The formulation of a six-node solid-shell called SHB6, which is a linear, isoparametric element, is discussed. An eigenvalue analysis of the element stiffness matrix is first carried out. Several modifications are introduced into the formulation of the SHB6 element following the assumed strain method adopted by Belytschko and Bindeman. SHB6's coordinates and displacements are related to the nodal coordinates and displacements through the linear shape functions. Applying the simplified form of the Hu-Washizu nonlinear mixed variational principle, in which the assumed stress field is chosen to be orthogonal to the difference between the symmetric part of the displacement gradient and the assumed strain field, the formula is obtained. The newly developed SHB6 element was implemented into the finite element codes INCA and ASTER. It represents some improvement since it converges well and performs much better than the PRI6 six-node three-dimensional element in all of the benchmark problems tested

ABED-MERAIM, Farid

COMBESCURE, Alain

STRUCTURAL ENGINEERING AND MECHANICS

ABED-MERAIM, Farid 

SAM : Science Arts et Métiers

Science Arts & Métiers (SAM)is an open access repository that collects the work of Arts et Métiers ParisTechresearchers and makes it freely available over the web where possible.This is an author-deposited version published in: http://sam.ensam.euHandle ID: .http://hdl.handle.net/10985/10198To cite this version :Farid ABED-MERAIM, Alain COMBESCURE - New prismatic solid-shell element : Assumed strainformulation and hourglass mode analysis - Structural Engineering and Mechanics - Vol. 37, n°2,p.253-256 - 2011Any correspondence concerning this service should be sent to the repositoryAdministrator : archiveouverte@ensam.euNew prismatic solid-shell element: assumed strain formulation and hourglass mode analysis Farid Abed-Meraim1,†, Alain Combescure2 1 LEM3, CNRS, Arts et Métiers ParisTech, 57078 Metz Cedex 3, France 2 LaMCoS, UMR CNRS 5259, INSA de Lyon, 69621 Villeurbanne Cedex, France Keywords:    assumed-strain solid–shell SHB6; six-node prism; shear and thickness locking; hourglass. 1. IntroductionIn three-dimensional analysis of structural problems, the development of effective eight-node solid-shell finite elements has been a major objective over the last decade (Belytschko and Bindeman 1993, Hauptmann and Schweizerhof 1998, Abed-Meraim and Combescure 2002, Legay and Combescure 2003). However, to be able to mesh complex geometries and with the advent of free mesh generation tools not generating only hexahedrons, the development of prismatic elements is made necessary. This paper presents the formulation of a six-node solid-shell called SHB6. It represents a thick shell obtained from a purely 3D approach. The assumed strain method is adopted together with an in-plane reduced integration scheme with five integration points along the thickness direction. The 3D elastic constitutive law is also modified so that shell-like behavior is intended for the element and in order to alleviate shear and membrane locking. A detailed eigenvalue analysis of the element stiffness matrix is first carried out. We demonstrate that the kernel of this stiffness matrix only reduces to rigid body movements and hence, in contrast to the eight-node solid-shell element (SHB8PS), the SHB6 element does not require stabilization. On the other hand, to attenuate locking phenomena, several modifications are introduced into the formulation of the SHB6 element following the assumed strain method adopted by Belytschko and Bindeman (1993). Finally, one example, among the variety of benchmark problems performed, is shown to illustrate the performance of the new element. 2. Formulation of the SHB6 finite elementThe SHB6 is a solid-shell with only 3 displacement DOF per node, and it has a special direction called “thickness”. It is integrated with five integration points along this direction and only one point in the in-plane directions. Fig. 1 shows the SHB6 reference geometry and its integration points. 2.1 Kinematics and interpolation The SHB6 is a linear, isoparametric element. Its coordinates ix  and displacements iu  are related to the nodal coordinates iIx  and displacements iIu  through the linear shape functions IN  as ∑===61),,(),,(IIiIIiIi NxNxx ζηξζηξ   and ∑===61),,(),,(IIiIIiIi NuNuu ζηξζηξ  (1) The standard tri-linear shape functions IN  are: † Corresponding author, Associate Professor, Ph.D., E-mail: Farid.Abed-Meraim@ensam.eu  Farid Abed-Meraim and Alain Combescure  ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )[ ] [ ] [ ]1, , 1 1 1 1 1 1 1 120,1 ;     0,1 ;     1,1IN ξ η ζ ζ ξ η ζ ξ ζ η ζ ξ η ζ ξ ζ ηξ η ξ ζ= − − − − − + − − + +  = = − = − (2)       Fig. 1 Reference geometry of the SHB6 element and location of its integration points 2.2 Discrete gradient operator Using some mathematical derivations, similarly to the SHB8PS formulation (Abed-Meraim and Combescure 2002), we can explicitly relate the strain field to the nodal displacements as  ( ),,,,,,, ,, ,, ,, ,, ,, ,0 00 00 0000T Tx xx x T Ty yy yT T xz zz zs T T T Tx y y x y xy xT T T Tx z z xz xz xy z z y T T T Tz yz yb hub hudb huu du u b h b hu ub h b hu ub h b hα αα αα αα αα αα αα αα αα αγγγγ γγ γγ γ+++∇ = = ⋅+ + +++ +++ +                       yzB dd= ⋅      (3) where functions hα are: 1 2,  h hηζ ξζ= = ; vectors ib  are: , 0(0) ;  1, 2,3Ti i ib N N x iξ η ζ= = == = ∂ ∂ =  and the constant vectors αγ  are given by: ( )311, 21 ;  2Ti iih h x bα αααγ=== − ⋅   ∑ . In this latter expression, ( )1 0, 0, 1, 0, 0, 1 ;Th = −  ( )2 0, 1, 0, 0, 1, 0Th = −  and vectors ix  denote the nodal coordinates. 