64 Failure of beams and panels

(a) Isotropic solids

The longitudinal (or 'fiber') stress, a, at a point, y, from the neutral axis of a uniform beam loaded elastically in bending by a moment, M, is o _M _ /I 1

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

where ym is the perpendicular distance from the neutral axis to the outer surface of the beam. If this stress reaches the yield strength, ay, of the material 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, af (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 = I/ym; full plasticity involves the quantity H (see Figure 6.3).

(b) Metal foams

The strength of open-cell metal foams scales as (p/ps)3/2, that of closed-cell foams has an additional linear term (Table 4.2). When seeking bending strength at low weight, the material index characterizing performance (see Appendix) is a^2/p (beams) or aly/2/p (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.

Plasticity t


Plastic hinge f





y y y y

y y








y y y y

y iv/


(Full plasticity)

mf = Failure moment (Nm)

sf = Modulus of rupture (N/m2)

s* = oy (Plastic material) = of (Brittle material)

Figure 6.3 Failure of beams and panels

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