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x 8.175 = .061 = 82.29, the value of Mw, as in Table 62: for the value of My for Beams, see (322).
Col. 11 of Table 55 has been calculated by the rule (308); it will be observed that the actual experimental strain in col. 12, is often less than the wrinkling strain in col. 11 ; this is due to the fact that the strains in col. 12 are complicated by flexure (306), and are affected by the length of the pillar, as shown for example by Nos. 27, 28, and 29.
TABLE 62.-Of the Value of Mw for the RESISTANCE of THIN
WROUGHT-IRON PLATES to WRINKLING.
(313.) It should be observed that the wrinkling strain is independent of the length of the plate; this is shown by the
same experiments, for although with lengths of 10 and 2 feet respectively, the actual compressive strains in col. 12 varied from 4.902 to 5.537 tons per square inch, when correction is made for flexure, and the effect of the length is thus eliminated, col. 8 shows the same error in both, or 22• 2 and 22: 4 per cent. respectively.
Another proof that the wrinkling strain is independent of length, is that even with long pillars, the plate often fails near the end ; for instance, No. 2 was 10 feet long, but failed by wrinkling 14 inches from one end. No. 8 was 5 feet long, but gave way at 7 inches from one end. Now the crushing strain due to flexure is a maximum at the centre, and is reduced progressively towards the ends, where it becomes nil; but the crushing strain due to direct pressure is the same from end to end. These two facts, that the length has so little effect on the strength of the pillar; and that failure by wrinkling takes place indifferently in any part of the length, show that the wrinkling strain is independent of the length of the plate.
(314.) An obvious and economical method of increasing the strength of a plate in resisting wrinkling, is by adding vertical ribs as at A, B, &c., in Fig. 62, which in effect reduce the breadth, and thereby increase the strength in a much higher ratio than the weight:—thus one central rib reduces the width to half, and the wrinkling strain is increased in the ratio of v 2 to 71, or from 1.0 to 1.41, or 41 per cent. Similarly two ribs give an increase of N3 = 1.73, or 73 per cent., &c.
(315.) It should be observed that the total wrinkling strain increases in a much higher ratio than the square-root of the thickness which governs the resistance per square inch only: thus for thicknesses in the ratio
the wrinkling strain per square inch follows the ratio tw and becomes,
1 1.41 1.73 2 2.24 2.45
But the areas are also increased in the simple proportion of the
thickness, and the total strains are therefore increased in the ratio, w tw x tw, or tw?-5, and become :
See (396) and Tables 74 and 75.
The total strain due to flexure is practically simply proportional to the thickness of the plate :—thus with a pillar 12 inches square externally, and to inch thick, therefore 113 inches internally, D3-6 – d3-6 becomes 284:—with } inch thick, therefore 11inches internally, D366 - 036 = 559, which is nearly proportional to the thickness.
(316.) Another result of the rules is that in order to obtain equality of strength in a rectangular pillar other than square, the thickness of the plates should be simply proportional to the breadths :—thus, if the ratio of the sides is 3 to 1, the thicknesses should be in the same ratio (472).
The absolute crushing strength of wrought iron in pillars is 19 tons per square inch, and in order to obtain the full value of the material, the wrinkling strain should not be less than that. For example, No. 4 in Table 55 failed with 5.926 tons per square inch, col. 12, being very nearly the calculated Wrinkling strain in col. 11, whereas by crushing it would not have failed with less than 19 tons, so that 5.926 • 19 = • 31, or 31 per cent. only of the strength of the material is utilised and 69 per cent. is wasted. We can easily find the ratio of the breadth to the thickness which is necessary to secure that equality of strength :-say we take 1 inch thick, then the breadth of a plate supported at both edges to give Ww = 19 tons will be given by the rule (310), which becomes bw = (Vi x 80 - 19) 17.72, or say 18 inches. The same thickness of plate supported at one edge only, would have a breadth of 18 : 4 = 41 inches projecting beyond the angle-iron, as P in Fig. 93. These dimensions apply only to plates subjected to direct pressure as in a pillar, and they may be taken as ratios applicable to all thicknesses.
(317.) For plates 1 inch thick forming part of a plate-iron tubular beam or girder, and supported at both edges as in Fig. 58, the value of My = 104, and the rule (310) gives the breadth
for Ww = 19 tons, bw = ( 71 x 104 - 19) = 29.96, say 30 inches :—for the top flange of a girder, supported at one edge only, and measured as in Fig. 60, the breadth would be 30 - 4 = 74 inches.
Calculating in this way we may find the breadths of wroughtiron plates under different conditions, such that the Wrinkling Strain shall be equal to the Crushing strain, or 19 tons per square inch in all cases : for thicknesses of :
1 inch, the breadth of plate supported at both edges and forming part of a pillar = 21 41 63
114 13) 151 18 inches. The same plates forming the top of a Tubular beam would have breadths of:
31 77 111 15 189 221 261 30 inches. The breadths for plates supported at one edge only in pillars become : ਚ 11 14 27 23
3; 44 inches, and in plates forming the top flange of a plate-iron girder, Fig. 93, we have :
1 13 21 31 44 55 61 7) inches. With breadths greater than those given above the Wrinkling strain would be less than 19 tons per square inch: with less breadths the wrinkling strain by calculation would come out more than 19 tons, but this would not be realised; in that case the strength of the plate would be limited by the crushing strength.
(318.) Let Fig. 62 be the section of a short pier for a bridge, &c., 6 feet square, of wrought-iron plate inch thick, strengthened with T ribs A, B, &c., giving 11} inches between their edges, we should then have 19 tons per square inch wrinkling strain, and should thus have obtained the utmost possible effect from the material. If in this case we dispense with the ribs,