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.

B:I,

we have for the larger beam the ratio -

W: bd2

l

Therefore, putting these ratios in an expressed proportion, we have -

B:I:: W: bd2.

l

From which (the product of the means equalling the product of the extremes; see Art. 373) we have -

W = Bbd2 (19.)

l

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.

##### Limit Of Weight At Middle

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

al

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.