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Failure of Beams and Panels

The longitudinal (or 'fibre') stress s at a point y from the neutral axis of a uniform beam loaded elastically in bending by a moment M is


Figure A3.1

where I is the second moment of area (Section A1), E is Young's modulus, Ro is the radius of curvature before applying the moment and R is the radius after it is applied. The tensile stress in the outer fibre of such a beam is


Figure A3.2

where ym is the perpendicular distance from the neutral axis to the outer surface of the beam. If this stress reaches the yield strength sy of the material of the beam, small zones of plasticity appear at the surface (top diagram, Fig. A3). The beam is no longer elastic and, in this sense, has failed. If, instead, the maximum fibre stress reaches the brittle fracture strength, sf (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 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).

The failure moments and failure loads for each of these three types of failure and for each of several geometries of loading are given on the diagram. The formulae labelled 'onset' refer to the first two failure modes; those l abelled '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 S. Both are listed in Fig. A1 and defined in the text which accompanies it.

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Figure A3 Failure of Beams