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MINISTRY OF EDUCATION AND TRAINING NATIONAL UNIVERSITY OF CIVIL ENGINEERING Duong Thanh Huan STATIC AND VIBRATION ANALYSIS OF FGM DOUBLY-CURVED SHELLS IN THERMAL ENVIRONMENT Major: Engineering Mechanic Code: 9520101 SUMMARY OF DOCTORAL DISSERTATION Hanoi - 2019 The work was completed at: NATIONAL UNIVERSITY OF CIVIL ENGINEERING (NUCE) Academic supervisor : Assoc Prof PhD Tran Huu Quoc – NUCE Assoc Prof PhD Tran Minh Tu – NUCE Peer reviewer 1: Prof PhD Nguyen Tien Khiem Institute of Mechanics Peer reviewer 2: Prof PhD Nguyen Thai Chung Military Techical Academy Peer reviewer 3: Prof PhD Nguyen Tien Chuong Thuyloi University The doctoral dissertation will be defended at the level of the State Council of Dissertation Assessment's meeting at the National University of Civil Engineering at hour .', day month year 2019 The dissertation is available for reference at the libraries as follows: - National Library of Vietnam; - Library of National University of Civil Engineering; LIST OF PUBLISHED WORKS BY AUTHOR RELATED TO THIS DISSERTATION’S TOPIC Tran Minh Tu, Tran Huu Quoc, Duong Thanh Huan, and Nguyen Van Long (2014), Vibration analysis of functionally graded plates using various shear deformation plate theories, Proceedings of the 3rd International Conference of Engineering Mechanics and Automation ICEMA3, University of Engineering and Technology – Vietnam National University, ISBN: 978604-913-367-1, trang 580-587 Trần Minh Tú, Trần Hữu Quốc, Dương Thành Huân (2015), Phân tích tĩnh động Panel trụ làm vật liệu có tính biến thiên (FGM) theo lý thuyết biến dạng cắt bậc (FSDT), Tuyển tập cơng trình Hội nghị Khoa học tồn quốc Cơ học vật rắn biến dạng lần thứ XII, Thành phố Đà Nẵng, ISBN: 978-604-82-2028-0, Tập 2, trang 1506-1513 Trần Hữu Quốc, Dương Thành Huân, Trần Minh Tú, Nghiêm Hà Tân (2017), Phân tích Panel trụ FGM chịu uốn có xét đến ảnh hưởng nhiệt độ - Lời giải giải tích Lời giải số, Tạp chí Khoa học Công nghệ Xây dựng, ISSN: 1859-2996, Tập 11 số 2, trang 38 – 46 Duong Thanh Huan, Tran Minh Tu and Tran Huu Quoc (2017), Analytical solutions for bending, buckling and vibration analysis of functionally graded cylindrical panel, Vietnam Journal of Science and Technology 55(5): p 587-597, DOI: 10.15625/2525-2518/55/5/8843 Duong Thanh Huan, Tran Huu Quoc, Tran Minh Tu and Le Minh Lu (2017), Free vibration analysis of functionally graded doubly-curved shallow shells including thermal effect, Vietnam Journal of Agricultural Sciences, Vol 15, No 10: p 1410-1422, ISSN: 1859-0004 Tran Huu Quoc, Duong Thanh Huan and Tran Minh Tu (2018), Dynamic behavior analysis of FGM doubly curved panels considering temperature dependency of material properties, Tuyển tập Hội nghị Khoa học toàn quốc Cơ học Vật rắn lần thứ XIV, Trường Đại học Trần Đại Nghĩa, Thành phố Hồ Chí Minh (Đã chấp nhận đăng) Tran Huu Quoc, Duong Thanh Huan and Tran Minh Tu (2018), Free vibration analysis of functionally graded doubly curved shell panels resting on elastic foundation in thermal environment, International Journal of Advanced Structural Engineering, 10(3): p 275-283, DOI: 10.1007/s40091-018-0197-x Duong Thanh Huan, Tran Huu Quoc and Tran Minh Tu (2018), Free vibration analysis of functionally graded shell panels with various geometric shapes in thermal environment, Vietnam Journal of Mechanics, 40(3): p 199-215, DOI: https://doi.org/10.15625/08667136/10776 INTRODUCTION The necessity of the topic Functionally graded materials (FGMs) are the advanced composites, microscopically inhomogeneous, whose volume fraction of constituents vary gradually from one surface to the other As a result, the mechanical properties vary smoothly and continuously in the preferred direction, thus eliminating interface problems and mitigating thermal stress concentrations In order to optimally design the functionally graded plates and shells, the mechanical behaviors of FG structures need to be well understood Therefore, a development of computational models and methods has attracted the attention of many international and local researchers To enrich the studies on mechanical behaviors of FG structures in both sides: computational models and methods, the subject of dissertation is chosen as “Static and vibration analysis of FGM doubly-curved shells in thermal environment” Aims and content of the research Use the first shear deformation theory (FSDT) to develop the relations, the constitutive equations to analyze static, and vibration response of FG doubly-curved shells by analytical method Use the 3D degenerated shell element to develop the algorithm and finite element model to analyze static, and vibration response of FG doubly-curved shells Build a Matlab’s code to investigate the effect of material parameters, geometric dimensions, temperature and boundary conditions on deflection, stress field, and dynamic behavior of FG doubly-curved shells Object and scope of the research Object: Functionally graded doubly-curved shell panels with constant thickness and various boundary conditions The projection of the panel on the xy-plane is a rectangle Scope: Linear analysis – predicting deflection, stress components, and frequencies of FG doubly-curved shell panels under mechanical and thermal loads with various boundary conditions Scientific basis of the thesis The functionally graded materials (FGM) have been widely used in many technical fields due to many advantages compared to traditional composite materials, especially the ability to work in high temperature environments, prevents cracking or debonding, stress concentration at the interface between two materials, Based on the first shear deformation theory (FSDT), the analytical solutions, algorithms and finite element models has been developed for static and vibration analysis of doubly-curved shells in thermal environment The effect of material parameters, geometric dimension of structure and temperature on the mechanical behavior of FGM shells are investigated, and the drawn notes are useful references for analysis, design