Being in possession of the foregoing facts and deductions, the carpenter is enabled to determine the exact strain to which the parts of any piece of framing will be subjected.

Suppose it be required to determine the strain exerted on each of the pieces shown at Fig. 3. From the point where the pieces meet, draw the vertical line ab of any convenient length; from b, draw cb parallel to AC. Assuming the weight C to be five hundred pounds, we proceed as follows: Let the line ab represent the weight. Divide it into ten equal parts, and one of these again into ten others. Each one of the divisions last named - being a hundredth part of the line ab, which represents the whole weight of five hundred pounds - will represent five pounds. Measure the line cb by this scale; and as many parts as it contains, D|ultiplied by five, will be the number of pounds the piece AC must support Proceed in the same way with ca, and the result will be the weight supported by BC*

The horizontal strain exerted on the tie-beam may be determined as follows: From the point c, Fig. 4, draw a line parallel to the tie-beam. The line cd, measured by the scale as before described, will represent the pressure, or strain, exerted thereon by each piece. If the pieces be unequally inclined, as in Fig. 3, proceed as before described; and the parallel hues will represent the horizontal strain, as in Fig. 4. of will represent the vertical strain on A; and ad, the strain on B.

If a weight be applied to any part of an inclined beam, as W, Fig. 8, the direct transverse strain may be determined on the same principles. From a point beneath the centre of the weight, draw the line ab of any convenient length. From the end of the line at 6, draw cb at right angles with the beam. Having divided the line ab into a scale of parts representing the number of pounds weight at W, we have, by measuring the line cb with this scale, the number of pounds weight exerted as transverse or cross strain on the beam AB.

It may also be observed, that ac will give the force with which the ball would move down the beam; or, in other words, if the ball be fixed, it would show the force exerted in the direction of the beam, dc will represent the strain exerted on the wall, should the beam rest against it. Those strains to which the several timbers of a crane are subjected are identical with those exerted on the various timbers of most examples of framing.

* It should be borne in mind, that the particular inclination of the pieces determines the aggregate pressure; and, although the sum of the two may amount to more than the weight applied, it does not necessarily follow that the calculation is wrong.

While the crane is an exceedingly simple machine, it fully illustrates every point under consideration; and has, in consequence, become with most authors a favorite model for illustration.

Fig. 5 illustrates the nature and amount of strain a weight will exert on both a tie and a strut at the same time. Instead of the beams HE and JE of Fig. 2, as substitutes for the ropes AE and BE of Fig. 1, we may substitute, in place of rope EB, the strut CE, Fig. 5, and permit a rope AE to remain. The weight is supported in this example precisely as it was in each of the others, and the method of procedure to ascertain the respective strains is the same. Eo, as a scale representing the weight, is the scale for measuring po, which is the strain on the strut CE; and os, the strain on the tie, or rope.

Fig. 6 represents the same principle, and, in like manner, illustrates the means of determining the strain on inclined timbers. Ab represents the scale of weight. The line Ac, measured by the scale, will give the strain on AB; and be, that on CA. Should the projecting timbers be inclined downwards, the method of calculation would be the same.

If a beam be inclined against a wall, as at Fig. 7, and the inclination be less than forty-five degrees, there will be a tendency to slide; but there is an angle to which the end of the beam may be cut, so that this tendency will be entirely overcome.

This discovery is of great value to the carpenter, since the ends of truss-rafters, struts, etc, formed in accordance with the rule, will exert no lateral strain on the wood against which it abuts.

From the centre of the beam at d, draw the line ab parallel to AC. From a, draw af to the centre of the beam at/; then, from a, draw ag to the centre of the inclined beam at the lower end. ag will represent the direction in which the beam presses upon the abutment at B or g; and the parts should therefore be cut at right angles to the line named.

If we divide the line ab into a scale representing the weight on the centre of the beam, and draw be perpendicular thereto, be will represent the pressure against the abutment, or the tensile strain exerted on the beam AB.

The foregoing comprehends all the important principles relative to the strains exerted on the timbers of a frame. In calculating these, however, it is to be remembered, that the simpler a piece of framing is, so is the resolution of the forces exerted upon it; and vice versa. Although, in most instances, strains are more or less complicated, interfering with, and at times counteracting, each other, still the exact strain upon each part is susceptible of calculation; and any one who has sufficient curiosity and perseverance may, by following the rules, determine the nature and quantity of the strain exerted on any specimen of framing, however complicated.