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After reading Beams One and Beams Two, you know something about beams. But something is missing from those pages - everything in them is two-dimensional. Beams Three takes you into the third dimension.
Why is this important? It is important because very few structures are even approximately two-dimensional, and even if they were, they would possess the potential to become three-dimensional by oscillation and by buckling. In fact, the slimmer the structure, the less it resists oscillation or vibration.
This picture shows a garden fence. Originally it comprised a number of panels, attached to vertical posts, which are held in metal sockets at the bottom. The sockets are not deep, and afford little resistance to lateral movements. The panels were not connected to each other. Strong winds made individual panels of the fence oscillate. To reduce the amplitude of the movements, a long beam was added along the top of the fence, effectively creating one long panel. Although this had no effect on the supports at the bottom, it greatly reduced the motions, because any local force was distributed along the fence, and to some extent resisted by all the panels. At two places on the fence, the top was tied to the house by struts. The rigidity is now such that the fence barely moves in a wind, and a gate could be hung at one end of the fence with no risk of sagging.
What has this to do with bridges? During the last fifty years, the need for fast roads has led to many bridges being built across deep and wide valleys. Many of the designs are bold box girders, slender and elegant, on tall slim piers. The bridge is anchored only at the two ends and at the bases of the piers. The effect of wind along the valley is an important design consideration. Lateral oscillation could be unpleasant or even dangerous for traffic. Furthermore, a long bridge produces large thermal expansion and contraction, difficult to accommodate if the beam is continuous. Looking at some of these amazingly slender structures, it is hard not to think of the Tay railway bridge, the high girders of which were blown away in a high wind. Unfortunately, the Tay bridge was far ahead of its time in this respect. The right materials were not available, a thorough understanding of wind pressure was not available, and adequate quality control in this case was also not always applied.
Let's look first at oscillation. The curves below show qualitatively the stress-strain curves for a straight beam and a curved beam under the influence of a lateral force, such as a wind.
We must remember also that a bridge may be subject to forces from the traffic. For example, if a fast or heavy vehicle swerves, the bridge must supply the necessary forces to it. The reaction on the road will tend to make the bridge sway. The sway, in a straight beam, can be resisted by two effects, firstly the lateral stiffness of the deck, acting as a beam, and secondly the stiffness of the piers, acting as cantilevers. In a structure which carries all roadways on a single bridge, the stiffness will be high, but splitting the structure into two bridges, carrying the two carriageways, will reduce the stiffness.
The force on the straight beam is always tensile, which tends to restore the beam to its original position. But because of the inertia of the beam, an oscillation will occur after an impulse. For even quite large deflections, the force is small. In contrast, for even small deflections of the curved beam, the force is proportional to the deflection, and can be compressive or tensile: the beam can act as arch or cable. So the curved beam is constrained from oscillating in its fundamental mode with an antinode in the middle, though it can still oscillate with one or more nodes, in the same way as the cables of a suspension bridge.
The next picture shows qualitatively the effect of temperature on the stresses in a straight beam and in a curved beam, both fixed at both ends. It is obvious why long pipelines often include curved sections, often in the form of sinuous loops. Long continuous sections of railway track or road barrier are constructed so that they are in tension at any likely temperature.
The next two pictures show the lowest modes of oscillation for a straight beam and a curved beam.
The second picture shows clearly why the lowest mode is suppressed in an arch and a curved beam, and why suspension bridges oscillate, if at all, with a central node. The variation in length during one cycle of the fundamental oscillation would be very much greater than in the higher modes. Furthermore, the variation of length with displacement in the fundamental mode is different from that in all other modes. In what way? (Plot a graph of length versus deflection to see how.) Note that transverse oscillations are unaffected by the curvature - it is only the vertical mode that is suppressed.
pictures show another foray into the third dimension. The first
shows and apple resting on a slab of expanded polystyrene, which is
clearly not a good material for making beams. But it is very good for showing
the principles of beams.
In the next picture a central support has been added. It is a narrow column rather than a full with support. The apple has been moved towards one edge of the beam, and we can see that the beam is now supported at three points only. This tells us that supports do not only push upwards: sometimes they have to pull downwards. At the two ends of the main part of the Forth Bridge, massive weights inside the piers pull the cantilevers down to balance the weight of the suspended spans.
In the third picture, instead of gluing the beam to the supports, two oranges have been used. In addition, the apple has been moved as far from the centre line as possible, much further than it was before. The two yellow lines show that the beam is still distorted from a plane surface, and in a rather subtle way. But not by much - considering the material we are using. With a well designed beam, especially a box girder, the bridge would deflect very little.
What is the point of this demonstration? Why not just make a wide pier in the middle? Why not have a narrow pier on each side of the beam? There can be various practical reasons. The wide pier might be more expensive. If the road over the top is very wide, the people underneath might experience a sort of tunnel effect. With a wide pier or two columns, vertical alignment needs to be accurate, to achieve the required stresses. If the bridge has numerous spans, a row of wide piers can look very heavy and boring, while a forest of narrow columns can look very untidy.
In order to use this technique, we need to ensure that the bridge is rigid enough to transmit the torsional forces to the two ends. The piers can help by being wider at the top than the bottom. A simple taper can look somewhat peculiar, but a smooth curving out at the top can look very good.
For a completely different solution to the problem of the multi-span bridge click here.
go back to the small scale. The picture shows a hole in a Cotswold
wall where the river Coln flows through, near its source at Brockhampton.
A large stone acts as a beam to carry the wall.
Close to the hole in the wall, a spring forms a puddle from which a tiny stream runs into the Coln, which is in the trough at the back. A very short tributary. To get the track across the boggy ground, some hollow bricks have been put down to form a narrow bridge for the track, and a tunnel for the water. The top of the brick looks like a pair of beams, but the thickness and rigidity mean that the action is more like that of a frame.
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