- Email: [email protected]
Linear static analysis of Functionally Graded Plate using Spline Finite Strip Method
Composite Structures 117 (2014) 309–315
Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Linear static analysis of Functionally Graded Plate using Spline Finite Strip Method K.P. Beena ⇑, U. Parvathy College of Engineering, Trivandrum, India
a r t i c l e
i n f o
Article history: Available online 10 July 2014 Keywords: Functionally Graded Plates Classical Plate Theory Spline Finite Strip Method Idealisation techniques
a b s t r a c t Spline Finite Strip Method has been found to be very efficient in the analysis of plates, shells and stiffened plates/shells for their linear static, linear stability, non-linear static, non-linear instability and vibration behaviour. The present work explores the extension of the Spline Finite Strip Method to the analysis of functionally graded material (FGM) plates. Here the modulus of elasticity of FGM plate varies along the thickness direction and the variation is idealised by power, sigmoid and exponential functions. The Poisson’s ratio of the FGM plate is assumed to be constant throughout the thickness direction. The analysis is done for moderately thin plates subjected to uniformly distributed, central concentrated and line loads and the deflections and stresses are obtained using the Classical Plate Theory. A comparative study of the three idealisation techniques is also done. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Functionally graded materials (FGM) are the advanced materials in the family of engineering composites which are developed with a view to tailor the material architecture at microscopic scales. They are made of two or more component characterised by a compositional gradient from one component to the other throughout the thickness. FGM’s were initially developed in the late 1980’s for use in high temperature applications by a group of Japanese scientists as ultrahigh temperature resistant materials for aircraft, space vehicles and other engineering applications. These advanced materials with engineering gradients of composition, structure and/or specific properties in the preferred direction/orientation are superior to homogeneous material composed of similar constituents [1,7]. The concept of FGM, initially developed for super heat resistant materials to be used in space planes or nuclear fusion reactors, is now of interest to designers of functional materials for energy conversion, dental and orthopedic implants, sensors and thermo-generators and wear resistant coatings. An FGM can be prepared by continuously changing the constituents of multi-phase materials in a pre-determined volume fraction of the constituent material. Due to the continuous change in material properties of an FGM, the interfaces between two materials disappear but the characteristics of two or more materials of the composite are preserved. Subsequently the stress singularity at ⇑ Corresponding author. Tel.: +91 9645923362. E-mail address: [email protected] (K.P. Beena). http://dx.doi.org/10.1016/j.compstruct.2014.07.002 0263-8223/Ó 2014 Elsevier Ltd. All rights reserved.
the interface of a composite can be eliminated and thus the bonding strength is enhanced [2,3]. In view of the wide material variations and applications of FGMs, many research works have already been done for the bending [2,3,5] and buckling analysis [4,6,8–15]. FGMs have heterogeneous microstructure with material properties varying smoothly and continuously in preferred direction. To simplify the complicated heterogeneous microstructure of FGM, different homogenisation schemes are already developed. Power law function (PFGM), exponential function (EFGM) and sigmoid function (SFGM) are widely used to describe the variation of material properties of FGM. Various well established theories are available for the analysis of isotropic and composite plates which can be extended to FGM plates also. In this work, the Spline Finite Strip Method is used for the bending analysis of FGM plates using Classical Plate Theory. The three idealisation techniques power-law, sigmoid and exponential functions are used to show the variation of Young’s modulus along the thickness direction. The Poisson’s ratio is assumed to be a constant in all directions. The material properties are varied continuously in the thickness direction according to the volume fraction of the constituent materials. The results are validated using the closed form solution developed by Chi and Chung [2]. 2. Spline Finite Strip Method Although closed form analytical method may be possible in simple cases of idealised structure and loading, various numerical approaches like FEM, CFSM, SFSM, etc. are usually resorted to
310
K.P. Beena, U. Parvathy / Composite Structures 117 (2014) 309–315
complex systems and loading conditions. The Finite Element Method (FEM) has been extensively used for the analysis of plated structures. The computational requirement of FEM, in terms of storage space and time is very high, especially in linear prismatic members wherein some of the elements have small width. Hence, this method has only limited application in the stability and nonlinear analysis of linear prismatic members modelled using plates and shells, especially when iterative non-linear analysis is needed, as in optimum design. The Classical Finite Strip Method (CFSM), on the other hand, allows more efficient modelling of such prismatic members using strips as elements along the length of the member. This method works well for simple boundary conditions (simply supported, clamped, etc.), but fails to effectively deal with complex boundary conditions and partial and concentrated loads, since the trigonometric functions used to model displacements in the longitudinal direction are infinitely continuous. The continuity and discontinuity requirements can be satisfied by replacing the classical trigonometric function by a spline function as is done in Spline Finite Strip Method (SFSM). The Spline Finite Strip Method has been found to be very efficient in the analysis of plates, shells and stiffened plates/shells for their linear static, linear stability, non-linear static, non-linear instability and vibration behaviour [17–19]. The spline function is defined as a piecewise polynomial of nth degree which is smoothly connected to the adjoining spline functions which has n-1 continuous derivatives. There is variety of splines namely natural spline, cardinal spline, basic spline, etc. B3 spline (cubic basic) is most common and is continuous over only four consecutive sections. The B3 spline series is a piecewise cubic polynomial, which is an ideal approximation of the bending behaviour. Another property of B3 spline is its localised behaviour that makes the stiffness matrix highly banded. Owing to this property, incorporating the boundary conditions is easy, and only three splines adjacent to the constraint need to be modified. Equal and unequal spaced spline series have been used by many researchers [16–19] to analyse thin and thick plate structures. The unequal splines are more efficient when the structure is subjected to concentrated loads and reactions, when the support of members are either isolated or at irregular locations and when cut-outs are present. Hence here unequal B3 splines are used to model the variation of deflection in the longitudinal direction. 3. Problem formulation The functionally graded material (FGM) can be produced by continuously varying the constituents of multi-phase materials in a predetermined profile. The most distinct features of an FGM are the non-uniform microstructures with continuously graded macro properties. An FGM can be defined by varying the volume fractions using the power-law, exponential or sigmoid function. In the rectangular FGM plate shown in Fig. 1, coordinates x and y define the plane of the plate, and the z-axis originated at the middle surface of the plate is in the thickness direction. The material properties, Young’s modulus and the Poisson’s ratio, on the upper and lower surfaces are different but are pre-assigned according to the performance demands and the Young’s modulus and Poisson’s ratio of the plates vary continuously only in the thickness direction (z-axis) i.e., E = E(z), m = m(z). Delale and Erdogan [19] indicated that the effect of Poisson’s ratio on the deformation is much less than that of Young’s modulus. Hence here Poisson’s ratio of the plate is assumed to be a constant. and the variation of Young’s moduli is idealised using any one of the three functions i.e. power-law functions (P-FGM), exponential functions (E-FGM), or sigmoid functions (S-FGM).
