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VIII.3-1 Timoshenko Beams (1)

x 3. q. x 3. x 1. x 2. VIII.3-1 Timoshenko Beams (1). In both beam theory, only stress resultants (sum over cross section area) are considered. 3D problems  1D problems !!. Elementary beam theory (Euler-Bernoulli beam theory). Timoshenko beam theory.

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VIII.3-1 Timoshenko Beams (1)

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  1. x3 q x3 x1 x2 VIII.3-1 Timoshenko Beams (1) In both beam theory, only stress resultants (sum over cross section area) are considered. 3D problems  1D problems !! Elementary beam theory (Euler-Bernoulli beam theory) Timoshenko beam theory • A plane normal to the beam axis in the undeformed state remains normal in the deformed state. Assume: r b=0 symmetric axis • A plane normal to the beam axis in the undeformed state remains plane in the deformed state. • All the points on a normal cross-sectional plane have the same transverse displacement. • There is no stretch along the beam axis.  no thickness stretch neglect shear deformation!!

  2. x3 q x3 x1 x2 B.C. VIII.3-2 Timoshenko Beams (2) prismatic beam: n1 = 0:   = equations of equilibrium: symmetric axis strain field: stress field: Ok!! geometry, loading: symmetric w.r.t. x3-axis  s12 odd function of x2

  3. VIII.3-3 Timoshenko Beams (3) Summary: prismatic beam: n1 = 0:   =

  4. Approximations: • Neglect VIII.3-4 Timoshenko Beams (4) • Replace m by 2m •  : shear factor, a correction factor  is used to adjust the approximate theory to agree with the 3D theory. When n = 0.3,  = 0.850 for rectangular cross-section and 0.886 for circular cross-section. Timoshenko beam equation

  5. VIII.3-5 Remarks • Euler-Bernoulli beam theory neglects shear deformation • The Timoshenko beam theory accounts for flexural as well as shear deformation. While the Euler-Bernoulli beam theory accounts only for flexural deformation. • Two B.C.s are required at both ends • either w or Q • either dw/dx1 or M

  6. x3 q x1 L B.C.s: VIII.3-6 Example (1) cross-sectional area A moment of inertia I correction factor

  7. B.C.s: VIII.3-7 Example (2)

  8. VIII.3-8 Example (3)

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