CORRUGATED BAR COMPANY, INC. 



MOMENTS AND SHEARS FOR CONTINUOUS 

 BEAMS 



The moment factors commonly specified for continuous beams assume equal spans 

 and uniform loads. While these factors are within safe limits for the usual conditions 

 met with in building design, cases arise where it is advisable to investigate the actual 

 moments and shears produced, through inequality of span and load, by the theorem of 

 three moments. 



This theorem may be employed in problems involving either uniform or concentrated 

 loads or combinations of the two, but a full discussion of the theory involved would 

 be out of place in a book of this character and, therefore, only a brief statement cov- 

 ering its application will be given. 



In the formulas and diagrams which follow it is assumed that the moment of inertia 

 is constant throughout the length of the beam and that the supports retain their same 

 relative position after the beam is loaded as before. 



UNIFORM LOAD 



For uniform load the theorem is expressed by the formula, 



M,h + 2M^{h + /2) + M3/2 = - ^ - ^ 



Fig. 4 



The equation as written applies to the moments at the supports in span h and k; 

 by increasing all the subscripts by one it will apply to spans h and h and so on for as 

 many spans as there are in the structure. It will thus be seen that there may be ob- 

 tained as many equations as there are unknoijvn moments, assuming that the moments 

 at the first and last support are zero or their values are known. Having obtained the 

 moments at the supports, the shears and moments at any other section of the beam 

 may be found by the following equations. 



Consider for example, spans h and h in Fig. 4. 



„ Mi— Ml toili 

 ' h ^ 2 



V'i = Vi-wA 



-, M3 — M2 , Ws 



V'3=V2-wd2 



The reaction at any support (Ri, R2, etc.) will be equal to the shear on its right 

 plus that on its left with the sign reversed. 



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