128 



EQUATION OF THE ELASTIC LINE. 



Art. 81. 



, which according to equations (2) and (12) 

 j> i Mv / v 



deformation is 



ux 



gives 



This reduces to 



In order to get < in terms of x and y, the radius of curvature 

 p, of the elastic line is introduced; this is the distance from the 

 neutral plane to the intersection of AB and CD produced. Since 



dx 



od(j> = dx, d<f> = and equation (b) becomes 

 P 



I M ,, El 



t \ 

 (c) 



From the calculus l 



dx 2 dx 2 



Since the deflections are very small, -^ the tangent of the 



angle which the elastic line makes with the axis of x, is a very 

 small quantity, and its square is negligible. 

 Equation (c) becomes 



El^ = -M (39) 



The minus sign is gotten by adopting the following conven- 

 tions with regard to the moment and the deflection. It is usual 

 to call a moment positive when it bends the beam so that it will 

 be concave on its upper side, as is the case in a beam supported 

 at its ends. The deflection is always downward and is usually 

 called positive, that is, y is positive downward. Now, in a beam 



supported at its ends, -^ will be plus at the origin (the left 

 support), will decrease to zero and then become minus, that is, 

 it decreases over the whole length of the beam, and -^ must 

 therefore be negative. M is positive over the whole length of tin 

 beam; therefore the two sides of equation (39) have opposite 

 signs. This will be found to be true in any case, including 

 beams in which the moment changes from positive to negative, 

 or vice versa. 



Equation (39) is called the differential equation of the elastic 



1 See Edwards' Differential Calculus, page 137. 



