6 FORCES IN THE SAME PLANE. '[CHAP. 1. 



As this method is not as well known as it deserves to be, it 

 will perhaps be of advantage to pause for a moment in the 

 development of our subject, and make this direct application of 

 the principles already established. 



BRACED SEMI-ARCH. 



9. Stoney, in his "Theory of Strains," Yol. I., page 123, 

 gives the following example of a " braced semi-arch," repre- 

 sented by Fig. 5, PI. 1. The dimensions are as follows : pro- 

 jecting portion, 40 ft. long, 10 ft. deep at wall. Lower flange, 

 circular, with a horizontal tangent 2 ft. below the extremity of 

 girder. Radius of lower flange, 104 ft. Load uniform and 

 equal to one ton per running foot supposed to be collected into 

 weights of 10 tons at each upper apex, except the end one, 

 which has only 5 tons. 



Fig. 5 shows the arch drawn to a scale of 10 ft. to an inch. 



This scale is too small in this case to ensure good results ; in 

 general the larger the scale to which the frame can be drawn, 

 the better; but for the purpose of illustration it will answer 

 well enough. With a large scale for the frame diagram, a 

 scale of 10 tons to an inch will in general be found to answer 

 well. Fig. 5 (a) gives the strains in the varkms members to a 

 scale of 10 tons to an inch and Fig. 5 (b) 20 tons to an inch ; 

 the first for the load at the extremity alone, the second for a 

 nniform load. 



Fig. 5 (a) is thus obtained. "We first lay off the weight, 5 

 tons, to scale, in the direction in which it acts ; i.e., down- 

 wards. Now this weight and the strains in diagonal 1, and 

 flange A, are in equilibrium; therefore by article (4) the force 

 polygon must close. Drawing lines therefore from the ends of 

 the line representing the weight of 5 tons, parallel to these 

 pieces and prolonging them to their intersection, we obtain 

 the strains in A and 1. Commencing with the beginning of 

 the weight line and following down around the triangle thus 

 formed, we find that A acts from right to left, as shown by the 

 arrow. A acts then away from the apex / it is therefore in 

 tension. Diagonal 1 acts towards the apex and is hence com- 

 pressed. 



We pass now to apex 0, of the frame. Here we have the 

 strains in E and diagonals 1 and 2, and these three strains hold 

 each other in equilibrium. The strain in 1 we have already, 



