Buckling column with varying moment of inertia

[XYZpu]

I am trying to predict the failure of a axially loaded rod and clevis. The clevis' moment is much larger than the rod. Assuming the rod's moment across the length provides an overly conservative conclusion. Simplified, the system is a long column with halves that have different moments of inertia. How can I predict the critical load?

Pcr--> =====----- <---Pcr

 

[FFIELD]

Buckeling load for a column with a verying radius of gyration is a very difficult problem. Although there are theoretical ways to do this I guess, I have never seen any. Might try load testing an actual specimen. Compression on round bars is not a very good idea anyways as once the load does not coincide with the centerline of the rod your theoretical strength goes out the window.

 

[VoyageofDiscovery]

Hi XYZpu,

See Timoshenko's &quot;Theory of Elastic Stability&quot;, it has all you need.

 

[jhardy1]

Try the perennial favourite “Formulas for Stress and Strain” by Roark and Young. Table 34 (in my 5th edition) has numerous formulae for cantilever columns, fixed-ended, pin-ended etc, with varying moments of inertia.

 

[ishvaaag]

What follows is a way of finding the anelastic buckling load manually.

There are as said closed form formulae (or charts) that solve the elastic buckling of members like yours.

To get the critical load accounting the true material behaviour you will need to introduce as well the effect of material nonlinearity, which is usually portraited as a function of concurrent axial stress.

For a purely elastic case you would enter a chart with Ithin/Ibigger in a chart by Pfluger to get a Fi factor that multiplied by EIbigger/L^2 would give the buckling load.

To get the anelastic buckling load for the same, you need to enter with (Ethin·Ithin)/(Ebigger·Ibigger), then multiply by EIbigger/L^2 to get the anelastic buckling load.

However, both Ethin and Ebigger are functions of a surmised standing stress -you can use CEC formulation for the reduction of the bending stiffness on concurrent axial stress-, or if you want, of a surmised standing (testing) axial load. This means that you will need to chart for a number of test loads. Its maximum not exceeding the squashing stress of the thinner part is the anelastic buckling load you are searching for.

As you see from this process, for any surmised standing axial load there is a statement of an anelastic critical load. As long as within the anelastic realm, this measures with the standing load what its corresponding buckling load is, or if you want, the safety factor to buckling failure. As you grow your standing axial load, the necessary reductions in axial stiffnesses provoke that the standing load is more and more closer to its limit anelastic buckling load. If what you are searching is an absolute maximum for the load placeable in the column, by following such iterative progression of the stated standing load it will get to coincide with its corresponding limit anelastic buckling load, which would be the absolute one for the member.

This makes to note how, if anelastic buckling is what controls the true safety factor to buckling is actually more when scarcely loaded than its proportion to such limit anelastic load (established this way) would show. This because the above named lesser reductions (if any) in bending stiffnesses at such axial stresses.

 

[ishvaaag]

Of course when considering any case for the Pfluger chart all the inputs to the chart and values in the formulation must be consistent to the level of axial compression present, if considering anelasticity.

 

[ChipB]

You could try and treat it as a stepped column. AISC published some info on these in their engineering journal a while back (30 years ago)....