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A linear quadrilateral shell element with fast stiffness computation
1
F. GruttmannW. Wagner
Institut f¨ ur Werkstoffe und Mechanik im Bauwesen
Technische Universit¨ at Darmstadt
Petersenstraße 12
64287 Darmstadt
Germany
Institut f¨ ur Baustatik
Universit¨ at Karlsruhe (TH)
Kaiserstraße 12
76131 Karlsruhe
Germany
Abstract
A new quadrilateral shell element with 5/6 nodal degrees of freedom is presented.
Assuming linear isotropic elasticity a Hellinger–Reissner functional with independent
displacements, rotations and stress resultants is used. Within the mixed formulation
the stress resultants are interpolated using five parameters for the membrane forces as
well as for the bending moments and four parameters for the shear forces. The hybrid
element stiffness matrix resulting from the stationary condition is integrated analytically.
This leads to a part obtained by one point integration and a stabilization matrix. The
element possesses the correct rank, is free of locking and is applicable within the whole
range of thin and thick shells. The in–plane and bending patch tests are fulfilled and
the computed numerical examples show that the convergence behaviour of the stress
resultants is very good in comparison to comparable existing elements. The essential
advantage is the fast stiffness computation due to the analytically integrated matrices.
Keywords: Reissner–Mindlin shell theory, Hellinger–Reissner variational principle, quadri
lateral shell element, effective analytical stiffness evaluation, one point integration and stabi
lization matrix, in–plane and bending patch test
1 Introduction
Computational shell analysis is based on stress resultant theories e.g. [1, 2] or on the so–
called degenerated approach [3]. New developments in this field are discussed in e.g. [4,
5]. In the following only the main computational aspects are considered. Although the
hypothesis underlying the degenerated approach and classical shell theory are essentially the
same, the reduction to resultant form is typically carried out numerically in the former, and
analytically in the latter, [6]. Many of the computational shell models consider transverse
shear deformations within a Reissner–Mindlin theory [7], [8] to by–pass the difficulties caused
by C1–requirements of the Kirchhoff–Love theory, see e.g. [9, 10, 11].
Generally, shell behaviour is extremely sensitive to initial geometry and imperfections, thus
a successful correlation between theory and analysis is achieved only after including specific
details of these quantities. Low order elements like quadrilaterals based on standard dis
placement interpolation are usually characterized by locking phenomena. In shells two types
of locking occur: transverse shear locking in which bending modes are excluded and nearly
all energy is stored in transverse shear terms, and; membrane locking in which all bend
ing energy is restrained and energy is stored in membrane terms. Elements which exhibit a
1published in: Comp. Meth. Appl. Mech. Engng., 194, 4279–4300, 2005.
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locking tendency lead to unacceptable stiff results when reasonable finite element meshes are
employed.
In attempting to avoid locking, reduced integration methods have often been advocated, see
e.g. [12]. Use of reduced (or selective reduced) integration is often accompanied by spurious
zero energy modes. Hence, authors have developed stabilization techniques to regain the
correct rank of the element stiffness matrix, e.g. [13, 14, 9]. In some cases, however, results
computed using these formulations turned out to be sensitive to the ad hoc hourglass control
parameters. Furthermore these elements do not fulfill the bending patch test.
An effective method to avoid transverse shear locking is based on assumed shear strain fields
first proposed in [15], and subsequently extended and reformulated in [16, 17, 18, 19].
Mixed variational principles provide the basis for the discussed finite element techniques.
Assuming linear elasticity a Hellinger–Reissner functional has been used in e.g. [20, 21].
Ref. [20] describe a quadrilateral element with assumed stresses for membrane, bending and
shear parts whereas in Ref [21] explicit stabilization matrices for a nine node element have
been derived. For general nonlinear material behaviour a three field variational functional
with independent displacements, stresses and strains is more appropriate. Within the so–
called enhanced strain formulations the independent stresses are eliminated from the set of
equations using orthogonality conditions and a two field formulation remains, [22]. For shells
this method has been applied enhancing the Green–Lagrangean membrane strains e.g. in [23].
The drawbacks in [13, 14, 9] have been overcome in [24]. The theory is based on a Hu–Washizu
three–field variational principle with independent displacements, stresses and strains. The
stabilization matrix is derived from the orthogonality between the constant part of the strain
field and the nonconstant part using 5 degrees of freedom at each node. In this context we
also refer to [25] where an updated Lagrangian approach is used. Further developments for
different boundary value problems are considered in [26], where stabilization matrices on basis
of the enhanced strain method have been derived.
An important issue within the context of developing a finite shell model is the number and
type of rotational parameters on the element. Mostly general shell theories exclude explicit
dependence of a rotational field about the normal to the shell surface which leads to a five
parameter model (three displacements and two local rotations). Use of 5 degree–of–freedom
frame requires construction of special coordinate systems for the rotational parameters. Con
sidering the so–called drilling degreeof–freedom leads to a finite element discretization with
six nodal parameters. This has some advantages since both displacement and rotation pa
rameters are associated with a global coordinate frame. On the other hand a larger set of
algebraic equations has to be solved. In this context we mention the four–node shell element
according to [27] with three global displacements and three global rotations at each node.
The element employs a membrane interpolation field with drilling degrees–of–freedom. The
bending stiffness is based on the discrete Kirchhoff theory. For arbitrary shaped elements
a transformation of the stiffness matrix, which considers the warping effects, leads to good
results also for a non–flat geometry. The element [27] has been widely used in the literature
for comparisons, e.g. in [10] and in the present paper.
