502 



Thcorv. 



BRIDGE. 



PLATE 

 I.XXXT. 



>!?.$. 



pressure at the abutment N . Take NM the weight 

 of the section N, TM is the pressure on the joint of 

 M, and making MT m similar to NT n', M w will also 

 represent the extreme friction in that joint, and T 

 its load, and so on successively. Wherefore, if T m, 

 T I, &c. be found, the joints of the arch may be drawn 

 at right angles to these lines respectively, and every 

 stone will be exactly in the predicament of N, that 

 is, just kept by its friction from sliding away. 



The positions of Tm, T/may be readily discovered ; 

 for the angle n' T m must be equal to NTM. If, 

 therefore, we make T a equal to TN, draw the tangent 

 n IP. and making a fi=\M, and joining T b, we have 

 HTfiirXTM. And, in this manner, taking a b, b c, 

 &c. for the weights of the successive sections from 

 the scale, and drawing lines from T, the joints may be 

 formed perpendicular to the lines thus drawn. 



Upon the same principles, we readily find a con- 

 struction for the extreme weights of the sections, 

 when the positions of the abutments, &c. are given. 

 This is so evident, that we- shall not stop to point it 

 out. 



But a more convenient construction perhaps would 

 be, to take the horizontal thrust, or quantity of fric- 

 tion in the vertical line C d, Fig. 5. Lay oft" the 

 weight of the semi-aYch d a, draw C a, make C x 

 equal to it, also x z, mark off the weight of the sec- 

 tions along .r z, and through the divisions draw lines 

 from the centre; the joints required are parallel to 

 these lir.es. 



II. Let it be required, in the next place, to deter- 

 mine the other limit to the position of the joints, or 

 that in which each section is just prevented from sli- 

 ding in, by the friction on its lower bed. 



Here it is evident, that as the friction acts precise, 

 ly opposite to its direction in the former case, the 

 joints may have, on the opposite side, exactly the 

 same degree of obliquity to the position of equilibrium. 

 Draw, therefore, the tangent vy parallel to a c, cut it 

 with C v equal to a C, lay off the weights of the sec- 

 tions along vy, and draw lines from C ; these lines will 

 exhibit the positions of the joints, which of course may 

 be drawn parallel to them.- We have marked these 

 two limits of position in three joints of the half arch 

 above the same figure, assuming the friction at one- 

 third, and taking the first section of 30 as equal to 

 the thrust: and any other arch might have been in- 

 troduced as well as the circular. Any of the lines in 

 the triangle C da makes with the corresponding line in 

 Cy >',or in C zx, an angle equal to aCx, that is, when 

 the friction is one-third uf the pressure, equal to 18 

 26' ; and when the friction is one- half, this angle 

 is 26 34*'. The position of any joint, therefore, may 

 vary in the former 18 26', and in the latter case 2(i 

 31', on either side of the position of equilibrium, be- 

 fore any sliding can take place among the sections. 

 Nay, the friction of polished freestone is even more 

 than one half, perhaps it is two-thirds of the pres- 

 sure, which would give 33 V. And it is proper to 

 observe, that this is not confined to the aimulus of 

 archstoues, but holds equallywith whatever weight 

 the sections may be loaded. We may observe then, 

 that in any arch, the. posit, on of t/ut joints may tic va- 

 ried about 20* , perhaps 30 from that of equilibrium, 



before any derangement can arise from the sliding of Thee 



the arch stones. 



This is a most important conclusion, and leads to 

 extensive practical consequences. It affords a true 

 explanation of the facility with which arches are every 

 where constructed, even by the common country ma- 

 son. The equilibration theory has shewn us, that by 

 adjusting the inclination of the joint to the weight of 

 incumbent matter, we may suit an arch to any given 

 circumstances ; and we here find in the friction of the 

 parts a powerful addition to its stability. We trust, 

 therefore, that the reader now sees the propriety of 

 the observation, which we made above respecting 

 the inutility of searching very minutely into the exact 

 position of these joints. It is in common cases scarce- 

 ly possible to go wrong. But it must be observed, 

 that the variation of position above mentioned, is to 

 be reckoned from the position of equilibration, not 

 from the common joints radiating all from one centre, 

 or perpendicular to the curve, unless where such an 

 arch is equilibrated by the superincumbent weight. 

 For in an annulusof archstones, with radiating joints, 

 which is the most common mode of construction, 

 those towards the vertex can be drawn only a very lit- 

 tle lower, and those towards the springing only a very- 

 little higher than the original centre, though either 

 of them admits of a considerable variation in the op- 

 posite direction. 



For this reason therefore we approve highly of the 

 practice, which we believe is very general among arti- 

 ficers, we mean that of backing up the arch v.-ith so- 

 lid masonry, for several courses above the springing. 

 For granting that the friction on the horizontal bed 

 be fully equal to the thrust, yet as the tendency to 

 slide off is greatest there, it is well that it should be 

 effectually resisted. This is readily done by the solid 

 backing, which increases the mass of friction ; and 

 in the case of a bridge of several arches, enables us to 

 set the contrary thrusts of adjoining arches in opposi- 

 tion to each other. The materials, therefore, in that 

 part, ought to be laid close up to the spring courses, 

 and also bonded into the inferior part of the abut- 

 ment or pier, which will act as a sort of doweilmg, 

 and does not preclude the employment of that means 

 also. If great security is thought necessary, cement, 

 being a compiesbible substance, ought to be sparing- 

 ly employed in the vertical joints at the back of the 

 archstones. 



The friction of the sections of the arch, as it per- 

 mits a considerable variation to take place in the po- 

 sition of the joints, will also admit a considerable de- 

 viation from the load, which is necessary for equili- 

 brium over any point of the curve. 



It would not be difficult to investigate the extent 

 to which this variation of weight might be earned. 

 But we shall at present only remind the reader, that 

 as we find a variation of 20 practicable in the posi- 

 tion of the joints, he may conclude, that each section 

 will admit of its load being altered to that which 

 would suit a point in the curve 20 on either side of 

 it. 



But in speaking of this alteration of weight, it 

 must be observed, that we consider it only so far us it 

 is likely to cause the sections of the arch to slide OR 



