The formula of solution are (1), (2), (3), Article 122. In The weakest joint makes an angle with the vertical which is nearly equal to 60°. Let us take this angle = = 60°; then 0 = 10472, sin2 = 2, cos 0=, sin 10, cos 0=866. Hence, by formulæ (1), (2), (3), ♪ being the weight of a unit of volume, = 3094 8 (3). r2 sin 10 {3 (r+ h) 0 cos 0 – 2r sin 0} S Wherefore, if y be the lever arm of N, the horizontal thrust, we have, by taking (3) from the sum of (1) and (2), It may be interesting to give the values of N for different joints of the arch. These are given in the following examples 2, 3. Substituting the values in the formulæ of (1), (2), (3), Art. 122, and collecting the quantities together with their proper signs, we get 2. In ex. 1 find the thrust on the joint which makes an angle of 30° with the vertical. In this case sin, sin2 10 = ⚫067, cos 0 = ·866, 0=5236. Hence by the preceding formulæ, This is considerably less than the thrust on the joint, which makes an angle of 60° with the vertical. 3. In ex. 1 find the thrust on the springing line of the arch, at which the angle the joint makes with the vertical is 70°. In this case, the calculation of the respective terms of the preceding equation, in ex. 1, will be best effected by logarithms. 4. At Bishop Auckland, in the county of Durham, there is a segmental arch over the river Wear, which dates about 1388; the rise or height of the segment is 22 feet, the span or chord of the arc is 100 feet, and the thickness at the key is 15 of a foot. Find the horizontal thrust, neglecting the surcharge, the line of pressure being supposed to pass through the centres of the voussoirs. Obs. The semi-angle at the centre being less than 60°, the greatest thrust will be nearly at the joint of the springing line (Art. 123). Ans. 2 x 0635 8. 5. The span of a segmental arch called the "Hutcheson Bridge" at Glasgow (Stephenson, Architect), is 79 feet, its rise or height of segment 13 feet, and thickness at the key 3 feet; find the horizontal thrust of this arch before any surcharge was placed upon it. Ans. 166.7 8. 6. The span of a segmental arch is 50 feet, rise 9 feet, thickness at the key 2 feet; find the thrust on the supposition that the line of level of the horizontal surcharge on the arch passes through the highest point of the extrados. Ans. 122·8 S. 7. In the last example, find the thrust by supposing the horizontal surcharge to rise to a height of 2 feet above the extrados at the key. Ans. 265.2 d. 8. In example 1, determine the resultant pressure on the weak joint of the arch. Ans. R=296 & r2. 9. In the last example find the angle which the resultant pressure makes with the vertical. Ans. 27° 45'. Section 6. THE THICKNESS OF THE PIER IN GENERAL. 122. In Art. 116, the thickness of the pier for semicircular arches has been determined on the supposition that the resultant pressure of all the forces which act on the base of the pier passes through the extrados of the pier, the curve of pressure in this extreme case being supposed to pass through the extrados at the key, the intrados of the weakest joint at the reins, and the extrados at the pier. The following expressions for the thickness of the pier are applicable to an arch of any form, and to any curve of pressure. The surcharge of the arch, as explained in Art. 118, is supposed to be reduced in height in the proportion of its density to the density of the arch. 123. Let H be the height of the pier from its base to the reduced surcharge C' I' (Fig. 56); h the height of the reduced ridge C' above the level of the springing line; h' the depth of the arch and its reduced surcharge at the springing line; A the area of that part of the face of the arch, and its reduced surcharge immediately above the half-span ; m the moment of the area A with respect to an axis through the interior edge of the joint at the springing line; 7 the lever arm of the horizontal thrust on the arch, measured from the base of the pier to the centre of pressure of the joint at the key; N the horizontal thrust; r the length of the half-span, whatever may be the curve of the intrados; Р the distance between the exterior edge of the base of the pier, and the centre of pressure of the base; x the required thickness of pier; |