Metal Foams: A Design Guide

6.4 Failure of beams and panels
(a) Isotropic solids
Design formulae for simple structures 69
The longitudinal (or ‘fiber’) stress, , at a point, y, from the neutral axis of a
uniform beam loaded elastically in bending by a moment, M, is
�
M 1
D D E
y I R
�
1
R0
where I is the second moment of area (Section 6.2), E is Young’s modulus,
R0 is the radius of curvature before applying the moment and R is the radius
after it is applied. The tensile stress in the outer fiber of such a beam is
D Mym
I
where ym is the perpendicular distance from the neutral axis to the outer
surface of the beam. If this stress reaches the yield strength, y, ofthematerial
of the beam, small zones of plasticity appear at the surface (top diagram,
Figure 6.3). The beam is no longer elastic, and, in this sense, has failed. If,
instead, the maximum fiber stress reaches the brittle fracture strength, f (the
‘modulus of rupture’, often shortened to MOR) of the material of the beam,
a crack nucleates at the surface and propagates inwards (second diagram in
Figure 6.3); in this case, the beam has certainly failed. A third criterion for
failure is often important: that the plastic zones penetrate through the section
of the beam, linking to form a plastic hinge (third diagram in Figure 6.3).
The failure moments and failure loads for each of these three types of
failure and for each of several geometries of loading are given in Figure 6.3.
The formulae labeled ONSET refer to the first two failure modes; those labeled
FULL PLASTICITY refer to the third. Two new functions of section shape
are involved. Onset of failure involves the quantity Z D I/ym; full plasticity
involves the quantity H (see Figure 6.3).
(b) Metal foams
The strength of open-cell metal foams scales as ⊲ / s⊳3/2 , that of closedcell
foams has an additional linear term (Table 4.2). When seeking bending
strength at low weight, the material index characterizing performance (see
Appendix) is 3/2
y / (beams) or 1/2
y
/ (panels). Used as beams, foams have
approximately the same index value as the material of which they are made;
as panels, they have a higher one, meaning that, for a given bend strength,
foam panels can be lighter. Clamping metal foams requires special attention:
see Section 6.7.