Section 4.6 Solid Mechanics Part I Kelly1364.6 The Elementary Beam Theory In this section, problems involving long and slender beams are addressed. As with pressure vessels, the geometry of the beam, and the specific type of loading which will be considered, allows for approximations to be made to the full three-dimensional linear elastic stress-strain relations. 4.6.1 The Beam The term beamhas a very specific meaning in engineering mechanics: it is a component that is designed to support transverse loads, that is, loads that act perpendicular to the longitudinal axis of the beam, Fig. 4.6.1. The beam supports the load by bendingonly. Other mechanisms, for example twisting of the beam, are not allowed for in this theory. Figure 4.6.1: A supported beam loaded by a force and a distribution of pressure It is convenient to show a two-dimensional cross-section of the three-dimensional beam together with the beam cross section, as in Fig. 4.6.1. The cross section of this beam happens to be rectangular but it can be any of many possible shapes. It will assumed that the beam has a longitudinal plane of symmetry, with the cross section symmetric about this plane, as shown in Fig. 4.6.2. Further, it will be assumed that the loading and supports are also symmetric about this plane. With these conditions, the beam has no tendency to twist and will undergo bending only1. Figure 4.6.2: The longitudinal plane of symmetry of a beam Imagine now that the beam consists of many fibres aligned longitudinally, as in Fig. 4.6.3. When the beam is bent by the action of downward transverse loads, the fibres near the top of the beam contract in length whereas the fibres near the bottom of the beam extend. Somewhere in between, there will be a plane where the fibres do not change length. This is called the neutral surface. The intersection of the longitudinal plane of symmetry and the neutral surface is called the axis of the beam, and the deformed axis is called the deflection curve. 1certain special cases, where there is nota plane of symmetry for geometry and/or loading, can lead also to bending with no twist, but these are not considered here longitudinal plane of symmetry roller support pin support applied force applied pressure cross section
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