If the ends of things are fixed...when it expands just a little, why is there a huge buckle? why does it bow?

so much...like if you have a railroad track and the two ends are secure and it expands by the heat by just a foot, why is the buckling of the traack soooo dramatic?

If the ends of things are fixed...when it expands just a little, why is there a huge buckle? why does it bow?
Well, do the math.





Assume you have a piece of track that is of length L and is initially flat.





When it heats up a little, its length will be L+D. At this point, it has to bulge a bit because the ends are fixed, so suppose it bulges as an arc of a circle with radius R and center O.








Let A and B be the two ends of the track, and let @ be the angle AOB in radians.





Let C be the point half-way between A and B so AOC is a right triangle with the right angle at C, an angle at O of @/2, and a distance OA of R.





Then we know that L+D = R@ and


R sin (@/2) = L/2





Since we know L and D and have two equations, we should be able to solve for the two unknowns, R and @.





But what you are really interested in is how high the track bows and that is R(1 - cos (@/2))





Now if @ is small, then sin @ is very well approximated by @ - (@^3)/6, and cos @ is well approximated by 1 - (@^2)/2.





Substituting these approximations, we get:





L+D = R@


L = 2R sin (@/2) = 2R((@/2) - ((@/2)^3)/6) = R(@ - (@^3)/24)





This gives us:


D = R(@^3)/24) so


D/(L+D) = (@^2)/24





Since you know D and L, you can compute @^2, then @ then R and then R(1 - cos (@/2))





But there is a shortcut. Since cos @ = 1 - (@^2)/2 and we want R(1 - cos(@/2)) we have:





R(1 - cos(@/2)) = R(1 - (1 - (@^2)/8)) = R ((@^2)/8) or


R(1 - cos(@/2)) = (L+D)(@/8) = (L+D)sqrt(24D/(L+D) )/8 = sqrt(6D(L+D))/4





Why don't you try it with numbers you think reasonable? As you can see, the larger L is, the bigger the bulge, even with D equal to 1.
Reply:Without getting into all the math....





When something linear (such as railroad tracks) expands, it in effect becomes longer. If the ends of that linear track are fixed in place, the extra length of the track has to go somewhere -- and the track buckles as the result.





Try it with a piece of string: Hold the ends of the string in either hand (one end in one hand, the other end in the other hand) and stretch out the string; this is the "track" before being heated. Now, bring the two ends together a bit; this represents the "track" being heated. While the ends remain fixed in place (yes, I know, you moved the ends - but that's only because it's a piece of string and you can't heat it up like a piece of metal; assume the distance between the ends remains the same and the length of the string changes). Because the effective length of the "track" became longer, and that extra length is now occupying the same space as the original length, it has to "bulge" (or even break, if the expansion is great enough) or the ends will be forced to move outward -- but the ends can't move because the ends are fixed in place.





I hope this was clear enough.



credot siosse