1848.] 



THE CIVIL ENGINEER AND ARCIIITECrS JOURNAL. 



133 



arrow ; continue this direction till the line intersects the vertical, 

 throue;h the centre of gravity ; from the point of intersection of 

 these two lines measure off, on the vertical line, a distance equal 

 to 15 parts of the scale, making the side AV, of the parallelogram ; 

 and on the line in the direction of the pressure on the block, m ea- 



Diagram S 



iO 50 W W 30 ',0 50 60 W gO 90 100 

 Scale of equal parts, representing Certs. 



sure off a distance equal to 80 parts of the scale, making the second 

 side P, of the parallelogram ; draw the other two sides ; and the 

 diagonal R, W, P,, will represent the direction of the resultant 

 of these pressures ; its length, equal to 81ij parts of the scale, wiU 

 give the amount of its pressure, 81^ cwt., and the line, continued 

 till it intersects the joint No. 1, will represent the point of applica- 

 tion of this resultant pressure on the second block ; that is to say, 

 that point will be the centre of pressure of all the pressures, com- 

 municated throughout the surfaces of contact, from the first block 

 to the second, and the amount of the resultant, 8I5 cwt., will be 

 the aggregate of these pressures. 



If the line W,, representing the weight of the block, is drawn 

 from the point of intersection, in the direction in which it acts, 

 that is, vertically downwards, then the line P,, representing the 

 pressure on the block, must be drawn in the direction in which it 

 acts, that is, from right to left. If, however, as in the present 

 case, it is more convenient, the lines may be drawn each in the 

 direction opposite to that in which the pressures act, that is, the 

 weight represented by a line vertically upwards, and the pressure 

 by a line from left to right, in which case the resultant pressure 

 will act in the direction of the diagonal, but towards the point of 

 intersection of the two lines, that is, from right to left in the pre- 

 sent example. 



The resultant pressure on the third block is determined in a 

 manner precisely similar to that described above, with regard to 

 the second ; a vertical line is drawn through the centre of gravity 

 of the second block, and the direction of the resultant pressure on 

 the same, P^, is continued till the lines intersect ; 8I5 parts mea- 

 sured from this intersection on the latter line, form one side of the 

 parallelogram, and 15 parts measured on the vertical line from the 

 other, for the amount of the resultant pressure on the second block 

 is 81^ cwt., and the weight of the second block is 15 cwt. : the 

 parallelogram being completed, the diagonal produced determines 

 the position and direction of the resultant pressure on the third 

 block, and its length, measured by the scale, determines the amount 

 of the pressure to be 86 cwt. In like manner, the pressure on the 

 fourth block is 100 cwt., and its position and direction are shown 

 by the arrow P4 ; also P5 P^ P,, show the position and direction 

 of the resultant pressures on the fifth, sixth, and seventh blocks, 

 and their respective amounts are determined by the length of the 

 diagonals of the fourth, fifth, and sixth parallelograms, and if any 

 of the blocks were removed, and replaced by a prop, in the position 

 and direction shown by the arrow : as for example, if the seventh 

 block were removed, and replaced by the prop there shown, then 

 all tlie remaining portion of the structure would be balanced on 

 the point of the prop. Each of these arrows are tangents to the 

 line of resistance, which can be drawn from point to point by the 

 eye, or by means of a piece of whalebone, or a metal spring. 



If, instead of the pressure on the first block, the pressure on 

 any other block be given, the resultant pressure on all the others 

 may be found in a similar manner. Thus, if the pressure on the 

 fourth block is known, the pressure on the fifth, sixth, and seventh 

 will be found in precisely the same manner as above described. 

