210 Mr. 0. H. Lander on Struts and Tie-Rods 



But if Rj be the point where PR cuts the base-line ac 

 we have : — 



PR 1= |(a£) +*(<*>) 



EI7 r =PR-PR 1 =-RRi, 



26 

 L 



RR.L 



or EI0--= 



In other words, the length RR t when multiplied by the 

 span o£ the beam is proportional to the slope of the beam at 

 the support k. The same reasoning may be applied to the 

 span lying to the left of k, and we shall get another point 

 which is such that the distance by which the base-line pusses 

 over or under it, when multiplied by L is proportional to 

 the slope of the beam over the support. Since if the slope 

 of the beam is positive when proceeding from left to right 

 it will be negative when proceeding from right to left, it 

 follows that if the spans be equal the base-line must pass 

 over the characteristic point en one side of the support and 

 under the characteristic point on the other side of the 

 support by equal distances. If the spans be unequal these 

 distances will be inversely as the two spans. In order then 

 to draw the complete bending moment diagram for a con- 

 tinuous beam, it is necessary firstly to draw the diagram for 

 each span as though it were discontinuous. Secondly to 

 find the position of each pair of characteristic points for 

 each support and then complete the figure by drawing a 

 broken line such that it passes over and under the charac- 

 teristic points at each side of each support by equal distance 

 or by distances inversely proportional to the spans. For 

 uniform loads the method is very simple in application since 

 the heights of the characteristic points above the free-ended 

 base-line are equal to two-thirds the maximum height of the 

 parabolic bending moment diagram for the span to which 

 they belong. 



Consider now the same beam having a load equal to P 

 applied axially at each end. The bending moment diagram 

 for the beam under the lateral load only may be drawn, and 

 hence the curve of the bended beam under this loading only 

 be obtained. This curve will not, as in the case of a dis- 

 continuous beam, exactly represent the bending moments 

 due to the end load even supposing the beam to bend exactly 

 to this shape. In order to obtain a diagram of bending 

 moment due to the end loading it will be necessary to take 



