Talk:Timoshenko–Ehrenfest beam theory

Discussion on last equation
Something isn't right with the final equation on this page. The condition



\frac{HEI}{L^3 A \kappa G} \ll 0 $$ doesn't make any sense, as it cannot be a negative quantity. By dimensional analysis, I think it should be



\frac{EI}{L A \kappa G} \ll 1 $$. —The preceding unsigned comment was added by 128.83.39.182 (talk • contribs).

Timoshenko shear coefficient
The Timoshenko shear coefficient κ is within the range [0.5, 1].

For rectangular section beam, κ = 5/6 (instead of 6/5).

Moreover, for circular section beam, κ = 9/10.

Comments

The shear coefficient is dependent to the Poisson's Ratio. For rectangular cross-section,

\kappa = \cfrac{10(1+\nu)}{12+11\nu} $$

For circular cross-section,

\kappa = \cfrac{6(1+\nu)}{7+6\nu} $$

Guanshi Li (talk) 19:00, 1 August 2011 (UTC)
 * Can you add this information into the body of the article? Also, can you check the equations for damped vibrations?  It'll also be nice to see an example in the article to be able to compare the Timoshenko solution with the Euler-Bernoulli solution for an equivalent case. Bbanerje (talk) 22:04, 1 August 2011 (UTC)

Axial Effects and Damping
I am expanding this equation to include the effects of constant axial forces and damping. By including the axial force the equation results:

EI~\cfrac{\partial^4 w}{\partial x^4} + N~\cfrac{\partial^2 w}{\partial x^2} + m~\frac{\partial^2 w}{\partial t^2} - \left(J+\cfrac{mEI}{\kappa AG}\right)~\cfrac{\partial^4 w}{\partial x^2 \partial t^2} + \cfrac{mJ}{\kappa AG}~\cfrac{\partial^4 w}{\partial t^4} = q + \cfrac{J}{\kappa AG}~\frac{\partial^2 q}{\partial t^2} - \cfrac{EI}{\kappa A G}~\frac{\partial^2 q}{\partial x^2}\quad\square $$

Damping is a bit more complicated. I assumed the damping force proportional to the velocity with the form:



\eta(x)~\cfrac{\partial w}{\partial x} $$

and after following the same process explained in the same article this is what we get:

EI~\cfrac{\partial^4 w}{\partial x^4} + N~\cfrac{\partial^2 w}{\partial x^2} + m~\frac{\partial^2 w}{\partial t^2} - \left(J+\cfrac{mEI}{\kappa AG}\right)~\cfrac{\partial^4 w}{\partial x^2 \partial t^2} + \cfrac{mJ}{\kappa AG}~\cfrac{\partial^4 w}{\partial t^4} - \cfrac{J \eta(x)}{\kappa AG}~\cfrac{\partial^3 w}{\partial t^3}+EI~\cfrac{\partial^2}{\partial x^2}\left(\eta(x)\cfrac{\partial w}{\partial t}\right) - \eta(x)\cfrac{\partial w}{\partial t} = q + \cfrac{J}{\kappa AG}~\frac{\partial^2 q}{\partial t^2} - \cfrac{EI}{\kappa A G}~\frac{\partial^2 q}{\partial x^2}\quad\square $$

This equations can be obtained by following the same procedure as the main article but introducing the following changes in the original equations:



m \frac{\partial^{2}w}{\partial t^{2}} + \eta(x)~\cfrac{\partial w}{\partial x} = \frac{\partial}{\partial x}\left[ \kappa AG \left(\frac{\partial w}{\partial x}-\varphi\right)\right] + q(x,t) $$



J \frac{\partial^{2}\varphi}{\partial t^{2}} = N\frac{\partial w}{\partial x} + \frac{\partial}{\partial x}\left(EI\frac{\partial \varphi}{\partial x}\right)+\kappa AG\left(\frac{\partial w}{\partial x}-\varphi\right) $$

Armonmar (talk) 03:44, 20 April 2010 (UTC)


 * Thanks for the addition Armonmar. Please add it into the main article if you're confident that it's correct.  A reference that can be checked will also be useful.  Also, please sign your comments with four tildes so that the date of the comment can be easily identified. Bbanerje (talk) 21:59, 11 April 2010 (UTC)

Comments
The damping force is supposed to be proportional to velocity, which is first derivative of time t, rather than position x.

\eta(x)~\cfrac{\partial w}{\partial t} $$ It can be verified by conducting dimension analysis.

Guanshi Li (talk) 18:52, 1 August 2011 (UTC)

Quasistatic Timoshenko beam
I believe the relation between first derivative of curvature, actual rotation and shear strain is incorrect and should be as below

\frac{\mathrm{d} w}{\mathrm{d} x} = \varphi + \frac{1}{\kappa AG} \frac{\mathrm{d}}{\mathrm{d} x}\left(EI\frac{\mathrm{d} \varphi}{\mathrm{d} x}\right) $$

It should be "plus" rather than "minus", because the deflection will be larger due to the shear strain. However, I went through the derivation but didn't find the problem. Anyone can check with this?

Guanshi Li (talk) 23:53, 5 September 2011 (UTC)
 * The sign depends on the choice of coordinate system (whether it's left handed or right handed). Bbanerje (talk) 21:56, 6 September 2011 (UTC)

The illustration is wrong
The illustration titled "Deformation of a Timoshenko beam (blue) compared with that of an Euler-Bernoulli beam (red)." is wrong.

In this drawing, the red outline (Euler-Bernoulli, perpendicular section remains perpendicular after bending) is correct.

However, Timoshenko's theory taking into account the longitudinal shear of a beam, the blue outline should be on the other side: The top fibre of the beam is longer in Timoshenko's theory than in Euler-Bernoulli theory, not shorter. The same applies in reverse to the bottom fibre.

The illustration should be changed, because it is incomprehensible as it is now.

2406:5600:76:29E1:6042:719F:C3A9:F44E (talk) 06:18, 31 July 2015 (UTC) 2406:5600:76:29E1:6042:719F:C3A9:F44E (talk) 06:18, 31 July 2015 (UTC)
 * From the figure, I don't believe that the actual loading of the beam was provided, or supposed to be provided. The main point of that figure is to illustrate that Timoshenko beam theory, unlike Euler beam theory, does support shear deformation.  That's what has been depicted on the figure.  As no loading was actually presented, there is no objective way to tell if the shear deformation should go one way or the other. Maybe the shear force induces a ccw rotation, and maybe the positive moment applied to the end of the beam induces a cw rotation.  Who knows.  Most importantly, who cares?  This article is about Timoshenko beam theory, and the figure clearly represents the main difference between Timoshenko and Euler beam theory.  That's what really  matters. -- Mecanismo | Talk 00:14, 2 August 2015 (UTC)

Physically, really?
"Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, while the result is a larger deflection under a static load and lower predicted eigenfrequencies for a given set of boundary conditions." Don't you mean mathematically? Using a different model does nothing to a real beam. 131.180.113.238 (talk) 14:37, 18 January 2016 (UTC)

Dubious caption
The caption of the third diagram is not very good. It seems to be equating "slope" with "angle".Lathamibird (talk) 03:08, 5 February 2018 (UTC)