# Most efficient way to build a toothpick bridge

In my stem class we are going to have a competition where we have to build a toothpick bridge that can support as must weight as possible, while using only toothpicks and glue. These are the rules:

• Each toothpick costs \$10,000. • Each end of a toothpick glued to another end of a toothpick costs \$10,000.
• The bridge must span at least 1 foot.
• We can't build beams down to the floor for support.
• The base of the bridge, the part cars would drive on if it were a real bridge, can't have any large gaps in it, although I'm not entirely how much this matters.
• We will have a time limit when building it, but I don't know what it will be. We can bring a hairdryer to help the glue dry faster.
• It must hold a lot of weight, and be cheap. I don't know how much each of those factors matter, but I'd imagine the weight it can hold is much more important.

My teacher wants us to do lots of research before the project and have a clear plan, so I have been looking into what kind of bridge to build. It seems like the obvious choice is a beam bridge, but I could be wrong. I found this site talking about how a truss can help a beam bridge hold more weight, but I don't quite understand how it helps.

If a beam bridge is my best option, would a truss significantly help at this scale? And if so, which kind of truss? The warren looks more efficient as far as costs are concerned, but I don't know if the pratt or any other kind has a major advantage. Any advice on this would be greatly appreciated, thanks!

• This is a pretty broad question so I don't think I'll attempt an answer. I recommend you look into truss design. Your toothpicks will be relatively strong in axial compression. Something like a Warren Truss maybe since it utilized equilateral triangles? – William S. Godfrey- S.E. Feb 20 '16 at 13:59

As you are spanning 10 feet and the length of the toothpicks is small in comparison you have quite a lot of options in terms of the design of the structure you use. In fact one approach would be to design the bridge as if it was a monolithic material and construct that as a mesh of toothpicks as if it wa a finite element analysis model.

At some point you will want to refine your specification to prioritise between cost and load bearing capacity. In this case cost and weight are pretty much the same considerations you have a fixed price for materials (and presumably the design and construction are 'free'). As strong as possible and as cheap as possible is an open ended problem and at some point you need to pick a specific target to aim for.

Given the materials you have to work with a lot depends on the properties of the glue joints. In particular, how flexible are they and how strong are they relatively in compression and tension.

My first thought would be a girder arch bridge as this gives you an efficient overall structure which can be constructed from relatively short members. One example is the Ponte dell'Accademia in venice and looking at the early industrial and pre-industrial wooden bridges is likely to be a useful line of research.

Another key consideration is how your 'foundations' are situated you will notice that building from bank to bank across a river required different designs to crossing a gorge or valley. Similarly if you don't need to have a completely flat road deck you have a lot more options.

Update :

Having clarified that the span is 1 foot not 10 foot (which I probably should have questioned in the first place). Then a box girder construction does become quite a sensible approach. In terms of detailed design it is really a case of optimisation and as you have no control over the basic materials (toothpicks and glue) then it is a question of generating a design which meets the strength requirements using the fewest toothpicks. By extension this means looking at how the design will be evaluated, for example will the 'best' bridge be the one which holds the most weight or the cheapest one which holds a specific minimum weight.

A lot of real world engineering ultimately comes down to this ability to properly evaluate and define a problem and find a good solution ie the ability to determine exactly what problem you are trying to solve and how success is determined.

