332 TITE INVERTED AKCH. [CHAP. XVT. 



equal angle with the vertical. "We have thus acting upon the 

 tower simply the half load ; and the tension in the rear chain 

 is equal to that in the last link, O d. 



If from O we draw parallels to the other links, we have at 

 cnce the strains in these links, O c, O b, O a, etc. 



Now, if the anchorage is a solid block of masonry, its con- 

 dition of stability is, of course, very easily determined. The 

 moment of the tension in the rear cable, with reference to the 

 edge of rotation, must be more than balanced by the moment 

 of the weight of the block acting at its centre of gravity, with 

 reference to this edge. The case is too simple to need further 

 notice. 



It is, however, more economical to make the anchorage hol- 

 low that is, in the form of an arch. The preceding method 

 for determining the stability of the arch has then here direct 

 application. 



Thus, laying off along the vertical through the centre of the 

 tower the weights of segments of the arch, we form with these 

 segment weights and the double tension in the chain an equi- 

 librium polygon. For this we have the pole O,, A O, being 

 double the tension Od already found. We then draw O,l, Oj2, 

 O,3, etc., and then from A parallels to these to the segment 

 verticals 1, 2, 3, etc. We thus have the polygon A 1, 2, 3, 4, 5. 

 \_N~ote. We take the double tension, as before, for the arch, we 

 took 2 H instead of H, in order to ensure stability.] 



The last line of this polygon 4 5 prolonged must, for sta- 

 bility, pass within the pier abutment, and its resultant, when it 

 is combined with the weight of the pier and pier abutment, 

 must pass within the abutment foundation. Through its inter- 

 section with the vertical line through the axis of the tower the 

 curve of pressure for the arch must pass. 



Drawing now 0, 4 parallel to the rear chain, and making it 

 also equal to the double tension, or twice O d, we find the pole 

 O,, and from it draw 0, 1, 0, 2, O a 3, etc., and then construct 

 the pressure line for the arch. It must, for stability, lie within 

 the middle third. 



To ensure stability when the tension in the rear chain dimin- 

 ishes, or when the bridge is unloaded, the arch must be stable 

 by itself. We must, therefore, construct the curve of pressure 

 f< >r the arch alone, neglecting the tension of the rear chains, 

 as explained in the preceding chapter. 



