In this work, a new physically stabilized and locking-free formulation of the SHB8PS element is presented. This is a solid-shell element based on a purely 3D formulation. It has eight nodes as well as five integration points, all distributed along the “thickness” direction. Consequently, it can be used for the modeling of thin structures, while providing an accurate description of the various through-thickness phenomena. The reduced integration has been used in order to prevent some locking phenomena and to increase computational efficiency. The spurious zero-energy deformation modes due to the reduced integration are efficiently stabilized, whereas the strain components corresponding to locking modes are eliminated with a projection technique following the Enhanced Assumed Strain (EAS) method.Contrat EDF R&

ABED-MERAIM, Farid

COMBESCURE, Alain

English

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Farid ABED-MERAIM, Alain COMBESCURE - A new locking-free formulation for the SHB8PS
solid–shell element: non-linear benchmark problems - 2007
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Science Arts & Métiers (SAM)
is an open access repository that collects the work of Arts et Métiers ParisTech
researchers and makes it freely available over the web where possible.
This is an author-deposited version published in: http://sam.ensam.eu
Handle ID: .http://hdl.handle.net/null
To cite this version :
Farid ABED-MERAIM, Alain COMBESCURE - A new locking-free formulation for the SHB8PS
solid–shell element: non-linear benchmark problems - In: Int. Conf. on Computational Methods for
Coupled Problems in Science and Engineering: COUPLED PROBLEMS 2007, Espagne, 2007-
05-21 - Int. Conf. on Computational Methods for Coupled Problems in Science and Engineering:
COUPLED PROBLEMS 2007 - 2007
Any correspondence concerning this service should be sent to the repository
Administrator : archiveouverte@ensam.eu
A NEW LOCKING-FREE FORMULATION FOR THE SHB8PS SOLID-
SHELL ELEMENT: NON-LINEAR BENCHMARK PROBLEMS 
FARID ABED-MERAIM* AND ALAIN COMBESCURE†
* Laboratoire de Physique et Mécanique des Matériaux (LPMM)
ENSAM CER de Metz, UMR CNRS 7554, 4 rue Augustin Fresnel, 57078 Metz, France 
E-mail: farid.abed-meraim@metz.ensam.fr 
† Laboratoire de Mécanique des Contacts et des Solides (LaMCoS) 
INSA de Lyon, UMR CNRS 5514, 18-20 rue des Sciences, 69621 Villeurbanne, France 
Key words: SHB8PS Solid-shell, Hourglass, Shear and Membrane Locking, Assumed Strain 
Method, Orthogonal Projection, Benchmark Problems. 
Summary. In this work, a new physically stabilized and locking-free formulation of the 
SHB8PS element is presented. This is a solid-shell element based on a purely 3D formulation. 
It has eight nodes as well as five integration points, all distributed along the “thickness” 
direction. Consequently, it can be used for the modeling of thin structures, while providing an 
accurate description of the various through-thickness phenomena. The reduced integration 
has been used in order to prevent some locking phenomena and to increase computational 
efficiency. The spurious zero-energy deformation modes due to the reduced integration are 
efficiently stabilized, whereas the strain components corresponding to locking modes are 
eliminated with a projection technique following the Enhanced Assumed Strain (EAS) method. 
1 INTRODUCTION 
Over the last decade, considerable progress has been made in the development of three-
dimensional finite elements capable of modeling thin structures (see references 1, 2, 3, 4, 5). The 
coupling between solid and shell formulations is a good way to provide continuum finite 
element models that can be efficiently used for structural applications. These solid-shell 
elements have numerous advantages for the analysis of various complex structural forms that 
are common in many industrial applications. Their main advantage is to allow the meshing of 
complex structural forms without the classical problems of connecting zones meshed with 
different element types (continuum and structural elements for instance). Another important 
benefit of solid-shell elements is the avoidance of tedious and complex pure-shell element 
formulations. In this work, a new locking-free formulation for the SHB8PS solid-shell 
element is performed. More specifically, this work focuses on the elimination of the residual 
membrane and shear locking phenomena persisting in the previous formulations. By using 
orthogonal projection of the discrete gradient operator, these severe shear and membrane 
locking modes are removed. Several numerical experiments in linear and non linear 
benchmark problems show that this new formulation of the SHB8PS element is effective and 
allows fast convergence without locking phenomena. 
2 FORMULATION OF THE SHB8PS ELEMENT 
The element’s coordinates ix  and displacements  are interpolated using the 
isoparametric trilinear shape functions 
( 1,...,3)i iu =
( ), , ( 1, ..., 8) I IN ξ η ζ = . By introducing the Hallquist’s 
vectors ( ib , i = 1,..., 3), defined in ref. 
6 as: 
, (0,0,0)       1, 2,3         
T
i ib N i Hallquist Form= = (1)
where , / iiN N x∂ ∂= , one can show that the discrete gradient operator, which relates the linear 
deformations to the nodal displacements (i.e., ( )S u B d∇ = ⋅ ), is given by Eqn. (2). This B -
matrix uses the following variables: 
1,3
1 2 3 4
1
8 ( )
,  ,  ,  
T
j jj h x bh
h h h h
α α αγ
ηζ ζξ ξη ξηζ
= ⋅⎧ ⎡ ⎤= −⎪ ⎣ ⎦⎨ = = = =⎪⎩
∑
; 
1 2 3, 8 1 2 3 8
1 2
3 4
( , , ....., ),    ( , , ,....., )  
(1,1, 1, 1, 1, 1,1,1),   (1, 1, 1,1, 1,1,1, 1)
(1, 1,1, 1,1, 1,1, 1),   ( 1,1, 1,1,1, 1,1, 1)
T T
i i i i i i i i i i
T T
T T
d u u u u x x x x x
h h
h h
⎧ = =⎪ = − − − − = − − − −⎨⎪ = − − − − = − − − −⎩
( )
,
,
,
, ,
, ,
, ,
 
