III European Conference on Computational Mechanics
Solids, Structures and Coupled Problems in Engineering
C.A. Mota Soares et.al. (eds.)
Lisbon, Portugal, 5–8 June 2006
DIFFERENTIAL QUADRATURE SOLUTION FOR PARABOLIC
STRUCTURAL SHELL ELEMENTS
Francesco Tornabene1, Erasmo Viola2
1
DISTART - Department, Faculty of Engineering, University of Bologna
Viale Risorgimento 2, 40136 Bologna, Italy
e-mail: [email protected]
2
DISTART - Department, Faculty of Engineering, University of Bologna
Viale Risorgimento 2, 40136 Bologna, Italy
e-mail: [email protected]
Keywords: GDQ method, sampling point distribution, free vibrations, parabolic shells, FSD
theory.
Abstract. This work deals with the dynamical behaviour of complete parabolic shells of revolution and parabolic shell panels. The First-order Shear Deformation Theory (FSDT) is used
to analyze the above moderately thick structural elements. The treatment is conducted within
the theory of linear elasticity, when the material behaviour is assumed to be homogeneous
and isotropic. The governing equations of motion, written in terms of internal resultants, are
expressed as functions of five kinematic parameters, by using the constitutive and the congruence relationships. The boundary conditions considered are clamped (C) and free (F) edge.
Numerical solutions have been computed by means of the technique known as the Generalized
Differential Quadrature (GDQ) Method. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. At the moment it
can only be pointed out that by using the GDQ technique the numerical statement of the problem does not pass through any variational formulation, but deals directly with the governing
equations of motion. Referring to the formulation of the dynamic equilibrium in terms of harmonic amplitudes of mid-surface displacements and rotations, in this paper the system of second-order linear partial differential equations is solved, without resorting to the onedimensional formulation of the dynamic equilibrium of the shell. The discretization of the system leads to a standard linear eigenvalue problem, where two independent variables are involved. Several examples of parabolic shell elements are presented to illustrate the validity
and the accuracy of GDQ method. The convergence rate of the natural frequencies is shown
to be very fast and the stability of the numerical methodology is very good. The accuracy of
the method is sensitive to the number of sampling points used, to their distribution and to the
boundary conditions. The effect of the distribution choice of sampling points on the accuracy
of GDQ solution is investigated. GDQ results, which are based upon the FSDT, are compared
with the ones obtained using commercial programs such as Ansys, Femap/Nastran, Abaqus,
Straus, Pro/Engineer.
Francesco Tornabene, Erasmo Viola
1
INTRODUCTION
Structures of shell revolution type have been widespread in many fields of engineering,
where they give rise to optimum conditions for dynamical behaviour, strength and stability.
Pressure vessels, cooling towers, water tanks, dome-shaped structures, dams, turbine engine
components and so forth, perform particular functions over different branches of structural
engineering.
The purpose of this paper is to study the dynamic behaviour of structures derived from
shells of revolution. The equations given here incorporate the effects of transverse shear deformation and rotary inertia.
The geometric model refers to a moderately thick shell. The solution is obtained by using
the numerical technique termed GDQ method, which leads to a generalized eigenvalue problem. The main features of the numerical technique under discussion, as well as its historical
development, are illustrated in section 3. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. Numerical results will also be computed by using commercial programs.
It should be noted that there are various two-dimensional theories of thin shells. Any twodimensional theory of shells is an approximation of the real three-dimensional problem. Starting from Love’s theory about the thin shells, which dates back to 100 years ago, a lot of contributions on this topic have been made since then. The main purpose has been that of seeking
better and better approximations for the exact three-dimensional elasticity solutions for shells.
In the last fifty years refined two-dimensional linear theories of thin shells have developed
including important contributions by Sanders [1], Flügge [2], Niordson [4]. In these refined
shell theories the deformation is based on the Kirchhoff-Love assumption. In other words, this
theory assumes that normals to the shell middle-surface remain normal to it during deformations and unstretched in length.
It is worth noting that when the refined theories of thin shells are applied to thick shells,
the errors could be quite large. With the increasing use of thick shells in various engineering
applications, simple and accurate theories for thick shells have been developed. With respect
to the thin shells, the thick shell theories take the transverse shear deformation and rotary inertia into account. The transverse shear deformation has been incorporated into shell theories by
following the work of Reissner [4] for the plate theory.
Several studies have been presented earlier for the vibration analysis of such revolution
shells and the most popular numerical tool in carrying out these analyses is currently the finite
element method. The generalized collocation method based on the ring element method has
also been applied [5,6]. With regard to the latter method each static and kinematic variable is
transformed into a theoretically infinite Fourier series of harmonic components, with respect
to the circumferential co-ordinates.
In this paper, the governing equations of motion are a set of five bi-dimensional partial differential equations with variable coefficients. These fundamental equations are expressed in
terms of kinematic parameters and can be obtained by combining the three basic sets of equations, namely balance, congruence and constitutive equations.
Referring to the formulation of the dynamic equilibrium in terms of harmonic amplitudes
of mid-surface displacements and rotations, in this paper the system of second-order linear
partial differential equations is solved. The discretization of the system leads to a standard linear eigenvalue problem, where two independent variables are involved. In this way it is possible to compute the complete assessment of the modal shapes corresponding to natural
frequencies of structures.
2
Francesco Tornabene, Erasmo Viola
2
BASIC GOVERNING EQUATIONS
The notation for the co-ordinates is shown in Figure 1. The co-ordinates along the meridional and circumferential directions are α1 = αϕ and α 2 = αϑ , respectively. The distance of each
point from the shell mid-surface along the normal is ζ . The total thickness of the shell is represented by h . The angle formed by the extended normal to the surface and the axis of rotation z or the geometric axis z ' of the meridian curve is defined as the meridional angle ϕ and
the angle between the radius of the parallel circle and the x axis is designated as the circumferential angle ϑ as shown on Figure 2. The parametric co-ordinates ( ϕ ,ϑ ) define, respectively, the meridional curves and the parallel circles upon the middle surface of the shell.
The geometry of the shell considered hereafter is a surface of revolution with a parabolic
curved meridian, that can be described with the following equation:
( R0 − c )
2
−kz =0
(2.1)
where k = ( s12 − d 2 ) S is the characteristic parameter of the parabola and d ∈ [0, s1[ . The horizontal radius R0 (ϕ ) of a generic parallel of the shell represents the distance of each point from
the axis of revolution and assumes the form:
R0 (ϕ ) = c +
k
tan ϕ
2
(2.2)
where c is the shift of the geometric axis of the parabolic meridian with reference to the axis
of revolution.
ζ
ζ ,w
Mid-surface
Reference
surface
O
βϑ
O
βϕ
α1 = αϕ
h
αϕ ,uϕ
α 2 = αϑ
αϑ ,uϑ
Figure 1: Co-ordinate system of the shell and reference surface.
The position of an arbitrary point within the shell material is known by the co-ordinates ϕ
( ϕ0 ≤ ϕ ≤ ϕ1 ), ϑ ( 0 ≤ ϑ ≤ 2π ) upon the middle surface, and ζ directed along the outward normal and measured from the reference surface ( −h 2 ≤ ζ ≤ h 2 ).
For the considered shell of revolution, the radii of curvature Rϕ (ϕ ) , Rϑ (ϕ ) in the meridional and circumferential directions, the first derivative of Rϕ (ϕ ) respect to ϕ and the GaussCodazzi relation can be expressed, respectively, as follows:
Rϕ (ϕ ) =
Rϕ ,ϕ
k 1
,
2 cos3 ϕ
3 k sin ϕ
,
=
=
dϕ
2 cos 4 ϕ
dRϕ
Rϑ (ϕ ) =
R0,ϕ
3
R0
sin ϕ
dR
= 0 = Rϕ cos ϕ
dϕ
(2.3)
Francesco Tornabene, Erasmo Viola
In developing a moderately thick shell theory we make certain assumptions. They are outlined below:
− The transverse normal is inextensible:
εn ≈ 0
− Normals to the reference surface of the shell before deformation remain straight but
not necessarily normal after deformation (a relaxed Kirchhoff-Love hypothesis).
− The transverse normal stress is negligible so that the plane assumption can be invoked:
σ n = σ n (α1 , α 2 , ζ , t ) = 0
c
O
O'
x
d s
0
R0 (ϕ )
ϕ
S
R0 (ϕ )
n
t 2 = tϑ
t1 = tϕ
dϕ
ϑ
O
Rϑ
n
x
Rϕ
y
s1
z
Axis of
revolution
Geometric
axis
(a)
z'
(b)
Figure 2: Geometry of parabolic shell.
Consistent with the assumptions of a moderately thick shell theory, the displacement field
assumed in this study is that of the First-order Shear Deformation Theory (FSDT) and can be
put in the following form:
(
(
(
) (
) (
) (
)
)
(
(
⎧Uϕ αϕ , αϑ , ζ , t = uϕ αϕ , αϑ , t + ζβϕ αϕ , αϑ , t
⎪
⎪
⎨Uϑ αϕ , αϑ , ζ , t = uϑ αϕ , αϑ , t + ζβϑ αϕ , αϑ , t
⎪
⎪⎩W αϕ , αϑ , ζ , t = w αϕ , αϑ , t
)
)
)
(2.4)
where uϕ , uϑ , w are the displacement components of points lying on the middle surface ( ζ = 0 )
of the shell, along meridional, circumferential and normal directions, respectively. βϕ and βϑ
are normal-to-mid-surface rotations, respectively.
