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A semi-analytical three-dimensional free vibration analysis of functionally graded curved panels
International Journal of Pressure Vessels and Piping 87 (2010) 470e480
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A semi-analytical three-dimensional free vibration analysis of functionally graded curved panels P. Zahedinejad a, P. Malekzadeh b, c, *, M. Farid a, G. Karami d a
Department of Mechanical Engineering, Islamic Azad University, Branch of Shiraz, Shiraz, Iran Department of Mechanical Engineering, Persian Gulf University, Persian Gulf University Boulevard, Bushehr 75168, Iran c Center of Excellence for Computational Mechanics, Shiraz University, Shiraz, Iran d Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Fargo, ND 58105-5285, USA b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 February 2009 Received in revised form 29 May 2010 Accepted 11 June 2010
Based on the three-dimensional elasticity theory, free vibration analysis of functionally graded (FG) curved thick panels under various boundary conditions is studied. Panel with two opposite edges simply supported and arbitrary boundary conditions at the other edges are considered. Two different models of material properties variations based on the power law distribution in terms of the volume fractions of the constituents and the exponential distribution of the material properties through the thickness are considered. Differential quadrature method in conjunction with the trigonometric functions is used to discretize the governing equations. With a continuous material properties variation assumption through the thickness of the curved panel, differential quadrature method is efficiently used to discretize the governing equations and to implement the related boundary conditions at the top and bottom surfaces of the curved panel and in strong form. The convergence of the method is demonstrated and to validate the results, comparisons are made with the solutions for isotropic and FG curved panels. By examining the results of thick FG curved panels for various geometrical and material parameters and subjected to different boundary conditions, the influence of these parameters and in particular, those due to functionally graded material parameters are studied. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Free vibration Curved panels Functionally graded material
1. Introduction Layered composite materials, due to their thermal and mechanical merits compared to single-composed materials, have been widely used for a variety of engineering applications. However, owing to the sharp discontinuity in the material properties at interfaces between two different materials, there may exist stress concentrations resulting in severe material failure [1,2]. Functionally graded materials (FGMs) are heterogeneous composite materials, in which the material properties vary continuously from one interface to the other. The advantage of using these materials is that they can survive the high thermal gradient environment, while maintaining their structural integrity. Typically, an FGM is made of a ceramic and a metal for the purpose of thermal protection against large temperature gradients. The ceramic material provides the high-temperature resistance due to
* Corresponding author at: Department of Mechanical Engineering, Persian Gulf University, Persian Gulf University Boulevard, Bushehr 75168, Iran. Tel.: þ98 771 4222150; fax: þ98 771 4540376. E-mail addresses: [email protected], [email protected] (P. Malekzadeh). 0308-0161/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2010.06.001
its low thermal conductivity, while the ductile metal constituent prevents fracture due to its greater toughness. FGMs are now developed for general use as structural elements in extremely high temperature environments. A listing of different applications can be found in Ref. [3]. Most of the studies on FGMs have been restricted to thermal stress analysis, thermal buckling, fracture mechanics and optimization [4e10]. Little attention has been given to vibration analysis of FGM plates, shells and panels. Despite the evident importance in practical applications, investigations on the dynamic characteristics of FGM shell structures are still limited in number. Some researches [11e14], are based on the classical shell theories, i.e., neglecting the effect of transverse shear deformation. The application of such theories to moderately thick or thick shell structures can lead to serious errors. In order to account the effect of transverse shear, some other studies have been conducted based on the first order shear deformation theory [15e17]. Consequently, the higher order shear deformation theories were developed to improve the first order shear deformation theory [18e20]. Yang and Shen [18] gave a semi-analytical approach based on Reddy’s higher order shear deformation shell theory, for free vibration and dynamic instability of simply supported FGM
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
Nomenclature Axij, Azij b {b} Bxij, Bzij C Co [C] {d} E(z) EC EM Eo G h I Lx [M] Nx Nz p r Rm
The first order DQ weighting coefficients along the xand the z-directions Arc length of panel (¼Rmqo) Vector of boundary degrees of freedom The second order DQ weighting coefficients along the x-and the z-directions A typical FGM material property constant A typical FGM material property constant at plane z ¼ 0 Material stiffness matrix Vector of domain degrees of freedom Young’s modulus Young’s modulus of ceramic Young’s modulus of metal Young’s modulus at plane z ¼ 0 Shear modulus Thickness of panel pffiffiffiffiffiffiffi Imaginary number ð ¼ 1Þ Axial length of panel Mass matrix Number of the DQ discretization points in the xdirection Number of the DQ discretization points in the zdirection The power law index Radius at an arbitrary point of panel Mean radius of panel
cylindrical panels under combined static and periodic axial forces. Free vibration and stability of simply supported functionally graded shallow shells according to a 2-D higher-order shear deformation theory was proposed by Matsunaga [19]. In another work, based on the 2-D higher order shear deformation theory, free vibration and stability of functionally graded circular cylindrical shells is recently studied by Matsunaga [20]. In these studies the variation of the radius through the thickness is not considered and the problem formulations are based on constant radius of curvature through the thickness of the panel. Two-dimensional theories reduce the dimensions of problems from three to two by introducing some assumptions in mathematical modeling which leads to simple expressions and derivation of solutions. However, these simplifications inherently bring errors and therefore may lead to unreliable results for relatively thick curved panels. As a result, three-dimensional analysis of thick curved panels not only provides realistic results but also allows further physical insights, which cannot otherwise be predicted by two-dimensional analysis. There have been some studies on free vibration analysis of isotropic and composite panels and shallow shells based on the three-dimensional elasticity formulation [21,22], but to the author’s knowledge, no work has been published on the free vibration analysis of thick FGM curved panels. Due to the intrinsic complexity of the problem based on the three-dimensional elasticity, powerful numerical methods are needed to solve the governing equations with related boundary conditions. Hence, the differential quadrature method (DQM) is used in this work. The method has been widely used for static and free vibration analysis of beams, plates, shells and panels [23e33]. In the application of DQM for such problems, it was concluded that accurate results with less computations can be obtained.
471
½Sdd ; ½Sbb ; ½Sdb ; ½Sbd Stiffness matrices T Kinetic energy u Displacement component along the axial direction V Total linear elastic strain energy v Displacement component along the circumferential direction Volume fraction [¼(z/h)p] Vf w Displacement component along the thickness direction x Axial coordinate variable z Radial coordinate variable (thickness direction) 3ij Strain tensor components (i, j ¼ r, q, z) g Material property graded index n Poisson’s ratio q Circumferential coordinate variable qo Panel angle r(z) Mass per unit volume rC Mass per unit volume of ceramic rM Mass per unit volume of metal ro Mass per unit volume at plane z ¼ 0 sij Stress tensor components (i, j ¼ r, q, z) u Natural frequency of the panel Non-dimensional frequency parameter 6mn pffiffiffiffiffiffiffiffi Umn Non-dimensional frequency parameter ½ ¼ umn h r=G Umn Non-dimensional frequency parameter ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ ¼ umn Lx rð1 n2 Þ=E Means the function value at the DQ grid point z ¼ zj ($)j
2. Theoretical formulations 2.1. FGM material properties An FGM cylindrical panel with its coordinate system (x, q, z) is shown in Fig. 1, where x and q are the axial and circumferential coordinate variables of the panel and z is the coordinate variable of the thickness coordinate. The origin of the coordinate system is located at the corner of the panel. The panel has mean radius Rm, thickness h, axial length Lx, total panel angle qo and arc length Ly(¼Rmqo). It is assumed that the panel is made of a mixture of ceramics and metals, and the material composition is continuously varied such that the top surface (z ¼ h) of the panel is ceramic only,
Fig. 1. Geometry of functionally graded curved panel.
472
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
whereas the bottom surface (z ¼ 0) is metal only. In the first model the modulus of elasticity E and mass density r are assumed to be in terms of a simple power law distribution of volume fraction as follows
EðzÞ ¼ EM þ ECM Vf ; nðzÞ ¼ n; rðzÞ ¼ rM þ rCM Vf
where ið ¼ 0; 1; 2; .Þ is circumferential wave number. Ui, Vi and Wi are unknown displacement functions in the x, q, and z-directions, respectively. Using Eq. (6) in conjunction with Hamilton’s principle, one obtains, Governing equations:
dUi :C55
where
ECM ¼ EC EM ; rCM ¼ rC rM ; Vf ¼ ðz=hÞp
(2)
p is the power law index which takes values greater than or equal to zero. The value of p equal to zero represents a fully ceramic panel and infinite p shows a fully metallic panel. Subscripts M and C refer to the metal and ceramic constituents, which denote the material properties of the bottom and top surfaces of the curved panel, respectively. In the second model the variation of the material properties is assumed to have exponential distribution along the thickness of the curved panel as follows
C ¼ Co eðgz=hÞ
! ip vUi v2 V i C44 dC44 vVi þ þ C44 þ r dz vz r qo vx vz2 ! 2 2 C44 dC44 ip v Vi Vi C22 þ Vi þ C66 rdz r qo r2 vx2 C22 þ C44 dC44 ip ip vWi þ Wi þ ðC23 þ C44 Þ þ r dz r qo r qo vz ! v2 V i r ¼ 0 ð8Þ vt 2 v 2 Ui C12 C13 dC13 vUi ðC23 þ C44 Þ vxvz r dz vx ! ip vVi C22 þ C44 dC23 ip v2 Wi Vi þ C33 þ r dz r qo vz r qo vz2 2 C33 dC33 vWi C22 dC23 ip Wi C44 Wi þ þ r dz vz rdz r qo r2 ! ! v2 Wi v2 Wi r ¼ 0 ð9Þ þ C55 2 vx vt 2
dWi :ðC13 þ C55 Þ
Based on the three-dimensional small deformation theory of elasticity, the strainedisplacement relations can be expressed as
3xx ¼
(4)
Also, the three-dimensional constitutive relations for the FGM curved panel can be written as
8 9 sxx > > > > > > > s > > > > < qq > =
2
C11 6 C12 6 6 C13 szz ¼ 6 6 0 s > > zq > > 6 > > > > 4 0 s > > xz > > : ; s xq 0
C12 C22 C23 0 0 0
C13 C23 C33 0 0 0
0 0 0 C44 0 0
0 0 0 0 C55 0
0 0 0 0 0 C66
9 38 3xx > > > > > > > 7> > > 3qq > 7> 7< 3zz = 7 7> gzq > > 7> > > 5> g > > > > ; : xz >
qo
ipq ; ¼ Vi ðx; z; tÞcos
qo
ipq ¼ Wi ðx; z; tÞsin
qo
The boundary conditions along the free surfaces at z ¼ 0 and z ¼ h become,
vUi vWi vVi Vi ip Wi ¼ 0; þ ¼ 0; C44 þ vz vx vz r r qo vUi ip W vWi C23 ¼ 0 V i þ C33 C13 vx r vz r qo i C55
(5)
gx q
ð10Þ
Also, the natural and geometrical boundary conditions along the edges x ¼ 0 and x ¼ Lx become,
Using Eqs. (4) and (5) in conjunction with Hamilton’s principle [34], the equations of motion for free vibration analysis of FGM curved panel can be obtained. For panels with two opposite edges simply supported, the displacement components can be expanded in terms of normal mode functions in the direction normal to these edges. In this study, it is assumed that the edges q ¼ 0 and q ¼ qo are simply supported. Hence,
ipq ; uðx; q; z; tÞ ¼ Ui ðx; z; tÞsin
þ
2.2. The basic formulations
vu 1 vv w vw þ ; 3zz ¼ ; 3qq ¼ ; gxq vx r vq r vz vv 1 vu vw vu 1 vw vv v ¼ ; gxz ¼ þ þ þ ; g zq ¼ vx r vq vx vz r vq vz r
v2 Ui vz2
dVi :ðC12 þ C66 Þ
(3)
where C stands for the material stiffness matrix components Cij ði; j ¼ 1; .; 6Þ and the mass density r. Co is the corresponding values at the plane z ¼ 0 (Metallic layer) and g is the material property graded index. In both models the Poisson’s ratio n is assumed to be constant.
