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So it is taught that arches are good to support structure and most bridges are build out of arches, so I am curious to know how we can determine the internal forces of such complex shapes. Is it only done by numerical methods such as FEA or there are analytical equations that help prove it.
Can you point to any famous analytical equations?
That is a good question.
The mechanical wisdom of arches manifested in the way the Loads flow, or directed, through the structure. If it is designed correctly, there are only compression loads all along the arch. In this case, when you design such a bridge, you most probably deal with the stability of the structure. I.e. studying its sensitivity to modifications in the estimated load distribution - which may come up with internal loads having moments or tensile components.
Please note that each load distribution has it own unique shape that "idealize" the internal loads (for a specific geometry like the span and the height of the bridge).
Designing arches and bridge is an ancient art, it has been established way before FEA was even a wild dream. One of the tricks of "seeing" it is done by a rope. a rope is quite stiff when you try to stretch it, But will deform easily in case you apply compression forces or try to bend it (apply a moment). Take a rope, pin it on both ends and hang weights all along. The rope will deform in a way it "feels" only tension. Put only one weight in the middle of the rope and you will get a triangle. Arrange a uniform distributed weights all along the rope and you will get the classical hyperbolic shape.
If you arrange the weights in a way that represents the loads your upcoming bridge shall be subjected to - you can snapshot the rope shape, flip it, and here is your bridge. Now, since your bridge shape is inverted - It will "feel" only compression.
