72 



STRENGTH OF MATERIALS 



Inserting in this expression the value of s just found, = E ; 

 whence 



Let the radius of curvature CF of the elastic curve be denoted by p. 

 Then pdfi = x, and inserting this value of dfi in the above equation, 



it becomes 



El 



From the differential calculus, the radius of curvature of any . ui\f 

 can be expressed by the formula 



.. hfflT 



da* 



But since the deformation of the beam is assumed to be small, tin- 

 slope of the tangent at any point of the elastic curve is small; that 



is to say, -/- is infinitesimal, and consequently ( ) can be neglected 

 dx , tfy W 1 



. . <Py 

 in comparison with 



El I 



- = o = ; whence 



Under this assumption p = -3-* and there- 



(Py 



,/ , - 



(37) 



\\hich is the reqi; 

 differential equation of 

 the elastic curve. 



In what follows tin- 

 #1 external bending mo- 

 ment M is assumed to 

 be negative if it tends 



FIG. 55 to revolve the portion 



of the beam under 



consideration in a clockwise direction, and positive if the revolution 

 is counter-clockwise. 