The essential features and new contributions of the present formulation are summarized as
follows:
(i) Assuming linear elasticity the variational formulation of the shear–elastic shell is based
on the Hellinger–Reissner principle. We specify the shape functions for the independent
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stress resultants, where the interpolation of the membrane forces and of the bending
moments corresponds to the approach in [10]. Here, the new contribution is the ana
lytical integration of the matrices, which leads to a one–point integrated part and an
explicit stabilization matrix. This requires the replacement of the variable base vectors
and director vectors by those of the element center, which corresponds to a projection
on a flat surface, see also [27]. For warped elements the above mentioned transformation
according to [27] is implemented.
(ii) The interpolation of the shear forces along with assumed shear strains and the explicit
matrix representation of the stabilization matrix is a further new contribution.
(iii) The element possesses with six zero eigenvalues the correct rank. No control parameters
have to be chosen to prevent locking or to avoid hour–glassing. The in–plane and
bending patch tests are fulfilled. Especially the convergence behaviour of the stress
resultants is superior to comparable four–node shell elements. The main advantage is
the fast stiffness computation. In our implementation the present formulation requires
only about 60% of the computing time to setup the global stiffness matrix compared
with the element [27].
(iv) The element formulation allows the analysis of shells with intersections. At all nodes
which are not positioned on intersections the drilling degree of freedom is fixed. Thus,
the nodal degrees of freedom are: three global displacements components, three global
rotations at nodes on intersections and two local rotations at other nodes.
2 Kinematics and Variational Formulation
Let B0be the three–dimensional Euclidean space occupied by the shell in the un–deformed
configuration with boundary ∂B0. The position vector Φ of any point P ∈ B0is associated
with the global coordinate frame ei
Φ(ξ1,ξ2,ξ3) = Φiei= X + ξ3D(ξ1,ξ2)
with
D(ξ1,ξ2) = 1 and
−h
2≤ ξ3≤h
2
(1)
with the position vector X(ξ1,ξ2) of the shell mid–surface Ω, the shell thickness h, and ξi
the convected coordinate system of the body. A director D(ξ1,ξ2) is defined as a vector
perpendicular to the shell mid–surface. The usual summation convention is used, where
Latin indices range from 1 to 3 and Greek indices range from 1 to 2. Commas denote partial
differentiation with respect to the coordinates ξi.
Hence, the geometry of the deformed shell space B is described by
φ(ξ1,ξ2,ξ3) = φiei= x(ξ1,ξ2) + ξ3d(ξ1,ξ2)withd = D + ∆d. (2)
With the kinematic assumption (2) shear deformations are accounted for and thus d is not
normal to the deformed shell mid–surface.
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Inserting the position vectors (1) and (2) in the linear strain tensor ¯ ε one obtains
¯ ε = ¯ εijGi⊗ Gj
¯ εαβ= εαβ+ ξ3καβ
2 ¯ εα3= γα
¯ ε33= 0, (3)
where Gidenote the contravariant base vectors. The membrane strains εαβ, curvatures καβ
and shear strains γαread
εαβ
=
1
2(u,α·X,β+u,β·X,α)
1
2(u,α·D,β+u,β·D,α+X,α·∆d,β+X,β·∆d,α)
γα = u,α·D + X,α·∆d.
and are organized in a vector ε = [ε11,ε22,2ε12,κ11,κ22,2κ12,γ1,γ2]T.
καβ
=
(4)
The variational formulation is based on a Hellinger–Reissner functional, where the displace
ment field and the stress resultants are independent.
Ω and by boundary loads¯t on a part of the boundary Γσ. The potential is a function
of the displacement field v = [u,∆d]Twith u = x − X and the stress resultants σ =
[n11,n22,n12,m11,m22,m12,q1,q2]Twith membrane forces nαβ, bending moments mαβ and
shear forces qα
The shell is loaded by loads ¯ p in
ΠHR(v,σ) =
?
(Ω)
(εTσ −1
2σTC−1σ)dA −
?
(Ω)
uT¯ pdA −
?
(Γσ)
uT¯tds → stat.(5)
Assuming linear isotropic elasticity the constitutive matrix reads
C =
Cm
0
0
00
0Cb
0Cs
withCm=
Eh
1 − ν2
1 ν
ν
0
01
0 0
1 − ν
2
,
Cb
=
h2
12Cm
Cs
= κGh12
, (6)
with the second order unit matrix 12, Young´s modulus E, shear modulus G, Poisson´s ratio
ν and shear correction factor κ =5
The stationary condition yields
?
(Ω)
6.
δΠHR(v,σ,δv,δσ) =[δεTσ + δσT(ε − C−1σ) − δuT¯ p]dA −
?
(Γσ)
δuT¯tds = 0 (7)
with virtual displacements δv = [δu,δd]Tand virtual stress resultants
δσ = [δn11,δn22,δn12,δm11,δm22,δm12,δq1,δq2]T. The virtual shell strains read
δεαβ
=
1
2(δu,α·X,β+δu,β·X,α)
1
2(δu,α·D,β+δu,β·D,α+X,α·δd,β+X,β·δd,α)
δγα = δu,α·D + X,α·δd
and are summarized in vector notation δε = [δε11,δε22,2δε12,δκ11,δκ22,2δκ12,δγ1,δγ2]T.
δκαβ
=
(8)
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3 Finite Element Equations
3.1Mid–Surface and Displacement Interpolation
For a quadrilateral element we exploit the isoparametric concept with coordinates ξ and η
defined in the unit square {ξ,η} ∈ [−1,1]. Hence the position vector and the director vector
of the shell mid–surface are interpolated using the bi–linear functions
NI=1
4(1 + ξIξ)(1 + ηIη) = a0I+ a1Iξ + a2Iη + hIξη
ξI∈ {−1,
a0I=1
1,1,−1}
a1I=1
ηI∈ {−1,−1,
a2I=1
1,1}
44ξI
4ηI
hI=1
4ξIηI
(9)
as follows
Xh=
4
?
I=1
NIXI
Dh=
4
?
