A BEAM UNIFORMLY LOADED.
405. We have up to the present only considered the case where the load is suspended from the centre of the beam. But in the actual employment of beams the load is not generally applied in this manner. See in the rafters which support a roof how every inch in the entire length has its burden of slates to bear. The beams which support a warehouse floor have to carry their load in whatever manner the goods are disposed: sometimes, as for example in a grain-store, the pressure will be tolerably uniform along the beams, while if the weights be irregularly scattered on the floor, there will be corresponding inequalities in the mode in which the loads are distributed over the beams. It will therefore be useful for us to examine the strength of a beam when its load is applied otherwise than at the centre.
406. We shall employ, in the first place, a beam 40" span, 0"·5 broad, and 1" deep; and we shall break it by applying a load simultaneously at two points, as may be most conveniently done by the contrivance shown in the diagram, [Fig. 53]. a b is the beam resting on two supports; c and d are the points of trisection of the span; from whence loops descend, which carry an iron bar p q; at the centre r of which a weight w is suspended. The load is thus divided equally between the two points c and d, and we may regard a b as a beam loaded at its two points of trisection. The tray and weights are employed which we have used in the apparatus represented in [Fig. 58].
Fig. 53.
407. We proceed to break this beam. Adding weights to the tray, we see that it yields with 117 lbs., and cracks across between c and d. On reference to [Table XXIV]. we find from experiment No. 2 that a similar bar was broken by 77 lbs. at the centre; now ³/₂ × 77 = 115·5; hence we may state with sufficient approximation that the bar is half as strong again when the load is suspended from the two points of trisection as it is when suspended from the centre. It is remarkable that in breaking the beam in this manner the fracture is equally likely to occur at any point between c and d.
408. A beam uniformly loaded requires twice as much load to break it as would be sufficient if the load were merely suspended from the centre. The mode of applying a load uniformly is shown in [Fig. 54].
Fig. 54.
In an experiment actually tried, a beam 40" × 0"·5 × 1" placed edgewise was found to support ten 14 lb. weights ranged as in the figure; one or two stone more would, however, doubtless produce fracture.
409. We infer from these considerations that beams loaded in the manner in which they are usually employed are considerably stronger than would be indicated by the results in [Table XXIV].
EFFECT OF SECURING THE ENDS OF A BEAM
UPON ITS STRENGTH.
410. It has been noticed during the experiments that when the weights are suspended from a beam and the beam begins to deflect, the ends curve upwards from the supports. This bending of the ends is for example shown in [Fig. 54]. If we restrain the ends of the beam from bending up in this manner, we shall add very considerably to its strength. This we can do by clamping them down to the supports.
411. Let us experiment upon a beam 40" × 1" × 1". We clamp each of the ends and then break the beam by a weight suspended from the centre. It requires 238 lbs. to accomplish fracture. This is a little more than half as much again as 152 lbs., which we find from [Table XXIV]. was the weight required to break this bar when its ends were free. Calculation shows that the strength of a beam may be even doubled when the ends are kept horizontal by more perfect methods than we have used.
412. When the beam gives way under these circumstances, there is not only a fracture in the centre, but each of the halves are also found to be broken across near the points of support; the necessity for three fractures instead of one explains the increase of strength obtained by restraining the ends to the horizontal direction.
413. In structures the beams are generally more or less secured at each end, and are therefore more capable of bearing resistance than would be indicated by [Table XXIV]. From the consideration of [Arts. 408] and [411], we can infer that a beam secured at each end and uniformly loaded would require three or four times as much load to break it as would be sufficient if the ends were free and if the load were applied at the centre.
BEAMS SECURED AT ONE END AND
LOADED AT THE OTHER.
414. A beam, one end of which is firmly imbedded in masonry or otherwise secured, is occasionally called upon to support a weight suspended from its extremity. Such a beam is shown in [Fig. 55].
In the case we shall examine, a b is a pine beam of dimensions 20" × 0"·5 × 0"·5, and we find that, when w reaches 10 lbs., the beam breaks. In experiment No. 8, [Table XXIV]., a similar beam required 36 lbs.; hence we see that the beam is broken in the manner of [Fig. 55], by about one-fourth of the load which would have been required if the beam had been supported at each end and laden in the centre.
Fig. 55.
We shall presently have occasion to apply some of the results obtained by the experiments made in the lecture now terminated.
LECTURE XIII.
THE PRINCIPLES OF FRAMEWORK.
Introduction.—Weight sustained by Tie and Strut.—Bridge with Two Struts.—Bridge with Four Struts.—Bridge with Two Ties.—Simple Form of Trussed Bridge.