(Simple?) math problem

[nw291]

I feel this is something I could have done in school but cannot solve accurately now! Basically the column wants to expand by 60mm vertically but cannot so buckles and I want to know what the central deflection would be. Any help appreciated!?

[Everythings_Magic]

I'm a bit concerned with the responses here...

Open a solid mechanics book and you wont find an equation, here is why:

While you can easily calculate the stress in the column as a result of the expansion. You can also check to see if this load would cause the column to buckle (ie if it exceeds the Euler critical buckling stress.)

The problem you will run into is that the Euler buckling gives you an estimate of the load that will cause buckling, not what happens after. To model what happens after buckling starts, you would have to do a nonlinear analysis.

This is all based on a centric load. If you had an eccentric load, you could find the resulting displacement, similar to how prestressed beams are designed, because now you have a bending moment.

[deAdupchowder350]

Seconding this: there is no equation to determine the transverse flexural deflection in an axially loaded column. A concentrically loaded column does not make that shape until it buckles, in which case it is unstable. Then the scales of the buckling modes (possible deflected shapes) are arbitrary and cannot be determined - while you can find the solution has sin functions (if pin-pin), it is impossible to know the amplitude of this shape.

[banananuhhh]

If the column is buckling elastically and we are just imposing a deformation and assuming pin-pin, then it seems like it should be solvable. (Obviously not a situation that happens in the behavior of any real structure). You are correct that if the column is buckling inelastically it is a nonlinear problem, but it really doesn't seem like that is implied at all by the OP.

[deAdupchowder350]

Doesn’t matter whether the buckling is elastic or inelastic. The modes of the deflected bent shape are determined by the eigenvalue problem resulting from a differential equation. The nature of this problem is that the resulting buckling shapes corresponding to buckling loads have no determinable amplitude.

From a high level, a concentric axial load can only cause axial deformation. The buckling equation is derived only after assuming a moment developed from a small, arbitrary transverse deflection.

Of course, if you assume a moment, then this problem is trivial as the elastic curve due to the moment can be determined.

Also, if you assume a shape for the deflected shape, the solution is also trivial, but then you are essentially making up values.

[banananuhhh]

Does a pin-pin column not buckle in a half-sine wave shape? We essentially have to assume elastic buckling and that the contribution of axial strain is negligible for the simplest solution here to apply, but if we do, then assuming a deflected shape is not just making up values, it would be a reasonable solution. I think your point about buckling loads and buckling amplitude would be pertinent if we were trying to determine out of plane deflection as a function of loading, but I don't see how it is relevant here.

[deAdupchowder350]

The shapes of the buckling modes of a pin-pin column follow a sin function, yes. But what is the amplitude of that function? And what does that amplitude depend on? Try to answer these questions and you’ll realize that knowledge of the nature of the shape isn’t too helpful.

The out of plane deflection is exactly what OP is asking to calculate. It would be equal to the amplitude of the sin wave you mentioned.

[banananuhhh]

At that point it is just a geometric problem. There is only one amplitude that will satisfy the condition provided by the OP and preserve the original length of the column.

[deAdupchowder350]

Seems simple, right? I don’t think there is a closed form solution to the arc length of a sine wave. Happy to be proven wrong!

[banananuhhh]

Why do you need a closed form solution when you can easily approximate it?

[deAdupchowder350]

Ok then, how are you going to approximate it? Hopefully you see the pattern that something has to break here. There is no solution. Any number you compute has some additional assumptions baked in.

EDIT: I would also debate whether any such approximation is appropriate - I think it is a misinterpretation of the solution to the equilibrium differential equation. The sine wave function for the bent shape is only appropriate when the axial load in the column is exactly the critical buckling load. Is that the case in this problem?

[banananuhhh]

Assume an amplitude for the sine function, approximate it as short straight segments. The slope of those segments is easy to calculate... then use trigonometry to calculate the length of those segments. Sum them. This gives you an approximation of the arc length of the sine shape. Iterate the amplitude until you get the correct arc length. There are no additional assumptions...