I=1
NIDI,(10)
where the index h denotes the finite element approximation. The position vectors XI and
the local cartesian basis systems akI, k = 1,2,3 are generated within the mesh input. Here,
DI= a3Iis perpendicular to Ω and a1I, a2Iare constructed in such a way that the boundary
conditions can be accommodated. With (10)2the orthogonality is only given at the nodes.
Furthermore, a local cartesian basis tiis introduced at the element center
¯d1 = X3− X1
¯d2 = X2− X4
d1 =
¯d1/¯d1
¯d2/¯d2
d2 =
t1 = (d1+ d2)/d1+ d2
t2 = (d1− d2)/d1− d2
t3 = t1× t2
(11)
The shell mid–surface described by (10)1is in general a non–planar surface Ωh, whereas the
flat projection introduced by t1and t2is denoted by Ωh
0, see Fig. 1 and Ref. [27].
h
A
D
C
B
4
3
2
1
midsurface ( =0)?
h?
?
p
?
3
?
?
X
3
X
2
X
1
1
4
3
2
h
h
h
h
t3
t2
t1
?
h
0
?
h
Figure 1: Quadrilateral shell element
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The displacements and rotations are interpolated using also the bi–linear functions
uh=
4
?
I=1
NIuI
∆dh=
4
?
I=1
NI∆dI. (12)
Here, uI= uIkekdescribes the nodal displacement vector and ∆dI= ϕI× DIis given with
the nodal rotation vector ϕI= ϕIkekwhere ϕIkare rotations about global cartesian axes.
The virtual displacements δu and rotations δd are approximated in the same way.
3.2Transverse Shear Strains
The fulfillment of the bending patch test is discussed in [28], where for a plate it is shown, that
with the transverse shear strains (4)3the patch test can not be fulfilled within the present
mixed formulation. The non constant part of the shear strains according to (4)3leads for a
constant stress state to a contribution of the shear energy on the element level.
For this reason we approximate the shear strains with independent interpolation functions
proposed in [18, 19] as follows
?γ1
γ2
?
= J−1
?γξ
γη
?
where
γξ
=
1
2[(1 − η)γB
1
2[(1 − ξ)γA
ξ+ (1 + η)γD
ξ]
γη
=
η+ (1 + ξ)γC
η]
(13)
The strains at the midside nodes A,B,C,D according to Fig. 1 are specified as follows
γM
ξ
γL
= [u,ξ·D + X,ξ·∆d]M
= [u,η·D + X,η·∆d]L
M
L = A,C
= B,D
η
(14)
where the following quantities are given with the bilinear interpolation (10)
DA
DB
DC
DD
XA,η
XB,ξ
XC,η
XD,ξ
=
=
=
=
=
=
=
=
1
2(D4+ D1)
1
2(D1+ D2)
1
2(D2+ D3)
1
2(D3+ D4)
1
2(X4− X1)
1
2(X2− X1)
1
2(X3− X2)
1
2(X3− X4)
∆dA
∆dB
∆dC
∆dD
uA,η
uB,ξ
uC,η
uD,ξ
=
=
=
=
=
=
=
=
1
2(∆d4+ ∆d1)
1
2(∆d1+ ∆d2)
1
2(∆d2+ ∆d3)
1
2(∆d3+ ∆d4)
1
2(u4− u1)
1
2(u2− u1)
1
2(u3− u2)
1
2(u3− u4)
(15)
Remark:
An alternative three field variational formulation based on a Hu–Washizu principle for the
shear part, which would be the appropriate variational formulation for an independent shear
interpolation, leads to identical finite element matrices due to the fact that the shear stiffness
matrix is diagonal.
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3.3 Discrete Strain Displacement Matrix
Considering (4) and the finite element equations (10)  (15) the approximation of the strains
is now obtained by
4
?
with
NI,2XT,2
εh=
I=1
BIvI,vI= [uI,ϕI]T, (16)
BI=
NI,1XT,1
0
0
NI,1XT,2+NI,2XT,1
0
NI,1DT,1
NI,1bT
1I
NI,2DT,2
NI,2bT
2I
NI,1DT,2+NI,2DT,1
NI,1bT
2I+ NI,2bT
NI,ξξIbT
1I
J−1
NI,ξDT
M
NI,ηDT
L
J−1
M
NI,ηηIbT
L
(17)
and bαI= DI× X,α= WIX,α, bM= WIXM,ξ, bL= WIXL,η. The allocation of the mid–
side nodes to the corner nodes is given by (I,M,L) ∈ {(1,B,A);(2,B,C);(3,D,C);(4,D,A)}.
The skew–symmetric matrix WIis associated to DI= DIkekas follows
Furthermore, the derivatives of the position vectors X,αand director vectors D,αare obtained
from (10) in a standard way using
WI= skewDI=
0 −DI3
DI3
DI2
0 −DI1
DI1
−DI2
0
.
(18)
?
NI,1
NI,1
?
= J−1
?
NI,ξ
NI,η
?
J =
xL,ξ
yL,ξ
xL,η
yL,η
=
Gξ· t1
Gη· t1 Gη· t2
Gξ· t2
.
(19)
Here, J denotes the Jacobian matrix where the local coordinates xL= (X − X0) · t1 and
yL= (X−X0)·t2are computed with the position vector of the element center X0. The base
vectors are obtained from
Gξ
= G0
ξ+ η G1
G0
ξ
=
4
?
4
?
4
?
I=1
a1IXI
Gη
= G0
η+ ξ G1
G0
η
=
I=1
a2IXI
G1
=
I=1
hIXI.
(20)
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The determinant of J yields
detJ = j0+ ξ j1+ η j2
j0 = (G0
j1 = (G0
j2 = (G1· t1)(G0
ξ· t1)(G0
ξ· t1)(G1· t2) − (G1· t1)(G0
η· t2) − (G0
η· t2) − (G0
η· t1)(G0
ξ· t2)
ξ· t2)
η· t1)(G1· t2).
