NTP Equations

From our initial idea that forces propagate along a structure producing stresses in the form of torques, we have built an NTP, a network that has all the information needed to compute the possible paths along which the loads could ``flow'' and the torques they would generate along the way.

For each force F we consider the network of all the joints in the structure as a flow network that will transmit it to the ground. Each joint j can support a certain fraction of such a force, given by the formula

$$\alpha _{j,F}=\max \left\{ 1,\left\vert \frac{K_{j}}{\delta (j,F)\vert\vert F\vert\vert}\right\vert \right\}$$ (2.1)

where K_j is the maximum capacity of the joint, the distance between the line generated by the force vector and the joint, and ||F|| the magnitude of the force. Thus if the torque generated is less than the joint maximum K, then (the joint fully supports F); otherwise is K divided by the torque. The arm of the torque can have a positive or negative sign depending on whether it acts clockwise or counterclockwise.

If one given force F is fixed and each joint on the graph is labeled with the corresponding according to eq. 2.1, a network flow problem (NFP) [27] is obtained where the source is the node to which the force is applied and the sink is the ground. Each joint links two nodes in the network and has a capacity equal to _j,F. A net flow represents a valid distribution of the force F throughout the structure: F can be supported by the structure if there is a solution to the NFP with a net flow of 1.

With more than one force, a solution for the entire network can be described as a set of flows, one for each force, all valued one. But as multiple forces acting on one joint are added, the capacity constraint needs to be enforced globally instead of locally, that is, the combined torques must be equal to or less than the capacity of the joint:

$$\left\vert \sum _{F}\phi _{F}(j)\delta (j,F)\left\Vert F\right\Vert \right\vert \leq K_{j}$$ (2.2)

This problem is not solvable by standard NFP algorithms, due to the multiplicity of the flow (one flow per force) and the magnification of magnitudes due to the torque arm (so the capacity of a joint is different for each load). Equation 2.2 is equivalent to a multicommodity network flow problem [2, ch. 17].