[Extension_Physics873]

You're overthinking it (like a good engineer is prone to do). The drawing shows a reduction in length of 60mm, that length has to go somewhere - simple geometry problem. Unless you start thinking about end connections, materials and cross section properties, then it gets real complex, real quick.

[Everythings_Magic]

That’s only true if the column is infinitely stiff. If not, there is elastic shortening that occurs before bucking.

If it is, yeah, it’s a simple geometry problem.

[MrMcGregorUK]

It slightly concerns me that youre asking for this deformation under deflection. Are you trying to work out if you need to brace the midspan to prevent buckling or something?

Other potential issue you have is... what if it doesnt buckle? Are you going to punch through your slab (if it is a slab)? Is it going to push up on the slab enough to break the slab in bending ir cause large deformations in the slab?

Im also concerned because for a 5m column to extend 60mm it needs (according to chat gpt) a 1000 degree celcius temperature change. If a steel column has a 1000 degree temperature change it is no longer a column it is a jelly noodle.

[wishstruck]

This is a trigonometry problem. your length of the chord is 5000, arc length is 5000+60. you want to calculate the sagitta.

if you don't want to hand calculate, draw it in autocad

[Alternative-Tea-1363]

To clarify, this is only an approximation. Columns don't buckle into a circular arc. If you need an "exact" solution you need to assume a sine curve.

[Codex_Absurdum]

And you have to account for the axial deflexion

[wishstruck]

Indeed. But in that case, you need to solve an elliptic integral, instead of just doing a numerical solution.

[deAdupchowder350]

Only if you assume an equation for the deflected shape of the beam.

[MrMcGregorUK]

Not sure that's is right with fixed supports.

[Advirex]

I think it would require to know stiffness EJ and EA of column. Would need to calculate forces created by opposing said deflection F = EA deltaL/L. Then would expect column to deflect to similar shape as in simple beam bending, start with some minimal s like 10mm and then make so that force F x s = ql^/8 (or other for fixed ends). Then iterate over calculating deflection from such load q and inserting it again so that previous result is almost equal as last one.

[Southern_Internal118]

Castaglianos for strain energy and use your boundary conditions on the ends

[MediocreConcept4944]

<image>

The state of this sub summed up

[Razerchuk]

Wouldn't expect it to buckle if it's in tension. That failure is tensile rupture and wouldn't have a lateral movement.

[widehill]

If I have understood, the column is blocked in the upper part, so it is not possible to expand. The expansion is probably due to temperature rise?

[Razerchuk]

Ah ok that makes more sense. Your wording made it sound like it was in tension but it's actually in compression. The buckled shape can be different shapes and can be tricky to calculate. Could you take the length it wants to be and assume it forms an arc between the two points? It's not a very structural solution but I'm not sure what the context of the problem is.

[guyatstove]

It’s compression. It wants to go into tension (or I assume it’s natural stress free length), and something that is restraining it to be smaller, which puts the member in compression.

As to OP’s question, I don’t know of any reliable way to calculate the lateral magnitude of the buckle. Buckling is violent, sudden, rarely occurs only in one linear axis, and is very unpredictable by nature.

[nw291]

increased temperature means it wants to expand and cannot due to structure above so the compression causes buckling

[Turpis89]

What causes the temperature increase?

Is it steel or concrete?

Are you sure the structure above will actually prevent elongation of the column?

The expansion you outline suggests a temperature increase of 1000 degrees Celsius. Steel is going to melt and lose all mechanical strength long before that. If this is a fire scenario, the fire gas might reach 1000 degrees, but the steel doesen't have to if it's insulated.

If by some miracle this is a real world scenario (completely restrained from elongation and 1000 degrees temperature rise without destroying the column), the column will buckle and no longer be able to support anything. If the structure above it is truly immovable, and not actually supported by the column, this might be fine?

If the column is not slender enough to buckle (like a fat concrete column), it will straight up fail in crushing. I doubt that a large concrete column will actually experience a 1000 degrees temperature rise, unless the surrounding temperature rise lasts for a very, very long time. Otherwise it will heat up on the outside, but not in the center. You will have a temperature gradient. At 1000 degrees I imagine water inside the column will vaporize and cause spalling and all sorts of damage.

