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

This relation between lines and pressures is applicable in ascertaining the pressures induced by known weights throughout any system of framing. The parallelogram b e df is called the Parallelogram of Forces; the two lines b e and bf being called the components, and the line b d the resultant. Where it is required to find the components from a given resultant (Fig. 22-B), the fourth line df need not be drawn, for the triangle b d e gives the desired result. But when the resultant is to be ascertained from given components (Fig. 28), it is more convenient to draw the fourth line.

73. - The Resolution of Forces: is the finding of two or more forces which, acting in different directions, shall exactly balance the pressure of any given single force. To make a practical application of this, let it be required to ascertain the oblique pressure in Fig. 22-B. In this figure the line bd measures half an inch (0.5 inch), and the line be three tenths of an inch (0.3 inch). Now if the weight W be supposed to be 1200 pounds, then the first stated proportion above, b d: W:: b e: P, becomes 0.5: 1200::0.3: P.

And since the product of the means divided by one of the extremes gives the other extreme, this proportion may be put in the form of an equation, thus -

1200x0.3/0.5 = P.

Performing the arithmetical operation here indicated - that is, multiplying together the two quantities above the line, and dividing the product by the quantity under the line - the

quotient will be equal to the quantity represented by P, viz., the pressure resisted by the timber C. Thus -

1200 0.3

0-5)360.0

720 = P.

The strain upon the timber C is, therefore, equal to 720 pounds; and since, in this case (the two timbers being inclined equally from the vertical), the line e d is equal to the line b e, therefore the strain upon the other timber D is also 720 pounds.

Fig. 23.

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