under the action of moving force are also discussed in Chapter 4.1 Validation and convergence of FE models 4.1.1 Convergence of FE models 14 The convergence of four FE models developed in the thesis in evaluating the fundamental frequency parameter µ of a simply supported 2DFGM beams with constant cross-section (c = 0) is examined in the thesis The effect of temperature is not considered herein (∆T = 0K) Some comments can be drawn as follows: - The fundamental frequency parameters of 2D-FGM beams received from four FE models developed in the thesis are very close - Three of the four FE models have high convergence rate, namely FBKo model, FBHi model and TBSγ model When using these three models to calculate, the fundamental frequency parameters of the 2DFGM beam converges to the same value with only 16 or 18 elements However, TBSθ model converges very slowly, requiring up to 70 elements - Values of the grading indexes pairs (nx , nz ) not affect to the convergence rate of the FE models From the convergence of the above-mentioned FE models, the thesis will only use models with good convergence to calculate and compare numerical results The convergence of FBHi model in evaluating the fundamental frequency parameter of the tapered 2D-FGM beam is also carried out by the thesis In calculating the fundamental frequency parameter, convergence rate of FBHi model of the tapered 2D-FGM beam is slower than a constant cross-section ones It requires up to 30 elements to achieve the convergence rate 4.1.2 Validation of FE models Since there is no data on the vibration of the 2D-FGM beam with the power-law variations of the material properties as considered in the thesis, the comparison will be carried out for the 1D-FGM beam, a special case of the 2D-FGM beam The fundamental frequency parameter and the dynamic response obtained in the thesis are compared with the data available in the literature The effect of temperature and change of the cross-section are considered Comparative results show that the FE models developed in the thesis are reliable and it can be used to study vibration of the 2D-FGM beam 4.2 Free Vibration 4.2.1 Constant cross-section beams 4.2.1.1 Influence of material distribution 15 Fig 4.1 illustrates the influence of grading indexes on the first four natural frequency parameters of S-S beams with ∆T = 50K 20 µ2 µ1 2 1.5 n x 0.5 0 0.5 1.5 n 15 10 2 1.5 n z 40 x 0.5 0 0.5 1.5 n z 60 µ4 µ3 50 30 40 20 1.5 nx 0.5 0 0.5 1.5 30 nz 1.5 n x 0.5 0 0.5 1.5 n z Fig 4.1 Influence of grading indexes on the first four natural frequency parameters of S-S beams with ∆T = 50K From Fig 4.1 ones can see that: - At a given value of the index nx , the fundamental frequency parameter µ1 tends to decreased by the increase in the index nz The decrease of µ1 is more significant for the beam with a higher index nx The effect of the index nx on the fundamental frequency parameter is different from that of the index nz , and µ1 increases with the increase of the nx index However, the increase of µ1 is more significant for the beam associated with a lower index nz - The fundamental frequency parameter attains a maximum value at nx = and nz = 0, and this is the special case when the beam degrades to the axially FG beam made of the two ceramics - At the given value of the temperature rise, the effect of the grading indexes on the higher frequency parameters is similar to the case of the fundamental frequency parameter, they are also decreased by increasing the index nz and they are increased by increasing index nx 4.2.1.2 Influence of temperature rise Fig 4.2 illustrates the influence of grading indexes on the fundamental frequency parameters of S-S beams for various temperature rise ∆T Some comments can be drawn from Fig 4.2 as follows: 16 4 µ µ1 3 2 1.5 nx 0.5 0 0.5 1.5 2 1.5 nx nz 0.5 (a) ∆T=0 K 0.5 1.5 n z (a) ∆T=20 K 5 µ1 µ1 2 0 1.5 n x 0.5 0 0.5 1.5 nz 2 1.5 n 0.5 x (c) ∆T=40 K 0 0.5 1.5 n z (d) ∆T=80 K Fig 4.2 Influence of grading indexes on µ1 of S-S beams for various temperature rise ∆T - The relation between the grading indexes and frequency parameters unchanged when the value of ∆T increases That means, the frequency parameters decrease when increasing the index nz and they increased with increasing the index nx However, this relation is affected by the temperature rise In particular, when nz increases from to 2, the fundamental frequency parameters of the beam is significantly decrease, especially when the index nx is large - The fundamental frequency parameters of beams is significantly decrease when the value of the ∆T increases 4.2.1.3 Influence of the boundary conditions Some comments can be drawn from this section as follows: - The frequency parameters of the C-C beam is highest while that one of C-F beams is lowest At the reference temperature (∆T = 0K), the variation of the frequency parameters with the grading indexes of the C-C beam and the C-F beam is similar to that of the S-S beam However, the C-F beam is more sensitive to the change in the index nx than the S-S and C-C beams, especially when nz is small 17 - The effect of the grading indexes on the higher frequency parameters of the C-C beam and the C-F beam is similar to that of the S-S beam - The variation of the frequency parameters of C-C and C-F beams with values of the temperature rise are similar to S-S beams However, this variation is strongly influenced by boundary conditions Specifically, C-C beams are less affected by temperature rise In contrast, C-F beams are very sensitive to the rise of temperature 4.2.1.4 Influence of the aspect ratio 4.5 3.5 2.5 1.5 µ µ1 The effect of the beam aspect ratio, L/h, on the fundamental frequency parameters of the beam is illustrated in Fig 4.7, where the variations of the fundamental frequency parameter with the grading indexes of the SS beam are depicted for two values of the aspect ratio, L/h = 10 and L/h = 30, and for a temperature rise ∆T = 50K In Fig 4.7, the relation between the grading indexes and frequency parameters unchanged when the value of L/h increases, means an increase in the aspect ratio leads to a significantly decrease of the fundamental frequency parameter It should be noted that previous studies have shown that when beams are placed at reference temperature, an increase in the aspect ratio leads to a significantly increase of the fundamental frequency parameter However, as shown in Fig 4.7, this is no longer true when the effect of temperature is considered This can be explained by the fact that when beams are placed in temperature environments, the stiffness of the beams with high aspect ratio is significantly decrease than that of beams with low aspect ratio 1.5 nx 0.5 0.5 0 (a) ∆T=50 K, L/h=10 1.5 n z 4.5 3.5 2.5 1.5 1.5 n x 0.5 0 0.5 1.5 nz (b) ∆T=50 K, L/h=30 Fig 4.7 Variation of fundamental frequency parameter with grading indexes of S-S beam in thermal environment with different values of aspect ratio 18 4.2.1.5 Mode shapes Fig 4.8 illustrates the first three mode shapes for u0 , w0 and γ0 of SS beams with two pairs of the grading indexes: (nx , nz ) = (0.0, 0.5) and (nx , nz ) = (0.5, 0.5), in the reference temperature (∆T = 0) 1.5 w mode 1.5 mode u γ0 0.5 0.5 n =0, n =0.5 x −0.5 n =0.5, n =0.5 z 0.25 0.5 0.75 −0.5 mode 0.5 0.75 0.25 0.5 0.75 0.25 0.5 0.75 0.5 0 −0.5 −0.5 −1 −1 −1.5 1.5 0.25 0.5 0.75 −1.5 1.5 mode 1 0.5 0.5 0 −0.5 −0.5 −1 −1.5 z 0.25 mode 0.5 (a) x 1.5 1.5 mode −1 0.25 0.5 0.75 −1.5 (b) Fig 4.8 The first three mode shapes for u0 , w0 and γ0 of S-S beams with ∆T = 0K: (a) (nx , nz ) = (0, 0.5), (b) (nx , nz ) = (0.5, 0.5) As can be seen from the figure, the mode shapes of the 2-D FGM beam as depicted in Fig 4.8(b) are very different from that of the unidirectional transverse FGM beam as depicted in Fig 4.8(a) While the first and third modes of the transverse displacement w0 of 1D-FGM beam are symmetric with respect to the mid-span, that of the 2D FGM beam are not The figure also shows the difference in the mode shape of u0 and γ0 of the 2-D FGM beam with that of the 1D beam, and the asymmetric of the second mode for γ0 with respect to the mid-span is clearly seen from Fig 4.8(b) Thus, the variation of the constituent materials in the longitudinal directions has a significant influence