# Can the strength of a strut be determined by deflections?

I am interesting in replacing a solid strut with tubing, but I do not want to destructively test the strut to determine its yield point. I do not want to make assumptions about the yield point based on material, because heat treatment and other factors can affect the yield point.

Can I cantilever suspend the strut and record its deflection due to a weight placed on the end, to infer the strength of the strut?

In other words lets say the strut deflects 5 inches when it is cantilevered and a weight is placed on its end. The tubing under test is placed the same way and only deflects 4 inches. Can I assume that the tubing will be an adequate replacement for the strut (ignoring factors like fatique, age microcracking, etc etc)?

I can do math that would suggest the replacement tubing will be significant lighter and stronger than the bar strut, but I am trying to find a way to actually physically test it.

• I was writing an answer when a doubt crossed my mind: what do you mean by "heat treatment"? Do you mean a history of relevant temperature fluctuations (which tend to weaken steel) or actual treatments such as quenching and tempering (which tend to strengthen it in some way)? – Wasabi Jan 21 '16 at 19:24
• Heat treatment is deliberate exposure of steel to particular temperatures for particular amounts of time in order to change the chemistry and structure of the steel, and thereby its mechanical properties. – Wallace Park Jan 21 '16 at 19:35
• @WallacePark, heat treat usually doesn't effect the strength of low or mild carbon steels too much. – grfrazee Jan 21 '16 at 21:13

The other answers say that this isn't possible in the way that you want to do it, but they don't provide the equations that show why.

### Deflection of a Cantilever

The equation for the deflection of a cantilever beam:

$$\delta = \frac{FL^3}{3EI}$$

Where: $F=Force \\ L= \text{Length of cantilever} \\ E=\text{Modulus of Elasticity} \\ I=\text{Moment of Inertial}$

As you can see, none of these components have anything to do with strength. The only variable that is affected by the material is $\text{E}$. This is basically constant for steels.

### Column Strength

You didn't necessarily ask about the material strength of your tube though. You asked about replacing a strut. A strut is a column. The compressive capacity of a column is ultimately controlled by Euler Buckling (hint: it rhymes with "oiler").

$$F=\frac{\pi^2EI}{(KL)^2}$$

Where: $\text{K is dependent on the end connections} \\ \text{(well, see above, everything else is the same)}$

This also shows that buckling capacity isn't dependent on material strength. Assuming that the original solid bar is the same material as your replacement tube, there is only one parameter to compare $\text{I}$, (Moment of Inertia).

Take the diameter of your original solid bar and use that to determine the the required diameter and thickness of the wall so that the moments of inertia match.

### What if the materials are not the same?

It the materials are not the same, then the E value will change. Everything else in the equations stays the same, so EI will need to stay the same. The moment of inertia may go up or down depending on how the modulus of elasticity changes.

This ends up meaning that a stiffer strut will have a greater EI. This also contributes to a higher buckling force. If buckling is the controlling failure mode, then a strut that deflects less will be stronger.

Note: Insert standard warnings about there being lots of other factors that could affect this. Don't use this for things related to human life or anything that you care about.

The deflection will tell you the rigidity of the material but this doesn't vary much for steels and so it doesn't tell you anything about the yield stress.

It's fairly unlikely that something like a solid strut has been heat treated.

Also given that it's solid it's unlikely to be anything other than mild steel, because as you have identified, if the designer was sufficiently concerned about its strength to use a high strength steel then they would also have used tube.

What size of tube would be an equivalent replacement depends a lot on loading. For something mostly under bending stress a tube will save a lot of weight, for something in pure tension there is no real advantage to using tube.

The above is a bit of a generalisation but if it is an application sufficiently critical that a solid, high strength steel strut is required then I would leave it alone if I were you.

• if the designer was sufficiently concerned about its strength to use a high strength steel then they would also have used tube. You lost me here. Why would they use a tube if they needed high-strength steel? They make high-strength plain bars that are cheaper and much easier to weld. – grfrazee Jan 21 '16 at 21:15

Can I cantilever suspend the strut and record its deflection due to a weight placed on the end, to infer the strength of the strut?

Steels all have (more or less) the same modulus of elasticity. Therefore, two different steel samples of the same cross-section but with differing yield strengths will have essentially the same deflection when loaded. So, no, you can't use deflection to infer the material strength of the steel.

Another thing to consider is that struts typically only resist compressive or tensile forces. Thus, the cross-sectional area of the strut is what matters (not so much its section modulus). We don't know what your layout looks like, so I can't say for sure what function yours serves.

I do not want to make assumptions about the yield point based on material, because heat treatment and other factors can affect the yield point.

You don't say what this strut is for or what machine, structure, etc. it will be mounted on. However, most low and mild carbon steels are not heat treated to achieve their structural properties (stainless is a different matter). A safe assumption for a lower-end yield point for mild steel is probably 30 ksi, which I base on ASTM A36 (36 ksi yield) and ASTM A1011 (33 ksi), which are both common steels in the industry.

In other words lets say the strut deflects 5 inches when it is cantilevered and a weight is placed on its end. The tubing under test is placed the same way and only deflects 4 inches. Can I assume that the tubing will be an adequate replacement for the strut (ignoring factors like fatigue, age microcracking, etc etc)?

We have established that steel has a typical lower-end yield around 30 ksi and that two similar cross-sections will have the same deflection, even if their yield strengths differ. If your new tubing member deflects less than the existing strut, you can say that the tube is stiffer than the strut, but you cannot truly say it is stronger since you don't know the yield strength of the existing strut's metal. You could always err on the conservative side and assume a higher yield strength for the strut (say 100 ksi, as a very high end value) and back-figure the needed tube size based on a lower yield stress for it.

Note: this should not be construed as any sort of approved engineering advice. It is merely my take on the matter and should be accorded as such. Do not modify a structure if you are afraid doing so will have the potential to cause injury or death.