the use of ansys to calculate the behaviour of

THE USE OF ANSYS TO CALCULATE THE BEHAVIOUR OFSANDWICH STRUCTURES

Vincent Manet*

EÂcole des Mines de Saint-Etienne, Material and Mechanical Department, 158, cours Fauriel, 42023 Saint-Etienne Cedex 2, France

(Received 8 September 1997; accepted 9 January 1998)

AbstractIn this article, we use di�erent models to compute dis-placements and stresses of a simply supported sandwichbeam subjected to a uniform pressure. 8-node quad-rilateral elements (Plane 82), multi-layered 8-nodequadrilateral shell elements (Shell 91) and multi-layered20-node cubic elements (Solid 46) are used. The in¯uenceof mesh re®nement and of the ratio of Young's moduli ofthe layers are studied. Finally, a local Reissner method ispresented and assessed, which permits an improvement inthe accuracy of interface stresses for a high ratio ofYoung's moduli of the layers with Plane 82 elements.# 1998 Elsevier Science Ltd. All rights reserved

Keywords: ANSYS, sandwich structure, interfacestresses, local Reissner method, post-processing

1 INTRODUCTION

Sandwich materials are currently much valued inindustry, and especially in the ®elds of transport (auto-motive, aeronautics, shipbuilding and railroads) andcivil engineering. It is therefore important to determinewhich elements should be used to model such structures.A sandwich structure is composed of three layers, viz.two surfaces made of rigid layers, working as mem-branes, which represent the skins, and a thick and softcentral layer, the core, with low rigidity and density andessentially submitted to transverse shear loading, sand-wiched between the faces. In the design process, inter-face stresses can be of great importance, since they playa crucial role in failure modes.1,2

The core being essentially subjected to transverse shearstress, this component, of which the rigidity is generallyvery much lower than that of the others, must not beneglected: in some cases, e�ects arising from shear e�ectsexceed other phenomena (¯exural e�ects for example).3±6

The determination of transverse shear stress at inter-faces is therefore of particular importance in the designof new optimized materials.

If we assume that the three layers remain perfectly

bonded, then at interfaces, the displacement ®eld mustbe continuous and the normal trace of the stresstensor must be continuous.

In this article we shall study a very simple case usingthe well-known ®nite element software ANSYS 5.2. Weshall not discuss special elements based on hybrid,7,8

mixed9±12 or modi®ed13±15 formulations nor shall weconsider pre- and post-processing methods.16,17 Solu-tions obtained with di�erent models (complex or sim-ple) are compared. Particular emphasis is put on theirrespecting of continuity requirements. By modifying thesti�ness of the core, we shall see which model should bepreferred by designers. Finally a method, based onReissner's formulation, is developed to improve theaccuracy for new sandwich structures.

2 DESCRIPTION OF THE SANDWICH BEAMSTUDIED

One of simplest examples is the case of the simply sup-ported sandwich beam subjected to a uniform pressureon its top face. Such a beam is shown in Fig. 1.

2.1 CharacteristicsThe total length of the beam is L=24mm, its totalheight H=2mm and the core represents 80% of thetotal height of this symmetrical sandwich. The appliedpressure is q=ÿ1N/mm. The thickness of the beam inthe y direction is taken to be equal to 1. By symmetry,only one half of the beam is modelled.

2.2 Parameters of the studyIn this study, we are interested in determining thestructural response at point A (at the interface betweenthe top skin and the core and located at x=L/4) whendi�erent parameters vary.

The skins are made of aluminum (Es=70GPa andvs=0.34) and the core will be one of the following:

1. carbon/epoxy (Ec=3.4Gpa and vc=0.34);2. foam (Ec=0.34Gpa and vc=0.40);3. soft foam (Ec=70Mpa and vc=0.40);4. other material: vc=0.4 is ®xed and Es/Ec varies.

Composites Science and Technology 58 (1998) 1899±1905# 1998 Elsevier Science Ltd. All rights reserved

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1899

*Fax: 0033 4 77 420000; e-mail: [email protected]

2.3 The models

We shall use the following models:

(a) The reference model: 2D with 8-node quad-rilateral elements (Plane 82); 4 elements throughthe thickness of each skin (nskin=4), 32 throughthe thickness of the core (ncore=32), and 400longitudinal cuts (ncuts=400) in the beam's axisdirections (16000 elements for the half beam).

(b) A planar model using the plane element Plane 82:One element is used to model each layer(nskin=ncore=1), i.e. 3 elements through thethickness of the sandwich;

(c) A model using the multi-layered cubic elementSolid 46: One element through the total thicknessrepresenting all the layers of the sandwich struc-ture.

