EXTERNAL FORCES 



41 







] ff , "~" 7 f _"~.T..j'. J 



due to the cantilever end of the beam. The straight line under 

 it shows moment due to the concentrated load on the extreme 

 end. The lower slightly curved lines, AC and BD, represent the 

 combined moments under the cantilever ends. This line at each 

 point is the sum of the two moments, so is a mean between the 

 parabola and straight line. 



The bending moment at the middle of the span between the 

 two supports is found as follows, there being a positive and a nega- 

 tive moment to consider: 



-M = 250 x 10.5 + (10.5 x 20) x 5.25 = -3827.5 ft. Ibs. 

 +M = 460 X 7.5 = 3450 ft. Ibs. 

 Actual 



M - +M - M = -3827.5 + 3450 = -377.5 ft. Ibs. 



This negative mo- 

 ment is set off at the 

 middle of the span 

 measuring down from 

 the line A B. The pa- 

 rabola CED is drawn. 

 The bending moment 

 at any point is found 

 by scaling the length 

 intercepted between 

 the line AB and the 

 bounding line ACEDB 

 of the bending moment 

 diagram. All lengths 

 measured horizontally 

 are distances and all 

 lengths measured ver- 

 tically are forces. 



When the positive moment is greater than the negative moment 

 the point E is set off above the line AB, so the parabola in such 

 case is partly above and partly below this line. The curve above 

 indicates positive and the curve below indicates negative moment. 

 The maximum moment is where the shear changes sign. Where 

 the moment curve crosses the line AB there is no moment, this 

 point being termed the " point of reverse moment " or " point of 

 contraflexure," or " point of inflection." 



Shear Diagram 



Fig. 37 Graphical Method for Beam with Two 

 Overhanging Ends 



