This section is from the book "The American House Carpenter", by R. G. Hatfield. Also available from Amazon: The American House Carpenter.

A comparison of pressures in timbers, according to their position, may be readily made by drawing various designs of framing and estimating the several strains in accordance with the parallelogram of forces, always drawing the triangle b d e so that the three lines shall be parallel with the three forces or pressures respectively. The length of the lines forming this triangle is unimportant, but it will be found more convenient if the line drawn parallel with the known force is made to contain as many inches as the known force contains pounds, or as many tenths of an inch as pounds, or as many inches as to is, or tenths of an inch as tons; or, in general, as many divisions of any convenient scale as there are units of weight or pressure in the known force. If drawn in this manner, then the number of divisions of the same scale found in the other two lines of the triangle will equal the units of pressure or weight of the other two forces respectively, and the pressures sought will be ascertained simply by applying the scale to the lines of the triangle.

For example, in Fig. 23, the vertical line b d, of the triangle, measures fifty-five hundredths of an inch (0.55 inch); the line be, fifty hundredths (0.50 inch); and the line e d, forty (0.40 inch). Now, if it be supposed that the vertical pressure, or the weight suspended below b d, is equal to 55 pounds, then the pressure on A will equal 50 pounds, and that on B will equal 40 pounds; for, by the proportion above stated,

b d: W::b e: P, 55:55:50:50;

and so of the other pressure.

If a scale cannot be had of equal proportions with the forces, the arithmetical process will be shortened somewhat by making the line of the triangle that represents the known weight equal to unity of a decimally divided scale, then the other lines will be measured in tenths or hundredths; and in the numerical statement of the proportions between the lines and forces, the first term being unity, the fourth term wvill be ascertained simply by multiplying the second and third terms together.

For example, if the three lines are 1, 0.7, and T.3, and the known weight is 6 tons, then

b d:W:: be : P becomes 1:6:: 0.7: P = 4.2,

equals four and two tenths tons. Again -

b d: W:: e d: Q becomes 1: 6:: 1.3: Q = 7.8,

equals seven and eight tenths tons.

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