 Then with regard to the third block, it will be acted on by its own 

 weight, and the pressure from the second block, and the given 

 pressure on the fourth block, is the resultant of these two pres- 

 sures ; if, therefore, a vertical line is drawn through the centre of 

 gravity of the third block, and another line is drawn in the direc- 

 tion of the given jiressure on the fourth block, and from the point 

 of intersection of these two lines there is measured off on the ver- 

 tical, as many parts of the scale as there are cwts. in the weight 

 of the block, and on the other line, as many parts as there are 

 cwts. in the given pressure on the fourth block ; then there have 

 been measured the side W^ of the parallelogram and the diagonal 

 R, W3 P., and these two lines determine the parallelogram, the 

 second side of which, from the point of intersection, represents 

 the pressure on tlie third block. This pressure on the third block 

 being determined, that on the second and that on the first block 

 may be found in the same manner, the lines drawn being the same 

 as those in the example. 



Art. 4. — In nearly all cases of arched structures, the pressure on 

 any one of the voussoirs is unknown, and this constitutes the diffi- 

 culty of the subject : the point of application, the direction, and 

 the amount of the resultant pressure on any of the voussoirs being 

 determined, the conditions of stability of the whole structure are 

 found by the application of the foregoing problem. To determine 

 the stability of the arch with regard to the first condition of failure, 

 diagram 2, that is, supposing failure to take place, by the slipping 

 of one vouissoir on another, the direction only of the resultant 

 pressures is required ; but to determine whether the arch will fail 

 (as in diagrams 3 or 4), by the voussoirs turning on their edges, or 

 by the material failing, not only the direction, but the points of 

 application and the amount of the pressures must be determined. 

 1 he theories of the arch, which preceded tliat of Professor Alose- 

 ley, take into consideration only the first condition of failure (Art. 

 1, diagram 2), it being supposed tliat if the arch faUed, it would 

 he by one of the voussoirs slipping on another. The experiments 

 of Rennie, Morin, and others, had not then been made, and the 

 resistance of the friction of one stone on another was much under- 

 rated, so that it was considered necessary for stability, that the 

 direction of the pressures should always be perpendicular to the 

 joints; of course this could only be the case for one particular 

 system of pressures, and if the weights on the voussoirs and other 

 pressures were so arranged, that tlie resultant pressure on each 

 joint acted in a direction perpendicular to it, then if any weight 

 were added to the system, or any taken away, the positions and 

 directions of the resultant pressures would, of course, vary also, 

 and their directions be no longer perpendicular to the joint. It 

 seems to have been the practice of bridge-builders, to take the 

 weight of the arch-stones and backing for the fixed system of 

 pressures ; and this weight being very great in proportion to that 

 of the wagons, carriages, and people passing over, the eft'ect of the 

 latter was not an important consideration, and the old problems 

 sufficiently answered the purpose. In the case, however, of a light 

 railway bridge, traversed by a heavy train, which, coming upon it 

 suddenly, has twice the effect of a stationary pressure of the same 

 weight, the effect of such traffic must not be omitted from the cal- 

 culation ; but if the arch is designed and the weights on the vous- 

 soirs arranged, so that the resultant pressures shall be perpendi- 

 cular to the joints when the train is on the bridge, then, when the 

 train has moved off, all the resultant pressures will have taken new 

 positions and directions, no longer perpendicular to the joints; so 

 that, according to the theories themselves, the arch would fail. 

 These theories are also quite useless in determining the stability 

 of vaults on high walls ; there is not, perhaps, a single vaulted roof 

 now standing, that does not prove their fallacy. 



Art. 5. — VVithout taking into consideration the adhesion of the 

 cement in the joints, the limiting angle of resistance for the sur- 

 faces of all materials used in arches, is so large, that it would be 

 difficult to design an arch and loading, in which the first condition 

 of failure would be fulfilled ; in the pier or abutment, however, 

 such failure is likely to occur, and must be carefully guarded 

 against. 



The second condition of failure, diagram 3, is, strictly speaking, 

 impossible, for no block will turn on its edge upon another, with- 

 out some abrasion, or elastic yielding, of the surfaces, in which 

 case it becomes that shown in diagram 4, or the third condition of 

 failure ; however, as the failure takes place from the tendency of 

 the pressures to turn the blocks on their edges, it seems that tb« 



