328 TUB STONE ABCH. [CHAT. XV. 



2<l. Width of abutment. Since we have laid off the arch 

 wi-ighte to scale in their true value, the pressure line thus ob- 

 tained is the true pressure line for the arch. But we have laid 

 off tlu abutment lainiiuc I), 10, and 11, one-half their true value, 

 and the pressure line thus ohtained with the same thrust and 

 pole O is the same as if we had taken their true value and twice 

 H. Its intersection with the foundation gives us, then, the 

 proper width of the abutment for stability, according to our 

 assumption of 2 for the coefficient of stability (Art. 177). 



179. Thus we can easily determine for any given case of arch 

 and surcharge the horizontal thrust and the proper width of abut- 

 ment, and then from the pressure line can easily so dispose the 

 joints as to prevent sliding. If the dimensions of the arch as 

 given are not such as to be stable, it will be found impossible 

 to inscribe, as above, a pressure line which shall lie within the 

 middle third, and the curve of extrados or intrados will have to 

 be altered so that this shall be the case. The pressure line thus 

 obtained, it is true, does not exactly correspond with the true 

 one, as it is still possible to inscribe another which shall deviate 

 still less from the true line. We have also taken the double 

 thrust for the abutment laminae alone, instead of for all laminae 

 from the joint of rupture of the arch. Both deviations are made 

 on account of the far greater ease and rapidity of construction. 

 It would be found very tedious to take first the force polygon up 

 to somewhere about the section of rupture, then by long trial 

 find the innermost support line, and finally, after the section of 

 rupture is by this line determined, to lay off the remainder of the 

 force polygon, and prolong the pressure line through the abut- 

 ments. 



It is far simpler to proceed, as above, by assuming the point 

 of application of the horizontal thrust, as also temporarily the 

 section of rupture. We obtain thus a somewhat smaller value 

 for the width of abutment, but, on the other hand, we have 

 taken the coefficient of stability at 2 instead of 1.9, as assumed 

 in Petifs tables. 



Moreover, the widths of abutment thus obtained are greater 





 the edge as near a* the strength of the material witt allow. In many arches the 



pressure line does pass outside of the middle third. The condition is not 

 essential for stability, except in the absence of data determining the proper 

 limits. If the pressure line will not lie within the required limits, whatever 

 they are, the dimensions of arch must be changed, so that it will. 



