132 PERIOD OF VIBRATION OF STEAM VESSELS. 



BEAM UNIFORMLY LOADED, WITH SUPPORTS NOT AT ENDS. 



Fig. 9, Plate 59. 



The formula for the bending moment changes at the point of support. For 

 points outside of the support, the bending moment is given by the formula 



M= -VM'AL-xy (53) 



For points between supports, 



M = y2wL{y2L^A-x)-y2w{y2L-xY (54) 



By plotting these values, and integrating graphically for the values of i and y, 

 formulas (34) and (35), we can obtain the deflections relative to the lowest point 

 of the beam. By subtracting these values of y from y^, the deflection of the point of 

 support, we obtain the deflection relative to the points of support. Integrating graph- 

 ically these latter in accordance with formula (45), we obtain the value of 8. If 

 we square the deflections relative to the points of support, and integrate graphically 

 in accordance with formula (48), we obtain the value of ^^ For points of support 



one-eighth length from end, ^ works out approximately equal to 







•0034-^ (55) 



It will be noticed that for a uniformly loaded beam of half length = I, if sup- 

 ported: — At ends, 



^j = .i64o|^* from (52) 



At y^ length from ends. 



At mid length, 



^=.0546|^* from (55) 



? = .0802 ^* from (44) 



d EI 



BEAM IRREGULARLY LOADED, WITH SUPPORTS NOT AT ENDS . 



This case can be solved by plotting the load curve, integrating for shear, and 

 integrating again for bending moment. The process of obtaining deflections 5 

 and k^ would be the same as in the preceding case. 



If the supports are not rigid, but are capable of movement, as for instance, a 

 beam supported from balance arms with counterweights, a further complication will 

 result. 