Fig. 1. The geometry of an FGM Plate.
3.1. Power law idealisation The volume fraction of the P-FGM is assumed to obey a power law function:
gðzÞ ¼
p z þ h=2 h
ð1Þ
where p is the material parameter and h is the thickness of the plate, the material properties of a P-FGM can be determined by the rule-of-mixture as
EðzÞ ¼ gðzÞE1 þ ½1 gðzÞE2
ð2Þ
where E1 and E2 are the Young’s moduli of the lowest (z = h/2) and top surfaces (z = h/2) of the FGM plate, respectively. 3.2. Sigmoid idealisation The material property variation across the thickness of the FGM plate using the sigmoid function is defined using two power-law functions to ensure smooth distribution of stresses among all the interfaces. The two power-law functions are defined by,
p 1 h=2 z for 0 6 z 6 h=2 2 h=2 p 1 h=2 þ z for h=2 6 z 6 0 g 2 ðzÞ ¼ 1 2 h=2
g 1 ðzÞ ¼
ð3Þ ð4Þ
Thus the Young’s modulus of the SFGM can be calculated by;
EðzÞ ¼ g 1 ðzÞE1 þ ½1 g 1 ðzÞE2 for 0 6 z 6 h=2
ð5Þ
EðzÞ ¼ g 2 ðzÞE1 þ ½1 g 2 ðzÞE2 for h=2 6 z 6 0
ð6Þ
3.3. Exponential idealisation The exponential function used to describe the material properties of FGMs is
Fig. 2. Configuration of a simply supported square FGM plate.
311
K.P. Beena, U. Parvathy / Composite Structures 117 (2014) 309–315 Table 1 Dimensionless maximum deflection (w/h) of simply supported FGM plate for various loadings (E1/E2 = 3 and p = 2). Type of load
PFGM
SFGM
EFGM
SFSM
Chi and Chung [2]
SFSM
Chi and Chung [2]
SFSM
Chi and Chung [2]
Load 1 Load 2 Load 3
0.247 0.703 0.410
0.247 0.706 0.410
0.227 0.646 0.377
0.227 0.649 0.377
0.230 0.656 0.382
0.230 0.659 0.382
Table 2 Dimensionless tensile stress (rx/q0) at the centre of simply supported FGM plate for various loadings (E1/E2 = 3 and p = 2). Type of load
Load 1 Load 2 Load 3
PFGM
SFGM
EFGM
SFSM
Chi and Chung [2]
SFSM
Chi and Chung [2]
SFSM
Chi and Chung [2]
1051.10 8374.44 2853.92
1057.71 8370.23 2855.23
953.68 7588.82 2588.65
955.72 7592.34 2591.72
1009.55 8002.92 2732.44
1010.23 8007.65 2738.43
Fig. 3. Dimensionless deflection (w/h) of simply supported PFGM, SFGM and EFGM plates under uniform load along the width of plate for various E1/E2.
EðzÞ ¼ AeBðzþh=2Þ where A ¼ E2
1 E1 and B ¼ ln h E2
ð7Þ ð8Þ
3.4. Stiffness matrix formulation using Classical Plate Theory The Classical Plate Theory is extended for the analysis of FGM Plates. Here both transverse shear and transverse normal stresses are neglected. The deformation is entirely due to bending and inplane stretching. The normal stresses rx, ry and shear stress sxy acting in the XY plane are derived. The stress resultants are
obtained by integrating stress along the thickness. Thus the axial forces and the bending moments are obtained in terms of coefficients Aij, Bij and Cij. 9 2 8 A11 > > < Nx = 6 Ny ¼ 4 A12 > > ; : N xy 0
A12 A11 0
9 2 38 0 > B11 = < ex0 > 7 e 6 0 5 þ 4 B12 y0 > ; :c > A66 0 xy0
B12 B11 0
38 @ 2 w 9 0 > @x2 > > > = < 2 7 0 5 @@yw2 > > > > B66 : 2 @ 2 w ; @x@y
8 9 2 9 2 38 38 @ 2 w 9 2 > B11 B12 0 > C 11 C 12 0 > > > < Mx > < ex0 > < @x2 > = = = 6 7 6 7 M y ¼ 4 B12 B11 0 5 ey0 þ 4 C 12 C 11 0 5 @@yw2 > > > > > > : :c ; > ; > Mxy 0 0 B66 0 0 C 66 : 2 @ 2 w ; xy0 @x@y
ð9Þ
ð10Þ
312
K.P. Beena, U. Parvathy / Composite Structures 117 (2014) 309–315
Fig. 4. Dimensionless stress (rx/q0) of simply supported PFGM, SFGM and EFGM plates under uniform load across the thickness direction for varying E1/E2.
Fig. 5. Comparison of dimensionless maximum deflection of PFGM, SFGM and EFGM for (a) uniformly distributed load (b) central concentrated load (c) line load.