!
2 C55 dC55 vUi ip Ui þ C66 r dz vz r qo ! v2 Ui ip vVi v2 Wi ðC þ C11 þ C Þ þ ðC13 þ C55 Þ 12 66 2 q vx vxvz r vx o ! C12 þ C55 dC55 vWi v 2 Ui þ ¼ 0 ð7Þ þ r r dz vx vt 2
(1)
Either Ui ¼ 0
or
C11
vUi ip W C12 Vi i vx r r qo vWi þ C13 ¼ 0 vz
Either Vi ¼ 0
or
C66
vðx; q; z; tÞ
Either Wi ¼ 0
wðx; q; z; tÞ (6)
or
C55
ip vV Ui þ i ¼ 0 vx r qo
vUi vWi þ vz vx
ð11Þ
(12)
¼ 0
(13)
Various boundary conditions along the edges at x ¼ 0 and x ¼ Lx can be obtained by combining the conditions stated in Eqs. (11)e(13) as,
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
Simply supported (S):
C11
vUi ip W vWi Vi i þC13 C12 ¼ 0; Vi ¼ 0; Wi ¼ 0 vx r vz r qo
Table 2 pffiffiffiffiffiffiffiffi Comparison of the frequency parameters Uim ¼ uim h r=G for isotropic curved panel under simply supported boundary condition (Lx ¼ Ly).
U11
(14)
Lx/h
Lx/R
CST
FSDT
Matsunaga
Present
2
0.2 0.4 0.2 0.4
2.0189 1.9983 0.3739 0.3760
1.4927 1.4891 0.3415 0.3442
1.5140 1.5088 0.3429 0.3454
1.5100 1.4933 0.3423 0.3429
0.2 0.4 0.2 0.4
2.2306 2.2545 0.8896 0.8927
2.2252 2.2362 0.8896 0.8926
2.2253 2.2365 0.8896 0.8926
2.2272 2.2437 0.8897 0.8931
5
Clamped (C):
Ui ¼ 0;
Vi ¼ 0;
Wi ¼ 0
(15)
U12
2 5
Free (F):
vUi ip W vWi C12 ¼ 0; Vi i þ C13 vx r vz r qo ip vV vUi vWi C66 Ui þ i ¼ 0; C55 þ ¼ 0 vx vz vx r qo
C11
2.3. Differential quadrature discretization
ð16Þ
For the free vibration analysis, the following solutions may be assumed for the displacement components
Ui ðx; z; tÞ ¼ U i ðx; zÞeIui t ;
Vi ðx; z; tÞ ¼ V i ðx; zÞeIui t ;
Wi ðx; z; tÞ ¼ W i ðx; zÞeIui t
ðC55 Þj
Nz X n¼1
473
ð17Þ
At this stage, the circumferentially discretized governing differential equations and the related boundary conditions are transformed into algebraic equations via the DQ method. Using the DQ discretization rules for spatial derivatives (please see Appendix A) and Eq. (17), the DQ analogs of the governing differential equations are obtained as Eq. (7):
! ! !2 X Nz Nz Nx X 1X dC55 ip z z x þ þ ðC11 Þj þ A U A U Bkm U imj ðC66 Þj U rj n ¼ 1 jn ikn dz j n ¼ 1 jn ikn rj qo m¼1 ikj ! # " Nz Nx Nx X N x X X X ðC12 þ C55 Þj ip dC55 Ax V þ ðC13 þ C55 Þj Axkm Azjn W imn þ þ Axkm W imj þ rj u2i U ikj ¼ 0 ð18Þ dz r rj qo m ¼ 1 km imj j j m¼1 n¼1 m¼1
Bzjn U ikn
ðC12 þ C66 Þj
Eq. (8):
ðC12 þ C66 Þj þ ðC22 Þj
! ! " X Nz Nz ðC44 Þj 1 dC44 1X dC44 z z þ ðC44 Þj þ Ajn V ikn þ Ajn V ikn þ dz j n ¼ 1 rj n ¼ 1 rj dz j rj2 m¼1 n¼1 !2 # ! # ! " N N z x X ðC22 þ C44 Þj ip ip X dC44 ip x z V irs þ ðC66 Þj Bkm V imj þ ðC23 þ C44 Þj A W þ þ W ikj rj dz j rj qo rj qo rj qo n ¼ 1 jn ikn m¼1
ip rj qo
!
Nx X
Axkm U imj
Nz X
Bzjn V ikn
þ rj u2i V ikj ¼ 0
ð19Þ
Table 1 Convergence behavior of the frequency parameters ð6im Þ for FGM curved panel with two opposite edges simply supported and other edges free, based on the first FGM material model formulation (Lx ¼ 1 m, Rm ¼ 1 m, p ¼ 1, qo ¼ 2p=3). Material property
h/Rm
Power law
0.3
0.5
Exponential
0.3
0.5
Nz ¼ 7
Nz ¼ 9
Nz ¼ 13
Nx ¼ 9
Nx ¼ 13
Nx ¼ 17
Nx ¼ 9
Nx ¼ 13
Nx ¼ 17
Nx ¼ 9
Nx ¼ 13
Nx ¼ 17
611 612 621 622 611 612 621 622
0.0196 0.0309 0.1332 0.1898 0.0513 0.0683 0.3204 0.3963
0.0196 0.0308 0.1332 0.1895 0.0513 0.0682 0.3204 0.3961
0.0196 0.0308 0.1332 0.1894 0.0514 0.0682 0.3204 0.3961
0.0197 0.0311 0.1337 0.1904 0.0519 0.0687 0.3222 0.3974
0.0197 0.0309 0.1336 0.1899 0.0519 0.0686 0.3222 0.3973
0.0197 0.0309 0.1336 0.1899 0.0519 0.0686 0.3222 0.3973
0.0197 0.0310 0.1337 0.1899 0.0519 0.0685 0.3225 0.3969
0.0197 0.0310 0.1337 0.1900 0.0519 0.0687 0.3225 0.3974
0.0197 0.0309 0.1337 0.1900 0.0519 0.0687 0.3225 0.3974
611 612 621 622 611 612 621 622
0.0254 0.0379 0.1721 0.2328 0.0666 0.0816 0.4112 0.4659
0.0254 0.0378 0.1720 0.2325 0.0666 0.0815 0.4112 0.4658
0.0254 0.0378 0.1720 0.2325 0.0666 0.0815 0.4112 0.4658
0.0254 0.0380 0.1722 0.2331 0.0669 0.0818 0.4125 0.4667
0.0254 0.0378 0.1722 0.2327 0.0669 0.0817 0.4125 0.4666
0.0254 0.0378 0.1722 0.2326 0.0669 0.0817 0.4125 0.4666
0.0254 0.0379 0.1722 0.2327 0.0669 0.0815 0.4126 0.4660
0.0254 0.0378 0.1722 0.2327 0.0669 0.0817 0.4126 0.4666
0.0254 0.0378 0.1722 0.2326 0.0669 0.0817 0.4126 0.4666
474
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
Eq. (10):
Table 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Comparison of the fundamental frequency parameter Uim ¼ uim Lx rð1 n2 Þ=E for isotropic curved panel under simply supported boundary condition (R/h ¼ 10, i ¼ m ¼ 1). Lx/h
Method
10
Present FSDT [36] 3-D Elasticity [22]
0.5
1
1.5
2
1.3166 1.3360 1.3174
0.5526 0.5563 0.5505
0.4066 0.4044 0.3998
0.3566 0.3505 0.3461
Table 4 pffiffiffiffiffiffiffiffiffiffiffiffi Comparison of the fundamental frequency parameter 6im ¼ uim h rc =Ec for FGM curved panel with four edges simply supported (Lx ¼ Ly, i ¼ m ¼ 1). Lx/R
Method
p 0
0.5
1
4
10
N
2
0.5
Present HSDT [19] Present HSDT [19]
0.9187 0.9334 0.8675 0.9163
0.8013 0.8213 0.7578 0.8105
0.7260 0.7483 0.6870 0.7410
0.5797 0.6011 0.5475 0.5960
0.5245 0.5460 0.4940 0.5392
0.4770 0.4752 0.4496 0.4665
Present HSDT [19] Present HSDT [19]
0.2113 0.2153 0.2164 0.2239
0.1814 0.1855 0.1852 0.1945
0.1639 0.1678 0.1676 0.1769
0.1367 0.1413 0.1394 0.1483
0.1271 0.1328 0.1286 0.1380
0.1109 0.1096 0.1133 0.1140
5
0.5 1
"
n¼1
Azjn U ikn
þ
Nx X m¼1
! Axkm W imj
¼ 0;
! # V ikj ip þ W ikj ¼ 0 rj rj qo n¼1 " ! # N x X W ikj ip x Akm U imj ðC23 Þj V ikj ðC13 Þj rj rj qo m¼1 ðC44 Þj
Nz X
Azjn V ikn
Nz X
þ ðC33 Þj
Lx/h
1
Nz X
ðC55 Þj
Ly/Lx
n¼1
Azjn W ikn ¼ 0
ð21Þ
Eq. (11):
Either U ikj ¼ 0 W ikj rj
or
ðC11 Þj
Nx X m¼1
# þ ðC13 Þj
Nz X n¼1
" Axkm U imj ðC12 Þj
Azjn W ikn ¼ 0
! ip V ikj rj qo ð22Þ
Eq. (9):
! # X Nz Nx dC13 ip X Axkm U imj ðC23 þ C44 Þj Azjn V ikn q r dz r o j j j m¼1 n¼1 m¼1 n¼1 ! ! ! " # X Nz Nz Nz X X ðC22 þ C44 Þj dC23 ip 1 dC 33 z z z þ þ V ikj þ ðC33 Þj Bjn W ikn þ A W A W rj dz j rj qo dz j n ¼ 1 jn ikn rj n ¼ 1 jn ikn n¼1 !2 # " Nx X ðC22 Þj 1 dC23 ip þ ðC55 Þj Bxkm W imj þ 2 ðC44 Þj W ikj þ rj u2i W ikj ¼ 0 q r dz r r o j j j m¼1 j
ðC13 þ C55 Þj
Nz Nx X X
Axkm Azjn U imn
" ðC12 C13 Þj
where Nx and Nz are the number of the DQ discretization points through the longitudinal and thickness directions respectively and Axij , Bxij , Azij , Bzij are the first and second order weighting coefficients of the differential quadrature method through the x-and z-directions, respectively. Also, ðÞj means the function value at the DQ grid point z ¼ zj . The DQ analogs of the boundary conditions become,
ð20Þ
Eq. (12):
" Either V ikj ¼ 0
or
ðC66 Þj
! # Nx X ip x U ikj þ Akm V imj ¼ 0 rj qo m¼1 (23)
Table 5 pffiffiffiffiffiffiffiffiffiffiffiffi Comparison of the frequency parameters 6im ¼ uim h rc =Ec for FGM curved panel with four edges simply supported based on constant radius of curvature and variable radius formulation through the thickness (Lx ¼ 1 m, p ¼ 1).
