Research Article | Open Access

Volume 2018 |Article ID 2804123 | https://doi.org/10.1155/2018/2804123

Aref Mehditabar, Gholam H. Rahimi, Kerameat Malekzadeh Fard, "Thermoelastic Analysis of Rotating Functionally Graded Truncated Conical Shell by the Methods of Polynomial Based Differential Quadrature and Fourier Expansion-Based Differential Quadrature", Mathematical Problems in Engineering, vol. 2018, Article ID 2804123, 19 pages, 2018. https://doi.org/10.1155/2018/2804123

# Thermoelastic Analysis of Rotating Functionally Graded Truncated Conical Shell by the Methods of Polynomial Based Differential Quadrature and Fourier Expansion-Based Differential Quadrature

Revised18 Feb 2018
Accepted28 Feb 2018
Published20 May 2018

#### Abstract

This paper focuses on the three-dimensional (3D) asymmetric problem of functionally graded (FG) truncated conical shell subjected to thermal field and inertia force due to the rotating part. The FG properties are assumed to be varied along the thickness according to power law distribution, whereas Poisson’s ratio is assumed to be constant. On the basis of 3D Green-Lagrange theory in general curvilinear coordinate, the fundamental equations are formulated and then two versions of differential quadrature method (DQM) including polynomial based differential quadrature (PDQ) and Fourier expansion-based differential quadrature (FDQ) are applied to discretize the resulting differential equations. The reliability of the present approach is validated by comparing with known literature where good agreement is reached using considerably few grid points. The effects of different mechanical boundary conditions, temperature fields, rotating angular speed, and shell thickness on the distributions of stress components and displacement in thickness direction for both axisymmetric and asymmetric cases are graphically depicted.

#### 1. Introduction

In this paper 3D asymmetric thermoelastic analysis of rotating FG truncated conical shell under temperature gradient through thickness and centrifugal load duo to rotation of the shell is investigated. The variations of material properties are assumed to be varied along the thickness according to power law rule, whereas Poisson’s ratio is supposed to be constant. Due to intrinsic complexity of existing equations, two versions of DQM technique including PDQ and FDQ in conjunction with Newton-Raphson iteration technique are employed to obtain the numerical results. The validity of the present approach is proved by comparing with that published literature that the good consistency using considerably few grid points is achieved. In this study, the sensitivities of displacement and stress components to different values of thermal loading, thickness of shell, rotating angular speed, and boundary conditions under both axisymmetric and asymmetric loading conditions are graphically plotted and discussed.

#### 2. Governing Equations

The FG truncated conical shell with the origin of the coordinate at vertex of whole cone, as illustrated in Figure 1, is considered. According to Figure 1, the set of orthogonal curvilinear coordinates , , and lie on meridian, circumferential, and thickness directions, respectively. The thickness of the shell is defined as , , and indicate the inner and outer radii of the cone at its small and large edge, respectively, denotes distance from the vertex to its small edge, L is length of the cone along the generator, determines the semivertex angle, and is rotating angular speed. Moreover, , , and indicate the total centrifugal force and its components along and directions, respectively.

In this study, the elastic modulus, heat conductivity and coefficients of thermal expansion are assumed to be varied only along the thickness direction according to the selected power law function as follows :

, , and are the elastic modulus, coefficient of thermal expansion, and thermal conductivity at the inner surface of shell, respectively, and also represents the FG in-homogeneity constant. Consider , , and as corresponding displacement components in s, , and , respectively; the relationships between strain and displacement based on 3D elasticity formulations in the conical coordinate are expressed as where .

The stress-strain relation according to constitutive equations for isotropic material in matrix form is defined as follows:where and indicate the normal and shear stress components, respectively, and denotes the strain components, represents the temperature distribution through thickness obtained from the steady-state heat conduction equation, and also is the coefficient of thermal expansion. The elastic coefficients for isotropic materials in terms of elastic stiffness and Poisson’s ratio are expressed as follows:

Assuming quasistatic state, the acceleration terms can be eliminated from the equilibrium equations . With this assumption, the equilibrium equations in , , and directions in the presence of body force are given in (5a)–(5c), respectively, asCentrifugal body force components in designated conical coordinate knowing that the cone radius at any point along its generator is are calculated by the following formulations:

