Figure the three pieces, A C, A D, and A B, are Ces on the stretch. This is the complete inversion of Figure 13 ; and the dotted position of A induces the same changes in the forces A A as ID that figure.
All calculations about the strength of carpentry are redu cible to this case ; for when more ties, or braces, meet in a poin t (a thing that rarely happens), they are to lie reduced to three, by substituting for any two the force resulting from their combination, and combining this with another; and so tm.
The young artist must be particularly careful not to mis take the kind of strain exerted on any piece of the framing, and suppose a piece to be a brace which is really a tie. It is very easy to avoid all mistakes in this matter by the following rule, which has no exception.
Take notice of the direction wherein the piece acts that produces the strain. Draw a line in that direction from the point on which the strain is exerted ; and let its length (measured on some scale of equal parts) express the magni tude of this action in pounds, hundreds. or tons. From its remote extremity, draw lines parallel to the pieces on which the strain is exerted. The line parallel to one piece will necessarily cut the other, or its direction produced ; if it cut the piece itself, that piece is compressed by the strain, and it is performing the office of a strut, or brace: if' it cut its direction produced, the piece is stretched, and it is a tie. In short, the strains on the pieces A C, A U. are to be estimated in the direction of the points F and it from the strained point A. Thus, in Figure 13, the upright piece, B A, loaded with the weight a, presses the point A in the direction A E ; so does the rope, A B, in tire other figures, or the batten A B, in Figure 15.
In generxl, if the straining piece be within the angle formed by the pieces which are strained. the strains sustained by them are of the opposite kind to that which it exert:. If it be pushing, they are drawing; lint if' it be within the angle formed by their directions produced, the strains sus tained by them are of the same kind, All the three are either drawing or pressing. If the straining piece lie within the angle formed by one piece and the produced direction of the other, its oss 0 strain, whether compression or extension, is o1 the same kind with that of the remote of the other two, and opposite to that of the nearest. Thus, in Figure
19, where A 0 is drawing. the remote piece, A c, is also draw ing, while A D is plishing or resisting eompression.
In these calculations the resistance of joints has no share; and they must not be supposed to exert any tends to prevent the angles from changing. The joints are supposed to be perfectly flexible, or like compass joints; the pin of which only keeps the pieces together when one more of the pieces draws or pulls; of which description the trpenter must always suppose all joints to be, when he cal culates the thrusts and draughts of the dilli•ent pieces of his frames. The strains on and their power to produce or balance thent, arc of a differetit kind, and require a very different examination.
Seeing that the angles which the pieces make with each other are of such importance to the Magnitude and the pro portion of the excited strains, it is proper to find out Some way of readily and compendiously conceiving and expressing this analogy.
First, in general, the strain on any piece is proportional to the straining tierce.
Secondly, the strain on any piece, A C, is proportional to the sine of the angle which the straining force makes with the other piece directly, and to the site of the angle which the pieces make with each other inversely.
For it is plain that the three pressures, A E, A F. and A G. which are exerted at the point A. are in the proportion of the lines A E, A le, and F E (because F E 1,1011011 to A o). lint because the sides of a triangle are proportional to the sines, of the oppo siteatigles, the strains are proportional to the sines of the an gles A F F., A E F, and F A E.. [lig the sine of E is the same with the sine of the angle C A D, which the two pieces. A C a-101 A D, make with each other ; and the sine of A E. F is the same with the sine of E A D, which the straining piece n A inakes with the piece A C. Therefore we have this sine c A D: sine F. A sine E A D = AE;Aleand Ale=AF, X . • Now, sine C A D the sines of angles are most conveniently conceived as deci mal fractions of the radius. which is considered as unity. Thus sine 30" is the same thing with 0.5 or ; and so of others. 'Therefore, to have the strain on A c, arising hunt any load, A E, acting in the direction A F., Multiply A E Lv the sine of E A D, and divide the product by the sine of c A D.