qo
h/Rm
Case
p=3
0.1
a b
0.3
a b
0.5
a b
2p=3
0.1
a b
0.3
a b
0.5
a b
a b
Frequency parameters 611
612
613
621
622
623
631
632
633
0.0596 0.0468 0.3115 0.3113 0.6671 0.6731
0.1166 0.1165 0.4624 0.4555 0.7651 0.7535
0.1547 0.1536 0.6855 0.6850 1.1621 1.1558
0.0916 0.0931 0.6170 0.6151 1.2482 1.2505
0.1604 0.1587 0.9105 0.9084 1.4744 1.5055
0.2535 0.2515 0.9133 0.9109 1.6862 1.7343
0.1809 0.1811 1.0318 1.0251 1.9442 1.9523
0.2325 0.2322 1.2512 1.2509 2.0924 2.2536
0.3165 0.3154 1.3396 1.3551 2.2241 2.3353
0.0720 0.0669 0.2319 0.2279 0.3858 0.3768
0.0773 0.0768 0.2738 0.2734 0.5576 0.5551
0.1141 0.1021 0.5633 0.5634 0.9487 0.9466
0.0596 0.0468 0.3115 0.3113 0.6671 0.6731
0.1166 0.1165 0.4624 0.4555 0.7651 0.7535
0.1547 0.1536 0.6855 0.6850 1.1621 1.1558
0.0672 0.0675 0.4405 0.4392 0.9247 0.9302
0.1386 0.1279 0.6901 0.6825 1.1303 1.1298
0.2267 0.2234 0.7792 0.7804 1.4166 1.4266
Constant radius of curvature formulation. Variable radius formulation.
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
Eq. (13):
Either W ikj ¼ 0 or ðC55 Þj
Nz X n¼1
Azjn U ikn þ
Nx X
! Axkm W imj
¼ 0 ð24Þ
8 9 8 9 < = < fU g = ; d ¼ fVg : ; : ; fWg domain
475
9 8 9 8 < = < fU g = b ¼ g : ; : fV ; fW g boundary
(25)
m¼1
Using Eq. (25), the discretized form of the equations of motion in the matrix form can be rearranged as To obtain the eigenvalue system of equations, the degrees of freedom are separated into the domain and the boundary degrees of freedom as
h in o n o Sdb b þ ½Sdd d u2i ½Mfdg ¼ 0
a
(26)
0.9
θο=π/3 θο=π/2 θο=2π/3
0.72
ϖ 11
0.54 0.36 0.18 0 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
L x /R m
b
1.3
θο=π/3 θο=π/2 θο=2π/3
1.06
ϖ 12
0.82 0.58 0.34 0.1 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
L x /R m
c
1.6
θο=π/3 θο=π/2 θο=2π/3
1.28
ϖ 13
0.96 0.64 0.32 0 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
L x /R m Fig. 2. (a)e(c) Variation of the frequency parameters verses length-to-mean radius ratios with different values of total panel angle for FGM curved panel with four edges simply supported ðh=Rm ¼ 0:3; p ¼ 1Þ.
Fig. 3. (a)e(c) Variation of the frequency parameters verses length-to-mean radius ratios with different values of total panel angle for FGM curved panel with two opposite edges simply supported and other edges clamped ðh=Rm ¼ 0:3; p ¼ 1Þ.
476
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where ½Sdb and ½Sdd are the stiffness matrices and ½M is the mass matrix. In a similar manner, the discretized form of the boundary conditions becomes
½Sbb fbg þ ½Sbd fdg ¼ 0
(27)
where ½Sbb and ½Sbd are the stiffness matrices. In the above equations, the elements of the stiffness and the mass matrices are
obtained based on the definition of the vectors of domain and boundary degrees of freedom from the DQ discretized form of the equations of motion and the boundary conditions. Using Eq. (27) to eliminate the boundary degrees of freedom fbg from Eq. (26), the result becomes,
h i n o S u2i ½M d ¼ 0
(28)
where
a
0.14
a
θο=π/3 θο=π/2 θο=2π/3
0.112
0.8 θο=π/3 θο=π/2 θο=2π/3
0.64
ϖ 11
0.084
ϖ 11
0.48
0.056
0.32
0.028 0.16
0 0.5
b
1
1.5
2
2.5 3 3.5 L x /R m
4
4.5
5
0.25
0 0.5
b
θο=π/3 θο=π/2 θο=2π/3
0.2
2
2.5
3
3.5
4
4.5
5
L x /R m 0.9 θο=π/3 θο=π/2 θο=2π/3
ϖ 12
ϖ 12
0.58
0.1
0.42
0.05
c
1.5
0.74
0.15
0 0.5
1
0.26
1
1.5
2
2.5 3 L x /R m
3.5
4
4.5
0.1 0.5
5
1
1.5
2
2.5
3
3.5
4
4.5
5
L x /R m
0.45
c
θο=π/3 θο=π/2 θο=2π/3
0.37
1.4 θο=π/3 θο=π/2 θο=2π/3
1.12
0.29
ϖ 13
ϖ 13
0.84
0.21
0.56
0.13
0.28
0.05 0.5
1
1.5
2
2.5 3 L x /R m
3.5
4
4.5
5
Fig. 4. (a)e(c) Variation of the frequency parameters verses length-to-mean radius ratios with different values of total panel angle for FGM curved panel with two opposite edges simply supported and other edges free ðh=Rm ¼ 0:3; p ¼ 1Þ.
0 0.5
1
1.5
2
2.5
3
L x /R m
3.5
4
4.5
5
Fig. 5. (a)e(c) Variation of the frequency parameters verses length-to-mean radius ratios with different values of total panel angle for FGM curved panel with three edges simply supported and one edge clamped ðh=Rm ¼ 0:3; p ¼ 1Þ.
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
h i S ¼ ½Sdd ½Sdb ½Sbb 1 ½Sbd The above equations can be solved to find the natural frequencies as well as the mode shapes of curved panels. 3. Numerical results In this section, first, the convergence behavior of the method is investigated and then comparisons with other available solutions are made to verify the accuracy of the results. The material properties for functionally graded panel are as follows,
Ec ¼ 380 109 Pa; 3
¼ 3800 kg=m ;
Em ¼ 70 109 Pa; 3
rm ¼ 2702 kg=m ;
rc n ¼ 0:3:
In all solved examples for functionally graded curved panels the non-dimensional frequency parameters ð6im Þ are defined as
a
1 0.8
0 0.1
0.2
0.3
0.4
0 0.1
0.5
h/R m
b
1 p=0 p=1 p=5 p=10
0.4
0.5
0.4
0.5
0.4
0.5
1.8 p=0 p=1 p=5 p=10
ϖ 12
ϖ 12
0.72 0.36
0.2 0 0.1
0.2
0.3
0.4
0 0.1
0.5
h/R m
c
1.5 p=0 p=1 p=5 p=10
0.2
0.3
h/R m 1.8 1.44
p=0 p=1 p=5 p=10
1.08
ϖ 13
ϖ 13
0.9
0.72
0.6
0.36
0.3 0 0.1
0.3
1.08
0.4
1.2
0.2
h/R m
1.44
0.6
c
0.72
0.24
0.2
0.8
p=0 p=1 p=5 p=10
0.48
0.4
b
1.2 0.96
ϖ 11
0.6
p=0 p=1 p=5 p=10
ϖ 11
a
477
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ½uim h rc =Ec for the first model and ½uim h ro =Eo for the second model of material properties variation. In all cases under consideration, the convergence behaviors against the number of DQ grid points along the length and the thickness directions of the curved panel were examined. However, for brevity purpose, here only those for functionally graded curved panels with two opposite edges simply supported and the two edges free are presented. In Table 1, the convergence behaviors of the frequency parameters for FGM curved panel based on the first and second models of the material properties variation are demonstrated. The results are presented for two different values of the thickness-to-mean radius ratios. It is obvious from this table that converged results are achieved with Nz ¼ 13 and Nx ¼ 17. In Table 2, comparisons of the frequency parameters are made between the present method and those obtained from other shell theories [35] for isotropic simply supported curved panel. Thepnonffiffiffiffiffiffiffiffi dimensional frequency parameter is defined as Uim ¼ uim h r=G.
0.2
0.3
0.4
0.5
h/R m Fig. 6. (a)e(c) Variation of the frequency parameters verses thickness-to-mean radius ratios with different values of material properties exponent for FGM curved panel with four edges simply supported ðLx ¼ 1 m; qo ¼ 1 radÞ.
0 0.1
0.2
0.3
h/R m Fig. 7. (a)e(c) Variation of the frequency parameters verses thickness-to-mean radius ratios with different values of material properties exponent for FGM curved panel with two opposite edges simply supported and other edges clamped ðLx ¼ 1 m; qo ¼ 1 radÞ.
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
Also in Table 3, comparison of the fundamental frequency parameter for isotropic simply supported curved panel are made between the results obtained using the present method and those obtained based on the three-dimensional elasticity formulation [22] and the first order shear deformation theory [36]. Here the non-dimensional formpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the frequency parameters is defined as ffi Uim ¼ uim Lx rð1 n2 Þ=E. According to these tables, close agreements exist between the present results and those obtained from other works using the higher order shear deformation theory and the three-dimensional elasticity theory. In Table 4, the results are compared for functionally graded simply supported curved panel with those obtained based on the higher order shear deformation theory using the power series expansion method [19]. The effects of variation of the material exponent and geometrical parameters on the fundamental frequency are demonstrated. Here again close agreement exists between the results. It should be mentioned that since in the
p=0 p=1 p=5 p=10
0.8 0.6 0.4 0.2 0 0.1
b
p=0 p=1 p=5 p=10
ϖ 11
c 0.2
0.3
0.4
ϖ 12
0.2
0.3
0.4
0.5
1.8 1.44
0.5
p=0 p=1 p=5 p=10
1.08
ϖ 13
0.6 p=0 p=1 p=5 p=10
0.72 0.36
0.3
0 0.1
0.2
0.2
0.3
h/R m
0.1 0 0.1
0.2
0.3
0.4
0.5
h/R m 1 p=0 p=1 p=5 p=10
ϖ 13
0.6 0.4 0.2 0 0.1
0.5
p=0 p=1 p=5 p=10
h/R m
h/R m
0.8
0.4
1.4
0 0.1
0 0.1
c
0.5
0.28
0.1
0.4
0.4
0.56
0.2
0.5
0.3
h/R m
1.12
0.3
b
0.2
0.84
0.5 0.4
1
ϖ 12
a
a
ϖ 11
478
0.2
0.3
0.4
0.5
h/R m Fig. 8. (a)e(c) Variation of the frequency parameters verses thickness-to-mean radius ratios with different values of material properties exponent for FGM curved panel with two opposite edges simply supported and other edges free ðLx ¼ 1 m; qo ¼ 1 radÞ.
