360. Hence the breaking load of a rod 20" long is more than double the breaking load of a rod of 40" long the same section; from this we learn that the sections being equal, short pillars are stronger than long pillars. It has been ascertained by experiment that the strength of a square pillar to resist compression is proportional to the square of its sectional area. Hence a rod of pine, 40" long and 1" square, having four times the section of the rod of the same length we have experimented on, would be sixteen times as strong, and consequently its breaking weight would amount to nearly a ton. The strength of a rod used as a tie depends only on its section, while the strength of a rod used as a strut depends on its length as well as on its section.
CONDITION OF A BEAM STRAINED BY A TRANSVERSE FORCE.
361. We next come to the important practical subject of the strength of timber when supporting a transverse strain; that is, when used as a beam. The nature of a transverse strain may be understood from [Fig. 51], which represents a small beam, strained by a load at its centre. [Fig. 52] shows two supports 40" apart, across which a rod of pine 48" × 1" × 1" is laid; at the middle of this rod a hook is placed, from which a tray for the reception of weights is suspended. A rod thus supported, and bearing weights, is said to be strained transversely. A rafter of a roof, the flooring of a room, a gangway from the wharf to a ship, many forms of bridge, and innumerable other examples, might be given of beams strained in this manner. To this important subject we shall devote the remainder of this lecture and the whole of the next.
362. The first point to be noticed is the deflection of the beam from which a weight is suspended. The beam is at first horizontal; but as the weight in the tray is augmented, the beam gradually curves downwards until, when the weight reaches a certain amount, the beam breaks across in the middle and the tray falls.
Fig. 51.
For convenience in recording the experiments the tray chain and hooks have been adjusted to weigh exactly 14 lbs. ([Fig. 52]). a b is a cord which is kept stretched by the little weights d: this cord gives a rough measure of the deflection of the beam from its horizontal position when strained by a load in the tray. In order to observe the deflection accurately an instrument is used called the cathetometer (g). It consists of a small telescope, always directed horizontally, though capable of being moved up and down a vertical triangular pillar; on one of the sides of the pillar a scale is engraved, so that the height of the telescope in any position can be accurately determined. The cathetometer is levelled by means of the screws h h, so that the triangular pillar on which the telescope slides is accurately vertical: the dotted line shows the direction of the visual ray when the centre c of the beam is seen by the observer through the telescope.
Fig. 52.
Inside the telescope and at its focus a line of spider’s web is fixed horizontally; on the bar to be observed, and near its middle point c, a cross of two fine lines is marked. The tray being removed, the beam becomes horizontal; the telescope of the cathetometer is then directed towards the beam, so that the lines marked upon it can be seen distinctly. By means of a screw the telescope may be raised or lowered until the spider’s web inside the telescope is observed to pass through the image of the intersection of the lines. The scale then indicates precisely how high the telescope is on the pillar.