The aim of this work is to propose a new finite element modeling for vibration of sandwich structures with soft core. Indeed, several approaches have been adopted in the literature to accurately model these types of structures, but show some limitations in certain configurations of high contrast of material properties or geometric aspect ratios between the different layers. In these situations, it is generally well-known that the use of higher-order or three-dimensional finite elements is more appropriate, but will generate a large number of degrees of freedom, and thereby, large CPU times. In this work, an alternative method is followed by considering the linear hexahedral solid-shell element previously developed by Abed-Meraim and Combescure [1]. This element is implemented into the commercial software ABAQUS Via a User Element (UEL) subroutine. Numerical tests on various cantilever sandwich beams are performed to show the efficiency of this approach

ABED-MERAIM, Farid

BOUDAOUD, Hakim

CHALAL, Hocine

DAYA, El Mostafa

KPEKY, Fessal

English

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Fessal KPEKY, Hakim BOUDAOUD, Hocine CHALAL, Farid ABED-MERAIM, El Mostafa DAYA -
Vibration modeling of sandwich structures using solid-shell finite elements - In: 11th World
Congress on Computational Mechanics, WCCM 2014, the 5th European Conference on
Computational Mechanics, ECCM 2014 and the 6th European Conference on Computational
Fluid Dynamics, ECFD 2014, Espagne, 2014-07-20 - World Congress on Computational
Mechanics; 5th European Conference - 2014
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11th World Congress on Computational Mechanics (WCCM XI) 
5th European Conference on Computational Mechanics (ECCM V) 
6th European Conference on Computational Fluid Dynamics (ECFD VI) 
F. Kpeky, H. Boudaoud, H. Chalal, F. Abed-Meraim and E.M. Daya 
 
 
 
 
VIBRATION MODELING OF SANDWICH STRUCTURES USING 
SOLID-SHELL FINITE ELEMENTS 
F. KPEKY
*
, H. BOUDAOUD
*
, H. CHALAL
†
, F. ABED-MERAIM
†
 AND E.M. 
DAYA
*
,‡ 
  
*
 Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux (LEM3) 
Université de Lorraine 
Ile du Saulcy, 57045 Metz, Cedex 1, France 
email: fessal.kpeky@univ-lorraine.fr, hakim.boudaoud@univ-lorraine.fr, 
el-mostafa.daya@univ-lorraine.fr, www.lem3.fr 
 
†
 Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux (LEM3) 
Arts et Métiers ParisTech, Campus de Metz 
4 rue A. Fresnel, 57078 Metz, Cedex 03, France 
email: hocine.chalal@ensam.eu, farid.abed-meraim@ensam.eu, www.lem3.fr 
 
‡
 Laboratory of Excellence on Design of Alloy Metals for low-mAss Structures (DAMAS) 
 
Key Words: Finite elements, Solid-Shell, Vibrations, Sandwich structures. 
Abstract. The aim of this work is to propose a new finite element modeling for vibration of 
sandwich structures with soft core. Indeed, several approaches have been adopted in the 
literature to accurately model these types of structures, but show some limitations in certain 
configurations of high contrast of material properties or geometric aspect ratios between the 
different layers. In these situations, it is generally well-known that the use of higher-order or 
three-dimensional finite elements is more appropriate, but will generate a large number of 
degrees of freedom, and thereby, large CPU times. In this work, an alternative method is 
followed by considering the linear hexahedral solid-shell element previously developed by 
Abed-Meraim and Combescure [1]. This element is implemented into the commercial 
software ABAQUS via a User Element (UEL) subroutine. Numerical tests on various 
cantilever sandwich beams are performed to show the efficiency of this approach.  
 