The kinematics hypothesis expressed by equations (2.4) should be supplemented by the
statement that the shell deflections are small and strains are infinitesimal, that is
w ( α ϕ , αϑ , t ) h .
It is worth noting that in-plane displacements Uϕ and Uϑ vary linearly through the thickness, while W remains independent of ζ . The relationships between strains and displacements along the shell reference (middle) surface ζ = 0 are the following:
4
Francesco Tornabene, Erasmo Viola
⎛ ∂uϕ
⎞
1
+ w ⎟ , εϑ =
⎜
∂
ϕ
R
0
⎝
⎠
1 ∂βϕ
1 ⎛ ∂βϑ
, κϑ =
κϕ =
Rϕ ∂ϕ
R0 ⎜⎝ ∂ϑ
εϕ =
γ ϕn =
1
Rϕ
⎞
1 ∂uϑ
1 ⎛ ∂uϕ
⎛ ∂uϑ
⎞
⎜ ∂ϑ + uϕ cos ϕ + w sin ϕ ⎟ , γ ϕϑ = R ∂ϕ + R ⎜ ∂ϑ − uϑ cos ϕ ⎟
⎝
⎠
0 ⎝
ϕ
⎠
⎞
1 ∂βϑ
1 ⎛ ∂βϕ
⎞
+ βϕ cos ϕ ⎟ , κϕϑ =
+
− βϑ cos ϕ ⎟
⎜
Rϕ ∂ϕ R0 ⎝ ∂ϑ
⎠
⎠
(2.5)
⎞
1 ⎛ ∂w
1 ⎛ ∂w
⎞
− uϕ ⎟ + βϕ , γ ϑ n =
− uϑ sin ϕ ⎟ + βϑ
⎜
⎜
Rϕ ⎝ ∂ϕ
R0 ⎝ ∂ϑ
⎠
⎠
In the above equations (2.5), the first three strains ε ϕ , εϑ , γ ϕϑ are in-plane meridional,
circumferential and shearing components, κϕ , κϑ , κϕϑ are the analogous curvature changes.
The last two components are transverse shearing strains.
The shell material assumed in the following is a mono-laminar elastic isotropic one. Accordingly, the following constitutive equations relate internal stress resultants and internal
couples with generalized strain components on the middle surface:
(
)
(
)
(1 −ν )
(
)
(
)
(1 −ν )
Nϕ = K ε ϕ + νεϑ , M ϕ = D κϕ + νκϑ , Qϕ = K
Nϑ = K ε ϑ + νε ϕ , M ϑ = D κϑ + νκϕ , Qϑ = K
Nϕϑ = Nϑϕ = K
(1 −ν )
2
γ ϕϑ , M ϕϑ = M ϑϕ = D
2χ
(1 −ν )
2
2χ
γ ϕn
(2.6)
γ ϑn
κϕϑ
where K = Eh (1 − ν 2 ) , D = Eh3 (12(1 −ν 2 )) are the membrane and bending rigidity, respectively. E is the Young modulus, ν is the Poisson ratio and χ is the shear factor which for
isotropic materials is usually taken as χ = 6 5 . In equations (2.6), the first three components
Nϕ , Nϑ , Nϕϑ are the in-plane meridional, circumferential and shearing force resultants,
M ϕ , M ϑ , M ϕϑ are the analogous couples, while the last two Qϕ , Qϑ are the transverse shears.
Following the direct approach or the Hamilton’s principle in dynamic version and remembering the Gauss-Codazzi relations for the shells of revolution dR0 / dϕ = Rϕ cos ϕ , five equations of dynamic equilibrium in terms of internal actions can be written for the shell element:
Qϕ
1 ∂Nϕ
1 ∂Nϕϑ cos ϕ
Nϕ − Nϑ +
+
+
= I 0uϕ + I1βϕ
Rϕ ∂ϕ
R0 ∂ϑ
R0
Rϕ
(
)
1 ∂Nϕϑ
1 ∂Nϑ
cos ϕ
sin ϕ
Nϕϑ +
Qϑ = I 0uϑ + I1βϑ
+
+2
Rϕ ∂ϕ
R0 ∂ϑ
R0
R0
Nϕ sin ϕ
1 ∂Qϕ
1 ∂Qϑ cos ϕ
Qϕ −
Nϑ = I 0 w
+
+
−
Rϕ ∂ϕ
R0 ∂ϑ
R0
Rϕ
R0
(2.7)
1 ∂M ϕ
1 ∂M ϕϑ cos ϕ
M ϕ − M ϑ − Qϕ = I1uϕ + I 2 βϕ
+
+
Rϕ ∂ϕ
R0 ∂ϑ
R0
(
)
1 ∂M ϕϑ
1 ∂M ϑ
cos ϕ
+
+2
M ϕϑ − Qϑ = I1uϑ + I 2 βϑ
R0 ∂ϑ
R0
Rϕ ∂ϕ
where:
⎛
h2
I 0 = μ h ⎜1 +
⎜ 12 Rϕ Rϑ
⎝
⎞
⎛ 1
1
+
⎟ , I1 = μ h3 ⎜
⎟
⎜ 12 Rϕ 12 Rϑ
⎠
⎝
⎞
⎛ 1
h2
⎟ , I 2 = μ h3 ⎜ +
⎟
⎜ 12 80 Rϕ Rϑ
⎠
⎝
⎞
⎟
⎟
⎠
(2.8)
are the mass inertias and μ is the mass density of the material per unit volume. The first three
equations (2.7) represent translational equilibriums along meridional, circumferential and
5
Francesco Tornabene, Erasmo Viola
normal directions, while the last two are rotational equilibrium equations about the ϕ and ϑ
directions.
The three basic sets of equations, namely the kinematic, the equilibrium and the constitutive equations may be combined to give the fundamental system of equations, also known as
the governing system equations. Substituting the definition equations (2.5) into the constitutive equations (2.6) and the result of this substitution into the equilibrium equations (2.7), the
complete equations of motion in terms of displacements can be written in the extended form
as:
2
2
K ∂ uϕ (1 − ν ) K ∂ uϕ (1 + ν ) K ∂ 2uϑ
K ⎛ cos ϕ Rϕ , ϕ
⎜
+
+
+
− 2
2
2
2
2
2 R0 ∂ϑ
2 Rϕ R0 ∂ϕ ∂ϑ Rϕ ⎜⎝ R0
Rϕ ∂ϕ
Rϕ
K ⎡ 1 ⎛ (1 − ν ) ⎞ ν sin ϕ ⎤ ∂w ( 3 − ν ) cos ϕ ∂uϑ
+
−
+
K 2
⎢ ⎜1 +
⎥
⎟+
Rϕ ⎢⎣ Rϕ ⎝⎜
2 χ ⎠⎟
R0 ⎥⎦ ∂ϕ
2
R0 ∂ϑ
⎡ 1
⎡ 1 ⎛ ν sin ϕ cos 2 ϕ ⎞ 1 (1 − ν ) ⎤
⎥ uϕ + K ⎢
−K ⎢ ⎜
+
⎟+ 2
R0 ⎟⎠ Rϕ 2 χ ⎥
⎢ Rϕ
⎢⎣ R0 ⎜⎝ Rϕ
⎦
⎣
⎛
K (1 − ν )
h2
βϕ = μ h ⎜1 +
⎜ 12 Rϕ Rϑ
Rϕ 2 χ
⎝
+
⎛ cos ϕ Rϕ , ϕ
⎜
− 2
⎜ R0
Rϕ
⎝
⎞
⎛ 1
1
+
⎟ uϕ + μ h3 ⎜
⎟
⎜ 12 Rϕ 12 Rϑ
⎠
⎝
⎞ ∂u
⎟ ϕ +
⎟ ∂ϕ
⎠
⎞ sin ϕ cos ϕ ⎤
⎥w+
⎟−
⎟
R02
⎥
⎠
⎦
⎞
⎟ βϕ
⎟
⎠
(1 − ν )
2
(1 − ν ) K ⎜⎛ cos ϕ − Rϕ ,ϕ
K ∂ 2uϑ
K ∂ 2uϑ (1 + ν ) K ∂ uϕ
+
+
+
2
2
2
2
2 Rϕ ∂ϕ
2 Rϕ R0 ∂ϕ ∂ϑ
2 Rϕ ⎜⎝ R0
R0 ∂ϑ
Rϕ2
( 3 − ν ) K cosϕ ∂uϕ + K ⎢⎡ ν + sin ϕ ⎛1 + (1 − ν ) ⎞ ⎥⎤ ∂w +
+
⎜
⎟