The convention of implied summation
  of repeated subscripts is adopted
0 0
0 0
0 0
0
0
0
T T
x x
T T
y y
T T
z z
T T T T
y y x x
T T T T
z z y y
T T T T
z z x x
b h
b h
b h
B
b h b h
b h b h
b h b h
α α
α α
α α
α α α α
α α α α
α α α α
α
γ
γ
γ
γ γ
γ γ
γ γ
=
+
+
+
+ +
+ +
+ +
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
(2)
Despite the geometry of the element (eight-node hexahedron), several modifications are 
introduced in order to provide it with shell features. Among them, a shell-like behavior is 
intended for the element, by modifying the three-dimensional constitutive law so that the plane-
stress conditions are approached and by aligning all the integration points along a privileged 
direction, called the thickness. The stiffness matrix is then obtained by Gauss integration: 
5
1
( ) ( ) ( ) ( ) 
e
TT
e I I
I
K B C B v J B C Bd I Iω ζ ζ ζ ζ
=Ω
= ⋅ ⋅ = ⋅ ⋅∑∫ (3)
Because the ,ihα  functions 3, 4;  1, 2, 3( i )α = =  vanish at the five Gauss points, of coordinates 
 the 0,  0,I I Iξ η ζ= = ≠ B -matrix Eqn. (2) reduces at its 12B  part, with only the ,ihα  terms 
1, 2;  1, 2, 3( i )α = = . This leads to six hourglass modes generated by 3 4and h h . These spurious 
modes are stabilized following the approach given in ref. 7. Moreover, we apply an assumed 
strain method in order to eliminate locking. The B -matrix is thus projected onto B  as: 
12 34B B B= + (4)
Consequently, the stiffness matrix, Eqn. (3), can be rewritten as: 
12e SK K K= + TAB  (5)
The first term 12K  is obtained by Gauss integration, Eqn. (3). The second term STABK
represents the stabilization stiffness: 
12 34 34 12 34 34
e e e
T T T
STABK B C B dv B C B dv B C B dv
Ω Ω Ω
+ += ⋅ ⋅ ⋅ ⋅ ⋅ ⋅∫ ∫ ∫ (6)
3 NUMERICAL RESULTS AND DISCUSSIONS 
The performance of this new formulation has been tested through a variety of linear and 
non linear mechanical problems. In all of these tests, the new version showed better 
performance than the previous formulation. In particular, the improvement is significant in the 
pinched hemispherical shell test. This test has become very popular and is used by many 
authors. It is severe since the shear and membrane locking phenomena are very important and 
emphasized by the problem geometry (distorted, skewed elements). As reported by many 
authors, in this doubly-curved shell problem, the membrane locking is much more severe than 
shear locking. Fig. 1 shows the geometry, loading and boundary conditions for this problem. 
Figure 1: Schematic of hemispherical shell 
The radius is R=10, the thickness t=0.04, the Young’s modulus E=6.825×107, Poisson’s 
ratio ν =0.3. Using the symmetry, only a quarter of the hemisphere is meshed using a single 
element through the thickness and with two unit loads along directions Ox and Oy. The 
analytical solution for the radial displacement at the load point is 0.0924. The convergence 
results are reported in Tab. 1 in terms of the normalized displacement at the load point. The 
new version of the SHB8PS element is compared to the former one and to the three elements 
HEX8, HEXDS and H8-ct-cp. The HEX8 element is a standard, 8-node, full integration solid 