 
1 INTRODUCTION 
Problems involving vibration occur in many areas of mechanical, civil and aerospace 
engineering. These vibrations are undesirable because they lead to noise and system 
dysfunction. An efficient passive solution to reduce vibrations is the use of sandwich 
structures with elastic faces and viscoelastic core [2,3]. 
Various kinematic models and numerical methods have been devoted to determine 
accurately the damping properties of viscoelastic sandwich structures under vibration. Hu et 
al. [3] have presented a review and assessment of existing models. Indeed, earlier classical 
F. Kpeky, H. Boudaoud, H. Chalal, F. Abed-Meraim and E.M. Daya. 
 2 
theories of laminated thin shell and plate approximations are based on the Kirchhoff–Love 
model. With these approaches, the deformation due to shear is neglected as compared to other 
strains. Then Reissner [4] and Mindlin [5] established first order theories that take into 
account this shear deformation. However, these studies have shown that the deformation 
varies at least in a quadratic form with the shear stress zero on the outer surfaces of the skins. 
Subsequent studies, Reddy [6] and Touratier [7] (to name only these), have made major 
contributions by proposing higher-order theories of the displacement field in the thickness 
(cubic and sinusoidal, respectively). The major advantage of these is to allow a parabolic 
description of the shear stress while ensuring the condition of zero shear stress on the free 
surface of the sandwich structures.  
All of these studies provide estimates of the accuracy, under various loads, of the overall 
stiffness, frequencies, loss factor and many other properties. However, the sandwich structure 
is treated as a single layer to facilitate analysis. Unfortunately, this assumption does not 
correctly describe some phenomena in a structure exhibiting high contrast of stiffness 
between different layers. To compensate these shortcomings, zigzag theories assuming 
continuity (IC-ZZT) or not (ID-ZZT) have been developed. These theories describe layer-by-
layer the displacement field ensuring continuity conditions of the displacement field imposed 
at the interfaces between the core and the faces (see, e.g., Boudaoud et al. [8], Bilasse et al. 
[9], Abdoun et al. [10]). 
These models have shown their limitations and an alternative approach could be the use of 
three-dimensional finite element assemblies, but this generally leads to a large number of 
degrees of freedom. Another approach proposed in this work can be the use of a solid-shell 
element based on a fully three-dimensional formulation. Such a solid-shell element has been 
developed in order to correctly take into account the through-thickness phenomena, while 
maintaining the CPU time at reasonable levels [1,11,12]. This is a linear isoparametric 
hexahedral element having only nodal displacements as degrees of freedom and provided with 
a set of integration points distributed along the thickness direction. To avoid locking 
phenomena, the fully three-dimensional elastic constitutive matrix was also modified in order 
to approach shell-like behavior. To eliminate the zero-energy hourglass modes due to the 
reduced integration, an effective stabilization technique was used following the “Assumed 
Strain” method of Belytschko and Bindeman [13]. Several benchmark tests were analyzed to 
show the effectiveness of this solid-shell element in linear and non-linear problems. Recently, 
Salahouelhadj et al. [14] successfully simulated sheet metal forming processes using the 
SHB8PS solid-shell element coupled with an anisotropic large strain elastic-plastic model. 
The purpose of the current work is to combine this solid-shell concept with sandwich 
structure modeling in order to evaluate its capabilities in analyzing vibration of sandwich 
structures. A number of representative applications will be shown. 
2 FORMULATION OF THE PROBLEM 
In this work, we consider the free vibration problem of sandwich structures with soft core 
pictured in Figure 1. 
The basic equilibrium equations are obtained by using the virtual work principle as 
follows:  
F. Kpeky, H. Boudaoud, H. Chalal, F. Abed-Meraim and E.M. Daya. 
 3 
acc ext intP P P     (1) 
where 
accP  is the virtual work associated with the kinetic energy, and extP  and intP represent 
the virtual work of external loads and internal forces, respectively, for each layer.  
 
Figure 1: Sandwich beam structure 
One can show that Eq. (1) may be transformed into Eq. (2) to obtain the vibration 
eigenmodes, frequencies and equivalent loss factors: 
     2( ) 0e e eK M U    (2) 
where [Me] and [Ke] denote the element mass and stiffness matrices, respectively. Note that 
[Ke] is more detailed in [1,11,12,14] and [Me] is identical to that of a standard linear brick 
element. 
Unlike models in the literature of sandwich structures, we propose in this work a new finite 
element method to solve (2) for any geometrical and material configurations of the sandwich 
structures. To achieve this goal, the formulation of the SHB8PS solid-shell element is 
considered. More details of this kind of element can be found in [1,11,12,14]. As mentioned 
above, this is an eight-node hexahedral element with only displacement degrees of freedom. 
The associated integration points are arranged in a preferred direction (thickness) in the local 
coordinate frame (Figure 2). The classical plane-stress constitutive law, usually adopted in 
shell formulations, has been amended to take into account shear and membrane effects. 
Accordingly, the resulting elasticity matrix C
ele
 is expressed in this local coordinate frame in 
terms of the Young modulus E  and the Poisson ratio   as follows: 
F. Kpeky, H. Boudaoud, H. Chalal, F. Abed-Meraim and E.M. Daya. 
 4 
2 0 0 0 0
2 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
ele E
  