(21)
Note, that zL= (X − X0) · t3 does not enter in (19), which makes clear that the below
computed matrices are defined in Ωh
0.
3.4 Interpolation of the Stress Resultants
The independent field of stress resultants σ is interpolated as follows
σh
= Sβ
S =
?
13
0
0
0
13
0
0
0
12
Sm
0
0
0
Sb
0
0
0
Ss
Sm= Sb
=
J0
J0
J0
11J0
12J0
11J0
11(η − ¯ η) J0
12(η − ¯ η) J0
12(η − ¯ η) J0
21J0
22J0
21J0
21(ξ −¯ξ)
22(ξ −¯ξ)
22(ξ −¯ξ)
?
Ss
=
J0
J0
11(η − ¯ η) J0
12(η − ¯ η) J0
21(ξ −¯ξ)
22(ξ −¯ξ)
.
(22)
Here, we denote by 12,13second and third order unit matrices, respectively. The vector β
contains 8 parameters for the constant part and 6 parameters for the varying part of the stress
field, respectively. The interpolation of the membrane forces and the bending moments in
(22) corresponds to the procedure in Ref. [10]. In this context see also the original approach
for plane stress of Pian and Sumihara [29] with¯ξ = ¯ η = 0 and the text book Zienkiewicz
and Taylor, part 1, [30]. Finally we mention Ref. [20], where the shear approximation is
performed in a more complicated way.
The constants¯ξ and ¯ η are introduced to obtain decoupled matrices in the below defined
matrix H and denote the coordinates of the center of gravity of the element.
¯ξ =
1
Ae
?
(Ωe)
ξ dA =1
3
j1
j0
¯ η =
1
Ae
?
(Ωe)
η dA =1
3
j2
j0
(23)
The element area is given by Ae = 4j0. The transformation coefficients in (22) are the
components of the Jacobian matrix J according to (19), evaluated at the element center
J0
αβ= Jαβ(ξ = 0,η = 0). The coefficients have to be constant in order to fulfill the patch test,
see e.g. [30].
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3.5 Analytical Integration of the Element Matrices
Inserting the finite element equations (9) (23) and the corresponding equations for the virtual
stresses and virtual strains into the stationary condition (7) yields
δΠh
HR=
numel
?
e=1
?
δβ
δv
?T
e
??
−H G
GT
0
? ?
β
v
?
−
?
0
p
??
e
= 0, (24)
where numel denotes the total number of shell elements to discretize the problem. Here,
v = [v1,v2,v3,v4]Tis the element displacement vector and and δv,δβ the corresponding
virtual element vectors, respectively. The element load vector p = [p1,p2,p3,p4]Twhich
follows from the external virtual work is identical with a pure displacement formulation.
Furthermore the matrices H and G are introduced with B = [B1,B2,B3,B4],
?
(Ωe)
H =STC−1SdA,G =
?
(Ωe)
STBdA. (25)
Since the integrant in (25)1involves only polynomials of the coordinates ξ and η the integration
can be carried out analytically. In this context we also refer to the expressions for a plate in
[28]. Due to the introduced constants¯ξ and ¯ η one obtains a matrix H only with diagonal
entries
H =
AeC−1
0
h0
withh =
hm
0
0
0
hb
0
0
0
hs
(6×6)
. (26)
The components of the symmetric submatrices hm, hb= 12hm/h2and hsare given with
hm
11
=
Aef11
3Eh(J02
Aef22
3Eh(J02
= hm
11+ J02
12)2
hm
22
=
21+ J02
22)2
hm
12 21=Aef12
Aef11
3κGh(J02
Aef22
3κGh(J02
21=Aef12
3Eh
?
(J0
11J0
21+ J0
22J0
12)2− ν (J0
11J0
22− J0
12J0
21)2?
hs
11
=
11+ J02
12)
hs
22
=
21+ J02
22)
hs
12
= hs
3κGh(J0
?j2
j0
?j1
j0
j2
j0
11J0
21+ J0
22J0
12).
f11 = 1 −1
3
?2
?2
f22 = 1 −1
f12 = −1
3
j1
j0
3
(27)
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The matrix G according to (25)2can only be integrated analytically in Ωh
we replace in (17) the base vectors X,αby tα, introduce b0
D0,αevaluated at the element center
These assumptions have consequences on the element behaviour, which are discussed in the
next section. With the simplifications G0is obtained by analytical integration, where the
subscript 0 refers to the flat projection Ωh
0
0. For this reason
αI= WItαand replace D,αby
D0T,1
D0T,2
=1
4J0−1
−DT
−DT
1+ DT
2+ DT
3− DT
3+ DT
4
1− DT
2+ DT
4
, J0= J(ξ = η = 0). (28)
G0= [G01,G02,G03,G04]G0I=
AeB0
0I
g0I
(29)
where
B0
0I=
N0
I,1tT
N0
1
0
I,2tT
2
0
N0
I,1tT
N0
2+ N0
I,2tT
1
0
I,1D0T,1
N0
N0
I,1b0T
N0
1I
I,2D0T,2
I,2b0T
2I
N0
I,1D0T,2+N0
I,2D0T,1
a1IDT
N0
I,1b0T
2I+ N0
I,2b0T
bT
M
bT
L
1I
J0−1
M
a2IDT
L
1
4J0−1
?
N0
N0
I,1
I,2
?
= J0−1
?
a1I
a2I
?
(30)
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and
g0I
=
Ae
3γI
J0
11tT
J0
1+ J0
12tT
2
0
21tT
1+ J0
22tT
2
0
J0
11D0T,1+J0
J0
12D0T,2
J0
11b0T
J0
1I+ J0
12b0T
2I
21D0T,1+J0
γ11
22D0T,2
21b0T
1I+ J0
22b0T
2I
IDT
γ21
M+ γ12
IDT
L
γ11
IξIbT
γ21
M+ γ12
IηIbT
L
IDT
M+ γ22
IDT
LIξIbT
M+ γ22
IηIbT
L
γI
= hI−j2
= (hI−j2
= −j2
j0a1I−j1
j0a2I
γ11
I
j0a1I)/γI
γ12
I
= −j1
= (hI−j1
j0a2I/γI
γ21
I
j0a1I/γI
γ22
I
j0a2I)/γI.
