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I am working on project involving designing a recoil spring coil of diameter 10.2 mm and wire diameter 1.25 mm. I tried using a solid core wire but it failed due inadequate tensile strength. Dimensions can only be altered by 5 mm, which is not sufficient to make the solid core wire work.

My professor suggested looking into braided wire rather than a solid core wire. I considered 6x7 wire rope but they are too big. After some googling, I found out there are recoil springs with braided wires with 3 strands. I know it is going to be somewhat less than 3x the strength individual wires. I searched handbooks, reference books, etc. but I could not find any established relations/formulae between strands and tensile strength. How can I determine the tensile strength of a braided steel wire based on the number of strands?

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First off, the tensile strength is a property of the material, not the spring itself. Strength, often denoted as $\sigma = F/A$, can be considered how much force a material can withstand for a given cross-section. If you want something to be able to withstand more force the simplest approach is to increase the cross-section or get a stronger material.

Second, I think you are mistaking the purpose of braided wire. Generally braiding wires doesn't increase their ability to withstand forces. Assuming the net diameter of the braided wire is equal to the diameter of a solid one, the cross-sectional area of the solid cable will be larger, suggesting that it can withstand more force. What braiding does do is make the wire more resistant to wear.

Think of it this way: once a crack forms in a solid wire, it can spread across the whole thing, resulting in failure. If a crack forms in a single strand of wire in a braided wire, then only the individual strand will fail, leaving the other strands to take up the load. Keep in mind this doesn't prevent the braided wire from failing, it just gives you a bit more use (and visible signs of damage) before the wire fails completely.

Check out this page about stranded recoil springs. The author says the stranded springs have an improved lifetime in applications with high impact stresses and velocities. The author also claims there are currently no accurate programs for calculating the stresses on braided wire.

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This is because the strands are continually curved and thus the stress distribution inside of them is higher near the centerline (higher curvature) than on the outside.

In addition, the contact forces add stresses in the off-axis directions which raise the internal stress level beyond $\sigma = \frac{T}{A}$.

The math is quite complex, but not unwieldy. Stranded cables offer superior strength for their weight compared to a solid wire of the same overall diameter.

PS. The curvature of a wire whose centerline prescribes a circle of $r$ and has a helix with pitch (distance/angle) of $p$ is $$\rho = \frac{r^2+p^2}{r}$$

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