K.P. Beena, U. Parvathy / Composite Structures 117 (2014) 309–315
Fig. 6. Comparison of dimensionless maximum tensile stress of PFGM, SFGM and EFGM for (a) uniformly distributed load (b) concentrated load (c) line load.
Fig. 7. Comparison of dimensionless deflection (w/h) and dimensionless stress (rx/q0) of simply supported PFGM plate for E1/E2 = 10.
Fig. 8. Comparison of dimensionless deflection (w/h) and dimensionless stress (rx/q0) of simply supported SFGM plate for E1/E2 = 10.
313
314
K.P. Beena, U. Parvathy / Composite Structures 117 (2014) 309–315
Fig. 9. Comparison of dimensionless deflection (w/h) and dimensionless stress (rx/q0) of simply supported EFGM plate for E1/E2 = 10.
They are the stiffness matrix coefficients which are obtained by the integration of material properties of the FGM plate and are defined as shown:
A11 ¼
Z
h=2
h=2
A66 ¼
Z
h=2
h=2
B11 ¼ B66 ¼
Z Z
h=2 h=2 h=2 h=2
C 11 ¼
Z
h=2
h=2
C 66 ¼
Z
h=2
h=2
EðzÞ 1 tðzÞ
2
dz;
A12 ¼
Z
h=2
EðzÞtðzÞ
h=2
1 tðzÞ2
1 tðzÞ dz 2 1 tðzÞ2
dz; 4.1. Effect of ‘E1/E2’ ratio on deflections and stresses of FGM Plate
EðzÞ
ð11Þ
Z h=2 zEðzÞtðzÞ dz; B ¼ dz; 12 2 2 1 tðzÞ h=2 1 tðzÞ zEðzÞ 1 tðzÞ dz 2 1 tðzÞ2 zEðzÞ
z2 EðzÞ 1 tðzÞ
dz; 2
C 12 ¼
Z
h=2 h=2
1 tðzÞ dz 2 1 tðzÞ
It is seen that the SFSM results on FGM plate under simply supported boundary conditions agree very well with the exact value obtained from closed form solutions. The largest error obtained is less than 0.5% for all the three idealisation techniques.
z2 EðzÞtðzÞ 1 tðzÞ2
dz;
z2 EðzÞ
2
These stiffness matrix coefficients are used to derive the equilibrium equations which in turn gives the deflection and stresses of FGM plate. 4. Results and discussions A simply supported square FGM plate as shown in Fig. 2 with width to thickness ratio equal to 50 is analysed for various E1/E2 ratios and p values. The Poisson’s ratio of the FGM plate is assumed to be constant throughout the whole plate as t = 0.3. The Young’s modulus at any point on the FGM plate varies continuously in the thickness direction based on the volume fraction of the constituents. The Young’s modulus at the bottom surface of the FGM plate, E1 is kept as a constant as 2.1 106 kg/cm2. The Young’s modulus at the top surface of the FGM plate E2 varies with the ratio of E1/E2. The analysis has been carried out for uniformly distributed load, central concentrated load and line loads. Due to the symmetry about the x- and y-axes, only one quarter of the full plate is considered for analysis. The quarter plate is divided into 5 strips in the longitudinal direction and 5 nodal lines in the transverse direction. The plate is analysed for uniformly distributed load (Load 1, q0 = 1 kg/cm2), concentrated load (Load 2, P = 10,000 kg) and line load (Load 3, P0 = 100 kg/cm) with simply supported boundary conditions. The deflections and stresses are obtained for all the three idealisation techniques, considering equal spacing for the knots and nodal lines. The values of dimensionless maximum deflection and dimensionless tensile stresses at the centre of FGM plate for the three loadings are validated in Tables 1 and 2.
The effect of E1/E2 ratio on PFGM, SFGM and EFGM plates subjected to three loading conditions are analysed. The dimensionless deflections along the x-direction and dimensionless stresses across the thickness direction at the centre of FGM plate are plotted graphically for each loading pattern for various E1/E2 ratios. A comparison between the three idealisation techniques was done for the dimensionless maximum deflection and dimensionless maximum tensile stress for all the loadings. 4.1.1. Square plate subjected to uniformly distributed load First, consider that the FGM plate is subjected to a uniform load with the magnitude q0 = 1 kg/cm2. The dimensionless deflections and dimensionless stresses acting on the FGM plate under the action of uniform load is evaluated using the three idealisation techniques for different E1/E2 values, assuming the material parameter ‘p’ as 2. The results are plotted in Figs. 3 and 4. 4.1.2. Comparison of PFGM, SFGM and EFGM plate for various ‘E1/E2’ ratios The dimensionless deflections and dimensionless stresses of the FGM plate using power-law, sigmoid and exponential (PFGM, SFGM and EFGM) functions are compared for various E1/E2 values. Here the deflection refers to the maximum deflection acting at the centre of the plate and the stress refers to the maximum tensile stress acting at the bottom most surface of the FGM plate. The three idealisation techniques are thus compared for the three loading cases. The comparison of maximum deflection and maximum tensile stresses of PFGM, SFGM and EFGM plates for the three loadings are plotted graphically as shown in Figs. 5 and 6. From the graphs, it can be observed that the pattern of deflection and stresses for all the three types of loading for various ‘E1/ E2’ ratio is same. As E1/E2 increases, the deflection and stresses of the EFGM plate increases than that of PFGM as well as SFGM plate. This is because of the exponential function used in EFGM plate to describe the variation of Young’s modulus throughout the thickness direction. The SFGM plate undergoes less deflection and stress for all the loading cases. 4.2. Comparison of uniformly distributed, concentrated and line loads on simply supported FGM plate The deflections and stresses are obtained for a square plate with E1/E2 = 10 and material parameter p = 2. For comparison a uniform
K.P. Beena, U. Parvathy / Composite Structures 117 (2014) 309–315