on the vibration modes of the beam The mode shapes for u0 , w0 and γ0 of the S-S 2D-FGM beam in thermal environment are also considered for various values of the grading indexes The grading indexes and temperature rise have a significant influence on the vibration modes of the beam, and not only vibration amplitude but also the position of the critical point is changed 4.2.2 Tapered beams 19 4.2.2.1 Influence of material distribution The influence of the grading indexes on the frequency parameter received for tapered 2D-FGN beams is similar to constant cross-section beams However, the longitudinal index nz have less effect on the fundamental frequency parameter of tapered beams compared to constant cross-section beams, especially with C-F boundary beam tapered beams 4.2.2.2 Influence of taper ratio and taper case The taper ratio versus the fundamental frequency parameter of the tapered 2D-FGM beam with nz = 0.5 and different values of nx is depicted in Figs 4.14-4.16 for the C-F, S-S and C-C beams, respectively As can be seen from the figures, the variation of the frequency parameter with the taper ratio is governed by the boundary conditions and the taper case as well While the frequency parameter of the C-F beam increases by increasing the taper ratio, that of the S-S and C-C beams decreases with the increase of the taper ratio, regardless of the taper case For a given boundary condition, the dependence of the frequency parameter upon the taper ratio c is, however significantly influenced by the taper case The rate of the variation of µ1 with is the most significant for the type C of the C-F and S-S beams, while that is occurred for the type B of the C-C beam 4.2.2.3 Influence of aspect ratio Some comments can be drawn from this section as follows: - The effect of the aspect ratio on the frequency of the tapered beam is less significant than that of the uniform beam - The effect of the aspect ratio on the frequency is also influenced by the boundary condition, and the increase of the fundamental frequency of the S-S beam is more significant than that of the C-F beam, regardless of the grading indexes and the taper ratio 4.3 Forced vibration 4.3.1 Influence of the moving load speed In Fig 4.17, the time histories for normalized mid-span deflection, w0 (L/2,t)/wst , of the 2D-FGM beam are depicted for various values of the moving load speed v and the indexes nx and nz In the Figure, the midspan deflection is normalized by the static deflection of the isotropic beam 20 3 Case A Case B Case C µ µ 1.5 1.5 (a) n =0, n =0.5 x 0.3 z 0.6 0.9 c 0.3 0.6 0.9 0.6 0.9 c Case A Case B Case C 2.5 µ 1.5 1.5 (c) n =2, n =0.5 (b) nx=0.5, nz=0 Case A Case B Case C 2.5 µ1 Case A Case B Case C 2.5 1 2.5 x 0.3 0.6 (d) n =0.5, n =2 z 0.9 x z 0.3 c c Fig 4.14 Taper ratio versus fundamental frequency parameter of C-F beam with different taper cases and grading indexes:: (a) (nx , nz ) = (0, 0.5), (b) (nx , nz ) = (0.5, 0), (c) (nx , nz ) = (2, 0.5), (d) (nx , nz ) = (0.5, 2) 5 (a) n =0, n =0.5 x (b) n =0.5, n =0 z x Case A Case B Case C z µ µ Case A Case B Case C 0.3 0.6 0.9 0.3 x x 0.9 z µ 0.6 (d) n =0.5, n =2 z µ 0.9 (c) n =2, n =0.5 Case A Case B Case C 0.6 c c Case A Case B Case C 0.3 0.6 c 0.9 0.3 c Fig 4.15 Taper ratio versus fundamental frequency parameter of S-S beam with different taper cases and grading indexes:: (a) (nx , nz ) = (0, 0.5), (b) (nx , nz ) = (0.5, 0), (c) (nx , nz ) = (2, 0.5), (d) (nx , nz ) = (0.5, 2) made of aluminum The moving load speed, as seen from the Figure, affects both the dynamic deflection and the way the beam vibrates For 21 7 µ µ1 (a) n =0, n =0.5 x (b) n =0.5, n =0 x z Case A Case B Case C 0.3 0.6 0.9 z Case A Case B Case C 0.3 c 0.6 0.9 0.6 0.9 c 9 µ µ 7 (c) n =2, n =0.5 x (d) n =0.5, n =2 z x Case A Case B Case C 0.3 0.6 0.9 z Case A Case B Case C c 0.3 c Fig 4.16 Taper ratio versus fundamental frequency parameter of C-C beam with different taper cases and grading indexes:: (a) (nx , nz ) = (0, 0.5), (b) (nx , nz ) = (0.5, 0), (c) (nx , nz ) = (2, 0.5), (d) (nx , nz ) = (0.5, 2) the given values of the grading indexes, the beam shows more vibration