212 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[July, 



The condition, that the curve must lie within the thickness at every joint 

 was also tested in the following manner. A semicircular arch, of which 

 the thickness was one-ninth of the radius, was constructed in four pieces, 

 having the joints e and/, Br. 2, at the points of contact of the curve of equal 

 horizontal thrust with the intrados and extrados. A similar arch was also 

 made in six pieces, having the joints at a, b, d, c, where the curve lies 

 within the thickness. In the first case, yielding took place, by the crown 

 descending and the haunches going out, and in the second, though composed 

 of a greater number of pieces, perfect stability was obtained. 



Fig. 3. 



Lastly, it being obvious, that these conditions, if correct, must apply to 

 any form of structure whose stability depended on equilibrium ; the curve 

 of equal horizontal thrust was ascertained, in a series of rectangular pieces, 

 as in fig. 3, and it was found, that when they were placed inclined to each 

 other at an angle of 45°, the thickness must be -1404 of the length to con- 

 tain the curve, and that the point of contact was -3535 of the length, from 

 the upper extremity ; also, that whether the inclination was greater or less 

 than 45°, the curve fell within the thickness. Then taking two rectangular 



quently turned, in order to ascertain, by actual experiment, 

 thrust to the weight, in a semicircular arch. 



the ratio of the 



Kg. *■ 

 pieces of wood of this form, and dividing them where the curve touches the 

 extradosal line at a, they will yield by the apex going upwards, wheu 

 placed at 45°; but when the angle of inclination is made greater or less 

 than 45° as in fig. 4, stability is obtained ; and at the inclination of 45°, if 

 the divisions be made at c and ,;, instead of at «, fig. 3, although composed 

 of a greater number of pieces, stability is also obtained. 



Before leaving this part of the subject, it may not be out of place to men- 

 lion another experiment, which exhibits the analogy between the catenary 

 and the curve of horizontal thrust.— On a vertical plane surface, an inverted 

 semicircular arch was drawn, and divided into eighteen voussoirs of equal 

 dimensions. Through the centre of gravity of each voussoir, a vertical 

 line was drawn, as in fig. 5. From two pins, fixed at p and p', a strong 

 fine silk cord was hung, and eighteen pieces of chain, of equal weight, 

 were attached to if, representing the equal weights of the voussoirs. This 

 species of catenary was then adjusted, so that each of ihe chains hung op- 

 posite the vertical lines, and the apex fell just within the thickness of the 

 arch as shown on the figure. The similarity of the curve thus produced, 

 to that of the curte of equal horizontal thrust, was immediately apparent.' 

 Next, one of the pins at p was withdrawn, and the cord was lengthened 

 and attached to another pin at V, so as to retain the partp c p' in'its ori- 

 ginal position. The line I' p thus represented the resultant of all the forces 

 aclmg at p, and completing the triangle V a p ; a p ibe weight or vertical 

 force, was to P a the horizontal force as 275 to 1, which result was found 

 to accord perfectly with that exhibited in a brick arch which was subse- 



Fig. 5. 



Having now, it is presumed, given sufficient practical evidence of the 

 existence of the line of equal horizontal thrust, it only remains to notice, in 

 this part of the subject, that as well as the position of the point of rupture 

 being denoted by it, the direction in which yielding will take place, may 

 also be known. That is to say, it will be outwards, |when the curve of 

 equal horizontal thrust touches the intrados, and inwards when it touches 

 the extrados, and before actual rupture, the approach of the curve to either 

 extremity of the voussoirs, indicates the tendency to yield. 



Numerous other experiments, of which it is unnecessary Ic give the de- 

 tails, have shown, that the conditions of equilibrium are the same for the 

 arch and the abutment as for the arch itself; in fact, that the arch and the 

 abutment, when together, may be considered as an arch. 



" On the Geometrical Construction of the Curve of equal Horizontal 

 Thrust." 

 The two half arches being assumed to be symmetrical, the apex of the 

 curve will be in a vertical line equidistant from the springings, and for the 

 present purpose, it will be sufficient to assume one of the two points (sup- 

 posed to be given), to be in this line. The construction of the curve then 

 resolves itself into two problems. 



Fig. 6. 



1st. To find a third point in the curve, at any joint between the two 

 points given. 



2ndly. To find a third point in the curve, at any joint beyond the two 

 points given. 



