J = reaction of right abutment = 18.75 tons, The horizontal line through J, meeting K, K, represents the horizontal component of the reaction of each abutment. This also represents the stress on the pin b at the crown of the arch. KI B, In finding the stresses by the diagram we start at the points a and c, fig. 163, in succession, and find the stresses on each member from these points towards the apex. This is clearly shown in the diagram. As a check on the accuracy of the diagram, the last line drawn for the left half of the arch, viz., HJ, which is drawn parallel to the diagonal H J, should come to the point J, so as to form a closed polygon. Similarly, the last line drawn for the right half, viz., Ĥ1 Л parallel to H1 J in fig. 163, should also pass through J. = K, A, K x K KA O E H, Case 3.-Fig. 166 represents the stress-diagram when each apex is loaded with 15 tons. It is constructed by drawing the vertical line K K = 105 tons, the total load resting on the arch. Set off K K1 = K1 K2 = K2 Kg, &c., 15 tons. Each half of the arch is loaded with 52.5 tons. The vertical proportion of the load on the left half borne by the abutment at a= 22.5 tons, and at the apex b is 30 tons. Similarly, the vertical reaction of the right abutment at c = 22.5 tons, and that on the pin at the apex=30 tons. K Fig. 166. Set off, therefore, on the vertical line K K2, Kæ1 =K, x= 22.5 tons; then x1 = 0 = 30 tons. Draw xJ, J parallel to the dotted lines cb and b. Join KJ, K, J. These lines will represent the reactions at the left and right abutments respectively, both in direction and magnitude. The horizontal line J O will represent the horizontal thrusts at the abutments, and also the stress on the pin at the apex of the arch. Knowing the abutment reactions the stresses are easily found, as shown in the diagram. 219. Stresses on the Braced Arch by the Principle of Moments. -The stresses on a braced arch with only one system of triangulation, may be also determined analytically by the aid of the principle of moments. Taking the case we have just been considering namely, the arch loaded at each apex with 15 tonslet it be required, for example, to determine the stress in the horizontal bay FK2 Through a and c draw az and ez parallel to JK and J K (fig. 166); a ≈ and c z will represent the directions of the abutment reactions. Draw ff, perpendicular to the bay F K, and ƒƒ1⁄2 perpendicular to a z. The portion of the arch to the left of the line ff, is held in equilibrium by three external forces and the stress on the bay FK2 The external forces are the abutment reaction equal to 131 tons acting along the line a 2, and the two vertical loads of 15 tons acting vertically downwards at the two apices. The abutment reaction tends to lift the segment upwards round the point ƒ as a hinge, while the vertical loads at the apices tend to turn it in the opposite direction. Consequently the moment of the stress in FK, must be equal to the difference of the moments of these external forces. SFK2 ׃ƒ1 = 131 ׃ƒ2 − 15 (6·25 + 18·75) = 131 × 4·8 – 15 × 25; or SF K2 = 63.4 tons. CHAPTER XVIII. ROOFS. STRESSES ON ROOF TRUSSES. 220. Roofing may be divided into two parts, namely:— 1. The Framework. 2. The Covering. The architect has principally to do with roofs made of timber, while those made of iron or steel usually fall within the province of the engineer. Timber roofs, as a rule, are employed only in covering buildings of small span, such as houses, churches, and the smaller warehouses and mills. The sizes of the rafters, purlins, and other members of such roofs are generally fixed by the light of practical experience; the architect in too many cases not troubling himself with the stresses coming upon them. In those roofs in which a combination of timber and iron is used, the main rafters and struts of the principals are made of timber, while the members in tension are made of wrought-iron or steel rods. Trusses of this composite structure are not, as a rule, to be recommended, for reasons which have been referred to in Art. 18. The framework of a roof consists of 1. The main trusses or principals. 2. The purlins or scantlings connecting the principals together. 3. The sash bars, intermediate rafters, wind ties, &c. Under the head of framework are sometimes included the girders and columns for supporting the roof. 221. Main Principals. The main trusses or principals may be supported on walls, columns, or girders, and as far as their design is concerned may be classed under two main heads, viz. : 1. Complete trusses or those in which the pressure on the supports acts in a vertical direction. 2. Arches, braced or otherwise, which produce an outward pressure on the supports. Trusses of the first-class may be subdivided into two groups: (a.) Those with straight rafters, examples of which are given in figs. 164 and 184. (b.) Those with curved rafters (Bowstring trusses), like that shown in figs. 190 and 192. Fig. 167a, is an example of the second-class, and represents a braced arch. Principals of the first-class are, like an ordinary braced girder, self-contained, and exert only a vertical pressure on the supports, except it be a side thrust arising from the pressure of the wind. The various members of these principals are exposed to compressive and tensile stresses; the main rafters being in compression while the bottom members or main ties are in tension. The intermediate bracings connecting these together may be exposed to compressive or tensile stresses according to their position. The load on a roof being constant, or nearly so, it follows that the amount of stress on any particular member does not vary, or varies only in very small degree, except in the case of a light roof when a strong wind is blowing, or when a large accumulation of snow takes place. It is not often, however, that these events occur, so that practically speaking the stresses on the different members are constant. In principals of the second-class, the stability of the roof depends not only on that of the main ribs themselves, but also on the suitability of the supports to resist a lateral pressure. Generally speaking, arched principals are used only in very large spans, or where a clear headway is required, or to produce architectural effect. 222. Simple Forms of Roof Trusses.-The simplest form of roof truss is the isosceles triangle, shown in fig. 167. The height of the apex above the abutments varies from one-fifth to one-third of the span. The most common being one-fourth or a little more. AC and B C are the rafters, and A B the main tie. Let W = distributed weight on principal, l=span, h=height of apex above tie bar or the abutment. A uniform load W, so far as the direct stresses on the truss W are concerned, is equivalent to acting vertically downwards. at the apex; this produces a vertical reaction at each of the abutments = W Taking moments about the apex, we get— where angle of inclination of the rafters to the horizontal. In addition to this compressive stress on the rafters, they will be exposed to transverse stresses with a uniform load; if their length be considerable they become deflected, thereby resembling long bent struts. In this form they do not possess much W Fig. 167. B Fig. 167a. strength; hence the desirability of stiffening them by introducing intermediate bracing. The truss shown in fig. 167 is for this reason only suitable for very small spans. If the horizontal tie be omitted, there will be an outward horizontal thrust on the abutments = WI When the span of the roof is too great for the form of truss shown in fig. 167, further bracing should be introduced, as shown in figs. 168 and 171. In these examples the deflection of the rafters is prevented by the introduction of struts which are attached to their central points e and f. The introduction of these struts, de and df, necessitates the addition of the vertical tension member b d to balance the downward thrusts. If one of these principals be loaded with a distributed weight W W, this weight will be equivalent to a load of on each bay, 4 each resting at the points e, b, and ƒ, and W or to three loads of |