To find out whether a structure does or does not exceed its limit state, engi- neers must perform a remarkably large number of calculations. For a girder bridge, they must determine the future strengths of girders, piers, abutments, and connections (such as bolts) under the loads that will be placed on them.
To illustrate what the engineer must do to compute the various stresses as against the strain limits, let us consider a 400-foot three-span steel girder bridge supported on abutments at each end and two sets of two-column concrete piers, as shown in figures 5.1A and B. Note that each pier consists
A
A 400 ft
125 ft 150 ft 125 ft
Section A-A Traffic Barrier Concrete Deck
I-Girder
Cap Beam Column
Figure 5.1A. A three-span steel girder bridge viewed from the side.
Figure 5.1B. The same bridge viewed in cross section, revealing a two-column pier, the columns connected with a cap beam, on which five girders rest, supporting the deck.
of two columns connected to each other on top by a cap beam. Composed of five parallel girders, each taking the “I” shape, the superstructure supports a deck four lanes wide.
Let us confine ourselves just to the measurement of load and resistance on one girder—the middle girder over the middle span. Since stresses will vary along the length of the girder, we will seek the stress maxima—the points along the girder at which stress is greatest under given loads.
We begin by measuring the expected dead load, for which we shall consider just the weight of the girder, and that of the concrete deck, parapet (railings or wall along the sides), and light fixtures and any other utilities on the structure. Given the length of the girder and the dimension of its cross section, we can readily check dead weight in a professional manual, such as that published by AASHTO, from which we may project a dead load per linear foot of traffic lane.
Recall that the girder must first of all resist its own dead load. Even at this bridge’s modest main span, the girders will sag an inch or so in the middle.
Now we go on to vehicular live load. The standard practice is to esti- mate the effect of routine car traffic plus one heavy truck over a particular linear foot of deck. From studies that have been conducted, routine traffic is estimated to exert 0.64 kips per linear foot per lane. Professional standards also provide a model truck (figure 5.2), which consists of a cab weighing 8 kips over the axle, plus a truck bed carrying a container.
So far, we have assumed that the vehicular live load is static—as dur- ing a traffic jam. Now we include the fact that it is likely to be moving, and thereby bumping up and down, exerting additional downward loads. To account for this dynamic load, AASHTO requires an additional allowance of 33 percent above the normal vehicular load and the truck axle loads.
Figure 5.2. A model truck for estimating bridge loads.
6 ’
32.0 kips 32.0 kips
8.0 kips
14 ’ -30 ’ 14 ’
There is an additional allowance for braking forces, but since these are longitudinal, they do not affect the vertical resistance of the bridge.
Here we are looking only at vertical effects. We have left much else out as well. Pedestrian load is excluded because our bridge has no sidewalks.
Waterway effects are excluded. Special effects of any bridge curvature are no complicating factor, since our bridge is straight. Thankfully, we do not get into these complications here.
Given these simplified assumptions, the engineer has to estimate points of maximal stress. Typically, shear stresses are highest where the girder intersects the abutment and the pier—and these shear stresses will be larger or smaller depending on the position of the model truck on the span. At any point along the girder, the truck will exert a downward force, push- ing the girder down. The bending exerts maximum tension effects on the bottom flange of the I-beam, maximum compression on the top flange, and shear effects in the web. Bending moment is typically greater the farther we are from the supports, so it is typically highest in the middle of the girder.
Upon such calculations, the engineer can discover the points along the span of maximal shear stress and maximal bending stress. Then, she can determine whether the I-girder can at these points resist these stresses.
The resistance depends on the construction material, the dimensions of components, and how they are connected. Steel has fairly consistent quali- ties, so engineers can consult tables of materials and cross-section properties to help them estimate whether the expected loads would damage a girder of given dimensions.
Let’s say our study shows that the girder we have chosen as an example can indeed resist the projected loads at maximum points of stress. The engineer’s job is far from done. There are many more contingencies to be considered. What if the two piers, which have been driven into the riverbed, eventually undergo settling, burrowing down by two inches? Now the girders they support will sag two inches more than originally calculated. How will this affect the girder’s ability to resist the given truck loads at the points of highest stress? More calculation is needed.
To be sure, the engineer could have recommended a girder so thick that it could have held up against almost any conceivable load, even sev- eral tanks rolling on it at once. But that may not to be a good solution.
For one thing, the girder’s dead weight would grow, requiring piers with greater carrying capacity, and increasing downward pressure, forcing greater downward settlement on the pier. What is more, the cost would escalate. As much as it always seem to be an unmitigated good thing to increase safety, we must remember that ever-increased safety for one bridge deducts from the amount to be invested in other bridges, or in other public concerns, whether highway safety or public health.
So the engineer’s job is to keep the bridge cost to the amount needed to make it strong enough to meet anticipated loads, plus some safety factors to account for unanticipated stresses. That brings up the next problem: just how is the proper safety factor determined?