Metal Foams: A Design Guide

6.3 Elastic deflection of beams and panels
(a) Isotropic solids
Design formulae for simple structures 67
When a beam is loaded by a force, F, ormoments,M, the initially straight
axis is deformed into a curve. If the beam is uniform in section and properties,
long in relation to its depth and nowhere stressed beyond the elastic limit, the
deflection, υ, and the angle of rotation, , can be calculated from elastic beam
theory. The differential equation describing the curvature of the beam at a
point x along its length for small strains is
EI d2y D M⊲x⊳
2
dx
where y is the lateral deflection, and M⊲x⊳ is the bending moment at the point
x on the beam. E is Young’s modulus and I is the second moment of area
(Section 6.2). When M is constant, this becomes
� �
M 1 1
D E
I R
R0
where R0 is the radius of curvature before applying the moment and R the
radius after it is applied. Deflections, υ, and rotations, , are found by integrating
these equations along the beam. Equations for the deflection, υ, and
end slope, , of beams, for various common modes of loading are shown
below.
The stiffness of the beam is defined by
S D F
υ
D B1EI
ℓ 3
It depends on Young’s modulus, E, for the material of the beam, on its length,
l, and on the second moment of its section, I. ValuesofB1 are listed below.
(b) Metal foams
The moduli of open-cell metal foams scales as ⊲ / s⊳ 2 , that of closedcell
foams has an additional linear term (Table 4.2). When seeking bending
stiffness at low weight, the material index characterizing performance (see
Appendix) is E 1/2 / (beams) or E 1/3 / (panels (see Figure 6.2)). 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 the foam
panel is potentially lighter for the same bending stiffness. Their performance,
however, is best exploited as cores for sandwich structures (Chapter 10).
Clamping metal foams requires special attention: (see Section 6.7).