A bit of thought, or modelling with a computer-aided design program, shows that the bridge can be reduced to a simple triangle. The force required to break a well-constructed bridge is orders of magnitude greater than any other forces acting on it, such as its own weight, the weight of the toy car, “wind load” etc.
This is not the case for a real bridge, of course, which must be designed for a variety of vehicle loads, wind loading, snow or ice buildup, earthquakes and so on. Also, because of the power law (mass increases as the cube of the size, while strength increases as the square of the size), small structures are much much stronger than their full-size counterparts. A bit of simple physics (or CAD software) will put numbers to the forces. Simple analysis treats the sides as rigid bars, and the corners as free pivot points. One lower corner is fixed to the support, while the other is allowed to slide. The base of the triangle is in tension, while the sides are in compression. The higher the triangle, the less tension in the base. The limiting case for an infinitely high triangle is zero tension in the base, and half the test weight in compression in each side. If the triangle is made lower, the forces increase. In the limit of a zero-height triangle, they become infinite.
Forces in Simple Triangle: 200kg weight on apex
Stresses in trianglular element, from “Felt” software. Red is under tension, blue is in compression.
So the optimal shape to minimize the forces on the bridge is an infinitely high triangle. Two problems – we have only 100 sticks, and the test jig is less than 40cm high.
The bridge is contructed of two compression elements and one tension element. A bit of experiment reveals that failure of a tension element is typically due to shearing of an overlap joint, while failure of a compression element is typically due to buckling.
Design of a tension element for the base is relatively simple – a series of sticks overlapped a suitable amount performs well. Design of a compressive element is more difficult. The element must resist buckling, and must be designed so that the stress is distributed evenly across the individual sticks. This may be acheived in part by careful assembly – the element should be perfectly straight, and all the sticks should align exactly at the ends so that they all touch the supports. In real life, elements are often created with a complex cross-section in order to resist buckling. Three of the most common shapes are the I-beam, box section, and tube. Most real-world structures are made of these shapes.
For the stick bridge, the requirement to not cut sticks makes it difficult to create these common sections, though it is possible (though not the tube, of course). Instead, stiff elements may be made by laminating together pairs of sticks. This also guards against weakness in individual sticks – depending on the supplier, some sticks may have grain diagonally across the stick. Since wood will split along the grain, this makes them much weaker. In this case, pairs of sticks should be laminated so that the grains cross each other.
When these designs are tested, providing the joints are well made and sufficiently overlapped, the element will typically fail by buckling. Once the element starts to buckle, failure is progressively more rapid. As the sticks depart from perfect alignment, the inside of the curve becomes more stressed than the outside, taking the inside sticks beyond their breaking strength. The joint may become delaminated, a stick may split along the grain, or a stick break across the grain. To prevent buckling, it is necessary to make the element stiffer.
This can be done by making it thicker, but the finite number of sticks puts a limit on this. Another technique that may be used is the stayed mast, borrowed from sailboat design. In a sailboat, there are one or more masts (shown below on its side) which are under compression and subject to sideways force from the sail (this force can be many tons in strong winds). To stiffen the mast, steel cables are used together with “spreaders” to convert bending in the mast into tension in the cables which is more easily resisted.
This concept may be used in the stick bridge, to resist bending of the compressive members by staying them against the bottom tension member. This idea is shown in the third design.
Typically, bridge elements are built first, then glued together to make two or more trusses, The trusses are then joined with cross members, and finally the paper deck is glued on. Since at each step the glue must dry, it is important to allow enough time for all the steps. At least 3 days is required, and typically much more. When glueing elements, better results will be obtained if the sticks are clamped while the glue dries. Since you want to glue many elements at the same time, you need a lot of clamps. Fortunately, good spring clamps can be obtained at a “dollar store”. For single joins, clothes pegs may be used. For laminating, pieces of thick metal or wood and steel G-clamps allow many pairs of sticks to be laminated at once. Pairs of sticks may be arranged in two layers between the metal plates to give e.g. 24 pairs in 2 layers. It is important to make sure the sticks are exactly aligned and do not slip when pressure is applied.
It is important that the final elements should be exactly straight, or they will buckle. This means they must be glued together against a straight edge such as a long piece of wood. Elements must be measured carefully and overlaps glued to bring them to the designed length. For final assembly, a setsquare should be used to make sure that the bridge is exactly vertical and that the top load-bearing elements are exactly flat and horizontal. Any deviation – one stick protruding slightly, for instance – will concentrate stress under load and be a point of failure. Since sticks cannot be cut, any small errors in alignment may be corrected by adding glue.
The load-bearing points at the bottom corners and apex can be set up on flat metal plates (which the glue won’t stick to) and glue added to build up the round end of the sticks to give a flat bearing surface. The bridge should be constructed to spread the load equally to all elements. Just thinking about it helps – imagine what happens when the weight is applied, and each stick starts pushing on the next to transfer the load to the base. Are there any sticks that aren’t doing anything ? Any sticks that are doing more than their fair share of work ?
Testing your design is a good idea – it helps eliminate poor designs early before you have spent too much time on them. Also, it’s fun. The Richmond APEG test jig uses a car jack, cable and springs to pull evenly on the load plate, with an electronic load cell to measure the force. My test jig uses a set of bathroom scales and two threaded rods. Pieces of 2×4 are used for the cross-pieces. The upper crosspiece had to be reinforced with a metal plate as sticks would be driven into the soft wood when testing joints in pairs of sticks. Force is applied by turning the nuts on the screwed rods with a pair of wrenches.
Courtney from Study Moose
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