2.3 Variational principle Applying the simplified form of the Hu-Washizu nonlinear mixed variational principle, in which the assumed stress field is chosen to be orthogonal to the difference between the symmetric part of the displacement gradient and the assumed strain field, we obtain:  ( ),  0eT T extVu dV d fδpi ε δ ε σ δ= ⋅ − ⋅ =∫  (4) Replacing the assumed strain field, with its expression ( ) ( ) ( ),x t B x d tε = ⋅ , in Eq. (4) leads to the following expressions for the element internal force vector and stiffness matrix:  ( )int  eTVf B dVσ ε= ⋅∫ ,          TeeVB C BK dV⋅ ⋅= ∫  (5) 0 ζ η ξ 4 3 5 1 2 6 New prismatic solid-shell element: assumed strain formulation and hourglass mode analysis  For a standard displacement approach, B  is simply replaced by B  leading classically to   eTeVK B C B dV= ⋅ ⋅∫  (6) 2.4 Hourglass mode analysis for the SHB6 element Hourglass patterns are spurious zero-energy modes that are induced by reduced integration. The analysis of hourglass modes is equivalent to that of the stiffness matrix kernel, namely searching for zero-strain modes d  that satisfy:  ( ) 0IB dζ ⋅ = ,       int1, ...,I n=  (7) where intn  represents the number of integration points. To this end, we can build a basis for the vector space of the discretized displacements, given by the eighteen column vectors below, in which ( )1, 1, 1, 1, 1, 1TS = . We show then that only the first six column vectors in Eq. (8) verify Eq. (7), which corresponds to rigid body modes. This reveals that there are no hourglass modes for the SHB6 element, and thus no hourglass control is required.  −−−212121000000000000000000000000000000hhyxzyxShhzxyzxShhzyxzyS (8) 2.5 Assumed strain formulation for the SHB6 discrete gradient operator Among several treatments for alleviating shear and membrane locking, in the present formulation the discrete gradient will be appropriately modified. This consists first of decomposing the matrix B  into two parts: 21BBB += , then of projecting the second part onto an assumed strain operator such that 1 2B B B= + . As a result, the stiffness matrix becomes  1 1 1 2 2 1 2 2    e e e eT T T Te V V V VK B C B dV B C B dV B C B dV B C B dV= ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅∫ ∫ ∫ ∫  (9) The subsequent steps consist of choosing an adequate assumed strain field. This is a key point in the formulation and special care has been exercised in this regard. Finally, the above additive decomposition of the stiffness matrix is computed using a reduced integration scheme with five Gauss points, all located along the thickness direction. Note that the choice of an assumed strain field is mainly guided by the elimination of strain components that are responsible for shear and membrane locking. The effectiveness of this assumed strain is assessed through benchmark problems.  ,,,1 2, ,, ,, ,0 0 0 0 00 0 00 00 0 00 0 ;  0 0 0000 0 000 0 0T Tx xT Ty yT Tz zT T T TT T T Ty xy xz xz xT T T Tz yz yb hb hb hB Bb h b hb h b hb h b hα αα αα αα αα αα αα αα αα αγγγγ γγ γγ γ+++= =+ ++ ++ +                                 Farid Abed-Meraim and Alain Combescure 3.  Numerical results and comparison Several popular benchmark problems were performed to illustrate the element performance. One of these test problems, which is frequently used to test warping effects, is shown here. E = 29×106ν = 0.22P = 1L = 12l = 1.1h = 0.32 Fig. 2 Twisted beam (reference vertical displacement of point A at tip is 5.424x10-3) Table 1 Normalized vertical displacement at point A of the twisted cantilever beam problem Mesh layout PRI6 (3D solid element) SHB6 (without assumed strain) SHB6 (assumed strain) (6x2x1)x2 0.061 0.234 0.496 (12x4x1)x2 0.202 0.470 0.784 (24x4x1)x2 0.485 0.779 0.935 (36x8x1)x2 0.489 0.875 0.972 4.  Conclusions This newly developed SHB6 element was implemented into the finite element codes INCA and ASTER. It represents some improvement since it converges well and it performs much better than the PRI6 six-node three-dimensional element in all of the benchmark problems tested. Furthermore, it shows very good performances in problems using mixed meshes composed of SHB6 and SHB8PS elements. Thus, we can couple the SHB6 with other finite elements to mesh complex geometries, which could be obtained by free mesh generation tools. References Abed-Meraim, F. and Combescure, A. (2002), “SHB8PS a new adaptive, assumed-strain continuum mechanics shell element for impact analysis”, Comput. Struct., 80, 791-803. Belytschko, T. and Bindeman, L.P. (1993), “Assumed strain stabilization of the eight node hexahedral element”, Comput. Meth. Appl. Mech. Eng., 105, 225-260. Hauptmann, R. and Schweizerhof, K. (1998), “A systematic development of solid-shell element formulations for linear and non-linear analyses employing only displacement degrees of freedom”, Int. J. Num. Meth. Eng., 42, 49-69. Legay, A. and Combescure, A. (2003), “Elastoplastic stability analysis of shells using the physically stabilized finite element SHB8PS”, Int. J. Num. Meth. Eng., 57, 1299-1322. 