R0 ⎜⎝
2
2 χ ⎟⎠ ⎥⎦ ∂ϑ
R02 ∂ϑ R0 ⎢⎣ Rϕ
(1 − ν ) K ⎡ sin ϕ − 1 ⎛ cos2 ϕ + sin 2 ϕ ⎞ ⎤ u + K (1 − ν ) sin ϕ β =
+
⎢
⎜
⎟⎥ ϑ
ϑ
R0 ⎜⎝
χ ⎟⎠ ⎥⎦
R0
2 R0 ⎢⎣ Rϕ
2χ
⎛
⎛ 1
h2 ⎞
1 ⎞ = μ h ⎜1 +
+
⎟ u + μ h3 ⎜
⎟β
⎜ 12 Rϕ Rϑ ⎟ ϑ
⎜ 12 Rϕ 12 Rϑ ⎟ ϑ
⎝
⎠
⎝
⎠
(1 − ν )
2χ
K ∂ 2 w (1 − ν ) K ∂ 2 w K ⎡ 1
+
−
⎢
2 χ R02 ∂ϑ 2 Rϕ ⎢⎣ Rϕ
Rϕ2 ∂ϕ 2
(1 − ν )
⎞ ∂u
⎟ ϑ +
⎟ ∂ϕ
⎠
(2.10)
⎛ (1 − ν ) ⎞ ν sin ϕ ⎤ ∂uϕ
+
⎥
⎜⎜ 1 +
⎟+
R0 ⎦⎥ ∂ϕ
2 χ ⎟⎠
⎝
⎛ cos ϕ Rϕ , ϕ ⎞ ∂w (1 − ν ) K ∂βϕ K ⎡ ν
sin ϕ ⎛ (1 − ν ) ⎞ ⎤ ∂uϑ
⎜
− 2 ⎟
+
−
+
+
⎢
⎜1 +
⎟⎥
⎜ R0
R0 ⎜⎝
2χ
2 χ Rϕ ∂ϕ R0 ⎣⎢ Rϕ
2 χ ⎟⎠ ⎦⎥ ∂ϑ
Rϕ ⎟⎠ ∂ϕ
⎝
(1 − ν ) K ∂βϑ − K ⎢⎡ cos ϕ ⎜⎛ sin ϕ + ν ⎟⎞ + (1 − ν ) 1 ⎜⎛ cos ϕ − Rϕ ,ϕ ⎟⎞ ⎥⎤ u +
+
ϕ
Rϕ ⎟⎠
2 χ R0 ∂ϑ
2 χ Rϕ ⎜⎝ R0
Rϕ2 ⎟⎠ ⎥
⎢ R0 ⎜⎝ R0
⎣
⎦
+
⎡ 1
−K ⎢
⎢⎣ Rϕ
(2.9)
K
Rϕ
⎛ 1
(1 − ν ) cosϕ β = μ h ⎛⎜1 + h2
2ν sin ϕ ⎞ sin 2 ϕ ⎤
⎥w+ K
+
⎜
⎟+
ϕ
2
⎜ Rϕ
⎟
⎜ 12 Rϕ Rϑ
R0 ⎠
R0
2χ
R0 ⎥⎦
⎝
⎝
(2.11)
⎞
⎟w
⎟
⎠
2
2
D ∂ βϕ (1 − ν ) D ∂ βϕ (1 + ν ) D ∂ 2 βϑ (1 − ν ) K ∂w
+
+
−
+
2
2
2
2 R0 ∂ϑ 2
2 Rϕ R0 ∂ϕ ∂ϑ
2 χ Rϕ ∂ϕ
Rϕ ∂ϕ
+
D
Rϕ
⎛ cos ϕ Rϕ ,ϕ
⎜
− 2
⎜ R0
Rϕ
⎝
⎞ ∂βϕ ( 3 − ν ) cos ϕ ∂β
(1 − ν ) K u +
ϑ
⎟
−
+
D 2
ϕ
⎟ ∂ϕ
∂
ϑ
2
2 χ Rϕ
R
0
⎠
⎡ D ⎛ ν sin ϕ cos 2 ϕ ⎞
(1 − ν ) ⎤⎥ β = μ h3 ⎛⎜ 1 + 1
−⎢ ⎜
+
⎟+K
ϕ
⎜ 12 Rϕ 12 Rϑ
R ⎜ Rϕ
R0 ⎟⎠
2χ ⎥
⎝
⎣⎢ 0 ⎝
⎦
6
⎞
⎛ 1
h2
⎟ uϕ + μ h3 ⎜ +
⎜ 12 80Rϕ Rϑ
⎟
⎝
⎠
(2.12)
⎞
⎟ βϕ
⎟
⎠
Francesco Tornabene, Erasmo Viola
(1 − ν )
2
(1 − ν ) D ⎛⎜ cos ϕ − Rϕ ,ϕ
D ∂ 2 βϑ
D ∂ 2 βϑ (1 + ν ) D ∂ βϕ
+ 2
+
+
2
2
2
2 Rϕ ∂ϕ
2 Rϕ R0 ∂ϕ ∂ϑ
2 Rϕ ⎜⎝ R0
R0 ∂ϑ
Rϕ2
(1 − ν ) K ∂w + ( 3 − ν ) D cos ϕ ∂βϕ + (1 − ν ) K sin ϕ u +
−
ϑ
2 χ R0 ∂ϑ
2
2χ
R0
R02 ∂ϑ
⎡ (1 − ν ) D ⎛ sin ϕ cos 2 ϕ ⎞
(1 − ν ) ⎤⎥ β = μ h3 ⎜⎛ 1 + 1
+⎢
−
⎜
⎟−K
ϑ
⎜ 12 Rϕ 12 Rϑ
2χ ⎥
R0 ⎟⎠
⎢⎣ 2 R0 ⎜⎝ Rϕ
⎝
⎦
⎞ ∂β
⎟ ϑ +
⎟ ∂ϕ
⎠
(2.13)
⎞
⎛ 1
h2
⎟ uϑ + μ h3 ⎜ +
⎟
⎜ 12 80 Rϕ Rϑ
⎠
⎝
⎞
⎟ βϑ
⎟
⎠
In the following, two kinds of boundary conditions are considered, namely the fully
clamped edge boundary condition (C) and the free edge boundary condition (F). The equations describing the boundary conditions can be written as follows:
Clamped edge boundary condition (C):
uϕ = uϑ = w = βϕ = βϑ = 0
at ϕ = ϕ 0 or ϕ = ϕ 1 , 0 ≤ ϑ ≤ ϑ0
(2.14)
uϕ = uϑ = w = βϕ = βϑ = 0
at ϑ = 0 or ϑ = ϑ0 , ϕ 0 ≤ ϕ ≤ ϕ 1
(2.15)
Free edge boundary condition (F):
Nϕ = Nϕϑ = Qϕ = M ϕ = M ϕϑ = 0
at ϕ = ϕ 0 or ϕ = ϕ 1 , 0 ≤ ϑ ≤ ϑ0
(2.16)
Nϑ = Nϕϑ = Qϑ = M ϑ = M ϕϑ = 0
at ϑ = 0 or ϑ = ϑ0 , ϕ 0 ≤ ϕ ≤ ϕ 1
(2.17)
In addition to the external boundary conditions, the kinematical and physical compatibility
should be satisfied at the common meridian with ϑ = 0, 2π , if we want to consider a complete
parabolic dome of revolution. The kinematical compatibility conditions include the continuity
of displacements. The physical compatibility conditions can only be the five continuous conditions for the generalized stress resultants. To consider a complete revolute parabolic dome
characterized by ϑ0 = 2π , it is necessary to implement the kinematical and physical compatibility conditions between the meridians with ϑ = 0 and with ϑ0 = 2π :
Kinematical compatibility conditions:
uϕ (ϕ , 0, t ) = uϕ (ϕ , 2π , t ), uϑ (ϕ , 0, t ) = uϑ (ϕ , 2π , t ), w(ϕ , 0, t ) = w(ϕ , 2π , t ),
βϕ (ϕ , 0, t ) = βϕ (ϕ , 2π , t ), βϑ (ϕ , 0, t ) = βϑ (ϕ , 2π , t )
ϕ 0 ≤ ϕ ≤ ϕ1
(2.18)
Physical compatibility conditions:
Nϑ (ϕ , 0, t ) = Nϑ (ϕ , 2π , t ), Nϕϑ (ϕ , 0, t ) = Nϕϑ (ϕ , 2π , t ), Qϑ (ϕ , 0, t ) = Qϑ (ϕ , 2π , t ),
M ϑ (ϕ , 0, t ) = M ϑ (ϕ , 2π , t ), M ϕϑ (ϕ , 0, t ) = M ϕϑ (ϕ , 2π , t )
3
ϕ 0 ≤ ϕ ≤ ϕ1
(2.19)
GENERALIZED DIFFERENTIAL QUADRATURE METHOD
The GDQ method will be used to discretize the derivatives in the governing equations and
the boundary conditions. The GDQ approach was developed by Shu [7] to improve the Differential Quadrature technique [8,9] for the computation of weighting coefficients, entering
into the linear algebraic system of equations obtained from the discretization of the differential equation system, which can model the physical problem considered. The essence of the
differential quadrature method is that the partial derivative of a smooth function with respect
to a variable is approximated by a weighted sum of function values at all discrete points in
that direction. Its weighting coefficients are not related to any special problem and only depend on the grid points and the derivative order. In this methodology, an arbitrary grid distribution can be chosen without any limitation.
