Mr. J. M. Heppel on the Theory of Continuous Beams. 67 



We have then the following data : — 

 In spans (1 . 2) and (1 . 0), 



a = 92, b = 2a, c=3a, d=4a, l=5a, 



a =76-7, b=2a, l' = 3a; 



1 1 = T_ _ 1 1 



€l 1132E' 62 1520E' 63 1746E' * 4 1664E' e5 ~T857E' 



, = 1_ e , = 1_ = 1_ 



61 1100E' 63 960E' 63 720E 5 



^ = 2-89, fx 2 = 3Sl, /i 3 = 3-57, ju 4 =3'49, ju 5 = 3'65, 



^' = 2-84, /* 2 ' = 2-67, ^' = 2-32; 



T + T' = 0, E = 1440000. 

 In span (2 , 1), 



0=92, 5 = 2a, c = 3a, d=4a, l=5a ; 



= 1__ = 1_ _ 1 l_ _ 1 



61 1857E' * 2 1664E' 63 1746E' * 4 1520E' e5 ~TT32E ; 



^ = 3-65, ^ = 3-49, /u 3 =3'57, ^=3-31, A < 5 = 2-89; 



and from symmetry of loading T= - t= — 0" 002035. 



2 



Applying equation (30) and (32) to spans (1.0) and (1.2) respectively, 

 and eliminating T and T' by adding them, we obtain 



0-1888^ + 0-04827^- 10481 = 0; 

 and applying equation 32 to span (2 . 1), 



O-O48270 1 + O-O87650 2 - 5420 = 0, 

 whence ^=46206, <p 2 = 36387. 



Taking these values of X and (f> 2 , and applying equation (33) to the 

 calculation of the deflection at the middle of the large span, 



Y=0-375ft. = 4*5 inches. 



If, now, the values of <£ x , </> 2 , and Y be calculated from equations (26) 

 and (19), on the supposition that the moments of inertia of the section 

 and the loads are constant throughout each span and equal to their 

 mean values, they are 



1= 47O3O, 2 =3561O, Y=4'62, 

 which are almost identical with the values ascertained by Mr. Pole. 



If the variation of section alone be considered, the load being taken at 

 its mean value, 



</) x = 46382, </> 2 =34465, Y=4-52. 

 It therefore appears that the amount of variation in the section and 

 load which occurs in each span of the Britannia Bridge, when taken 

 strictly into account, produces scarcely any effect on the values of the 

 bending moments and deflections, which are practically the same as those 

 resulting from their mean values considered as constant ; and it may be 



f 2 



