160 DEFLECTION OF BEAMS 



(a), the points I, 2 and 3 being at a distance dx apart. The distortion 

 will be assumed to be so small that d I = dx. Now, the triangle 2-O-3 

 may be taken as similar to triangle 3-2-3 1 for small distortions, and 



O-2 : 2-3 :: 2-3 : 3-3 1 



but O-2 equals R, 2-3 equals dx, and 3-3 1 equals d 2 y; and, therefore, 



d 2 y i 



. T^=R (4) 



Now, in a beam as in (e) Fig. I, the stresses at any point in the 

 beam will vary as the distance from the neutral axis, and from similar 

 triangles we have 



R : dx :: c : A 

 and 



R = cdx (5) 



Now, if 5 is the fiber stress on the extreme fiber, and E is the 

 modulus of elasticity, we have 



A : 5 : : dx : E 



&E = Sdx (6) 



and, solving (5) and (6) for R, we have 



But from the common theory of flexure we have Me SI, and 

 substituting 



El 



R = - (7) 



M 



Substituting the value of R in (7) and (4) we have 

 d 2 y M 



The preceding discussion gives the following simple graphic method 

 for constructing the elastic curve of a beam : 



Construct the bending moment polygon for the given loading on 

 the beam. Load the beam with this bending moment polygon, and 



