## T

Figure 10.1 The geometry of sandwich structures: (a) a beam, (b) a circular column, (c) a square column, (d) a circular panel and (e) a shell element

F/2L

Jk_i

t |
F ac y | |||

* 4* | ||||

HF/2 |
(b) |
F/2 |

Figure 10.2 Sandwich beam under (a) four-point bending and (b) three-point bending loaded by flat-bottomed indenters of dimension a. The total load applied to the beam is F in each case; for four-point bending the two inner indenters are spaced a distance s apart. Both the core and face-sheets are treated as isotropic, elastic-plastic solids, with a Young's modulus Ef for the face-sheet and Ec for the core.

The elastic deflection S of the indenters on the top face relative to those on the bottom face is the sum of the flexural and shear deflections (Allen, 1969),

Ft3 Ft

48 (El)eq 4 (AG)eq for a four-point bend. Here, the equivalent flexural rigidity (EI)eq is Efbtd2 Efbt3 Ecbc3

_ Efbtd2 2

and (AG)eq, the equivalent shear rigidity, is:

c in terms of the shear modulus Gc of the core, the cross-sectional area A of the core, and the spacing d = c + t of the mid-planes of the face-sheets. The longitudinal bending stresses in the face and core are (Allen, 1969)

eq where M is the moment at the cross-section of interest and y is the distance from the neutral axis. The maximum moment is given by

for three-point bending, and by

for four-point bending (here s is the spacing of the inner load-points, Figure 10.2).

### 10.2 The strength of sandwich beams

In the design of sandwich beams, the strength is important as well as the stiffness. Consider again the sandwich beams under four-point bending and under three-point bending, as sketched in Figure 10.2. Simple analytical formulae can be derived by idealizing the foam core and solid face-sheets by rigid, ideally plastic solids of uniaxial strength of and oc, respectively.

### Face yield

When the skins of a sandwich panel or beam are made from a material of low yield strength, face yield determines the limit load Ffy. The simplest approach is to assume that plastic collapse occurs when the face sheets attain the yield strength oyf while the core yields simultaneously at a stress level of oy. For both three- and four-point bending, the collapse load is determined by equating the maximum bending moment within the sandwich panel to the plastic collapse moment of the section, giving

for three-point bending, and

for four-point bending.

These relations can be simplified by neglecting the contribution of the core to the plastic collapse moment, given by the second term on the right-hand sides of equations (10.9) and (10.10). On labeling this simplified estimate of the collapse load by F / v we find that

Ffy t c oy

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