# Pretension a truss mechanism (using the direct stiffness method)

Currently, I'm working on a mechanical mechanism where nodes are connected via beams. This is very comparable to planar truss mechanism analysis, but in my case, the deformations are large (relative to the mechanism). Instead of deforming beams, for which Hooke's law is often used, my beams act as (linear) springs, which can have positive or negative stiffness. I'll include an example with my question.

In the end, I want to know what force is needed (in x,y-direction) to have the purple node (N4, see figure) at every point on the grid. The location of N4 is therefore known and independent of stiffnesses/prestresses.

The green beams act as the springs, the yellow nodes are fixed and the blue node is free. The purple node (N4) is displaced over the grid, and the rest of the free nodes are moving accordingly, which is calculated via the well-known $$\mathbf{f} = \mathbf{K}\mathbf{d}$$:

Herein, the unknown displacement of the free nodes will be calculated first. In this example, we've two unknown forces and two unknown displacements, which can be solved with the four equations. When all the displacements are known, the same formula will be used to calculate the forces in all nodes.

I'm interested in the force needed (in x,y-direction) of the node which got the predefined translation. Because the deformations are relatively high, this process is done incrementally. The stiffness matrix $$\mathbf{K}$$ will therefore be changed at every step, corresponding to the new configuration of the mechanism.

Now, I want to include a pre-tension in the mechanism. As trivial as it sounds, I didn't find much information about that in various papers.

From intuition, I would suggest that the total force in the nodes is just a summation of $$\mathbf{f_{ext}} = \mathbf{K} \Delta \mathbf{d}$$ and the global pretension vector: $$\mathbf{f_{int}} = \mathbf{T}^T \mathbf{f_{loc}}$$, so that:

$$\mathbf{f_{ext}} = \mathbf{K} \Delta \mathbf{d} + \mathbf{T}^T \mathbf{f_{loc}}$$ for the first step (the incremental process is out of scope, for now).

However, the global stiffness matrix should be modified by the pretension, since the pretension makes the whole mechanism stiffer. But using the aforementioned reasoning, this is not the case, since the stiffness matrix stays unaffected.

Therefore, I'm wondering if I make any mistake in this reasoning. Any help would be greatly appreciated. In addition to that, any tips about papers/books involving pre-tensioning using the direct stiffness method would be very helpful.

• Do I understand correctly that the pre-tension in some members is due to this deformation analysis that you describe? I assume that some members are in tension and some are in compression after you finish prescribing these displacements? Are all the members in your mechanism (truss?) joined by essentially pin joints? It would be very helpful if you can include a picture of the initial geometry and one after the displacements have been prescribed/calculated. Jun 1 at 18:21
• Thank you for your suggestion, @BillGreene, I've added two figures now. I have the intention to add the pretension before deformation happens. All members are indeed coupled with (frictionless) pin joints. Jun 1 at 19:36