Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

826 ELIA EFRAIM AND MOSHE EISENBERGER
after transformation to the nondimensional coordinate ξ, we obtain the equations of motion in term of
the displacements as
K (0) (ξ, ω)U(ξ) + K (1) (ξ)U ′ (ξ) + K (2) (ξ)U ′′ (ξ) = 0, (14)
where primes refer to derivatives with respect to ξ, and the terms in the matrices K (0) , K (1) , and K (2) are
given in the Appendix. The solution is obtained using the exact element method algorithm [Eisenberger
1990] by assuming the solution as infinite power series
U(ξ) =
∞�
uiξ i , (15)
i=0
and following the procedure in [Eisenberger 1990] we get the five basic shape functions for each case
of unit displacement on the shell edges. Based on the values of the shape functions and their derivatives
at the two edges of the segment (ξ = 0; ξ = 1) we get the terms in the dynamic stiffness matrix as the
resultant forces along the unit angle segment of the perimeter of the shell, as shown in Figure 2, as
S1
S6
S2
S7
S3
S8
S4
S9
S5
S10
�
�
�
�
�
�
�
�
= Nφ�
ξ=0
ξ=1
�
�
�
= Nφθ � ξ=0
ξ=1
�
�
�
= Qφ�
ξ=0
ξ=1
�
�
�
= Mφ�
ξ=0
ξ=1
= Mφθ
�
�
�
� ξ=0
ξ=1
= E
1 − µ 2
=
= κ E
E
2(1 + µ)
2(1 + µ)
� R ′ p �
µu +
R0
R p�
u
R0
′ � � � �
R p
���
+ µnv + (1 + µ) w ,
R0
ξ=0
ξ=1
�
−nu − R′ p� � �
���
,
ξ=0
�
− R p
R0
R0
v + R p�
v
R0
′
u + R p�
w
R0
′ + ψs
E
=
(1 − µ 2 h
)
3 �
µ
12
R′ p� ψφ +
R0
R p�
R0
E h
=
2(1 + µ)
3 �
−nψφ −
12
R′ p� R0
ψ ′ φ
ψθ +
ξ=1
� �
���
,
ξ=0
ξ=1
+ µnψφ
� R p
R0
�
�ψ ′ θ
� �
���
,
ξ=0
ξ=1
� �
���
.
ξ=0
ξ=1
The dynamic stiffness matrix for a segment, having ten degrees of freedom, five on each edge, is
then assembled for the structure in the usual procedure of structural analysis. The natural frequencies of
vibration are found as the values of the frequency that will cause the assembled dynamic stiffness matrix
of the structure to become singular.
When the cut-outs size becomes relatively small (R p,in/R p,out < 0.1) the shape functions series converges
rather slowly and have relatively large number of terms. Therefore, in order speed the convergence
process one can divide the shell into sections with ratio R p,in = 0.3R p,out for each section. So, by adding
a small number of elements one can solve for shells with very small cut-outs (e.g., three elements for
R p,in = 0.03R p,out and four elements for R p,in = 0.01R p,out).
4. Numerical examples
In order to obtain a high-precision solution for vibration problems of thick spherical shells, numerical
calculations have been performed for a spherical shells with different thickness-radius ratios, and various
(16)