7
Francesco Tornabene, Erasmo Viola
The GDQ method is based on the analysis of a high-order polynomial approximation and
the analysis of a linear vector space [10]. For a general problem, it may not be possible to express the solution of the corresponding partial differential equation in a closed form. This solution function can be approximated by the two following types of function approximation:
high-order polynomial approximation and Fourier series expansion (harmonic functions). It is
well known that a smooth function in a domain can be accurately approximated by a highorder polynomial in accordance with the Weierstrass polynomial approximation theorem. In
fact, from the Weierstrass theorem, if f ( x) is a real valued continuous function defined in the
closed interval [a, b] , then there exists a sequence of polynomials Pr ( x) which converges to
f ( x) uniformly as r goes to infinity. In practical applications, a truncated finite polynomial
may be used. Thus, if f ( x) represents the solution of a partial differential equation, then it
can be approximated by a polynomial of a degree less than or equal to N − 1 , for N large
enough. The conventional form of this approximation is:
N
f ( x ) ≅ PN ( x ) = ∑ d j p j ( x )
(3.1)
j =1
where d j is a constant. Then, it is easy to show that the polynomial PN ( x) constitutes an
N -dimensional linear vector space VN with respect to the operation of vector addition and
scalar multiplication. Obviously, in the linear vector space VN , p j ( x) is a set of base vectors.
It can be seen that, in the linear polynomial vector space, there exist several sets of base
polynomials and each set of base polynomials can be expressed uniquely by another set of
base polynomials in the space. Using vector space analysis, the method for computing the
weighting coefficients can be generalized by a proper choice of base polynomials in a linear
vector space. For generality, the Lagrange interpolation polynomials are chosen as the base
polynomials. As a result, the weighting coefficients of the first order derivative are computed
by a simple algebraic formulation without any restriction on the choice of the grid points,
while the weighting coefficients of the second and higher order derivatives are given by a recurrence relationship.
When the Lagrange interpolated polynomials are assumed as a set of vector space base
functions, the approximation of the function f ( x) can be written as:
N
f ( x) ≅ ∑ p j ( x) f ( x j )
(3.2)
j =1
where N is the number of grid points in the whole domain, x j , j = 1, 2,..., N , are the coordinates of grid points in the variable domain and f ( x j ) are the function values at the grid
points. p j ( x) are the Lagrange interpolated polynomials, which can be defined by the following formula:
p j ( x) =
L ( x)
(x − x )L (x )
(1)
j
,
j = 1, 2,..., N
(3.3)
j
where:
L ( x) =
N
∏
( x − xi ),
L (1) ( x j ) =
i =1
N
∏ (x
i =1,i ≠ j
8
j
− xi )
(3.4)
Francesco Tornabene, Erasmo Viola
Differentiating equation (3.2) with respect to x and evaluating the first derivative at a certain point of the function domain, it is possible to obtain:
N
N
j =1
j =1
f (1) ( xi ) ≅ ∑ p (j1) ( xi ) f ( x j ) = ∑ ς ij(1) f ( x j ),
i = 1, 2,..., N
(3.5)
where ς ij(1) are the GDQ weighting coefficients of the first order derivative and xi denote the
co-ordinates of the grid points. In particular, it is worth noting that the weighting coefficients
of the first order derivative can be computed as:
(1)
p (1)
j ( xi ) = ς ij =
L (1) ( xi )
, i, j = 1, 2,..., N ,
( xi − x j ) L (1) ( x j )
i≠ j
(3.6)
From equation (3.6), ς ij(1) ( i ≠ j ) can be easily computed. However, the calculation of ς ii(1) is
not easy to compute. According to the analysis of a linear vector space, one set of base functions can be expressed uniquely by a linear sum of another set of base functions. Thus, if one
set of base polynomials satisfy a linear equation like (3.5), so does another set of base polynomials. As a consequence, the equation system for determining ς ij(1) and derived from the Lagrange interpolation polynomials should be equivalent to that derived from another set of base
polynomials, i.e. p j ( x ) = x j −1 , j = 1, 2,..., N . Thus, ς ij(1) satisfies the following equation, which is
obtained by the base polynomials p j ( x ) = x j −1 , when j = 1 :
N
∑ ς ij(1) = 0 ⇒ ς ii(1) = −
j =1
N
∑
ς ij(1) ,
i, j = 1, 2,..., N
(3.7)
j =1, j ≠ i
Equations (3.6) and (3.7) are two formulations to compute the weighting coefficients ς ij(1) .
It should be noted that, in the development of these two formulations, two sets of base polynomials were used in the linear polynomial vector space VN . Finally, the nth order derivative
of function f ( x) with respect to x at grid points xi , can be approximated by the GDQ approach:
d n f ( x)
dx n
=
N
∑ς
(n)
ij
f ( x j ),
i = 1, 2,..., N
(3.8)
j =1
x = xi
where ς ij( n) are the weighting coefficients of the nth order derivative. Similar to the first order
derivative and according to the polynomial approximation and the analysis of a linear vector
space, it is possible to determine a recurrence relationship to compute the second and higher
order derivatives. Thus, the weighting coefficients can be generated by the following recurrent formulation:
⎛
⎜
⎝
ς ij( n ) = n ⎜ ς ii( n −1)ς ij(1) −
N
ς ij( n −1) ⎞
⎟ , i ≠ j , n = 2,3,..., N − 1, i, j = 1, 2,..., N
xi − x j ⎟⎠
∑ ς ij(n) = 0 ⇒ ς ii(n) = −
j =1
N
∑
ς ij( n ) , n = 2,3,..., N − 1, i, j = 1, 2,..., N
(3.9)
(3.10)
j =1, j ≠ i
It is obvious from the above equations that the weighting coefficients of the second and
higher order derivatives can be determined from those of the first order derivative. Further-
9
Francesco Tornabene, Erasmo Viola
more, it is interesting to note that, the preceding coefficients ς ij( n) are dependent on the derivative order n , on the grid point distribution x j , j = 1, 2,..., N , and on the specific point xi , where
the derivative is computed. There is no need to obtain the weighting coefficients from a set of
algebraic equations which could be ill-conditioned when the number of grid points is large.
Furthermore, this set of expressions for the determination of the weighting coefficients is
so compact and simple that it is very easy to implement them in formulating and programming, because of the recurrence feature.
3.1
Grid distributions
Another important point for successful application of the GDQ method is how to distribute
the grid points. In fact, the accuracy of this method is usually sensitive to the grid point distribution. The optimal grid point distribution depends on the order of derivatives in the boundary condition and the number of grid points used. The grid point distribution also plays an
essential role in determining the convergence speed and stability of the GDQ method. In this
paper, the effects of the grid point distribution will be investigated for the vibration analysis
of parabolic shells. The natural and simplest choice of the grid points through the computational domain is the one having equally spaced points in the co-ordinate direction of the computational domain. However, it is demonstrated that non-uniform grid distribution usually
yields better results than equally spaced distribution. Quan and Chang [11,12] compared numerically the performances of the often-used non-uniform meshes and concluded that the grid
points originating from the Chebyshev polynomials of the first kind are optimum in all the
cases examined there. The zeros of orthogonal polynomials are the rational basis for the grid
points. Shu [10] have used other choice which has given better results than the zeros of Chebyshev and Legendre polynomials. Bert and Malik [13] indicated that the preferred type of
grid points changes with problems of interest and recommended the use of Chebyshev-GaussLobatto grid for the structural mechanics computation. With Lagrange interpolating polynomials, the Chebyshev-Gauss-Lobatto sampling point rule proves efficient for numerical reasons [14] so that for such a collocation the approximation error of the dependent variables
decreases as the number of nodes increases.
In this study, different grid point distributions are considered to investigate their effect on
the GDQ solution accuracy, convergence speed and stability. The typical distributions of grid
points, which are commonly used in the literature, in normalized form are reported as follows:
Equally spaced or uniform distribution
ri =
i −1
,
N −1
(3.11)
i = 1, 2,..., N
Roots of Chebyshev polynomials of the first kind (C I°)
ri =
gi − g1
⎛ ⎛ 2i − 1 ⎞ ⎞
, gi = cos ⎜ ⎜
⎟ π ⎟ , i = 1, 2,..., N
g N − g1
⎝ ⎝ 2N ⎠ ⎠
(3.12)
Roots of Chebyshev polynomials of the second kind (C II°)
ri =
gi − g1
⎛ iπ ⎞
, gi = cos ⎜
⎟ , i = 1, 2,..., N
g N − g1
⎝ N +1⎠
(3.13)
Roots of Legendre polynomials (Leg)
ri =
gi − g1
1
1 ⎞
⎛
⎛ 4i − 1 ⎞
π ⎟,
, gi = ⎜1 −
cos ⎜
+
2
3 ⎟
8
8
g N − g1
N
N
⎝
⎠
⎝ 4N + 2 ⎠
10
i = 1, 2,..., N
(3.14)
Francesco Tornabene, Erasmo Viola
Quadratic sampling points distribution (Quad)
⎧ ⎛ i − 1 ⎞2
⎪2 ⎜
⎟
⎪ ⎝ N −1 ⎠
ri = ⎨
2
⎪⎛ −2 ⎛ i − 1 ⎞ + 4 ⎛ i − 1 ⎞ − 1⎞
⎜
⎜
⎟ ⎟
⎪⎜ ⎜⎝ N − 1 ⎟⎠
⎝ N − 1 ⎠ ⎟⎠
⎩⎝
i = 1, 2,....,
N +1
2
(3.15)
⎛ N +1⎞
i=⎜
⎟ + 1,..., N
⎝ 2 ⎠
Chebyshev-Gauss-Lobatto sampling points (C-G-L)
⎛ i −1 ⎞
1 − cos⎜
⎟π
⎝ N −1 ⎠ ,
ri =
2
(3.16)
i = 1, 2,..., N
where N is the total number of sampling points used to discretize each direction.