Fig. 9. (a)e(c) Variation of the frequency parameters verses thickness-to-mean radius ratios with different values of material properties exponent for FGM curved panel with three edges simply supported and one edge clamped ðLx ¼ 1 m; qo ¼ 1 radÞ.
previous works [19,22,35,36] the variation of radius of curvature were not considered, the same assumption is employed here to obtain the results in Tables 2e4. In Table 5, comparison of the nine frequency parameters for simply supported FGM curved panels based on the constant radius of curvature and the variable radius of curvature formulation through the thickness are presented. The effects of various thickness-to-mean radius ratios and the panel angle on the frequency parameters are demonstrated. Hence, in the following examples, variable radius of curvature formulation is used to obtain the results. Based on the solved examples above, the convergence behavior and validity of the presented approach are demonstrated. Hereafter, the effects of the material properties and geometrical parameters on the frequency parameters of functionally graded curved panels under various boundary conditions are presented. In Figs. 2e5, variations of the frequency parameters versus length-to-mean radius ratios for different values of the panel angle of FGM curved panels under various boundary conditions are
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
479
Table 6 pffiffiffiffiffiffiffiffiffiffiffiffiffi The nine frequency parameters ½6im ¼ uim h ro =Eo for FGM curved panel with two opposite edges simply supported and other edges free based on the second material model formulation (Lx ¼ 1 m).
qo
h/Rm
g
Frequency parameters 611
612
613
621
622
623
631
632
633
p=3
0.3
1 2 3 1 2 3
0.1722 0.1574 0.1397 0.4125 0.3731 0.3296
0.2326 0.2162 0.1966 0.4665 0.4347 0.4013
0.4778 0.4625 0.4479 0.8667 0.8218 0.7770
0.6275 0.5801 0.5240 1.2750 1.1663 1.0548
0.6996 0.6527 0.5967 1.3087 1.2299 1.1450
0.9532 0.9078 0.8485 1.6546 1.5752 1.5065
1.1500 1.0688 0.9774 2.0886 1.9337 1.7834
1.2034 1.1256 1.0367 2.0929 1.9809 1.8598
1.3781 1.3105 1.2277 2.3031 2.2201 2.1345
1 2 3 1 2 3
0.0253 0.0230 0.0203 0.0669 0.0598 0.0521
0.0378 0.0346 0.0311 0.0817 0.0745 0.0671
0.3000 0.2921 0.2845 0.5323 0.5103 0.4887
0.1722 0.1574 0.1397 0.4125 0.3731 0.3296
0.2326 0.2162 0.1966 0.4665 0.4347 0.4013
0.4778 0.4625 0.4479 0.8667 0.8218 0.7770
0.3836 0.3529 0.3163 0.8390 0.7642 0.6840
0.4591 0.4277 0.3898 0.8911 0.8350 0.7749
0.7151 0.6946 0.6761 1.2562 1.1927 1.1332
0.5
2p=3
0.3
0.5
Table 7 pffiffiffiffiffiffiffiffiffiffiffiffiffi The nine frequency parameters ½6im ¼ uim h ro =Eo for FGM curved panel with two opposite edges simply supported and other edges clamped based on the second material model formulation (Lx ¼ 1 m).
qo
h/Rm
g
Frequency parameters 611
612
613
621
622
623
631
632
633
p=3
0.3
1 2 3 1 2 3
0.4840 0.4605 0.4305 0.9505 0.9058 0.8535
0.9323 0.8969 0.8485 1.6734 1.6515 1.5953
1.0281 1.0207 1.0157 1.7023 1.6830 1.6679
0.8030 0.7514 0.6900 1.5592 1.4490 1.3370
1.1617 1.1058 1.0363 2.0891 2.0125 1.9166
1.3429 1.3216 1.3050 2.2550 2.1800 2.1309
1.2702 1.1851 1.0900 2.2919 2.1281 1.9751
1.5423 1.4550 1.3563 2.7053 2.5652 2.4240
1.7812 1.7503 1.7248 2.8786 2.8087 2.7601
1 2 3 1 2 3
0.4506 0.4340 0.4121 0.8489 0.8229 0.7893
0.7620 0.7544 0.7480 1.2585 1.2426 1.2291
0.8912 0.8575 0.8131 1.6188 1.5757 1.5169
0.4840 0.4605 0.4305 0.9505 0.9058 0.8535
0.9323 0.8969 0.8485 1.6734 1.6515 1.5953
1.0281 1.0207 1.0157 1.7023 1.6830 1.6679
0.6117 0.5756 0.5313 1.2156 1.1395 1.0571
1.0204 0.9775 0.9210 1.8354 1.7901 1.7210
1.1616 1.1462 1.1347 1.9870 1.9201 1.8753
0.5
2p=3
0.3
0.5
presented. According to the results presented in these figures, increasing the panel length, in nearly all cases the frequency parameters decrease. This is because the flexural stiffness of the panel decreases by increasing its length. Also, one can see that increasing the panel angle, the frequency parameters decrease due to decrease of overall stiffness of the panel. The effects of thickness-to-mean radius ratios for different values of material power law index (p) and under various boundary conditions are presented in Figs. 6e9. It is found that by increasing the thickness-to-mean radius ratio, the frequency parameters increase while the material index has reverse effect. Since by increasing the material property exponent (p) the material of the panel changes from ceramic rich to metal rich, the stiffness of the panel reduces and hence the frequency parameters should decrease. This phenomenon can be seen in Figs. 6e9 for panels with different boundary conditions. Based on the exponential form of the material properties variation, the effects of length-to-mean radius ratios, the panel angle and different values of the material property index ðgÞ on the nine frequency parameters are studied in Table 6, for FGM curved panel with two edges simply supported and the other edges free. Also, in the case of FGM curved panel with two opposite edges simply supported and the other edges clamped, the results are presented in Table 7.
predictions of the frequency parameters, the variational effect of the radius through the thickness of the curved panel was considered. Trigonometric functions in circumferential direction and DQM in longitudinal and thickness directions were used to discretize the governing equations and related boundary conditions as a powerful numerical method. The material properties variation through the thickness was easily taken into account using this method. The convergence of the method was studied for different values of geometrical parameters and the accuracy of the results was verified by comparing the results with those of other shell theories such as higher order shear deformation theories and the three-dimensional elasticity, obtained using conventional methods. The effects of various geometrical and material parameters on frequency parameters were studied. It was shown that increasing the panel length, in nearly all cases the frequency parameters decrease. Also, it is found that decreasing the panel angle, the frequency parameters increase due to increase of overall stiffness of the panel. On the other hand, increasing the material property exponent (p) since the material of the panel changes from ceramic rich to metal rich, the stiffness of the panel reduces and hence the frequency parameters decrease. The presented results can be used as a benchmark solution to validate the other numerical methods and also to study the effectiveness and accuracy of the two-dimensional theories such as the classical and the first order shear deformation theories.
4. Conclusion
Appendix A. A brief review of the differential quadrature method
Based on the three-dimensional elasticity theory, free vibration analysis of thick functionally graded curved panels under various boundary conditions was studied. In order to achieve accurate
The basic idea of the differential quadrature method is that the derivative of a function, with respect to a space variable at a given
480
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
sampling point, is approximated as a weighted linear sum of the sampling points in the domain of that variable. In order to illustrate the DQ approximation, consider a functionf ðx; hÞ having its field on a rectangular domain 0 x a and 0 h b. Let, in the given domain, the function values be known or desired on a grid of sampling points. According to DQ method, the rth derivative of the function f ðx; hÞ can be approximated as Nx Nx X X vr f ðx; hÞ xðsÞ xðsÞ j ¼ Aim f xm ; hj ¼ Aij fmj ; r vx ðx;hÞ¼ðxi ;hj Þ m¼1 m¼1
for i
¼ 1; 2; .; Nx and s ¼ i ¼ 1; 2; .; Nx 1 (A.1) From this equation one can deduce that the important components xðsÞ of DQ approximations are the weighting coefficients ðAij Þ and the choice of sampling points. In order to determine the weighting coefficients a set of test functions should be used in Eq. (A.1). For polynomial basis functions DQ, a set of Lagrange polynomials are employed as the test functions. The weighting coefficients for the first-order derivatives in xi -direction are thus determined as [23]
1 Mðxi Þ 8
for isj a x xj M xj > i < Nx P Axij ¼ ; > Axij for i ¼ j : j ¼ 1 is j
i; j ¼ 1; 2; .; Nx ;
(A.2)
QN where Mðxi Þ ¼ j ¼x 1; isj ðxi xj Þ. The weighting coefficients of the second order derivative can be obtained as [23],
h i h ih i h i2 Bxij ¼ Axij Axij ¼ Axij :
(A.3)
In a similar manner, the weighting coefficients for the h-direction can be obtained. A simple and natural choices of the grid distribution is the uniform grid spacing rule, however, it was found that non-uniform grid spacing yields results with better accuracy. Hence, in this study, the ChebysheveGausseLobatto quadrature points are used, that is [23],
xi a
¼
( " #) h 1 ði 1Þp ; j 1 cos b 2 N 1 x
1 ðj 1Þp
1 cos ¼ 2 Nh 1 ¼ 1; 2; .; Nh
for i ¼ 1; 2; .; Nx ; j (A.4)
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[5] Praveen GN, Reddy JN. Nonlinear transient thermal elastic analysis of functionally graded ceramicemetal plates. Int J Solids Struct 1998;35:4457e76. [6] Reddy JN, Chin CD. Thermomechanical analysis of functionally graded cylinders and plates. J Therm Stress 1998;21:593e626. [7] Zimmerman RW, Lutz MP. Thermal stresses and thermal expansion in a uniformly heated functionally graded cylinder. J Therm Stress 1999;22:177e88. [8] Cho JR, Oden JT. Functionally graded material: a parameter study on thermalstress characteristics using the CrankeNicolsoneGalerkin scheme. Comput Meth Appl Mech Eng 2000;188:17e38. [9] Chunyu L, Weng GJ, Duan Z. Dynamic behavior of a cylindrical crack in a functionally graded interlayer under torsional loading. Int J Solids Struct 2001;38:7473e85. [10] Lanhe WU. Thermal buckling of a simply supported moderately thick rectangular FGM plate. Compos Struct 2004;64:211e8. [11] Loy CT, Lam KY, Reddy JN. Vibration of functionally graded cylindrical shells. Int J Mech Sci 1999;41:309e24. [12] Pradhan SC, Loy CT, Lam KY, Reddy JN. Vibration characteristics of functionally graded cylindrical shells under various boundary conditions. Appl Acoustics 2000;61:119e29. [13] Ng TY, Lam KM, Liew KM, Reddy JN. Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading. Int J Solids Struct 2001;38:1295e309. [14] Han X, Liu GR, Xi ZC, Lam KY. Transient waves in a functionally graded cylinder. Int J Solids Struct 2001;38:3021e37. [15] Pradhan SC. Vibration Suppression of FGM shells using embedded magnetostrictive layers. Int J Solids Struct 2005;42:2465e88. [16] Kadoli R, Ganesan N. Buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified boundary condition. J Sound Vib 2006;289:450e80. [17] Sheng GG, Wang X. Thermomechanical vibration analysis of a functionally graded shell with flowing fluid. European J Mech A/Solids 2008;27:1075e87. [18] Yang J, Shen HS. Free vibration and parametric resonance of shear deformable functionally graded cylindrical panels. J Sound Vib 2003;261:871e93. [19] Matsunaga H. Free vibration and stability of functionally graded shallow shells according to a 2-D higher-order deformation theory. Compos Struct 2008;84:132e46. [20] Matsunaga H. Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory. Compos Struct 2009;88:519e31. [21] Bhimaraddi A. Three-dimensional elasticity solution for static response of orthotropic doubly curved shallow shells on rectangular platform. Compos Struct 1993;24:67e77. [22] Chern YC, Chao CC. Comparison of natural frequencies of laminates by 3Dtheory, part II: curved panels. J Sound Vib 2000;230:1009e30. [23] Bert CW, Malik M. Differential quadrature method in computational mechanics: a review. Appl Mech Rev 1996;49:1e27. [24] Bert CW, Malik M. Differential quadrature method: a powerful new technique for analysis of composite structures. Compos Struct 1997;39:179e89. [25] Karami G, Malekzadeh P. A new differential quadrature methodology for beam analysis and the associated DQEM. Comput Meth Appl Mech Eng 2002;191:3509e26. [26] Karami G, Malekzadeh P. Application of a new differential quadrature methodology for free vibration analysis of plates. Int J Numer Meth Eng 2003;56:847e67. [27] Chio ST, Chou YT. Vibration analysis of non-circular curved panels by the differential quadrature method. J Sound Vib 2003;259:525e39. [28] Malekzadeh P, Karami G. In-plane free vibration analysis of circular arches with varying cross section. J Sound Vib 2004;274:777e99. [29] Civalek Ö, Ülker M. Harmonic differential quadrature (HDQ) for axisymmetric bending analysis of thin isotropic circular plates. Int J Structural Eng Mech 2004;17:1e14. [30] Karami G, Malekzadeh P, Mohebpour SR. 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Contents lists available at ScienceDirect