and maintenance Research methodology Analytical method: Using the first shear deformation theory to develop the constitutive equations, algorithm, and Matlab’s codes for analyzing the static and vibration response of doublycurved shells in thermal environment with some common boundary conditions Finite element method: Develop an algorithm, finite element model (3D-Degenerated shell element), and Matlab’s codes to analyze the static and vibration response of doubly-curved shells in thermal environment with various geometrical shapes and boundary conditions Significant contributions of the dissertation Develop analytical solutions based on the first-order shear deformation theory (FSDT) for static and vibration linear analysis of FGM doubly-curved shell panels in thermal environment with some common boundary conditions Develop algorithms, finite element model using 3D degenerated shell element for linear static and vibration analysis of FGM doubly-curved shell panels in thermal environment Built Matlab’s codes to investigate the effect of material parameters; geometric dimensions; boundary conditions; various temperature distribution through the thickness; damping rate; ratio of forced/natural frequencies (ratio Ω / ω) on the static and dynamic behaviors of doublycurved shells in thermal environment CHAPTER OVERVIEW OF THE RESEARCH TOPIC This chapter gives a brief introduction about the functionally graded material (FGM) - the mechanical properties of materials; FGM structures and applications; Overview of national and international researches on static and dynamic analysis of FGM shell structures in thermal environments The review study shows that the above-mentioned studies are mainly focused on FGM plate structure For the FGM shell structure, studies on the thermal influences mainly focus on closed shell objects, such as cylindrical shell, conical shell For shell panel types: doubly-curved shell, cylindrical panel, spherical panel, the studies on their nonlinear behavior are limited with analytical solutions and some simple boundary conditions and shell geometrical shapes Studies of the shell structure with complex geometrical shapes, as well as the investigation of their work in the thermal environment is relatively rare, especially the analysis of thermal stress should be studied further Published studies on the dynamic problems can be classified into two approaches: The first approach (Type 1): The temperature changes only affect the mechanical properties of the materials The second approach (Type 2): The temperature changes not only changes the mechanical properties of the materials but also causes initial thermal stresses CHAPTER STATIC AND VIBRATION ANALYSIS OF FG DOUBLY-CURVED SHELL IN THERMAL ENVIRONMENT USING THE FIRST SHEAR DEFORMATION THEORY BY ANALYTICAL SOLUTION 2.1 Introduction In this chapter, the constitutive equations are summarized based on the first order shear deformation theory (FSDT) to derive the equations of motion for FGM doubly-curved shell in the thermal environment according to the second approach (Type 2): the temperature is not only changes the material properties but also causes the initial stress before the shell oscillates Galerkin method and Newmark integration method are used to solve the differential equation system and determine deflection, stress, natural frequency and dynamic response of FGM doubly-curved shell with some popular boundary conditions 2.2 Functionally graded doubly-curved shell Consider a FG doubly-curved shallow shell (Figure 2.1) of constan thickness (h), the middle surface is x-y surface, and the z axis is perpendicular to the mid-surface The projection of shell in x-y plane is the rectangle with the dimensions in the x and y directions, a and b respectively; The principal radii of shell in the x and y directions are Rx and Ry, respectively y z MỈt trung b×nh x b a Ry Rx Figure 2.1 Geometrical shape and dimensions of FG doubly-curved shell 2.3 The assumptions The FG doubly-curved shell considered in this thesis is an open shell; the projection of shell in x-y plane is a rectangular shape that satisfies Mindlin's assumptions The steady heat conduction is considered Therefore, the thermal load is considered as static load The FG shells are subjected to transverse uniformly distributed mechanical load The structure is firstly subjected to static mechanical and temperature loads, then the dynamic load are applied The linear analysis is applied, so the displacements of the shell subjected to mechanical load and thermal load can be written as: u p ut u d Thus, the temperature is not only causes the change of the mechanical properties of the materials (E, ν) but also causes deformation and static initial stress After deformation, the shell will be oscillated about the new equilibrium position To investigate the effect of temperature distribution on the behavior of shells, the thesis proceeds with the following steps: - Step 1: Analyze static problems when the shell is subjected to the thermal load and the other static loads to predict the deflection and stress of the shell - Step 2: Stress calculated from the above static problem is considered as the pre-stress of the shell when the shell oscillates These pre-stressed components will therefore be taken into account when setting the dynamic problem equation of the shell 2.4 Static analysis of FG doubly-curved shells in thermal environment In this section, the first-order shear deformation Reissner-Mindlin theory is used to establish the constitutive equations for FG doubly-curved shells subjected to static mechanical loads and thermal load The displacement components, and static stress u t , t are determined 2.4.1 Displacement field Based on the first order shear deformation theory, the displacement field of the doublycurved shells is assumed as [88]: tt t t u x, y, z, t u0 x, y, t z x x, y, t v tt tt w x, y, z, t v0t x, y, t z yt x, y, t (2.2) x, y, z, t w0t x, y, t 2.4.2 Strain field Nonzero strain components developed from the displacement field given