For the numerical computations presented in this paper, the co-ordinates of grid points
(ϕi , ϑ j ) are chosen as:
ϕi = ri (ϕ1 − ϕ0 ) + ϕ0 ,
ϑ j = rjϑ0 ,
i = 1, 2,..., N ,
j = 1, 2,..., M ,
for ϕ ∈ [ϕ0 , ϕ1 ]
(3.17)
for ϑ ∈ [ 0,ϑ0 ] (with ϑ0 ≤ 2π )
where ri , rj are two grid distributions of previous ones and N , M are the total number of sampling points used to discretize the domain in ϕ and ϑ directions, respectively, of the parabolic shell.
1
ϑ =0
ϕ = ϕ0
i
ϑ = ϑ0
(ϕ , ϑ )
i
j
αϑ
N
M
1
αϕ
j
ϕ = ϕ1
Figure 4: C-G-L Grid distribution on a parabolic shell.
4
NUMERICAL IMPLEMENTATION
A novel approach in numerically solving the governing equations (2.9), (2.10), (2.11),
(2.12) and (2.13) is represented by the Generalized Differential Quadrature (GDQ) method.
This method, for the problem studied herein, demonstrates its numerical accuracy and extreme coding simplicity.
11
Francesco Tornabene, Erasmo Viola
In the following, only the free vibration of parabolic dome or panel will be studied. So, using the method of variable separation, it is possible to seek solutions that are harmonic in time
and whose frequency is ω ; then, the displacements and the rotations can be written as follows:
uϕ (αϕ , αϑ , t ) = U ϕ (αϕ , αϑ )eiωt
uϑ (αϕ , αϑ , t ) = U ϑ (αϕ , αϑ )eiωt
(4.1)
w(αϕ , αϑ , t ) = W ζ (αϕ , αϑ )eiωt
βϕ (αϕ , αϑ , t ) = Bϕ (αϕ , αϑ )eiωt
βϑ (αϕ , αϑ , t ) = Bϑ (αϕ , αϑ )eiωt
where the vibration spatial amplitude values ( U ϕ (αϕ , αϑ ) , U ϑ (αϕ , αϑ ) , W ζ (αϕ , αϑ ) , Bϕ (αϕ , αϑ ) ,
Bϑ (αϕ , αϑ ) ) fulfill the fundamental differential system.
The basic steps in the GDQ solution of the free vibration problem of parabolic shell type
structures are as in the following:
− Discretization of independent variables ϕ ∈ [ϕ0 , ϕ1 ], ϑ ∈ [ 0, ϑ0 ] (with ϑ0 ≤ 2π ) .
The spatial derivatives are approximated according to GDQ rule.
− The differential governing systems (2.9), (2.10), (2.11), (2.12), and (2.13) are transformed into linear eigenvalue problems for the natural frequencies. The boundary conditions are imposed in the sampling points corresponding to the boundary. All these
relations are imposed pointwise.
− The solution of the previously stated discrete system in terms of natural frequencies and
mode shape components is worked out. For each mode, local values of dependent variables are used to obtain the complete assessment of the deformed configuration.
The simple numerical operations illustrated here, applying the GDQ procedure, enable one
to write the equations of motion in discrete form, transforming any space derivative into a
weighted sum of node values of dependent variables. Each of the approximated equations is
valid in a single sampling point.
The governing equations can be discretized as:
−
K
Rϕ2i
N
∑ς
ϕ ( 2)
U kjϕ +
ik
⎡ 1
−K ⎢
⎢⎣ R0i
K
R02i
2
k =1
M
∑ς
ϑ ( 2)
⎛ ν sin ϕi cos 2 ϕi
+
⎜
⎜
R0i
⎝ Rϕ i
ϕ
U mj
+
jm
N
∑ς
2
+
+
+
K
Rϕ2i
(1 − ν )
2
K
R0i
N
∑ς
(1 − ν )
2
U kjϑ +
k =1
K
R02i
⎛
K ⎜ cos ϕi Rϕ , ϕ
− 2
Rϕ i ⎜ R0i
Rϕ i
⎝
⎡ν
sin ϕi
+
⎢
R0i
⎢⎣ Rϕ i
K
R0i
ϕ (1)
Wkjζ −
ik
K
Rϕ i R0i
i
M
ϑ ( 2)
jm
ϑ
+
U im
m =1
N
ϕ (1)
ik
k =1
R02i
2
Rϕ , ϕ
⎛ M ϑ (1) ϑ ⎞ K ⎛ cos ϕ
i
⎜
⎜
ς jm U km ⎟ +
− 2
⎜
⎟ Rϕ i ⎜ R0i
Rϕ i
⎝ m =1
⎠
⎝
∑
M
∑ς
∑ς
ϕ (1)
ik
U kjϑ +
k =1
2
K
Rϕ i R0i
N
∑ς
jm
N
∑ς
ϕ (1)
ik
R02i
i
k =1
ϑ
U im
+
(4.2)
⎤
⎞
⎟ − sin ϕi cos ϕi ⎥ W ζ +
2
⎥ ij
⎟
R0i
⎥⎦
⎠
ϕ (1)
ik
M
∑ς
⎛ M ϑ (1) ϕ ⎞
⎜
ς jm U km ⎟ +
⎜
⎟
⎝ m =1
⎠
∑
ϑ (1)
jm
ϕ
U im
+
(4.3)
m =1
⎛ (1 − ν ) ⎞ ⎤ M ϑ (1) ζ
(1 − ν ) sin ϕi Bϑ +
ς jm Wim + K
⎜⎜ 1 +
⎟⎟ ⎥
ij
R0i
2χ
2
χ
⎥
⎝
⎠ ⎦ m =1
⎡ sin ϕ
1
i
−
⎢
R0i
⎣⎢ Rϕ i
∑
⎛ 2
sin 2 ϕi
⎜⎜ cos ϕi +
χ
⎝
⎛
⎞⎤ ϑ
h2
2
⎟⎟ ⎥ U ij = −ω μ h ⎜⎜ 1 +
⎠ ⎦⎥
⎝ 12 Rϕ i Rϑ i
12
U kjϕ +
⎞ ϕ
⎟ Bij
⎟
⎠
k =1
( 3 − ν ) K cos ϕi
2
⎞
⎟
⎟
⎠
ϑ (1)
⎛
R
⎜ cos ϕi − ϕ , ϕ
⎜ R
Rϕ2i
⎝ 0i
(1 + ν )
i
m =1
⎛ 1
⎞
1
+
⎟U ijϕ − ω 2 μ h3 ⎜
⎜ 12 Rϕ i 12 Rϑ i
⎟
⎝
⎠
∑ς
⎞
⎟
⎟
⎠
N
∑ς
( 3 − ν ) K cosϕi
⎡
⎞ 1 (1 − ν ) ⎤ ϕ
1
⎥U + K ⎢
⎟+ 2
⎟ Rϕ i 2 χ ⎥ ij
⎢ Rϕ i
⎠
⎦
⎢⎣
ϕ ( 2)
ik
2
k =1
⎛
K (1 − ν ) ϕ
h2
Bij = −ω 2 μ h ⎜1 +
⎜ 12 Rϕ i Rϑi
Rϕ i 2 χ
⎝
(1 − ν )
(1 + ν )
m =1
⎛ (1 − ν ) ⎞ ν sin ϕi ⎤
⎥
⎜⎜1 +
⎟+
2 χ ⎟⎠
R0i ⎥⎦
⎝
K ⎡ 1
+
⎢
Rϕ i ⎢⎣ Rϕ i
+
(1 − ν )
⎞ ϑ
⎛ 1
1
+
⎟ U − ω 2 μ h3 ⎜
⎟ ij
⎜ 12 Rϕ i 12 Rϑ i
⎠
⎝
⎞ ϑ
⎟B
⎟ ij
⎠
Francesco Tornabene, Erasmo Viola
(1 − ν )
2χ
+
K
Rϕ2i
(1 − ν ) K
ϕ 2
ς ik ( )Wkjζ +
2 χ R02i
k =1
N
∑
⎛
K ⎜ cos ϕi Rϕ ,ϕ
− 2
Rϕ i ⎜ R0i
Rϕ i
⎝
(1 − ν )
2χ
i
N
−
+
(1 − ν )
2
Bkj +
2
k =1
(1 − ν )
2χ
(1 − ν )
2χ
D
Rϕ2i
(1 − ν )
ϕ ( 2) ϕ
N
∑
K
Rϕ i
N
∑ς
Wkjζ +
k =1
K ϕ ⎡ D
U ij − ⎢
Rϕ i
⎢⎣ R0i
D
ϕ 2
ς ik ( ) Bkjϑ + 2
R0i
k =1
D
R02i
ϕ (1)
ik
M
∑ς
ϑ ( 2) ϕ
jm
Bim +
2
⎛
D ⎜ cos ϕi Rϕ , ϕ
− 2
Rϕ i ⎜ R0i
Rϕ i
⎝
⎛ ν sin ϕi cos ϕi
+
⎜
⎜ Rϕ i
R0i
⎝
∑
(1 + ν )
m =1
2
M
∑
ς ik ( ) Bkjϕ −
ϕ1
k =1
N
∑ς
K ⎡ν
sin ϕi
+
⎢
R0i ⎣⎢ Rϕ i
R0i
i
ϑ
+
ς ϑjm( 2) Bim
m =1
N
∑ς
ϕ (1)
ik
k =1
∑
k =1
⎛
ς ik ( ) ⎜
ϕ1
M
∑ς
⎜
⎝ m =1
∑
⎞ ζ
⎟Wij
⎟
⎠
M
∑ς
(4.5)
ϑ (1) ϑ
jm
Bim +
m =1
⎞ ϕ
⎛ 1
h2
