respectively under tension and compression, be parallel to one another, then the stress upon these flanges will vary as the bending-moments, and will accordingly be very severe in the middle and will dwindle towards the ends. But if we make the depth of the girder everywhere proportional to the bending-moments, that is
to say if we copy in the girder the outlines of the bending-moment diagram, then our design will automatically meet the circumstances of the case, for the horizontal stress in each flange will now be uniform throughout the length of the girder. In short, in {697} Professor Fidlerãs words, ãEvery diagram of moments represents the outline of a framed structure which will carry the given load with a uniform horizontal stress in the principal members.ã
In the following diagrams (Fig. 342, a, b) (which are taken from the original ones of Culmann), we see at once that the loaded beam or bracket (a) has a ãdanger-pointã close to its fixed base, that is to say at the point remotest from its load. But in the parabolic bracket (b) there is no danger-point at all, for the dimensions of the structure are made to increase pari passu with the bending-moments: stress and resistance vary together. Again in Fig. 340, we have a simple span (A), with its stress diagram (B); and in Fig. 341 we have the corôÙreôÙsponôÙding parabolic girder, whose stresses are now uniform throughout. In fact we see that, by a process of conversion, the stress diagram in each case becomes the structural diagram in the other22. Now all this is but the modern rendering of one of Galileoãs most famous propositions. In the Dialogue which we have already quoted more than once23, Sagredo says ãIt would be a fine thing if one could discover the proper shape to give a