to Transverse Strain in Beams. 417 



Integrating between a; and ^-' 



WJ WP 



(2-*--) 



4 16r* ' 



therefore, adding (13), 



EI ^_w^ 2 Wl 2 W/ 2 (r-l) 

 d# 4 16r 2 4r 2 (3 — r) 

 Integrating again between the same limits, 



(y yi) ~ 12 96^ I6r* + 32i* 4r 2 (3-r) + Sr\3-r) ; 



therefore, adding (15), 



m , _, Wx* WP WPz Wl 2 {r-l)x , 3W/ 3 (r-l) 

 * 1 (y + o)=ir + Jq-3 - T7T3- "" ^Q_.A + 



12 ' 48r 8 16r 2 4r 3 (3-r) 8r 3 (3-r) 



-4^ h - L 3^ ; * * (16) 

 and when x = 0, y = 0, 



FT£- W 3WZ 3 (r-l) WZ 3 2 



■" 48r* + 8r 3 (3-r) 4r 3 3-V 



._12W/J_ 3(r-l) ^L_ 2 \ 



E&d 3 V48r 3 + 8r 3 (3-r) 4r 3 3-rJ' * > ' 



where ^ is the normal deflexion on the supposition of perfect 

 elasticity, and 8 the increased deflexion consequent upon over- 

 strain. 



Equations (14) and (16) are the equations to the curve for 

 the portions of the beam between the centre and the distance 



3- from the point of support, and between that and the point 



of support respectively, the value of the constant 8 in both 

 being determined by (17). 



Before proceeding further, it will be well to recapitulate the 

 leading features of the theory above developed, which may be 

 termed the theory of overstrain. 



In any beam or bar of rectangular section subjected to. 

 transverse strain, the form of the areas of equal resistance, 

 shown in fig. 6, and the coincidence of the neutral axis with 

 the centre of gravity, presuppose a condition of perfect elas- 

 ticity ; that is, that the resistance to the extension or com- 