⎟U ij − ω 2 μ h3 ⎜ +
⎟
⎜ 12 80 Rϕ i Rϑ i
⎠
⎝
⎞ ϕ
⎟ Bij
⎟
⎠
⎞
Bkm ⎟ +
⎟
⎠
ϑ (1) ϕ
jm
(1 − ν ) D ⎛⎜ cosϕi − Rϕ ,ϕ i ⎞⎟ N ς ϕ (1) Bϑ − (1 − ν ) K M ς ϑ (1)W ζ + ( 3 − ν ) D cosϕi M ς ϑ (1) Bϕ +
+
ik
kj
jm
im
jm
im
2 Rϕ i ⎜ R0i
2 χ R0i m =1
2
Rϕ2i ⎟ k =1
R02i m =1
⎝
⎠
(1 − ν ) K sin ϕi U ϑ + ⎡⎢ (1 − ν ) D ⎛⎜ sin ϕi − cos2 ϕi ⎞⎟ − K (1 − ν ) ⎤⎥ Bϑ = −ω 2 μ h3 ⎛⎜ 1 + 1 ⎞⎟U ϑ − ω 2 μ h3 ⎛⎜ 1 + h2
+
ij
ij
⎜ 12 Rϕ i 12 Rϑi ⎟ ij
⎜ 12 80 Rϕ i Rϑ i
R0i
R0i ⎟⎠
2χ ⎥
2χ
⎢⎣ 2 R0i ⎜⎝ Rϕ i
⎝
⎠
⎝
⎦
∑
(4.4)
∑
∑
N
⎛ (1 − ν ) ⎞ ⎤ M ϑ (1) ϑ
ς jm U im +
⎜⎜1 +
⎟⎥
2 χ ⎟⎠ ⎦⎥ m =1
⎝
⎛ M ϑ (1) ϑ ⎞
⎜
ς jm Bkm ⎟ +
⎜
⎟
⎝ m =1
⎠
⎞ N
( 3 − ν ) D cosϕi
ϕ1
⎟
ς ik ( ) Bkjϕ −
⎟
2
R02i
⎠ k =1
D
Rϕ i R0i
U kjϕ +
⎞⎤
⎟⎥ U ϕ +
⎟ ⎥ ij
⎠ ⎦⎥
⎞
(1 − ν ) ⎤⎥ Bϕ = −ω 2 μ h3 ⎛⎜ 1 + 1
⎟+K
ij
⎟
⎜ 12 Rϕ i 12 Rϑi
2χ ⎥
⎠
⎝
⎦
(1 + ν )
2
i
D
Rϕ i R0i
ϕ (1)
ik
k =1
⎞ sin 2 ϕi ⎤ ζ
(1 − ν ) cosϕi Bϕ = −ω 2 μ h ⎛⎜1 + h2
⎥ Wij + K
⎟+
ij
2
⎟
⎜ 12 Rϕ i Rϑi
2χ
R0i
R0i ⎥⎦
⎠
⎝
⎛ 1
2ν sin ϕi
+
⎜
⎜ Rϕ i
R0i
⎝
ik
N
⎛
⎛ sin ϕi
ν ⎞ (1 − ν ) 1 ⎜ cos ϕi Rϕ , ϕ
+
− 2
⎜
⎟+
⎜ R0i
2 χ Rϕ i ⎜ R0i
Rϕ i ⎟⎠
Rϕ i
⎝
⎝
∑
∑ς
m =1
K ⎡ 1 ⎛ (1 − ν ) ⎞ ν sin ϕi ⎤
⎢
⎥
⎜1 +
⎟+
2 χ ⎟⎠
Rϕ i ⎣⎢ Rϕ i ⎜⎝
R0i ⎦⎥
∑
(1 − ν ) K ς ϑ (1) Bϑ − K ⎡⎢ cosϕi
+
jm
im
⎢ R0i
2 χ R0i m =1
⎣⎢
D
Rϕ2i
∑
ς ϑjm( 2)Wimζ −
⎞ N
(1 − ν ) K
ϕ1
⎟
ς ik ( )Wkjζ +
⎟
2 χ Rϕ i
⎠ k =1
M
⎡ 1
−K ⎢
⎢⎣ Rϕ i
M
∑
(4.6)
∑
⎞ ϑ
⎟ Bij
⎟
⎠
where i = 2,3,..., N − 1 , j = 2,3,..., M − 1 and ς ikϕ(1) , ς ϑjm(1) , ς ikϕ( 2) and ς ϑjm( 2) are the weighting coefficients
of the first and second derivatives in ϕ and ϑ directions, respectively. On the other hand,
N , M are the total number of grid points in ϕ and ϑ directions.
Applying the GDQ methodology, the discretized forms of the boundary conditions are
given as follows:
Clamped edge boundary condition (C):
U ajϕ = U ajϑ = Wajζ = Bajϕ = Bajϑ = 0
for a = 1, N and j = 1, 2,..., M
U ibϕ = U ibϑ = Wibζ = Bibϕ = Bibϑ = 0
for b = 1, M and i = 1, 2,..., N
(4.7)
Free edge boundary condition (F):
⎧ 1
⎪
⎪ Rϕ a
⎪
⎪ 1
⎪R
⎪ ϕa
⎪⎪ 1
⎨
⎪ Rϕ a
⎪
⎪ 1
⎪ Rϕ a
⎪
⎪ 1
⎪
⎩⎪ Rϕ a
⎛
⎜⎜
⎝
N
∑ς
⎞ ν ⎛ M ϑ (1) ϑ
⎞
ϕ
ζ
U kjϕ + Wajζ ⎟⎟ +
⎜⎜ ς jm U am + U aj cos ϕ a + Waj sin ϕ a ⎟⎟ = 0
⎠ R0 a ⎝ m =1
⎠
k =1
N
∑ς
ϕ (1)
ak
U kjϑ +
k =1
⎛
⎜⎜
⎝
N
∑ς
∑ς
N
k =1
∑
⎞
Wkjζ − U ajϕ ⎟⎟ + Bajϕ = 0
⎠
ϕ (1)
ak
Bkjϕ +
M
⎞
ϑ (1) ϑ
ϕ
⎜ ς jm Bam + Baj cos ϕ a ⎟⎟ = 0
⎜
R0 a ⎝ m =1
⎠
Bkjϑ +
M
⎞
1 ⎛
ϑ (1) ϕ
ϑ
⎜ ς jm Bam − Baj cos ϕ a ⎟⎟ = 0
⎜
R0 a ⎝ m =1
⎠
k =1
∑ς
M
⎞
1 ⎛
ϑ (1) ϕ
ϑ
⎜ ς jm U am − U aj cos ϕ a ⎟⎟ = 0
⎜
R0 a ⎝ m =1
⎠
ϕ (1)
ak
k =1
N
∑
ϕ (1)
ak
ϕ (1)
ak
ν ⎛
for a = 1, N and j = 1, 2,..., M
∑
∑
13
(4.8)
Francesco Tornabene, Erasmo Viola
⎧ 1
⎪
⎪ R0i
⎪
⎪ 1
⎪R
⎪ ϕi
⎪⎪ 1
⎨
⎪ R0i
⎪
⎪ 1
⎪ R0i
⎪
⎪ 1
⎪
⎪⎩ Rϕ i
⎛ M ϑ (1) ϑ
⎞ ν ⎛
ϕ
ζ
⎜⎜ ς bm U im + U ib cos ϕi + Wib sin ϕi ⎟⎟ +
⎜⎜
⎝ m =1
⎠ Rϕ i ⎝
∑
N
∑ς
ϕ (1)
ik
ϑ
+
U kb
k =1
1
R0i
N
∑ς
⎞
ϕ
+ Wibζ ⎟⎟ = 0
U kb
⎠
ϕ (1)
ik
k =1
⎛
⎞
ϑ (1) ϕ
ϑ
⎜⎜ ς bm U im − U ib cos ϕi ⎟⎟ = 0
⎝ m =1
⎠
M
∑
⎛ M ϑ (1) ζ
⎞
ϑ
ϑ
⎜⎜ ς bm Wim − U ib sin ϕi ⎟⎟ + Bib = 0
⎝ m =1
⎠
∑
⎛ M ϑ (1) ϑ
⎞ ν
ϕ
⎜⎜ ς bm Bim + Bib cos ϕi ⎟⎟ +
⎝ m =1
⎠ Rϕ i
∑
M
∑ς
ϕ (1)
bm
ϑ
Bim
+
m =1
1
R0i
⎛
⎜⎜
⎝
N
∑ς
k =1
ϑ (1)
ik
for b = 1, M and i = 1, 2,..., N
N
∑ς
ϕ (1)
ik
(4.9)
ϕ
Bkb
=0
k =1
⎞
ϕ
Bkb
− Bibϑ cos ϕi ⎟ = 0
⎟
⎠
Kinematical and physical compatibility conditions:
ϕ
ϑ
ϕ
ϑ
U iϕ1 = U iM
,U iϑ1 = U iM
,Wi1ζ = WiMζ , Biϕ1 = BiM
, Biϑ1 = BiM
⎧ 1
⎪
⎪ R0i
⎪
⎪
⎪
⎪
⎪ 1
⎪
⎪⎪ Rϕ i
⎨
⎪ 1
⎪R
⎪ 0i
⎪
⎪ 1
⎪ R0i
⎪
⎪ 1
⎪
⎪⎩ Rϕ i
⎛ M ϑ (1) ϑ
⎞ ν ⎛
ϕ
ζ
⎜⎜ ς 1m U im + U i1 cos ϕi + Wi1 sin ϕi ⎟⎟ +
⎜⎜
⎝ m =1
⎠ Rϕ i ⎝
∑
=
1
R0i
N
∑ς
⎞
U kϕ1 + Wi1ζ ⎟ =
⎟
⎠
ϕ (1)
ik
k =1
⎛
⎞ ν ⎛
ϑ (1) ϑ
ϕ
ζ
⎜⎜
⎜⎜ ς bm U im + U iM cos ϕi + WiM sin ϕi ⎟⎟ +
⎝ m =1
⎠ Rϕ i ⎝
M
∑
N
∑ς
k =1
⎞
ϕ
U kM
+ WiMζ ⎟⎟
⎠
ϕ (1)
ik
M
⎛
⎞
⎞
1
1 ⎛
ϑ (1) ϕ
ϑ (1) ϕ
ϑ
ϑ
ϑ
ς ikϕ (1)U kM
+
⎜⎜ ς 1m U im − U i1 cos ϕi ⎟⎟ =
⎜⎜ ς Mm U im − U iM cos ϕi ⎟⎟
R0i ⎝ m =1
k =1
⎝ m =1
⎠ Rϕ i k =1
⎠
M
M
⎛
⎞
⎞
1 ⎛
ϑ (1) ζ
ϑ (1) ζ
ϑ
ϑ
ϑ
ϑ
⎜⎜ ς 1m Wim − U i1 sin ϕi ⎟⎟ + Bi1 =
⎜⎜ ς Mm Wim − U iM sin ϕi ⎟⎟ + BiM
R0i ⎝ m =1
⎝ m =1
⎠
⎠
N
∑
ς ikϕ (1)U kϑ1 +
1
R0i
M
N
∑
∑
∑
(4.10)
∑
⎛ M ϑ (1) ϑ
⎞ ν N ϕ (1) ϕ
ϕ
ς ik Bk1 =
⎜⎜ ς 1m Bim + Bi1 cos ϕi ⎟⎟ +
⎝ m =1
⎠ Rϕ i k =1
M
N
⎞
1 ⎛
ϑ (1) ϕ
ϑ
ϑ
ς 1ϕm(1) Bim
+
⎜⎜ ς ik Bk1 − Bi1 cos ϕi ⎟⎟ =
R0i ⎝ k =1
m =1
⎠
∑
∑
∑
∑
∑
⎛ M ϑ (1) ϑ
⎞ ν N ϕ (1) ϕ
ϕ
ς ik BkM
⎜⎜ ς Mm Bim + BiM cos ϕi ⎟⎟ +
⎝ m =1
⎠ Rϕ i k =1
N
M
⎞
1
1 ⎛
ϕ (1) ϑ
ϑ (1) ϕ
ϑ
Bim +
ς Mm
⎜⎜ ς ik BkM − BiM cos ϕi ⎟⎟
Rϕ i m =1
R0i ⎝ k =1
⎠
for i = 2,..., N − 1
1
R0i
∑
∑
∑
∑
Applying the differential quadrature procedure, the whole system of differential equations
has been discretized and the global assembling leads to the following set of linear algebraic
equations:
⎡ K bb
⎢K
⎣⎢ db
K bd ⎤ ⎡ δb ⎤
0 ⎤ ⎡ δb ⎤
⎡0
= ω2 ⎢
⎥
⎢
⎥
⎥⎢ ⎥
K dd ⎦⎥ ⎣⎢δ d ⎦⎥
⎣⎢ 0 M dd ⎥⎦ ⎣⎢δ d ⎦⎥
(4.11)
In the above matrices and vectors, the partitioning is set forth by subscripts b and d, referring to the system degrees of freedom and standing for boundary and domain, respectively. In
