# Boundary condition: displacement

I have a controversial case. I have a rod, which is fixed from one end (constraint). From another end, I apply a compressive force, by pressing the rod down. So in a way I have a constraint, but at the same time an external force. How do I model it? Do I give a boundary condition (displacement = 0) from both ends, or just from one?

• Controversial ? – Qmechanic Sep 21 '18 at 17:34
• Well, to me it is, as I don't understand how it could be constraint and have a displacement at the same time. Now I understand that it is not a constraint in terms of displacement. – Ekaterina Wiktorski Sep 21 '18 at 21:40

You have different types of boundary conditions:

• Dirichlet boundary conditions. Which specify the value of the solution to the differential equation. In this case you are solving for displacements, so this kind of BC is the one that you used to constraining the fixed end of the rod.
• Neumann boundary conditions. These specify the value of the derivative of your solution is going to take. In this case the derivative of the displacements is related with the strain tensor, which is related to the appliance of external forces, which is what you are doing at the end of the rod.
• Robin boundary conditions. I dont know much about these...so according to wikipedia: When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the boundary of the domain.

So: Do I give a boundary condition (displacement = 0) from both ends, or just from one?

(for an axially loaded bar) Displacement u = 0 just in one end, and σ(L) = 0, meaning EA du/dx at (x=L) = 0, at the other (L being the length of the bar).

• Thank you for the answer. I am solving a system of equations for nodal displacements and reaction forces. From what I understand now, the node where I apply the compressive force is not really the constraint, right? – Ekaterina Wiktorski Sep 21 '18 at 14:49
• @EkaterinaWiktorski That's right. Both are boundary conditions but 'constraint' usually refers to Dirichlet boundary conditions (specially in some FEM codes). – user190081 Sep 21 '18 at 14:59

You have a boundary condition of zero displacement at the truly restrained end (assuming that the restrained end is a totally rigid wall). The rod is elastic (like a very stiff spring), so there will be a displacement at the end where you supply the force. The displacement will vary linearly from 0 at the restrained end to $$u_{max}$$ at the end where you apply the displacement. The strain in the rod will be uniform at $$u_{max}/L$$, where L is the original length of the rod, and the force will be $$F=EA\frac{u_{max}}{L}$$, where A is the cross sectional area of the rod and E is the Young's modulus of the rod metal.

• Thank you, it has become clear now. Actually, there is another thing I am wondering about (since you mentioned a totally rigid wall). I calculate the natural frequencies of my rod (using M and K matrix as eig(M^(-1)*K)), and draw the modal shapes. The 1st natural frequency is 79 Hz. Then I do an experiment: apply an initial displacement and let it go to measure the dynamic force/displacement using load cells. They I apply FFT, and I see a different frequency, 30 Hz. Could it be because my string is in reality not constrained that well at one end or are there other reasons? – Ekaterina Wiktorski Sep 21 '18 at 14:45
• Are you now not talking about a rod, but rather plucking a string? – Chet Miller Sep 21 '18 at 14:51
• It is actually a hollow aluminum pipe of 93 cm + 40 cm of steel on the bottom (where I apply force), diameter is around 1 cm, a big bigger for the steel part. The load cell is mounted on top of a flat plate to which the pipe is attached. The pipe can be rotated by the way. – Ekaterina Wiktorski Sep 21 '18 at 15:01
• I am unable to picture your model and your experiment from your description. – Chet Miller Sep 21 '18 at 15:17
• Should I describe it in more details or I could even send a picture if you are interested? – Ekaterina Wiktorski Sep 21 '18 at 15:23