International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
A semi-analytical three-dimensional free vibration analysis of functionally graded curved panels P. Zahedinejad a, P. Malekzadeh b, c, *, M. Farid a, G. Karami d a
Department of Mechanical Engineering, Islamic Azad University, Branch of Shiraz, Shiraz, Iran Department of Mechanical Engineering, Persian Gulf University, Persian Gulf University Boulevard, Bushehr 75168, Iran c Center of Excellence for Computational Mechanics, Shiraz University, Shiraz, Iran d Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Fargo, ND 58105-5285, USA b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 February 2009 Received in revised form 29 May 2010 Accepted 11 June 2010
Based on the three-dimensional elasticity theory, free vibration analysis of functionally graded (FG) curved thick panels under various boundary conditions is studied. Panel with two opposite edges simply supported and arbitrary boundary conditions at the other edges are considered. Two different models of material properties variations based on the power law distribution in terms of the volume fractions of the constituents and the exponential distribution of the material properties through the thickness are considered. Differential quadrature method in conjunction with the trigonometric functions is used to discretize the governing equations. With a continuous material properties variation assumption through the thickness of the curved panel, differential quadrature method is efficiently used to discretize the governing equations and to implement the related boundary conditions at the top and bottom surfaces of the curved panel and in strong form. The convergence of the method is demonstrated and to validate the results, comparisons are made with the solutions for isotropic and FG curved panels. By examining the results of thick FG curved panels for various geometrical and material parameters and subjected to different boundary conditions, the influence of these parameters and in particular, those due to functionally graded material parameters are studied. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Free vibration Curved panels Functionally graded material
1. Introduction Layered composite materials, due to their thermal and mechanical merits compared to single-composed materials, have been widely used for a variety of engineering applications. However, owing to the sharp discontinuity in the material properties at interfaces between two different materials, there may exist stress concentrations resulting in severe material failure [1,2]. Functionally graded materials (FGMs) are heterogeneous composite materials, in which the material properties vary continuously from one interface to the other. The advantage of using these materials is that they can survive the high thermal gradient environment, while maintaining their structural integrity. Typically, an FGM is made of a ceramic and a metal for the purpose of thermal protection against large temperature gradients. The ceramic material provides the high-temperature resistance due to
* Corresponding author at: Department of Mechanical Engineering, Persian Gulf University, Persian Gulf University Boulevard, Bushehr 75168, Iran. Tel.: þ98 771 4222150; fax: þ98 771 4540376. E-mail addresses: [email protected], [email protected] (P. Malekzadeh). 0308-0161/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2010.06.001
its low thermal conductivity, while the ductile metal constituent prevents fracture due to its greater toughness. FGMs are now developed for general use as structural elements in extremely high temperature environments. A listing of different applications can be found in Ref. [3]. Most of the studies on FGMs have been restricted to thermal stress analysis, thermal buckling, fracture mechanics and optimization [4e10]. Little attention has been given to vibration analysis of FGM plates, shells and panels. Despite the evident importance in practical applications, investigations on the dynamic characteristics of FGM shell structures are still limited in number. Some researches [11e14], are based on the classical shell theories, i.e., neglecting the effect of transverse shear deformation. The application of such theories to moderately thick or thick shell structures can lead to serious errors. In order to account the effect of transverse shear, some other studies have been conducted based on the first order shear deformation theory [15e17]. Consequently, the higher order shear deformation theories were developed to improve the first order shear deformation theory [18e20]. Yang and Shen [18] gave a semi-analytical approach based on Reddy’s higher order shear deformation shell theory, for free vibration and dynamic instability of simply supported FGM
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
Nomenclature Axij, Azij b {b} Bxij, Bzij C Co [C] {d} E(z) EC EM Eo G h I Lx [M] Nx Nz p r Rm
The first order DQ weighting coefficients along the xand the z-directions Arc length of panel (¼Rmqo) Vector of boundary degrees of freedom The second order DQ weighting coefficients along the x-and the z-directions A typical FGM material property constant A typical FGM material property constant at plane z ¼ 0 Material stiffness matrix Vector of domain degrees of freedom Young’s modulus Young’s modulus of ceramic Young’s modulus of metal Young’s modulus at plane z ¼ 0 Shear modulus Thickness of panel pffiffiffiffiffiffiffi Imaginary number ð ¼ 1Þ Axial length of panel Mass matrix Number of the DQ discretization points in the xdirection Number of the DQ discretization points in the zdirection The power law index Radius at an arbitrary point of panel Mean radius of panel
cylindrical panels under combined static and periodic axial forces. Free vibration and stability of simply supported functionally graded shallow shells according to a 2-D higher-order shear deformation theory was proposed by Matsunaga [19]. In another work, based on the 2-D higher order shear deformation theory, free vibration and stability of functionally graded circular cylindrical shells is recently studied by Matsunaga [20]. In these studies the variation of the radius through the thickness is not considered and the problem formulations are based on constant radius of curvature through the thickness of the panel. Two-dimensional theories reduce the dimensions of problems from three to two by introducing some assumptions in mathematical modeling which leads to simple expressions and derivation of solutions. However, these simplifications inherently bring errors and therefore may lead to unreliable results for relatively thick curved panels. As a result, three-dimensional analysis of thick curved panels not only provides realistic results but also allows further physical insights, which cannot otherwise be predicted by two-dimensional analysis. There have been some studies on free vibration analysis of isotropic and composite panels and shallow shells based on the three-dimensional elasticity formulation [21,22], but to the author’s knowledge, no work has been published on the free vibration analysis of thick FGM curved panels. Due to the intrinsic complexity of the problem based on the three-dimensional elasticity, powerful numerical methods are needed to solve the governing equations with related boundary conditions. Hence, the differential quadrature method (DQM) is used in this work. The method has been widely used for static and free vibration analysis of beams, plates, shells and panels [23e33]. In the application of DQM for such problems, it was concluded that accurate results with less computations can be obtained.
471
½Sdd ; ½Sbb ; ½Sdb ; ½Sbd Stiffness matrices T Kinetic energy u Displacement component along the axial direction V Total linear elastic strain energy v Displacement component along the circumferential direction Volume fraction [¼(z/h)p] Vf w Displacement component along the thickness direction x Axial coordinate variable z Radial coordinate variable (thickness direction) 3ij Strain tensor components (i, j ¼ r, q, z) g Material property graded index n Poisson’s ratio q Circumferential coordinate variable qo Panel angle r(z) Mass per unit volume rC Mass per unit volume of ceramic rM Mass per unit volume of metal ro Mass per unit volume at plane z ¼ 0 sij Stress tensor components (i, j ¼ r, q, z) u Natural frequency of the panel Non-dimensional frequency parameter 6mn pffiffiffiffiffiffiffiffi Umn Non-dimensional frequency parameter ½ ¼ umn h r=G Umn Non-dimensional frequency parameter ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ ¼ umn Lx rð1 n2 Þ=E Means the function value at the DQ grid point z ¼ zj ($)j
2. Theoretical formulations 2.1. FGM material properties An FGM cylindrical panel with its coordinate system (x, q, z) is shown in Fig. 1, where x and q are the axial and circumferential coordinate variables of the panel and z is the coordinate variable of the thickness coordinate. The origin of the coordinate system is located at the corner of the panel. The panel has mean radius Rm, thickness h, axial length Lx, total panel angle qo and arc length Ly(¼Rmqo). It is assumed that the panel is made of a mixture of ceramics and metals, and the material composition is continuously varied such that the top surface (z ¼ h) of the panel is ceramic only,
Fig. 1. Geometry of functionally graded curved panel.
