### # Expanding Bridge

Bridges normally have expansion joints built in so that as the material changes from heat, for example, the change is absorbed without disturbing the surface. Suppose a bridge could be built with rigid material so that expansion is absorbed by a hinge action in the middle of the bridge. The ends are firmly anchored at A and C. Let the bridge be one mile (5280 ft) long.

If the coefficient of expansion is (1/2640) the bridge surface would increase in length by 2 ft.

Discussion: What is the intuition? Would a person 6 ft tall still be able to see a person 6 ft tall at the other end of the bridge at maximum expansion?

Question: How high would the center of the bridge be at maximum expansion?

Let's explore that a bit further. Letting h be the height we have a right triangle with legs of length h and 2640 and hypotenuse of 2641. Using the Pythagorean relation, we find that the height is between 70 and 80 feet!

Does this seem reasonable?

Hypothesis: Maybe the size of h would be much smaller if we considered the bridge expansion to be in the arc of a circle.

See Bridge 2 for this bridge expansion in the arc of a circle.

## Related application:

The steel rails of a railroad track are subject to expansion and contraction from heat and cold. The technology of railroad tracks has evolved over the years to improve the quality of the steel and the         construction. When tracks were constructed from 30 ft long units of track, the small shim was placed at the joints to absorb the expansion and contraction. For reason of safety as well as reduction of the noise of steel train wheels rolling across the joints (the 'clickity clack' sound train riders could hear) railroad construction moved to continuous welded rail. When rail expands beyond its capacity, a condition called sun kink can occur in which the alignment of the rails is disturbed by one or both rails bowing out or buckling from the heat.

The technology for installing continuous welded rail involves heating the rails as they are laid and/or stretching them with hydraulics. The rails are fastened into place under tension and the tension absorbs the expansion or contraction.

The expansion factor in going from -30 F. to 130 F. may be in the range of 6 feet for a mile of rail.