Caspian Journal of Applied Sciences Research, 2(4), pp. 17-23, 2013 Available online at http://www.cjasr.com ISSN: 2251-9114, ©2012 CJASR
Full Length Research Paper Application of Differential Transformation Method (DTM) for bending Analysis of Functionally Graded Circular Plates Somayeh Abbasi1, Fatemeh Farhatnia2, Saeid Rasouli Jazi3 1,2,3
Department of Mechanical Engineering, Faculty of Engineering, Islamic Azad University, Khomeinishahr branch, Isfahan, Iran *Corresponding Author:
[email protected] Received 24 November 2012; Accepted 21 February 2013
In this paper, based on classical plate theory, circular functionally graded plates (FGPs) are investigated using transformation method (DTM). The Poisson’s ratios of the FGM plates are assumed to be constant whereas modulus of elasticity defined in power-law form. The material properties vary arbitrarily along the thickness of the plate. In this study, DTM as relatively new techniques is implemented to investigate the influence of the material gradients, transverse loading and boundary conditions on static response of FG circular plates. The plate is exposed to uniform, linear and parabolic transverse loading. The derived formulation in this paper has capability for analyzing the bending behavior of circular plates, in case of transverse symmetrical loadings with any arbitrary distribution. Some numerical examples are given and comparisons are made between present results and the known results in the literature.The numerical results demonstrate that DTM is quite accurate and efficient for this kind of problems. Key words: Circular plate, Bending analysis, Differential transformation method.
and metal as the constituent of FGM that prevents failure when the structure exposed to thermal and mechanical loadings. It has many application areas such as thermal barrier coatings for turbine blades (electricity production), armor protection for military applications, fusion energy devices, and biomedical materials including bone and dental implants, space/aerospace industries, automotive applications, avoid corrosion resistance, fatigue, fracture and stress corrosion cracking (Inala and Mohanty, 2012). In the following, we review some relevant works concerned to the present subject. The axisymmetric bending of FG circular plates based on first shear deformation Mindlin plate theory was studied by Reddy et al., (1999). Yun et al. (2010) investigated the axisymmetric bending of transversely isotropic and functionally graded circular plates subject to arbitrarily transverse loads using the direct displacement based on threedimensional theory. They expanded the transverse loading in the Fourier-Bessel series. Gandhi et al. (1985) studied the nonlinear axisymmetric static analysis of elastic orthotropic thin circular plates with elastically restrained edges for rotation and in-plane displacement. They presented numerical
1. INTRODUCTION Plates are one of the important structures in modern engineering applications. They are widely used in marines, mechanical and civil engineering. Investigation and analysis of this kind of structures is therefore continuously interest to the designers and engineers. Materials can be categorized as homogenous and non-homogenous materials. For instance, the concrete material is nonhomogeneous one whether with or without presence of fibers (Haido, 2012). Also, FGMs are categorized as non-homogenous materials, too. This kind of materials is a new class of composite ones that was introduced for the first time by a group of Japanese scientists in the mid-1980s (Koizumi, 1997). FGMs could overcome to difficulties that arise when traditional composite materials exposed to extremely high temperature gradients. The main problem was the weakness of interfaces between adjacent layers in extreme high temperature gradient that leads to fail. Typically, FGMs are composed of mixture of ceramics that provides the high thermal barriers when the structure exposed to high temperature gradients
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Abbasi et al. Application of Differential Transformation Method (DTM) for bending Analysis of Functionally Graded Circular Plates
results for orthotropic plates with different boundary conditions. A closed form solutions was proposed for axisymmetric bending of circular plates having non-linear variable thickness based on two independent integrals of the hypergeometric differential equation describing the rotation field, by Vivio and Vullio, (2010). They evaluated the elastic stresses and deformations according to a power of a linear function, subjected to symmetrical bending due to lateral loads either distributed on upper surface or distributed along the inner or the outer edges. They compared their results with Finite element analysis. Chen (2012) presented an innovative technique for nonlinear differential equation of bending problem. In his study he combined the state-space method with DQ method. Sahraee and Saidi (2009) studied the bending and buckling of the thick FG plates based on unconstrained-order shear deformation plate theory. They considered the effect of material distributions through the thickness on maximum displacement and critical buckling load under axisymmetric loading. In this study, we employ differential transformation method (DTM) for bending analysis of functionally circular plates under clamped and simply supported edges. Differential transformation method is an iterative procedure that based on Taylor series expansion. Whereas it is not known method for analyzing the bending response of plates, the purpose of this paper is to