472
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
whereas the bottom surface (z ¼ 0) is metal only. In the first model the modulus of elasticity E and mass density r are assumed to be in terms of a simple power law distribution of volume fraction as follows
EðzÞ ¼ EM þ ECM Vf ; nðzÞ ¼ n; rðzÞ ¼ rM þ rCM Vf
where ið ¼ 0; 1; 2; .Þ is circumferential wave number. Ui, Vi and Wi are unknown displacement functions in the x, q, and z-directions, respectively. Using Eq. (6) in conjunction with Hamilton’s principle, one obtains, Governing equations:
dUi :C55
where
ECM ¼ EC EM ; rCM ¼ rC rM ; Vf ¼ ðz=hÞp
(2)
p is the power law index which takes values greater than or equal to zero. The value of p equal to zero represents a fully ceramic panel and infinite p shows a fully metallic panel. Subscripts M and C refer to the metal and ceramic constituents, which denote the material properties of the bottom and top surfaces of the curved panel, respectively. In the second model the variation of the material properties is assumed to have exponential distribution along the thickness of the curved panel as follows
C ¼ Co eðgz=hÞ
! ip vUi v2 V i C44 dC44 vVi þ þ C44 þ r dz vz r qo vx vz2 ! 2 2 C44 dC44 ip v Vi Vi C22 þ Vi þ C66 rdz r qo r2 vx2 C22 þ C44 dC44 ip ip vWi þ Wi þ ðC23 þ C44 Þ þ r dz r qo r qo vz ! v2 V i r ¼ 0 ð8Þ vt 2 v 2 Ui C12 C13 dC13 vUi ðC23 þ C44 Þ vxvz r dz vx ! ip vVi C22 þ C44 dC23 ip v2 Wi Vi þ C33 þ r dz r qo vz r qo vz2 2 C33 dC33 vWi C22 dC23 ip Wi C44 Wi þ þ r dz vz rdz r qo r2 ! ! v2 Wi v2 Wi r ¼ 0 ð9Þ þ C55 2 vx vt 2
dWi :ðC13 þ C55 Þ
Based on the three-dimensional small deformation theory of elasticity, the strainedisplacement relations can be expressed as
3xx ¼
(4)
Also, the three-dimensional constitutive relations for the FGM curved panel can be written as
8 9 sxx > > > > > > > s > > > > < qq > =
2
C11 6 C12 6 6 C13 szz ¼ 6 6 0 s > > zq > > 6 > > > > 4 0 s > > xz > > : ; s xq 0
C12 C22 C23 0 0 0
C13 C23 C33 0 0 0
0 0 0 C44 0 0
0 0 0 0 C55 0
0 0 0 0 0 C66
9 38 3xx > > > > > > > 7> > > 3qq > 7> 7< 3zz = 7 7> gzq > > 7> > > 5> g > > > > ; : xz >
qo
ipq ; ¼ Vi ðx; z; tÞcos
qo
ipq ¼ Wi ðx; z; tÞsin
qo
The boundary conditions along the free surfaces at z ¼ 0 and z ¼ h become,
vUi vWi vVi Vi ip Wi ¼ 0; þ ¼ 0; C44 þ vz vx vz r r qo vUi ip W vWi C23 ¼ 0 V i þ C33 C13 vx r vz r qo i C55
(5)
gx q
ð10Þ
Also, the natural and geometrical boundary conditions along the edges x ¼ 0 and x ¼ Lx become,
Using Eqs. (4) and (5) in conjunction with Hamilton’s principle [34], the equations of motion for free vibration analysis of FGM curved panel can be obtained. For panels with two opposite edges simply supported, the displacement components can be expanded in terms of normal mode functions in the direction normal to these edges. In this study, it is assumed that the edges q ¼ 0 and q ¼ qo are simply supported. Hence,
ipq ; uðx; q; z; tÞ ¼ Ui ðx; z; tÞsin
þ
2.2. The basic formulations
vu 1 vv w vw þ ; 3zz ¼ ; 3qq ¼ ; gxq vx r vq r vz vv 1 vu vw vu 1 vw vv v ¼ ; gxz ¼ þ þ þ ; g zq ¼ vx r vq vx vz r vq vz r
v2 Ui vz2
dVi :ðC12 þ C66 Þ
(3)
where C stands for the material stiffness matrix components Cij ði; j ¼ 1; .; 6Þ and the mass density r. Co is the corresponding values at the plane z ¼ 0 (Metallic layer) and g is the material property graded index. In both models the Poisson’s ratio n is assumed to be constant.
!
2 C55 dC55 vUi ip Ui þ C66 r dz vz r qo ! v2 Ui ip vVi v2 Wi ðC þ C11 þ C Þ þ ðC13 þ C55 Þ 12 66 2 q vx vxvz r vx o ! C12 þ C55 dC55 vWi v 2 Ui þ ¼ 0 ð7Þ þ r r dz vx vt 2
(1)
Either Ui ¼ 0
or
C11
vUi ip W C12 Vi i vx r r qo vWi þ C13 ¼ 0 vz
Either Vi ¼ 0
or
C66
vðx; q; z; tÞ
Either Wi ¼ 0
wðx; q; z; tÞ (6)
or
C55
ip vV Ui þ i ¼ 0 vx r qo
vUi vWi þ vz vx
ð11Þ
(12)
¼ 0
(13)
Various boundary conditions along the edges at x ¼ 0 and x ¼ Lx can be obtained by combining the conditions stated in Eqs. (11)e(13) as,
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
Simply supported (S):
C11
vUi ip W vWi Vi i þC13 C12 ¼ 0; Vi ¼ 0; Wi ¼ 0 vx r vz r qo
Table 2 pffiffiffiffiffiffiffiffi Comparison of the frequency parameters Uim ¼ uim h r=G for isotropic curved panel under simply supported boundary condition (Lx ¼ Ly).
U11
(14)
Lx/h
Lx/R
CST
FSDT
Matsunaga
Present
2
0.2 0.4 0.2 0.4
2.0189 1.9983 0.3739 0.3760
1.4927 1.4891 0.3415 0.3442
1.5140 1.5088 0.3429 0.3454
1.5100 1.4933 0.3423 0.3429
0.2 0.4 0.2 0.4
2.2306 2.2545 0.8896 0.8927
2.2252 2.2362 0.8896 0.8926
2.2253 2.2365 0.8896 0.8926
2.2272 2.2437 0.8897 0.8931
5
Clamped (C):
Ui ¼ 0;
Vi ¼ 0;
Wi ¼ 0
(15)
U12
2 5
Free (F):
vUi ip W vWi C12 ¼ 0; Vi i þ C13 vx r vz r qo ip vV vUi vWi C66 Ui þ i ¼ 0; C55 þ ¼ 0 vx vz vx r qo
C11
2.3. Differential quadrature discretization
ð16Þ
For the free vibration analysis, the following solutions may be assumed for the displacement components
Ui ðx; z; tÞ ¼ U i ðx; zÞeIui t ;
Vi ðx; z; tÞ ¼ V i ðx; zÞeIui t ;
Wi ðx; z; tÞ ¼ W i ðx; zÞeIui t
ðC55 Þj
Nz X n¼1
473
ð17Þ
At this stage, the circumferentially discretized governing differential equations and the related boundary conditions are transformed into algebraic equations via the DQ method. Using the DQ discretization rules for spatial derivatives (please see Appendix A) and Eq. (17), the DQ analogs of the governing differential equations are obtained as Eq. (7):
! ! !2 X Nz Nz Nx X 1X dC55 ip z z x þ þ ðC11 Þj þ A U A U Bkm U imj ðC66 Þj U rj n ¼ 1 jn ikn dz j n ¼ 1 jn ikn rj qo m¼1 ikj ! # " Nz Nx Nx X N x X X X ðC12 þ C55 Þj ip dC55 Ax V þ ðC13 þ C55 Þj Axkm Azjn W imn þ þ Axkm W imj þ rj u2i U ikj ¼ 0 ð18Þ dz r rj qo m ¼ 1 km imj j j m¼1 n¼1 m¼1
Bzjn U ikn
ðC12 þ C66 Þj
Eq. (8):
ðC12 þ C66 Þj þ ðC22 Þj
! ! " X Nz Nz ðC44 Þj 1 dC44 1X dC44 z z þ ðC44 Þj þ Ajn V ikn þ Ajn V ikn þ dz j n ¼ 1 rj n ¼ 1 rj dz j rj2 m¼1 n¼1 !2 # ! # ! " N N z x X ðC22 þ C44 Þj ip ip X dC44 ip x z V irs þ ðC66 Þj Bkm V imj þ ðC23 þ C44 Þj A W þ þ W ikj rj dz j rj qo rj qo rj qo n ¼ 1 jn ikn m¼1
ip rj qo
!
Nx X
Axkm U imj
Nz X
Bzjn V ikn
þ rj u2i V ikj ¼ 0
ð19Þ
Table 1 Convergence behavior of the frequency parameters ð6im Þ for FGM curved panel with two opposite edges simply supported and other edges free, based on the first FGM material model formulation (Lx ¼ 1 m, Rm ¼ 1 m, p ¼ 1, qo ¼ 2p=3). Material property
h/Rm
Power law
0.3
0.5
Exponential
0.3
0.5
Nz ¼ 7
Nz ¼ 9
Nz ¼ 13
Nx ¼ 9
Nx ¼ 13
Nx ¼ 17
Nx ¼ 9
Nx ¼ 13
Nx ¼ 17
Nx ¼ 9
Nx ¼ 13
Nx ¼ 17
611 612 621 622 611 612 621 622
0.0196 0.0309 0.1332 0.1898 0.0513 0.0683 0.3204 0.3963
0.0196 0.0308 0.1332 0.1895 0.0513 0.0682 0.3204 0.3961
0.0196 0.0308 0.1332 0.1894 0.0514 0.0682 0.3204 0.3961
0.0197 0.0311 0.1337 0.1904 0.0519 0.0687 0.3222 0.3974
0.0197 0.0309 0.1336 0.1899 0.0519 0.0686 0.3222 0.3973
0.0197 0.0309 0.1336 0.1899 0.0519 0.0686 0.3222 0.3973
0.0197 0.0310 0.1337 0.1899 0.0519 0.0685 0.3225 0.3969
0.0197 0.0310 0.1337 0.1900 0.0519 0.0687 0.3225 0.3974
0.0197 0.0309 0.1337 0.1900 0.0519 0.0687 0.3225 0.3974
611 612 621 622 611 612 621 622
0.0254 0.0379 0.1721 0.2328 0.0666 0.0816 0.4112 0.4659
0.0254 0.0378 0.1720 0.2325 0.0666 0.0815 0.4112 0.4658
0.0254 0.0378 0.1720 0.2325 0.0666 0.0815 0.4112 0.4658
0.0254 0.0380 0.1722 0.2331 0.0669 0.0818 0.4125 0.4667
0.0254 0.0378 0.1722 0.2327 0.0669 0.0817 0.4125 0.4666
0.0254 0.0378 0.1722 0.2326 0.0669 0.0817 0.4125 0.4666
0.0254 0.0379 0.1722 0.2327 0.0669 0.0815 0.4126 0.4660
0.0254 0.0378 0.1722 0.2327 0.0669 0.0817 0.4126 0.4666
0.0254 0.0378 0.1722 0.2326 0.0669 0.0817 0.4126 0.4666
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Eq. (10):
Table 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Comparison of the fundamental frequency parameter Uim ¼ uim Lx rð1 n2 Þ=E for isotropic curved panel under simply supported boundary condition (R/h ¼ 10, i ¼ m ¼ 1). Lx/h
Method
10
Present FSDT [36] 3-D Elasticity [22]
0.5
1
1.5
2
1.3166 1.3360 1.3174
0.5526 0.5563 0.5505
0.4066 0.4044 0.3998
0.3566 0.3505 0.3461
Table 4 pffiffiffiffiffiffiffiffiffiffiffiffi Comparison of the fundamental frequency parameter 6im ¼ uim h rc =Ec for FGM curved panel with four edges simply supported (Lx ¼ Ly, i ¼ m ¼ 1). Lx/R
Method
p 0
0.5
1
4
10
N
2
0.5
Present HSDT [19] Present HSDT [19]
0.9187 0.9334 0.8675 0.9163
0.8013 0.8213 0.7578 0.8105
0.7260 0.7483 0.6870 0.7410
0.5797 0.6011 0.5475 0.5960
0.5245 0.5460 0.4940 0.5392
0.4770 0.4752 0.4496 0.4665
Present HSDT [19] Present HSDT [19]
0.2113 0.2153 0.2164 0.2239
0.1814 0.1855 0.1852 0.1945
0.1639 0.1678 0.1676 0.1769
0.1367 0.1413 0.1394 0.1483
0.1271 0.1328 0.1286 0.1380
0.1109 0.1096 0.1133 0.1140
5
0.5 1
"
n¼1
Azjn U ikn
þ
Nx X m¼1
! Axkm W imj
¼ 0;
! # V ikj ip þ W ikj ¼ 0 rj rj qo n¼1 " ! # N x X W ikj ip x Akm U imj ðC23 Þj V ikj ðC13 Þj rj rj qo m¼1 ðC44 Þj
Nz X
Azjn V ikn
Nz X
þ ðC33 Þj
Lx/h
1
Nz X
ðC55 Þj
Ly/Lx
n¼1
Azjn W ikn ¼ 0
ð21Þ
Eq. (11):
Either U ikj ¼ 0 W ikj rj
or
ðC11 Þj
Nx X m¼1
# þ ðC13 Þj
Nz X n¼1
" Axkm U imj ðC12 Þj
Azjn W ikn ¼ 0
! ip V ikj rj qo ð22Þ
Eq. (9):
! # X Nz Nx dC13 ip X Axkm U imj ðC23 þ C44 Þj Azjn V ikn q r dz r o j j j m¼1 n¼1 m¼1 n¼1 ! ! ! " # X Nz Nz Nz X X ðC22 þ C44 Þj dC23 ip 1 dC 33 z z z þ þ V ikj þ ðC33 Þj Bjn W ikn þ A W A W rj dz j rj qo dz j n ¼ 1 jn ikn rj n ¼ 1 jn ikn n¼1 !2 # " Nx X ðC22 Þj 1 dC23 ip þ ðC55 Þj Bxkm W imj þ 2 ðC44 Þj W ikj þ rj u2i W ikj ¼ 0 q r dz r r o j j j m¼1 j
ðC13 þ C55 Þj
Nz Nx X X
Axkm Azjn U imn
" ðC12 C13 Þj
where Nx and Nz are the number of the DQ discretization points through the longitudinal and thickness directions respectively and Axij , Bxij , Azij , Bzij are the first and second order weighting coefficients of the differential quadrature method through the x-and z-directions, respectively. Also, ðÞj means the function value at the DQ grid point z ¼ zj . The DQ analogs of the boundary conditions become,
ð20Þ
Eq. (12):
" Either V ikj ¼ 0
or
ðC66 Þj
! # Nx X ip x U ikj þ Akm V imj ¼ 0 rj qo m¼1 (23)
Table 5 pffiffiffiffiffiffiffiffiffiffiffiffi Comparison of the frequency parameters 6im ¼ uim h rc =Ec for FGM curved panel with four edges simply supported based on constant radius of curvature and variable radius formulation through the thickness (Lx ¼ 1 m, p ¼ 1).
