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shapes.

Equivalent parameters and modeling

Fig.1.Unit cell of regular hexagonal honeycomb structure

The theory started from Gibson is sandwich plate theory, and only the core is equivalent when

using it, the equivalent parameters of regular hexagonal honeycomb core are as below:

34 2

, .3 3

x y s z s

t tE E E E E

l l

= = =

33 3

, , .2 23

xy s xz s yz s

t t tG E G G G G

l l l

= = =

(1)

2 1

, = .33 s

t

l =

in which, ,s s

E G ands

are the Young modulus, shear modulus and density of the panel core

material, respectively. is the Poisson ratio , is a correction factor, which depends on the

manufacture, and we set it 1.0 in this paper.

In order to study the validity of the equivalent models, we must compare the calculation results

given by these models with that of the 3D models, which are considered as reference results.

As in the fig.2, the 3D model is made up of many regular hexagonal cylinders, which are empty

inside. The left model shown above has been meshed with shell element and the right one is

unmeshed.

Fig.2. 3D model of a honeycomb sandwich panel

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Fig.3. Equivalent layered plate model [4]

From Eq.1, we can get the parameters for the sandwich core, i.e. plate 1 in fig.3. Plate 2 is as

same as that in 3D model.

In the study of structural of panels on radio telescopes, we mainly focus on the displacement

situation, which can lead to changes of the electrical performance of telescopes, such as gains and

efficiencies. Through the comparison, we have defined two relative errors in % given by the Eq.2;

the first one is in the term of maximal displacement while the second is in terms of RMS (root of

mean square) value of displacement.

3 3

3 3

100%, = 100%.e D e D

D D

U U RMS RMS

U RMS

= (2)

in which,3

,e DU U are respectively the maximal displacements of equal model and 3D model in Z

direction. 3,e DRMS RMS are the RMS value of displacement of nodes within the uppermost layer,the meaning of subscript is similar to the former.

Examples and analysis

1) Example 1

These calculations are performed with square panels, whose areas range from 20.002 0.9m , butthe thickness of panels is kept to0.02m . The original sandwich panel is made of aluminum with the

density of 3 32.73 10 /kg cm and elastic modulus of 70.71 10 MPa .10 times of self-weigh is

considered to make the displacements more obvious. Also, four angle nodes of the panels in thelowest layer are constrained with all 6 DOFs, as is shown in fig.2.

As shown in fig.4, the both errors decrease with the area increases. The values of these errors

when the area reaches 20.9m is less than5.5%.

2) Example 2

We fix the areas of panels to 20.81m , but the1 2

L L= ratio varies from 0.05 9 , where the

1 2,L L respectively stand for the length in the ,x y directions. The thickness, loads, constraint and

material are the same as example 1.

From fig.5, the values of error reach the worst when is near 1; whether the increases or

decreases from 1.0, the errors both are better.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

10

20

30

40

50

60

area of panel (m2)

,

(%)

Fig.4. relative errors (%) vs area of panel

10-2

10-1

100

101

102

-3

-2

-1

0

1

2

3

4

5

6

7

the ratio of length to width

,

(%)

Fig.5. relative errors (%) vs the ratio of length to width

The regular hexagonal is not central symmtry figure, i.e., the figure doesnt coincide with the

original one after rotating about the geometric center for 90 . So, the calculation results are

unequal with the same loads and constraints even when1 2 2 1

L L L L= , which can explain why the

curves in the fig. 5 are asymmetric.

Summary

Two groups of panels are calculated to study the validity of the equal model, which are createdwith the formula listed in Eq.1, in terms of maximal displacement and RMS of displacement.

It is found that the equivalent layered model can produce a rather good accuracy in general. But

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when the area of panel gets too small, such as below 20.1m , the errors could be too big to be

accepted. The relative errors reach the worst when the panel is almost square with the different ratio

of length to width and equal area, surprisingly.

The paper provides a reference for engineers, especially large radio telescope designers, who can

take the changed accuracy into account when dealing with different shapes of panels.

Acknowledgement

The work described in this paper is carried out at the Research Institute on Mechatronics, Xidian

University. This work is supported by the Fundamental Research Funds for the Central

Universities (No. K50510040008).

References

[1] Gibson L J, Ashby M F, Schajer G S:The mechanics of two-dimension cellular materials. Proc.

R. Soc. A382(1982), p25-42.[2] FU Ming-huiYIN Jiu-ren: Equivalent elastic parameters of the honeycomb core. Acta

Mechanic Sinica. Vol. 31(1999), p113-118(in Chinese)

[3] Wang Ying-jian: Deformation models of honeycomb cell under in-plane shear. ActaScientiarum Naturalium Universitatis Pekinensis, Vol. 27(1991), p301-307(in Chinese).

[4] Saidi A, Coorevits P, Guessasma M: Homogenization of a sandwich structure and validity ofthe corresponding two-dimensional equivalent model. Journal of Sandwich Structures and

Material, Vol. 7(2005), p7-30.

[5] Xia Li-juan, Jin Xian-ding, Wang Yang-bao: Equivalent Analysis of Honeycomb SandwichPlates for Satellite Structure. Journal of Shanghai Jiaotong University. Vol. 37(2003),

p999-1001(in Chinese).

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