# How do I calculate how many newtons a hollow rod can carry without buckling?

how can I calculate how many newtons a 600[mm] long rod with a known yield strength (300 [MPa]), young modulus (190 [GPa]) and hollow (outer diameter = 16[mm], inner diameter = 10[mm]) will carry without buckling?

Thanks.

Edit: My main question is "How do I use the yield strength in the Euler buckling formula? What does it do and how do I determine "Factor Counting for End Conditions" for suspension pushrods?

• quick google search gives : engineeringtoolbox.com/euler-column-formula-d_1813.html Oct 2, 2020 at 19:29
• Hi there. This looks like homework. You need to tell us what you tried and we'll help from there.
– NMech
Oct 2, 2020 at 19:49
• Yes, you are right. This is my first question, my mistake. My main question is "How do I use the yield strength in the Euler buckling formula? What does it do and how do I determine "Factor Counting for End Conditions" for suspension pushrods? Oct 2, 2020 at 20:04
• Yield strength has nothing to do with Euler buckling, except that if the yield strength is less than the axial compressive stress from the buckling load, the rod will yield before it buckles. Oct 2, 2020 at 21:03
• @alephzero ... and that if the yield stress is less than the maximum (wrt position in the rod and wrt choice of axes) shear stress in the bent configuration, the rod will yield immediately after it buckles. Oct 4, 2020 at 14:59

For Euler Buckling the critical load is:

$$P_{cr} = \left(\frac{\pi}{K L}\right)^2 E I$$ where:

• $$P_{cr}$$ is the critical load
• $$E$$ is Young's modulus (for steel assume 200 [GPa])
• $$I$$ is the second moment of area. ($$\frac{\pi}{64}(d_o^4- d_i^4)$$
• $$d_o, d_i$$ is the outer and inner diameter respectively
• $$L$$ is the length of the rod
• $$K$$: is a parameter that depends of how each end of the beam is constrained. K is the most difficult to determine. For your case, would be 1.

Additionally, regarding the question of the yield stress $$\sigma_y$$. The way you use it, is that you need to check whether, the $$P_{cr}$$ is greater than the $$\sigma_y\cdot A=\sigma_y\cdot \frac{\pi}{4}\cdot(d_o^2 - d_i^2)$$. If it is greater then yield occurs earlier than buckling. A few things missing here:

We need to determine the young's modulus (yield strength is irrelevant).

Construct two circles of radius x and y and get the area moment with x^4 - y^4 times pi/2.

Assume column effective length is 1.

Then plug those into the above formula and you'll get the force your column will sustain without buckling. Factor counting isnt needed.

Note that once you do this for one rod you can linearly extrapolate based on the radius values and modulus for other rods.