Boundary condition, Node moving on a curve

[janoskias]

Hey guys,

I'm new in FEM and stuck with the following problem (TLTR: question is if 2B is theoretically correct?, I am grateful for thoughts about the other ones (1 and 2A)):

I have a (geometric) non-linear 2D analysis (lets say a pulled rope) and I want to use TRUSS elements.

-----------------------------------------
Problem:
Let's consider a very simple case A |>o------o B, single TRUSS element, where A is fixed and B is moving.
I want B to move along a prescribed curve lets say c(x) = [x, f(x)].

I am not interested in the dynamics, only the equilibrium.
I struggle with the appropriate boundary condition. I have basically 2 ideas in my mind.
------------------------------------------

1. Instead of using the conventional truss element stiffness matrix, derive one with different shape function, which matches with c(x). I haven't tried it yet, and I want my trusses only carry axial loads,
it might also bring some non axial loads in? And it also should be recalculated for each curve.


2. Use the conventional stiffness matrix for the truss. (Geometric stiffness is also calculated and incremental force is added, c is linearized around B_x0, so c(x) = (x, n*x))
Then my calculation basically becomes:

K*b = F_B​

The A corresponding elements disappear, because A is fixed, so I have [k11,k12; k21,k22]*[bx; by] = [f_Bx; f_By] (1)
Possibilities that come in my mind:

2A
Prescribe a condition for f_Bx/f_By = 1/n, this is not so good, because the B coordinates also modify the force components, so I need more iteration
​

2B
Modify K before inverting. If I write (1), and use that k12 = k21 then after a short derivation, if k12 = (f_By*k11-n*f_Bx*k22)/(f_Bx-n*f_By) then the solution for b will be [x, nx], so my condition is basically filled. But I don't know if my approach is right, or if there is a standard method for a problem like that.
​

Thanks in advance.

 

[FEA way]

In FEA software we achieve such complex motion by breaking it into time-dependent boundary conditions with amplitude (various types: tabular, smooth step, periodic and so on) for each DOF. Maybe you could follow the same approach here. At least for comparison with other solutions.

 

[rb1957]

I would model as a roller contact on a track and run large displacement.

The other approach might be graphical ... like what we've done before computers. You know the path you want the free end to follow, a track. The reaction is normal to the track. If the other end is pinned and the applied load is a point vector (maybe a sum of a distributed load) then the beam is a three force member and you can draw a force polygon (in this case, a triangle). Now, whilst you don't know the two reaction forces, this triangle ensures force equilibrium, so check moment equilibrium to find the solution.

another day in paradise, or is paradise one day closer ?

 

[janoskias]

Hey,

thanks for the answers.

@FEA way, do you have perhaps some links to online docs or something where I could read more about them?

@rb1957, it is a good idea. Am I interpreting right, that the geometric stiffness matrix does something similar? And I am just wondering, that the procedure I wrote in 2B does not accidentally do the same?

Anyways, great insights, thanks!