(31)
3.6Transformation of the Element Matrices
Since the interpolation of the stress resultants are discontinuous at the element boundaries and
with δv ?= 0, δβ ?= 0, the parameters β can be eliminated on the element level considering
(24)
β = H−1Gv.(32)
Hence the stationary condition (24) reads
δΠh
HR=
numel
?
e=1
δvTS =
numel
?
e=1
δvT
0S0= 0,S = GTH−1Gv − p
S0 = GT
0H−1G0v0− p0.
(33)
The assembly can be performed as within a pure displacement formulation. However, at first
the nodal force vector S0= [S01,S02,S03,S04]Thas to be transformed considering the distance
vector rIbetween Ωh
0and Ωhat the nodes, see Fig. 1 and Ref. [27]
rI= zIt3,zI= (XI− X0) · t3= ±¯h,X0=1
4
4
?
I=1
XI. (34)
The equilibrium equations read with¯ WI= skewrIdefined by rI× fI=¯ WIfI
m0I
f0I
=
S0I
13
0
¯ WI
13
fI
mI
= TSISI.
(35)
11
Page 12
The corresponding transformations for the displacements and virtual displacements can be
derived from the virtual work (33)
u0I
ϕ0I
=
v0I
13
¯ WI
013
uG
I
ϕG
I
= T1IvG
I.
(36)
It should be noted that without transformation (36) the element is unacceptable stiff for
warped configurations, see also [27].
At the nodes which are not positioned on intersections no drilling stiffness is available and a
second transformation of the stiffness and the load vector is necessary:
uG
I
ϕG
I
=
vG
I
13
0
0T3I
uG
I
ϕL
I
= T2IvI
?13
T3I
=
for nodes on shell intersections
for all other nodes[a1I,a2I](3×2)
(37)
At all nodes which are not positioned on intersections the drilling degree of freedom is fixed.
Thus the element possesses six degrees of freedom at all nodes on intersections and five at all
other nodes. In this context we also refer to [31, 32].
12
Page 13
Combining (36) and (37) with (33) yields the total transformation TI= T1IT2Iand
?
gI
GI
= G0ITI=
AeB0
I
?
B0
I
=
N0
I,1tT
N0
1
N0
I,1¯bT
N0
1I
I,2tT
2I,2¯bT
2I
I,2¯bT
N0
I,1tT
N0
2+ N0
I,2tT
1
N0
I,1¯bT
2I+ N0
N0
1I
I,1D0T,1
N0
I,1˜bT
N0
1I
I,2D0T,2
I,2˜bT
2I
N0
I,1D0T,2+N0
I,2D0T,1
a1IDT
N0
I,1˜bT
2I+ N0
I,2˜bT
˜bT
M
˜bT
L
1I
J0−1
M
a2IDT
L
J0−1
gI
=
Ae
3γI
J0
11tT
J0
1+ J0
12tT
2
J0
11¯bT
J0
1I+ J0
12¯bT
22¯bT
2I
21tT
1+ J0
22tT
2 21¯bT
J0
1I+ J0
2I
J0
11D0T,1+J0
J0
12D0T,2
11˜bT
J0
1I+ J0
12˜bT
2I
21D0T,1+J0
22D0T,2
21˜bT
1I+ J0
22˜bT
2I
γ11
IDT
˜DT
M+ γ12
IDT
˜DT
L
γ11
IˆbT
M+ γ12
IˆbT
L
γ21
IM+ γ22
IL
γ21
IˆbT
M+ γ22
IˆbT
L
¯bαI
= TT
4Itα
˜bαI
= TT
4ID0,α+TT
3Ib0
αI
˜bM
=
1
4(ξITT
4IDM+ TT
3IbM)
˜bL =
1
4(ηITT
4IDL+ TT
3IbL)
ˆbM
= TT
4IDM+ ξITT
3IbM
ˆbL = TT
4IDL+ ηITT
3IbL.
(38)
with T4I=¯ WIT3I. For a constant load ¯ p = ¯ pieiin Ω we obtain
pI= TT
Ip0I= Ae(a0I+1
3
j1
j0a1I+1
3
j2
j0a2I)
¯ p
TT
4I¯ p
.(39)
Thus considering (26), (33) and (38) the element stiffness matrix reads
ke
= GTH−1G = k0+ kstab
kIK
= GT
IH−1GK= AeB0T
ICB0
K+ gT
Ih−1gK.
(40)
13
Page 14
Here, k0denotes the stiffness of a one–point integrated Reissner–Mindlin shell element with
assumed shear strains and kstab the stabilization matrix. The matrix h according to (26)
consists of three submatrices of order two and thus can easily be inverted. The element
stiffness matrix possesses with six zero eigenvalues the correct rank.
4 Examples
The derived element formulation has been implemented in an extended version of the general
purpose finite element program FEAP, see Zienkiewicz and Taylor [30].
4.1Membrane and bending patch test
Here we investigate a rectangular plate under membrane forces and bending moments accord
ing to [33]. Both, membrane and bending patch test are fulfilled by the present element.
4.2Corner supported square plate
A corner supported plate with edge length 2a subjected to uniform load is discussed. Con
sidering symmetry the mesh consists of 8 × 8 elements for a quarter of the plate, see Fig. 2.
