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.

The stiffness of thebeam in bending is calculated from the

Mode of Loading | B_{1} | B_{2} | B_{3} | B_{4} |

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 c

4. Substitute c

5. Using the objective function(the weight equation) determine the weight of the beam

6. Ask yourself... are c

7. Modify c

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.

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).

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....