Joints in two dimensions

We began considering two-dimensional structures, assuming that all the bricks are of width 1, assembled in a plane. A fulcrum effect, which is the angular torque exerted over two joined bricks, constitutes the principal cause for the breakage of a stressed structure of Lego bricks. We designed our model around this idea, describing the system of static forces inside a complex structure of Lego bricks as a network of rotational joints located at each union between brick pairs and subject to loads (fig. 2.2).

**Figure 2.1:** *Fulcrum effect*: a $2\times 1$ union resists more than twice the load of a $1\times 1$ because the second knob is farther away from the axis of rotation.
$\resizebox*{0.6\textwidth}{!}{\includegraphics{lego/graph/fulcrum3.eps}}$

Table 2.1: Estimated minimal torque capacities of the basic types of joints. Note: these values correct the ones on [48, table 1].

Joint size ( $\omega$ )	Approximate torque capacity ( $\kappa _{\omega }$ )
knobs	N-m $\times 10^{-3}$
1	12.7
2	61.5
3	109.8
4	192.7
5	345.0
6	424.0

Bricks joined by just one knob resist only a small amount of torque; bigger unions are stronger. The resistance of the joint depends on the number of knobs involved. We measured the minimum amount of stress that different linear ( $1\times 1$ , $2\times 1$ , $3\times 1$ , etc.) unions of brick pairs support (table 2.1).

From a structure formed by a combination of bricks, our model builds a network with joints of different capacities, according to the table. Each idealized joint is placed at the center of the area of contact between every pair of bricks. A margin of safety, set to 20% in our experiments, is discounted from the resistances of all joints in the structure, to ensure robustness in the model's predictions.

All forces acting in the structure have to be in equilibrium for it to be static. Each brick generates, by its weight, a gravitational force acting downwards. There may be other forces generated by external loads.

Each force has a site of application in one brick -- each brick's weight is a force applied to itself; external forces also ``enter'' the structure through one brick -- and has to be canceled by one or more reaction forces for that brick to be stable. Reaction forces can come from any of the joints that connect it to neighbor bricks. But the brick exerting a reaction force becomes unstable and has to be stabilized in turn by a reaction from a third brick. The load seems to ``flow'' from one brick to the other. Thus by the action-reaction principle, a load is propagated through the network until finally absorbed by a fixed body, the ``ground''.

**Figure 2.2:** Model of a 2D Lego structure showing the brick outlines (rectangles), centers of mass (circles), joints (diagonal lines, with axis located at the star), and ``ground'' where the structure is attached (shaded area). The thickness of the joint's lines is proportional to the strength of the joint. A distribution of forces was calculated: highly stressed joints are shown in light color, whereas those more relaxed are darker. Note that the x and y axis are in different scales.
$\resizebox*{0.95\textwidth}{!}{\includegraphics{lego/graph/bridsce2.eps}}$

The principle of propagation of forces described, combined with the limitations imposed to each individual joint, generates a set of equations (section 2.2.6). A solution means that there is a way to distribute all the forces along the structure. This is the principle of our simulator: as long as there is a way to distribute the weights among the network of bricks such that no joint is stressed beyond its maximum capacity, the structure will not break.