Understanding bridges

Some years ago, I co-authored with my late friend Larry Kline an article about prototype railroad bridges: how they work, how the prototypes evolved, and how one may choose a bridge on model layouts. Here’s a citation: Larry E. Kline and Anthony W. Thompson, “The Evolution of American Railroad Bridges, 1830–1994,” Symposium on Railroad History, Volume 3, A.C. Kalmbach Memorial Library, National Model Railroad Association, Chattanooga, TN, 1994.

My purpose in the present post is to introduce the topic by suggesting a simple way of understanding what a bridge does, and what affects its performance. I will begin with an analogy the most people will understand from experience. Consider  a 1 x 10-inch board, say ten feet long. It can readily be imagined that if it is laid flat, that is, the wide side horizontal, and supported only at the extreme ends, it would be quite springy and flexible if an adult tried to walk its length. 

But now imagine it set up on edge, so that the wide side is vertical, only end-supported, and braced so that it is maintained vertical. Stepping onto it (a delicate balance problem, to be sure) would reveal that in this orientation, it’s very stiff; even a heavy man would scarcely cause any deflection. Yet it’s the same wooden plank.

The reason for this major dependence on thickness, or if you will, depth of the beam, is expressed in the formula for stiffness of a beam. I won’t address the math, except to point out that the stiffness varies as the cube of the thickness, that is, thickness to the third power. That 1 x 10 on edge is a thousand times stiffer than the same plank lying flat. Note that this is not a material property of the plank, but depends on orientation only.

Bridges essentially follow this fact in most bridge designs; the key is the thickness of the “plank,” with everything else much less important. Of course, the stiffness of the material itself matters; wood is hardly one-thousandth as stiff per unit size as is steel. But for any given material, it is all about thickness or depth.

In essence, a bridge is a beam across a gap in the terrain. And in fact, very short bridges over culverts or tiny creeks can be simple wood beams under the track. Longer bridges of that kind require deeper and deeper beams, but of course it is often more practical, rather than increase the beam size, to simply subdivide the gap. Trestle bents at suitable intervals permit using short-span beams under track, from bent to bent.

(The photo is by J.R. Knoll on the Apache Railway south of Holbrook, Arizona, my collection.)

Of course, the trestle bents need not be as short as shown above; the identical principle is illustrated with far taller trestle bridges, still with wood beams under the track, as shown in this famous Richard Steinheimer photo on the San Diego & Arizona Eastern, with a Baldwin road-switcher leading a mixed freight across Goat Canyon trestle in Carriso Gorge in 1952. (used with permission, DeGolyer Library)

And steel is a far more suitable material for substantial loads than wood. The familiar girder bridge, with girders beneath the rails or alongside them, uses this principal of a beam under or alongside the rails, of course with sturdy crossbeams connecting the side girders.

These bridges, though simple in appearance, do in fact have very specific design characteristics. Of course, the most basic is the depth of the girder, relative to its length. This again references the third-power dependence of girder stiffness on depth, so naturally the depth will increase together with length. 

There is extensive prototype information on this topic in Paul Mallery’s outstanding book, Bridge and Trestle Handbook, first published by Simmons-Boardman in 1958. I have the second or revised edition, published in 1976 by Boynton and Associates.  For the present subject, Chapter 9 on plate-girder bridges is applicable. It contains a table of typical length and depth of girders, Figure 2 in this chapter, which appear to range between 7 and 9 times longer than they are deep, in other words, a length-to-depth ratio between 7:1 and 9:1. 

The photo below shows a deck girder bridge in the process of construction, and its length to depth ratio is indeed about 8:1. This is the proportion identified in Mallery’s book, as just mentioned. This is the Butte Slough bridge of the Sacramento Northern, east of Colusa, California, on November 1, 1912 (Harre DeMoro collection, courtesy Kristin DeMoro). 

I realized that this same topic was important when looking at a bridge on my layout, originally built very simply by just cutting down the Atlas commercial girder bridge to the appropriate length to span the gap on my layout. But as soon as I looked at prototype bridge photos, I could see the difference: the proportions of my short bridge were way off. Once I recognized that, I replaced the bridge with one of the correct proportions, following Mallery’s information. I described that project in a trio of posts. Here are links:

https://modelingthesp.blogspot.com/2013/05/a-new-sp-bridge-for-shumala.html

http://modelingthesp.blogspot.com/2013/08/a-new-sp-bridge-for-shumala-part-2.html

https://modelingthesp.blogspot.com/2013/09/a-new-sp-bridge-for-shumala-part-3.html

This concludes what I want to say about simple bridges. But there are more complex designs, particularly the widely-used truss bridge, and I will turn to bridges of that type in a future post.

Tony Thompson