Study of a composite sandwich-beam with an uniform load.

1. INTRODUCTION

The behavioral study, the calculus of the stresses and deformations of the composite elements and advanced composite structures, are complex problems. A lot of assembly elements of the mechanical structures are beams or can be assimilated as a beam. The approach consists of an original method originally developed by Mr. Daniel GAY (2002)

2. ASSUMPTIONS FOR CALCULATING

For the calculus, we utilize the following working hypotheses: we consider the composite-beams with constant geometrical characteristics in all transverse sections, which have certain boundaries and are supposed to be perfect cleaved between them, with constant material characteristics in all sections and with isotropic characteristics for the phases.

3. WORK PREMISES

We need, for the arrow calculus, to determine the equivalent rigidities, noted respectively, with:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

1/k<GS> = 1/k [summation over (i)][G.sub.i][S.sub.i] (2)

For the equivalent rigidities (1) and (2), the approximate calculus is enough precise for the sandwich-beams (the sandwich-structures) for which the thicknesses of the inferior parts, respectively superior parts, are much less than the thickness of the heart.

We suppose that the cutting stresses on the section have a linear variance of an element, after the y axis, under the cutting load (T) effect. The materials which make up the sandwich-beam are noted with 1 and 2 and are supposed isotropic, or transverse isotropic. We note with [U.sub.[tau]] the deformation energy due to the cutting stresses. For a sandwich-beam structure (Fig. 7), we can write:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where A is the transversal section area and [[tau].sub.xy] is the cutting stress in the superior layer:

[[tau].sub.xy] = [h.sub.1] - 2y/[h.sub.1] - [h.sub.2] [[tau].sub.0] (4)

[FIGURE 1 OMITTED]

T, the cutting load, is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[[tau].sub.0], the maximum tangential stress is:

[[tau].sub.0] = T 2 1/[h.sub.1] + [h.sub.2] b (6)

The linear deformation energy of (3) will be:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

We obtain, after calculus:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

or, with the relation (3), we can write:

[dU.sub.[tau]]/dx = 1/2 k[T.sup.2]/<GS> = 2[T.sup.2]/b[([h.sub.1] + [h.sub.2]).sup.2] ([h.sub.2]/[G.sub.2] + [h.sub.1] - [h.sub.2]/3[G.sub.1])

We obtain for the coefficient k:

k = 4<GS>/b[([h.sub.1] + [h.sub.2]).sup.2] ([h.sub.2]/[G.sub.2] + [h.sub.1] - [h.sub.2]/3[G.sub.1])

<GS> = [G.sub.1] ([h.sub.1] - [h.sub.2]) + [G.sub.2][h.sub.2] (12)

The expression (11) is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

For a sandwich-structure, with equal thicknesses for superior and inferior layers, noted by [h.sub.d], with the material characteristic [G.sub.d] and the heart thickness [h.sub.p] with the material characteristic [G.sub.p].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

In the right parenthesis, the term which is added to 1, is much less than the unit. We have for k:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

We obtain the next simplified form for [G.sub.p] [much less than] [G.sub.d] and [h.sub.d] [much less than] [h.sub.p]:

k/<GS> = 1/b[G.sub.p]([h.sub.p] + [2.sub.hd]) (17)

4. APPLICATION

We consider a beam, made of duralumin, articulated at the right edge, simple rested at the left edge, with an uniform load and thickness h (Fig. 2).

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

We propose the arrow calculus at the middle.

The classical calculus for the Strength of Materials is (Young, 2002):

f = 5 q [l.sup.4]/384[EI.sub.z] (18)

[I.sub.z] = [bh.sup.3]/12 (19)

We have for duralumin GAY (1997):

[E.sub.d] = 7,5 x [10.sup.4] MPa (20)

For the Fig. 2, we consider the numerical values:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

and the uniform load q is:

q = 1N/mm (22)

After calculus, we obtain the arrow:

f = 10,416667mm (23)

We cut now the duralumin beam off, after the meridian plan, in two equal parts, the parts have the same thickness of [h.sub.d] = 2,5mm (Fig. 3). Each part will be cleaving on a polyurethane parallelepiped, shaped as a sandwich-beam, having approximatively the same mass with the initial beam. The total polyurethane thickness is [h.sub.p] = 25 mm. This sandwich-beam is underpinned and stressed in same conditions as well as the beam of Fig. 2. We calculate the arrow for the middle of the sandwich-beam from Fig. 3. We note with Up the energy of elastic deformation at the bending and with [U.sub.[tau]], the energy of elastic deformation at the cutting. The total energy of elastic deformation is (Gay, 2002):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

where [G.sub.p] = 20 MPa (Gay, 1997).

With the Castigliano's theorem, for the arrow f' we can write (Young, 2002):

f' = [partial derivative]U/[partial derivative]F (26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

For the Fig. 3, in a section x, we can write M and T and we calculate the arrow f', with F=0:

f' = [5ql.sup.4]/384<[EI.sub.z]> + [ql.sup.2] k/8<GS> (28)

The approximate calculus for the equivalent rigidities is given by:

<[EI.sub.z]> = [E.sub.d][h.sub.d]b[([h.sub.p] + [h.sub.d]).sup.2]/2 + [E.sub.p][H.sup.3.sub.p]b/12 (29)

where [E.sub.p] = 60 MPa. We obtain:

<[EI.sub.z]> = 7097,6562 x [10.sup.6] (30)

k/<GS> = 1/b[G.sub.p] ([h.sub.p] + 2[h.sub.d]) = 1,6666667 x [10.sup.-5] (31)

The arrow f' will be:

f'= 0.6354916 mm (32)

The comparison between the arrows acquired in the two occurrences (Fig. 2 and Fig. 3), leads to the following:

f/f' [congruent to] 16.4 (33)

which signifies that the sandwich-beam admits an arrow reduction of 16.4 in the ratio of the beam in one material, without a significant augmentation of the beam mass.

5. CONCLUSION

The sandwich-beams admit a significant arrow reduction (function of the supports and loadings of beams). The composite-beams calculus is very important for the computer programs and the optimizing algorithms.

6. REFERENCES

Barrau, J. J. & Laroze, S. (1987). Calculation of composite structures, Ed. Eyrolles-Masson, ISBN 9782225811432 Paris

Gay, D. (1997). Composite materials, Ed. Hermes, ISBN 1587160846 Paris

Gay, D. & Hoa, S.V. (2002). Composite materials-Structure and Applications, Ed. CRC Press, ISBN 9781420045192, Boca Raton, Florida

Young, W. C. (1989). ROARK'S--Formulas for Stress & Strain, 6-th edition, Ed. McGraw-Hill Book Company, ISBN 0070725411, New-York

Young, W. C. (2002). ROARK'S--Formulas for Stress & Strain, 7-th edition, Ed. McGraw-Hill Book Company, ISBN 0070725423, New-York
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