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In engineering a rigid structure is something that is analyzed as if none of the components deform, that is bend, compress or stretch. For example when analyzing a bridge made out of riveted beams to see how much force is on the joints, the assumption is often made that when there's a load on it, it doesn't deform, that is all the beams stay the same length and don't bend. This isn't exactly true in real life, but it's a good enough approximation that the force on the joints can be pretty accurately calculated.

The reason this approximation is used is that it would be extremely difficult to calculate the exact deformations and for things like bridges the margin for error in doing the design is very large. The errors caused by the rigid assumption are much smaller than the margin for safety engineered into the structure.

For designing things like a bungee jumping tower, treating the cord as a rigid structure is probably not a very good idea.

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