order to make the computation more efficient, kinematic condensation of non-domain degrees
of freedom is performed:
(K
)
− K db ( K bb ) K bd δ d = ω 2 M dd δ d
-1
dd
(4.12)
The natural frequencies of the structure considered can be determined by making the following determinant vanish:
(K
)
− K db ( K bb ) K bd − ω 2 M dd = 0
-1
dd
14
(4.13)
Francesco Tornabene, Erasmo Viola
5
APPLICATIONS AND RESULTS
Based on the above derivations, in the present paragraph some results and considerations
about the free vibration problem of parabolic panels and parabolic domes are presented. The
analysis has been carried out by means of numerical procedures illustrated above. The mechanical characteristics for the considered structures are listed in Table 1. In order to verify
the accuracy of the present method, some comparisons have also been performed. The first
ten natural frequencies of a parabolic panel and a parabolic dome are reported in Tables 2, 3
and 4. The details regarding the geometry of the structures are indicated below:
1. Parabolic panel: k = 0.5, c = 0 m, h = 0.1m, s0 = 0.3m, s1 = 1m, S = 2 m, ϑ0 = 90° (Table 2);
2. Parabolic panel: k = 4.5, c = 9 m, h = 0.1m, s0 = −3m, s1 = 3m, S = 2 m, ϑ0 = 90° (Table 3);
3. Parabolic dome: k = 0.5, c = 0 m, h = 0.1m, s0 = 0.3m, s1 = 1m, S = 2 m, ϑ0 = 360° (Table 4).
The geometrical boundary conditions for the parabolic panel are identified by the following convention. For example, the symbolism C-F-C-F indicates that the edges ϕ = ϕ 1 , ϑ = ϑ0 ,
ϕ = ϕ0 , ϑ = 0 are clamped, free, clamped and free, respectively. In particular, we have considered the parabolic panels characterized by C-F-F-F and C-F-C-F boundary conditions (Tables
2 and 3). For the parabolic dome, for example, the symbolism C-F indicates that the edges
ϕ = ϕ 1 and ϕ = ϕ0 are clamped and free, respectively. In this case, the missing boundary conditions are the kinematical and physical compatibility conditions that are applied at the same
meridian for ϑ = 0 and ϑ = 2π . In this work the parabolic dome that we have examined is
characterized by C-F boundary conditions (Tables 4).
One of the aims of this paper is to compare results from the present analysis with those obtained with finite element techniques and based on the same shell theory. In Tables 2, 3 and 4
we have compared the 2D shell theory results obtained by the GDQ Method with the FEM
results obtained by some commercial programs using the same 2D shell theory. For the GDQ
results reported in Tables 2, 3 and 4, we have considered the grid distribution with
N = M = 21 , while we have used shell elements with 8 nodes for the commercial programs.
It is noteworthy that the results from the present methodology are very close to those obtained by the commercial programs. As can be seen, the numerical results show excellent
agreement. Furthermore, it is significant that the computational effort in terms of time and
number of grid points is smaller for the GDQ method results than for the finite element
method. To obtain converging and accurate solutions with the FEM codes, the 50 × 50 mesh is
used in Tables 2, 3 and 4.
In Figures 5 and 6, we have reported the first mode shapes for the parabolic panels characterized by C-F-F-F and C-F-C-F boundary conditions, while in Figure 7 the mode shapes for
the parabolic dome characterized by C-F boundary conditions are illustrated. In particular, for
the parabolic dome there are some symmetrical mode shapes due to the symmetry of the problem considered in 3D space. In these cases, we have summarized the symmetrical mode
shapes in one figure.
The convergence and the stability of some natural frequencies for the structures under consideration with various grid distributions are shown in Figs. 8, 9 and 10. Well converging results for the frequencies can be obtained, if non-uniform grid point distribution is considered.
In fact, the uniform grid distribution always presents less accurate results compared to nonuniform grids [14]. It can be seen from the figures that the Chebyshev-Gauss-Lobatto (C-G-L)
grid point distribution has the most rapid converging speed and provides more accurate solutions. Instead, the solutions obtained by using Chebyshev I° (C I°), Chebyshev II° (C II°),
Legendre (Leg) and Quadratic (Quad) grid point distributions oscillate much more. It is
shown that the solution accuracy of the non-uniform grid distributions stays steady with in15
Francesco Tornabene, Erasmo Viola
creasing N = M and does not decrease due to the numerical instabilities even if N = M becomes too large. For all the treated cases the non-uniform distributions are stable if the number of grid points increases. As shown in all the figures under consideration, to obtain
accurate results for the higher frequencies the number of sampling points must not be too
large.
