# Why Honeycomb Structures Deflect Less Than Solid Structures

Sandwich structures for vibration isolation tables are stiffer than solid tables of equal weight. For the same weight, the fundamental frequency of a sandwich table is higher than that of a solid table. Although the bending behavior of a sandwich table is complex, the following simple approximations are useful.
The fundamental frequency of a vibration isolation table acted on by only self-weight is given by:

Where:
f is the fundamental frequency (Hz)
δ is the self-weight induced deflection
g is the acceleration due to gravity
C is estimated to be 1.13–1.26, depending on the table geometry*
The self-weight induced deflection of a rectangular table with simple supports at its corners is given by:

Where:
δ is the self-weight induced deflection
P is the table weight per unit area
L is the table length
b is the table width
D is the static flexural rigidity of the table

The static flexural rigidity of a solid table is given by:

Where:
DSOLID is the static flexural rigidity of the solid table,
E is the elastic modulus of the solid material
h is the table thickness
υ is the Poisson’s ratio for the solid material

For a sandwich table, the flexural rigidity is given approximately by:

Where:
DSANDWICH is the flexural rigidity of the sandwich table
E is the elastic modulus of the face sheets of the sandwich
tF is the thickness of the face sheets of the sandwich
h is the table thickness.

(The previous equation assumes that the top and bottom face sheets have the same thickness and the shear stiffness of the core is not significant.)
Using the equations for deflection and static flexural rigidity, the ratio of the solid to sandwich deflection for equally thick tables is given by:

Where:
δSOLID is the deflection of the solid table
δSANDWICH is the deflection of the sandwich table
PSANDWICH is the weight per unit area of the sandwich table
ρ is the density of the solid material

An example of the above equation is the performance of an 8 in. thick Newport table compared with a solid 8 in. thick carbon steel table. The properties of the two tables are:

 Solid Sandwich 1. Elastic modulus 29 x 106 psi 29 x 106 psi 2. Skin density 0.28 lb-in.-3 0.28 lb-in.-3 3. Poisson’s ratio 0.33 0.33 4. Face sheet thickness N/A 0.19 in. 5. Weight per unit area 2.24 lb/in.2 0.17 lb/in.2

The static flexural rigidity of the solid table is:

The static flexural rigidity of the sandwich table is:

But the ratio of the self-weight deflections is:

This example shows that the self-weight deflection of a sandwich table is about half that of a solid table of the same thickness. Indeed, experience has shown that the self-weight deflection of a sandwich table compared with a solid table of equal weight is significantly better than this example.

* See C.W. Bert, Journal of Sound and Vibration, 1993, 162 (3), 547-557.

Note: We greatly appreciate the mathematical treatment supplied by Daniel Vukobratovich of the Optical Sciences Center at the University of Arizona.