Art. 115-COMBINATIONS OF BUCKLING AND BENDING. 201 



To evaluate (7 X and C/ we have the conditions y=y l} when 



n dy dyi 

 x=l 1} and oJi=*2> and -^-=-3- when x=l and #!=,. 



Differentiating equations (a) and (a^), imposing the condi- 

 tions, eliminating, and reducing 



_ P t sin h 

 1 ~~OPsin (Ii6+ltf) 

 Substituting this value in equation (a), differentiating and 



putting -~ ~0, that value of x may be found which will make 



t/ 2/ max . The resulting equation is, however, too complicated for 

 practical use. Only a slight error will be committed if it is as- 

 sumed that the maximum deflection occurs at the same point that 

 it does when the longitudinal load is not acting ; and to be on the 

 safe side, the maximum moment from transverse load may also 

 be taken as coming at this point. Under these assumptions, 



Ttf T-k ^2 



and 



(67) 



U I/max 



From (a) and (&) 

 2/max = - 



when 



JL=* 



(68) 



As in the case of Fig. 146, it will usually be found accurate 

 enough to neglect that part of the deflection due to P, in which 



y mmx =P t HjL^ (2 k+li } approx. (69) 



There is, however, not much difference in the ease with which 

 equations (68) and (69) are applied. 



When P t is in the middle l 1 =l 2 =y 2 L=x' and equation (68) 



bec mes ' y M =^^(toniL-i) (70) 



An equation similar to equation (64) may be gotten if 

 tan 1/2.QL is developed according to the formula 



tan x=x-}-'y^x s -\-^x s -\- etc. 



powers higher than the fifth being negligible in ordinary cases. 

 Equation (70) becomes, 



