lec2 - Euler-Bernoulli Beams: Bending, Buckling, and...

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Unformatted text preview: Euler-Bernoulli Beams: Bending, Buckling, and Vibration David M. Parks 2.002 Mechanics and Materials II Department of Mechanical Engineering MIT February 9, 2004 Linear Elastic Beam Theory • Basics of beams –Geometry of deformation –Equilibrium of “slices” –Constitutive equations •Applications: –Cantilever beam deflection –Buckling of beams under axial compression –Vibration of beams Beam Theory: Slice Equilibrium Relations Axial force balance : • q(x): distributed load/length • N(x): axial force • V(x): shear force • M(x): bending moment Transverse force balance : Moment balance about ‘x+dx’: Euler-Bernoulli Beam Theory: Displacement, strain, and stress distributions Beam theory assumptions on spatial variation of displacement components: Axial strain distribution in beam: 1-D stress/strain relation: Stress distribution in terms of Displacement field: y Axial strain varies linearly Through-thickness at section ‘x’ ε ε- κ h/2 ε xx (y) ε + κ h/2 Slice Equilibrium:...
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lec2 - Euler-Bernoulli Beams: Bending, Buckling, and...

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