Using (2), (3), and (6) and substituting them into (5a)–(5c), the following equilibrium equations in terms of displacement components and temperature in the three directions of , , and are derived in (7a)–(7c), respectively, as follows:The steady-state heat conduction equation without heat generation in the conical coordinate can be written asThe mathematical expressions of mechanical and thermal boundary conditions for the two considered cases including clamped and simply supported at top and bottom of truncated conical shell are described as follows:For both axisymmetric and asymmetric cases, the inner surface of conical shell is kept at zero and the outer surface is varied at constant temperature . Thus, the following mechanical and thermal boundary conditions must be satisfied for the inner and outer surfaces:

#### 3. The Method of Solution

The DQ method as powerful discretization tool is applied to transform a system of resulting differential equations to a system of algebraic equations. The general rule of DQM postulates that derivative of any continuous function with respect to space variable at any discrete point in the domain can be expressed as a linear weighted linear summation of all the functional values in the whole domain as follows :where , , and denote the weighting coefficients for the mth-order derivatives of the function in thickness direction at mesh point , along the generator at mesh point and circumferential direction at mesh point , respectively. Furthermore, , , and are the numbers of sampling points along the s, θ, and , respectively. In this study, polynomial differential quadrature (PDQ) is employed to determine the weighting coefficients along and directions and the Fourier expansion-based differential quadrature (FDQ) is used to compute the weighting coefficient along the circumferential direction. The weighting coefficients are calculated according to following explicit formulations :Using (11) and applying relations in (12a) and (12b), the DQ analogs of the governing, related boundary conditions, and heat transfer equations are achieved. The discretized forms of equations using DQ technique are given in detail in Appendix. The system of algebraic equations discretized by DQM is effectively solved based on the semismooth Newton-Raphson iteration procedure as follows. Consider the vector of unknown variables and the resulting governing and boundary equations : The system of equation with unknown variable vector can be solved using the following Newton-Raphson iteration scheme:where the applied notation kth in the definition of Newton technique is iteration number. It should be noted that the solution starts from . In (14), is successively replaced by previous iteration and this procedure continues until difference between two successive iterations equals designated tolerance or .

#### 4. Numerical Results and Discussion

The coordinates of the grid points are chosen based on the Chebyshev-Gauss-Lobatto relation as follows :

In the s-direction,In the -direction,In the –direction,The geometrical properties of truncated conical shell are considered to be , , , and . The truncated conical shell is assumed to be made up of aluminum with the following material properties :Here, the presented results are based on the nondimensional forms of thickness displacement, temperature, and stresses as defined in the following:In order to illustrate the convergence and accuracy of presented method, the thermoelastic problem of thick-walled FG spherical shell carried out by Eslami et al.  is chosen to validate the results obtained by the present approach. The problem considered was the thermoelastic analysis of 1D thick-walled FG spherical shell. Reference  assumed the material properties except Poisson’s ratio to be varied along the thickness according to following power law rule:where , , and are the modulus of elasticity, the coefficient of thermal expansion, and thermal conduction coefficient at inner radius, respectively, and is the in-homogeneity constant. The inner and outer radii were assumed to be and , respectively. The used material properties were  GPa and °C and also the thermal and mechanical boundary conditions were supposed to be °C, °C,MPa, and . Furthermore, for comparison, the values of circumferential stress and radial displacement for are selected. Figure 2(a) shows the variations of nondimensional values of circumferential stress along radial direction computed by present method and . It is observed from Figure 2(a) that the present numerical technique yields convergence at and also the converged results are in an excellent agreement with those obtained by . For more validation of the present approach, the variations of dimensionless radial displacement along thickness calculated by DQM and with those achieved by  are demonstrated in Figure 2(b). According to Figure 2(b), for some values of , the dimensionless radial displacement predicted by present approach is little higher than the values obtained by  and with other ranges of is smaller. It can be concluded from Figure 2(b) that the good consistency (maximum discrepancy of 2.5%) between present numerical technique and  is depicted that confirms the efficiency and reliability of the present numerical method. For more assurance of the accuracy of present method, another pressure vessel component under thermomechanical loading modelled by  is performed in Figures 3(a) and 3(b).