(d) A model done with the multi-layered shell elementShell 91, with sandwich option (keyopt(9)=1):One element through the total thickness repre-senting all the layers of the sandwich structure.

2.4 Results of interestIn our studies, we shall focus on the following results ofparticular interest:

1. the maximum displacement of the structure in thez-direction, denoted Uz in results;

2. the discontinuous components of stresses, �xx atpoint A in the skin and in the core, and the con-tinuous component �zz;

3. interlaminar stress: this is the continuous compo-nent �xz at point A.

3 STUDY OF THE SANDWICH BEAM

We now present results obtained with ANSYS 5.2 andcorresponding to di�erent materials and di�erentmeshes.

3.1 In¯uence of ncuts on the di�erent modelsIn this section, we are interested in the structuralresponses to the di�erent models, for values of Ec of3.4GPa, 0.34GPa and 70MPa.

Results for the case Ec=340MPa are plotted in Fig. 2for displacements, Figs 3 and 4 for the two continuouscomponents �zz and �xx and Figs 5 and 6 for �xx in thecore and in the skin, respectively.

Fig. 1. Sandwich beam.

Fig. 2. Uz for Es/Ec&200: in¯uence of ncuts.

Fig. 3. �zz for Es/Ec&200: in¯uence of ncuts.

Fig. 4. �xz for Es/Ec&200: in¯uence of ncuts.

Fig. 5. �xx in the core for Es/Ec&200: in¯uence of ncuts.

1900 V. Manet

Tables 1±3 present numerical results and errorpercentages after convergence for these 3 cases.

From these Figures and Tables, the followingconclusions can be drawn:

1. Plane 82 is very much better than other models.Nevertheless, it should be noticed that it seems todiverge for displacements (with the coarse meshused: nskin=ncore=1);

2. Solid 46 is the worst model. It never convergestowards the correct values (for any component ofstresses nor for displacements);

3. Shell 91 is particularly interesting for continuouscomponents of stresses �zz and �xz;

4. Plane 82 is the only model leading to a correctdetermination of the discontinuous component �xxin the skin and the core;

5. It seems that errors increase with the ratio Es/Ec.This point will be studied in the next section.

3.2 In¯uence of the ratio Es/Ec for ncuts=20Since every material which can be obtained in a thinskin shape is acceptable for the skins and every materialwith low density is acceptable for the core, sandwichmaterials cover an extremely wide domain.

A parameter of interest is therefore the ratio ofYoung's moduli, Es/Ec. This parameter can vary from 4(old sandwiches, so to speak, very close to laminates) to1000 (new high-technology sandwiches go up to 1500).But we must note that sandwiches now generally exhibita ratio between 200 and 1500.

In this section we shall study the in¯uence of thisratio on the di�erent modellings when the utilized meshis ®xed to ncuts=20.

Results relating to displacements are plotted in Fig. 7.Continuous components �zz and �xz are shown in Figs 8and 9. The discontinuous component �xx is illustrated inFigs 10 and 11 in the core and in the skin, respectively.

Table 2. Case Ec=0.34Gpa

Uz �zz �xz �xxskin

�xxcore

ncuts

Ref 0.51353 0.95461 3.2956 103.71 0.8262P82 0.52618 0.96321 3.3091 102.38 0.8701 400

2.463% 0.901% 0.41% 1.28% 5.31%S91 0.55712 0.90000 3.3024 132.09 0.6599 1000

8.488% 5.721% 0.22% 27.4% 20.1%S46 0.45994 0.50000 3.0000 130.68 1.2042 1000

10.44% 47.62% 8.97% 26.0% 45.7%

Table 1. Case Ec=3.4Gpa


�xxcore

ncuts

Ref 0.21596 0.94624 3.1635 123.13 6.2844P82 0.21527 0.95033 3.2616 123.08 6.2843 400

0.319% 0.432% 3.10% 0.04% 0.001%S91 0.21388 0.90000 3.1587 126.35 6.1369 1000

0.963% 4.887% 0.15% 2.61% 2.35%S46 0.20355 0.50000 3.0000 123.28 8.4202 1000

5.746% 47.16% 5.17% 0.12% 34.0%

Table 3. Case Ec=70Mpa


�xxcore

ncuts

Ref 1.740 0.96121 3.2926 20.43 0.3614P82 1.741 0.96847 3.1561 20.77 0.3301 400

0.06% 0.755% 4.15% 1.66% 8.66%S91 1.987 0.90000 3.3161 132.64 0.1364 1000

14.2% 6.368% 0.71% 549% 62.2%S46 1.506 0.50000 3.0000 131.43 0.4802 1000

13.4% 47.98% 8.89% 543% 32.9%

Fig. 6. �xx in the top skin for Es/Ec&200: in¯uence of ncuts.