New prismatic solid-shell element : Assumed strain formulation and hourglass mode analysis

The formulation of a six-node solid-shell called SHB6, which is a linear, isoparametric element, is discussed. An eigenvalue analysis of the element stiffness matrix is first carried out. Several modifications are introduced into the formulation of the SHB6 element following the assumed strain method adopted by Belytschko and Bindeman. SHB6's coordinates and displacements are related to the nodal coordinates and displacements through the linear shape functions. Applying the simplified form of the Hu-Washizu nonlinear mixed variational principle, in which the assumed stress field is chosen to be orthogonal to the difference between the symmetric part of the displacement gradient and the assumed strain field, the formula is obtained. The newly developed SHB6 element was implemented into the finite element codes INCA and ASTER. It represents some improvement since it converges well and performs much better than the PRI6 six-node three-dimensional element in all of the benchmark problems tested.International audienceThe formulation of a six-node solid-shell called SHB6, which is a linear, isoparametric element, is discussed. An eigenvalue analysis of the element stiffness matrix is first carried out. Several modifications are introduced into the formulation of the SHB6 element following the assumed strain method adopted by Belytschko and Bindeman. SHB6's coordinates and displacements are related to the nodal coordinates and displacements through the linear shape functions. Applying the simplified form of the Hu-Washizu nonlinear mixed variational principle, in which the assumed stress field is chosen to be orthogonal to the difference between the symmetric part of the displacement gradient and the assumed strain field, the formula is obtained. The newly developed SHB6 element was implemented into the finite element codes INCA and ASTER. It represents some improvement since it converges well and performs much better than the PRI6 six-node three-dimensional element in all of the benchmark problems tested