in Eq (2.2) are given as: t mu t t c (2.3) t with mu is the membrane - bending strains: u t wt xt x x Rx xxt t t yt t v0 w0 t mu yy y R z y y t xy t t t t u v y x 0 y x y x t t 0 x x t t 0 y z y t t xy xy (2.4) and ct is the shear strains: w0t v0t t y yzt y Ry t c t t t xz w0 u0 xt x R x (2.5) 2.4.3 Stress field For FG doubly-curved shells in thermal environment, the constitutive relations can be written as xxt Q11 t yy Q12 t xy t xzt yz Q12 Q22 0 0 Q66 0 0 Q44 0 xxt xxT( z ) t yy T( z ) yy t xy xzt Q55 yzt (2.7) 2.4.4 Internal force resultants The stress resultants are expressed in the form: N xxt A11 t N yy A21 N xyt t M xx B11 t M yy B21 M xyt t Qxz Qt yz A12 A22 B12 B22 0 0 A66 0 B66 0 B11 B21 D11 D21 0 B12 B22 D12 D22 0 0 B66 0 D66 0 0 0 0 A44 0 0xt N xxnd 0t y N yynd 0t xy xt M xxnd yt M yynd xyt xzt A55 yzt (2.14) 2.4.5 Equations of motion The principle of minimum total potential energy is used to derive the equations of motion of FG doubly-curved shells The principle can be stated in mathematical form as: Lt U t W t (2.15) in which: L , U , W are variations of the strain energy, of the potential energy of the external loads, and of the kinetic energy respectively Substituting the equations expressed strain energy, potential energy of the external loads, kinetic energy in to Eq (2.15) Using the generalized displacement–strain relations and stress–strain relations, and the fundamentals of calculus of variations and collecting the coefficients of u0t , v0t , w0t , xt , yt , the equations of motion of FG doubly-curved shells are obtained as: t t t t N xxt N xy Qxzt 0; u : x y Rx v : t t w0t : : t y t Nt Qxzt Qyz N xxt yy qzt x y Rx Ry M xyt x M yyt y N xyt x N yyt y Qyzt Ry 0 t M xxt M xy : Qxzt x y t x (2.18) Qyzt Now Eq (2.3) is substituted into Eq (2.14) to obtain internal force resultants in terms of kinematic displacements, which is further substituted in Eq (2.18) to give the equations of motion in term of displacements and rotations u t u0t , v0t , w0t , xt , yt 2.4.6 Analytical solution The analytical solutions for the present formulation are developed for different boundary condition Three following common boundary conditions are considered: Simply supported (SSSS): v0t w0t yt N xxt M xxt at x = 0, a t t u0t w0t xt N yy M yy at y = 0, b Simply supported - Clamped - Simply supported - Clamped (SCSC) : v0t w0t yt N xxt M xxt at x = 0, a u0t v0t w0t xt yt at y = 0, b (2.20) (2.21) Clamped (CCCC) : u0t v0t w0t xt yt at x = 0, a and y = 0, b (2.22) SSSS SCSC CCCC Fig 2.4 Three common boundary conditions The displacement fields satisfying the boundary conditions in terms of undetermined t t t t t coefficients (U mn , Vmn , Wmn , Xmn , Ymn ) and trigonometric functions are assumed as [69]: X m x Y y t Yn y ; v0t x, y Vmn X m x n x y m 1 n 1 m 1 n 1 X m x t t w0t x, y Wmn X m x Yn y ; xt x, y Xmn Yn y ; x m 1 n 1 m 1 n 1 Y y t yt x, y Ymn X m x n y m 1 n 1 t u0t x, y U mn (2.23) The assumed trigonometric functions X m x and Yn y functions satisfying the boundary conditions are broadly classified as SSSS, SCSC, and CCCC as listed in Table 2.1 [69] Transverse load and thermal load are also expanded in the form of Fourier series as follows: t qt x, y Qmn sin m x sin n y ; T z Tmn z sin m x sin n y m 1 n 1 (2.24) m 1 n 1 in which a b q x, y sin m x sin n y dxdy ; ab 0 0 a b Tmn z T z sin m x sin n y dxdy ab 0 0 t Qmn (2.25) By substituting equations (2.23) and (2.24) into the equations of motion in term of displacements Applying the Galerkin method, we obtained the system of algebraic equations for different boundary conditions in the following form: (2.27) K t F t F mn ch nd Solve the system of equations (2.27) with m, n = 1, , the five unknowns t t t t (U , Vmn , Wmn , Xmn , Ymn ) are determined Replacing these quantities into the Eq (2.23), we get t mn the displacement components ( u0t , v0t , w0t , xt , yt ) 2.5 Dynamic analysis of FGM doubly-curved shell in thermal environment From above mentioned, the displacements of the shell can be expressed in the form: p t u u u d Static displacement of the shell is determined through solving static problem Now, the equations of motion are developed and are solved to analyze dynamic response of the panel in the thermal environment according to the second approach (Type 2) According to this approach, stress components caused by temperature have been calculated in Section 2.4 and are considered as initial stresses when developing the equations of motion of the shell Hamilton’s principle is used herein to derive the equations of motion appropriate to the displacement field and the constitutive equations The principle can be stated in analytical form as: T T L dt T d d d W d dt (2.33) where U U in which U d is the strain energy due to additional stresses caused by dynamic loads U0 is the strain energy due to initial stress and can be written as: d d d d d d 2 w0 w0 w0 w0 U xx yy xy dzdxdy h x x y y A in which xx , 0yy , xy0 are stress components caused by thermal and static loads h2 (2.36) The work done by the external forces is calculated as the formula (2.17) in which dynamic load is considered as the applied load The kinetic energy is determined through the time derivatives of the displacement and density of the material Determining the equations of total strain energy External work, and kinetic energy into Eq (2.33), using the generalized displacement–strain relations, stress– strain relations, and the fundamentals of calculus of variations and collecting the coefficients of u0d , v0d , w 0d , xd , dy , the equations of motion are obtained as d N xxd N xy Qxzd 2u0d 2xd I0 I1 x y Rx t t 2v0d I0 I1 2y x y Ry t t N xyd N yyd Qyzd d d d d d d Qxzd Qyz N xxd N yy w0d w0d w0 w0 w0 d N xx0 2N N q I xy yy z x y Rx Ry x x y y t d M xxd M xy 2u0d 2xd d Qxz I1 I 2 x y t t d d M xy M yy 2 yd 2v0d d Qyz I1 I 2 x y t t (2.38) Substituting the displacement fields satisfying the boundary conditions in terms of d d d d d , Vmn , Wmn , Xmn , Ymn ) and trigonometric functions assumed in undetermined coefficients (U mn similar form