246 



SUPPLEMENT TO CHAP. XHT. 



[ART. 4. 



4. Determination of the Moments Supports all on 

 level. When all the supports are in the same horizontal, the ordinates 

 ftii A,, ^r, etc., are equal; and hence the term involving 21 disappears, 

 and we have simply 



M r _i Zr_i + 2 M r (Zr_i + Zr) + Mr +1 ^ = 

 P r _i Zr'_i (k - #) +P r Zr (2 - 3 * + *), 

 as already given in Art. 144. 



Now let s = number of spans, and let a single load P be placed on the 

 rth span. [PI. 23, Fig. 89.] 



From the above theorem, since Mi and M a+ i are zero, we may write the 

 following equations : 



2 M a (Z, + Za) + M, Za = ; 



M, I, + 2 M, (Za + Z 8 ) + M Z, = 0. 



M r _! Zr.! + 2 M r (Zr_i + Zr) + M r+1 ^ = 

 F r ZT (2 & - 3 &' + F) = A; 



Mr Z, + 2 M r+l (Z,. + Zr +1 ) + M r+2 Zr+l = 

 P r Z,. 8 (A - F) = B. 



(6) 



M g _ 2 Z s _ 2 + 2 M 8 _! (Z B _2 + Z s _!) + MS Zg.! = Oj 



M 8 _! Z g _! + 2 M 8 (Z 8 _! + Z 8 ) =-. 0. 



The solution of these equations can be best effected by the method of 

 indeterminate coefficients, as referred to in Art. 136. 



Thus we multiply the first equation by a number <? 2 , whose value we 

 shall hereafter determine, so as to satisfy desired conditions. The second 

 we multiply by c t , the third by c 4 , the rth by Cf+i, etc., the index of c cor- 

 responding always to that of M in the middle term. Having performed 

 these multiplications, add the equations, and arrange according to the co- 

 efficients of M 2 , Ms, etc. We thus have the equation 

 [2 c (Zi + Z a ) 4- Ci It] M a + [c, Z a + 2 c, (Z a + Z,) + c 4 Z 8 ] M s + . . . 



+ [Cg_2 Z 8 _ 2 + 2 Cg_i (Z8_3 + Z s _i) + C B Zs_i] M B _i 



4- [Cg_i J 8 -i + 2 CB (k-i + k)] M, = A <v + B Cr+1 . 

 Now suppose we wish to determine M s . We have only to require that 

 such relations shall exist among the multipliers c that all the terms in the 

 first member of the above equation, except the last, shall disappear. We 

 have then evidently, for the conditions which these multipliers must 

 satisfy, 



2 Ct (Zi + Z a ) + c, Z, = ; 



Ca Za + 2 C, (Za +Z 8 ) + C 4 Z, = ; 



r-l Zr_! + 2 <V (Z P _i + Zr) + 



= 0; 



._ 2 Z B _j + 2 C8_i (Z B _ 8 + Z B _i) 



= 0; 