Abed-Meraim, Farid

Combescure, Alain

HAL Descartes

SAM;  the Arts et Métiers ParisTech open access repository

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: https://sam.ensam.eu
Handle ID: .http://hdl.handle.net/10985/10198
To cite this version :
Farid ABED-MERAIM, Alain COMBESCURE - New prismatic solid-shell element : Assumed strain
formulation and hourglass mode analysis - Structural Engineering and Mechanics - Vol. 37, n°2,
p.253-256 - 2011
Any correspondence concerning this service should be sent to the repository
Administrator : archiveouverte@ensam.eu
New prismatic solid-shell element: assumed strain formulation 
and hourglass mode analysis 
Farid Abed-Meraim1,†, Alain Combescure2 
1 LEM3, CNRS, Arts et Métiers ParisTech, 57078 Metz Cedex 3, France 
2 LaMCoS, UMR CNRS 5259, INSA de Lyon, 69621 Villeurbanne Cedex, France 
Keywords:    assumed-strain solid–shell SHB6; six-node prism; shear and thickness locking; hourglass. 
1. Introduction
In three-dimensional analysis of structural problems, the development of effective eight-node solid-shell finite 
elements has been a major objective over the last decade (Belytschko and Bindeman 1993, Hauptmann and 
Schweizerhof 1998, Abed-Meraim and Combescure 2002, Legay and Combescure 2003). However, to be able 
to mesh complex geometries and with the advent of free mesh generation tools not generating only hexahedrons, 
the development of prismatic elements is made necessary. This paper presents the formulation of a six-node 
solid-shell called SHB6. It represents a thick shell obtained from a purely 3D approach. The assumed strain 
method is adopted together with an in-plane reduced integration scheme with five integration points along the 
thickness direction. The 3D elastic constitutive law is also modified so that shell-like behavior is intended for 
the element and in order to alleviate shear and membrane locking. 
A detailed eigenvalue analysis of the element stiffness matrix is first carried out. We demonstrate that the 
kernel of this stiffness matrix only reduces to rigid body movements and hence, in contrast to the eight-node 
solid-shell element (SHB8PS), the SHB6 element does not require stabilization. On the other hand, to attenuate 
locking phenomena, several modifications are introduced into the formulation of the SHB6 element following 
the assumed strain method adopted by Belytschko and Bindeman (1993). Finally, one example, among the 
variety of benchmark problems performed, is shown to illustrate the performance of the new element. 
2. Formulation of the SHB6 finite element
The SHB6 is a solid-shell with only 3 displacement DOF per node, and it has a special direction called 
“thickness”. It is integrated with five integration points along this direction and only one point in the in-plane 
directions. Fig. 1 shows the SHB6 reference geometry and its integration points. 
2.1 Kinematics and interpolation 
The SHB6 is a linear, isoparametric element. Its coordinates ix  and displacements iu  are related to the nodal 
coordinates iIx  and displacements iIu  through the linear shape functions IN  as 
∑
=
==
6
1
),,(),,(
I
IiIIiIi NxNxx ζηξζηξ   and ∑
=
==
6
1
),,(),,(
I
IiIIiIi NuNuu ζηξζηξ  (1) 
The standard tri-linear shape functions IN  are: 
†
 Corresponding author, Associate Professor, Ph.D., E-mail: Farid.Abed-Meraim@ensam.eu 
 Farid Abed-Meraim and Alain Combescure 
 
( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )
[ ] [ ] [ ]
1
, , 1 1 1 1 1 1 1 1
2
0,1 ;     0,1 ;     1,1
IN ξ η ζ ζ ξ η ζ ξ ζ η ζ ξ η ζ ξ ζ η
ξ η ξ ζ
= − − − − − + − − + +  
= = − = −
 (2) 
 
 
 
 
 
 
Fig. 1 Reference geometry of the SHB6 element and location of its integration points 
2.2 Discrete gradient operator 
Using some mathematical derivations, similarly to the SHB8PS formulation (Abed-Meraim and Combescure 
2002), we can explicitly relate the strain field to the nodal displacements as 
 ( )
,
,
,
,
,,
, ,
, ,
, ,
, ,
, ,
, ,
0 0
0 0
0 0
0
0
0
T T
x x
x x T T
y y
y y
T T x
z zz z
s T T T T
x y y x y xy x
T T T Tx z z x
z xz x
y z z y T T T T
z yz y
b h
u
b h
u
db hu
u d
u u b h b h
u u
b h b h
u u
b h b h
α α
α α
α α
α αα α
α αα α
α αα α
γ
γ
γ
γ γ
γ γ
γ γ
+
+
+
∇ = = ⋅
+ + +
+
+ +
+
+ +
 
  
  
  
  
  
  
  
  
    
  
y
z
B d
d
= ⋅
 
 
 
  
 (3) 
where functions h
α
 are: 1 2,  h hηζ ξζ= = ; vectors ib  are: , 0(0) ;  1, 2,3Ti i ib N N x iξ η ζ= = == = ∂ ∂ =  and the 
constant vectors 
α
γ  are given by: ( )3
1
1, 2
1
 ;  
2
T
i i
i
h h x b
α αα
αγ
=
== − ⋅
 
 
 
∑ . In this latter expression, 
( )1 0, 0, 1, 0, 0, 1 ;Th = −  ( )2 0, 1, 0, 0, 1, 0Th = −  and vectors ix  denote the nodal 
coordinates. 
2.3 Variational principle 
Applying the simplified form of the Hu-Washizu nonlinear mixed variational principle, in which the assumed 
stress field is chosen to be orthogonal to the difference between the symmetric part of the displacement gradient 
and the assumed strain field, we obtain: 
 ( ),  0
e
T T ext
V
u dV d fδpi ε δ ε σ δ= ⋅ − ⋅ =∫  (4) 
Replacing the assumed strain field, with its expression ( ) ( ) ( ),x t B x d tε = ⋅ , in Eq. (4) leads to the 
following expressions for the element internal force vector and stiffness matrix: 
 ( )int  
e
T
V
f B dVσ ε= ⋅∫ ,          
T
e
e
V
B C BK dV⋅ ⋅= ∫  (5) 
0 
ζ η 
ξ 
4 3 
5 
1 
2 
6 
New prismatic solid-shell element: assumed strain formulation and hourglass mode analysis 
 
For a standard displacement approach, B  is simply replaced by B  leading classically to 
 
 
e
T
e
V
K B C B dV= ⋅ ⋅∫  (6) 
2.4 Hourglass mode analysis for the SHB6 element 
Hourglass patterns are spurious zero-energy modes that are induced by reduced integration. The analysis of 
hourglass modes is equivalent to that of the stiffness matrix kernel, namely searching for zero-strain modes d  
that satisfy: 
 ( ) 0IB dζ ⋅ = ,       int1, ...,I n=  (7) 
where intn  represents the number of integration points. 
To this end, we can build a basis for the vector space of the discretized displacements, given by the eighteen 
column vectors below, in which ( )1, 1, 1, 1, 1, 1TS = . We show then that only the first six column 
vectors in Eq. (8) verify Eq. (7), which corresponds to rigid body modes. This reveals that there are no hourglass 
modes for the SHB6 element, and thus no hourglass control is required. 
 