load of 1 kg/cm2 and a line load of 100 kg/cm are applied which are equivalent to a concentrated load of 10,000 kg. The dimensionless deflection and dimensionless stresses for the PFGM, SFGM and EFGM plate subjected to uniform, concentrated and line loads are plotted as shown in Figs. 7–9. It is shown in Figs. 7–9 that the three curves corresponding to uniform, point and line loads intersect at the point of z = 0.0625 h, which is the location of neutral surface of the FGM plate. The position of the neutral surface of the Functionally Graded Plate remains same for the three loading conditions. 5. Conclusion In this study, the bending analysis of the Functionally Graded Plate was carried out using power-law, sigmoidal and exponential function for simply supported boundary conditions. The analysis was done for thin and moderately thick plates using Classical Plate Theory. The Spline Finite Strip Method was implemented in Visual C++ programming language. Deflection and stress values obtained from Spline Finite Strip Method were validated using the closed form Fourier series solution developed by Chi and Chung [2] for simply supported Functionally Graded Plate. The results lead to the following conclusions: 1. The more the E1/E2, the larger the deflection ‘w’ and larger the tensile stress ‘rx’. This is because of the decrease in stiffness of the FGM plate for larger E1/E2. It was observed that the compressive stress does not increase with increase in E1/E2. 2. A rapid increase in deflection and tensile stress occurs in the case of EFGM plates for E1/E2 > 10. This is due to the exponential function used to define the variation of Young’s modulus in the EFGM plates. 3. The simply supported SFGM plates under all the three loadings show only a gradual increase in deflection compared to PFGM and EFGM plates. This is due to the use of two power-law functions to define the variation of Young’s modulus across the thickness direction. Thus a smooth distribution of stresses was ensured. 4. The stress at the centre of PFGM, SFGM and EFGM plate for all the three loading conditions was found to be maximum at the bottom edge (z = h/2) of the plate. This maximum stress which was located at the bottom surface was found to be tensile in nature. 5. The maximum tensile stress increases with increase in ‘E1/E2’ ratio. The compressive stress is maximum at the top surface (z = h/2) for small ‘E1/E2’. As the ratio of E1/E2 increases, the maximum compressive stress moves towards the inner side of the plate. 6. The neutral surface of PFGM, SFGM and EFGM plates under uniformly distributed, central concentrated and line loads get shifted towards the positive z-direction with increase in E1/E2.
315
Moreover, it was observed that the position of the neutral surface remain same for FG plate under all the three loadings as well as under different idealisation techniques for a constant E1/E2 ratio. Thus the position of neutral surface of the Functionally Graded Plate depends on the ratio of ‘E1/E2’ and is independent of the loading conditions.
References [1] Jha DK. Tarun Kant, Singh RK. A critical review of recent research on functionally graded plates. Composite Structures 2013;96:833–49. [2] Chi Shyang-Ho, Chung Yen-Ling. Mechanical behaviour of functionally graded material plates under transverse load-Part I, analysis. Int J Solids Struct 2006;43(13):3657–74. [3] Chi Shyang-Ho, Chung Yen-Ling. Mechanical behaviour of functionally graded material plates under transverse load-Part II, numerical results. Int J Solids Struct 2006;43(13):3675–91. [4] Mokhtar Bouazza, Abedlouahed Tounsi, El Abbas Adda Bedia, Abdelkader Megueni. Buckling analysis of functionally graded plates with simply supported edges. Leonardo J Sci 2009;15:21–32. [5] Kashtalyan M. Three dimensional elasticity solution for bending of functionally graded rectangular plates. Eur J Mech Solids 2004;23:853–64. [6] Reddy JN. Mechanics of laminated composite plates. New York: CRC Press; 1997. [7] Abrate Serge. Functionally graded plates behave like homogenous plates. Compos B Eng 2008;39(1):151–8. [8] Na Kyung-Su, Kim Ji-Hwan. Three-dimensional thermal buckling analysis of functionally graded materials. Compos B Eng 2004;35:429–43. [9] Thai Huu-Tai, Choi Dong-Ho. An efficient and simple refined theory for buckling analysis of functionally graded plates. Appl Math Model 2012;36:1008–22. [10] Samsam Shariat BA, Javaheri R, Eslami MR. Buckling of imperfect functionally graded plates under in-plane compressive loading. Thin Walled Struct 2005;43:1020–36. [11] Samsam Shariat BA, Eslami MR. Buckling of thick functionally graded plates under mechanical and thermal loads. Compos Struct 2007;78:433–9. [12] Majid Badiey, Kouchakzadeh MA. Buckling of a functionally graded plate (FGP) under shear and in-plane directional loading. In: 27th International congress of the aeronautical sciences, 2010. [13] Wu Tsung-Lin, Shukla KK, Huang Jin H. Post-buckling analysis of functionally graded rectangular plates. Compos Struct 2007;81:1–10. [14] Wang S, Dawe DJ. Spline finite strip analysis of the buckling and vibration of composite prismatic plate structures. Int J Mech Sci 1997;39(10): 1161–80. [15] Parvathy U, Beena KP. Spline finite strip bending analysis of functionally graded plate using power-law function. Int J Sci Eng Res 2013;4(5):166–9. [16] Beena KP, Kalyanaraman V. Adaptive finite strip analysis of layered fibre composite plates. In: 17th International conference on composites/nano engineering (ICCE-17), Hawaii, USA; 26th July–1st August, 2009. [17] Beena KP, Kalyanaraman V. Adaptive modelling of non-symmetric composite plates using spline finite strip method. In: Third international congress on computational mechanics and simulation (ICCMS09), IIT Bombay: Powai, Mumbai; on 1–5 December, 2009. [18] Cheung YK, Kong J. Linear elastic stability analysis of shear-deformable plates using a modified spline finite strip method. Comput Struct 1993;47(2): 189–92. [19] Delale F, Erdogan F. The crack problem for a nonhomogeneous plane. ASME J Appl Mech 1983;50:609–14.
Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Linear static analysis of Functionally Graded Plate using Spline Finite Strip Method K.P. Beena ⇑, U. Parvathy College of Engineering, Trivandrum, India
a r t i c l e
i n f o
Article history: Available online 10 July 2014 Keywords: Functionally Graded Plates Classical Plate Theory Spline Finite Strip Method Idealisation techniques
a b s t r a c t Spline Finite Strip Method has been found to be very efficient in the analysis of plates, shells and stiffened plates/shells for their linear static, linear stability, non-linear static, non-linear instability and vibration behaviour. The present work explores the extension of the Spline Finite Strip Method to the analysis of functionally graded material (FGM) plates. Here the modulus of elasticity of FGM plate varies along the thickness direction and the variation is idealised by power, sigmoid and exponential functions. The Poisson’s ratio of the FGM plate is assumed to be constant throughout the thickness direction. The analysis is done for moderately thin plates subjected to uniformly distributed, central concentrated and line loads and the deflections and stresses are obtained using the Classical Plate Theory. A comparative study of the three idealisation techniques is also done. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Functionally graded materials (FGM) are the advanced materials in the family of engineering composites which are developed with a view to tailor the material architecture at microscopic scales. They are made of two or more component characterised by a compositional gradient from one component to the other throughout the thickness. FGM’s were initially developed in the late 1980’s for use in high temperature applications by a group of Japanese scientists as ultrahigh temperature resistant materials for aircraft, space vehicles and other engineering applications. These advanced materials with engineering gradients of composition, structure and/or specific properties in the preferred direction/orientation are superior to homogeneous material composed of similar constituents [1,7]. The concept of FGM, initially developed for super heat resistant materials to be used in space planes or nuclear fusion reactors, is now of interest to designers of functional materials for energy conversion, dental and orthopedic implants, sensors and thermo-generators and wear resistant coatings. An FGM can be prepared by continuously changing the constituents of multi-phase materials in a pre-determined volume fraction of the constituent material. Due to the continuous change in material properties of an FGM, the interfaces between two materials disappear but the characteristics of two or more materials of the composite are preserved. Subsequently the stress singularity at ⇑ Corresponding author. Tel.: +91 9645923362. E-mail address: [email protected] (K.P. Beena). http://dx.doi.org/10.1016/j.compstruct.2014.07.002 0263-8223/Ó 2014 Elsevier Ltd. All rights reserved.
the interface of a composite can be eliminated and thus the bonding strength is enhanced [2,3]. In view of the wide material variations and applications of FGMs, many research works have already been done for the bending [2,3,5] and buckling analysis [4,6,8–15]. FGMs have heterogeneous microstructure with material properties varying smoothly and continuously in preferred direction. To simplify the complicated heterogeneous microstructure of FGM, different homogenisation schemes are already developed. Power law function (PFGM), exponential function (EFGM) and sigmoid function (SFGM) are widely used to describe the variation of material properties of FGM. Various well established theories are available for the analysis of isotropic and composite plates which can be extended to FGM plates also. In this work, the Spline Finite Strip Method is used for the bending analysis of FGM plates using Classical Plate Theory. The three idealisation techniques power-law, sigmoid and exponential functions are used to show the variation of Young’s modulus along the thickness direction. The Poisson’s ratio is assumed to be a constant in all directions. The material properties are varied continuously in the thickness direction according to the volume fraction of the constituent materials. The results are validated using the closed form solution developed by Chi and Chung [2]. 2. Spline Finite Strip Method Although closed form analytical method may be possible in simple cases of idealised structure and loading, various numerical approaches like FEM, CFSM, SFSM, etc. are usually resorted to
310
K.P. Beena, U. Parvathy / Composite Structures 117 (2014) 309–315
complex systems and loading conditions. The Finite Element Method (FEM) has been extensively used for the analysis of plated structures. The computational requirement of FEM, in terms of storage space and time is very high, especially in linear prismatic members wherein some of the elements have small width. Hence, this method has only limited application in the stability and nonlinear analysis of linear prismatic members modelled using plates and shells, especially when iterative non-linear analysis is needed, as in optimum design. The Classical Finite Strip Method (CFSM), on the other hand, allows more efficient modelling of such prismatic members using strips as elements along the length of the member. This method works well for simple boundary conditions (simply supported, clamped, etc.), but fails to effectively deal with complex boundary conditions and partial and concentrated loads, since the trigonometric functions used to model displacements in the longitudinal direction are infinitely continuous. The continuity and discontinuity requirements can be satisfied by replacing the classical trigonometric function by a spline function as is done in Spline Finite Strip Method (SFSM). The Spline Finite Strip Method has been found to be very efficient in the analysis of plates, shells and stiffened plates/shells for their linear static, linear stability, non-linear static, non-linear instability and vibration behaviour [17–19]. The spline function is defined as a piecewise polynomial of nth degree which is smoothly connected to the adjoining spline functions which has n-1 continuous derivatives. There is variety of splines namely natural spline, cardinal spline, basic spline, etc. B3 spline (cubic basic) is most common and is continuous over only four consecutive sections. The B3 spline series is a piecewise cubic polynomial, which is an ideal approximation of the bending behaviour. Another property of B3 spline is its localised behaviour that makes the stiffness matrix highly banded. Owing to this property, incorporating the boundary conditions is easy, and only three splines adjacent to the constraint need to be modified. Equal and unequal spaced spline series have been used by many researchers [16–19] to analyse thin and thick plate structures. The unequal splines are more efficient when the structure is subjected to concentrated loads and reactions, when the support of members are either isolated or at irregular locations and when cut-outs are present. Hence here unequal B3 splines are used to model the variation of deflection in the longitudinal direction. 3. Problem formulation The functionally graded material (FGM) can be produced by continuously varying the constituents of multi-phase materials in a predetermined profile. The most distinct features of an FGM are the non-uniform microstructures with continuously graded macro properties. An FGM can be defined by varying the volume fractions using the power-law, exponential or sigmoid function. In the rectangular FGM plate shown in Fig. 1, coordinates x and y define the plane of the plate, and the z-axis originated at the middle surface of the plate is in the thickness direction. The material properties, Young’s modulus and the Poisson’s ratio, on the upper and lower surfaces are different but are pre-assigned according to the performance demands and the Young’s modulus and Poisson’s ratio of the plates vary continuously only in the thickness direction (z-axis) i.e., E = E(z), m = m(z). Delale and Erdogan [19] indicated that the effect of Poisson’s ratio on the deformation is much less than that of Young’s modulus. Hence here Poisson’s ratio of the plate is assumed to be a constant. and the variation of Young’s moduli is idealised using any one of the three functions i.e. power-law functions (P-FGM), exponential functions (E-FGM), or sigmoid functions (S-FGM).
