As the breaking weight of a beam at the centre is double that when distributed over its whole surface, and as we have in a special diagram illustrated how the pressure on. a beam decreases as the weight from the centre increases, the depth of beams in practice may be made to decrease from the centre to the ends ; the line, where this is adopted, of the upper edge of beam being a parabolic curve, a good proportion being for the ends, two-thirds of the depth of the middle. But as in buildings beams are generally made to carry materials above them, they are formed of equal depth throughout; but the breadth of the beam may be reduced where circumstances will admit of it, the ends being one-half the breadth of the middle, the outline assuming the form of a parabolic curve, which gives to the breadth of the bottom flange a direct proportion to the strength of the beam at any point. The thickness of the web of beams is about equal to that of the bottom flange, practically a little less at the point where it joins the latter, which it is made to do with a slight curve; the web, however, is not uniformly thick, but tapers as it approaches the upper flange, at the junction of which it is a trifle less in thickness than that of the latter, and to which also the junction is made by slight curves at the sides. Such is the improved arrrangements in the form of beams, that a saving of nearly one-third in the material is effected as compared with the old forms.
We have stated that the breaking weight of a beam at the centre is double that when the weight is distributed over its whole surface; but as some of our readers may not know why this is so, we may state this in the words of an authority : " The centre of gravity of each half span of a uniformly distributed weight is but half as far from its corresponding support at the end of the beam as is the centre of gravity ; if a weight suspended at the middle, and the effect of a given weight upon a beam, is necessarily and directly as the distance of its centre of gravity from the point of support." The formula used for finding the area, in square inches, of a cast-iron beam or cantilever, which projects from a wall, in which it is supported at one end, and over which the weight is uniformly distributed, as in the case of a flagstone for the landing of an outside stair, is as follows: where a represents the area of the flange, in square inches; W the weight to be supported; I the length of the cantilever, measured from the surface of wall to its extremity; and d its depth in inches, the length also is taken in inches : a = W x l/12 x d ; where the weight is placed at the outer extremity of the cantilever, the formula is the same as above, only the divisor is six times the depth in place of twelve. In calculating the dimensions of wrought-iron beams, the usual form of which for spans, of what may be called ordinary length, and known as built beams, is shown in a preceding section. The constant used is .75 generally, although in the form of solid wrought-iron beams, of the section shown in a preceding paragraph, a constant as high as double this amount has been proved by the result of experiments made in the case of beams rolled at Belgian iron-works, which, singular to say, have long excelled ours in the production of this class of beams. The formula for finding the breaking weight is the same as that for cast-iron, the constant, however, being .75 instead of '25, as there stated; or the constant may be higher according as the experiments of any given section or form manufactured by any particular firm may indicate. The rule may be, however, here more particularly stated. Take the sectional area of the bottom flange in inches, and multiply the depth of the girder by this, the depth being also in inches ; multiply the result by .75, divide the product by the bearing or the span, taken in inches; the result will be breaking weight in tons at the centre of the beam. But if the load is to be distributed over the surface of the beam it will bear twice the amount, as shown by the above rule. To save the trouble attendant upon the calculation, as per above rule - we give here a statement of a few sizes of beams and the load which they will bear, with this (the load) uniformly distributed. To avoid repetition in the following statement, let the student note that the spans or bearings of the beams increase two feet in each case, the span in all the cases beginning at 6 and ending with 30 feet. The figures marked (s. d.) denote the sectional dimensions of each beam, and this is followed by the spans or bearings of the beams, thus : -
(s.d.) 12" x 5" x 5" - 36 - 27 - 2112/20 - 18,
Would read thus - Sectional Dimensions, 12 inches by 5 inches by 5 inches - 36 tons with a span of 6 feet - 27 tons with a span of 8 feet and - 2112/20 with a span of 10 feet, the span, as before stated, increasing two feet in each case stated.
(s.d.) 12 x 5 x 5-36 - 27 - 21 12/20 - 18 - 15 8/20 - 13 10/20 - 12 - 10 16/20 - 9 16/20-
9 - 8 5/20 - 7 5/20 - 6 10/20.
(s.d.) 10 x 5 x 5 - 27 10/20 - 20 12/20-16 10/20 - 13 10/20 - 12 - 10 5/20 - 9 2/20 - 8 -7 2/20
6 10/20 - 5 l2/20 - 5 - 4 8/20.
(s.d.) 8x5x5 - 18 10/20 - 13 15/20 - 11 - 9 2/20.
(s.d.) 7 16/20 - 6 15/20 - 6-5 8/20 - 5 - 4 12/20..
(s.d.) 4 5/20 - 3 18/20- 3 12/20.
In the case of trussed or latticed beams, which we have elsewhere illustrated, and which have no central web, as in the case of built or solid rolled beams; and supposing the top and bottom flanges to have equal powers of resistance respectively to compression and extension; " then the constant, whatever it is ascertained to be, is exactly four times the breaking weight per square inch of the bottom flange, the weight being taken in tons. Thus, where the beam fails, by tearing across the bottom flange, a constant of 80 will show that the breaking weight of the iron is 20 tons per square inch, a constant of 90 shows 22½ tons, etc. In this case the beam is supposed to fail only by breaking the bottom flange, and sinking vertically. Most forms of beams, excepting box beams, are double trussed, held together side by side, are much weakened however by lateral flexure; or, in other words, by twisting out of shape, instead of sinking at the middle of their length. In this case - the almost invariable case with single beams - the real breaking strength may be more than is represented by one-fourth of the constant; that is, with a constant of 80, the real breaking strength of the material, instead of being 20 tons only, may be perhaps 22 or 25 tons. It need scarcely be said, however, what we have elsewhere in a preceding paragraph stated, that much of the strength of a beam, other than a solid rolled one, depends not only upon its design, but upon its construction, the punching or drilling of the rivet holes, and the accurate adjustment of these to one another, the riveting itself, and the disposition of the angle-irons, covering or butting plates, etc.