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L31-BeamDeflections-fall 10

L31-BeamDeflections-fall 10 - EAS 209-Fall 2010 Instructors...

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EAS 209-Fall 2010 Instructors: Christine Human Lecture 31 Beam Deflections We have previously designed beams for strength by insuring that the stresses caused by the applied loads are less than the allowable stresses. Another important criterion for design is to limit the beam deflection. Floor beams in buildings need to be designed to limit deflections, as large deflections may result in cracks (even in the elastic range) and sagging that may make occupants nervous. For design of machines and aircrafts, deflections need to be controlled in order to minimize undesirable vibrations. Today’s Lecture: Develop a general equation for the beam deflection call the deflection curve or elastic curve. Today we will only discuss statically determinate beams. Today’s Homework: 12/20/2010 1 Lecture 31-Beam Deflections
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EAS 209-Fall 2010 Instructors: Christine Human Curvature of Beams in Pure Bending The dominant source of beam deflections is bending Curvature of Beams Under Transverse Loading For cantilever, κ varies linearly with x At the free end, κ A =0 At the support, κ B max =-PL/EI 12/20/2010 2 Lecture 31-Beam Deflections For a beam in pure bending, the deformed shape forms an arc of a circle of radius ρ . The curvature is constant and given by EI M = = ρ κ 1 We will assume the relationship between bending moment and curvature remains valid for general transverse loadings by ignoring shear deformations. Px x M - = ) ( x Instantaneous radius or curvature changes along length EI Px EI x M x - = = ) ( ) ( κ
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EAS 209-Fall 2010 Instructors: Christine Human Equation of Elastic Curve Since for a prismatic beam ( EI =constant), the curvature is proportional to the moment, the curvature diagram will take the same shape as the bending diagram. We can use the curvature to sketch the deflected shape of the beam. 12/20/2010 3 Lecture 31-Beam Deflections Consider the overhanging beam with the applied loads Using equilibrium, we can determine the reactions and bending moment diagram as shown Noting the relationship between
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