lec18 - 2.001 MECHANICS AND MATERIALS I Lecture#20 Prof...

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2.001 - MECHANICS AND MATERIALS I Lecture 11/20/2006 Prof. Carol Livermore Beam Bending Consider a ”slender” (long and thin) beam Q: What happens inside when we bend it? Assume: Cross-section and material properties are constant along the length Symmetric cross-section about x-y plane Pure Bending 1 #20
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What is radius of curvature ( ρ ) when M is applied? If it is in compression on one side and tension on the other, there must be a plain with no strain. This is called the neutral axis. Note: The coordinate system is fixed such that y = 0 is on the neutral axis. Compatibility L 0 onneutralaxis = Undeformed Length L 0 = ρ Δ ϕ L ( y ) = ( ρ y ϕ Change in length in x Δ L L ( y ) L 0 ( ρ y ϕ ρ Δ ϕ Δ ϕy xx = = = = = Original length in x L 0 L 0 ρ Δ ϕ ρ Δ ϕ y xx = ρ Note: This is just a result of compatibility. It is purely geometric. Note: ρ → ∞ : Flat Beam 2
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ρ 0: Very Sharp Curve Define curvature ( κ ) 1 κ = ρ κ 0: Beam is flat κ → ∞ : Beam is highly curved xy = xz = 0 due to symmetry.
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