176 GRAPHIC AND ANALYTIC [CHAl 1 . XII. 



Inserting this value of % in the value for x above, we have 



I (2 l-a) 



x = - 



5 I a 



for the distance of the inflection vertical to the left of the left- 

 hand support, which is supposed fixed. 



Now this is the equation of an hyperbola, as shown in PL 21, 

 Fig. 81, whose vertex is at,/, the distance Ay being 2 I, whose 

 assymptotes are respectively parallel and perpendicular to the 

 span, the perpendicular distance of E above the span A B be- 



5 2 



ing - I, and which intersects A B at - I from A. The ordinate 

 2i 5 



d <?, A d being equal to I, is - I. The diameter passes through 



E and /", and E f is, therefore, the semi-transverse axis. The 

 hyperbola can, therefore, be easily constructed. We need only 

 to construct that portion between A B and the point e. 



The construction for the point of inflection \ is, therefore, 

 simply as follows : 



Lay off A h vertically upwards and equal to the distance of 

 the weight P from A, and draw the horizontal h 1) to intersection 

 b with the curve. Now make A a = I and draw a, b to inter- 

 section G. Draw b A to intersection P with weight, and then 

 P c intersects A B at the point of inflection i^ Decomposing 

 P along P B and P c, as in Art. 114, we have at once the re- 

 actions at A and B. Here also we see that, by a construction 

 purely graphic and abundantly exact, we can find the inflection 

 point and the reactions. 



The method detailed in Art. 114 can then be applied to de- 

 termine the various strains in the different pieces. It is un- 

 necessary to give an example, as the process is precisely similar. 

 We have simply in this case to start with the reaction at the 

 free end B and follow it through. Observe only that, as this 

 reaction must be less than for a girder with free ends for the 

 same position of P, the point h will lie nearer the force line 

 B' A B (Fig. 30, ), hence I m will not pass exactly through B, 

 but will lie to the right of it, giving thus a reversal of strain in 

 the flanges, as by reason of the inflection point should be the 

 case. 



Instead of constructing the hyperbola, we may calculate its 

 ordinates from the equation for x above, for different values 

 of a. 