Fig. 1. The geometry of an FGM Plate.
3.1. Power law idealisation The volume fraction of the P-FGM is assumed to obey a power law function:
gðzÞ ¼
p z þ h=2 h
ð1Þ
where p is the material parameter and h is the thickness of the plate, the material properties of a P-FGM can be determined by the rule-of-mixture as
EðzÞ ¼ gðzÞE1 þ ½1 gðzÞE2
ð2Þ
where E1 and E2 are the Young’s moduli of the lowest (z = h/2) and top surfaces (z = h/2) of the FGM plate, respectively. 3.2. Sigmoid idealisation The material property variation across the thickness of the FGM plate using the sigmoid function is defined using two power-law functions to ensure smooth distribution of stresses among all the interfaces. The two power-law functions are defined by,
p 1 h=2 z for 0 6 z 6 h=2 2 h=2 p 1 h=2 þ z for h=2 6 z 6 0 g 2 ðzÞ ¼ 1 2 h=2
g 1 ðzÞ ¼
ð3Þ ð4Þ
Thus the Young’s modulus of the SFGM can be calculated by;
EðzÞ ¼ g 1 ðzÞE1 þ ½1 g 1 ðzÞE2 for 0 6 z 6 h=2
ð5Þ
EðzÞ ¼ g 2 ðzÞE1 þ ½1 g 2 ðzÞE2 for h=2 6 z 6 0
ð6Þ
3.3. Exponential idealisation The exponential function used to describe the material properties of FGMs is
Fig. 2. Configuration of a simply supported square FGM plate.
311
K.P. Beena, U. Parvathy / Composite Structures 117 (2014) 309–315 Table 1 Dimensionless maximum deflection (w/h) of simply supported FGM plate for various loadings (E1/E2 = 3 and p = 2). Type of load
PFGM
SFGM
EFGM
SFSM
Chi and Chung [2]
SFSM
Chi and Chung [2]
SFSM
Chi and Chung [2]
Load 1 Load 2 Load 3
0.247 0.703 0.410
0.247 0.706 0.410
0.227 0.646 0.377
0.227 0.649 0.377
0.230 0.656 0.382
0.230 0.659 0.382
Table 2 Dimensionless tensile stress (rx/q0) at the centre of simply supported FGM plate for various loadings (E1/E2 = 3 and p = 2). Type of load
Load 1 Load 2 Load 3
PFGM
SFGM
EFGM
SFSM
Chi and Chung [2]
SFSM
Chi and Chung [2]
SFSM
Chi and Chung [2]
1051.10 8374.44 2853.92
1057.71 8370.23 2855.23
953.68 7588.82 2588.65
955.72 7592.34 2591.72
1009.55 8002.92 2732.44
1010.23 8007.65 2738.43
Fig. 3. Dimensionless deflection (w/h) of simply supported PFGM, SFGM and EFGM plates under uniform load along the width of plate for various E1/E2.
EðzÞ ¼ AeBðzþh=2Þ where A ¼ E2
1 E1 and B ¼ ln h E2
ð7Þ ð8Þ
3.4. Stiffness matrix formulation using Classical Plate Theory The Classical Plate Theory is extended for the analysis of FGM Plates. Here both transverse shear and transverse normal stresses are neglected. The deformation is entirely due to bending and inplane stretching. The normal stresses rx, ry and shear stress sxy acting in the XY plane are derived. The stress resultants are
obtained by integrating stress along the thickness. Thus the axial forces and the bending moments are obtained in terms of coefficients Aij, Bij and Cij. 9 2 8 A11 > > < Nx = 6 Ny ¼ 4 A12 > > ; : N xy 0
A12 A11 0
9 2 38 0 > B11 = < ex0 > 7 e 6 0 5 þ 4 B12 y0 > ; :c > A66 0 xy0
B12 B11 0
38 @ 2 w 9 0 > @x2 > > > = < 2 7 0 5 @@yw2 > > > > B66 : 2 @ 2 w ; @x@y
8 9 2 9 2 38 38 @ 2 w 9 2 > B11 B12 0 > C 11 C 12 0 > > > < Mx > < ex0 > < @x2 > = = = 6 7 6 7 M y ¼ 4 B12 B11 0 5 ey0 þ 4 C 12 C 11 0 5 @@yw2 > > > > > > : :c ; > ; > Mxy 0 0 B66 0 0 C 66 : 2 @ 2 w ; xy0 @x@y
ð9Þ
ð10Þ
312
K.P. Beena, U. Parvathy / Composite Structures 117 (2014) 309–315
Fig. 4. Dimensionless stress (rx/q0) of simply supported PFGM, SFGM and EFGM plates under uniform load across the thickness direction for varying E1/E2.
Fig. 5. Comparison of dimensionless maximum deflection of PFGM, SFGM and EFGM for (a) uniformly distributed load (b) central concentrated load (c) line load.