as (2.23) Applying the Galerkin method (similar to the one presented in the static problem), from equation (2.38), we obtain the system of equations in the brief form as follows: M mnd K Kini mnd Fchd ( t ) with Kini is the stiffness matrix caused by initial stress 2.5.1 Free vibration analysis (2.45) From equation (2.45), set the load vector equal to zero, we get the equation of the FG doublyd it curved shallow shells for free-vibration analysis SetU mn e ; Vmnd t Vmn0 e i t ; t U mn d d Wmnd t Wmn0 e i t ; Xmn ei t ; Ymn ei t we have the equations of free vibration t Xmn t Ymn problem which can be written as standard eigenvalue matrix, with unknown displacements 0 0 0 as follows: mn U mn , Vmn , Wmn , Xmn , Ymn K K M 0 ini mn (2.51) The natural frequency of shell is determined from the equation: det K Kini M (2.52) Solve eigenvalue equations (2.52) we obtain natural frequencies mn ; the fundamental natural frequency is predicted by cb mn 2.5.2 Forced vibration analysis The equation (2.45) is the forced vibration equation of FG doubly-curved shallow shells in thermal environment without damping Effect of temperature and other static loads which considered in the static problem are expressed through the stiffness matrix [Kini] Therefore, the right side of the equation is dynamic load vector only When taking into account the damp, the equation (2.45) becomes: M mnd C mnd K Kini mnd F d t (2.54) where [C] is damping matrix [98]: C a1 M a2 K Kini (2.55) in which a1 and a2 are mass damping factors and Rayleigh damping factors, respectively Using the Newmark- integral method to solve the system of equations (2.54) obtained the time-shifting response of the FG doubly-curved shallow shells subjected to mechanical loads in the thermal environment 2.7 Validation examples of analytical solution Based on the present formulation, three Matlab’s codes for PC are built, including: ShellpanelStatic(GT) program: predict deflection and stress of the FG doubly-curved shallow shells subjected to mechanical loads in the thermal environment ShellpanelStatic(GT) program: determinate natural frequency and vibration mode of FG doubly-curved shallow shells in the thermal environment ShellpanelForcedvibration(GT) program: investigate dynamic response of FG doublycurved shallow shells in the thermal environment The above mentioned programs are implemented in steps as shown on the flowchart in Figure 2.6, Figure 2.7 and Figure 2.8 in the thesis Thus, in this chapter, the constituve relations, the equations of motion are summarized for the FG doubly-curved shells in thermal environment based on the first-order shear deformation theory Using Galerkin method and Newmark method, analytical solutions are developed to calculate deflection, stress, natural frequency, and to analyze dynamic response for doubly curved FG shell in thermal environment with various common boundary conditions Computer programs have been built for numerical investigations However, the limits of the analytical solutions can be pointed as following: - It is only applicable for some simple types of curved shell structure such as cylindrical shell, spherical shell and hyperbolic paraboloid shell - It is only applicable for few common boundary conditions The goal of chapter is developing finite element model for static and dynamic analysis of FGM shells with different shapes in thermal environment The finite element model will overcome the limitations of the analytical model 10 u' x' v' x' y' y' u' v' ' x' y' y' x' x' z' u' w' y' z' z' x' v' w' z' y' (3.16) These components can also be calculated in the general coordinate system through the following transfer matrix: u' u x' x v v x' x y y' y' u' v' y u v ' x' y' y' x' T x y T y x T y' z' v' w' yz v w x' z' x z z' y' z y w' u' w u x' z' x z (3.25) These strain components can be represented and calculated in natural coordinates through the Jacobean matrix of transformations as follows: dV dxdydz det J d d d (3.26) Combining the Eq (3.25), (3.30) and (3.31), we get the strain components in the general coordinate system as follows: Bi ui B ue where B B1 ue T (3.32) i 1 B2 u1T B8 is the deformation properties matrix, u2T u8T is the node displacement vector of the element 3.5.4 Stress field The stress components are calculated through the strain components as described in Eq (2.8) and (2.9), which can be written in reduced form as follows: ' D' ' nd (3.34) 3.5.5 Static analysis of FGM shell Apply the total potential energy minimization principle to establish a static equilibrium equation for the FGM shell subjected to mechanical load and thermal load as in Eq (2.16) The strain energy of element: 11 1 1 1 Ue T T T ue B T D' T B det J d d d ue 1 1 1 1 1 1 T T T ue B T D' nd det J d d d 1 1 1 (3.37) The work done by the external forces on element: We ue T 1 1 N 1 1 NB T A px p y H d d p z (3.39) Substituting Eq (3.37) and Eq (3.39) into the equation of the total potential energy Apply the total potential energy minimization principle and simplified, we get: Ke ue Fech Fend (3.40) By assembling the element matrices, the global equations of motion for the FGM shell subjected to mechanical and thermal loads can be obtained as: K u F ch F nd (3.44) Solve the system of equations (3.44) to get the nodal displacement vector of the structure, thereby calculating the stress components at any point in the shell 3.5.6 Dynamic analysis of FGM shell In this section, the equations of motion are derived using Hamilton’s principle for analyzing dynamic response for FG doubly-curved shell in thermal environment according to the second approach (Type 2) by the finite element method The total strain energy is expressed as: (3.45) U e U ed U e0 with U ed is the additional strain energy due to dynamic loads, 1 1 1 1 T T T T U d dV ue B Te D' Te B det J d d d ue Ve 1 1 1 d e (3.46) U e0 is the strain energy due to the initial thermal stresses d d d d w0 w0 w0 w0 dzdxdy U xx (3.47) yy xy h x y x y A 0 where, xx , yy , xy are the initial stress components caused by thermal and static loads, which are