How do I calculate the bending energy for a cross-linked beam network?

[jjyoung]

Hello, I'm working on a project involving a cross-linked network of beam-like objects. We take these networks and apply extensional and shear strains to it, then apply an energy minimization procedure to find the optimal positions for all the cross-links in the new state. Therefore I have to calculate the total energy of the system. Right now, my computation includes only extensional/compression energy so kx^2/2, but I need to add bending energy. And I'm stuck here. I know my beams are supposed to be Bernoulli beams. The bending energy (as I understand it) is the bending moment:
M = EI *(u'') (second derivative of the deflection). In order to compure this I need the defection. To get this, I need to solve the Bernoulli equation:
EI*(u'''') = w where u'''' is the fourth order deriv of deflection and w is the force.
I'm not sure what my "force" should be. Basically at a crosslink, the two beams are attached together at a certain angle. If after the whole network has moved, this angle has changed between the two beams, then a force has been imposed there because the beams are no longer at their optimal angle.

So: 1. How do I calculate this force? is it just as simple as the difference in the angle times a spring constant? or something like this

2. Once I have the force, I have to solve the Bernoulli equation to get the deflection, so being a fourth order equation I need four boundary conditions. 2 at each end of the beam, so in this situation these wouldn't be "clamped" right? Because technically they can move, even though its not very energetically favorable for them to do so. If it's not clamped, then any suggestions on what the boundary conditions should be in this situation? Because the ends are certainly not free either...

Any suggestions would be most helpful. If this is the incorrect section to post this under, I apologize!

 

[nvn]

jjyoung: Is your beam network coplanar, with all displacements coplanar? Or is this a 3-D network? The transverse applied load w on each beam is zero. The end boundary conditions (BCs) of each beam depend on the response of the entire network. Therefore, you will need to run a finite element analysis program to obtain the translational and rotational displacements of each node. Then you will have the end BCs for each beam.

 

[jjyoung]

It is a 3D calculation. So these beams can translate and rotate in 3D.

So you are saying my Bernoulli equation should be EI*u'''' = 0, and I should be able to get boundary conditions like u and u' for the two ends of each beam, which will give me the four conditions to solve for u

thank you for the advice!

 

[nvn]

Yes, but there will be six boundary conditions per node, twelve for each beam.