K.P. Beena, U. Parvathy / Composite Structures 117 (2014) 309–315
Fig. 6. Comparison of dimensionless maximum tensile stress of PFGM, SFGM and EFGM for (a) uniformly distributed load (b) concentrated load (c) line load.
Fig. 7. Comparison of dimensionless deflection (w/h) and dimensionless stress (rx/q0) of simply supported PFGM plate for E1/E2 = 10.
Fig. 8. Comparison of dimensionless deflection (w/h) and dimensionless stress (rx/q0) of simply supported SFGM plate for E1/E2 = 10.
313
314
K.P. Beena, U. Parvathy / Composite Structures 117 (2014) 309–315
Fig. 9. Comparison of dimensionless deflection (w/h) and dimensionless stress (rx/q0) of simply supported EFGM plate for E1/E2 = 10.
They are the stiffness matrix coefficients which are obtained by the integration of material properties of the FGM plate and are defined as shown:
A11 ¼
Z
h=2
h=2
A66 ¼
Z
h=2
h=2
B11 ¼ B66 ¼
Z Z
h=2 h=2 h=2 h=2
C 11 ¼
Z
h=2
h=2
C 66 ¼
Z
h=2
h=2
EðzÞ 1 tðzÞ
2
dz;
A12 ¼
Z
h=2
EðzÞtðzÞ
h=2
1 tðzÞ2
1 tðzÞ dz 2 1 tðzÞ2
dz; 4.1. Effect of ‘E1/E2’ ratio on deflections and stresses of FGM Plate
EðzÞ
ð11Þ
Z h=2 zEðzÞtðzÞ dz; B ¼ dz; 12 2 2 1 tðzÞ h=2 1 tðzÞ zEðzÞ 1 tðzÞ dz 2 1 tðzÞ2 zEðzÞ
z2 EðzÞ 1 tðzÞ
dz; 2
C 12 ¼
Z
h=2 h=2
1 tðzÞ dz 2 1 tðzÞ
It is seen that the SFSM results on FGM plate under simply supported boundary conditions agree very well with the exact value obtained from closed form solutions. The largest error obtained is less than 0.5% for all the three idealisation techniques.
z2 EðzÞtðzÞ 1 tðzÞ2
dz;
z2 EðzÞ
2
These stiffness matrix coefficients are used to derive the equilibrium equations which in turn gives the deflection and stresses of FGM plate. 4. Results and discussions A simply supported square FGM plate as shown in Fig. 2 with width to thickness ratio equal to 50 is analysed for various E1/E2 ratios and p values. The Poisson’s ratio of the FGM plate is assumed to be constant throughout the whole plate as t = 0.3. The Young’s modulus at any point on the FGM plate varies continuously in the thickness direction based on the volume fraction of the constituents. The Young’s modulus at the bottom surface of the FGM plate, E1 is kept as a constant as 2.1 106 kg/cm2. The Young’s modulus at the top surface of the FGM plate E2 varies with the ratio of E1/E2. The analysis has been carried out for uniformly distributed load, central concentrated load and line loads. Due to the symmetry about the x- and y-axes, only one quarter of the full plate is considered for analysis. The quarter plate is divided into 5 strips in the longitudinal direction and 5 nodal lines in the transverse direction. The plate is analysed for uniformly distributed load (Load 1, q0 = 1 kg/cm2), concentrated load (Load 2, P = 10,000 kg) and line load (Load 3, P0 = 100 kg/cm) with simply supported boundary conditions. The deflections and stresses are obtained for all the three idealisation techniques, considering equal spacing for the knots and nodal lines. The values of dimensionless maximum deflection and dimensionless tensile stresses at the centre of FGM plate for the three loadings are validated in Tables 1 and 2.
The effect of E1/E2 ratio on PFGM, SFGM and EFGM plates subjected to three loading conditions are analysed. The dimensionless deflections along the x-direction and dimensionless stresses across the thickness direction at the centre of FGM plate are plotted graphically for each loading pattern for various E1/E2 ratios. A comparison between the three idealisation techniques was done for the dimensionless maximum deflection and dimensionless maximum tensile stress for all the loadings. 4.1.1. Square plate subjected to uniformly distributed load First, consider that the FGM plate is subjected to a uniform load with the magnitude q0 = 1 kg/cm2. The dimensionless deflections and dimensionless stresses acting on the FGM plate under the action of uniform load is evaluated using the three idealisation techniques for different E1/E2 values, assuming the material parameter ‘p’ as 2. The results are plotted in Figs. 3 and 4. 4.1.2. Comparison of PFGM, SFGM and EFGM plate for various ‘E1/E2’ ratios The dimensionless deflections and dimensionless stresses of the FGM plate using power-law, sigmoid and exponential (PFGM, SFGM and EFGM) functions are compared for various E1/E2 values. Here the deflection refers to the maximum deflection acting at the centre of the plate and the stress refers to the maximum tensile stress acting at the bottom most surface of the FGM plate. The three idealisation techniques are thus compared for the three loading cases. The comparison of maximum deflection and maximum tensile stresses of PFGM, SFGM and EFGM plates for the three loadings are plotted graphically as shown in Figs. 5 and 6. From the graphs, it can be observed that the pattern of deflection and stresses for all the three types of loading for various ‘E1/ E2’ ratio is same. As E1/E2 increases, the deflection and stresses of the EFGM plate increases than that of PFGM as well as SFGM plate. This is because of the exponential function used in EFGM plate to describe the variation of Young’s modulus throughout the thickness direction. The SFGM plate undergoes less deflection and stress for all the loading cases. 4.2. Comparison of uniformly distributed, concentrated and line loads on simply supported FGM plate The deflections and stresses are obtained for a square plate with E1/E2 = 10 and material parameter p = 2. For comparison a uniform
K.P. Beena, U. Parvathy / Composite Structures 117 (2014) 309–315