−−
−
21
21
21
0000000000
0000000000
0000000000
hhyxzyxS
hhzxyzxS
hhzyxzyS
 (8) 
2.5 Assumed strain formulation for the SHB6 discrete gradient operator 
Among several treatments for alleviating shear and membrane locking, in the present formulation the discrete 
gradient will be appropriately modified. This consists first of decomposing the matrix B  into two parts: 
21
BBB += , then of projecting the second part onto an assumed strain operator such that 
1 2
B B B= + . As a 
result, the stiffness matrix becomes 
 
1 1 1 2 2 1 2 2
    
e e e e
T T T T
e V V V V
K B C B dV B C B dV B C B dV B C B dV= ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅∫ ∫ ∫ ∫  (9) 
The subsequent steps consist of choosing an adequate assumed strain field. This is a key point in the 
formulation and special care has been exercised in this regard. Finally, the above additive decomposition of the 
stiffness matrix is computed using a reduced integration scheme with five Gauss points, all located along the 
thickness direction. Note that the choice of an assumed strain field is mainly guided by the elimination of strain 
components that are responsible for shear and membrane locking. The effectiveness of this assumed strain is 
assessed through benchmark problems. 
 
,
,
,
1 2
, ,
, ,
, ,
0 0 0 0 0
0 0 00 0
0 0 00 0 ;  0 0 0
0
0
0 0 0
00 0 0
T T
x x
T T
y y
T T
z z
T T T T
T T T Ty xy x
z xz x
T T T T
z yz y
b h
b h
b hB B
b h b h
b h b h
b h b h
α α
α α
α α
α αα α
α αα α
α αα α
γ
γ
γ
γ γ
γ γ
γ γ
+
+
+
= =
+ +
+ +
+ +
   
   
   
   
   
   
   
   
   
    
 
 Farid Abed-Meraim and Alain Combescure 
3.  Numerical results and comparison 
Several popular benchmark problems were performed to illustrate the element performance. One of these test 
problems, which is frequently used to test warping effects, is shown here. 
E = 29×106
ν = 0.22
P = 1
L = 12
l = 1.1
h = 0.32
 
Fig. 2 Twisted beam (reference vertical displacement of point A at tip is 5.424x10-3) 
Table 1 Normalized vertical displacement at point A of the twisted cantilever beam problem 
Mesh layout PRI6 (3D solid element) SHB6 (without assumed strain) SHB6 (assumed strain) 
(6x2x1)x2 0.061 0.234 0.496 
(12x4x1)x2 0.202 0.470 0.784 
(24x4x1)x2 0.485 0.779 0.935 
(36x8x1)x2 0.489 0.875 0.972 
4.  Conclusions 
This newly developed SHB6 element was implemented into the finite element codes INCA and ASTER. It 
represents some improvement since it converges well and it performs much better than the PRI6 six-node three-
dimensional element in all of the benchmark problems tested. Furthermore, it shows very good performances in 
problems using mixed meshes composed of SHB6 and SHB8PS elements. Thus, we can couple the SHB6 with 
other finite elements to mesh complex geometries, which could be obtained by free mesh generation tools. 
References 
Abed-Meraim, F. and Combescure, A. (2002), “SHB8PS a new adaptive, assumed-strain continuum mechanics shell 
element for impact analysis”, Comput. Struct., 80, 791-803. 
Belytschko, T. and Bindeman, L.P. (1993), “Assumed strain stabilization of the eight node hexahedral element”, 
Comput. Meth. Appl. Mech. Eng., 105, 225-260. 
Hauptmann, R. and Schweizerhof, K. (1998), “A systematic development of solid-shell element formulations for 
linear and non-linear analyses employing only displacement degrees of freedom”, Int. J. Num. Meth. Eng., 42, 
49-69. 
Legay, A. and Combescure, A. (2003), “Elastoplastic stability analysis of shells using the physically stabilized finite 
element SHB8PS”, Int. J. Num. Meth. Eng., 57, 1299-1322. 


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New prismatic solid-shell element : Assumed strain formulation and hourglass mode analysis

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