qo
h/Rm
Case
p=3
0.1
a b
0.3
a b
0.5
a b
2p=3
0.1
a b
0.3
a b
0.5
a b
a b
Frequency parameters 611
612
613
621
622
623
631
632
633
0.0596 0.0468 0.3115 0.3113 0.6671 0.6731
0.1166 0.1165 0.4624 0.4555 0.7651 0.7535
0.1547 0.1536 0.6855 0.6850 1.1621 1.1558
0.0916 0.0931 0.6170 0.6151 1.2482 1.2505
0.1604 0.1587 0.9105 0.9084 1.4744 1.5055
0.2535 0.2515 0.9133 0.9109 1.6862 1.7343
0.1809 0.1811 1.0318 1.0251 1.9442 1.9523
0.2325 0.2322 1.2512 1.2509 2.0924 2.2536
0.3165 0.3154 1.3396 1.3551 2.2241 2.3353
0.0720 0.0669 0.2319 0.2279 0.3858 0.3768
0.0773 0.0768 0.2738 0.2734 0.5576 0.5551
0.1141 0.1021 0.5633 0.5634 0.9487 0.9466
0.0596 0.0468 0.3115 0.3113 0.6671 0.6731
0.1166 0.1165 0.4624 0.4555 0.7651 0.7535
0.1547 0.1536 0.6855 0.6850 1.1621 1.1558
0.0672 0.0675 0.4405 0.4392 0.9247 0.9302
0.1386 0.1279 0.6901 0.6825 1.1303 1.1298
0.2267 0.2234 0.7792 0.7804 1.4166 1.4266
Constant radius of curvature formulation. Variable radius formulation.
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
Eq. (13):
Either W ikj ¼ 0 or ðC55 Þj
Nz X n¼1
Azjn U ikn þ
Nx X
! Axkm W imj
¼ 0 ð24Þ
8 9 8 9 < = < fU g = ; d ¼ fVg : ; : ; fWg domain
475
9 8 9 8 < = < fU g = b ¼ g : ; : fV ; fW g boundary
(25)
m¼1
Using Eq. (25), the discretized form of the equations of motion in the matrix form can be rearranged as To obtain the eigenvalue system of equations, the degrees of freedom are separated into the domain and the boundary degrees of freedom as
h in o n o Sdb b þ ½Sdd d u2i ½Mfdg ¼ 0
a
(26)
0.9
θο=π/3 θο=π/2 θο=2π/3
0.72
ϖ 11
0.54 0.36 0.18 0 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
L x /R m
b
1.3
θο=π/3 θο=π/2 θο=2π/3
1.06
ϖ 12
0.82 0.58 0.34 0.1 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
L x /R m
c
1.6
θο=π/3 θο=π/2 θο=2π/3
1.28
ϖ 13
0.96 0.64 0.32 0 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
L x /R m Fig. 2. (a)e(c) Variation of the frequency parameters verses length-to-mean radius ratios with different values of total panel angle for FGM curved panel with four edges simply supported ðh=Rm ¼ 0:3; p ¼ 1Þ.
Fig. 3. (a)e(c) Variation of the frequency parameters verses length-to-mean radius ratios with different values of total panel angle for FGM curved panel with two opposite edges simply supported and other edges clamped ðh=Rm ¼ 0:3; p ¼ 1Þ.
476
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
where ½Sdb and ½Sdd are the stiffness matrices and ½M is the mass matrix. In a similar manner, the discretized form of the boundary conditions becomes
½Sbb fbg þ ½Sbd fdg ¼ 0
(27)
where ½Sbb and ½Sbd are the stiffness matrices. In the above equations, the elements of the stiffness and the mass matrices are
obtained based on the definition of the vectors of domain and boundary degrees of freedom from the DQ discretized form of the equations of motion and the boundary conditions. Using Eq. (27) to eliminate the boundary degrees of freedom fbg from Eq. (26), the result becomes,
h i n o S u2i ½M d ¼ 0
(28)
where
a
0.14
a
θο=π/3 θο=π/2 θο=2π/3
0.112
0.8 θο=π/3 θο=π/2 θο=2π/3
0.64
ϖ 11
0.084
ϖ 11
0.48
0.056
0.32
0.028 0.16
0 0.5
b
1
1.5
2
2.5 3 3.5 L x /R m
4
4.5
5
0.25
0 0.5
b
θο=π/3 θο=π/2 θο=2π/3
0.2
2
2.5
3
3.5
4
4.5
5
L x /R m 0.9 θο=π/3 θο=π/2 θο=2π/3
ϖ 12
ϖ 12
0.58
0.1
0.42
0.05
c
1.5
0.74
0.15
0 0.5
1
0.26
1
1.5
2
2.5 3 L x /R m
3.5
4
4.5
0.1 0.5
5
1
1.5
2
2.5
3
3.5
4
4.5
5
L x /R m
0.45
c
θο=π/3 θο=π/2 θο=2π/3
0.37
1.4 θο=π/3 θο=π/2 θο=2π/3
1.12
0.29
ϖ 13
ϖ 13
0.84
0.21
0.56
0.13
0.28
0.05 0.5
1
1.5
2
2.5 3 L x /R m
3.5
4
4.5
5
Fig. 4. (a)e(c) Variation of the frequency parameters verses length-to-mean radius ratios with different values of total panel angle for FGM curved panel with two opposite edges simply supported and other edges free ðh=Rm ¼ 0:3; p ¼ 1Þ.
0 0.5
1
1.5
2
2.5
3
L x /R m
3.5
4
4.5
5
Fig. 5. (a)e(c) Variation of the frequency parameters verses length-to-mean radius ratios with different values of total panel angle for FGM curved panel with three edges simply supported and one edge clamped ðh=Rm ¼ 0:3; p ¼ 1Þ.
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
h i S ¼ ½Sdd ½Sdb ½Sbb 1 ½Sbd The above equations can be solved to find the natural frequencies as well as the mode shapes of curved panels. 3. Numerical results In this section, first, the convergence behavior of the method is investigated and then comparisons with other available solutions are made to verify the accuracy of the results. The material properties for functionally graded panel are as follows,
Ec ¼ 380 109 Pa; 3
¼ 3800 kg=m ;
Em ¼ 70 109 Pa; 3
rm ¼ 2702 kg=m ;
rc n ¼ 0:3:
In all solved examples for functionally graded curved panels the non-dimensional frequency parameters ð6im Þ are defined as
a
1 0.8
0 0.1
0.2
0.3
0.4
0 0.1
0.5
h/R m
b
1 p=0 p=1 p=5 p=10
0.4
0.5
0.4
0.5
0.4
0.5
1.8 p=0 p=1 p=5 p=10
ϖ 12
ϖ 12
0.72 0.36
0.2 0 0.1
0.2
0.3
0.4
0 0.1
0.5
h/R m
c
1.5 p=0 p=1 p=5 p=10
0.2
0.3
h/R m 1.8 1.44
p=0 p=1 p=5 p=10
1.08
ϖ 13
ϖ 13
0.9
0.72
0.6
0.36
0.3 0 0.1
0.3
1.08
0.4
1.2
0.2
h/R m
1.44
0.6
c
0.72
0.24
0.2
0.8
p=0 p=1 p=5 p=10
0.48
0.4
b
1.2 0.96
ϖ 11
0.6
p=0 p=1 p=5 p=10
ϖ 11
a
477
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ½uim h rc =Ec for the first model and ½uim h ro =Eo for the second model of material properties variation. In all cases under consideration, the convergence behaviors against the number of DQ grid points along the length and the thickness directions of the curved panel were examined. However, for brevity purpose, here only those for functionally graded curved panels with two opposite edges simply supported and the two edges free are presented. In Table 1, the convergence behaviors of the frequency parameters for FGM curved panel based on the first and second models of the material properties variation are demonstrated. The results are presented for two different values of the thickness-to-mean radius ratios. It is obvious from this table that converged results are achieved with Nz ¼ 13 and Nx ¼ 17. In Table 2, comparisons of the frequency parameters are made between the present method and those obtained from other shell theories [35] for isotropic simply supported curved panel. Thepnonffiffiffiffiffiffiffiffi dimensional frequency parameter is defined as Uim ¼ uim h r=G.
0.2
0.3
0.4
0.5
h/R m Fig. 6. (a)e(c) Variation of the frequency parameters verses thickness-to-mean radius ratios with different values of material properties exponent for FGM curved panel with four edges simply supported ðLx ¼ 1 m; qo ¼ 1 radÞ.
0 0.1
0.2
0.3
h/R m Fig. 7. (a)e(c) Variation of the frequency parameters verses thickness-to-mean radius ratios with different values of material properties exponent for FGM curved panel with two opposite edges simply supported and other edges clamped ðLx ¼ 1 m; qo ¼ 1 radÞ.