element (8 Gauss points). The HEXDS element is an 8-node, four Gauss points solid element 
(see ref. 8). The H8-ct-cp element was developed in ref. 9. Tab. 1 shows that the new version 
of the SHB8PS element provides an excellent convergence and shows no locking. 
This new version has also been tested on a variety of non linear, elastic and elastic-plastic 
problems. We demonstrate that the projection adopted in this formulation better eliminates the 
locking phenomena. As shown particularly in the pinched hemisphere test, Tab. 1, this element 
also demonstrates an excellent efficiency and convergence through numerous other tests. 
SHB8PS
previous 
version
HEX8 HEXDS H8-ct-cp
SHB8PS 
new 
version Number of 
elements Ux/Uref Ux/Uref Ux/Uref Ux/Uref Ux/Uref 
12 0.0629 0.0005 0.05 0.8645 
27 0.0474 0.0011 1.0155 
48 0.1660 0.0023 0.408 0.35 1.0098 
75 0.2252 0.0030 0.512 0.58 1.0096 
192 0.6332 0.0076 0.701 0.95 1.0008 
363 0.8592 0.0140 0.800 1.0006 
768 0.9651 0.0287 1.0006 
1462  0.9910 0.0520 1.0009 
Table 1: Normalized displacement at the load point of the pinched hemispherical shell 
REFERENCES 
[1] F. Abed-Meraim and A. Combescure, “SHB8PS a new intelligent assumed strain 
continuum mechanics shell element for impact analysis on a rotating body”, First M.I.T. 
Conference on Comput. Fluid and Solid Mechanics, 12-15 June, U.S.A., (2001). 
[2] F. Abed-Meraim and A. Combescure, “SHB8PS- a new adaptive, assumed-strain 
continuum mechanics shell element for impact analysis”, Computers & Structures, 80, 
791-803 (2002). 
[3] Y.-I. Chen and G.-Y. Wu, “A mixed 8-node hexahedral element based on the Hu-Washizu 
principle and the field extrapolation technique”, Structural Engineering and Mechanics, 
17, 113-140 (2004). 
[4] A. Legay and A. Combescure, “Elastoplastic stability analysis of shells using the 
physically stabilized finite element SHB8PS”, International Journal for Numerical 
Methods and Engineering, 57, 1299-1322 (2003). 
[5] K.Y. Sze and L.Q. Yao, “A hybrid stress ANS solid-shell element and its generalization 
for smart structure modeling. Part I-solid-shell element formulation”, International 
Journal for Numerical Methods in Engineering, 48, 545-564 (2000). 
[6] J.O. Hallquist, Theoretical manual for DYNA3D, UC1D-19401 Lawrence Livermore 
National Lab., University of California, (1983). 
[7] T. Belytschko and L.P. Bindeman, “Assumed strain stabilization of the eight node 
hexahedral element”, Computer Meth. in Applied Mech. and Engng, 105, 225-260 (1993). 
[8] W.K. Liu, Y. Guo, S. Tang and T. Belytschko, “A multiple-quadrature eight-node 
hexahedral finite element for large deformation elastoplastic analysis”, Computer 
Methods in Applied Mechanics and Engineering, 154, 69-132 (1998). 
[9] D. Lemosse, Eléments finis iso-paramétriques tridimensionnels pour l’étude des 
structures minces, PhD Thesis, Ecole Doctorale SPMI/INSA-Rouen, (2000). 


A new locking-free formulation for the SHB8PS solid–shell element: non-linear benchmark problems

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A new locking-free formulation for the SHB8PS solid–shell element: non-linear benchmark problems

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