CHAP. Xm.] ANALYTICAL FOKMUL.E. 237 



and similarly on the other side, 



M 8 = 0, M^ = * M, +1 , M 8 _ 2 = B -f=* M^, etc. 



ff r+l VT+I ' 



We must remember always to give the proper signs to the 

 moments, viz., positive for extremities of loaded span, and 

 alternating each way from these for the others. From the 

 formulae of Art. 140 we can then easily find the shear at any 

 support. 



155. Continuous Girder tevel Supports Spans all dif- 

 ferent General Formulae.* The preceding formulae com- 

 prise the case of one or, at most, two variable end spans. We 

 give below the general formulae for all spans different. These 

 formulas include all the others as special cases. Thus, if we 

 make all spans equal, we have the formulae of Art. 149. If end 

 spans ^ and 1 9 are made zero, and we take the number of spans 

 equal to s 2, and first support 2, we have the continuous gir- 

 der with fixed ends, in which the intermediate spans may or 

 may not be equal, as we choose. If we make ^ = or l a = 

 alone, and s 1 = No. of spans, we have a continuous girder 

 fixed at one end only. In short, the formulae comprise the 

 entire case of level supports. They are as follows : 



Let 8 = number of spans, i r = length of loaded span, k = , 



4 



a being distance of load from left support ; ^, ^, 7 3 . . . . 4-i4j 

 the length of the various spans counting from left. 



Then, when m < r + 1, M m = c 



+ t 



the 



_ 



at supports of loaded span, S r = - - - + ^ S' r+1 = 



i t 



Mr + 1 ~ Mf + ?'. For tmloaded spans, S m = Mgt ~ Mm+1 , S' m = 



'm 1 



* These formulas were first given by Mr. Merriman, and may be found in 

 the London PML Magazine, Sept., 1875. 

 f We can put * 2 ^_ 1+ 2 fc+ ^ d a = - d^ and ^ 0^+ 2 (? g + ^ . 



= o I . These values may be used for supported ends. The above 



values are, however, the most general, and hold good not only for supported 

 ends, but tot faced ends as well. 



