Theory of Continuous Beams. 69 



Let — P be the upward shearing-force exerted close to the point of sup- 

 port (#=0), <I> the bending moment, and T the tangent of the inclination, 

 positive downwards, at the same point. Then, by the general theory of 

 deflection, we have, at any point x of the span I, the following equations : — 



moment, <X> = <!> — Voc-\-m; (2) 



deflection, y=Tx— Yq + ® n + F (3) 



Let 4> x be the moment at the further end of the span I, and suppose it 

 given. This gives the following values for the shearing-force P and slope 

 T at the point (x= 0) :— 



p = *o-*i + »ii ; (4) 



I 



and because y x —Q, 



I °\l 2 1) I 2 I 2 1' ' ' 



Consider, now, an adjacent span extending from the point of support 

 (x=0) to a distance (—x=l') in the opposite direction, and let the defi- 

 nite integrals expressed by the formulae (1), with their lower limits still 

 at the same point (x=0), be taken for this new span, being distinguished 

 by the suffix —1 instead of 1. Let — T' be the slope at the point of sup- 

 port (x= 0). Then we have for the value of that slope, 



■ ■ < 5A > 



Add together the equations (5) and (5 A), and let t=T—V denote the 

 tangent of the small angle made by the neutral layers of the two spans 

 with each other in order to give imperfect continuity. Then, after clearing 

 fractions, we have the following equation, which expresses the theorem of 

 the three moments in Mr. Heppel's theory :— 



0=^( 4l ^ + 9 _ 1 P- Hl «'=- M _ 1 W 2 )-4> 1 ? i r-*_ l? _ I 



''}■ 



+ m£f 2 + m^q_l 2 —Y x W 2 -¥_Vl 2 -tl 2 V 2 . 



That equation gives a linear relation between the bending moments 

 <£_i> $ , at any three consecutive points of support, and certain known 

 functions of known quantities. In a continuous girder of N spans there 

 are N — 1 such equations and N — 1 unknown moments ; for the moments 

 at the end most supports are each = 0. The moments at the interme- 

 diate points of support are to be found by elimination ; which having 

 been done, the remaining quantities required may be computed for any 

 particular span as follows : — The inclination T at a point of support by 

 equation (5) ; the shearing force P at the same point by equation (4) ; 

 the deflection y and moment $ at any point in that span by equations (3) 

 and (2). Points of maximum and minimum bending moment are of course 



found by making ^- =0 ; and points of inflection by making $=0. 



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