Linear_Elasticity_06_Beam_Theory

Linear_Elasticity_06_Beam_Theory - Section 4.6 4.6 The...

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Section 4.6 Solid Mechanics Part I Kelly 136 4.6 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 beam has 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 bending only . 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 only 1 . 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 . 1 certain special cases, where there is not a 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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Section 4.6 Solid Mechanics Part I Kelly 137 Figure 4.6.3: the neutral surface of a beam A conventional coordinate system is attached to the beam in Fig. 4.6.3. The x axis coincides with the (longitudinal) axis of the beam, the y axis is in the transverse direction and the longitudinal plane of symmetry is in the y x plane, also called the plane of bending . 4.6.2 Moments and Forces in a Beam Normal and shear stresses act over any cross section of a beam, as shown in Fig. 4.6.4. The normal and shear stresses acting on each side of the cross section are equal and opposite for equilibrium, Fig. 4.6.4b. The normal stresses σ will vary over a section during bending. Referring again to Fig. 4.6.3, over one part of the section the stress will
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 1 taught by Professor Staff during the Fall '11 term at Auckland.

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Linear_Elasticity_06_Beam_Theory - Section 4.6 4.6 The...

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