e h2 calculated in the static problem as above mentioned The kinetic energy of element: 1 1 1 T T Te ue N A N B N A N B det J d d d ue 1 1 1 (3.56) Substituting Eqs (3.39), (3.46), (3.49) and (3.56) into the Hamiltonian principle equation and simplified, we get the motion equation system of the element as follows: M e ue Ke Keg ue Pe (3.57) By assembling the element matrices, we get the global equations of motion of the shell: M u K K g u P (3.62) The equations (3.62) are the forced vibration equation without damping of the shell; When taking into account the damping effect, Eq (3.62) becomes: 12 M u C u K K g u P (3.63) with [C] is the damping matrix as shown in Chapter When damping and external load are neglected, we obtained the equations of motion for free vibration analysis: M u K K g u 0 (3.64) After imposing boundary conditions of the structure and solving Eqs (3.64), we get the natural frequency and the free vibration modes of the shell For forced vibration problem, we solve Eqs (3.63) by Newmark direct integration method and we will obtain dynamic response of FGM shell in thermal environment 3.6 Validation examples of finite element method Based on the theoretical formulation presented in Chapter 3, the author have built 03 computer programs based on finite element model using Matlab software to perform numerical investigations, including: ShellpanelStatic(PTHH) program: predicting deflection and stress of the FG doublycurved shallow shells subjected to mechanical loads in the thermal environment ShellpanelStatic(GT) program: determining frequency and vibration mode of the FG doubly-curved shallow shells in the thermal environment ShellpanelForcedvibration(GT) program: investigating dynamic response of the FG doubly-curved shallow shells in the thermal environment The flowcharts of each program are presented in Figure 3.5, Figure 3.6 and Figure 3.7 in the thesis As consequence, in this Chapter 3, the thesis has built a finite element model to analyze the FGM shell structures in the thermal environment: • The finite element model of the thesis allows calculations with various shaped shell structures (the shell surface is described by a mathematical function) • Considering the properties and initial stress caused by temperature affecting the dynamic behavior of the FGM shell • Computer programs that calculate deflection, stress, frequency and vibration mode, dynamic response of the FG doubly-curved shells in thermal environments have also been established In the next chapter, the thesis will examine the number and draw conclusions The dissertation has some publications related to the content of this chapter, including: (3), (5), (8) (in the "List papers of the author related to the thesis topic") CHAPTER NUMERCIAL RESULTS 4.1 Introduction In chapter and chapter analytical and finite element solutions are presented to analyze bending, and vibration behaviors of FGM doubly-curved shell in thermal environment In this chapter, using self-written Matlab’s codes the numerical investigations will be conducted to give the valuable notes for designers The developed finite element model can be applied for shell panels with various geometrical shaped as presented in Chapter However, in this section the author will investigate some common types of shells as shown in Table 4.1 13 Table 4.1 Surface equations and geometries of some types of shells Type of shell Surface equations z Cylindrical (CYL) Spherical (SPH) Hyperbolic paraboloid (HPR) Hypar (HYP) Conoid (CON) a x 2Rx 2 2Ry z 2Ry Rx ( Ry ) x 0a y b b y 2 a x 2Rx 2 2Ry b y 2 a z x 2Rx 2 Parameters Rx Ry x 0a y b b y 2 Rx Ry x 0a y b 4c z xy ab a a x 2 b b y 2 x z hl H h hl a y y b b x 0a y b Shape 14 Panel (P6) y k yb (kx = ky = 1.2) x z kx a 2 a a x 2 b b y 2 4.2 Validation examples The validation examples are conducted to verify the accuracy of the solution (Analytical solution and FEM) and the computer program The convergence of the FEM program is tested and mesh grid of 14 x 14 elements is selected to perform the numerical examples Material properties are shown in Table 4.2 Some validation results including: - The deflection of FGM doubly-curved shell (Al/ZrO2) subjected to uniformly distributed load: the results are compared with those of Kiani et al [56] - The deflection of FGM cylindrical shell (Al/ZrO2) subjected to thermal load: the results are compared with those of Kar and Panda [55] - The natural frequency of FG spherical shell (Si3N4/SUS304) with temperature-dependent properties (calculated according to the first approach – Type 1): the results are compared with those of Shen et al [93] - The natural frequency of FGM plate (Si3N4/SUS304) with temperature-dependent properties and take into account the initial deformation and stress caused by the temperature (calculated by the second approach – Type 2): the results are compared with those of Li et al [58] - The transient analysis of displacement of FGM plate (Al/ZrO2): the plot is compared with those of Reddy [85] 4.2.1 Example KC1 - The deflection of FGM doubly-curved shell subjected to mechanical load Table 4.3 Non-dimensional deflection w2 of FGM doubly-curved shell subjected to uniform distributed loads The power law index p Ratio (a/b; Rx/a; Model p=0 p = 0.5 p = 1.0 p = 2.0 p = 10 Ry/b; a/h) Kiani et al [56] 0.6107 0.7756 0.8689 0.9559 1.1133 Thesis (Analytical) 0.6118 0.7769 0.8704 0.9575 1.1152 (1; 5; 10; 5) 0.6079 0.7686 0.8596 0.9449 1.1038 Thesis (FEM) Difference (%) 0.46 0.90 1.07 1.15 0.85 Kiani et al [56] 1.4216 1.8016 2.0148 2.2102 2.5707 Thesis (Analytical) 1.4595 1.8516 2.0704 2.2688 2.6337 (0.5; -5; 5; 5) Thesis (FEM) 1.4355 1.8294 2.0493 2.2476 2.6008 Difference (%) 2.60 2.70 2.69 2.58 2.39 4.2.2 Example KC2 – The deflection of FGM cylindrical shell subjected to thermal load Table 4.4 Non-dimensional deflection w of FGM cylindrical shell subjected to thermal load Temperature ΔT = 100 K ΔT = 200 K ΔT = 300 K Model Kar and Panda [55] 0.0048 0.0096 0.0144 Thesis (Analytical) 0.0048 0.0097 0.0145 Thesis (FEM) 0.0048 0.0097 0.0145 