cycles when it is subjected to the lower moving speed load The grading indexes considerably affect the dynamic deflection of the beam, but they hardly affect curve shapes of the time histories 4.3.2 Influence of material distribution In Fig 4.18, the relation between the dynamic magnification factor and the moving load speed is illustrated for various values of the indexes nz and nx As seen from the Figure, the relation between Dd and v of the 2D-FGM beam is similar to that of an isotropic beam under a moving load, that is, the factor Dd both increases and decreases, and it then monotonously increases to a maximum value when increasing the moving load speed The repeated increase and decrease of the factor Dd for lower values of the moving load speed in Fig 4.18, as mentioned above, is associated with the oscillations of the beam under the load with the lower moving load speed to the critical speed ratios The effect of the grading index nz on the factor Dd is, however, different from that of the index nx The dynamic magnification factor steadily decreases as the index nx increases, whereas it increases by the increase in the index nz The effect of the two grading indexes on the factor Dd can be explained by 0.6 0.6 0.4 0.4 w0(L/2,t)/wst w0(L/2,t)/wst 22 0.2 v=20 m/s v=50 m/s v=100 m/s −0.1 0.2 0.2 0.4 0.6 v=20 m/s v=50 m/s v=100 m/s 0.8 (a) −0.1 (b) 0.2 t/∆T* st 0.4 0.1 v=20 m/s v=50 m/s v=100 m/s −0.2 0.2 w (L/2,t)/w w0(L/2,t)/wst 0.8 (c) 0.8 0.8 0.3 1.2 0.4 0.6 t/∆T* 0.2 0.4 t/∆T* 0.6 v=20 m/s v=50 m/s v=100 m/s 0.8 −0.05 (d) 0.2 0.4 0.6 t/∆T* Fig 4.17 Time histories for normalized mid-span deflection with different indexes nx and nz : (a) (nx , nz ) = (1/3, 1/3), (b) (nx , nz ) = (3, 3), (c) (nx , nz ) = (0, 3), (d) (nx , nz ) = (3, 0) the dependence of the rigidities on these indexes The beam associated with a higher index nx contains more C1 and M1, and thus, its rigidities are higher, whereas the rigidities of the beam with a higher index nz are lower The thickness distribution of the normalized axial stress at mid-span section of the 2D-FGM beam is depicted for v = 100 m/s and various values of the grading indexes The stress in the Fig 4.20 was computed at the time when the load arrives at the mid-span, and it was normalized as σ ∗ = σxx /σ0 , where σ0 = PLh/8I The thickness distribution of the stress of 2D-FGM beam, as seen from the Figure, is very different from that of isotropic beams, and the stress does not vanish at the mid-span, except for the case nz = 0, which corresponds to the axially FG beam composed of the two ceramics The influence of the index nz on the stress distribution is also very different from that of the index nx The maximum amplitude of both the compressive and tensile stresses decreases as the index nx increases, whereas it increases as the index nz increases Thus, by raising the index nx , we could decrease not only the dynamic magnification 23 0.9 0.9 nx=0 nx=1/3 nx=1 nx=3 nz=1/3 0.8 0.7 nx=1/3 0.8 0.7 0.6 D d Dd 0.6 0.5 0.5 0.4 0.4 0.3 0.3 (a) 0.2 50 100 150 200 250 300 nz=0 nz=1/3 nz=1 nz=3 (b) 350 0.2 50 100 150 200 250 300 350 v (m/s) v (m/s) Fig 4.18 Relation between dynamic magnification factor and moving load speed with different indexes: (a) nz = 1/3, nx is variable; (b) nx = 1/3, nz is variable 0.5 0.5 n =0 n =0 n =1/3 n =1/3 x z x z nx=1 0.25 nz=1 0.25 nz=3 z/h z/h nx=3 −0.25 −0.25 (b) n =1/3 (a) n =1/3 −0.5 −2 z −1 σ* −0.5 −2 x −1 σ* Fig 4.20 Thickness distribution of normalized axial stress at mid-span section for v = 100 m/s: (a) nz = 1/3, nx is variable, (b) nx = 1/3, nz is variable factor, but also the maximum amplitude of the axial stress Conclusion of Chapter On the basis of comparing the numerical results obtained in the thesis and the published results, Chapter has proved that all four FE models developed in the thesis are reliable in evaluating the vibration characteristics of FGM beams Three FE models are confirmed to have high convergence rate, namely FBKo, FBHi and TBSγ models, while TBSγ model has a much slower convergence rate Using the FE models and the numerical calculation program, Chapter analyzed free