STRESS DIAGRAMS 197 
a 
« 
and a funicular polygon, the latter having one angular point at O, as 
explained in the latter part of Art. 57, p. 38. 
ae 182. The Three-Hinged Arch.—If the ends of a roof or bridge truss 
re secured to foundations by hinged joints, and there is another hinged 
int at an intermediate 
‘point, say, at the middle mA A 
of the truss, such a truss § 
is known asa three-hinged i 
arch, and it is said to ' 
constructed on the three- 
hinged system. The. de- 
t ination of the stresses 
1 the various bars of such 
a truss may be proceeded 
with as in an ordinary 
_ tress as soon as the re- 
actions at the hinges are 
_ determined. 
One method of find- 
ing the reactions at the 
_ hinges is as follows. _ Aaa |e 2 
Fig. 268 shows a truss 
hinged at A, B,-and C. The resultant load on the part AB is the force 
P, and the resultant load on the part BC is the force Q. First neglect the 
Toad persing on the part BC. The part BC is then under the action of two 
forces only, viz. the reactions at B and C, and these forces must balance 
‘one Bdaother, and will therefore act in opposite directions along the straight 
‘line BC. The truss as a whole is now under the action of three forces, 
viz. the force P, the reaction T, at C, which acts along CB, and the 
reaction §, at A. Since these three forces are in equilibrium, ’and since 
“the lines of action of two of them, T, and P, meet at m, therefore the line 
of action of the third one, §,, must be Am. By means of the triangle of 
forces the magnitudes of 8, and T, can be determined. 
Next neglect the load on ‘the part AB, and consider the load Q on the part 
"Be. This load Q will cause reactions S and T, at A and B respectively, 
and these reactions may be found in the same way as §, and T, were 
po ‘When both loads P and Q act, it is evident that the reaction at A will 
an the resultant of S, and §,, and the reaction at B will be the resultant 
and T,. 
he reaction of the part AB on the part CB at B will be the force 
“which will balance the force Q and the reaction at C, and the reaction of 
hey CB on the part AB at B will be the force which will balance the 
e Pand the reaction at A, These two reactions will, of course, be 
B qual and opposite. 
— When the truss is symmetrical about a vertical centre line, and is 
.m mmetrically loaded, the reactions at B will be horizontal, and the. line 
f action of the reaction at A will be the line joining A with the point of 
itersection of the line of action of the resultant load on the half trass 
B with the horizontal line through B. The direction of the reaction at 
Cc Cis found in like manner. 