For example saying that it must carry 20lbs of weight and be as cheap as possible is a well defined problem as is cost no more than $500,000 and carry as much weight as possible. • This answer seems to ignore the cost placed on glued joints by assuming that cost only scales with the number of toothpicks. That strikes me as a pretty major flaw. As I understand the problem statement, a connection between three members would represent a cost of \$30,000 for toothpicks and an additional \$30,000 for the glued joint. – Air Apr 27 '16 at 20:06 You didn't specify cost of glue. If the glue is of no cost (other than end to end joints as you mentioned) then my inclination is to think outside the box here. You could shred the toothpicks into fibers (pull them apart, grind them up, whatever) and then use the glue to create a composite matrix using the fibers. You could get a lot of stringy fibers from very few toothpicks. I don't know all the rules, but If you can bring stuff in other than just a hairdryer, then I would bring in a food saver type vacuum bagger and some 10 or 20 lbs foam, and some Vaseline. Carve a negative form for the bridge into the foam and coat it with vaseline (as a mold release agent). Mix up the wood fibers and glue and spread this mixture out into the form. Slip the whole thing into a food saver bag and turn on the vacuum pump to apply pressure while it cures. Use the hair dryer to apply heat over the whole thing to help cure while it's in the bag. Finally, once your WFRG (wood-fiber-reinforced-glue) bridge is cured you would remove the whole thing from the bag, and pop the bridge out of the mold. Done. I didn't see time limit listed in your question, but I would imagine all of this would be way faster than building a more traditional bridge gluing individual toothpicks together, so I assume the time allotment would be sufficient. Again, depending on the rules about bringing stuff in, you might be able to make the mold ahead of time and bring it with you, so all you have to do during the construction phase is shred wood, mix with glue, apply, and cure. Something I learned at an early age is it's better to ask forgiveness than ask permission. If the rules don't specifically ban you from bringing in this outside stuff then I would say it's fair game. I believe this fiber-reinforced-matrix construction could work well. However I've never done it myself, so I'm not positive. It's similar in concept to fiberglass or carbon fiber, GRFC, or FRP, and conceptually would work the same way and offer the same benefits. Traditionally, fiber-reinforced products work because the fibers are suitably strong (overall, not individually) to resist great tensile loads, while the matrix binder (in this case the glue) provides adequate support to hold the fibers in place. As it's your homework assignment, I would do your research on FRP's (fiber reinforced plastics) and other composite materials to determine whether or not this will indeed work for you. An H-Beam or I-Beam form I think would prove suitable. You could also look at adding gussets and ribbing within your foam mold to increase strength and rigidity at areas of stress concentration. Looking at the rules you've written, I don't see how this would violate any of them. If you have access to 3d modeling software and FEA software at your school, you could do an optimization study to give you a good idea of how to build the bridge and where to add gussets to assure structural integrity. Many FEA suites offer the ability to apply composite materials. Also you can look at the way many plastic playsets and slides are constructed. These are typically roto-molded plastic, not composites, but the beam construction and ribbing could give you some hints as to the approach to take. Finally, You're mileage may vary, but I'll tell you this - if I was a teacher, who did this project every year and watched countless students create countless iterations of the same basic bridge design using girder construction, I would be pretty impressed if someone brought in a completely fresh approach to the problem. This attitude of thinking differently has served me well in my career, it may or may not work for you. Just a thought. • To the downvoter, could I ask why the downvote? – CBRF23 Apr 27 '16 at 19:57 • The cost of glue seems to be reflected by, "Each end of a toothpick glued to another end of a toothpick costs$10,000." In any case, there is no way pulping and reconstituting toothpicks is going to be feasible in a timed classroom competition. – Air Apr 27 '16 at 19:57
• @Air - I would challenge you to this. I think I could shred toothpicks and create a matrix faster than you could glue individual ones together. Have you ever participated in a contest like this? I have - it's time consuming. Lots of time to shave toothpicks into hairy fibers with a razorblade or similar tool. – CBRF23 Apr 27 '16 at 20:17
• @air - to me that says nothing of the cost of glue. End-to-end is a very specific joint. For example, a t-joint, where one toothpick was glued to the middle of another, would not be end to end in my book, and would not be subject to the \$10,000 cost penalty. In my answer, not a single toothpick is glued end-to-end, so the only cost would be however many toothpicks you had to shred. My gut tells me this would be significantly less than the number you would use assembling a bridge with individual sticks, as I'd get a pretty good size pile of fibers from a single toothpick. – CBRF23 Apr 27 '16 at 20:18
• However going with what was said above, instead of foam for mold and vaseline as release agent, you could built sheets/panels on a flat piece of wax paper. When cured, you could then build up panels into stronger shapes. – Forward Ed Jun 24 '16 at 4:13