72 HESISTAKCE OF MATERIALS 



and, substituting this value of S* in the above expression for D A , it 

 reduces to 



n . 



Similarly, by considering the span BC and calculating the deflec- 

 tion D c of the point C measured from the same tangent at B, we 

 obtain the equation 





 c ~ 



Also, forming a moment equation with C as center of moments, 

 we have 



and, eliminating Sf between these relations, the result is 



QEI 



Now, since these deflections lie on opposite sides of the tangent 

 at B, we have, from similar triangles, 



Therefore, substituting the expressions for D A and D c in this rela- 

 tion, combining terms, and transposing, we obtain the relation 



(66) JfA + 2 X, ^ + g + 



In this relation, M^ M^ and Jf g are stress couples acting on the beam. 

 The external moments at the supports are equal in amount but 

 opposite in sign to the stress couples, or internal moments. There- 

 fore, calling M^M^ M 8 the external moments at the supports, the sign 

 of the expression is changed ; that is 



(67) JTA + 2 JT.ft + I,) + M,l, = - WA ' + W ' 1 *. 



This is the required theorem of three moments for uniform loads. 



43. Theorem of three moments for concentrated loads. Consider 

 a continuous beam bearing a single concentrated load in each span. 



