122 



STRESSES IN THREE-HINGED ARCH 



bending moment at hinge C, the line of action of the reaction R 2 must 

 pass through the hinge at the crown. This determines the direction of 

 reaction R 2 , and since the three external forces P L , R 2 and P produce 

 equilibrium in the structure they must meet in a point. Therefore to 

 find the direction of 7^ produce B C to d and join d and A. 



FIG. 66. 



The values of R and R 2 may then be obtained from the force poly- 



gon. 



The reactions due to loads on the right segment may be found in 

 the same manner. The two operations may be combined in one as il- 

 lustrated in the solution of the dead load stresses in a three-hinged arch, 

 Fig. 67. 



Calculation of Dead Load Stresses. To find the reactions for 

 the dead loads in Fig. 67, the loads are laid off on the load line of the 

 force polygon in order, beginning at the left reaction A f and two equi- 

 librium polygons, one for each segment, are drawn using the same 

 force polygon. The vertical reactions at the crown, P G , and at abut- 

 ments, P R and Pg, are found by drawing a line through pole o of the 

 force polygon parallel to the closing lines of the equilibrium polygons. 

 The load P Q at the crown causes reactions R^ and RJ-, and combining 



