146 UNIVERSITY OF COLORADO STUDIES 



and 9 of Table II. Column 10 gives the interscepts of the trapezoid 

 mm' KB by adding those of columns 8 and 9.' 



Column 11 gives the interscepts between the closing line mm' 

 and the broken line BIJK found by taking the algebraic sum of 

 columns 10 and 3, considering the moments of column 10 negative 

 and those of column 3 po!?itive. 



Column 12 gives the interscepts between the closing line KjE.^ 

 and the neutral curve of the rib found by taking the algebraic sum 

 of each of the figures in column 2 and the constant interscept VK, 

 = V'Ko to the closing line K,K,, considering figures in column 2 

 positive and the common interscept negative. 



Columns 11 and 12 give lengths proportional to the true bending 

 moments at the various points of the arched rib and the trial poly- 

 gonal frame respectively. The algebraic sum of the figures in these 

 columns in each case should be zero since the closing lines K,K2 of 

 the rib and rmn' of the trial equilibrium frame each satisfy the con- 

 ditions, — = constant, EM = and EMa:'=0. 

 ' EI 



The true equivalent equilibrium frame however must also satisfy 

 with the rib the condition 2M?/ = 0. We must therefore change the 

 position of the assumed pole C to a point D such that the correspond- 

 ing frame KjEFGHK2 may have the closing line K1K2 and satisfy 

 the last condition. When this true equilibrium polygon for the rib 

 under the assumed loading has been determined the vertical inter- 

 scepts between it and the rib's neutral curve will be proportional 

 to the actual bending moments at all points in the rib. 



Through C draw CZ parallel to mm' to cut the load line AB in 

 the point Z. At Z erect the perpendicular ZD to the load line. 

 The pole of the true equilibrium frame must lie on this line ZD. 



Consider the equation; — 



5:^,M,y-2^,M„y=0. (18) 



2^, M,i2/=:0 because the rib closing line KjK.^ has been drawn 



(1) The interscepts of the trapezoid can be scaled directly from Plate 1; the above method 

 is simple and generally more accurate. 



