I am currently planning an Eagle Project in which I am building a pergola with a porch swing attached to one of the top pieces. In doing so I need to figure out if I will need a bigger piece of wood or alter the design in order make the swing usable. I've been searching the internet for about half an hour now and haven't been able to figure out a way to calculate how much weight it will be able to handle. I am hoping there is an equation that I am able to use in order to calculate the maximum load based on the type of wood length between the two chains, and with the wood being supported on both ends. Any help is greatly appreciated. Thank you!
In evaluating if the members in an assembly will be adequate, it is important to consider both the individual pieces, the connections and geometry of the entire project together to determine if it is adequate. Perhaps the word 'breaking' is not the best term to use, but rather 'failure' works better. The project won't necessarily fail because of a single piece suddenly breaks. Rather it will most likely experience excessive movement, bending or joint failure over a period of time due to unusual loading or repetitive stresses. Structural engineers study for years to understand the strength of materials, the distribution of forces, the capabilities of connections and put it all together in a complete design.
To directly answer your question, evaluating the strength of an individual board is quite inexact. There are plenty of websites that can help you determine the design capabilities of structural framing members (do a search for wood span calculators). Note that these have built in safety margins based upon industry standards. They do not tell you when the board will break, but rather what sizes are necessary to limit deflection (sagging) of the board. Woodworkers on this site often reference the 'sagulator' to evaluate the strength for smaller, non structural, project elements.
@Graphus entered a comment for your question which I believe offers the best approach, 'Don't reinvent the wheel'. Design your project using the examples that are all around us. I have never seen a woodworking project that does not borrow from the countless examples from the past. If you want to construct a porch swing, look at the many project available on the internet. Seek out pergolas in your area and evaluate the size and spans of the framing. If you can, test them out to see how they work for your application.
As you evolve your project design from other examples you may have specific questions. Please feel free to post new questions here to get focused answers.
In woodworking, as in Civil Engineering, the simplest solutions are often the best.
Go ahead and build your design. Then apply successively larger weights, until it fails. At that point, you will know how much weight it can support. Go back and re-build your design, or re-design your project, as appropriate!