CHAP. Vm.] CONTINUOUS GIRDERS. 127 



Moreover, 



(3\ dy = ?L= , 



' da H, H 



Differentiating : 



d? 2 y j> d x 



y. = s or 



a x 



(4) -- U d/- = *' 



Now, had we formed a force polygon by laying off the forces, 

 then taken a pole at distance H and drawn lines from pole to 

 ends of forces, the corresponding equilibrium polygon would, 

 as we have seen, Art. 43, be tangent to the curve A B D at the 

 points midway between the forces. The greater the number 

 of forces taken, the shorter, therefore, the sides of the polygon ; 

 the nearer it will approach the curve A B D. This curve is 

 therefore the equilibrium curve, found according to the graph- 

 ical method. Its equation is given above by (4). 



But the equation of the elastic line is, as is well known, 

 d? y M * 



where E is the modulus of elasticity of the material, M the 

 moment of the exterior forces, and I the moment of inertia of 

 the cross -section. 



Comparing now this equation with equation (4) above, we 

 see that the elastic line is an equilibrium curve whose horizon- 

 tal force H is E, and whose vertical load per unit of length p 



M 



is represented by the variable quantity --- 



This simple relation, first given by Mohr, renders possible 

 the graphical representation of the elastic line, and not only 

 solves graphically almost all problems connected with it, but in 

 many cases simplifies considerably the analytical discussion 

 also. 



81. Elastic Curve If we choose the pole distance H at -th 

 E instead of E, the ordinates of the elastic line will be n 

 times too great. If the scale of the figure is, however, - th the 



* Stoney Theory of Strains, p. 146. Wood Resistance of Materials, p. 98. 

 Also Supplement to Chap. VII , Art. 11. 



