1 vR2

where

Here, R is the shell radius, t the face sheet thickness, c the core thickness. The parameter [l measures the relative shear compliance of the core. It must be less than 1/V2 to avoid localized shear kinking of the shell wall. This result (10.45) applies if the length of the shell I is at least several times the axial buckle wavelength. It also assumes that sufficiently strong end support conditions are in effect. End conditions modify (10.45) slightly, but not as much as imperfections, to be discussed later.

The condition for face yielding of the perfect shell prior to (or simultaneous with) buckling is:

Yielding of the core does not directly affect the load-carrying capacity since it supports no significant load. However, it will affect the ability of the core to suppress face sheet wrinkling and to maintain the shear stiffness necessary for post-buckling load-carrying capacity. Yielding of the core prior to buckling is avoided if the axial strain in the unbuckled shell (s = N/(2tEf)) does not exceed the uniaxial compressive yield strain of the core. In other words, core yielding is excluded if its yield strain is larger than that of the face sheets.

If neither the core nor the face sheets yield, face sheet wrinkling may occur. It is governed by the condition introduced earlier (equation (10.24b)). In terms of the load per unit circumferential length, the onset of face sheet wrinkling in the perfect shell occurs when

For shells with core parent material identical to the face sheet material (ps = pf and Ef = Es), with core modulus and density still related by equation (10.22a), the weight index is

When the shell buckles elastically, uninfluenced by yielding or wrinkling, global optimization of the shell could be carried out analytically to obtain the values of t, c and pc which minimize the weight of the shell for prescribed N. This elastic optimization implies that the core weight is twice that of the two face sheets. However, it is of little practical value because face sheet yielding or wrinkling invariably intervene at the levels of load index wherein the sandwich shell is weight competitive.

Fixed core density

Consider shells with prescribed core density pc designed to carry load per circumferential length, N. Subject to the inequality, af /Ef < ka22/3(pc/ps)4/3, core yielding always excludes face sheet wrinkling, and vice versa. An optimal design having face sheet yielding coincident with buckling is obtained by using equation (10.46) to give t, and then using that expression in equation (10.45) to obtain c. The weight follows from equation (10.27). The procedure for design against simultaneous wrinkling and buckling follows the same steps, but now using equation (10.47) rather than (10.46).

The outcome from the above optimization is shown in Figure 10.16(a) in the form of a plot of the weight index as a function of load index, N/(EfR), for two values of core density. These plots are constructed for a core with stiffness at the low end of the range found for commercial materials (a2 = 1). The core has been assumed to remain elastic, and the yield strength of the face sheets is that for a structural aluminum alloy. The range of the load index displayed is that for which sandwich cylinders have a competitive edge over more conventional construction comprising axial stiffeners. Included in Figure 10.16(a) is the weight-optimized shell with hat-shaped axial stiffeners which buckles between rings spaced a distance R apart.

The sandwich results are independent of shell length, whereas the axially stiffened results do depend on the segment length, typically R. The optimized sandwich shells in Figure 10.16(a) experience simultaneous buckling and face sheet yielding, except in the range N/(EfR) 4 x 10 6 where the buckling is elastic. Note that the relative weight of the core to the total weight for these shells (Figure 10.16(b)) is very different from that predicted by the elastic global analysis. For shells in the mid-range of the structural index, the core weight comprises only about 25% of the total.

This example affirms that metal foam core sandwich shells can have a competitive advantage over established structural methods of stiffening, particularly at relatively low structural indices.

Global minimum

To pursue the subject further, the sandwich shells have been optimized with respect to relative core density pc, as well as t and c, allowing for all possible combinations of face sheet yielding and wrinkling. Simultaneously, the consequence of using a core with superior stiffness is addressed by assuming a core having properties comparable to the best commercial materials (a2 = 4 rather than a2 = 1). The results for the fully optimized foam-core sandwich shells are plotted in Figure 10.17 with accompanying plots for the optimal relative density of the core. The operative deformation modes in the optimal design are indicated in the plot of core density. Wrinkling and buckling are simultaneous at the lowest values of the load index; wrinkling, yielding and buckling in the intermediate range; and yielding and buckling at the high end. Again, the global elastic

N/EtR (x105)

Load Index

Load Index

N/EfR(x105)

Figure 10.16 (a) Weight index versus load index for cylindrical sandwich shells. (b) Relative weight versus load index for cylindrical sandwich shells

N/EfR(x105)

Figure 10.16 (a) Weight index versus load index for cylindrical sandwich shells. (b) Relative weight versus load index for cylindrical sandwich shells design has no relevance. For reference, the result for the optimally designed cylindrical shell with axial hat-stiffeners is repeated from Figure 10.16(a).

This comparison illustrates the weight superiority of foam metal-core sandwich shells over conventional shell construction as well as the potential benefit to be gained by using a core material with the best available stiffness.