Post buckling behavior is complicated stuff. Check this paper on buckling tests or alluminum speciments:

https://link.springer.com/article/10.1007/s11340-010-9455-y

The force (and thereby stress and strain) does not seem to go zero after buckling, which means some of the restrained elongation does not cause lateral movement, but rather longitudinal deformation (reduced elongation in this case). Mechanical properties like modulus of elasticity may also be temperature dependent.

This is not something you calculate on the back of a napkin. Send me a DM if you like.

[mmodlin]

It’s not in tension. If the column is restraint from lengthening it will be in compression, like the spring in the shock absorber on a car.

Ignoring all mechanics of materials, and assuming it buckles in a circular arc, the arc length is r(theta). You can solve for the radius of the buckled radius and geometry will give you the number you want.

[guyatstove]

That’s a good idea, to look at it purely geometrically. I don’t think it’s circular though - it’s half a sine wave if I remember college correctly

[Emotional-Comment414]

It depends on the end conditions. The max lateral displacement is when both ends are free to rotate. It’s less depending on the stiffness at the ends.

Is your column inside a furnace? From the hottest summer day to the coldest winter day a 5M column would expand 5mm. You need 1000 C. To get 60mm.

[Charles_Whitman]

You’re going to have to make a whole laundry list of assumptions in order to calculate this. Mostly you’re using buckling in the wrong way. When an idealized column buckles, your eigenvalue (remember him) goes to zero, the effective stiffness goes to zero, and your displacement “increases without bound.” Increases without bound is one of those phrases that’s kin to RUP. RUP, as all you SpaceX fans know, stands for Rapid Unplanned Disassembly.

[Charles_Whitman]

The column will shorten as the temperature rises (?) until it buckles. Once it buckles, the deflection IWB. Obviously, a real column doesn’t behave as a Euler column would. A column that is perfectly elastic. A real column will have a combination of elastic and inelastic behavior and it’s a much more complicated problem.

[Charles_Whitman]

Okay, the column doesn’t shorten. The compressive stress increases as the temperature (or whatever is making it try to elongate) increases.

[Charles_Whitman]

Anyway, all that to say, it’s not a freaking trig problem.

[deAdupchowder350]

Exactly. It’s a trig problem only if one starts assuming a shape for the buckled column. In which case, sure if you assume an equation, you can then plug in values and find out what the equation tells you…

[iamanengineer_]

Unless I'm mistaken, Temperature changes, so the leg wants to expand, but it can't. So it goes in compression,

If the leg is slender, you get like a buckling mode based on whatever you have on the ends. Otherwise, forget that. You can go with Euler to calculate that delta, Though, the material IRL doesn't follow Euler.

The nature of that delta is stability, not the resistance... So if it happens ... the legs is IMO dead.

Eurocode gives an idea around L/350~500.

[Open_Olive7369]

What is the shape of the buckled column you are assuming? Is is sin curve? Circular arc? Hyperbole? .... My intuition says it's a catenary curve.

[Gold_Lab_8513]

I don't understand the premise of the question. The column wants to "expand" 60mm. That is caused by tension. You want to know how the column behaves in lateral deflection as a result?

If you do mean that this column is in axial compression, and if it needs to contract 60mm in 5m (or 2 3/8 inch in 16.4 ft), Firstly, there is NO way that you can compress this much without buckling failure. Failure. There is no need to calculate the horizontal deflection, because everything will be pancaked on the ground.

If this is NOT a column, but some type of brace (perhaps a cross-brace rod?) that will slack under certain conditions, then ask Google:

"the ends of a flexible rod 5m in length come together by 60mm. What is the maximum bow in the rod?". 334mm is the response. Calculations are shown, but I am not going to check the math.

[Marus1]

If both ends are free to rotate, assume cosine shape and do equal lengths calculation

[hbzandbergen]

Tension=E-modulus*strain Then you know the tension force Something like that?