Substituting the modulus of rupture / (Table VIII.), for l in equation 3, 14, we can from the above values 21 of calculate the ultimate strength of any given cross (JL section to resist any given bending moment. Thus, if we wish to know what breadth we must give a bar of wrought iron 3 inches deep, supported at points 3 feet apart, so that it may break with 2000 K> at the centre, we find the maximum bending moment from equation 2, 16; and we find the value of p from equation 3, 14, by the help of table IX. Then equating M and p, we have—
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6WL 6x2000x36 . from which b = - - = 4x9x42000
Since the strength of the beam is directly proportional to 2I the values of -, table IX. shows us how to dispose of the ct material in the cross section so as to obtain the maximum strength. The expression for a rectangle shows that the strength is proportional to the breadth of a beam, but to the square of its depth ; and therefore, as appears in the second column, for the same quantity of material the strength of a rectangular beam increases simply as the depth, so that a deep narrow beam is preferable to a square or to a broad and flat one. The circular cross section is weak compared even with the square (the ratio for the same quantity of material is -846 to 1). The hollow tube is stronger than the thin rectangular plate of the same depth and cross section, but clearly the material is much best applied when whollyused in the form of two thin flat plates, separated from one another by a web of the maximum depth d which can be practically allowed. Thus the hollow rectangle is the form preferred for large girders, the material intended to resist the bending moment being placed .in the top and bottom members of the girder, and kept apart by vertical webs, which add somewhat to the moment //., but which are chiefly employed to resist shearing stress, as will be presently shown.
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Fig. 16.
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Fig. 17.
the moment of its elastic forces is exactly the same as that of a hollow rectangle, having the same values for b, d, and d v and having a value for b l equal to 2b. 2 in fig. 1C It is usual to neglect the strength to resist bending moment given by the vertical web, and to design the girder so that the top and bottom plates are alone sufficient to meet this stress. The model, fig. 10, shows at once that to obtain the full resistance from the material employed to resist tension or compression consistently with a given depth, it must all be placed in two horizontal plates at the extreme top and bottom of the beam ; moreover, if S is the sectional area of the top or bottom member, and /is the stress it will bear (assumed equal for tension and compression), the maximum moment which the elastic forces can exert is the sum of the moments due to the top and bottom members about the axis or tongue, i.e., 2(?/Sx?cZ) or J/ScZ; or dismissing the idea of an axis through the centre of gravity of the section, we may (to refer again to the illustrative model) take the moment of the elastic forces round the line running through the top row of india-rubber pieces looked upon as a fulcrum, and then the moment of the bottom row will be as before ir/Sc/, and will be the ivhole moment of the elastic forces. Similarly, by taking the bottom row as the fulcrum, we see that the moment due to the top row will be the same. This moment calculated in either way is the ivhole moment of the elastic forces. This latter mode of considering the moment in the simple case of an IE or hollow rect angle, enables us to find the value of the moment of elastic forces for those cases in which the ultimate strength may not be the same for tension and compression. Thus, assume that the strength of wrought iron to resist tension, or /, is 25 tons, and its strength to resist com pression, or /., is 20 tons, then calling S t the area of the bottom member at the section considered, and S c the area of the top member, we have for /u, the two values 25 S t d and 20 S c c; whichever happens to be the smaller will represent the available value of the moment of the elastic forces. Consequently we must, in order not to waste material, design the beam so that the ratio of the material in the upper member to that in the lower member shall be 5 to 4. That this ratio ought to be adopted is evident from the fact that the strength of the beam will be limited by that of the weaker member. No structure is perfectly designed unless it will when overstrained give way simul taneously in every part. The foregoing theory, the sound ness of which is borne out by experiment, tacitly assumes that, although the strength differs, the modulus of elasticity is constant for tension and compression. For example, if the flanges are made with sections bearing to one another the proportion of 4 to 5, the neutral axis (neglecting the web) will, assuming E constant, be at a distance from the top or larger flange equal to ^ths of the depth ; then the intensity of stress varying directly as the distance from the neutral axis will for the two flanges be in the desired ratio of 4 to 5. Thus we see not only that the I, or hollow rectangle, has the advantage of being the best form for a girder, but that it allows us easily to arrange the material to the best advantage, even when its strengths to resist tension and compression are dissimilar. Values of the modulus of rupture given above are therefore not to be applied to this design of beam, but the values of f t and/, are to be taken from Tables I. and II. In the case of cast-iron the member under tension is made with six times the cross section of the member under compres sion, the reason being the same as that for making the ratio of the upper and lower members of the wrought iron beam 5 to 4. When a cast-iron beam is thus designed, the moment which any section can exert will for a given depth be proportional to the area S, of the lower flange. Pro fessor Hodgkinson veri fied this theory experi mentally, and found that the ultimate value of the moment due to elastic forces expressed in Ibs and inches for beams thus designed was fji ? 16500S/Z. The value of the con stant agrees closely with the tensile strength of cast-iron. The ex periments by Professor Hodgkinson are there fore consistent with the assumption, that al though the strength of cast-iron is very different in resist
ing tension and compression, nevertheless the modulus of