506 



BRIDGE. 



Theory. It may be proper to' observe, that the French ar* 

 chitects Perronet and Sou/Hot, made an experi- 

 ment on the strength of the stone of which it was 

 composed. They found, that a cubic foot of it, which 

 weighs] 52 lb, required 240,000 tb. to crush it. In 

 the above investigation we have only taken it at 50,000. 

 The thickness at the crown of the arch, cannot, 

 with propriety, be reduced so much as we have sup- 

 posed, in the above examples. This part of the 

 structure is liable to be strained transversely. And 

 it has been found, that when stone, or other matter, 

 is bearing a great pressure longitudinally, its strength 

 against a transverse strain is thereby much diminished. 

 But, independent of that, there is another cause for 

 preserving the crown of a greater thickness. The 

 varying pressure of carriages would be apt to pro- 

 duce some motion among small stones; this would 

 chip away their angles, and accelerate the destruction 

 of the building. But there is seldom any need for 

 this reduction. In most cases, it would only be ad- 

 ditional labour. 



OF PIERS. 



Piers. THE piers and abutments of a bridge must be so 

 constructed, that each arch may stand independent 

 of its neighbours. For though, by the mutual abut- 

 ment of arch against arch, the whole may rest upon 

 very slender piers, if onee the structure is erected ; 

 yet, as they must be formed singly, and are exposed 

 to many accidents, it will be best to contrive them, 

 that the destruction of one arch may not involve in it 

 that of the whole. 



Some of the writers, on the principles of bridges, 

 in treating this department of their subject, have 

 Found it necessary, by the help of the higher calcu- 

 lus, to find the centre of gravity of the semi-arch. 

 The solution of the problem, we are convinced, so 

 far as it is ujcful in practice, lies much nearer the 

 surface. 



The reader lias already frequently seen, that the 

 ultimate pressure may, in every case, be reduced to 

 two others, viz. the weight of the semi-arch above, 

 and the horizontal thrust. In the equilibrated arch, 

 this pressure is directed perpendicularly to the joints 

 of the sections ; and these being usually drawn at 

 right angles to the curve, the pressure is in the direc- 

 tion of the tangent to the arch. Hence, we have 

 often called it the tangential pressure. Upon this 

 principle, however, when the curve springs at right 

 angles to the horizon, an infinite pressure is required 

 in the vertical direction, a supposition wliich cannot 

 have place in practice. We must accordingly call in 

 the assistance of friction in that case ; a force which 

 may be set in opposition to the horizontal thrust, and' 

 which, increasing with the superincumbent weight, 

 very fortunately keeps pace also with what it is in- 

 tended to oppose. 



Granting, then, that the friction is so contrived, 

 upon vhe principles already explained, that there is 

 no danger of any slide at the horizontal or springing 

 joint ; it will be readily admitted, that no slide is 

 likely to take place in any horizontal course below 

 that, till we arrive at the foundation ; for the disturb- 

 ing force is constant, but the friction increates as we 



descend. Our principal care then must be, that the 

 pier does not overset, by turning on the farther joint 

 E of its base, as a fulcrum. Take a in the horizon- 

 tal joint, Aa as the centre of pressure. Draw aV 

 to represent the weight of the semi arch, and VT the 

 horizontal thrust ; then Tais the ultimate pressure: 

 and if, when produced, it falls within the base of the 

 pier, it is perfectly obvious that it can never overturn 

 it. And this is altogether independent of the weight 

 of the pier; for if that were a mass of ice, immersed 

 to the springing in water, the case would be exactly 

 the same. 



But the pier itself has a considerable stability, arising 

 from its own weight; and even though the direction 

 of the ultimate pressure of the arch alone pass out of 

 the base, the tendency to overturn the pier may be 

 balanced by its weight. This weight may be sup- 

 posed concentrated in the centre of gravity of the 

 pier, and of course to act in the vertical line which 

 bisects it. 



Its effect will be nearly found by laying off in that 

 line from the point q, where the direction of illegiti- 

 mate pressure of the arch intersects it, gr=to the 

 weight of the pier, and taking gsrrthe ultimate pres- 

 sure=nT, and completing the parallelogram, the dia- 

 gonal drawn from q will re-present the direction and 

 magnitude of the united pressure of the arch and pier. 

 This is not strictly accurate ; it would be so if a and q 

 coincided, which is the case with a single arch standing" 

 on a pillar : but in general, the ultimate pressure is 

 still more favourable than this. Its direction <it any 

 point is in the tangent of a curve, wliich approaches 

 the vertical as we descend, since the proportion ari- 

 sing from the weight of the pier increases with its 

 height. 



In order to find analytical expressions for these 

 forces, let the horizontal thrust of the arch n/. The 

 weight of the half arch =rt, and that of the pier 

 =p, the height of the pier to the springing of the 

 arch =/i, the breadth at the base =6. 



1. Then the horizontal thrust acting in AG, 

 tends to overturn the pier, and its force round the 

 fulcrum E will be represented by multiplying it by 

 the perpendicular distance AD=z viz. Ax'- 



2. The weight of the pier acts in the direction EC, 

 and its effect will be represented by multiplying it by 

 the leverage CE, viz. pxib- 



3. The arch acts with the leverage EK, which is 

 not equal to the breadth of the pier, by the part 

 KD=AH, say -J of the depth of the joint at the, 

 springing. This will never exceed one-fourth of the 

 breadth, when two different rings of arch-stones rise 

 from the same pier, unless the pier widen below. 

 CallEK, thcrefoie =:J-6. 



We have now ht ^Itp + ^ba ; whence, 



fit 4.- /it . 



1st, 6= - - = - 5 , and consequently, 



4;J + -} 2p + 3 



To find the least breadth of the pier at its base, 

 divide the horizontal thrust by half the pier added 

 to three fourths of the half arch. Multiply the 

 height of the pierby the quotient. 



2d , k- 



, that is, 



The height of a pier to the springing, having 