load of 1 kg/cm2 and a line load of 100 kg/cm are applied which are equivalent to a concentrated load of 10,000 kg. The dimensionless deflection and dimensionless stresses for the PFGM, SFGM and EFGM plate subjected to uniform, concentrated and line loads are plotted as shown in Figs. 7–9. It is shown in Figs. 7–9 that the three curves corresponding to uniform, point and line loads intersect at the point of z = 0.0625 h, which is the location of neutral surface of the FGM plate. The position of the neutral surface of the Functionally Graded Plate remains same for the three loading conditions. 5. Conclusion In this study, the bending analysis of the Functionally Graded Plate was carried out using power-law, sigmoidal and exponential function for simply supported boundary conditions. The analysis was done for thin and moderately thick plates using Classical Plate Theory. The Spline Finite Strip Method was implemented in Visual C++ programming language. Deflection and stress values obtained from Spline Finite Strip Method were validated using the closed form Fourier series solution developed by Chi and Chung [2] for simply supported Functionally Graded Plate. The results lead to the following conclusions: 1. The more the E1/E2, the larger the deflection ‘w’ and larger the tensile stress ‘rx’. This is because of the decrease in stiffness of the FGM plate for larger E1/E2. It was observed that the compressive stress does not increase with increase in E1/E2. 2. A rapid increase in deflection and tensile stress occurs in the case of EFGM plates for E1/E2 > 10. This is due to the exponential function used to define the variation of Young’s modulus in the EFGM plates. 3. The simply supported SFGM plates under all the three loadings show only a gradual increase in deflection compared to PFGM and EFGM plates. This is due to the use of two power-law functions to define the variation of Young’s modulus across the thickness direction. Thus a smooth distribution of stresses was ensured. 4. The stress at the centre of PFGM, SFGM and EFGM plate for all the three loading conditions was found to be maximum at the bottom edge (z = h/2) of the plate. This maximum stress which was located at the bottom surface was found to be tensile in nature. 5. The maximum tensile stress increases with increase in ‘E1/E2’ ratio. The compressive stress is maximum at the top surface (z = h/2) for small ‘E1/E2’. As the ratio of E1/E2 increases, the maximum compressive stress moves towards the inner side of the plate. 6. The neutral surface of PFGM, SFGM and EFGM plates under uniformly distributed, central concentrated and line loads get shifted towards the positive z-direction with increase in E1/E2.
315
Moreover, it was observed that the position of the neutral surface remain same for FG plate under all the three loadings as well as under different idealisation techniques for a constant E1/E2 ratio. Thus the position of neutral surface of the Functionally Graded Plate depends on the ratio of ‘E1/E2’ and is independent of the loading conditions.
References [1] Jha DK. Tarun Kant, Singh RK. A critical review of recent research on functionally graded plates. Composite Structures 2013;96:833–49. [2] Chi Shyang-Ho, Chung Yen-Ling. Mechanical behaviour of functionally graded material plates under transverse load-Part I, analysis. Int J Solids Struct 2006;43(13):3657–74. [3] Chi Shyang-Ho, Chung Yen-Ling. Mechanical behaviour of functionally graded material plates under transverse load-Part II, numerical results. Int J Solids Struct 2006;43(13):3675–91. [4] Mokhtar Bouazza, Abedlouahed Tounsi, El Abbas Adda Bedia, Abdelkader Megueni. Buckling analysis of functionally graded plates with simply supported edges. Leonardo J Sci 2009;15:21–32. [5] Kashtalyan M. Three dimensional elasticity solution for bending of functionally graded rectangular plates. Eur J Mech Solids 2004;23:853–64. [6] Reddy JN. Mechanics of laminated composite plates. New York: CRC Press; 1997. [7] Abrate Serge. Functionally graded plates behave like homogenous plates. Compos B Eng 2008;39(1):151–8. [8] Na Kyung-Su, Kim Ji-Hwan. Three-dimensional thermal buckling analysis of functionally graded materials. Compos B Eng 2004;35:429–43. [9] Thai Huu-Tai, Choi Dong-Ho. An efficient and simple refined theory for buckling analysis of functionally graded plates. Appl Math Model 2012;36:1008–22. [10] Samsam Shariat BA, Javaheri R, Eslami MR. Buckling of imperfect functionally graded plates under in-plane compressive loading. Thin Walled Struct 2005;43:1020–36. [11] Samsam Shariat BA, Eslami MR. Buckling of thick functionally graded plates under mechanical and thermal loads. Compos Struct 2007;78:433–9. [12] Majid Badiey, Kouchakzadeh MA. Buckling of a functionally graded plate (FGP) under shear and in-plane directional loading. In: 27th International congress of the aeronautical sciences, 2010. [13] Wu Tsung-Lin, Shukla KK, Huang Jin H. Post-buckling analysis of functionally graded rectangular plates. Compos Struct 2007;81:1–10. [14] Wang S, Dawe DJ. Spline finite strip analysis of the buckling and vibration of composite prismatic plate structures. Int J Mech Sci 1997;39(10): 1161–80. [15] Parvathy U, Beena KP. Spline finite strip bending analysis of functionally graded plate using power-law function. Int J Sci Eng Res 2013;4(5):166–9. [16] Beena KP, Kalyanaraman V. Adaptive finite strip analysis of layered fibre composite plates. In: 17th International conference on composites/nano engineering (ICCE-17), Hawaii, USA; 26th July–1st August, 2009. [17] Beena KP, Kalyanaraman V. Adaptive modelling of non-symmetric composite plates using spline finite strip method. In: Third international congress on computational mechanics and simulation (ICCMS09), IIT Bombay: Powai, Mumbai; on 1–5 December, 2009. [18] Cheung YK, Kong J. Linear elastic stability analysis of shear-deformable plates using a modified spline finite strip method. Comput Struct 1993;47(2): 189–92. [19] Delale F, Erdogan F. The crack problem for a nonhomogeneous plane. ASME J Appl Mech 1983;50:609–14.