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VIII.2-1 Cantilever Beams Subjected to End loads (1)

M. M. x 2. x 2. prismatic. D. x 2. x 3. x 2. x 3. P. P. x 1. x 1. x 1. x 1. VIII.2-1 Cantilever Beams Subjected to End loads (1). Pure Bending. prismatic, not necessarily be circular. semi-inverse method:. assume. r b = 0. assume. r b = 0. there is shear force!!.

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VIII.2-1 Cantilever Beams Subjected to End loads (1)

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  1. M M x2 x2 prismatic D x2 x3 x2 x3 P P x1 x1 x1 x1 VIII.2-1 Cantilever Beams Subjected to End loads (1) Pure Bending prismatic, not necessarily be circular semi-inverse method: assume r b = 0 assume r b = 0 there is shear force!!

  2. Beltrami-Michell compatibility equation (r b = 0) x2 x3 P x1 VIII.2-2 Cantilever Beams Subjected to End loads (2) equations of equilibrium: compatibility conditions boundary conditions: strain field • lateral surface

  3. x2 x3 P x1 OK!! at x3= VIII.2-3 Cantilever Beams Subjected to End loads (3) OK!! • both two ends at x3=0, clamped: OK!! If x1-x2 axes are the principal axes.

  4. Beltrami-Michell compatibility equation Introduce a stress function , such that --- equation for stress function !! Equation for stress function ? VIII.2-4 Cantilever Beams Subjected to End loads (4) Solution for s13 & s23 : From Beltrami-Michell compatibility equation From eq. of equilibrium: Eq. of equilibrium is satisfied automatically!! What are f & c ??

  5. Consider D x2 P x1 the location of shear center: VIII.2-5 Cantilever Beams Subjected to End loads (5) c can be eliminated if the loading P induced no torque (i.e. is passing through the shear center). rotation of cross-section about the x3 axis, and is proportional to x3.  torsional deformation

  6. B. C.: on C VIII.2-6 Cantilever Beams Subjected to End loads (6) Summary B.C. on lateral surface (in terms of ) If along the boundary curve C on C --- equation for f (x2) If along the boundary curve C Not consistent with p.8-10 1/30/04 ?? Since constant j does not influence s

  7. If along C D x2 x2 x1 P let x3 P x1 • eigenfunction expansion • superposition + separation of variables VIII.2-7 Example (1) circular cross section boundary curve C:

  8. x2 x1 1.38P/A 1.33P/A 1.23P/A VIII.2-8 Example (2) If n = 0.3 compare: elementary theory

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