The deck-stiffened arch and the multi-hinged arch

If you look in books and web-sites, you probably won't find anything about multi-hinged arches, but that is not necessarily a reason for not mentioning them here.

What you can find in many places is a statement, mentioned in this web-site, that an arch can have no more than three hinges or pins if it is to remain stable. An arch with two or three pins usually has two of them at the springing, but this isn't essential.  If the angle of the arch is rigidly fixed at the springing, the outer hinges can be within the arch. This isn't actually ever done, but it is mentioned here for completeness, to show that the range of things that are built is not always the range of things that could be built, because some structures are disadvantageous. Nevertheless, it can be worth looking at these to see what the problems are, and to see if anything has been overlooked, or to see if perhaps changes in materials or methods have made new constructions possible.

Before looking into this, let's look into trusses, which have their own pages in this web-site. We are familiar with the idea that trusses have to be built from triangles in order to make them rigid enough, if all the joints are pinned. Yet we sometimes see steel framed buildings made with rectangles, except for one panel at one end, which is triangulated. The fact is that there is a hidden stiffening mechanism, namely the foundation on which the building rests, and as a result, the single triangle stabilises all the other panels.

Similarly, a three-hinged arch has no triangles, but the abutments are are fixed in position in all three dimensions, and this constrains the movement of the arch. If we look at the number of triangles needed by a truss for a given number of panels, we can see that the three pinned arch could be viewed as a degenerate (in the mathematical sense) case of a truss which needs no triangles.

This digression may seem to be irrelevant to arches, but let us look at what an arch has to do.

Firstly, it has to withstand the thrust generated by its own weight, and a perfectly funicular arch would be in pure compression throughout, like a curved strut.

Secondly, it has to be stiff, because it is inherently unstable: of all the shapes that could be made with an arch of uniform cross-section and a given length, the catenary arch has the highest centre of gravity, though not the highest crown.

Thirdly, it has to be stiff against live loads, taking on beam-like properties. Many structural members can be considered as superpositions of two "ideal" or "pure" members.

Returning to the foundation we discussed earlier, let us imagine a long stiff beam spanning the region above an arch, and let us imagine connecting the beam and the arch by some vertical piers. We can imagine making this beam so stiff that the arch can be thinned down until it is only stiff enough for the segments between the piers to be stable against buckling. The beam has rigidized the arch in the same way that the foundations rigidized the imaginary building mentioned earlier. One end of the beam should be fixed to the ground, while the other end of the beam can be free to slide.

What we now have is something that looks like a truss made of quadrilaterals, just as the imaginary building was made largely of rectangles. But whereas a truss generally rests on just two supports, here we have a beam resting on two supports, and an arch resting on two other supports, and that is how the stability is achieved.

What is the benefit of this? In an arch that has a separate deck above, the deck has to be locally stiffened against live loads, and it may as well be stiffened enough to protect the arch against instability as well. An aesthetic advantage is that the separation of functions means that the observer can clearly see the division of labour between the thin arch and the thicker deck, as in the diagram below.

archdefsa.gif (4718 bytes)

If you think that appearance is important, you might think about the three possible cases - thicker arch and thinner deck, thinner arch and thicker deck, and both roughly equal in thickness. In the third case, would we prefer both to be thick or both to be thin?

Furthermore, all the joints between the deck and the columns, and between the columns and the arch, could be pins or hinges, and so we could imagine a multi-hinged arch. This isn't often done: it is much simpler to cast concrete as a whole than to design and build actual hinges. Even the construction called a concrete hinge usually has no actual pin: it is simple a narrow neck which is less rigid than the main body of concrete. If all the joints were hinges, then the arch segments would clearly have to be almost straight. Why not perfectly straight? In actual deck-stiffened arches that have no hinges, the segments are either straight or slightly curved, just as the main cables of a suspension bridge are slightly curved between hangers.

Let us look at this type of arch another way. Imagine a thin arch with a large number of straight sections, joined by pins or hinges, and joined to the springing by hinges as well. Now imagine two deep stiff curved beams with the same shape as the arch. Imagine connecting these beams to the arch, one on each side, but not in any way connected to the springing. The connection is to be done at each hinge, in such a way that the arch can slide relative to the beam, along the axis of both. The internal hinges of the arch are now non-functional. The arch now takes all the thrust, and the beam deals with any bending moments imposed by live loads. In a real arch, both functions are performed by the arch. If we now imagine the two beams being disconnected from the arch, raised up and made straight, end then connected to the arch by struts to the hinges, we have a deck stiffened arch.

Thus we see that even a purely funicular arch becomes "impure" when a live load is added, for it takes on beam function as well. A dual function is not unusual in structures, and must be allowed for by considering all possible live loads, including wind, water and ice where necessary. Even a strut can be "impure" if the applied forces are eccentric, that is, not along the axis of symmetry. Such forces are especially dangerous if they increase the probability of buckling in plates.