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THEORY AND PRACTICE OF ENGINEERING.

Book II.

The sides of triangles are as the sines of the oppositeangles, and when three forces keep a body in equilibrio,each is proportional to the sine of the angle madeby the other two ; the weight W, for instance, is as thesine of the angle A E F, the weight b as the sine of theangle A F E, &c. &c. When the framing of a roof, or apair of principals, are drawn upon paper to a scale, theproportion of the forces may be correctly ascertained inthis manner, without further calculation, at least suffi-ciently near for all practical purposes. Tie-beams con-necting two rafters at the feet prevent their spreading ;and the horizontal strain given to the tie-beam is depo-sited with the entire weight, and without any thrust uponthe walls. The triangle is the strongest form that can begiven to framing; suppose three pieces of timberunited by pins, such a triangle will not alter its form ;but if the top pin be drawn out, the two inclined sideswill revolve round the other two pins; but as long as thethree pieces of timber are of the same length the figurewill retain its form. The pins, upon which any pairare held together, are free to revolve in circles, beforeno difference to its strength or its form: from this very simple principle two importantrules are elicited for framing timbers, and which show that all framing should have thefigure of a triangle, with the largest or most obtuse angle that can be obtained, or thatthe application will admit of, care being taken to prevent the sides from bending.

To properly brace a piece of carpentry, so as to resist any force that may be applied toit, and to give it the necessary stability, requires careful arrangement, and considerableknowledge in the art; but the chief object to be attended to in framing a roof is to provideagainst any alteration in the form of the triangle made by its principals, or the sagging orbending of the tie-beam, which is the longest side of it. In determining the scantlings ofthe various timbers, it must be borne in mind that the load the roof has to support is equallydistributed over their whole surface, and which never need be calculated at more than afloor has to carry. The timbers are, however, more apt to shrink, and alter their formby change of temperature, and oak is more subject to this than fir. The principals of aroof may be regarded as the girders of a floor, the common rafters as the bridging joistsbut there is the advantage of giving the principals additional strength by framing andtrussing them, which cannot be done so readily with the girder; and the more at rightangles to the rafters the strutting pieces are placed, the firmer and stronger the roofbecomes.

The most advantageous method of employing a piece of timber, so as to have its entirestrength, is to place it perpendicularly, as in a king-post, which not only holds up the tie-beam where it is the weakest and inclined to sag most, but, when properly strutted,forms with the principal rafters a complete truss. A collar in addition to the tie-beamplaced half-way up the principals, and acting as a coupling, has not the stability of the king-post with its braces, which transport the weight of the purlin to a point or points whichcontribute to its strength.

Some of the roofs used between the fifteenth and seventeeth centuries had their principalsset out in the form of an equilateral triangle ; a collar was framed into them, and on thisrested a king-post, into the head of which the tops of the principals were framed andpinned ; two short struts were placed above the collar, from the king to the rafters, andtwo others below the collar were also framed into the principals : on the principals werepinned blocks, to support the purlins without cutting into them, and the common raftershad their feet framed into a small plate, which laid fair with the outside of the wall ;they were also securely pinned to the purlins and the ridge piece, which rested on thetop of the king-post or tree-post in the middle.

This formed altogether a very solid roof, but required a quantity of timber, and ofcourse a considerable expense: the tie-beams generally, for roofs from 25 to 30 feet span,were in depth equal to an eighteenth part of the width between the walls, and all the restof the timbers in proportion.

In some early buildings, where it was required to improve the rooms obtained in theroof, the tie-beams, on which the floor rests, are placed two or three feet below the level ofthe pole-plate, upon which the common rafters pitch : by forming a semicircle over the tie-beam, composed of crooked or cut timbers, and introducing above it another tie framedinto the principals, and again strutted on to the ends of the lower tie-beam, a convenientarrangement for a room is obtained : above the upper tie was introduced the tree-post,with a collar and struts, which deposited the weight of the upper portion of the roof on tothe tie-beam, supported partly by the middle of the arch.

H

Fig. 2110.

the third is put in, but this makes