Fig. 7. Uz for ncuts=20: in¯uence of Es/Ec.

Fig. 8. �zz for ncuts=20: in¯uence of Es/Ec.

Behaviour of sandwich structures 1901

From these ®gures, the following conclusions can bedrawn:

1. Plane 82 is the best model for displacements, �zzand �xx in the core and the skin;

2. Shell 91 and Solid 46 are acceptable for displace-ments and in the core. They are acceptable for �xxin the skin for Es/Ec�50;

3. Shell 91 leads to an acceptable approximation of�zz and is very interesting for �xz;

4. Plane 82, which was exceptionally good in the lastsection, shows some di�culty here, especially athigh Es/Ec ratio for �xz. The in¯uence of the

meshing of the beam with Plane 82 elements isstudied in the next section.

3.3 Element Plane 82: in¯uence of mesh re®nementIn previous sections, the mesh corresponding to the 8-node quadrilateral element Plane 82 only used 1 elementto model 1 layer. We propose to see what happens whenthe number of elements through the thickness of the skins(nskin) and of the core (ncore) vary. Displacements areplotted in Fig. 12, �zz and �xz in Figs 13 and 14, and �xx inFigs 15 and 16 in the core and in the skin, respectively.These computations are done for a ratio Es/Ec=500.

Fig. 9. �xz for ncuts=20: in¯uence of Es/Ec.

Fig. 10. �xx in the core for ncuts=20: in¯uence of Es/Ec.

Fig. 11. �xx in the top skin for ncuts=20: in¯uence of Es/Ec.

Fig. 12. Uz for ncuts=20 and Es/Ec=500: in¯uence of nskinand ncore.

Fig. 13. �zz for ncuts=20 and Es/Ec=500: in¯uence of nskinand ncore.

Fig. 14. �xz for ncuts=20 and Es/Ec=500: in¯uence of nskinand ncore.

1902 V. Manet

From these ®gures, the following conclusions can bedrawn:

1. results are always accurate when ncore=8 nskin,i.e. when the meshing is regular through thethickness;

2. nskin and ncore do not have any in¯uence on theconvergence of displacements, essentially causedby ¯exure: the number of longitudinal cuts, ncuts,is therefore the most dominant parameter;

3. a very re®ned mesh (nskin=4) must be used inorder to converge towards the correct value of �xz;

4. a coarse mesh (nskin=1) does not permit toobtain an acceptable value �zz;

5. convergence towards �xx reference value in thecore is controlled by ncuts.

Results are not improved by increasing nskin norncore. The last point is also true for the convergencetowards �xx value in the skin.

4 LOCAL REISSNER: IMPROVING RESULTSFOR PLANE 82

As can be see from Fig. 9 and from Table 4 (whichsummaries results and gives the good `working zone' ofthe di�erent models), Plane 82 is not able to give accu-rately the interlaminar stress �xz with a coarse mesh.Since this component is very important in the designprocess, results must be improved.

A way of improving results is to re®ne the meshing.In Fig. 9, the curve `Plane 82/2' gives results obtainedwith nskin=1 and ncore=2 (instead of 1). This slightestmodi®cation of the mesh (4 elements through the thick-ness of the sandwich instead of 3) is su�cient to lead tovery good results for Es/Ec�200. But, as mentionedbefore, sandwiches nowadays exhibit ratios generallybetween 200 and 1500. In this range, the convergence isonly reached with a very re®ned meshing: nskin�3 andncore=8 nskin. Such a mesh yields an unacceptablecomputation time. In order to improve the accuracy ofstresses, we must answer the following question: howare nodal stresses computed?

Nodal stresses, �, are generally computed by using aminimization process. They are obtained from nodaldisplacements q using a least-squares method and byminimizing: �

��m ÿ �u�2d �1�

where sm denotes the mixed way to calculate stresses:

�m � N�� 2�

and su the displacements way:

�u � DLNuq �3�

or using the stress projection method18 by minimizing:�

��m ÿ �u�d �4�

Fig. 15. �xx in the core for ncuts=20 and Es/Ec=500: in¯u-ence of nskin and ncore.

Fig. 16. �xx in the top skin for ncuts=20 and Es/Ec=500:in¯uence of nskin and ncore.

