It is now clearly shown, after all this long discussion, how little can be done to protect land from effects of floods. It is shown that whatever is done is ineffectual in some respects, and not permanent in others: to seek to protect from floods is, in short, to seek for fixed conditions where nature insists on change, which in effect means a stand up fight between man and nature, wherein man is sure to get the worst of it, if not at once, at any rate in the long run. The discussion also seems most unprofitable, as it is impossible to reach any definite conclusions. and no rules can be laid down which may be relied on in every case. And so, after all this long consideration of the subject, all that has been achieved is to point out a few things to those who are looking for safe guidance in flood matters which should not be done, leaving what should be done to the good judgment of the sufferer from damage by floods.

ON A GRAPHIC METHOD OF

DETERMINING

THE CHANGE OF FORM OF FRAMED STRUCTURES UNDER STRESS.

By Professor W. C. KERNOT, M.A., M.C.E.

[Plates.]

. WHEN a framed structure, such as, for example, the girder or truss of a roof or bridge, or a trestle-tower for carrying a railway viaduct or elevated water-tank, is loaded, its form is altered by the elastic deprivation of its various elements. In well-designed structures these elements are usually exposed to longitudinal thrusts and pulls, all bending actions being eliminated as far as possible. From these calculable thrusts and pulls, the cross-sectional areas, lengths, and direct modulus of elasticity of the material, the change in length of each element is easily determined. The problem that next presents itself is this" Having given the change in length of the various elements, to determine the alteration in form of the whole structure." As such structures almost always consist of a series of triangles, this may be done by plane trigonometry. Taking the altered lengths of the sides of the triangles, the alteration in the angles may be computed, and then by a further calculation based on the alteration of both sides and angles, the movement of any point in the structure from its original position may be determined. The whole calculation involves but simple mathematics, but is very laborious. and consequently rarely attempted. To find a To find a rapid and convenient mode of arriving at the same result within such limits of approximation as are needed for practical purposes appeared therefore desirable, and after some consideration the writer was led to adopt a graphic method suited for rapid and convenient use in the drawing office. Of this, as of other graphic operations now popular with engineers, it may be remarked that, while subject to small errors due to imperfections of draughtsmanship, such errors are of no practical moment, being under ordinary office conditions less in magnitude than the inevitable uncertainty in the computed stresses and modulus of elasticity of the material. The system is best illustrated by a series of examples, commencing with a very simple case, and proceeding thence to more complex ones.

Let ABC in Fig. 1 represent a bracket attached to a rigid wall, and loaded at B. It is required to find the magnitude and direction of the movement of B when the load