  



 
 
 
 
 
 
 
 
  
C         with    
21
E




  and  
 2. 1
E




 (3) 
 
 
int
2
1
n
  
4   
2   
1  
7   
3   
      
8   
6   
5   
   
 
Figure 2: Reference element geometry and integration points 
 
3 NUMERICAL TESTS 
3.1 Validation of the present model 
Numerical analyses of a cantilever sandwich beam are first performed in order to validate 
the present model. Natural frequencies are computed, and the geometric and material 
properties considered in this preliminary study are reported in Table 1. 
The results of the SHB8PS element are compared with those of the Abaqus eight-node 
reduced integration element C3D8R and listed in Table 2. 
Table 1: Geometric and material properties 
E      L  h  b  
2.1E11 Pa 0.3 7800 kg.m
-3 
1 m 0.01 m 0.1 m 
 
 
 
 
 
 
F. Kpeky, H. Boudaoud, H. Chalal, F. Abed-Meraim and E.M. Daya. 
 5 
Table 2: Eigen frequencies of the cantilever beam 
Element 
type 
Mesh layout 
f1 ref = 8.4 f2 ref = 52.5 f3 ref = 83.8 f4 ref = 147.1 
f1/f1ref f2/f2ref f3/f3ref f4/f4ref 
C3D8R 
(30x3x1)=90 0.10 0.10 0.18 0.20 
(40x4x1)=160 0.10 0.10 0.18 0.20 
(80x8x1)=640 0.10 0.10 0.18 0.19 
(30x3x4)=360 0.94 0.97 0.94 0.97 
(40x4x4)= 640 0.98 0.98 0.96 0.98 
(80x8x4)=2560 0.98 0.98 0.99 0.99 
(30x3x5)=450 0.98 0.99 0.94 0.99 
(40x4x5)=800 0.99 0.99 0.96 1.00 
(80x8x5)=3200 1.00 1.00 0.99 1.00 
SHB8PS 
(2 Integration 
Points) 
(30x3x1)=90 1.00 1.01 1.00 1.01 
(40x4x1)=160 1.00 1.01 1.00 1.01 
 
3.2 Vibration analysis of sandwich beams 
In order to evaluate the solid-shell performance, finite element (FE) analyses of various 
cantilever sandwich beams have been performed using the FE code Abaqus/Standard. The 
results obtained with the SHB8PS element are compared to those given by other Abaqus 
formulations, namely a continuum plane-strain quadratic element CPE8R, a three-dimensional 
hexahedral quadratic element C3D20R, and a 3D hexahedral linear full-integration C3D8 
element. The results are also compared to those yielded by a 2D FE code of FSDT IC-ZZT 
developed in Matlab. The SHB8PS element has been implemented through an UEL 
subroutine. The sandwich beam consists of two face sheets made of aluminum and a core 
(Figure 2) whose mechanical and geometric properties are given in Table 3. 
 
Table 3: Sandwich beam parameters 
f
E  f  c  h  f  c  
6.9E10 Pa 2766 kg.m
-3
 1600 kg.m
-3
 0.05 m 0.3 0.49 
 
Three dimensionless beam parameters are used in this comparative study, namely, the ratio 
of core to face Young modulus (Ec/Ef), the ratio of beam length to beam total thickness (L/h), 
and the ratio of core to skin thickness (hc/hf). Let us remark that under these considerations 
and by using material parameters as listed in Table 1, all sandwich beam possible 
configurations can be represented. For these purposes, eigen frequencies evaluations are made 
in three configurations: 
Case 1. Thin/thick core: 0 1 100hc hf . /           (
520 2 10/ ; /L h Ec Ef    ) 
Case 2. Short/long beam: 4 100L h /              ( 51 2 10/ ; /hc hf Ec Ef    ) 
F. Kpeky, H. Boudaoud, H. Chalal, F. Abed-Meraim and E.M. Daya. 
 6 
Case 3. Soft/rigid core: 0 0001 100Ec Ef . /      ( 1 20/ ; /hc hf L h  ) 
The results of these cases are shown in Tables 4, 5, and 6, by providing the first eigen 
frequencies corresponding to different formulations, where NDOF and NIP denote the 
number of degrees of freedom per layer and the number of integration points per element, 
respectively. It appears from all of these tables that the solid-shell element formulation gives 
accurate results for the different configurations within reasonable CPU times. 
 
