results, and we shall find that this is due to our having taken the web in (b) as if it extended between the centre lines of the flanges, i.e. length II inches. Let us now repeat the first calculation, but give the web its true dimensions, then I=(10×1) 102+363, a result which is less than 1 per cent in error; hence the second method should be followed in all cases if possible. We will now consider how we should proceed to find the stress produced. We have and for our section y=h/2, where h is the total depth, and not the depth of the approximate section b. We can easily see that this is the proper value to take. For, accurately, Using the approximate value of I, but taking 12" as the depth, While treating the question as if (b) were the real shape, and hence II" the total depth, Plainly (2) is practically accurate, while (3) is not so; and, moreover, it errs in the wrong direction, since it makes the stress appear less than its real value. The latter inaccuracy would have been still worse had we found the value of I by the first method of approximation given; for this would give an error of 15.7 per cent on the unsafe side. We will examine the present case also to see what amount of accuracy can be obtained by the use of the formula Hh = M. Here, if we take ʼn the mean depth from centre to centre of flange—i.e. II ins.—we obtain HX 11=M, .'. H= But H is the total stress on 6 sq. ins., i.e. 6p an error of 11.4 per cent, but on the safe side; the real p being less than this. If we take, however, h= 12, then we get M p= 72' which is practically accurate; the small error of about 3 per cent being again on the safe side. The natural depth in this case would be II ins., that being the distance between the resultant stresses on the flanges, and then the 11.4 per cent error shows the amount of help afforded by the web. In the second case above, this is partially corrected by taking the outside value of h. We have thoroughly examined the value of the approximations in this case, and from this the student will be able in any given case to judge of the best method. The approximations are of least use when the flanges are thick; and of course they should be used only for rough or preliminary calculations, final values of the stress being always obtained by the exact method. We cannot always use the more correct approxima tion, because sometimes the areas are the quantities to be found, and we do not know the thickness of the flanges but only perhaps the total depth of the beam, or it may be the mean depth. In such a case we must of necessity treat the web as if its length were the full depth; but we now know what sort of an error that causes, and hence how to allow for it. Resisting Powers of Flange and Web.-We can obtain a measure of the relative value of metal in the web and flange as follows: Fig. 278 represents an I beam, the approximate form being drawn. A= area of each flange, C= Let N N h -A Fig. 278. Then for a flange For the web Thus while A is multiplied by h2/4, C is only multiplied by h2/12, and hence area for area the metal of the flanges offers three times as much resistance as that of the web. By addition we obtain a simple approximate formula for Io of the beam, h2 h2 + C 4 12 If we consider the error of the approximation we see that we underrate A and overrate C, so that the ratio of powers is something in excess of three (compare with derivation of I section from rectangular, page 372). Beams of Equal Strength.—We saw in the last chapter that for a section of equal strength or greatest resistance the neutral axis (Fig. 268) must be so situated ᄌ -B Fig. 279. We can now proceed to see how these areas A, B, and C can be calculated when we know the moment which the beam is required to resist. We must first find I, and since we do not know the thicknesses, we must in Fig. 279 take the approximate figure. Then about the neutral axis. I of top flange = Aye, bottom flange = By2, here we split C into two rectangles, each having the neutral axis as one end, and their areas are yeh · C and Yt/h. C. C ... I=Ay2+By?2 +↓ = (yê+y?). Substituting the values of ye and yt from page 362, The equation here found, combined with the equation of page 362, is not sufficient to determine the four quantities A, B, C and h; but, as we have seen, there are other conditions which limit the values of some of these quantities, e.g. h is limited as we have seen on page 373; then, if we are given a certain value of h, this limits the thickness of the web so that C would be determined, and then the two equations would determine what should be the proportions of A and B; we should call this the best beam under the given conditions. In wrought-iron the T shape is common, since this iron is strong against tension; hence this gives us the extra condition Bo; and so for other conditions, examples of which will be given. с B Beams of Uniform Strength.-If a beam be subjected to a constant B. M. its section everywhere should be the same; but in ordinary cases the B. M. varies from point to point, and the beams are accordingly made deeper or broader where the A greater B. M.'s come. Plainly this course is economical, for take the case of a beam AB (Fig. 280) carrying a single weight W at C. K B N D Fig. 280. The diagram of B. M. is then a triangle ADB, where The section at C then must be strong enough to withstand this moment. If then the section of the beam be uniform I and y will be constant all along AB; so that at K for instance the stress will be only KN/CD times what it is at C. But there is no gain by having only this small stress at K, because the beam will be injured if the stress at C pass the limit allowed, quite irrespective of what may be the stress at other points. Moreover there is an actual loss by making the section at K larger than it need be for strength; for the |