Difference (%)* 0.00 0.01 0.00 15 4.2.3 Example KC3 - The natural frequency of FGM doubly-curved shell Table 4.5 Dimensionless fundamental frequency of simply supported FGM doubly-curved shell versus the different power law index p Temperature Model Tc=500 K, Tm = 300 K Shen et al [93] Thesis (Analytical) Thesis (FEM) Difference (%) p=0 12.5478 12.8004 12.8055 2.01 The power law index p p = 0.5 p=1 p=2 8.6898 7.6262 6.8414 8.9214 7.7881 6.8991 8.9250 7.7912 6.9018 2.64 2.12 0.88 p=5 6.2032 6.1688 6.1713 0.52 4.2.4 Example KC4 - The natural frequency of FGM plate Table 4.6 Natural frequency parameters * for SSSS rectangular FGM plates subjected to linear temperature rise (ΔT = 300K) Ratio h/b Ratio a/b 0.5 0.2 1.0 Model Li et al [58] Thesis (Analytical) Thesis (FEM) Li et al [58] Thesis (Analytical) Thesis (FEM) p=1 5.3755 5.1955 5.3366 2.4526 2.4098 2.4438 The power law index p p=2 p=5 4.7955 4.3277 4.6531 4.2158 4.7803 4.3336 2.1893 1.9780 2.1616 1.9623 2.1910 1.9880 p = 10 4.1280 4.0168 4.1289 1.8890 1.8695 1.8931 4.2.5 Example KC5 – The dynamic response of FGM plate Figure 4.1 Dynamic response of simply supported FGM square plates subjected to a suddenly applied lateral load: central deflection versus time; 4.3 Static problem In this section, the effects of the power law index p, the side-to-thickness ratio of a/h, the law of heat transfer through the thickness and boundary conditions on the deflection and stresses of the FGM shells are investigated (as shown in Table 4.1) 4.3.1 Example 4.1 – The effect of power law index p a The effect of power law index p on the deflection 16 Comment: For isotropic material (p = 0) cylindrical shells (CYL), spherical (SPH), hyperbolic paraboloid (HPR) and hypar (HYP) have the largest central deflection, then the deflection decrease with increasing value of power law index p Whereas for CON and P6, the deflection increases significantly of when the volume index p increases in the range from to 0.5, then the deflection decreases with increasing value of power law index p Figure 4.2 The effect of power law index p on the central deflection of FG shell b The effect of power law index p on the stress components The variations of stresses through the thickness for different values of power law index p that performed by analytical solutions and FEM for types of shells are plotted in Figure 4.3 to Figure 4.5 In which, stress components are calculated at the following points: σxx (a/2; b/2; z); σyy (a/2; b/2; z); τxy (a; b; z) Figure 4.3 The effect of power law index p to stress xx [N/m2] Figure 4.4 The effect of power law index p to stress yy [N/m2] Figure 4.5 The effect of power law index p to stress xy [N/m2] 17 Comment: The graphs on the figures from 4.3 to 4.5 show that when p changes, the volume fraction of FGM constituents also change (the content of ceramic/metal content), caused the different distributions of stresses through the shell thickness When the power law index p = 0, the distribution of stress is linear through the thickness of shell, and the stresses reached to extremes at the upper and lower surfaces of the shell, this due to the shell is made of isotropic material With non-zero values of p, the distributions of stresses are nonlinear through the shell thickness, extreme stress may not reach at the top surface which may occur at any point depending on the shape of shell 4.3.2 Example 4.2 – The effect of a/h ratio Comment: Table 4.9 and Figure 4.6 show that: Figure 4.6 The effect of a/h ratio to deflection of FGM shell - When the ratio a/h increases gradually from 10 to 50, the deflection of CYL, SHP and P6 shells developed in the positive z-direction (curvature direction of the shell) Whereas the deflection of HPR, HYP and CON shells develop in the opposite direction (negative z-direction) It can be seen that the shape and ratio a/h (thickness of the shell) will make the forceresistance capacity of shell change when the shell is subjected to mechanical and thermal loads 4.3.3 Example 4.3 – The effect of heat transfer law according to shell thickness Comment: Figure 4.7 shows that: - For CYL, SPH, CON and P6 shells tend to deflect on the upper side when subjected to simultaneous effects of mechanical and thermal loads, whereas HPR and HYP shells tend to deflect downwards - For each type of different shell, when the different law of heat transfer according to thickness will give a different behavior of deflection, as follows: + CYL shell has the smallest deflection under uniform temperature rise; the deflection is Figure 4.7 The effect of heat transfer law to maximum with a linear temperature change across deflection of FGM shell the thickness + SPH shell the deflection has the smallest deflection with a nonlinear temperature change across the thickness and maximum deflection with a linear temperature change + HPR and HYP have the largest deflection under uniform temperature rise; Linear and nonlinear heat transfer for approximate deflection + CON and P6 shell have the smallest deflection when heat transfer is nonlinear; the uniform temperature rise give the biggest deflection 4.4 Free vibration problem Firstly, natural frequencies of SPH shell are calculated according to two above mentioned approaches and are compared with each other Then, the effect of the various parameters on natural frequencies of FGM shell in the thermal environment according to the second approach (Type 2) is investigated by both of analytical and finite element methods 4.41 Example 4.5 - Comparison of results calculated by two approaches (type1 and type 2) 18 Table 4.10 Comparison of natural frequency Ω1 of FGM shell calculated by two approaches ΔT (K) Heat transfer law Approach 100 200 300 400 500 Type 7.4482 7.3595 7.2442 7.0992 6.9201 Uniform temperature Type 7.0627 6.5242 5.8869 5.1336 4.2295 Difference (%) 5.18 11.35 18.74 27.69 38.88 Type 7.5032 7.4910 7.4757 7.4572 7.4355 Linear temperature Type 7.5074 7.4962 7.4784 7.4539 7.4224 Difference (%) 0.06 0.07 0.04 0.04 0.18 Type 7.4878 7.4640 7.4404 7.4167 7.3924 Type 7.4048 7.2922 