vibration and 24 forced vibration problems of the 2D-FGM beam The numerical results obtained in Chapter are illustrated with tables and graphs On the basis of the numerical results obtained, Chapter gave some comments regarding the effect of material distribution and geometric parameters on the vibration characteristics of the 2D-FGM beam The numerical results obtained in Chapter help to design and optimize 2D-FGM beam structures subjected to dynamic loads CONCLUSIONS The thesis has built four FE models used in vibration analysis of the 2D-FGM beam, of which there are two models based on first-order shear deformation theory (FSDT) and two models based on the improved thirdorder shear deformation theory (ITSDT) Effect of temperature and the change of the cross-section is also considered in the construction of the FE models Three of the four models provide very good convergence and it is used to compare and calculate specific problems Based on FE models, a computer code has been developed and employed to analyze free and forced vibration of the 2D-FGM beam Since then evaluated the effect of material parameters and geometric parameters, temperature rise and moving force on the vibration characteristics as well as dynamic response of beams NEW CONTRIBUTIONS OF THE THESIS The thesis has some new contributions as follows: A new material model for 2D-FGM beams is proposed for the first time in the thesis The model contains four distinct constituent materials, two ceramics and two metals, whose volume fraction varies in both the thickness and longitudinal directions by a power-law distribution Various finite element models based on the first- and third-order shear deformation theories are derived and employed in analyzing free and forced vibration of 2D-FGM beams Especially, the model based on the improved third-order shear deformation theory using the transverse shear rotation as a unknown function is proposed for the first time PUBLICATIONS OF THE AUTHOR NGUYEN DINH KIEN and TRAN THI THOM, Free vibration of tapered BFGM beams using an efficient shear deformable finite element model, Steel and Composite Structures, 2018, 29(3), 363377 (ISI Journal) TRAN THI THOM and NGUYEN DINH KIEN, Free vibration analysis of 2-D FGM beams in thermal environment based on a new third-order shear deformation theory, Vietnam Journal of Mechanics, 2018, 40(2), 121-140 TRAN THI THOM and NGUYEN DINH KIEN, Free vibration of two-directional FGM beams using a higher-order Timoshenko beam element, Journal of Science and Technology, 2018, 56(3), 380-396 NGUYEN DINH KIEN, NGUYEN QUANG HUAN, TRAN THI THOM and BUI VAN TUYEN, Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load, Acta Mechanica, 2017, 228, 141–155 (ISI Journal) TRAN THI THOM, NGUYEN DINH KIEN and NGUYEN DUC HIEU, Beam element based on a new third-order shear deformation theory for vibration analysis of 2-D FGM beams, Tuyển tập Hội nghị Cơ học toàn quốc lần thứ X, Hà Nội, 2017, 1165-1172 TRAN THI THOM, NGUYEN QUANG HUAN, NGUYEN DINH KIEN and BUI VAN TUYEN, Fundamental frequency analysis of FG porous beams in thermal environment using the improved thirdorder shear deformation theory, Proceedings of 4th International Conference on Engineering Mechanics and Automation (ICEMA4), Hanoi, 2016, 393-400 TRAN THI THOM, BUI VAN TUYEN and NGUYEN DINH KIEN, Vibration of functionally graded sandwich beams in high temperature environment, Tuyển tập Hội nghị Khoa học toàn quốc Cơ học Vật rắn biến dạng lần thứ XII, Đại học Duy Tân, Đà Nẵng, 2015, 13881395 ... deformation theories, namely the first shear deformation theory and the improved third-order shear deformation theory Chapter presents the construction of FEM models based on different beam theories... Beam theories Based on the pros and cons of the theories, this thesis will use Timoshenko’s first-order shear deformation theory (FSDT) [127] and the improved third-order shear deformation theory... up based on two kinds of shear deformation theories, namely the first shear deformation theory (FSDT) and the improved third-order shear deformation theory (ITSDT) proposed by Shi [40] In particular,