Note on Continuous Beams* 309 



deflexions in any practical case would not be visible on the 

 drawing-board. At this distance is set up on either side 

 of the centroid-vertical lengths representing the area of the 

 bending-moment curve for the span, supposing it to be a 

 simply-supported beam. These areas have already been 

 determined in order to find the centroid ; they are plotted 

 here to a scale of 1 in. = 6000 sq. foot tons, which is also the 

 scale of the second-vector figure. Through the ends of these 

 lengths the so-called cross-lines are drawn, being produced 

 back to meet the support verticals. They clearly must meet 

 on the centroid-vertical. The cross-lines prepare the way 

 for the fourth stage, the construction of the support tangents 

 to the elastic line and the " mid-links.''' This follows by 

 Cnlmann's theory of the " fixed points."" The terminals of the 

 girder being simply supported, the fixed points from which 

 we start are the points of support themselves F x and E 2 ". 

 Then by Oulraann's rules a fixed point on the other mid-link, 

 F 2 , is known, because F X F 2 is equal to the intercept made 

 by the cross-lines on the same vertical. Starting from F 2 

 we find F x ; by drawing any right-hand mid-link whatever 

 through F 2 till it meets the inverted third-line ; we join the 

 second point of support to the point in which this arbitrary 

 mid -link meets the right-hand third-line of its own span, and 

 produce this line backwards till it meets the left-hand third- 

 line of the second span ; this point of intersection is then 

 joined to the point in which the arbitrary mid-link through 

 F 2 meets the third-line ; F/ lies on the line so drawn ; but 

 it also lies on the line joining F 2 to the right-hand support 

 of its own span, thus F/ is found. This process takes 

 much less time to apply graphically than it does to de- 

 scribe in words. The cross-lines give F/ F 2 / and so F/ is 

 determined. F/' is found from F 3 ' precisely as F x ' from F 2 , 

 and F 2 " from F/' is determined by aid of the intercept of 

 the cross-lines on the vertical through F/'. A similar 

 process working from left to right gives the second series of 

 fixed points E 2 " E/', E/ E/, E 2 and E x . The construction 

 of these fixed points depends upon the theorems (i.) that the 

 vertical intercept between two mid-links is equal to that 

 between two cross-lines, (ii.) that the inverted third-line is the 

 locus of the intersection of any right-hand mid-link of one 

 span and the corresponding left-hand mid-link of the adjacent 

 span, (iii.) that the support-tangent passes through the points 

 in which the adjacent mid-links meet the adjacent third-lines. 

 The proofs of these propositions are due to Oulmann, but they 

 can be readily followed and the resulting methods used at 

 once by any draughtsman. The fifth stage can now be under- 

 PJiil. Mag. S. 5. Vol. 46. No. 280. Sept. 1898. Z 



