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Unformatted text preview: Mechanics & Materials 1 Mechanics & Materials 1 Chapter 12 Chapter 12 Bending Bending Bending Bending Assumptions for Analysis Assumptions for Analysis a) Transverse planes before bending remain transverse after bending, i.e. No warping. b) Beam material is homogeneous and isotropic and obeys Hooke's law with E the same in tension or compression. c) The beam is straight and has constant or slightly tapered cross section. d) Loads do not cause twisting or buckling. This is satisfied if the loading plane coincides with the section's symmetry axis. e) Applied load is pure bending moment. f) The definition for beams with applied positive and negative bending moments are as follows: Positive and Negative Moments Positive and Negative Moments Beams Deformation Geometry Beams Deformation Geometry • When a beam is subjected to a pure bending moment, it will deform into a curved shape and this shape is the arc of a circle with a very large radius compared to the size of the beam. • The fibers on the top surface are experiencing a compressive stresses, and those on the bottom a tensile stress. • Thus, at some point between these two surfaces, there must be a plane where the stresses and strains are ZERO. This is the 'Neutral Plane' (NP) or Neutral Axis (NA). Beams Deformation Geometry Beams Deformation Geometry Beams Deformation Geometry Beams Deformation Geometry Mark a section a distance y from the Neutral Axis as ij, and another section on the Neutral Axis as mn. These sections are of equal length as they define the length between two transverse planes. • The applied bending moment causes the segment ij and mn to deform into concentric arcs i 1 j 1 and m 1 n 1 with an angle d θ between the segments i 1 m 1 and j 1 n 1 . The distance between these two arcs is still y . • The strain of segment i 1 j 1 is defined as length i 1 j 1 minus the original length ij over the original length, such that: • Now length mn and ij are defined as: • The length i 1 j 1 is defined as: ij ij j i xx = 1 1 ε θ Rd ij mn = = θ d y R j i ) ( 1 1 = Beam Deformation Geometry Beam Deformation Geometry Stress Stress  Strain in Bending Strain in Bending • So the strain becomes • Which indicates that the strain is linearly varying with y. Note that this equation is relative to the neutral axis. It also doesn’t relate to the loading. And since stress is strain times Young's Modulus then the stress can be defined by the following equation and is also linearly varying with y. R y Rd Rd d y R xx = = θ θ θ ε ) ( R y E = σ • The drawing of the right hand end of the beam showing the stress distribution and applied bending moment. • Let σ xx dA be the component of force acting on the element of area dA....
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 Spring '09
 Schwarz
 Second moment of area, neutral axis, neutral plane

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