126 CONTINUOUS GIRDERS. [CHAP. VTO. 



curvature of the deflected beam, for any cross-section, is in- 

 versely proportional to the moment of the exterior forces. 

 Mow the deflection at any point is so small, and the radius of 

 curvature so great, that its construction is impracticable, and 

 will so remain until Geometry furnishes us with simple rela- 

 tions between the corresponding radii of curvature of pro- 

 jected figures whose projection centre lies in the vertical to the 

 horizontal axis of the beam. If such relations were known, we 

 could by projection exaggerate the deflection of the beam 

 until the radius of curvature became measurable. Since we 

 are not yet able to do this, we must have recourse to calcula- 

 tion" lie then enters into a somewhat abstruse analytic dis- 

 cussion of the continuous girder, and deduces formulae for the 

 reactions at the supports. These being thus known, the graph- 

 ical method is then applied. 



Concerning this difficulty, Mohr * remarks that it has but 

 little weight, and may be easily overcome if the same simplifi- 

 cation of the graphical method is made which is considered 

 allowable in the analytical investigation, viz., when we take in- 



. 



stead of the exact value of the radius of curvative 



da? 

 as given by the calculus, the approximate value -^ 



dtf 



Thus, let PI. 14, Fig. 50 represent a perfectly flexible cord 

 A B D loaded by arbitrary successive forces. The variation of 

 these forces per unit of horizontal projection dxwe represent by 

 p. Take the origin of co-ordinates at the lowest point B. If 

 the cord is supposed cut at B and D, we have at B a horizontal 

 force H, and at D a strain !-', which may be resolved into a 

 horizontal force H t and a vertical force V. Since these forces 

 are in equilibrium with the external forces, the conditions of 

 equilibrium are 



(1) H = H, 



and /# 



(2) V= / pdx. 



JO 



* Zeitsch. des hannov. Arch^-u. Ing. Vereins. Baud xiv., Heft 1. 