Global weight minima

Global weight minima

N/EfR(x105) Load index

Figure 10.17 Global weight minima for cylindrical sandwich shells Imperfection sensitivity

N/EfR(x105) Load index

Figure 10.17 Global weight minima for cylindrical sandwich shells Imperfection sensitivity

An important consideration for strength-limited thin-walled construction concerns the influence of imperfections. In most cases, imperfections reduce the buckling loads, sometimes considerably. In shells, imperfections cause out-of-plane bending, which lowers the maximum support load due to two effects: (1) by advancing non-linear collapse and (2) by causing premature plastic yielding, which reduces the local stiffness of the shell and, in turn, hastens collapse. Since they always exist, practical designs take this imperfection sensitivity into account. Generally, experimental results establish a knockdown factor on the theoretical loads that may be used as the design limit with relative impunity. Guidelines for such an experimental protocol are provided by two conclusions from a study of the interaction between plasticity and imperfections in optimally designed axially compressed sandwich shells (Hutchinson and He, 1999). When the perfect cylindrical shell is designed such that buckling and face sheet yielding coincide, buckling in the imperfect shell nearly always occurs prior to plastic yielding. Thus, knockdown factors obtained from standard elastic buckling tests are still applicable to the optimally designed shells. A similar statement pertains when core yielding in the perfect shell is coincident with buckling.

Thin sandwich structure

Thin sandwich structure n = P/sy eb(x1G3) Load index

Configuration 'P

Figure 10.18 Minimum weight axially compressed panels f

Figure 10.18 Minimum weight axially compressed panels

0.01

Optimum density, (pc/ps)

Wrinkling, buckling

Xefbl

Wrinkling, yielding, buckling

Yielding, buckling

Figure 10.19 Global weight minima for sandwich and hat-stiffened panels

Other configurations

Corresponding diagrams for panels and columns (Budiansky, 1999) are presented on Figures 10.18-10.20; associated buckling modes are indicated on the insets. Results for minimum weight flat sandwich panels at a fixed core density, pc/ps = 0.1 (Figure 10.18) are not especially promising. There is only a small domain of weight advantage, arising when sandwich construction is used

within the stringers, as well as the panels, of a stringer-stiffened configuration. This construction has lowest weight at small levels of load index.

Further minimization with core density leads to more pronounced weight savings (Figure 10.19). In this case, even flat sandwich panels can have lower weight than stringer-stiffened panels, especially at lower levels of load index. The failure modes governing the weight change as the load index changes and the minimum weights coincide with simultaneous occurrences of either two or three modes, as in the case of the optimally designed cylindrical shells. The challenge in taking advantage of the potential weight savings arises in manufacturing and relates to the low relative densities required to realize these performance levels (Figure 10.19) and the need for acceptable morphological quality.

Results for columns (Figure 10.20) indicate that thin-walled sandwich tubes are lighter than foam-filled and conventional tubes, but the beneficial load ranges are small.

Figure 10.20 Axially compressed columns

10.9 Recommendations for sandwich design

For those wishing to explore cellular metal core sandwich construction, the following recommendations are pertinent:

1. Determine the constraints that govern the structure and, in particular, whether it is stiffness or strength-limited.

2. If stiffness-limited, the procedure for determining the minimum weights is straightforward, using the formulae summarized in the tables. It is important to realize that there will always be lighter configurations (especially optimized honeycomb or waffle panels). Those configurations should be explicitly identified, whereupon a manufacturing cost and durability comparison can be made that determines the viability of sandwich construction. Other qualities of the cellular metal may bias the choice. It is important to calculate the domains wherein the weights based on elasticity considerations cannot be realized, because of the incidence of 'inelastic' modes: face yielding, core yielding, face wrinkling. Some help in assessing these limits has been provided.

3. When strength-limited (particularly when buckling-limited), the rules governing sandwich construction are less well formulated. In general, numerical methods are needed to compare and contrast this type of construction with stiffened systems. Some general guidelines are given in this Design Guide; these give insight into the loadings and configurations most likely to benefit from sandwich construction. Configurations unlikely to benefit are also described. It is recommended that where benefits seem likely, detailed simulations and testing should be used to assess the viability of sandwich construction.

References

Allen H.G. (1969) Analysis and Design of Structural Sandwich Panels, Pergamon Press, Oxford.

Andrews, E.H., Gioux, G., Onck, P. and Gibson, L.J. (1999) The role of specimen size, specimen shape and surface preparation in mechanical testing of aluminum foams. To appear in Mat. Sci. and Engineering A.

Bart-Smith, H. (2000) PhD thesis, Harvard University.

Budiansky, B. (1999) On the minimum weights of compression structures. Int. J. Solids and Structures 36, 3677-3708.

Deshpande, V.S. and Fleck, N.A. (1999) Isotropic constitutive models for metallic foams. To appear in J. Mech. Phys. Solids.

Gerard, G. (1956) Minimum Weight Analysis of Compression Structures, New York University Press, New York.

Gibson, L.J. and Ashby, M.F. (1997) Cellular Solids, Structure and Properties, 2nd edition, Cambridge University Press, Cambridge, Ch. 9, pp. 345 et seq.

Harte, A.-M. (1999) Private communication.

Hutchinson, J.W. and He, M.Y. (1999) Buckling of cylindrical sandwich shells with metal foam cores. Int. J. Solids and Structures (in press).

Shuaeib, F.M. and Soden, P.D. (1997) Indentation failure of composite sandwich beams.

Composite Science and Technology 57, 1249-1259. Soden, P.D. (1996) Indentation of composite sandwich beams. J. Strain Analysis 31(5), 353-360. Tennyson, R.C. and Chan, K.C. (1990) Buckling of imperfect sandwich cylinders under axial compression. Int. J. Solids and Structures 26, 1017-1036. Wilsea, M., Johnson, K.L. and Ashby, M.F. (1975) Int. J. Mech. Sci. 17, 457.

Chapter 11

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