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
Also in Table 3, comparison of the fundamental frequency parameter for isotropic simply supported curved panel are made between the results obtained using the present method and those obtained based on the three-dimensional elasticity formulation [22] and the first order shear deformation theory [36]. Here the non-dimensional formpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the frequency parameters is defined as ffi Uim ¼ uim Lx rð1 n2 Þ=E. According to these tables, close agreements exist between the present results and those obtained from other works using the higher order shear deformation theory and the three-dimensional elasticity theory. In Table 4, the results are compared for functionally graded simply supported curved panel with those obtained based on the higher order shear deformation theory using the power series expansion method [19]. The effects of variation of the material exponent and geometrical parameters on the fundamental frequency are demonstrated. Here again close agreement exists between the results. It should be mentioned that since in the
p=0 p=1 p=5 p=10
0.8 0.6 0.4 0.2 0 0.1
b
p=0 p=1 p=5 p=10
ϖ 11
c 0.2
0.3
0.4
ϖ 12
0.2
0.3
0.4
0.5
1.8 1.44
0.5
p=0 p=1 p=5 p=10
1.08
ϖ 13
0.6 p=0 p=1 p=5 p=10
0.72 0.36
0.3
0 0.1
0.2
0.2
0.3
h/R m
0.1 0 0.1
0.2
0.3
0.4
0.5
h/R m 1 p=0 p=1 p=5 p=10
ϖ 13
0.6 0.4 0.2 0 0.1
0.5
p=0 p=1 p=5 p=10
h/R m
h/R m
0.8
0.4
1.4
0 0.1
0 0.1
c
0.5
0.28
0.1
0.4
0.4
0.56
0.2
0.5
0.3
h/R m
1.12
0.3
b
0.2
0.84
0.5 0.4
1
ϖ 12
a
a
ϖ 11
478
0.2
0.3
0.4
0.5
h/R m Fig. 8. (a)e(c) Variation of the frequency parameters verses thickness-to-mean radius ratios with different values of material properties exponent for FGM curved panel with two opposite edges simply supported and other edges free ðLx ¼ 1 m; qo ¼ 1 radÞ.
Fig. 9. (a)e(c) Variation of the frequency parameters verses thickness-to-mean radius ratios with different values of material properties exponent for FGM curved panel with three edges simply supported and one edge clamped ðLx ¼ 1 m; qo ¼ 1 radÞ.
previous works [19,22,35,36] the variation of radius of curvature were not considered, the same assumption is employed here to obtain the results in Tables 2e4. In Table 5, comparison of the nine frequency parameters for simply supported FGM curved panels based on the constant radius of curvature and the variable radius of curvature formulation through the thickness are presented. The effects of various thickness-to-mean radius ratios and the panel angle on the frequency parameters are demonstrated. Hence, in the following examples, variable radius of curvature formulation is used to obtain the results. Based on the solved examples above, the convergence behavior and validity of the presented approach are demonstrated. Hereafter, the effects of the material properties and geometrical parameters on the frequency parameters of functionally graded curved panels under various boundary conditions are presented. In Figs. 2e5, variations of the frequency parameters versus length-to-mean radius ratios for different values of the panel angle of FGM curved panels under various boundary conditions are
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
479
Table 6 pffiffiffiffiffiffiffiffiffiffiffiffiffi The nine frequency parameters ½6im ¼ uim h ro =Eo for FGM curved panel with two opposite edges simply supported and other edges free based on the second material model formulation (Lx ¼ 1 m).
qo
h/Rm
g
Frequency parameters 611
612
613
621
622
623
631
632
633
p=3
0.3
1 2 3 1 2 3
0.1722 0.1574 0.1397 0.4125 0.3731 0.3296
0.2326 0.2162 0.1966 0.4665 0.4347 0.4013
0.4778 0.4625 0.4479 0.8667 0.8218 0.7770
0.6275 0.5801 0.5240 1.2750 1.1663 1.0548
0.6996 0.6527 0.5967 1.3087 1.2299 1.1450
0.9532 0.9078 0.8485 1.6546 1.5752 1.5065
1.1500 1.0688 0.9774 2.0886 1.9337 1.7834
1.2034 1.1256 1.0367 2.0929 1.9809 1.8598
1.3781 1.3105 1.2277 2.3031 2.2201 2.1345
1 2 3 1 2 3
0.0253 0.0230 0.0203 0.0669 0.0598 0.0521
0.0378 0.0346 0.0311 0.0817 0.0745 0.0671
0.3000 0.2921 0.2845 0.5323 0.5103 0.4887
0.1722 0.1574 0.1397 0.4125 0.3731 0.3296
0.2326 0.2162 0.1966 0.4665 0.4347 0.4013
0.4778 0.4625 0.4479 0.8667 0.8218 0.7770
0.3836 0.3529 0.3163 0.8390 0.7642 0.6840
0.4591 0.4277 0.3898 0.8911 0.8350 0.7749
0.7151 0.6946 0.6761 1.2562 1.1927 1.1332
0.5
2p=3
0.3
0.5
Table 7 pffiffiffiffiffiffiffiffiffiffiffiffiffi The nine frequency parameters ½6im ¼ uim h ro =Eo for FGM curved panel with two opposite edges simply supported and other edges clamped based on the second material model formulation (Lx ¼ 1 m).
qo
h/Rm
g
Frequency parameters 611
612
613
621
622
623
631
632
633
p=3
0.3
1 2 3 1 2 3
0.4840 0.4605 0.4305 0.9505 0.9058 0.8535
0.9323 0.8969 0.8485 1.6734 1.6515 1.5953
1.0281 1.0207 1.0157 1.7023 1.6830 1.6679
0.8030 0.7514 0.6900 1.5592 1.4490 1.3370
1.1617 1.1058 1.0363 2.0891 2.0125 1.9166
1.3429 1.3216 1.3050 2.2550 2.1800 2.1309
1.2702 1.1851 1.0900 2.2919 2.1281 1.9751
1.5423 1.4550 1.3563 2.7053 2.5652 2.4240
1.7812 1.7503 1.7248 2.8786 2.8087 2.7601
1 2 3 1 2 3
0.4506 0.4340 0.4121 0.8489 0.8229 0.7893
0.7620 0.7544 0.7480 1.2585 1.2426 1.2291
0.8912 0.8575 0.8131 1.6188 1.5757 1.5169
0.4840 0.4605 0.4305 0.9505 0.9058 0.8535
0.9323 0.8969 0.8485 1.6734 1.6515 1.5953
1.0281 1.0207 1.0157 1.7023 1.6830 1.6679
0.6117 0.5756 0.5313 1.2156 1.1395 1.0571
1.0204 0.9775 0.9210 1.8354 1.7901 1.7210
1.1616 1.1462 1.1347 1.9870 1.9201 1.8753
0.5
2p=3
0.3
0.5
presented. According to the results presented in these figures, increasing the panel length, in nearly all cases the frequency parameters decrease. This is because the flexural stiffness of the panel decreases by increasing its length. Also, one can see that increasing the panel angle, the frequency parameters decrease due to decrease of overall stiffness of the panel. The effects of thickness-to-mean radius ratios for different values of material power law index (p) and under various boundary conditions are presented in Figs. 6e9. It is found that by increasing the thickness-to-mean radius ratio, the frequency parameters increase while the material index has reverse effect. Since by increasing the material property exponent (p) the material of the panel changes from ceramic rich to metal rich, the stiffness of the panel reduces and hence the frequency parameters should decrease. This phenomenon can be seen in Figs. 6e9 for panels with different boundary conditions. Based on the exponential form of the material properties variation, the effects of length-to-mean radius ratios, the panel angle and different values of the material property index ðgÞ on the nine frequency parameters are studied in Table 6, for FGM curved panel with two edges simply supported and the other edges free. Also, in the case of FGM curved panel with two opposite edges simply supported and the other edges clamped, the results are presented in Table 7.
predictions of the frequency parameters, the variational effect of the radius through the thickness of the curved panel was considered. Trigonometric functions in circumferential direction and DQM in longitudinal and thickness directions were used to discretize the governing equations and related boundary conditions as a powerful numerical method. The material properties variation through the thickness was easily taken into account using this method. The convergence of the method was studied for different values of geometrical parameters and the accuracy of the results was verified by comparing the results with those of other shell theories such as higher order shear deformation theories and the three-dimensional elasticity, obtained using conventional methods. The effects of various geometrical and material parameters on frequency parameters were studied. It was shown that increasing the panel length, in nearly all cases the frequency parameters decrease. Also, it is found that decreasing the panel angle, the frequency parameters increase due to increase of overall stiffness of the panel. On the other hand, increasing the material property exponent (p) since the material of the panel changes from ceramic rich to metal rich, the stiffness of the panel reduces and hence the frequency parameters decrease. The presented results can be used as a benchmark solution to validate the other numerical methods and also to study the effectiveness and accuracy of the two-dimensional theories such as the classical and the first order shear deformation theories.
4. Conclusion
Appendix A. A brief review of the differential quadrature method
Based on the three-dimensional elasticity theory, free vibration analysis of thick functionally graded curved panels under various boundary conditions was studied. In order to achieve accurate
The basic idea of the differential quadrature method is that the derivative of a function, with respect to a space variable at a given
480
P. Zahedinejad et al. / International Journal of Pressure Vessels and Piping 87 (2010) 470e480
sampling point, is approximated as a weighted linear sum of the sampling points in the domain of that variable. In order to illustrate the DQ approximation, consider a functionf ðx; hÞ having its field on a rectangular domain 0 x a and 0 h b. Let, in the given domain, the function values be known or desired on a grid of sampling points. According to DQ method, the rth derivative of the function f ðx; hÞ can be approximated as Nx Nx X X vr f ðx; hÞ xðsÞ xðsÞ j ¼ Aim f xm ; hj ¼ Aij fmj ; r vx ðx;hÞ¼ðxi ;hj Þ m¼1 m¼1
for i
¼ 1; 2; .; Nx and s ¼ i ¼ 1; 2; .; Nx 1 (A.1) From this equation one can deduce that the important components xðsÞ of DQ approximations are the weighting coefficients ðAij Þ and the choice of sampling points. In order to determine the weighting coefficients a set of test functions should be used in Eq. (A.1). For polynomial basis functions DQ, a set of Lagrange polynomials are employed as the test functions. The weighting coefficients for the first-order derivatives in xi -direction are thus determined as [23]
1 Mðxi Þ 8
for isj a x xj M xj > i < Nx P Axij ¼ ; > Axij for i ¼ j : j ¼ 1 is j
i; j ¼ 1; 2; .; Nx ;
(A.2)
QN where Mðxi Þ ¼ j ¼x 1; isj ðxi xj Þ. The weighting coefficients of the second order derivative can be obtained as [23],
h i h ih i h i2 Bxij ¼ Axij Axij ¼ Axij :
(A.3)
In a similar manner, the weighting coefficients for the h-direction can be obtained. A simple and natural choices of the grid distribution is the uniform grid spacing rule, however, it was found that non-uniform grid spacing yields results with better accuracy. Hence, in this study, the ChebysheveGausseLobatto quadrature points are used, that is [23],
xi a
¼
( " #) h 1 ði 1Þp ; j 1 cos b 2 N 1 x
1 ðj 1Þp
1 cos ¼ 2 Nh 1 ¼ 1; 2; .; Nh
for i ¼ 1; 2; .; Nx ; j (A.4)
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