Truss Bridge Design

The history and principles of rational truss bridge design

Until the 19th century, bridges, and indeed all structures, were designed by methods familiar to Vitruvius, and set out by him in de Architectura, written in the 1st century BC. The key was proportion, established by experience with similar structures, and appropriate to the size and situation of the project. A unit length, or module, was established, and all parts were dimensioned with reference to it as the project was carefully drawn to scale. The drawings then served to guide the builders. If the proportions were correct, the materials of good quality, and the workmanship adequate, the result would be serviceable. Advances and changes were made gradually and cautiously in this system. These methods were still used by Smeaton and Telford at the time of the industrial revolution.

The new engineering material, iron, as it became inexpensive enough for use in structures, called for new methods. There was no easy guide to the proportioning of members such as the wrought iron chain links for a suspension bridge. Wrought iron bars were made, and then subjected to tension that was increased until they broke. From the proposed loads on the bridge, the tension on the supporting chains could be estimated, and from this the necessary amount of iron could be determined. Refinement of these procedures led on the one hand to the experimental determination of the unit stress, or force per unit area, that iron could safely and reliably resist, and on the other to methods of calculation based on statics to determine the forces acting on structural members. Dividing the force by the allowable unit stress then gave the necessary area of the member.

This method of rational design, which seems so obvious to us now, was adopted only very slowly. It was used at first mainly for tension members, such as links and rods, for which the forces were easy to determine, and the stress distributions uniform. The plastic nature of iron was a great help, as any part unduly stressed would stretch a little, and the stresses would be evened out without failure. Most structures, such as buildings and arch bridges, were still governed by compressive stresses, and proportion was a more reliable guide than unfamiliar and difficult calculations. Most stone structures are far stronger than necessary to support their working loads when designed for solidity and appearance, as evidenced by the survival of many early stone arch bridges for railways to the present day.

In the early United States, stone arches were rare because of their expense. The required money was not spent on roads at any rate, and cheaper substitutes had to be found. Wood was the obvious material, and was nearly universally used. Short spans could be trestles of bents and beams, or king- or queen-post trusses as were familiar. Major bridges were timber arches, made from laminated wooden arches from which the deck was suspended or upon which it was supported. There was usually some truss arrangment to stiffen the deck and main arches, according to the prejudices of the designer, so it was usually uncertain just what parts of the bridge were effective and which were not. If the bridge exhibited some distress under load, more wood was added to help prop it up. Combined with bad abutments, which moved and heaved, and seasonal torrents, these bridges were not generally trusted, though some proved remarkably serviceable.

When railways arrived after 1830, the question of bridges was again in the air. A few major masonry structures were made, but in general they were too costly, so wooden bridges were the only alternative. Those in use on roads proved inadequate and unsafe, so something different had to be found. Where it could be used, the timber trestle was completely adequate and reliable, if not permanent. Most engineers turned to iron bridges where trestles would be inappropriate, and a large number of designs were brought forward, for example by Fink and Bollman. These bridges were based on the usually erroneous conceptions of their designers. Where they proved strong enough, they were inefficient, and tended to fail abruptly when the stresses searched about and found a weak spot. Rational design of bridges was still in its infancy in 1860.

The idea then took hold of a bridge in which all the forces could be determined by the principles of statics, so they would not be altered by small inaccuracies of construction, or by changes in temperature or settlement of abutments. In a truss bridge, this meant a span supported at the ends, with members pinned together so they could rotate at least a little at the joints. The number of members meeting at a joint had to be small enough that the forces in each could be uniquely determined. There is a relatively small number of truss designs that satisfy this requirement.

The most popular design was the Pratt truss, which could be used in spans up to several hundred feet. As shown in the Figure, it consists of an upper chord, in compression, and a lower chord, in tension, connected by vertical and diagonal members. The loads w are applied to the truss at the panel joints, and the reactions R are applied at the ends. The principal job of the vertical posts is to keep the chords apart and brace them. The end posts carry only tension, but the others are designed as compression members. The diagonal members resist the shearing forces between the chords that arise when the loads tend to cause the centre of the span to sink. In the centre panel, there are diagonals in each direction, although only one direction is in tension at any one time, the other being slack. The reason is that a moving load is not applied evenly across the bridge, and as it moves one set or the other of the diagonals will find itself in tension. These counters are generally used in one or more of the central panels.

