Where is the Geometry Defined in the EBT and Timoshenko PDE's

[bugatti79]

Folks,

To date I have been reading about Euler Bernoulli Beam and Timoshenko Beam Theory desribed by the following equations respectively

EBT $\displaystyle \frac{d^2}{dx^2}\left( EI \frac{d^2 w}{dx^2}\right )+c_fw=q(x)$

Timoshenko $\displaystyle -\frac{d}{dx} \left[GAK_s \left(\Psi+\frac{dw}{dx}\right)\right]+c_fw=q$ and $\displaystyle - \frac{d}{dx} \left(EI \frac{d \Psi}{dx}\right)+GAK_s \left(\Psi+\frac{dw}{dx}\right)=0$

These expressions seem to be for straight beams. Where in the above PDE's is the geometry of the beam defined?

For example, if one wants to analyse a quadrant of a ring say (from $\pi/2$ to $\pi$) where it is constrained at $\pi$ position and a point load applied at $\pi/2$ position...

How is the PDE formulated?

 

[AlephZero]

These expressions seem to be for straight beams.

Correct.

Where in the above PDE's is the geometry of the beam defined?

The basic assumptions about the deflected shape of the beam, and the relation between the strains and the geometry when it deflects, all assume the beam was straight.

For a curved beam, you have to go back to first principles and reformulate the equations of motion. In general this is complicated, because (unlike straight beams) the axial, torsional, and bending of the beam are all coupled together by the curvature. Googling "curved beam equations of motion" gives lots of hits.

As a simple example of why it gets complcated, the "neutral axis" of a curved beam is not at is geometrical mid point, because that assumption would mean there was more material on the "outside" of the curve than on the "inside". But if the position of the neutral axis depends on the radius of curvature, you have to make some approximations if the curvature suddenly changes (e.g. two straight sections of beam are joined by a circular arc.)

In practice, you can often approximate a curved beam as an set of short straight beams joined end to end, to get a numerical solution (e.g. from a finite element model).

 

[Studiot]

As a simple example of why it gets complcated, the "neutral axis" of a curved beam is not at is geometrical mid point,

Just to clarify this means that:

The neutral surface does not, in general, coincide with the centroid of a section as it does with a straight beam.
Even with the usual assumption that plane sections remain plane after straining the stress strain relationship is non-linear (non hookean) as it is with a straight beam where the stress is assumed proportional to the distance from the neutral surface.

 

[bugatti79]

Correct.


The basic assumptions about the deflected shape of the beam, and the relation between the strains and the geometry when it deflects, all assume the beam was straight.

For a curved beam, you have to go back to first principles and reformulate the equations of motion. In general this is complicated, because (unlike straight beams) the axial, torsional, and bending of the beam are all coupled together by the curvature. Googling "curved beam equations of motion" gives lots of hits.

As a simple example of why it gets complcated, the "neutral axis" of a curved beam is not at is geometrical mid point, because that assumption would mean there was more material on the "outside" of the curve than on the "inside". But if the position of the neutral axis depends on the radius of curvature, you have to make some approximations if the curvature suddenly changes (e.g. two straight sections of beam are joined by a circular arc.)

In practice, you can often approximate a curved beam as an set of short straight beams joined end to end, to get a numerical solution (e.g. from a finite element model).

Just to clarify this means that:

The neutral surface does not, in general, coincide with the centroid of a section as it does with a straight beam.
Even with the usual assumption that plane sections remain plane after straining the stress strain relationship is non-linear (non hookean) as it is with a straight beam where the stress is assumed proportional to the distance from the neutral surface.

Thanks Guys

Can you recommend some books or online sources on how

1) the EBT and Timoshenko BT PDE's are formulated from first principles?
2) Good Introductory books on the principle of virtual work (minimum potential energy)

I would like to study the deformation of rings under load using theoretical/FE methods, so any recommendations to this end would be appreciated... Thanks