FORCES IN THE SAME PLANE. [CHAP. I. 



structures, the variation resulting from performing 

 the operation twice being inappreciable. Every symmetrical 

 frame gives also a symmetrical strain diagram, and the accu- 

 racy of the work is tested at every point by this double sym- 

 metry, and finally by the end or last point of the second half, 

 exactly coinciding with the last point of the first half. Thus 

 in Fig. 6 (a), if we had but one system of triangulation carried 

 through the frame, the strain diagram for the right half would 

 be precisely similar and symmetrical to that already found for 

 the first, and the end of the last line would fall, or should fall, 

 precisely upon the point b of the first. If it does not, and the 

 error is too great to be disregarded, then by checking corre- 

 sponding points in each half, we can find the point where the 

 error was committed. In any case errors do not accumulate. 

 Thus, armed with straight edge, scale, triangle, and dividers, 

 we can attack and solve the most intricate problems, without 

 calculation or tables, with ease, accuracy, and great saving of 

 time. 



METHOD OF SECTIONS. 



14. The results obtained by the above method are best 

 checked in general by Hitter's u method of sections/' or the 

 use of moments.* This consists in supposing the structure 

 divided by a section cutting only three pieces. We can then 

 take the intersection of two of these pieces as a centre of mo- 

 ments, and the sum (algebraic) of the moments of all the 

 exterior forces, such as reaction, loads, etc., upon one of the 

 portions into which the structure is divided by the section, with 

 reference to this centre of moments, must be balanced by the 

 moment of the strain in the third piece, with reference to this 

 same point. Thus in Fig. 6, PI. 2, required the strain in D. 

 Take a section through D, 7 and H (right half of Fig.), and let 

 a be the centre of moments. The moments of the strains in 7 

 and H are then, of course, zero, since these pieces pass through a. 

 The moment of the strain in D with reference to a must then 

 be balanced by the sum of the moments of all the outer forces 

 acting upon the portion to the left (or right) of the section. 



Thus, strain in D multiplied by its lever arm with respect to 

 a, is equal to moment of reaction at A, minus sum of the mo- 

 ments of loads between A and a, all with reference to a. If 



* Dock- urtd Brucken- Cowtructionen. Ritter. Hannover, 1873. 



