and the maximum stress due to the bending moment is (57) (58) whence, combining, we obtain the maximum and minimum intensities in the outer and inner flanges respectively, Another expression for the value of I, which neglects the thickness of the flanges and assumes their areas concentrated on a centre line in each case situated a distance, d, apart, with k as the area of the web, is k 6 (60) If only an approximation be required, may be ignored as very When the cross-section of the girder is not symmetrical about the centre of gravity, the position of the latter may be obtained by dividing the depth of the girder inversely as the ratio of the flange areas. Thus, if a be the area of the smaller flange and a, that of the larger, the distance of the centre of gravity from the larger flange will be where b is the horizontal dimension of the girder. Or it may be obtained graphically, thus :-Let A B (fig. 264) represent the web of the girder as a single line; set off horizontally AC = area of flange B, and BD = area of flange A. Join CD, and O is the required centre of gravity. Using the notation of (62) a fairly approximate value for I is For built girders, the moment of inertia will have to be calculated in detail from its component parts. Gates with Vertical Co-planar Girders.—For straight or flat gates with discontinuous horizontal members, a different method of stress investigation is necessary. The system of co-planar vertical girders which derive no support from each other, such as contiguous vertical voussoirs do, involves, as has already been pointed out, the use of two horizontal transoms, one at the head and the other at the sill, to afford them the necessary support. As the pressure increases with the depth, the total pressure, distributed unequally between these two members, in the proportion of wh2 1 to 2, the amount being at the top and wh2 at the sill. The verticals may accordingly be treated as independent beams, sustaining a uniformly increasing load. Under these conditions it is evident that the maximum bending moment cannot be at the centre. The bending moment at any point X may be found thus :-The pressure on the surface To obtain the maximum value of this expression it is only necessary to differentiate with respect to x and equate to zero. whence d M. dx (h2 = dx dx In other words, the maximum bending moment is situated, not at the h 3' h centre of pressure but at a point below the surface of the water. √3 and the dimensions of the girder can easily be calculated by any of the methods applied to instances of beams under similar conditions of loading. Subsidiary horizontal members are introduced between the verticals to transmit the pressure from the plating, which is made as thin as is consistent with durability and strength. Stresses in Panels.-For all practical purposes the pressure on each unsupported area of plating or planking between the gate framing may be taken as uniform, an assumption which is, of course, to a certain extent, erroneous. The variation from the truth is greatest in the case of the topmost panels, and the approach to accuracy increases with the depth. No account is taken, generally speaking, of the support derived from the fixture of the ends, nor, in cambered gates, of that due to the curvature, any excess of strength in these respects being set off against possible loss from corrosion or decay. Calling the shorter unsupported length of the panel l, and d the depth of the centre of the panel below the surface, the maximum bending moment wd 12 is 8 Then, if t be the thickness of the plate and ƒ the safe maximum fibre stress, the moment of resistance is whence ft2 ; and, equating, In the foregoing expression the unit is the foot. It will be, perhaps, more convenient to express t and l in inches. Giving w its numerical value, the expression then becomes Mr. Ivan C. Boobnoff, naval architect of the Imperial Russian Navy, proposes to calculate the thickness of plating for ships by a similar formula, deduced in a rather more elaborate manner These are theoretical thicknesses. (69) There is in practice a minimum of inch for iron and steel and 3 inches for wood, beyond which it is not safe to go, on account of the exceptionally rough usage to which the panels are subjected and their liability to corrosion and decay. Practical Illustrations. It will be useful at this stage to take an actual pair of gates and see how far their construction conforms to the theoretical requirements of the preceding formulæ. Examples of both wood and iron gates have been selected for this purpose, as representing two widely distinct types, the main dimensions of the entrances which they close being, as far as possible, alike, in order that a certain comparison may be instituted between them. For the plans and particulars relating to the metal gates the author is indebted to the courtesy of Mr. J. M. Moncrieff, of Messrs. Sandeman & Moncrieff, Newcastle-upon-Tyne. Case 1.-Wooden Gates.-A pair of gates at Liverpool, each leaf consisting of a series of curved horizontal ribs, built in two voussoirs with connecting pieces, as shown by the drawings in figs. 266, 267, and 268. With the |