ANALYSIS OF STKESS IN BEA.Ms 35 



which are neither lengthened nor shortened. The. horizontal line in 

 which tins strip intersects any cross section is called the neutral axis 

 of the section. 



41. Consequence of Bernoulli's assumption. From Fig. 19, it is 

 t- vident that as a consequence of Bernoulli's assumption the length- 

 ening or shortening of any longitudinal fiber is proportional to its 

 distance from the neutral axis. But, by Hooke's law, the stress is 

 proportional to the deformation produced. Therefore the stress on 

 any longitudinal fiber is likewise proportional to its distance from 

 the neutral axis. Navier was the first to deduce this result from 

 Bernoulli's assumption. 



If, then, the stresses are plotted for every point of a vertical strip 

 MN (Fig. 20), their ends will all lie in a straight line, and conse- 

 quently this distribution of stress is called the 

 straight-line law. 



42. Result of straight-line law. In rectan- 

 gular coordinates let the axis of Z coincide 

 with the neutral axis, and the axis of Y be 

 perpendicular to it and in the plane of the F 10^20" 

 cross section. Th-n if the normal stress at 



the distance y from the neutral axis is denoted by p, and that at a 

 distance y is denoted by p v from the straight-line law, 



(16) 



PO y 



Since in order to equilibrate the external bending moment the normal 

 stresses must also form a moment, the sum of the compress! ve stresses 

 must equal the sum of the tensile stresses. Therefore, since the tensile 

 ami compr-ssive stresses are of opposite sign, the algebraic sum of 

 all the normal stresses acting on the section must be zero, that is to 



I pdF= 0, where dF is the infinitesimal area on which p acts. 

 Inserting the value of p from (16), 



the strip* are free to slide longitudinally but are otherwise fixed. If the strips are of 

 the tame length as the beam before bending it will ). f.,uml that after bemlin- the upper 

 strip projects beyond the ends of the beam, while the lower strip dues not n-adi the ends. 

 H of this kind have been made by Morin and Tresca. See Unwin, The Testing 

 of Material* of Construction, p. X. 