Figs. 8 and 9 present the convergence characteristics of some frequencies for the parabolic
panels characterized by C-F-F-F and C-F-C-F boundary conditions, respectively, while Fig.
10 shows the same characteristics for the parabolic dome characterized by C-F boundary conditions. The boundary conditions influence the convergence and stability characteristics. In
fact, it can be seen from these figures that the solutions for the parabolic panel C-F-F-F have
the most rapid converging speed and provide more accurate results with a lower number of
sampling points. These case, when N = M = 19 , yield very accurate results for the considered
frequencies. The worked solutions for the parabolic panel C-F-C-F and parabolic dome C-F
oscillate much more and require a larger number of sampling points. In the last case, compatibility conditions are introduced and must be implemented to solve the complete shell of revolution problem. To obtain accurate results for the higher frequencies, it is necessary to use a
grid distribution with N = M = 21 sampling points.
6
CONCLUSIONS
A Generalized Differential Quadrature Method application to free vibration analysis of
parabolic shells has been presented to illustrate the versatility and the accuracy of this methodology. The adopted shell theory is the First-order Shear Deformation Theory. The dynamic
equilibrium equations are discretized and solved with the present method giving a standard
linear eigenvalue problem. The vibration results are obtained without the modal expansion
methodology. The complete 2D differential system, governing the structural problem, has
been solved. Due to the theoretical framework, no approximation (δ-point technique) is
needed in modelling the boundary edge conditions.
Various boundary conditions and grid point distributions have been considered. The GDQ
method provides a very simple algebraic formula for determining the weighting coefficients
required by the differential quadrature approximation without restricting in any way the
choice of mesh grids. Examples presented show that the generalized differential quadrature
method can produce accurate results utilizing only a small number of sampling points. The
present method provides converging results for all the cases as the number of grid points increases. Furthermore, discretizing and programming procedures are quite easy. Fast convergence and very good stability have been shown. The effect of grid point distribution on the
GDQ solution of the structural shell problem has been investigated. It was found that the Chebyshev-Gauss-Lobatto grid performs the best among the other four non-uniform typical grid
distributions for all cases analyzed.
Parameter
Density of mass μ
Young’s modulus E
Poisson coefficient ν
Shear factor χ
Value
7800 kg / m3
2.1·1011 Pa
0.3
6/5
Table 1: Physical parameters used in the analysis of free vibrations of the structures considered.
16
Francesco Tornabene, Erasmo Viola
Frequencies
[Hz]
f1
f2
f3
f4
f5
f6
f7
f8
f9
f10
GDQ Method
Abaqus
Ansys
Femap/Nastran
Straus
Pro/Engineer
77.73
126.69
264.87
347.35
362.72
515.43
563.17
658.37
803.89
845.74
77.76
124.27
268.08
351.77
361.29
514.46
566.54
660.50
804.01
850.82
77.77
124.52
268.64
352.22
362.36
520.69
571.33
665.73
803.95
850.51
77.74
124.67
268.16
352.05
363.21
515.24
568.21
665.61
803.85
848.83
77.65
124.63
267.28
351.86
363.06
518.65
567.89
665.34
803.29
847.74
77.81
124.41
268.36
351.99
361.73
515.98
567.62
664.38
804.38
850.27
Table 2: Shell theory for the parabolic panel C-F-F-F.
Frequencies
[Hz]
f1
f2
f3
f4
f5
f6
f7
f8
f9
f10
GDQ Method
Abaqus
Ansys
Femap/Nastran
Straus
Pro/Engineer
43.44
45.33
55.84
56.78
67.95
76.46
78.88
91.34
92.36
97.24
43.49
45.36
55.86
56.72
67.95
76.46
78.73
91.35
92.16
97.21
43.41
45.32
55.81
56.70
67.89
76.47
78.75
91.41
92.22
97.23
43.41
45.31
55.81
56.68
67.83
76.41
78.49
91.14
92.04
97.11
43.37
45.27
55.75
56.65
67.81
76.41
78.66
91.31
92.11
97.11
43.44
45.43
55.96
56.96
68.15
76.48
80.19
92.62
93.96
97.89
Table 3: Shell theory for the parabolic panel C-F-C-F.
Frequencies
[Hz]
GDQ Method
Abaqus
Ansys
Femap/Nastran
Straus
Pro/Engineer
f1
356.66
355.50
357.58
356.37
356.47
355.36
f2
356.66
355.50
357.58
356.37
356.74
355.48
f3
365.48
365.93
365.64
365.50
365.09
365.47
f4
365.48
365.93
365.64
365.50
365.16
365.47
f5
506.99
505.68
509.49
507.30
507.99
505.59
f6
506.99
505.68
509.49
507.30
508.69
505.59
f7
616.24
611.37
614.25
613.27
611.93
610.92
f8
616.24
611.37
614.25
613.27
612.08
610.98
f9
673.52
677.25
675.86
675.45
675.57
675.11
f10
716.37
716.07
722.67
719.04
720.95
720.03
Table 4: Shell theory for the parabolic dome C-F.
17
Francesco Tornabene, Erasmo Viola
Mode shape 1
Mode shape 2
Mode shape 3
Mode shape 4
Mode shape 5
Mode shape 6
Figure 5: Mode shapes for the parabolic panel C-F-F-F.
Mode shape 1
Mode shape 2
Mode shape 3
Mode shape 4
Mode shape 5
Mode shape 6
Figure 6: Mode shapes for the parabolic panel C-F-C-F.
Mode shapes 1-2
Mode shapes 3-4
Mode shapes 5-6
Mode shapes 7-8
Mode shape 9
Mode shape 10-11
Figure 7: Mode shapes for the parabolic dome C-F.
18
Francesco Tornabene, Erasmo Viola
Quad
C-G-L
C I°
C II°
Leg
10°
Frequency [Hz]
9°
8°
7°
6°
5°
4°
3°
2°
1° Freq.
N
Figure 8: Convergence and stability characteristics of the first ten frequencies for the parabolic panel C-F-F-F
using different typical grid distributions.
C-G-L
C-G-L
Quad
QuadC I°
CCI°II°
C
Leg
II°
Leg
Frequency [Hz]
10°
9°
8°
7°
6°
5°
4°
3°
2°
1° Freq.
N
Figure 9: Convergence and stability characteristics of the first ten frequencies for the parabolic panel C-F-C-F
using different typical grid distributions.
C-G-L
C-G-L
Quad
Quad C I°
C CI°II°
Leg
C
II°
Leg
10°
Frequency [Hz]
9°
7°, 8°
5°, 6°
3°, 4°
1°, 2° Freq.
N
Figure 10: Convergence and stability characteristics of the first ten frequencies for the parabolic dome C-F using
different typical grid distributions.
19
Francesco Tornabene, Erasmo Viola
ACKNOWLEDGMENT
This research was supported by the Italian Ministry for University and Scientific, Technological Research MIUR (40% and 60%). The research topic is one of the subjects of the Centre of
Study and Research for the Identification of Materials and Structures (CIMEST)“M.Capurso”.
REFERENCES
[1] J. L. Sanders, An improved first approximation theory of thin shells. NASA Report 24,
1959.
[2] W. Flügge, Stress in Shells. Springer, New York, 1960.
[3] F. I. Niordson, Shell Theory. North-Holland, Amsterdam, 1985.
[4] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates.
Journal of Applied Mechanics ASME 12, 66-77, 1945.
[5] E. Artioli, P. Gould and E. Viola, Generalized collocation method for rotational shells
free vibration analysis. The Seventh International Conference on Computational Structures Technology. Lisbon, Portugal, September 7 - 9, 2004.
[6] E. Viola and E. Artioli, The G.D.Q. method for the harmonic dynamic analysis of rotational shell structural elements. Structural Engineering and Mechanics 17, 789-817,
2004.
[7] C. Shu, Generalized differential-integral quadrature and application to the simulation of
incompressible viscous flows including parallel computation. PhD Thesis, University of
Glasgow, 1991.
[8] R. Bellman and J. Casti, Differential quadrature and long-term integration. Journal of
Mathematical Analysis and Applications 34, 235-238, 1971.
[9] R. Bellman, B. G. Kashef and J. Casti, Differential quadrature: a technique for the rapid
solution of nonlinear partial differential equations. Journal of Computational Physic 10,
40-52, 1972.
[10] C. Shu, Differential Quadrature and Its Application in Engineering. Springer, Berlin,
2000.
[11] J. R. Quan and C. T. Chang, New insights in solving distributed system equations by the
quadrature method – I. Analysis. Computers and Chemical Engineering 13, 779-788,
1989.
[12] J. R. Quan and C. T. Chang, New insights in solving distributed system equations by the
quadrature method – II. Numerical experiments. Computers and Chemical Engineering
13, 1017-1024, 1989.
[13] C. Bert and M. Malik, Differential quadrature method in computational mechanics. Applied Mechanical Reviews 49, 1-27, 1996.
[14] C. Shu, W. Chen, H. Xue and H. Du, Numerical study of grid distribution effect on accuracy of DQ analysis of beams and plates by error estimation of derivative approximation. International Journal for Numerical Methods in Engineering 51, 159-179, 2001.
20