7.1740 7.0498 6.9191 Nonlinear temperature Difference (%) 1.11 2.30 3.58 4.95 6.40 Comment: There is significant difference of the fundamental frequencies of spherical shell which are calculated by the first approach (Type 1) and the second approach (Type 2), the difference increases when the temperature changes increase When the temperature distribution is assumed to be uniform or nonlinear through the thickness, the fundamental frequency calculated by the second approach (Type 2) is significantly smaller than that calculated by the first approach (Type 1) Whereas, this difference is smaller when the temperature distribution is assumed to be linear through the shell thickness 4.4.2 Example 4.6 – Effect of power law index p, heat transfer laws and boundary condition Figure 4.10 Effect of heat transfer law on natural frequency Ω1 of FGM shell (SSSS) Figure 4.12 Effect of heat transfer law on natural frequency Ω1 of FGM shell (CCCC) 4.4.5 Example 4.9 – Effect of a/h ratio Figure 4.11 Effect of heat transfer law on natural frequency Ω1 of FGM shell (SCSC) Comment: The results show that as well as neglect the effect of temperature, the stiffness of FGM shells in the thermal environment is also significant influenced by the boundary conditions When the shell has more and more restraints, it becomes more stiffness, it can be seen that the fundamental frequency of shells with all edge in clamped conditions (CCCC) are always the largest Next is the frequency of twosided clamped, two-sided support (SCSC) and finally, the fundamental frequency of shells in the case of simply supported boundary condition (SSSS) is the smallest 19 Comment: When the ratio a/h increases, the shells become thinner and therefore the stiffness of the shell decreases As consequence, the natural frequency Ω2 of all shells decrease when the ratio a/h increases Figure 4.16 Effect of a/h ratio on natural frequency Ω2 of FGM shell 4.4.6 Example 4.10 – Effect of temperature changes Comment: The natural frequency of all shells decreases as the temperature difference (ΔT (K)) between two surfaces of shell increases This means that when the temperature difference (ΔT (K)) increases, the stiffness of shell will be decrease Figure 4.17 Effect of temperature (ΔT (K)) on natural frequency Ω1 of FGM shell (Nonlinear temperature) 4.4.7 Example 4.11 – Some free vibration modes of FGM shell In this section, the thesis investigates and calculates the five lowest natural frequencies of HYP, CON and P6 shells These are the shell types that are rarely mentioned Table 4.16 The five lowest natural frequencies (Ω1) of FGM shells Natural frequency Ω1 (x103) Boundary Type of shell condition ω1 ω2 ω3 ω4 ω5 SSSS 2.0341 2.0345 3.6399 4.3951 4.6808 HYP (c/a = 0.1) CCCC 4.3774 4.7996 4.8006 6.4161 6.7996 SSSS 2.0727 2.1043 3.2307 3.7603 4.6034 CON (a/Hh = 1; hl = 0.5Hh) CCCC 3.7003 3.7668 5.0569 5.4650 6.3319 SSSS 0.9266 2.1012 2.1013 4.0451 4.6699 P6 (kx = ky = 1.2) CCCC 1.5332 3.4303 3.4307 5.3815 6.4276 20 Comment: P6 shell has the second frequency equal to the third frequency, respectively; the second and third free vibration modes (mode (2, 1) and mode (1, 2)) are symmetrical For HYP and CON shells, there is no symmetric vibration and equal frequencies Figure 4.18 Some free vibration modes of FGM shell 4.5 Forced vibration In this section, the effect of material parameters, geometric dimensions, temperature and damping factor on displacement response of FGM shells will be examined Two types of impact load are considered: suddenly uniform load and harmonic load Load has the form: F(t) = P0.P(t) • Case 1: the shell subjected to suddenly uniform load: P(t) = {1, ≤ t}; • Case 2: the shell subjected to harmonic load: P(t) = sin (Ωt) in which, P0 is the amplitude of the forced force; Ω is the frequency of the forced force The displacement response is calculated at the center point of FGM shell: Km (a/2, b/2) 4.5.1 Example 4.12 - Case of FGM shell under suddenly applied uniform load Effect of power law index p, ratio of a/h, temperature difference ΔT (K) and damping factor on displacement response of FGM shells are presented in the figures from Figure 4.19 to Figure 4.22 Comment: As the power law index p increases, the period and amplitude of vibration of the shells increase This is perfectly consistent with the results in the free-vibration section of FGM shell, when the volume index p increases, the frequency decreases and so the period vibration increases that means when the power law index p increases, the Figure 4.19 Effect of power law index p on displacement stiffness of P-FGM shell is small response of FGM shells 21 Comment: The amplitude, frequency and period of vibration of the shell are not linearly proportional to the ratio of a/h of FGM shell Specifically, when the ratio a/h is small, the vibration amplitude of the shell is very small, and this amplitude increases rapidly when a/h ratio increases from 10 to 20 and 30 Figure 4.20 Effect of a/h ratio on displacement response of FGM shells Figure 4.21 Effect of temperature difference (ΔT (K)) on displacement response of FGM shells Comment: In the survey shells, the CYL shell has the largest difference in period and amplitude of vibration when ΔT increases from 100K to 300K, followed by HPR shell and the other shells have a smaller effect Thus, the shell has large curvature, the effect of temperature on dynamic response is greater than that of non-curved or small curvature Comment: The figures show that when the damping factor is small (0.01), the vibration of shell has also decreased quite quickly, for the case of damping factor is 0.07, the vibration of shell is quickly turned off This can be understood that under normal working conditions, the vibrations of shells will also turn off quickly due to the effect of stiffness and weight resistance of shell Figure 4.22 Effect of damping factor on displacement response of FGM shells 4.5.2 Example 4.13 - Case of FGM shell subjected to harmonic load Effect of power law index p, ratio of a/h, temperature difference ΔT (K) and