## Design and Properties of Sandwich Core Structures

Structural members made of two stiff, strong skins separated by a lightweight core are know as sandwich panels. The separation of the skins, which actually carry the load, by a low density core, increases the moment of inertia of the beam/panel with little increase in weight producing an efficient structure. Examples include skis, where a carbon fibre - epoxy skin is bonded to and separated by either a rigid polyurethane foam or by balsa wood, the fan blade of modern gas turbine jet engines where two titanium alloy skins (the front and rear surface of the blade are separated and bonded two a honycomb structure made from titanium sheet and the hulls of modern racing yachts where two thin layers of composite skin (glass/carbon/kevlar in vinyl ester/epoxy) are separated by a nomex core (nomex is a combination aluminium honeycomb filled with rigid polyurethane foam).

The mechanical properties of the sandwich depend on the core and face materials as well as the thickness of the core and faces. In most cases, the panels must have a minimum stiffness (deflection/unit load) and strength. The design process is essentially one of optimization wherein a property such as weight or stiffness/unit weight is minimized. A complete analysis can be found in “Cellular Solids - Structure and Properties” by Lorna Gibson & Mike Ashby.

## Design for Stiffness - Weight Optimization The simplest case to consider is that of point loading in 3-pt bend, though we shall show that the analysis is identical for other forms of bending load with the exception of the different values for the geometrical constants. The span of the beam is l, the width, b, the core thickness c, and the face thickness, t. The beam thickness is d where d = c+2t; we shall assume that t<<c and to a first approximation c~d. We shall denote the elastic modulus and density of the core as and where the * indicates a property of the foam rather than the material the foam is made from. Clearly the foam density and modulus are a function of the relative density of the core as well as the material the core is made from. To a first approximation these properties are given by where and are properties of the bulk (100% solid) core material. C1 (~1) and C2 (~0.4) are constants We can also define and as the density and longitudinal stiffness of the face material.

The stiffness of thebeam in bending is calculated from the equivalent flexural rigidity, (EI)eq, and the equivalent shear rigidity, (AG)eq. Using the parallel axis theorem The first and second terms describe the stiffness of the two face sheets and the core while the third term adds the stifness of the faces about the centre of the beam. In a good beam design, the third term is substantially larger than the first two so if d~c then The equivalent shear rigidity is When subject to a load, P, the deflection is the sum of the bending and shear components where B1 and B2 are constants dependent on the geometry of the plate and the type of loading (Table 1). The compliance of the beam is Table 1. Constants for bending and failure of beams
 Mode of Loading B1 B2 B3 B4 Cantilever, end load (P) 3 1 1 1 Cantilever, uniformly distributed load (P/l) 8 2 2 1 Three point Bend, central load (P) 48 4 4 2 Three point Bend, uniformly distributed load (P/l) 384/5 8 8 2 Ends built in, central load (P) 192 4 8 2 Ends built in, uniformly distributed load (P/l) 384 8 12 2

The key issue in design of most sandwich panels is the minimization of weight. The weight of the beam is where g is the acceleration due to gravity. This is known as the “objective function” since this is what we wish to minimize. The span,l, the width, b, and the stiffness, P/d, are fixed by the design, the free variables are c,t and . If the core density is fixed then the optimization is easy

1. Rewrite the stiffness equation in the form t=
2. Substitute the equation for t into the weight function
3. Differentiate the weight equation with respect to the other free variable, c and set equal to zero, find copt
4. Substitute copt into the “t” equation and find topt.
5. Using the objective function(the weight equation) determine the weight of the beam
6. Ask yourself... are copt and topt realistic/sensible ?
7. Modify copt and topt as necessary and reevaluate the weight.

Try this! The answers can be found here.

The weight minimization problem may be better understood by employing a graphical solution. Again, for simplicity, we shall assume that the core density os fixed. We now create a graph in which (t/l), thickness of the face /length of beam is plotted as a function of (c/l) core thickness to length (both of which are dimension less) and the two functions derived above, Weight and Stiffness are plotted. Before plotting, each equation must be rewritten such that (t/l) is a function of (c/l) ie. the stiffness constraint which plots as curve, and which plots as a straight line for any given weight, or a series of parallel straight lines for different weights.

Download the MathCad document in which these calculations and graphs are implemented.

The two functions are plotted below, one (1) curve for a specific stiffness (P/d) and a series of parallel lines for successively increasing weights (W). The optimum design point is found where the objective function (weight function) line is a tangent to the Stiffness Constraint. At this point the optimum (for minimum weight) values of face thickness, t, and core thickness (c) for a specified stiffness can be read off the graph.

If the core density were to be considered a free variable then a series of graphs are plotted for successively high core densities, the optimum values of c and t and hence weights determined then the actual minimum weight found. In reality, these ‘optimum’ values lead to too large core thickness' and too low core densities and some compromises are necessary for a realistic dsign.

Continue with strength and fracture of sandwich panels....