A more convergence of the method is investigated by obtaining dimensionless radial displacement for the thermoelastic behavior of thick-walled FG cylindrical shell carried out by . The same geometrical and material properties and also thermal and mechanical boundary conditions used in  were applied in . Table 1 is based on the behavior of FG cylindrical shell at the inner wall. According to the presented results in Table 1, for dimensionless radial displacement and for FG in-homogeneity constant , in (20), it is obvious that the converged results are achieved by . Moreover, for more certainty of applicability of the used numerical technique in the present paper, another comparison is performed for the thermoelastic analysis of thick-walled FG cylindrical shell carried out by . As one can see from Figures 3(a) and 3(b), the excellent agreements and inconsiderable discrepancy exist between present approach and .

 The number of grid points Present approach Present approach Present approach Jabbari et al.  0.00119 0.0011225 0.00124 0.00124
##### 4.1. Axisymmetric Problem

In this section, numerical results for axisymmetric case of FG truncated conical shell are calculated at the midpoint and graphically presented and discussed in Figures (4)–(6). Figures 4(a)4(f) demonstrate the effects of in-homogeneity constant , on the thickness distribution of the displacement component in thickness direction, stress components, and temperature field. FG truncated conical shell rotates with angular rotating speed  rad/s and is subjected to temperature gradient through thickness with the inner surface temperature of zero and the outer surface temperature of = 100°C. It can be observed from Figure 4(a) that the values of normalized thickness stress increase with changing of power law index from 1 to 3. It is also found that the prescribed mechanical boundary conditions of the problem at the inner and outer surfaces are satisfied. Moreover, the position that reaches the maximum value is closer to the outer surface at . It can be also found from Figure 4(b) that changing power law index from 1 to 3 causes increase of the values of dimensionless circumferential stress . It is interesting to note that the influences of are more pronounced on the regions closer to outer surface than those closer to the inner surface. Note that reaches maximum value at .

Figure 4(c) illustrates that the absolute values of increase as varies from 1 to 3. It is seen from Figure 4(d) that the absolute values of normalized shear stress with concave profile like in Figure 4(a) increase by increasing from 1 to 3. It is interesting to note that achieves its maximum values at . Moreover, from Figure 4(e) it is obvious that the amplitudes of dimensionless displacement increase through thickness due to increasing of . Temperature distribution along thickness for various is plotted in Figure 4(f). It can be inferred that the absolute values of temperature monotonically enhance with increasing power law index . The influences of temperature gradient through thickness on the thickness distributions of dimensionless stress component and thickness displacement are illustrated in Figures 5(a)5(e). It is assumed that FG truncated conical shell rotates with rotating angular speed  rad/s and also is held at zero and is varied. As Figure 5(a) reveals, the amplitudes of dimensionless thickness stress increase as temperature gradient through thickness is elevated and also thickness component of stress is tensile throughout the thickness. It is observed in Figure 5(b) that as temperature gradient through thickness rises, the absolute values of the circumferential stress increase that these increases are more obvious for the regions closer to the outer periphery of FG tube. It must be mentioned that maximum value of occurs at the inner wall. In Figure 5(c) same trend of Figure 4(b) is observed for the stress component in generator direction which means enhancing temperature gradient through thickness augments the compressive stresses. It is readily seen from Figure 5(d) that normalized shear stress is slightly affected by thermal load which can be neglected. It can be inferred from Figures 5(c) and 5(d) that the maximum value of occurs at middle of the thickness, whereas attains maximum value at the outer wall of the thickness. Figure 5(e) reveals that the greater the temperature gradient through thickness is, the greater the displacement is obtained.

Figures 6(a)6(e) illustrate the individual effects of two mechanical boundary conditions including clamped and simply supported on the parameters discussed in Figures 5(a)5(e). In this section, FG truncated conical shell rotates at the constant rotating angular speed  rad/s and the values of temperature at the inner and outer surfaces of the shell are = 0°C and = 100°C, respectively. It is viewed from Figure 6(a) that the effects of two boundary conditions on dimensionless stress component in thickness direction are small in the regions closer to the inner surface, whereas will be more pronounced in the regions closer to the outer surface. It is noteworthy to mention that the maximum value is shifted to the position closer to the outer surface. Figure 6(b) shows that the boundary conditions of clamped create higher values of circumferential stress than those of simply supported. It can be noted that the maximum value of for both mechanical boundary conditions occurs at the inner wall of the shell. The effects of two prescribed edge boundary conditions on thickness variation of are presented in Figure 6(c). It can be inferred from Figure 6(c) that changing boundary conditions from clamped to simply supported reduces the absolute values of dimensionless stress .

