i34 STRESSES IN TWO-HINGED ARCH 



calculated. The stresses given in column 4 are calculated as shown 

 in Fig. 71. Column 6, giving deformations of each member, and col- 

 umn 7, giving the order in which these deformations are used, are, how- 

 ever, the only values used in constructing the deformation diagrams. 



Deformation Diagram. The principle upon which the construc- 

 tion of the deformation diagram is based is as follows : Take the two 

 members a-c and c-b in (d) Fig. 72, meeting at the point c. Assume 

 that a-c is shortened and b-c is lengthened the amounts indicated. It 

 is required to find the new position, c r , of the point c. With center at a 

 and a radius equal to the new length of a-c = a-c', describe an arc. 

 The new position of c must be some place on this arc. Then with a 

 center at b and a radius equal to the new length of b-c = b-c', describe 

 an arc cutting the first are in c'. The new position of c must be some 

 place on this arc and will therefore be at the intersection of the two 

 arcs, c'. Since the deformations of the members are always very small 

 as compared with the lengths of the members, the arcs may be replaced 

 by perpendiculars, and the members themselves need not be drawn, (e) 

 Fig. 72. 



To draw the deformation diagram, (b) Fig. 72, for the arch 

 as a truss with one end on frictionless rollers and loaded with vertical 

 loads, proceed as follows : Begin with the member marked I, lay off its 

 deformation = + .076 inches (Table VII., column 6) to scale and 

 parallel to member i. Now lay off the deformation of 2 = .028 

 inches away, from the joint U 2 and parallel to the member 2, and lay 

 off deformation of 3 = .028 inches, away from the joint U' 2 and 

 parallel to the member 3. Perpendiculars erected at the ends of de- 

 formations 2 and 3 will meet in the new position of L 2 . The vertical 

 distance between the deformation I and point L 2 represents to scale the 

 change in position of L 2 relative to the member U 2 t/ 1 2 . At L 2 in the 

 deformation diagram lay off deformation of 4 = .069 inches, away 

 from the joint and parallel to the member 4, and at U 2 lay off deforma- 

 tion of 5 = -j- .054 inches, toward the joint and parallel to the member 

 5. The perpendiculars erected at the ends of the deformations 4 and 5 



