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

(1.) A heavy body always exerts a pressure equal to its own weight in a vertical direction. Example: Suppose an iron ball weighing 100 lbs. be supported upon the top of a perpendicular post (Fig. 22-A); then the pressure exerted upon that post will be equal to the Aveight of the ball, viz., 100 lbs. (2.) But if two inclined posts (Fig. 22-B) be substituted for the perpendicular support, the united pressures upon these posts will be more than equal to the weight, and will be in proportion to their position. The farther apart their feet are spread the greater will be the pressure, and vice versa. Hence tremendous strains may be exerted by a comparatively small weight. And it follows, therefore, that a piece of timber intended for a strut or post should be so placed that its axis may coincide, as nearly as possible, with the direction of the pressure. The direction of the pressure of the weight W (Fig. 22-B) is in the vertical line b d; and the weight W would fall in that line if the two posts were removed; hence the best position for a support for the weight would be inthat line. But as it rarely occurs in systems of framing that weights can be supported by any single resistance, they requiring generally two or more supports (as in the case of a roof supported by its rafters), it becomes important, therefore, to know the exact amount of pressure any certain weight is capable of exerting upon oblique supports. Now, it has been ascertained that the three lines of a triangle, drawn parallel with the direction of three concurring forces in equilibrium, are in proportion respectively to these forces. For example, in Fig. 22-B, we have a representation of three forces concurring in a point, which forces are in equilibrium and at rest; thus, the weight W is one force, and the resistances exerted by the two pieces of timber are the other two forces. The direction in which the first force acts is vertical - downwards; the direction of the two other forces is in the axis of each piece of timber respectively. These three forces all tend towards the point b.

Fig. 22.

Draw the axes a b and b c of the two supports; make b d vertical, and from d draw d e and d f parallel with the axes b c and b a repectively. Then the triangle b d e has its lines parallel respectively with the direction of the three forces; thus, bd is in the direction of the weight W,d e parallel with the axis of the timber D, and e b is in the direction of the timber C. In accordance with the principle above stated, the lengths of the sides of the triangle b d e are in proportion respectively to the three forces aforesaid; thus -

As the length of the line b d

Is to the number of pounds in the weight W,

So is the length of the line b e

To the number of pounds' pressure resisted by the timber C. Again -

As the length of the line b d

Is to the number of pounds in the weight W,

So is the length of the line d e

To the number of pounds' pressure resisted by the timber D. And again -

As the length of the line b e

Is to the pounds' pressure resisted by C,

So is the length of the line d e

To the pounds' pressure resisted by D.

These proportions are more briefly stated thus -

1 st. bd:W::be:P,

P being used as a symbol to represent the number of pounds' pressure resisted by the timber C.

2d. b d: W::d e: Q,

Q representing the number of pounds' pressure resisted by the timber D.

3d. b e: P:: de: Q.

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