While in trusses without superfluous members, or those supposed to be so, it is quite a simple matter to determine the stresses acting in their members, we are in much ignorance as to the nature and amount of the stresses in certain parts of plategirders, usually considered to be the simplest form of girders, although they have been the subject of elaborate mathematical investigations, especially by Prof. Airy and Mon. Bresse.

In trusses like ordinary lattice girders, any one can see at once that the bending moment produces two parallel stresses resisted by two chords, and that the shear produces in the web-members stresses whose directions correspond to those of the resisting members themselves, in order to form proper reactions at both ends of the girder. The amounts of all these stresses can be easily calculated by well-known methods. It it not so with continuous web-plate-girders. Even after we consider, as is most frequently done, the function of resisting flange stresses as taken off the web and consider the flanges alone as taking care of the bending moment, the ambiguity of the stresses in the webs still remains. Prof. Airy communicated the result of his investigation to the Royal Society in 1862, under the title, "On the Strains in the Interior of Beams." Starting from the consideration that a beam is composed of laminae in a vertical plane, and that any number of forces acting at a point may always be replaced by two forces at right angles to each other, he derived equations of equilibrium, with which he determined the magnitudes and directions of these two forces in several parts of a beam under various modes of loading and supporting. His investigation verified the already existing belief that stresses, compression and tension, take place in the web at the angle of 45°, a fact which Stephenson more than 40 years ago found from his experiments on the model of the Britannia Tubular Bridge. The days for Britannia and Conway tubes are now gone, but in those days there was much controversy as to the relative merits of plate and open-girders for large spans. Starting from the results of his experiment Stephenson argued that since a plate-girder is nothing more than a lattice-girder with web bars put close together, and that the bar in tension may at the same time be subjected to compression in the direction of its width, one-half the amount of metal can be saved if the web be a continuous plate. On the other hand, it has been rightly argued that, on account of the impossibility of the accurate determination of the stresses in the web, and of other practical causes, a larger amount of material is required to cover that ambiguity for plate-girders, than for lattice-girders, in which a more economical distribution of material is possible.

It is beyond the scope of our present work to enter into the discussion of mathematical investigations of continuous web strains, nor is it of much use when we consider in the light of experiments how how different from what the theory asserts the actual state of affairs in a piece of material subjected to stresses in different directions often is. Under such circumstances, the less strained part tends to help the more strained part, and the stresses change from one direction to another in manners depending upon the intensities of stresses and molecular arrangements of the materials, and thus defy all possibility of correct determination. On this point the experiments of Mr. Baker* on the strength of beams is of great interest.

In order, however, to form some notion of the nature of stresses in a continuous web, and to derive formulas necessary for proportioning the same, we shall view the matter in the simplest manner possible. In a beam supported at both ends, and subjected to the action of a vertical force, it is evident that at any of its vertical sections, by the virtue of the bending stress, that part of the beam above the neutral axis is subjected to compression, and that below to tension, both of which stresses attain maximum values at the outermost fibres of the beam, and decrease to zero at the neutral axis. This intensity of the stress at any point is at once obtained from the well-known equation of flexure:

* See Minutes of the Institution of Civil Engineers, 1880.

(M/I) y =f---------------(1)

i. e., the bending moment M, divided by the moment of inertia I of the section of the beam, multiplied by the distance y of the point from the neutral axis, gives the intensity f of the stress at that point.