demonstrate the capability and the computational efficiency of the proposed method. Zhou, (1986) proposed for the first time the differential transforms for solving linear and nonlinear initial value problems in electric circuit analysis. Afterwards, it was applied for solving some kinds of differential equations, for instance in integrodifferential equations (Arikoglu et al., 2005) and nonlinear and linear partial differential equations (Ayaz, 2004). (Arikoglu et al, 2010) studied the vibration of a three layered composite beam with a viscoelastic core. He solved the governing differential equations of motion by using differential transform method (DTM) in the frequency domain. He observed the results obtained using DTM are a good agreement with the findings of previous studies. This study will focus on considering DT method as reliable, simple and accurate one to analyze the deflection of FG circular plate under two kinds (uniform and linear) of transverse loading. Firstly, we described the differential transformation method and governing operation, as
well as to investigate the influence of FG volume fraction index and boundary conditions on deflection distribution. The resulting equations are employed to obtain the closed form solution for the bending of FG circular plate. 2. SOLUTION METHOD 2.1. Definition and operation of differential transform method (DTM) The differential transform method is a numerical method based on the Taylor series expansion that proposes the solution in the form of polynomials. The differential transform of the kth derivative of the function f(r) and differential inverse transform of F[k] are defined as follows (Ayaz, 2004): (1) Function f(r) is considered analytic in a domain R and r=r0 represent any point in R. It is represented by a power series whose center is located at r0. The fundamental mathematics operation of DTM can be found in Table 1. 2.2. The formulation of the problem and utilization of DTM In this section, we apply DT method on governing equation of bending of circular plate under transverse loading. Based on classical plate theory (CPT), the bending differential equation governing on circular plate can be given as follows (Venstel and Krauthammer, 2001): (2) where r ,w, n, D is radius of plate, out-plane displacement, loading power index ( n=0 for uniform and n=1 for linear loading) and flexural rigidity of plate, respectively. P is applied pressure on surface of the circular plate. Using the following non-dimensional parameters and employing into the differential Eq. 2, it can be rewritten as follows: (3)
where,
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Caspian Journal of Applied Sciences Research, 2(4), pp. 17-23, 2013
Table 1: Fundamental Theorems of DTM (Kaya, 2006) Original Function
Transformed Function
f (r) y(r) z(r)
F k Y k Z k
f r = y r
F k Y k
f r y r .z r
F k
k
Y[k ] Z[k k ] 1
1
k1 0
f r
dm y r dr
F k
m
m k ! k!
Y[k m]
0 if k n F k k n 1 if k n
f (r) rn
functionally graded material is assumed to be in terms of a simple power law distribution and Poisson’s ratio is assumed to be constant as follows (Najafizadeh et al., 2002):
In above relations, D is elastic flexural rigidity of FG plate that can be defined as follows (Venstel et al. 2001):
(5)
(4)
where Em and Ec are elasticity modulus of metal and ceramic (aluminum/alumina), respectively and g is volume fraction index, in which (g=0) represents a fully ceramic plate. Taking the differential transformation of Eq. 3, we exploited the following relation:
Whereas it is assumed that deformation is small in this paper, the in-plane displacement is ignored and only out-plane one, w, is calculated. In this paper, modulus of elasticity for
(6)
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Abbasi et al. Application of Differential Transformation Method (DTM) for bending Analysis of Functionally Graded Circular Plates
The following boundary conditions are considered for clamped and simply supported at the edges of the plate, whereas using differential transformation method into boundary conditions, we get:
(14)
For n=1:
(7)
-Simply Supported edge:
(15)
(8) For n=2: In both boundary conditions in case of axisymmetric bending of circular plate, the regularity condition leads to:
(16)
(9) 3. RESULTS AND DISCUSSION
By utilizing the appropriate theorems of DT method (see Table 1) and simplified form of the Eq. 6, we give the following recurrence relation:
The computer package Matlab is employed to code the expressions obtained by using DTM, to calculate the out-plane displacement, w(r) and sketch the relevant graphics. In the following figures, the constituent materials considered are Alumina (as ceramic) and Aluminium (as metal). The modulus of elasticity of Aluminium and Alumina are Em =70 GPa and Ec= 380 GPa, respectively and P=0.1MPa. The plate thickness is h=10 mm and the radius is l=0.6m. In this paper, we use convergence test to confirm 4 terms is sufficient to get precise value for determining the deflection of circular plate. As it is expected, by increasing g as FG power index, the elastic flexural rigidity reduces, consequently deflection increases too. Switching the boundary condition from simply supported to clamped edge, leads to make more resistance of plate against transverse loading, as a result the deflection decreases. Also, for n=1(linear distributed loading) and n=2 (parabolic distributed loading), the central deflection decreases, whereas the resultant transverse load decreases too. In order to validate the computed results, they are compared by Reddy et al., (1999) in Table 2. The values are taken as the same reference.
