How Much Does a Beam Bulge When it Expands?

Date: 2/24/96 at 17:32:8
From: Scott Firebaugh
Subject: Railroad beam problem

Dear Dr. Math,

Imagine a railroad beam 1/4 mile long fastened at both ends.  
The beam expands 2" one summer day, causing it to bulge up.  
How high does it go?  If you assume it forms 2 right triangles, 
it is easy to figure out using the pyth. Hmm. That is 126" high in 
the middle.  But what if it is an arc of a circle?  I got 3 
variables and 3 equations, but I could not solve it.  A friend 
said it is actually a catenary curve, which is less familiar to me 
than a circle.  Can you solve this?

Scott Firebaugh, Kokomo Christian School

Date: 6/26/96 at 9:34:15
From: Doctor Jerry
Subject: Re: Railroad beam problem

Dear Scott,

Please draw a circle with radius a (unknown).  Draw a central 
angle with measure t (in radians); please draw the angle so that 
its vertex is pointing directly towards the bottom of the page.  
Draw the horizontal line connecting the ends of the rays forming 
the sides of the angle.  The length of the horizontal line is 1/4 
mile.  The circle is the railroad track, with 2 inches added.  Let 
b be the length of the line from the center to the horizontal 
line.  We want a-b, which is the height of the circular track 
above the flat track.

Let the arc length cut from the circle by the angle t be s.  There 
is a formula connecting s, a, and t: s = a*t.  So,
a*t = 1/4+2*(1/12)*(1/5280)=7921/31680.
This is one equation in two unknowns.

Another equation is sin(t/2) = (1/8)/a.  This comes from one of 
the right triangles.

Eliminate a between these two equations, obtaining
sin(t/2)=31680*t/(8*7921).

This equation can be solved with a good calculator (set in 
radians).  You should find t is approximately 0.0550458.  From 
this you can find that a is approximately 4.54224.

Then, b = a*cos(t) = 4.53536.  Finally, a-b is 36.3 in feet.

You can see that I've assumed you know some trigonometry.  If you 
want an answer using algebra and geometry, please ask.

-Doctor Jerry,  The Math Forum