1. Institute of Applied Mechanics8-0 VIII.3-1 Timoshenko Beams (1) Elementary beam theory (Euler-Bernoulli beam theory) Timoshenko beam theory 1.A plane normal to the beam axis in the undeformed state remains plane in the deformed state. 2.All the points on a normal cross-sectional plane have the same transverse displacement. 3.There is no stretch along the beam axis.  no thickness stretch 4.A plane normal to the beam axis in the undeformed state remains normal in the deformed state. neglect shear deformation!! In both beam theory, only stress resultants (sum over cross section area) are considered. 3D problems  1D problems !! q x3x3 x1x1 Assume:  b  0 symmetric axis x3x3 x2x2

2. Institute of Applied Mechanics8-1 VIII.3-2 Timoshenko Beams (2) x3x3 x2x2 x3x3 q x1x1 strain field: stress field: equations of equilibrium: B.C. geometry, loading: symmetric w.r.t. x 3 -axis  symmetric axis prismatic beam: n 1 = 0:   = Ok!!  12 odd function of x 2

3. Institute of Applied Mechanics8-2 VIII.3-3 Timoshenko Beams (3) prismatic beam: n 1 = 0:   = Summary:

4. Institute of Applied Mechanics8-3 VIII.3-4 Timoshenko Beams (4)  is used to adjust the approximate theory to agree with the 3D theory. When = 0.3,  = 0.850 for rectangular cross-section and 0.886 for circular cross-section. Approximations: 1. Neglect 2. Replace  by  2   : shear factor, a correction factor Timoshenko beam equation

5. Institute of Applied Mechanics8-4 VIII.3-5 Remarks 1.Euler-Bernoulli beam theory neglects shear deformation 2.The Timoshenko beam theory accounts for flexural as well as shear deformation. While the Euler-Bernoulli beam theory accounts only for flexural deformation. 3.Two B.C.s are required at both ends either w or Q either dw/dx 1 or M

6. Institute of Applied Mechanics8-5 VIII.3-6 Example (1) q L cross-sectional area A moment of inertia I correction factor B.C.s: x3x3 x1x1

7. Institute of Applied Mechanics8-6 VIII.3-7 Example (2) B.C.s:

8. Institute of Applied Mechanics8-7 VIII.3-8 Example (3)