Hollow Tube Stronger than Solid bar of same Outside Diameter (O.D.)?

 https://physics.stackexchange.com/questions/12913/hollow-tube-stronger-than-solid-bar-of-same-outside-diameter-o-d

 I do not think there is much difference is strengh between hollow bars versus solid bars. Hollow bars are just mor rigid for the same weight. This is good for a car and has additional benefits in allowing the hollow spaces to be used for other things such as wiring conduits. 

 

http://www.physicsforums.com/archive/index.php/t-37701.html says

"Most of the strength of a cylinder comes from the outer portions. I think the contribution goes like the cube of the radial position. So, if you took a solid rod and drilled out a half the volume from the center, you do not lose half the strength. Strength to weight ratio is better for a hollow pipe than a solid rod."

The definition for the second moment of inertia Ic

for a filled and hollow cylinder can be found on http://en.wikipedia.org/wiki/Second_moment_of_area:

Ic=∫∫Ay2dxdy=∫R0∫2π0r2dϕ rdr=πr44

The surface area of the filled cylinder is:

A=πr2

Compare filled and hollow cylinder of equal mass:

sc=IcA=r24

, cylinder with fractional internal radius ri=xro and x<1:

sh=IhA=r4(1−x4)4r2(1−x2)=r2(1−x2)(1+x2)4(1−x2)=r2(1+x2)4>sc

.

This means a hollow cylinder is stronger than a rod of equal mass and the same material. A hollow cylinder with a bigger inside diameter is better. In the limit x→1

the hollow cylinder is twice as strong. Note that this limit isn't physically viable as it would be an cylinder with infinite radius and infinitesimally thin wall. However it is useful to define the upper limit of the second moment of inertia. I didn't expect the increase in strength only a factor of two.