This Demonstration shows some results of the Timoshenko beam theory: the deformation and slope of the centerline of a beam; the angular displacement of its cross sections; the internal moments and shear across the length of the beam with respect to specified load and end conditions. The primary purpose of this Demonstration is to provide images of beam deformation to help students understand the predictions of beam theory.

For the Timoshenko beam theory, the constitutive relations for moment and shear are:

. (1)

Here is the Young modulus, is the second moment of area of the cross section and is the shear modulus. Based on a slice of the beam, rewrite equations for the balance of linear momentum (2) and angular momentum (3) to give:

, (2)

. (3)

Substituting the constitutive relations for moment and shear and applying the nondimensionalizations from (4) gives the partial differential equations (5):

,

,

, (4)

, . (5)

For the static case, set the temporal derivatives (denoted by overdots) to 0; the code solves the resulting partial differential equation with the end conditions prescribed by the theory.

References

[1] S. M. Han, H. Benaroya and T. Wei, "Dynamics of Transversely Vibrating Beams Using Four Engineering Theories," Journal of Sound and Vibration, 225(5), 1999 pp. 935–988. doi:10.1006/jsvi.1999.2257.

[2] S. P. Timoshenko, "On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars," The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Series 6, 41(245), 1921 pp. 744–746. doi:10.1080/14786442108636264.