Table 4. Accuracy of results (ncuts�20 understood)

Uz �zz �xz �xx core �xx skin

P82 always good always good acceptable forEs/Ec2[100,400]

always good always good

improvement of resultsÐ use a ®ne meshing: nskin >3 and ncore 8=nskinÐ use local Reissners method

S91 always acceptable always acceptable always good good for Es/Ec�100 good for Es/Ec�20S46 always acceptable never acceptable acceptable for

Es/Ec2[8,15]always weak acceptable for

Es/Ec�20


In these equations, D is the generalized Hooke's matrixrelating stresses to strains, L the di�erential operatorrelating strains to displacements, Ns and Nu the matri-ces of shapes functions for stresses and displacements,and the volume or surface of interest.

It is to be noticed that these methods lead to con-vergence towards Reissner's (reference) solution.

As expressed in Ref. 19, the minimization process canbe global (done over the whole structure: being theentire structure) or local (done over one element: beingthe considered element). Since the local process con-verges towards the same limit as the global process, theminimization process chosen is generally the local one.

Nevertheless, instead of minimizing the di�erencebetween two solutions, it may be more convenient(simpler and faster) to directly ®nd the stress ®eld usingReissner's formulation.

In Reissner's solution, nodal stresses are related tonodal displacements by:20±23

� � Aÿ1Bq �5�

with:

A � ��

tN�SN�d �6�

and

B � ��

tN�LN ud �7�

S=Dÿ1 being the compliance matrix.In order to improve the stress computation at inter-

faces, we propose to use the last formulation on twoadjacent elements, located on each side of an interface.Doing so, we ensure the equilibrium state at interface ina better way. We shall refer to this method as `localReissner' method.

This kind of method is not more time consumingthan least squares methods generally used (in ANSYSfor example) to derive nodal stresses from nodaldisplacements.

Now looking at Fig. 17, which is a close-up view ofFig. 9 for sandwiches with Es/Ec�200, we can see that:

1. the use of the local Reissner method (denotedLocal Reissner/2 because we used ncore=2) per-mits to really improve the accuracy of �xz which isof great importance;

2. It is to be noticed that the same mesh with Plane82 (Plane 82/2) does not permit to improve resultswith this high Es/Ec ratio;

3. the solution given by Shell 91 is not so good as forlower Es/Ec ratio.

5 CONCLUSIONS

The reference solution has been obtained using a very®ne meshing and the 8-node quadrilateral element Plane82.

The ®rst study, in¯uence of ncuts on the di�erentmodel in section 3.1, seems to lead to the conclusionthat Plane 82 is the best model, especially when lookingat Figs 2±6 (obtained with a coarse mesh and in the caseEs/Ec&200). Nevertheless, the study of the in¯uence ofthe Es/Ec ratio in the previous section permits to seesome weaknesses of this model.

In terms for design quantities:

. all models lead to a correct value of displacements,but Plane 82 is the most accurate;

. �zz can be correctly given by Shell 91 and Plane 82,the latest being the most accurate;

. �xz is only very accurately computed with Shell 91,but for Es/Ec�200;

. �xx in the core can be calculated using any model,Plane 82 being the most accurate;

. �xx in the skin is very accurately computed withPlane 82 and with Shell 91 (but only for Es/Ec�20), and acceptable with Solid 46 (and only forEs/Ec�20).

A summary of results, and the `working zone' inwhich the di�erent elements can be used is given inTable 4.

Hence, from the previous results, we can say that:

1. for planar problems, Plane 82 is very well adapted.Nevertheless, it is to be noticed that this method isnot very stable for very coarse meshes (smallvalues of nskin, ncore and ncuts), and that inter-laminar stress �xz can only be reached with a ®nemeshing;

2. Shell 91 (with sandwich option) is a good way ofcomputing sandwich structures. Nevertheless if thedesigner must know �xx at interfaces, then thiselement can only be used for Es/Ec�50.

3. Solid 46 is not very accurate in the determinationof the design quantities. A model using this kindof element should be avoided.Fig. 17. �xz for ncuts=20: in¯uence of high Es/Ec ratio.

1904 V. Manet

The presented local Reissner method permits to reachexcellent results, especially for the interlaminar stress�xz and for Es/Ec�200 with a coarse mesh through thethickness of the sandwich.

This method is particularly interesting for the designof new sandwich materials.

Finally, we should like to emphasize the fact that thismethod is particularly easy to implement, as a stand-aloneprogram, but also in existing ®nite-element softwares.

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