I am interested in modeling a Timoshenko beam: https://en.wikipedia.org/wiki/Timoshenko%E2%80%93Ehrenfest_beam_theory

However, I would also like to add tension to the beam so that it can vibrate like a string as well as a beam.

The Timoshenko beam is described at that link by two coupled equations, where subscripts are derivatives of that variable:

$ρA w_{tt} = κAG(w_{xx}- φ_x) + q\tag{1}$

$ρIφ_{tt} = EIφ_{xx} + κAG(w_x- φ)\tag{2}$

Where $ρ$ is density, $I$ is second moment of area, $E$ is Young's Modulus, $G$ is the shear modulus, $κ$ is the Timoshenko shear coefficient, $A$ is cross sectional area, $w$ is transverse displacement, $φ$ is rotation of a segment, and $q$ is external force/load.

To solve a unified equation of motion, they isolate $φ_x$ from equation (1), take the $x$ derivative of equation (2), and then substitute in various derivatives of the $φ_x$ from equation (1) into (2).

This also helps to eliminate $φ$ which would be hard or impossible to solve for in a finite difference approach.

This gives a final equation of motion of:

$EIw_{xxxx} + ρA w_{tt} - (ρI + \frac{ρAEI}{κAG})w_{xxtt} + \frac{ρAρI}{κAG}w_{tttt} = q + \frac{ρI}{κAG}q_{tt} - \frac{EI}{κAG}q_{xx}\tag{3}$

Now let's imagine a very simple model for a vibrating string:

$ρA w_{tt} = (T + σA)w_{xx}\tag{4}$

where $T$ is basic tension and $σ$ is the extra stress on a given segment from lengthening (strain).

How could I combine these two systems? My idea is to just add these terms into equation (1) so I'd have:

$ρA w_{tt} = κAG(w_{xx}- φ_x) + q + (T + σA)w_{xx}\tag{5}$

Then I could just combine (5) with (2) to create a new version of (3).

But I'm wondering if this makes sense. Wouldn't these tension terms also affect equation (2)? Ie. Wouldn't I have to add something to equation (2) also?

Thanks for any ideas.

  • $\begingroup$ You need to learn enough nonlinear continuum mechanics to understand the idea that "total stiffness = elastic stiffness + stress stiffness + load stiffness". (You only need the case of infinitesimal strain plus arbitrary sized rigid body rotation.) The wiki page gives an equation in the section on "axial effects" but there is no attempt to explain where it came from. $\endgroup$
    – alephzero
    May 16 at 12:40
  • $\begingroup$ You will have to add the axial term to the external work on the beam to derive the combined equation. Examine the derivation of the Timoshenko beam equation for clues. It will have the form $\int_a^b f \delta u_0 dx$. $\endgroup$ May 17 at 6:05

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