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I have a model of a cantilever (fixed on one end, and free on all other boundaries) that bends in response to an internal stress field.

How can I calculate the amount of force this cantilever can exert on an external object?

The experimental analogue, as far as I know, is the blocking force--if you start with the cantilever in a flat state and physically obstruct the bending of the cantilever, how much force must you exert in order to keep the cantilever flattened out?

Additional Info

The model I'm using is a PDE that describes a non-linear, hyperelastic solid. For a given state of deformation, I can calculate the stress tensor and potential energy density throughout the cantilever.

So far, I have been using the finite element method to find the state of deformation in static equilibrium.

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  • $\begingroup$ Find what external force will produce the same deformation. This would be the force that opposes the internal stress field. $\endgroup$ – John Alexiou Jun 2 '16 at 21:55
  • $\begingroup$ @ja72 Since this is a trial-and-error solution, I was planning to use it as a verification method. $\endgroup$ – nnn Jun 3 '16 at 14:31
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Maybe you've tried this, but if you don't require an exact solution, you can simulate the experiment on finite element analysis, by restricting the other end of the beam and solving for the reaction force there. This reaction force is exactly what you want.

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  • $\begingroup$ I would like to do something like this, but I'm not sure how. I gather that if I approximate the stiffness matrix, I can work backwards to calculate the reaction force, but I don't know how to approximate the stiffness matrix in the first place. Would it just be the Jacobian matrix of the potential energy density? $\endgroup$ – nnn Jun 3 '16 at 14:37
  • $\begingroup$ You said you were running a finite element solution to find the state of deformation in static equilibrium. Use the same stiffness matrix, but now add a boundary condition that the displacement in the beam's end has to be zero. That eliminates some more lines in the stiffness matrix. You can solve then to find the displacement of the nodes (only your middle nodes will move now) , and calculate f = K*d to find the resulting force in each node. the resulting force on the restricted nodes will be the reaction force, thus the force the structure would exert on anything restricting the displacement $\endgroup$ – Lucas Franceschi Jun 3 '16 at 17:04
  • $\begingroup$ Right, so I have my PDE in weak form, and an equation for the residual, based on the current trial solution. The software I'm using (FEniCS) does the rest of the work for finite element, so I haven't calculated a stiffness matrix myself. I don't have a clear idea of how to find the stiffness matrix for my case, but perhaps that's worth opening a second question. $\endgroup$ – nnn Jun 5 '16 at 3:28
  • $\begingroup$ Right, you are using a software. In that case just modify your inicial configuration, adding a restriction on the restricted boundary, and run the solution again. The solution should now show you a reaction force on the nodes you restricted, that reaction is the one in question. No need to look to the internals of the calculations, just add a restriction on the nodes on the free end of the beam and run the solution to check the reaction force. $\endgroup$ – Lucas Franceschi Jun 8 '16 at 18:34
  • $\begingroup$ Okay. In my solution, I have the displacements and the stress tensor. So could I get the reaction force by doing a surface integral of dot(stress_tensor, normal_vector) on the fixed surface? (P.S., thanks for your patience so far) $\endgroup$ – nnn Jun 10 '16 at 14:17

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