Reactions at the Points of Support of Continuous Beams. 213 



if all the ordinates to the curves are taken at equal intervals, the 

 span being divided into an even number of the latter, the final 

 diagram AQCBRA may be plotted and its area obtained by any 

 planimeter, or by Simpson's rule. 



In tlie example appended the area has been found both ways. 



Elevation or Depression of Points of Support. 



Let c with the correct suffix represent the elevation of any point 

 of support above the line A A H , a depression being reckoned as 

 negative. 



Then in the equations to find the reactions, since the final deflec- 

 tion is not zero, we must write 



Sr 1 -hE5 1 = R l w 1 '+R 2 <'+ 



N 2 +E& = R 1 ^' + R.V + 



Some of the values of RiR 2 .... may be negative, in which case, 

 if the beam is not to be fastened down at the supports, a fresh solu- 

 tion must be sought by omitting one or more of the negative 

 reactions, until the remaining ones become positive. 



The mean fibre of the beam has hitherto been assumed a straight 

 line when under the action of no force. In certain girders this is not 

 the case, but the above methods may be applied with sufficient accu- 

 racy for practical purposes when the maximum distance of: the 

 external layer of the beam from the mean fibre is small compared 

 with the original radius of curva,ture of the mean fibre. 



For then = (_L_i ) approximately, 



ei \bi' r; 



where R is the original radius of curvature of the mean fibre at any 

 point and R' its curvature after loading. 



Hence, if v — F(#) is the original equation to the mean fibre, and 

 U = /( aj ) ^ ne equation after straining, 



then 



m 



d?y drv 

 EI ~~ dx l dx 2 



d 2 y _ d 2 v m 

 dx 2 ~ "^ + EI ' 



which shows that the final deflection curve is the result of super- 

 posing on the original curve of the mean fibre, the deflection curve 

 obtained under the given loading, for a beam of the same cross 