(10)
Finally the deflection equation is obtained for n=0, n=1 and n=2, respectively, as follows:
(11)
(12)
(13) W[0] and W[2] can be exploited from Eqs.(79) for n=0 n=1 and n=2, as follows: For n=0:
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Caspian Journal of Applied Sciences Research, 2(4), pp. 17-23, 2013
Fig.1: non-dimensional deflection of FG circular plates under uniform transverse loading (n=0) for (a) Clamped and (b) simply supported boundary conditions
Fig.2: non-dimensional deflection of FG circular plates under linear transverse loading (n=1) for (a) Clamped and (b) simply supported boundary conditions
Table 2: Comparison of present results with Reddy et al. (1999) and Li et al. (2008) P=1MPa ,l=0.1m ,h=0.03m ,ν=0.288 ,Em=110.25GPa ,Ec=278.41GPa. of FG circular plate for Clamped B.C.
g
0
2
10
Present
2.525
1.3882
1.1427
Reddy
2.525
1.388
1.143
Li
2.525
1.388
1.143
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Abbasi et al. Application of Differential Transformation Method (DTM) for bending Analysis of Functionally Graded Circular Plates
Fig.3: non-dimensional deflection of FG circular plates under linear transverse loading (n=2) for (a) Clamped and (b) simply supported boundary conditions core by using differential transform method. Comput. Struct. 92: 3031–3039. Ayaz F (2004). Solutions of the system of differential equations by differential transform method. Appl. Math. Comput. 147: 547–567. Chen Y Z (2012). Innovative iteration technique for nonlinear ordinary differential equations of large deflection problem of circular plates, Mech. Res. Commun. 43: 75-79. Gandhi M L, Dumir P C, Nath Y (1985). Nonlinear axisymmetric static analysis of orthotropic thin circular plates with elastically restrained edge, Comput. Struct.20: 841-853. Haido J H (2012). Prediction of static behavior for SFRC deep beams using new and simple nonlinear models. Caspian J. Appl. Sci. Res. 1:1-26. Inala R, Mohanty S C (2012). A Review on Free, Forced Vibration Analysis and Dynamic Stability of Ordinary and Functionally Grade Material Plates. Caspian J. Appl. Sci. Res. 1: 57-70. Koizumi M (1997). FGM activities in Japan. Compos. Part B: Eng. 28: 1–4. Kaya M (2006). Free vibration analysis of a rotating Timoshenko beam by differential transform method, Aircr. Eng. Aerosp Technol.: Int. J.78:194–203. Li X Y, Ding H J, Chen W.Q (2008). Elasticity solutions for a transversely isotropic functionally graded circular plate subject to
Table (2) presents a comparison study of the nondimensional deflection for simply supported and clamped from FG circular plate. It is obvious that the results of the present method are in excellent agreement with the results of Reddy et al. (1999) and Li et al. (2008). 4. CONCLUSION This present analysis exhibits the applicability of the differential transformation method to solve the governing differential equations of plate under transverse uniform and linear loadings. The results of this method are in good agreement with those obtained in the literature. It was showed that DTM is a reliable technique to handle differential equations governing on static analysis. REFERENCES Abrate S (2006). Free vibration, buckling and static deflections of functionally graded plates. Compos. Sci. Technol. 66: 2383–2394. Arikoglu A, Özkol I (2005). Solution of boundary value problems for integro-differential equations by using differential transform method. Appl. Math. Comput. 168:11451158. Arikoglu A, Ozkol I (2010). Vibration analysis of composite sandwich beams with viscoelastic
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Venstel V, Krauthammer T (2001). Thin plates and shells:Theory, Analysis and Applications, Marcel Dekker, Inc., New York. Vivio F, Vullo V (2010). Closed form solutions of axisymmetric bending of circular plates having non-linear variable thickness Int. J. Mech. Sci. 52:1234–1252. Wang Y, Rongqiao X, Haojiang D (2010). Threedimensional solution of axisymmetric bending of functionally graded circular plates, Compos. Struct. 92: 1683–1693. Yun W , Rongqiao X, Haojiang D,(2010). Threedimensional solution of axisymmetric bending of functionally graded circular plates, Comput. Struct. 92 (7):1683-1693. Zhou J K (1986). Differential transformation and its applications for electrical circuits, in Chinese, Huarjung University Press, Wuuhahn, China.
an axisymmetric transverse load , Int. J. Solids Struct. 45: 191-210. Najafzadeh M M, Eslami M R (2002). Buckling analysis of circular plates of functionally graded materials under uniform radial compression, Theor. Appl. Mech. A/Solids 44: 2479–2493. Reddy J N, Wang C.M, Kiti pornchai S (1999). Axisymmetric bending of functionally graded circular and annular plate, Theor. Appl. Mech. A/Solids. 18: 185–199. Sahraee S, Saidi A R (2009). Axisymmetric bending analysis of thick functionally graded circular plates using fourth-order shear deformation theory, Theor. Appl. Mech. A/Solids 28(5): 974-984
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