1. Sorry, I don't fully understand the first paragraph. What dose "the beam is taking all the rotation" mean ?

I also confuse that the moment in the beam is correct and it is not in equilibrium with the momemnt on the zerolength.

The zero-length elements and the beam are connected in series. For equilibrium they should have the same bending moment (bottom-end of the beam should have the same moment as the bottom hinge, and the top-end of the beam should have the same moment as the top hinge).

However in a nonlinear analysis you try to reach equilibrium convergence by iterations... And the solvers "thinks" that it found equilibrium when the error is smaller than the tolerance you provided in the test command.

But what happens if your tolerance is too high? OpenSees will accept an error that is not negligible, and so the equilibrium is not fullfilled. That's why in your previous (wrong) model, the beam had a correct momemnt, while the zero-length element had a much smaller momement.

2. The reason that the yield rotation in my hinge is so small(The correct value is about 0.03.) is I used a too large value for the initial stiffness of the hinge. Is it right?

Yes, of course the yield rotation is the yield momement / K0. In the image you provided, also the ETABS hinge has a very small yield rotation. Instead a value of 0.03 radians is reached only when the M-R curve drops to a residual value.

Don't be confused however! keep in mind that the ETABS is plotting Moment vs Plastic Rotaiton (not full rotation). That's why the initial stiffness is so high and the yield rotation is almost zero. In theory they should be K0 = inf, YieldPlasticRot = 0. But this is not possible due to numerical errors, so you use a large value as K0, and an almost 0 value for yield rotation.

But this is fine, because the zero-length element has a length = 0, so it should represent only the plastic part, since the elastic rotation is taken by the beam.

3. What does the component of S11 or S23 in Gauss point plot mean? What does the number 1 or 2 mean ?

That is just a side-effect of the fact that STKO and OpenSees are two distinct applications with different conventions. The zero-length element, when all 6 DOFs are used, gives an output called "stress". And it contains the following results:

- Fx
- Fy
- Fz
- Mx
- My
- Mz

However STKO assumes that a result called "stress" with 6 components is a 3D Stress Tensor in Voigt notation:

- S11
- S22
- S33
- S12
- S23
- S13

But you know that it is not a stress tensor but just the forces and moments of a 6-DOF spring, so the 5th (S23) and 6th (S13) components are actually My and Mz

4. If we analyze the plastic hinge, it will be better to choose the following setting. Is it right?

algorithm → Krylov-Newton

Test→normUnbalance

Because I don't really understand different kinds of algorithm and test and what is the best choice for us when we analyze

No, it is true only in this specific problem. The choise of the analysis settings depends on the problem type. This is quite complex to explain here. However our next free e-learning course will deal exactly with these kind of issues. I highly suggest you to attend it.

Here is the link to our webinar page:

https://asdeasoft.net/?stkowebinars-for-opensees
If you want I can take this one as an example and explain in details the steps I took to solve it.