ratio of natural frequency/forced force frequency (ratio Ω/ω) on displacement response of FGM shells are presented in the figures from Figure 4.19 to Figure 4.22 Comment: The power law index p causes the stiffness of shell to change, thus, natural frequency of shell changing When the natural frequency of shell is different from the frequency of the external load, the shell will fluctuate according to the external load Beat and resonance phenomenon occur when the natural frequency of shell is 22 Figure 4.23 Effect of power law index p Figure 4.24 Effect of a/h ratio close to and equal to the frequency of the forced force Comment: When the ratio a/h changes, the vibration amplitude of shell changes quickly, this once again confirms that the ratio a/h is a parameter which has greatly effect on the force-resistance capacity of FGM shells On the other hand, the thickness of shell is a very important factor for determining the stiffness of shell Comment: The effect of temperature on the stiffness of shell is very significant because in each specific case of size and material, temperature difference ΔT can cause the shell to fluctuate in harmonics, vibrating in beat form or resonance Figure 4.25 Effect of temperature difference ΔT (K) Figure 4.26 Effect of ratio of natural frequency/forced force frequency (ratio Ω/ω) Comment: When the shell subjects to harmonic load with a frequency far different from its natural frequency, the shell will vibrate according to the external load, the magnitude of the vibration amplitude depends on the shell stiffness and the amplitude of harmonic load When the frequency of harmonic load is close to the natural frequency of shell, it is the beat phenomenon (the boundary of the amplitude is also in the Sin figure form) and when the frequency of harmonic load is equal to the natural frequency of shell, resonance will be occur (vibration amplitude gradually increases to infinity) In the scope of the study, the thesis also investigates the dynamic response of conoid shell (CON) when it subjected to forced load with the frequency equal or near to its three lowest natural frequencies The results are shown in Figure 4.27 and Figure 4.28 with two cases of boundary conditions: CCCC and SSSS 23 Comment: The dynamic response of the conoid shell (CON) in Fig 4.27 shows that resonance and beat phenomenon occur when the frequency of the forced force is equal to and approximate to any natural frequency of shell Figure 4.27 Resonance and beat phenomenon of conoid shell (CON) occurs with the first three frequencies (SSSS boundary condition) Figure 4.28 Resonance and beat phenomenon of conoid shell (CON) occurs with the first three frequencies (CCCC boundary condition) Conclusion of the chapter: based on self-written Matlab programs, the following numerical examples are performed: - Investigate the variation of deflection and stress components through the shell thickness coordinates - Evaluate the effect of material parameters, geometric dimensions, temperature field and boundary conditions on deflection, stress components for some types of FGM shells subjected to uniform distributed mechanical load in thermal environment; - Evaluate the effect of material parameters, geometric dimensions, temperature and boundary conditions on fundamental frequency of FGM shell - Investigate the effect of material parameters, geometric dimensions, temperature, damping factor, ratio of natural frequency/forced force frequency (ratio Ω/ω) on dynamic response of FGM doubly-curved shell under suddenly applied uniform load and harmonic load - From the numerical examples, the useful comments and conclusions are drawn 24 CONCLUSION The new contributions of thesis 1) The constitutive equations for static, free vibration and forced vibration problems of FGM doubly-curved shells in thermal environment has been developed with some common boundary conditions In the dynamic problem, the shell is regarded as having the initial stress caused by the temperature 2) The degenerate 3D shell element is effectively applied to build the algorithm and finite element model for the shells have various geometrical shapes which are described by a mathematical function 3) Using self-written Matlab code to validate the reliability of algorithm and proposed computational model; Perform the numerical investigations to evaluate the effect of material parameters, geometric dimensions, boundary conditions; law of heat transfer through the thickness; damping factor; ratio of natural frequency/forced force frequency (ratio Ω/ω) on the deflection, stress field, natural frequency and transient analysis of displacement of FGM doubly-curved shell in the thermal environment Since, draw some significant scientific conclusions that help engineers to choose the suitable structural parameters in design The main results of dissertation are new, and published in 08 domestic and international papers including 01 Scopus paper The others papers are published in national scientific proceeding Further researches 1) Study nonlinear static, buckling and vibration behaviors of FG shells based on higherorder shear deformation theory 2) Study FG shells stiffened by stiffeners under mechanical and thermal loads using shear deformation theories 3) Analyze the static, buckling and vibration behaviors of FG shell with complicated geometric shapes working on thermal environment ... (2017), Phân tích Panel trụ FGM chịu uốn có xét đến ảnh hưởng nhiệt độ - Lời giải giải tích Lời giải số, Tạp chí Khoa học Cơng nghệ Xây dựng, ISSN: 1859-2996, Tập 11 số 2, trang 38 – 46 Duong Thanh... trang 580-587 Trần Minh Tú, Trần Hữu Quốc, Dương Thành Huân (2015), Phân tích tĩnh động Panel trụ làm vật liệu có tính biến thiên (FGM) theo lý thuyết biến dạng cắt bậc (FSDT), Tuyển tập cơng trình... graded material (FGM) - the mechanical properties of materials; FGM structures and applications; Overview of national and international researches on static and dynamic analysis of FGM shell structures
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