310 Messrs. H. J. Tomlinson and K. Pearson : 



taken, namely, join any pair of corresponding fixed-points, 

 e. g. F/ E/ and we have the real and not an arbitrary mid-link. 

 The mid-links being all known, the support-tangents can be 

 put in, and we have the second figure, i. e. that under the 

 bending-moment curves, completed. The necessity for the 

 mid-links meeting on the centroid-verticals, and for the 

 support-tangents passing through the points of support, gives 

 ample means of verifying the accuracy of the draughtmanship 

 up to this stage. 



In the sixth stage the areas of the reduced bending-moment 

 curves have been plotted down a vector line (the 2nd vector 

 polygon) as 1' 2' in the scale 1 m. = 6000 sq. ft. tons ; parallels 

 through 1' and W to the mid-links, determine the pole 0'. 

 This should be at a distance from V 2 / equal to the reduced 

 constant flexural rigidity used for the cross-lines, and this is 

 again a test of correctness. Parallels to the support-tangents 

 through 0', namely, O / / and 0'?> r cut off from the vector-lines 

 lengths O'l' and 2' 3', which determine by the following 

 process the unknown bending-moments at the points of sup- 

 port D and G. M/ and M 2 ' being the altered bending- 

 moments at these points, then O'l' and 2'3 ; for the second 

 vector polygon of the second span* are respectively equal to 

 •J (span x Mi') and \ (span x M 2 ), where care must be of course 

 taken to read off O'l' and 2'3' in the proper scale of sq. ft. 

 tons. Invert the process by which we altered the bending- 

 moment and M x and M 2 are found. Their numerical values 

 are 2316 ft. tons and 2000 ft. tons respectively. Plotting 

 these values and joining their tops to each other and to the 

 terminals of the beam, we have the broken line marked with 

 numbers in the drawing ; the vertical intercept between this 

 line and the three original parabolas measures the bending- 

 moment at anv cross-section of the girder on the scale of 

 1 in. = 6000 ft. "tons. 



The seventh stage is the determination of the reactions at 

 each point of support due to the load on that span. This is 

 done by the aid of the first vector polygon for each span. 

 On the scale of 1 in. = 100 tons the total load on each span is 

 plotted down a vector line as 01, parallels through and 1 to 

 the tangents to the parabolas at the points of support! give 

 the pole O ; through O a parallel Oe to the numbered base- 

 line of the true bending-moment curve for the span cuts off 



* The second vector polygons for the other spans give again means 

 of testing the accuracy of the results reached. 



t These tangents are found by joining the points of support to the 

 point reached when the maximum ordinate of the parabola is produced 

 its own length. 



