loadpaths

The purpose of many structures is to support an object, the load, in a position where it could not be without the structure. Examples are - a railway train moving high over a river, marathon runners on a suspension bridge, someone walking on an upper floor in a building, a package hanging from a crane.

The common feature of all these examples is the need for a means of transmitting the force that holds the object to some fixed place or places. Those places are in the ground, except for the cases of floating objects or aircraft. But even then, the weight eventually reaches the ground, though so diffused that it is not detectable.

Books tell us that when a boat or a ship passes on to an aqueduct that carries a canal, the structure carries no extra weight, because the vessel displaces its own weight of water. True, but to where is the water displaced? The water level does rise minutely in the presence of a vessel, but the volume of the canal is so much larger than that of the vessel that the rise in level is very small. It has in fact already occurred as soon as the vessel has passed through the last lock and entered the stretch of water that includes the aqueduct. No new rise happens at the aqueduct.

In the first picture, the load path runs from the owl down through the post and into the ground, where it spreads out. In the second picture, if the owl decides to take off, the load path is transferred to the air. Then where does it go? The air presses on the ground over the whole area of the earth, but any change in pressure cannot be transmitted faster than the speed of sound, so the weight of the owl is presumably taken quite locally. This is an unimportant example, but in real structures, the effects of moving and transient loads are extremely important. Aside from considerations of metal fatigue, oscillations have to be considered as well.

In this picture, the load path passes through the handler's arm, and then through the body, before reaching the ground. As in the previous examples, the load path is not a thin line: it fills the available volume. The stresses are distributed in the configuration of minimum energy. In this example, the arm is a cantilever, and there must be tensile stresses in the upper part, with compressive stresses below. In that sense, the load path splits into two.

In this picture we see the jib of a crane, with the load paths marked in red and blue. The path goes up the cables that hold the pulley, and along the top and bottom members of the jib. The connecting members cope with shear. The vertical members of the tower are shown in red and blue for compression and tension. Surely this is wrong; they are all in compression. Yes indeed, but the diagram is showing only the effects of the load, and not the constant stress from the self weight.

What actually happens is the compressive stress in the red vertical is increased from the static value, while that in the blue member is decreased.

Here is the load path from one rider on a ski-lift. The path goes up the supporting bar, which is mainly in tension, except at the two bends, where there is bending moment and therefore compression as well as tension. Then the path splits, going along the main cable in both directions. The ratio of the loads in the two parts is related to the ratio of the distances to the two nearest pillars. At a pillar, the load path runs along the cantilever arm (compression and tension again) and then down the pillar to the ground. Each rider adds some bending stress to the pillar, one way or the other. Because the riders are moving, the stresses are continually varying. In a high-speed situation, such as a roller-coaster, the supporting structures can be seen to move as the cars take the curves. Fatigue is definitely a consideration.

The following diagrams show the load paths for some typical bridges.

The diagram below shows a small part of a suspension bridge with a truck. The two hangers nearest the truck are coloured blue to indicate the extra tension in them. Because the deck has rigidity, some of the load is share with neighbouring hangers, as the darker blue suggests. The extent of the sharing depends on the rigidity of the deck and the weight of the load relative to the weight per unit length of the deck.

So the load path includes a part of the deck and some of the hangers. What next. The path clearly goes along the main cable, shown blue, in both directions. Both? How can that be? The cable to the left of the truck is pulling downwards, so it can't be helping to hold it up, can it? In fact, the tension in the main cable, coupled with its angular change at a hanger, must always balance the tension in that hanger. The changes in tension propagate to the towers, and the extra downward forces that the truck causes at the towers are in the inverse ratio of the distances of the truck from the towers.

The vertical components of the load paths then go down the towers into the ground, to be spread by the foundations, while the horizontal components go down the main cables to the anchorages.

The next picture shows a part of a cable-stayed bridge. Much the same considerations apply in as in the suspension bridge, except that unless the truck is very near the centre of the bridge, all the load is taken by one tower.