SIMPLE STRUCTURES 



177 



I 



It was also proved that the sum of the moments of any number of 

 forces lying in the same plane with respect to a point in this plane 

 is equal to the moment of their resultant with respect to this point. 



It remains to consider the case when the system of forces lie in 

 the same plane but are not concurrent, that is, do not all meet in 

 a point. This involves the properties of a force couple, defined as 

 two equal and opposite parallel forces F, F, not acting in the same 

 line (Fig. 116). 



For any couple F, F, let x denote the distance of any point in 

 its plane from the nearest force of the couple, and d the lever arm 

 of the couple (Fig. 116). Then the moment M of the couple with 

 respect to the point is 



x)-Fx = Fd. 



Therefore the moment of the 

 couple is constant and equal to 

 Fd with respect to any point in 

 its plane. Moreover, since the 

 moment of the couple involves FlG 116 



only the magnitude of the forces 



and their distance apart, it is evident that the couple can be revolved 

 through any angle without altering its value. A couple may, there- 

 fore, be moved about anywhere in its plane without altering its 

 numerical value or changing its effect in any way. 



It is also obvious that the forces of a couple may be altered hi 

 amount, provided that the lever arm is at the same time changed 

 so as to keep their product constant. Two or more couples may 

 therefore be combined by first reducing them to equivalent couples 

 having the same lever arm and then taking the algebraic sum 

 of the forces, thus giving a single resultant couple with this same 

 lever arm. 



Now consider any number of forces F^ F 2 , F s -, lying in the same 

 plane but not concurrent. At any arbitrary point 0(Fig. 117), intro- 

 duce two forces F^, F", opposite in direction, but each equal in 

 amount to F^ Since Ff and F" are equal and opposite they will 

 not disturb the equilibrium of the system. But F l and F{' together 

 form a couple of moment F' l d^ leaving the single force F^ equal 



