Metal Foams: A Design Guide

Sandwich structures 115
The elastic deflection υ 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),
υ D Fℓ3
48 ⊲EI⊳eq C Fℓ
4 ⊲AG⊳eq for a three-point bend, and
υ D F⊲ℓ s⊳2 ⊲ℓ C 2s⊳
48 ⊲EI⊳ eq
C
F⊲ℓ s⊳
4 ⊲AG⊳ eq
for a four-point bend. Here, the equivalent flexural rigidity ⊲EI⊳eq is
2
3 3
Efbtd Efbt Ecbc
⊲EI⊳eq D C C
2 6 12
³ Efbtd 2
2
and ⊲AG⊳eq, the equivalent shear rigidity, is:
⊲10.1⊳
⊲10.2⊳
⊲10.3⊳
⊲AG⊳eq D bd2
c Gc ³ bcGc
⊲10.4⊳
in terms of the shear modulus Gc of the core, the cross-sectional area A of the
core, and the spacing d D c C t of the mid-planes of the face-sheets.
The longitudinal bending stresses in the face and core are (Allen, 1969)
f MEf
D y ⊲10.5⊳
⊲EI⊳eq c MEc
D y ⊲10.6⊳
⊲EI⊳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
M D Fℓ
⊲10.7⊳
4
for three-point bending, and by
M D
F ⊲ℓ s⊳
4
⊲10.8⊳
for four-point bending (here s is the spacing of the inner load-points,
Figure 10.2).