# Is it valid to solve for the young's modulus required for a given critical load with Euler Columns?

I am working on a problem for research that is about material choices. I was wondering if the following rearrangement of Euler columns would be a valid way to check to see if a material exists for a given critical load and moment of inertia:

Original Equation:

$$P_{cr} = \frac{\pi^2EI}{l^2}$$

Where $$P_{cr}$$ is the critical load, $$E$$ is the young's modulus, $$I$$ is the moment of inertia, and $$l$$ is the length (note that I am omitting different end conditions).

Possible Reformulation to check for a valid young's modulus:

$$E = P_{cr}\frac{l^2}{\pi^2I}$$

Yes you can use the formula to solve for an acceptable modulus of elasticity (E) that prevents buckling.

Like any mathematical formula that relates 4 variables, you can calculate any variable based on the other three.

As others have indicated, the design may be limited by other factors which need to be checked after determining an acceptable E based on the buckling equation, and the appropriate safety factors need to be included.

No we could not.

colums buckle under a random load usually les than the Euler's buckling load due to many reasons, such as manufacturing defects or material residual stress.

Two columns with same section properties and same effective length will buckle under two different critical loads.

• Are you saying that the formula to calculate Pcr is wrong, and therefore this wrong equation cannot be used to calculate an acceptable E? What was Euler thinking when he came up with the equation! I think your answer should be "yes, you can use the equation if you include a reasonable factor of safety. The factor of safety is require because ". May 2, 2022 at 3:57
• no, I did not say that. you may want to read my answer again. Googling "manufacturing deficiencies or residual stresses" will hopefully clarify their meanings. May 2, 2022 at 6:05
• The original post is whether the Euler buckling formula can be used to calculate a required modulus when a load is known. You said "no". I would like you to clarify why the answer is no. The formula is valid for an ideal world, so the equation can be used to calculate either Pcr or E. Residual stress and manufacturing defects do not make the equation invalid. Those factors just require a reasonable factor of safety be used. When the factor of safety is used, the equation can be used to calculate any value from the other known values. Is this correct? May 2, 2022 at 14:51
• It would be nice to hear from @cj_cool May 2, 2022 at 14:57

As the expression suggests, Pcr is a function of (E, I, L), in other words, Pcr is can not be assumed as an arbitrary number, nor can be predetermined without knowing the parameters it is based on. So, the result of your expression is not meaningful, or useful. I suggest finding a subject that has a close/tighter relationship with the elastic modulus and with a broader data range, such as stresses vs strains, and/or deformation characteristics of various materials.

• I do not understand your reasoning. Let's say I choose steel and some structural shape. When I plug in E for steel, and I and L for the shape, let's assume the calculated Pcr is 10005. I happen to know that my load is 10000. This implies that the design is okay in buckling (but with a factor of safety of 1). Are you saying that if I use 10000 for Pcr, I and L for my shape and calculate E, I will not get a value that equals the modulus of steel? And based on the calculated E, are you saying that any material with an E equal or larger than the calculated value will not work in buckling? May 2, 2022 at 3:54
• If you need to plug an "E" to get Pcr (a derived limit), then what is the need to plug the same load to get back to the same "E"?
– r13
May 2, 2022 at 14:14
• True, you can use 10 different values of E, calculate 10 different values of Pcr, and see which Pcr is larger than the known load. From those 10 calcs, you know what E is acceptable. Or, you can use the known load, calculate the modulus of elasticity, and use the material that has a higher modulus of elasticity. Correct? That is what the OP is asking, and I want to know why you say the equation cannot be used to calculate that. My example in the comment is to show that it doesn't matter if you calculate Pcr or E from the equation -- the result is the same. May 2, 2022 at 14:57
• I didn't say he can't, but that expression does not make sense, or non-sense, as elastic modulus is a well-defined material property, which is a constant derived from the stress-strain relationship. Do we, on earth in general, need to calculate gravity for everybody because W = mg, thus g = W/m, and everybody weigh differently? You can rearrange many equations, but not all make sense in their end applications.
– r13
May 2, 2022 at 15:55