Here is your test model set up for explicit analysis:
Here is what I've done to make it work:
- I moved all the distributed masses defined in the elements (beams) in EdgeMass commands. They will produce the same result since the beam elements use a lumped mass apporach. But in this way, it's easier for me to explain the topic of translational and rotational masses.
- I added, for each translational mass you defined, the respective rotational mass. This is necessary in explicit dynamics because:
- If you use a mass-proportional damping your system matrix will be diagonal and thus singular at rotational dofs without mass
- Even if you use a mass- and stiffness-proportional damping, the time step decreases as the mass in one dof decreases, therfore a 0 mass in rotational DOFs will always lead to instabilities.
- How did I compute rotational masses?
First, you don't need to compute the exact value, you just need the order of magnitude to make sure you do not input un-realistic values. So I used the same values you used for translational masses scaled by a factor of 1.0e-3, because that is the order of magnitude of your beam's moment of intertia.
If you want to be more precise (not necessary as they do not influence the results), you can take the value in the EdgeMass (M=rho*Area), and compute the MR=rho*Iyy (Izz or Ixx) - Finally I set up the explicit analysis with the HHTexplicit_TP with alpha=1, so it is equal to the Central Difference, but it works also with element removal.
The critial time step is about L/c, where L is the characteristic length of your elements, and c is the wave speed.
- timestep.png (48.88 KiB) Viewed 1343 times
However using this value your analysis will soon go unstable. In fact, the equations for computing the critical time step are approximate, they assume elasticity, and they do not account for some special element formulations.
For example, your E changes during the analysis as your section go into nonlinear range, and the characteristic length of a cubic beam formulation (like the disp beam column in opensees) is less then the real length.
For this reason I lower it to 1.0e-6.
In fact, a common rule is that, for structural problems, the order of magnitude of the time step is of the order of micro-seconds!