24:3 SUPPLEMENT TO CHAP. XITJ. |_AJii8. 5, 6. 



Equations (7) and (8) are the general equations given in Art 155, which, 

 as we have seen, include the whole case of level supports. 



ft. Uniform Load. For uniform load the same equations hold 

 good. We have only to give a different value to A and B. 



Thus, for several concentrated loads we should have 

 A = 2 P l t * (2 Jfc - 8 W + V). 



For a uniform load over the whole span 7 n let w be the load per unit 

 of length, then 



A, r lr 



= I w d a ; or since a = klr, 2 P = I w 



Jo Jo 



Inserting this in place of S P above, and integrating, we have 



A = B = i w I*. 



Thus the equations of Art. 155 hold good for concentrated and uniform 

 load in any span, for any number and .any lengths of spans. 



The above formulae were first published in an article on the Flexure of 

 Continuous Girders, by Mansfield Merriman, C.E., in the tendon Phil. 

 Magazine, Sept., 1875. 



6. Formulae for the Tipper. The expressions for the reactions 

 in this case, already given in Art. 120, may be easily deduced. The solu- 

 tion is tedious by reason of lengthy reductions, but the process of deduc- 

 tion is simple. 



The construction in this case is indicated in Fig. 83, PI. 22. We sup- 

 pose, as shown there, a weight upon the first span only. Under the action 

 of this weight the beam deflects, and one centre support falls and the 

 other rises an equal amount. Thus, if we take the level line as reference, 

 A = h. Moreover, the reactions at these two supports must always be 

 equal. 



We have, then, as representing this state of things, A a = As, and calling 

 the supports 1, 2, 8 and 4, we have from Art. 1, since MI = M 4 = 0, and 



R, = S, = S, = - + P (1 - *), 



w - B' j. B M.-M, , M 



R = B | + 8, = r h -J-, 



R = S' = 



These reactions will evidently be known, if we can determine the mo- 

 ments. 

 Let T r = 6 E I f^r - Ar-i + AT ~ hr+l ~\. Then the gen- 



L ^r-l It J 



eral equation of three moments of Art 3 becomes, when we neglect P r , that 

 is, suppose only the first span loaded : 



M r _i ZT_I + 2 M r (lf_i + lf) + M r+1 IT = Y r 4 P r _i V-i (k 3 ). 



This expresses a relation between the moments at three consecutive 