The geometrical and material data are also given. An approximate ansatz according to [34]
reads
w(x,y) = c1+ c2x2+ c3y2+ c4x4+ c5x2y2+ c6y4,(41)
where the origin of the coordinate system lies in the center of the plate. The boundary
condition of vanishing bending moments at the edges can only be fulfilled in an integral
sense. The other boundary conditions and the partial differential equation can be fulfilled
exactly. The constants are determined and thus for y = 0 the approximate Kirchhoff solution
reads
w(x,y = 0) =
qa4
2Eh3[11 − 6ν − ν2+ (−5 + 4ν + ν2)(x
a)2+ (1 +ν
2−ν2
2)(x
a)4].(42)
a
h
q
E = 430000
ν = 0.38
= 12
= 0.375
= 0.03125
a
a
y
x
Figure 2: Corner supported plate
The deflections w(x,y = 0) obtained with different elements are plotted in Fig. 3. The
Belytschko/Tsay element [13] leads to hourglass modes for parameters rw < 0.02, optimal
results for 0.02 ≤ rw ≤ 0.05 and locking for rw > 0.05, see also [13] and Fig. 3. The
parameter rβ= 0.02 has been chosen constant in all cases.
14
Page 15
Corner Supported Plate
0,00
0,02
0,04
0,06
0,08
0,10
0,12
0,14
0,00 2,004,00 6,008,00 10,0012,00
coordinate x
displacement w
B/T r_w = 0.02
B/T r_w = 0.001
B/T r_w = 10
Present
Analytical (approx.)
Figure 3: Deflection w(x,y = 0) for the corner supported plate, comparison of different
elements
15
Page 16
4.3Hemispherical shell with a 18◦hole
The hemispherical shell with a 18◦hole under opposite loads is a standard example in linear
and nonlinear shell analysis. The material properties are E = 6.825 · 107and ν = 0.3,
the radius is R = 10 and the thickness is h = 0.04. Considering symmetry conditions a
quarter of the shell is modelled with a regular mesh, see Fig. 4. Table 1 presents results
for the displacement of the loaded node for different elements. The values are normalized
with respect to our converged solution w = 0.0935 for F = 1. Analytical solutions based on
asymptotic expansions are reported in [10] with w = 0.093. In Ref. [33] a value of w = 0.094
has been used for normalization. It can be seen that the results obtained with the present
element as well as the results using the elements [27], [23] converge against the same solution.
For this example the convergence behaviour of the investigated elements is practically the
same.
z
2F
y
x
2F
11
23
Figure 4: Hemispherical shell: Undeformed and deformed mesh (amplified by a factor 50)
Table 1: Normalized displacements for different elements
Nodes
per side
3
5
9
17
33
Simo[10]Taylor [27]Sauer [23]Present
91.4
99.9
99.3
99.4
∗66.1
92.5
100.7
100.0
∗100.0
106.7
103.8
100.3
99.8
100.0
106.2
103.8
100.4
99.8
100.0
∗Own results using the element of Taylor [27].
4.4Full hemispherical shell
The convergence behaviour using distorted and warped elements is investigated with the
hemispherical shell of the last section without the hole. The results for the deflections in load
direction are normalized with respect to our converged numerical solution w = 0.09227, see
16
Page 17
Table 2. In Ref. [10] a reference value of 0.0924 is given. The elements [23], [27] and the
present element converge against the same solution. The deformed mesh along with the radial
displacements is plotted in Fig. 5.
Table 2: Normalized displacements for different elements
Nodes
per side
Simo[10]
mixed
Simo[10] Taylor[27]∗
Sauer[23] Present
disp
46.9
93.4
98.9
5
9
65.2
96.9
99.4
77.0
100.8
100.8
100.2
100.0
50.5
95.5
100.0
100.0
100.0
57.4
97.2
100.3
100.1
100.0
17
33
65




∗Own results using the element of Taylor [27].
11
1
23
3
DISPLACEMENT 1
9.252E02 min
7.930E02
6.609E02
5.287E02
3.965E02
2.643E02
1.322E02
8.173E09
1.322E02
2.643E02
3.965E02
5.287E02
6.609E02
7.930E02
9.252E02 max
9.252E02 min
7.930E02
6.609E02
5.287E02
3.965E02
2.643E02
1.322E02
8.173E09
1.322E02
2.643E02
3.965E02
5.287E02
6.609E02
7.930E02
9.252E02 max
1
2
DISPLACEMENT 1
Figure 5: Radial displacements and deformed mesh (amplified by a factor 50)
4.5Twisted beam
This problem, a clamped beam twisted 90◦subjected to two different concentrated loads at
the tip, was originally introduced by MacNeal and Harder [33]. A more demanding thin
shell version was proposed by Jetteur [35] and is investigated in this paper. The example is
chosen to test the assess of warping on the performance of shell elements. Two load cases
are discussed. Load case 1 is a unit shear load in width direction whereas load case 2 is a
unit shear load in thickness direction, see Fig. 6. The computed tip displacements in load
direction are normalized with respect to our converged solutions 1.387 (load case 1) and 0.343
(load case 2) and are presented in Tab. 3. The displacements uzand uyare plotted for the
respective load case on the deformed configurations in Fig. 7.
