106
J. Chen et al.
Invoking δ
= 0 results in the following:
∂
/∂a =
K
e a
e
+ K
l a
l
+ K
b a
b
− F
e m
= 0,
(37)
where
K
e
=
H / 2
−H / 2
¨
e i
B
T DBh 1 h 2 dα 1 dα 2
dα 3 ;
(38)
K
l
=
H / 2
−H / 2
l / 2
−l / 2
N
l
T
D
l N
l d ¯
α 1 dα 3 /d ;
(39)
K
b
=
H / 2
−H / 2
l / 2
−l / 2
¯
N
r
T D
b ¯
N
r
+
¯
N
t
T D
b ¯
N
t
d ¯
α 1 dα 3 /d ;
(40)
F
e m =
¨
e m
N
e m
T f 0 dα 1 dα 2 .
(41)
The equilibrium equation for the shell is then obtained by assembling the above stiffness
matrices and force vectors in Eqs. (38)–(41) as
Ka = F.
(42)
The full integration is employed in the numerical computation of Eqs. (38)–(41). Thus,
there are 12 Hammer points for the triangular element in Eqs. (38) and (41), while
there are 4 Gaussian points for coordinate ¯
α 1 of the fictitious thin layers e l and e b in
Eqs. (39) and (40) [30], which is different from some other shell elements based on the
Reissner/Minlin assumption using the reduced/selected integration for the shear terms.
4 Convergence Tests
Six typical shell problems are tested below for the convergence study. The loads are
imposed on the middle surface. The indicator points are also located on the middle surface. The conversion from the three common constraints to the corresponding boundary
conditions is detailed as follows: ¯
u 10 , ¯
u 20 , ¯
θ 1 , ¯
θ 2 and u 30 of a fixed side take the value 0;
¯
u 10 , ¯
θ 1 and u 30 of a rigid supported side take the value 0; ¯
u 20 and ¯
θ 2 of a symmetrical
side take the value 0.
4.1 Square Plate
The plate is a special shell with a flat middle surface. Here, a four-side-fixed square plate
under a transverse load q is considered. The side length and the thickness of the plate
are L and H , respectively, as shown in Fig. 8. The material constants are E = 1 × 10 6
and ν = 0.3.
J. Chen et al.
Invoking δ
= 0 results in the following:
∂
/∂a =
K
e a
e
+ K
l a
l
+ K
b a
b
− F
e m
= 0,
(37)
where
K
e
=
H / 2
−H / 2
¨
e i
B
T DBh 1 h 2 dα 1 dα 2
dα 3 ;
(38)
K
l
=
H / 2
−H / 2
l / 2
−l / 2
N
l
T
D
l N
l d ¯
α 1 dα 3 /d ;
(39)
K
b
=
H / 2
−H / 2
l / 2
−l / 2
¯
N
r
T D
b ¯
N
r
+
¯
N
t
T D
b ¯
N
t
d ¯
α 1 dα 3 /d ;
(40)
F
e m =
¨
e m
N
e m
T f 0 dα 1 dα 2 .
(41)
The equilibrium equation for the shell is then obtained by assembling the above stiffness
matrices and force vectors in Eqs. (38)–(41) as
Ka = F.
(42)
The full integration is employed in the numerical computation of Eqs. (38)–(41). Thus,
there are 12 Hammer points for the triangular element in Eqs. (38) and (41), while
there are 4 Gaussian points for coordinate ¯
α 1 of the fictitious thin layers e l and e b in
Eqs. (39) and (40) [30], which is different from some other shell elements based on the
Reissner/Minlin assumption using the reduced/selected integration for the shear terms.
4 Convergence Tests
Six typical shell problems are tested below for the convergence study. The loads are
imposed on the middle surface. The indicator points are also located on the middle surface. The conversion from the three common constraints to the corresponding boundary
conditions is detailed as follows: ¯
u 10 , ¯
u 20 , ¯
θ 1 , ¯
θ 2 and u 30 of a fixed side take the value 0;
¯
u 10 , ¯
θ 1 and u 30 of a rigid supported side take the value 0; ¯
u 20 and ¯
θ 2 of a symmetrical
side take the value 0.
4.1 Square Plate
The plate is a special shell with a flat middle surface. Here, a four-side-fixed square plate
under a transverse load q is considered. The side length and the thickness of the plate
are L and H , respectively, as shown in Fig. 8. The material constants are E = 1 × 10 6
and ν = 0.3.