In order that the diminution of the strength of a beam by framing be as small as possible, all mortices should be located at or near the middle of the depth. There is a prevalent idea with some, who are aware that the upper fibres of a beam are compressed when subject to cross-strains, that it is not injurious to cut these top fibres, provided that the cutting be for the insertion of another piece of timber - as in the case of gaining the ends of beams into the side of a girder. They suppose that the piece filled in will as effectually resist the compression as the part removed would have done, had it not been taken out. Now, besides the effect of shrinkage, which of itself is quite sufficient to prevent the proper resistance to the strain, there is the mechanical difficulty of fitting the joints perfectly throughout; and, also, a great loss in the power of resistance, as the material is so much less capable of resistance when pressed at right angles to the direction of the fibres than when directly with them, as the results of the experiments in the tables show.
122. - Transverse Strains: Relation of Weight to Dimensions. - The strength of various materials, in their resistance to cross-strains, is given in Table III., Art. 96. The second column of the table contains the results of experiments made to test their resistance to rupture. In the case of each material, the figures given and represented by B indicate the pounds at the middle required to break a unit of the material, or a piece 1 inch square and 1 foot long between the bearings upon which the piece rests. To be able to use these indices of strength, in the computation of the strength of large beams, it is requisite, first, to establish the relation between the unit of material and the larger beam. Now, it may be easily comprehended that the strength of beams will be in proportion to their breadth; that is, when the length and depth remain the same, the strength will be directly as the breadth; for it is evident that a beam 2 inches broad will bear twice as much as one which is only 1 inch broad, or that one which is 6 inches broad will bear three times as much as one which is 2 inches broad. This establishes the relation of the weight to the breadth. With the depth, however, the relation is different; the strength is greater than simply in proportion to the depth. If the boards cut from a squared piece of timber be piled up in the order in which they came from the timber, and be loaded with a heavy weight at the middle, the boards will deflect or sag much more than they would have done in the timber before sawing. The greater strength of the material when in a solid piece of timber is due to the cohesion of the fibres at the line of separation, by which the several boards, before sawing, are prevented from sliding upon each other, and thus the resistance to compression and tension is made to contribute to the strength. This resistance is found to be in proportion to the depth. Thus the strength due to the depth is, first, that which arises from the quantity of the material (the greater the depth, the more the material), which is in proportion to the depth; then, that which ensues from the cohesion of the fibres in such a manner as to prevent sliding; this is also as the depth. Combining the two, we have, as the total result, the resistance in proportion to the square of the depth. The relation between the weight and the length is such that the longer the beam is, the less it will resist; a beam which is 20 feet long will sustain only half as much as one which is 10 feet long; the breadth and depth each being the same in the two beams. From this it results that the resistance is inversely in proportion to the length. To obtain, therefore, the relation between the strength of the unit of material and that of a larger beam, we have these facts, namely: the strength of the unit is the value of B, as recorded in Table III.;. and the strength of the larger beam, represented. by W, the weight required to break it, is the product of the breadth into the square of the depth, divided by the length; or, while for the unit we have the ratio -
we have for the larger beam the ratio -
Therefore, putting these ratios in an expressed proportion, we have -
B:I:: W: bd2.
From which (the product of the means equalling the product of the extremes; see Art. 373) we have -
W = Bbd2 (19.)
In which W represents the pounds required to break a beam, when acting at the middle between the two supports upon which the beam is laid; of which beam b represents the breadth and d the depth, both in inches, and / the length in feet between the supports; and B is from Table III., and represents the pounds required to break a unit of material like that contained in the larger beam.
123. - Safe Weight: Load at Middle. - The relation established, in the last article, between the weight and the dimensions is that which exists at the moment of rupture. The rule (19.) derived therefrom is not, therefore, directly practicable for computing the dimensions of beams for buildings. From it, however, one may readily be deduced which shall be practicable. In the fifth column of Table III. are given the least values of a, the factor of safety, explained in Art. 96. Now, if in place of B, the symbol for the breaking weight, the quotient of B divided by a be substituted, then the rule at once becomes practicable; the results now being in consonance with the requirements for materials used in buildings. Thus, with this modification, we have -
W = Bbd2
Therefore, to ascertain the weight which a beam may be safely loaded with at the centre, we have -
Rule XVI. - Multiply the value of B, Table HI., for the kind of material in the beam by the breadth and by the square of the depth of the beam in inches; divide the product by the product of the factor of safety into the length of the beam between bearings in feet, and the quotient will be the weight in pounds that the beam will safely sustain at the middle of its length.
Example. - What weight in pounds can be suspended safely from the middle of a Georgia-pine beam 4x10 inches, and 20 feet long between the bearings? For Georgia pine the value of B, in Table III., is 850, and the least value of a is 1.84. For reasons given in Art. g6, let a be taken as high as 4; then, in this case, the value of b is 4, and that of d is 10, while that of l is 20. Therefore, proceeding by the rule, 850 x 4 x 102 = 340000; this divided by 4 x 20 (= 80) gives a quotient of 4250 pounds, the required weight.
Observe that, had the value of a been taken at 3, instead of 4, the result by the rule would have been a load of 5667 pounds, instead of 4250, and the larger amount would be none too much for a safe load upon such a beam; although, with it, the deflection would be one third greater than with the lesser load. The value of a should always be assigned higher than the figures of the table, which show it at its least value; but just how much higher must depend upon the firmness required and the conditions of each particular case.