 
Table 4: Influence of hc/hf (L/h = 20, Ec/Ef = 2.10-5) 
 SHB8PS C3D8 C3D20R CPE8R 
FSDT IC-
ZZT 
NDOF/layer 2160 384000 48000 4320 3000 
NIP/element 2 8 8 4 - 
0 1c
f
h
h
 .  
21.186 21.140 21.205 21.205 21.505 
125.96 125.30 125.73 125.73 121.90 
349.99 346.24 347.46 347.46 335.25 
1c
f
h
h
  
13.006 13.146 13.015 13.015 13.032 
78.581 79.279 78.433 78.433 75.634 
219.38 220.12 217.73 217.73 209.08 
40c
f
h
h
  
3.0638 3.3325 3.0640 3.0640 4.2148 
9.3719 10.868 9.3607 9.3607 12.794 
16.256 20.835 16.193 16.193 21.958 
100c
f
h
h
  
2.9512 3.1409 2.9506 2.9506 4.0496 
8.900 9.8452 8.8891 8.8891 12.249 
15.134 17.798 15.086 15.086 21.108 
 
Table 5: Influence of L/h (hc/hf = 1, Ec/Ef = 2.10-5) 
 SHB8PS C3D8 C3D20R CPE8R 
FSDT IC-
ZZT 
NDOF/layer 2160 384000 48000 4320 3000 
NIP/element 2 8 8 4 - 
4
L
h
  
309.43 310.73 300.82 309.06 297.27 
1884.0 1772.1 1735.2 1792.0 1857.1 
10
L
h
  
47.823 50.224 48.509 50.294 48.636 
290.22 309.55 298.34 309.97 298.34 
40
L
h
  
3.6378 4.2195 3.6397 3.6399 0.4538 
20.157 24.134 20.118 20.120 1.0208 
100
L
h
  
0.83588 1.2253 0.83612 0.83617 0.2432 
3.6786 6.6111 3.6721 3.6724 0.4240 
F. Kpeky, H. Boudaoud, H. Chalal, F. Abed-Meraim and E.M. Daya. 
 7 
Table 6: Influence of Ec/Ef (hc/hf = 1, L/h = 20)  
 SHB8PS C3D8 C3D20R CPE8R 
FSDT IC-
ZZT 
NDOF/layer 2160 384000 48000 4320 3000 
NIP/element 2 8 8 4 - 
410c
f
E
E
  
16.939 16.908 16.947 16.947 16.815 
84.161 83.718 84.011 84.011 81.328 
224.48 222.11 222.97 222.97 214.41 
0 1.c
f
E
E
  
44.397 44.468 44.440 44.440 42.675 
263.24 262.97 262.79 262.79 260.79 
684.75 681.10 680.59 680.59 703.46 
100c
f
E
E
  
105.91 102.96 106.71 106.71 125.88 
662.01 642.25 665.74 665.74 788.19 
1847.8 1786.1 1851.6 1851.6 2203.9 
 
5 CONCLUSIONS 
An efficient analysis of sandwich beam vibrations has been proposed using a solid-shell 
formulation. The presented results show the interest in adopting solid-shell elements in this 
type of applications. The study can be extended to the modeling of multilayer structures using 
a single element with the possibility of assigning various material responses at different 
integration points. 
 
F. Kpeky, H. Boudaoud, H. Chalal, F. Abed-Meraim and E.M. Daya. 
 8 
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803. 
[2] Daya, E.M. and Potier-Ferry, M. A numerical method for nonlinear eigenvalue problems 
application to vibrations of viscoelastic structures. Comput. Struct. (2001) 79:533-541. 
[3] Hu, H., Belouettar, S., Potier-Ferry, M. and Daya, E.M. Review and assessment of 
various theories for modeling sandwich composites. Compos. Struct. (2008) 84:282-292. 
[4] Reissner, E. The effect of transverse shear of transverse shear deformation on the bending 
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[5] Mindlin, R.D. Influence of rotatory inertia and shear in flexural motions of isotropic 
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Vibration modeling of sandwich structures using solid-shell finite elements

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