198 COMBINATIONS OF BUCKLING AND BENDING. Art. 115. 



formula because the accidental eccentrities, etc., may conspire 

 to make M greater than that calculated from the deflection given 



by the formulas below. 

 In Practice M is usually 

 taken as dependent 

 ^ U p on the transverse 



|* ----- L ----- H load only, but to be 

 Fig. 146. exact both loads must 



be considered as acting 



simultaneously. The law of superposition is not accurate enough 

 in some cases. Fig. 146 shows a common case for which 



The equation of the elastic line is 

 =M*=\wx* \wLx-Py 



tf\0 Lx +fl w Ox+C&oa Ox 



when x=0, y=Q, hence 



w 



'OP 



when x=L, y=0, hence 



n w ( \ cos 6 L) 



Substituting the values of C^ and C., 

 ey=~(\ 0* x 2 i <9' J Lx 1+ cos Ox + tan \BL sin ftu) 



When x=^/ 2 L 



w 

 m.r=( 1 & L 2 \ 0* L 2 1+ cos I 6 L+ tan \ L sin 



W 



,_l_ip#) 



(63) 



The equation for ?/ max may be put into a simpler form by 

 developing sec y 2 0L into a series in accordance with the formula 



5 X A . 61 z 6 . 



. .etc 



24 



720 



l For the formula of integration see Johnson's " Differential Equations,' 

 Art. 82, or Boyd's " Differential Equations, 11 Arts. 33 and 41. 



