Design map of sandwich beams loaded in three-point bending.

Chincea, Ion ; Cernescu, Anghel ; Marsavina, Liviu 等

1. INTRODUCTION

Structural members made up of two stiff, strong skins separated by a lightweight core are known as sandwich panels. The mechanical behavior of a sandwich panel depends on the properties of the face and core materials and on its geometry. In most applications the panel must have some required minimum stiffness, it must not fail under some maximum service loading and it must be as light as possible (Gibson et al., 1997). The obvious attraction of sandwich structures is that they are light and stiff. The beam or panel must also have strength: it must carry the design loads without failing. At least five different failure modes are possible; a given sandwich will fail by the one which occurs at the lowest load, (Andrews et al., 2009), (Triantafillou et al., 1987), (Ley et al., 1999). With changing the geometry and loading the failure mode can change, too. So it is not enough to design against one mode; all must be considered, and the dominant mode--the one which determines failure--identified and evaluated.

In most cases, for sandwich beams loaded in three-point bending, the following failure modes can occurs, Fig. 1:

[FIGURE 1 OMITTED]

The dominant mechanism, for a given design, is the one giving failure at the lowest critical load. A transition in failure mechanism takes place when two mechanisms have the same failure load. This information can be displayed as a diagram or map, Fig. 2. The diagram is divided into fields, within which one failure mechanism is dominant. The fields are separated by field boundaries, which are the loci of design points for which two mechanisms have the same failure load.

Based on previous considerations, in this paper we determined a designed map for sandwich beam with aluminium faces and cork core loaded in three-point bending such in figure 3.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

2. DESIGN MAP OF A SANDWICH BEAM LOADED IN THREE-POINT BENDING

Applying the strength theory of sandwich beams we calculated that the critical stress and force for every failure mode, for different face thickness values and different cork densities, Fig. 4 and Tabel 1.

Face Sheet failure, occurs when the normal stress in the face equals the strength of the face material, [[sigma].sub.yf], or when:

[[sigma].sub.f] = Pl / [B.sub.3] b = [[sigma].sub.yf] (1)

Face wrinkling appears when the normal stress in the compressive face of the beam reaches the local instability stress. Wrinkling occurs when the compressive stress in the face is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Core shear failure, when the shear stress, [[tau].sub.c], equals the yield strength of the core in shear, [[tau].sup.*.sub.C]:

[[tau].sub.C] = [[tau].sup.*.sub.C] (3)

The yield strength of the core in shear depends on density in the same way as the uniaxial strength.

[FIGURE 4 OMITTED]

3. EXPERIMENTAL ANALYSIS OF THE SANDWICH BEAM LOADED IN THREE-POINT BENDING

The three-point bending tests were performed on tensile testing machine, model Zwick/Roell 5 kN, fig. 5, using specimens with dimensions from figure 3. Tests were carried out according with ASTM D 790-03 at a loading rate of 2 mm/min and at room temperature.

[FIGURE 5 OMITTED]

Five samples were tested and the results showed a failure load between 40 and 60 N for a core relative density equal with 0.15. The tests showed that the dominant failure mechanism of the tested sandwich beams in three-point bending is the cork core shear, Fig.6.

[FIGURE 6 OMITTED]

CRD--Core Relative Density; FM--Failure Mode; FW--Face wrinkling; FY--Face Yield; CS--Core Shear

4. CONCLUSION

This paper presents failure and design mode map for sandwich beams with cork core and aluminium faces used in some applications of railway vehicle. This design diagram provides information about the failure mode and critical forces (Tabel 1) of sandwich beams based on core density and dimensions of the analysed specimen (beam or panel). The failure mode observed on the tested sandwich beams was cork core shear (fig. 6) and was in agreement with the failure mode predicted from designed map, Fig. 4.

5. ACKNOWLEDGEMENTS

This study was partially supported by the project PERFORM ERA ID-57649, CONTRACT POSDRU/89/1.5/S/57649.

6. REFERENCES

L. Gibson, M. F. Ashby-"Cellular solids--Structures and Properties--second edition", Published by the Press Syndicate of the University of Cambridge, 1997

Zenkert D.,--"An introduction to sandwich construction", London, EAMS, 1995

Allen H.G.,--"Analysis and design of structural sandwich panels", Oxford, Pergamon Press, 1969

E. W. Andrews, N.A. Moussa--"Failure mode maps for composite sandwich panels subjected to air blast loading", International Journal of Impact Engineering, vol. 36, pp. 418-425, 2009

Triantafillou T.C., Gibson L.J.--"Failure mode maps for foam core sandwich beams ", Material Science and Engineering, 1987, 95, 37-53

Ley R.P., Lin W., Mbanefo U.-"Face sheet wrinkling in sandwich structures", Prepared for Langley Research Center under Contract NAS1-19347. El Segundo, CA: Northrop Grumman Corporation; 1999. NASA/CR-1999-208994

ASTM D 790-03-"Standard Test Methods for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials"

Swanson S.R.--"Introduction to design and analysis with advanced composite materials". Upper Saddle River, NJ: Prentice Hall; 1997

Wennhage P., Zenkert D.--"Testing of sandwich panels under uniform pressure". Journal of Testing and Evaluation, 1998; 26: 101-8

Tab.1. The critical force values of sandwich beams loaded in
three-point bending

 Analitical procedure

 t=0,02 t=0,1 t=0,5

CRD P[N] FM P[N] FM P[N] FM

0,01 0,77 FW
0,02 1,85 FW 3,07 CS 2,71 CS
0,03 3,24 FW 5,09 CS 5,09 CS
0,04 3,3 FY 7,68 CS 7,68 CS
0,05 3,3 FY 10,5 CS 10,5 CS
0,06 3,3 FY 14,1 CS 14,1 CS
0,07 3,3 FY 16,5 FY 17,9 CS
0,08 3,3 FY 16,5 FY 21,7 CS
0,09 3,3 FY 16,5 FY 25,7 CS
0,1 3,3 FY 16,5 FY 30,3 CS
0,15 3,3 FY 16,5 FY 56 CS
0,2 3,3 FY 16,5 FY 82,5 FY
0,3 3,3 FY 16,5 FY 82,5 FY
0,35 3,3 FY 16,5 FY 82,5 FY
0,37 3,3 FY 16,5 FY 82,5 FY
0,4 3,3 FY 16,5 FY 82,5 FY

 Analitical procedure

 t=1 t=1,5

CRD P[N] FM P[N] FM

0,01
0,02 2,71 CS 2,71 CS
0,03 5,09 CS 5,09 CS
0,04 7,68 CS 7,68 CS
0,05 10,5 CS 10,5 CS
0,06 14,1 CS 14,1 CS
0,07 17,9 CS 17,9 CS
0,08 21,7 CS 21,7 CS
0,09 25,7 CS 25,7 CS
0,1 30,3 CS 30,3 CS
0,15 56 CS 56 CS
0,2 82,5 CS 82,5 CS
0,3 157,7 CS 157,7 CS
0,35 165 FY 198,4 CS
0,37 165 FY 216,4 CS
0,4 165 FY 242,8 CS