17
Page 18
F=1.0
h
x
z
y
w
?
l
w
h
E = 29 · 106
ν = 0.22
F = 1.0
= 12
= 1.1
= 0.05
Figure 6: Twisted beam: geometrical and material data
Table 3: Load case 1 normalized displacement uzfor different elements
Mesh El.Simo[10]Taylor[27]Sauer[23]Present
1*6
2*12
4*24
8*48
699.4
100.0
100.1
100.2
100.1
100.2
100.1
100.0
99.5
99.7
99.9
100.0
102.0
100.6
99.3
100.0
24
96
384
Table 4: Load case 2 normalized displacement uyfor different elements
Mesh El.Simo[10]Taylor[27]Sauer[23]Present
1*6
2*12
4*24
8*48
695.1
98.6
99.7
100.0
102.1
101.0
100.2
100.0
94.0
98.4
99.6
99.9
104.3
100.5
99.3
100.0
24
96
384
18
Page 19
11
23
DISPLACEMENT 3
1.388E+00 min
1.289E+00
1.190E+00
1.090E+00
9.913E01
8.922E01
7.931E01
6.939E01
5.948E01
4.957E01
3.965E01
2.974E01
1.983E01
9.913E02
0.000E+00 max
11
23
DISPLACEMENT 3
1.388E+00 min
1.289E+00
1.190E+00
1.090E+00
9.913E01
8.922E01
7.931E01
6.939E01
5.948E01
4.957E01
3.965E01
2.974E01
1.983E01
9.913E02
0.000E+00 max
11
23
DISPLACEMENT 2
0.000E+00 min
2.450E02
4.900E02
7.351E02
9.801E02
1.225E01
1.470E01
1.715E01
1.960E01
2.205E01
2.450E01
2.695E01
2.940E01
3.185E01
3.430E01 max
11
23
DISPLACEMENT 2
0.000E+00 min
2.450E02
4.900E02
7.351E02
9.801E02
1.225E01
1.470E01
1.715E01
1.960E01
2.205E01
2.450E01
2.695E01
2.940E01
3.185E01
3.430E01 max
Figure 7: Deformed configurations for load case 1 and 2 and respective displacements uzand
uy
19
Page 20
4.6Hypar shell
f
f
h
?
?
z
x
y
l
f
h
E
ν
¯ pz
=
= l/32 m
=0.2 m
=108
=
=
−5 kN/m2
20 m
kN/m2
0
Figure 8: Hypar shell: geometrical and material data
The geometry of the considered hyperbolic paraboloid shell is described by the function z =
1/8xy. The shell is loaded by a constant load ¯ pzper shell middle surface in vertical direction.
Along the boundary the deflections are restrained in global z–direction. Furthermore the
boundary conditions ux(−l/2,0) = ux(l/2,0) = 0 and uy(0,−l/2) = uy(0,l/2) = 0 are
considered. An analytical Kirchhoff solution with slightly different boundary conditions using
Fourier series has been derived by Duddeck [36]. The shell with coordinates z(l/2,l/2) =
±f = l/32 is rather flat. Therefore the support perpendicular to the shell which has been
considered in [36] does not lead to significant different results. The geometrical and material
data as well as a typical finite element mesh are depicted in Fig. 8. The distribution of
the global displacement w = uz is symmetric with respect to x = 0 and y = 0. In Fig.
9 the deflection w (0 < x < l/2,y = 0) is depicted. The calculated results are in good
agreement with the solution of Duddeck. Furthermore the distribution of the bending moment
mxy(0 < x < l/2,y = l/2) is presented in this Figure. Differences occur along the edges due
to the fact that the analytical solution is based on a Kirchhoff theory whereas the numerical
solution is calculated using the Reissner–Mindlin theory. In Fig. 10 the distribution of the
bending moment mx(0 < x < l/2,y = 0) and mx(x = 0,0 < y < l/2) are shown. The good
agreement with the analytical solution is noted.
Table 5: Center displacement w(0,0) in cm for different elements
Nodes
per Side
w(0,0)
Taylor [27]∗
Sauer [23]Present
4.03
4.41
4.52
4.56
4.58
4.60
Duddeck [36]
2
4
8
4.05
4.42
4.51
4.55
4.56
4.57
2.84
4.39
4.51
4.56
4.58
4.60
17
33
654.6
∗Own results with the element of Taylor [27].
20
Page 21
Finally a convergence study is presented in Tables 5 and 6 for the center deflection w(0,0)
and the bending moment mx(0,0), respectively. It shows that the present element exhibits a
superior convergence behaviour for the bending moment.
Table 6: Center moment mx(0,0) in kNm/m for different elements
Nodes
per Side
mx(0,0)
Taylor [27]∗
Sauer [23] Present
53.6
64.0
64.3
64.9
65.1
65.3
Duddeck [36]
2
4
8
36.0
57.7
62.6
64.3
64.8
64.9
25.2
60.2
63.3
64.6
65.0
65.3
17
33
6563
∗Own results with the element of Taylor [27].
Hypar Duddeck
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
5
0246810
coordinate x [m]
displacement w [cm]
Present 4x 4 mesh
Present 8x 8 mesh
Present 32x32 mesh
Duddeck
[0,0][10,0]
Hypar Duddeck
0,0
10,0
20,0
30,0
40,0
50,0
60,0
70,0
80,0
90,0
0246810
coordinate x [m]
moment mxy [kNm/m]
Present 4x 4 mesh
Present 8x 8 mesh
Present 32x32 mesh
Duddeck
[0,10][10,10]
Figure 9: Displacement w(x,y  0 < x < l/2,y = 0) and bending moment mxy(x,y  0 < x <
l/2,y = l/2)
Hypar Duddeck
0,0
10,0
20,0
30,0
40,0
50,0
60,0
70,0
0246810
coordinate x [m]
moment mx [kNm/m]
Present 4x 4 mesh
Present 8x 8 mesh
Present 32x32 mesh
Duddeck
[0,0][10,0]
Hypar Duddeck
0,0
10,0
20,0
30,0
40,0
50,0
60,0
70,0
0246810
coordinate y [m]
moment mx [kNm/m]
Present 4x 4 mesh
Present 8x 8 mesh
Present 32x32 mesh
Duddeck
[0,0][0,10]
Figure 10: Bending moment mx(x,y  0 < x < l/2,y = 0) and mx(x,y  x = 0,0 < y < l/2)
21
Page 22
4.7 Steel frame structure
In the last example we discuss a symmetrical frame structure with welded cross–sections,
see Figs. 11, 12. Thus, different intersections of plates occur, which can be treated with
the present model. The frame is loaded by a constant vertical load ¯ p = 16 kN/m. The
problem with all geometrical data as well as the load distribution is presented in Fig. 11,
the underlying data for the cross–sections of a beam model are depicted in Fig. 12. Elastic
material behaviour is assumed using the parameters E = 21000 kN/cm2and ν = 0.3.
