CHAP. XV.] THE STONE ABCH. 327 



Laying off these weights in their order, we have the force 

 line (Fig. to left). The weights of the abutment laminae 9, 10 

 and 11 are laid off to same scale one-half 'of their proper in- 

 tensities. The reason will soon appear. 



1st. To determine the thrust H, and also the joint of rup- 

 ture. 



We first inscribe a pressure line by eye, and assume the 

 point of the intrados to which this line most nearly approaches 

 as the edge of the joint of rupture. Draw next from the cor- 

 responding point of the force line a line parallel to the assumed 

 pressure line at this point. This line will cut off from the 

 horizontal through the beginning of the force line our first- 

 approximate value of H. 



Thus, suppose we have inscribed by eye the pressure line 1, 



2, 3, 4, etc., which gives us the point a for the position of the 

 edge of the joint of rupture. Then a line drawn from 5 on 

 the force line, parallel to the side 4 5 of the pressure line, gives 

 us our first value for H. 



Now assuming this value of H, we erase the first assumed 

 pressure line, and proceed to construct the pressure line cor- 

 responding to this value of H, and the force line divisions 1, 2, 



3, 4, etc. If this pressure line lies always within the middle 

 third of the arch, it may be taken as the proper pressure line, 

 and H as the true thrust. In general, however, this will not 

 be the case. The pressure line thus determined may even pass 

 without the arch entirely. We then determine the new point 

 of rupture, as given by the point of exit of this pressure line, 

 and produce the side cut this point back to intersection with H 

 prolonged. From this point of intersection draw a line which 

 does lie within the middle third of the arch at the lamina of 

 rupture, and then in the force polygon from the corresponding 

 point of the force line draw a parallel to this line, thus cutting 

 off a new value for H. Erasing now the preceding pressure 

 line, we construct a third with this new value of H, producing 

 it right and left from the new line just drawn, which does lie 

 within the middle third, which will in general give us a pres- 

 sure line lying everywhere within the middle third of the arch. 

 If not, another approximation may be made. We thus find by 

 successive approximation the true joint of rupture and the cor- 

 responding thrust.* 



* Instead of the " middle third" we may allow the pressure line to approach 