Why the diagonals are in the direction they are can be deduced from the Figure on the left. When a beam consisting of two parts is bent, they slide on one another as shown. This is called shear, and must be resisted in all beams if they are to be strong. We can see that diagonals in the Pratt truss are directed so as to be in tension, preventing this shearing motion between the chords. If they were in the other direction, they would be in compression instead. In fact, a truss with diagonals in this direction is called a Howe truss, in which the verticals are now in tension. When trusses were made with as much wood and as little iron as possible, the Howe truss was popular, since only the verticals and the lower chord had to be iron rods. Pratt trusses could be built with timber upper chord and verticals, but they were mainly all-iron or all-steel bridges. Historically, trusses began as wooden bridges, since wood was the only building material that could resist tension reliably until iron became inexpensive.

The Pratt (and Howe) trusses are statically determinate, which means that the forces in each member can be found from the principles of statics. These principles state that the sum of the forces in any direction acting on a body, or the moments of the forces about any point, are zero in equilibrium. The first step is to find the reactions R at the abutments. The weight of the bridge itself, the dead load, is distributed between the panel joints, and the weight applied to the deck of the bridge, the live load, is assumed to be applied to the panel joints as well. These loads will be different for different positions of the live load, but the positions giving maximum stresses are known from experience. We can now start at a joint above a reaction, where there are only two unknown forces, and determine the unknown forces by the requirement that the sum of the horizontal and the sum of the vertical forces must each be zero. We can proceed joint by joint through the bridge, only picking up two new unkowns at each joint, until we reach the other reaction, which is the method of joints. It is somewhat faster to consider the forces acting on some larger part of the bridge, as in the Figure, where we now have three conditions at our disposal. Taking moments about point B eliminates all forces that pass through B, so we find an unknown immediately. This is called the method of sections.

It is now possible to proportion the members to resist the stresses we have found. The important members of the lower chord were known as eyebars because of the holes for the pins at each end, as shown in the Figure. If F is the tension in the bar, and σ is the maximum allowable tensile stress, then the area required is A = F/σ. The same area A is provided at the pin hole as in the main part of the bar, so the average stress will be the same. Although designed with a factor of safety, these eyebars sometimes failed, with serious results. Since the force in each member of a statically determinate truss is determined by statics, the failure of any element causes the failure of the whole bridge. The problem was that the stress was not uniformly distributed over the metal on each side of the pin, as it was in the main part of the bar. It was concentrated close to the hole, at the red areas, where the stress was three times the average value. Since eyebar design has taken this into account, they do not fail. This is an extremely valuable lesson, which still is occasionally not properly learned. In many cases, allowable stresses presuppose a certain configuration, and are not the actual stresses that occur, but values that are proved by test to be safe. Modern computer methods tend to encourage hubris in design, occasionally with embarrassing or tragic results.

The Pratt truss proved thoroughly reliable, never providing any surprises and capable of confident design. It is, however, not the most economical solution. Most of its dead load is in the middle of the span, and as the span increases it becomes increasingly more expensive to support. The depth of the truss increases with the span, which makes the members longer and more subject to buckling. There are modifications of the Pratt truss for longer spans that involve more bracing and other measures. It was usually more economical to break the bridge up into multiple spans supported on piers. The bridge is observed to be 'thinnest' at the piers, and 'thickest' between them. A more economical truss is designed like a continuous beam, which removes the joint at the pier, and allows the truss to bend over the pier. Now the bridge is thickest over the pier, with less material in mid-span. The ultimate is something like the Forth Bridge, with giant cantilevers over the piers, connected by light spans between the ends of the cantilevers. A continuous beam is not statically determinate, and the stresses depend on how much the members stretch. Nevertheless, the longest bridges are all of this type, since it is very advantageous.

With increased accuracy in field erection, it proved satisfactory to eliminate the expensive pin joints and to replace them with normal riveted or welded joints. The bridge is still designed as if it were statically determinate, which is approximately true. The upper chord was designed this way from the first, since the added stability rigid joints give to compression members is too valuable to throw away. For tension members, it does not matter.

The single truss that we have discussed here is, of course, only half the bridge. The two (or more) trusses in an actual bridge must be braced to prevent sideways buckling, for example by portal bracing at the ends. The roadway may run between the trusses, a through bridge, or on top of them, a deck bridge. Provision must be made for thermal expansion of the bridge, perhaps by some kind of roller at one end. It is no longer common to build steel truss bridges, especially in small sizes, since reinforced concrete trestles are normally less expensive.

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Composed by J. B. Calvert
Created 10 July 2000
Last revised