p
z
x
y
24 24
20
20
474
102
600
340
400
40
[cm]
1.2
1.2
1.2
1.6
Figure 11: Frame structure: system and loading
1.2
0.8
20
40
20
48
1.6
1.0
[cm]
frame column
A=
Iy
= 46094 cm4
Iz
=
IT
=
112.0 cm2
2137 cm4
70.6 cm4
horizontal member
A=
Iy
= 23472 cm4
Iz
=
IT
=
80.0 cm2
1602 cm4
29.9 cm4
Figure 12: Definition of cross sections
The finite element calculations are performed with the developed shell element using a mesh
with 530 nodes and 368 elements, see Fig. 13, and for comparison with 15 two–dimensional
beam elements, see Fig. 17. It can be seen from Tables 7 and 8 that the results of all
used three shell elements are very similar for the vertical displacement uz(0,0,400) in the
symmetry plane as well as for the stresses σ11in axial direction at the coordinates (0,0,420)
and (0,0,380). This could also be verified using the beam model with only little deviations.
In detail Fig. 13 show the un–deformed and deformed mesh, whereas the axial stresses σ11at
22
Page 23
the plate mid–surfaces are depicted in Figs. 15 and 16. Finally Fig. 17 presents the associated
results for the beam model. Here, normal forces of N = −72.6 kN and N = −96.0 kN occur
in the horizontal member and the column of the frame. It should be noted that only the shell
model is able to analyze the stress state in the corner of the frame (see Fig. 16).
Table 7: Vertical displacement uz(0,0,400) in cm
Displacement uz
Shell model
Beam model
Taylor[27]
1.818
Sauer[23]
1.817
Present
1.817
1.761
Table 8: Axial stresses σ11in symmetry plane in kN/cm2
Axial Stress
σ11(0,0,420)
σ11(0,0,380)
Taylor[27]Sauer[23]
8.77
Present
8.78
7.06
Beam
8.83
7.02
8.78
7.067.05
11
23
Figure 13: Undeformed and deformed mesh (amplified by a factor 20)
23
Page 24
11
23
DISPLACEMENT 2
1
1.817E+00 min
1.686E+00
1.554E+00
1.422E+00
1.291E+00
1.159E+00
1.027E+00
8.958E01
7.642E01
6.326E01
5.009E01
3.693E01
2.377E01
1.061E01
2.558E02 max
1
23
DISPLACEMENT 2
1.817E+00 min
1.686E+00
1.554E+00
1.422E+00
1.291E+00
1.159E+00
1.027E+00
8.958E01
7.642E01
6.326E01
5.009E01
3.693E01
2.377E01
1.061E01
2.558E02 max
Figure 14: Vertical deflection uzin cm
11
23
n_11
8.775E+00 min
7.645E+00
6.515E+00
5.384E+00
4.254E+00
3.124E+00
1.994E+00
8.635E01
2.667E01
1.397E+00
2.527E+00
3.657E+00
4.788E+00
5.918E+00
7.048E+00 max
11
23
n_11
8.775E+00 min
7.645E+00
6.515E+00
5.384E+00
4.254E+00
3.124E+00
1.994E+00
8.635E01
2.667E01
1.397E+00
2.527E+00
3.657E+00
4.788E+00
5.918E+00
7.048E+00 max
Figure 15: Axial stresses σ11at mid–surfaces in kN/cm2
24
Page 25
n_
Figure 16: Axial stresses σ11at mid–surfaces in kN/cm2in the frame corner
MOMENT M_y
1.801E+04 min
0.000E+00
9.502E+03 max
MOMENT M_y
1.801E+04 min
0.000E+00
9.502E+03 max
Figure 17: Beam model: undeformed mesh, deformed mesh (amplified by a factor 20) and
bending moment
25
Page 26
5Conclusions
The main aspect of the present work is to derive a four–node shell element with explicit
representation of the stiffness matrix. The formulation with 5 or 6 nodal degrees of freedom
is applicable for shell problems with intersections. The element possesses a correct rank, is
free of locking, and can be used for the structural analysis of thin and thick shells. The
computed results obtained for various shell problems with positive and negative Gaussian
curvature are very satisfactory. Especially the convergence behaviour of the stress resultants
is superior to comparable elements. The essential advantage is the fast stiffness computation
due to the analytically derived stiffness matrix.
References
[1] Koiter, W.T.: On the nonlinear theory of thin elastic shells, Proc. Kon. Ned. Ak. Wet.
B69, 1966, 154.
[2] Nagdhi, P.M.: The theory of shells, in Handbuch der Physik, Vol. VIa/2, Mechanics of
Solids II, C. Truesdell Ed., SpringerVerlag Berlin. 1972
[3] Ahmad, S., Irons B.M., Zienkiewicz, O.C.: Analysis of thick and thin shell structures by
curved finite elements, Int. J. Num. Meth. Engng., 2, 419–451, 1970.
[4] Papadrakakis, M., Samartin, A., O˜ nate, E. (eds.), IASSIACM 2000, Proceedings of the
Fourth International Colloquium on Computation of Shell & Spatial Structures, Chania,
Greece, 2000.
[5] Yang, H.T.Y., Saigal, S., Masud, A., Kapania, R.K.: A survey of recent shell finite
elements., Int. J. Num. Meth. Engng., 47, 101–127, 2000.
[6] B¨ uchter N., Ramm E.: Shell theory versus degeneration–A comparison in large rotation
finite element analysis, Int. J. Num. Meth. Engng., 34, 39–59, 1992.
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