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A linear quadrilateral shell element with fast stiffness computation

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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 non-constant 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 degree-of–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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