It should be noted that both boundary conditions under consideration in this paper exhibit symmetric response with respect to middle of the thickness. According to Figures 6(d) and 6(e), changing the boundary conditions from clamped to simply supported causes sharp reduction in the amplitudes of dimensionless shear stress and thickness displacement .

##### 4.2. Asymmetric Problem

For the asymmetric analysis, it is assumed that the FG truncated conical shell is under cosine type of internal pressure; that is,  MPa. In this section, the effects of the different angular velocity of ω, temperature gradient through thickness, and also the thickness of the shell on the stress components and thickness displacement for the three-dimensional problem of FG truncated conical shell are illustrated in Figures 7, 8, and 9, respectively. It must be noted that, in this section, geometrical parameters are given as in the previous section and all the circumferential variations of stress components and thickness displacement are calculated at location of and .

Figures 7(a)7(e) demonstrate the distributions of dimensionless stress components and thickness displacement through circumferential direction with various rotating angular speed . Thermal boundary conditions for the inner and outer surfaces are assumed to be = 0°C and = 100°C, respectively. From the plotted results, it can be well understood that as the angular velocity ω is augmented and centrifugal force is enhanced, the magnitudes of stress components become intensified, while increasing dimensionless shear stress is almost negligible and can be disregarded. Figures 8(a)8(e) reflect the sensitivities of dimensionless stress components and thickness displacement to temperature gradient through thickness. The rotating angular speed is assumed to be  rad/s and the temperature of inner surface is kept at zero and also the outer surface is varied. As noticed in Figures 8(a)8(e), increasing the temperature gradient through thickness causes increase of the absolute values of normalized thickness stress and thickness displacement along the circumferential direction, whereas it causes reduction in the absolute values of circumferential stress and stress component along the generator. Moreover, we can observe that thermal loading negligibly affects the dimensionless shear stress and can be disregarded.

In Figures 9(a)9(e), circumferential distributions of the normalized stress components and thickness displacement are computed for different values of the shell thickness with conditions of  rad/s, = 0°C, and = 100°C. It is obvious from Figures 8(a)8(e) that as the shell becomes thicker, the values of the stress components and displacement are reduced. It is also seen that the discrepancy of dimensionless shear stress for = 0.2, 0.3, and 0.4  is negligible. In addition, the shear stress varies in parabolic manner for = 0.2, 0,3 and 0.4 , while it is distributed almost constant along the circumferential direction when equals 0.5 .

#### 5. Conclusion

In this research, the thermoelastic behavior of FG rotating truncated conical shell under the individual effects of rotating angular speed, thermal loading, mechanical edge boundary conditions, and shell thickness were investigated. The material properties of FG were assumed to be graded in the thickness according to designated power law distribution, while Poisson’s ratio was considered to be constant. Two versions of DQM including PDQ and FDQ were utilized to discretize and convert the resulting differential equations to the algebraic equations and then Newton-Raphson iteratively computed the numerical results. High rate of convergence and accuracy of the presented computational procedure were confirmed by comparing with those achieved by available literature. Some important conclusions based on the obtained results can be inferred as follows:(A)Axisymmetric loading and boundary conditions:(1)The absolute values of dimensionless stress components, thickness displacement, and temperature significantly increased as in-homogeneity constant changed from 1 to 3. It was also concluded that, by controlling FG in-homogeneity constant , the values of stresses can be reduced and, consequently, contribute to lessen the failure-susceptibility and progressive rupture in components.(2)As temperature gradient through thickness increased, the magnitudes of thickness stress and displacement were augmented.(3)Changing the boundary conditions at edge from the clamped to simply supported caused reduction of the values of stress components and displacement.(B)Asymmetric loading and boundary conditions:(1)Stress components and displacement values increased as rotating angular speed increased.(2)Increasing temperature gradient though thickness caused increase in the thickness stress and thickness displacement, whereas the circumferential variations of hoop stress and also stress in generator direction decreased.(3)It can be inferred from the plotted results that the shell thickness has pronounced effects on the stresses and thickness displacement that, by increasing the shell thickness, the amplitudes of the thermoelastic parameters decreased.

#### Appendix

DQ form of (7a) is as follows:Applied DQ method (7b) is discretized asDQ form of (7c) is as follows:Discretized form of boundary conditions is as follows: