1879.] On Most General Problems in Continuous Beams. 497 



and simplifying, we find an equation connecting M , and l/ 2 , and 

 this is our form of the General Theorem of the Three Moments.* 



+ !Mi-^+^_A% + ^_^o. . . . (15). 



4. We have in the theorem of three moments a relation between the 

 bending moments M , M Y , lf 3 , at any three consecutive points of sup- 

 port ; therefore in a continuous girder of N spans there are N—l such 

 •equations, and N—l unknown moments, because the moments at the 

 end supports are each equal to nought. Hence the moments at all the 

 supports are found bf the easy solution of these simple simultaneous 

 equations. 



We may now suppose that the moments at the supports and Q at 

 the two ends of a span are found, then the shearing force at can be 

 calculated by equation (6), the bending moment at any point P by 

 equation (7), and the deflection at any point P by equation (10). 

 Points of maximum and minimum bending moment are obtained by 

 equating to nought the differential coefficient of M with respect to £»• 

 At a point of inflexion we know that the bending moment must be 

 equal to nought. 



We can prove that, in any span OQ of a continuous beam when we 

 know the angles of inclination of the girder at the two points of 

 support, we have sufficient data for making all the necessary calcula- 

 tions. For 



g . ftt o _ (M + H h ~M 1 ) g i-? 1 F 1 -M » 1 Z 1 ; (12 ) ? 



h 



at Q = (M +m 1 -M 1 ) g l -^ 1 -M oroi ; l+M ^_ go/i+gi ^ 



so that, knowing « and a l9 we can find M and Mj ; and then the 

 bending moment at any point P, if x equals OP, is 



M = M + m — x— — _JL r —l ( / ) , 



H 



so that we can draw the diagram of bending moment. Also the shear 

 at is 



So= M 2 -Mi ± m 1 (6)i 



h 



* Altliougli tlie theorem of the three moments as given in equation (15) involves 

 more expressions than Mr. Heppel's form as given by Professor Rankine (" Proc. Roy. 

 Soc.," vol. xviii, p. 178), still it will be observed that it is really more convenient. In 

 any case the summations d, f, g have to be calculated in order to obtain n, q, and F, 

 but with our form of the theorem there is this advantage, that when there are more 

 than two spans the summations have only to be made once for any span. 



