Why does a rope have different stiffness than a rod of the same diameter?

[crazie25]

Why would a rope have a lower bending and compressive stiffness that a rod of the same diameter, but would have the same axial tensile stiffness?

 

[Cyrus]

what do you mean by compressive and bending stiffness?

Do you mean the equivalent spring constant?

 

[Pyrrhus]

Why would a rope have a lower bending and compressive stiffness that a rod of the same diameter, but would have the same axial tensile stiffness?

Well a prismatic rod stifness is defined by k = \frac{EA}{L}.

Now let's suppose you have a rod and a rope with the same diameter and length. First notice that for the rope the cross sectional area A is equal to the cross sectional area of the individual wires, which will be less than the cross sectional are of our rod.

Now, the modulus of elasticity of our rope will be less than the modulus of elasticity of the material from which is made, due to the inherent property of the wires squeezing themselves.

Will our rope and rod of the same diameter and length have the same axial stiffness?

Theoretically, it'll depend solely on the rope effective modulus and the rod elasticity's modulus.

 

[Hacksaw]

It's actually really simple, a rope is made up of very small fibers. Imagine taking one fiber of the rope, and pulling on it, then pushing on it or bending it. It's really weak in bending, due to it's small second moment of area [a factor of the area and the geometry], and in compression it buckles easily because of the same factor. In tension however it is pretty strong for its thickness. A bigger rope is made up of many such stands and as such will share their properties.

A rod, made up of a material with the same properties as the rope, will have a similar resistance to tension. However it is MUCH harder to buckle because it has a very large second moment of area, and so it will have a much higher bending and compressive stiffness.

I know this is late for OP but it was the second google result of one of my searches so this might still help someone.