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
Fig. 341. The bridge constructed, as a parabolic girder.
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 Fidler’s words, “Every 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.”
Fig. 342.
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 “danger-point” 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 corresponding 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 other[629]. Now all this is but the modern rendering of one of Galileo’s most famous propositions. In the Dialogue which we have already quoted more than once[630], Sagredo says “It would be a fine thing if one could discover the proper shape to give a solid in order to make it equally resistant at every point, in which case a load placed at the middle would not produce fracture more easily than if placed at any other point.” And Galileo (in the person of Salviati) first puts the problem into its more general form; and then shews us how, by giving a parabolic outline to our beam, we have its simple and comprehensive solution.
In the case of our cantilever bridge, we shew the primitive girder {698} in Fig. [343], A, with its bending-moment diagram (B); and it is evident that, if we turn this diagram upside down, it will still be illustrative, just as before, of the bending-moments from point to point: for as yet it is merely a diagram, or graph, of relative magnitudes.
To either of these two stress diagrams, direct or inverted, we may fit the design of the construction, as in Figs. [343], C and 344.
