Couple Decomposition

Forces in a couple can be decomposed also. Couple in space can therefore resolved into three couple components of forces, Fx , Fy and Fz without change its moment arm at the point of application.

 

Therefore, the moment  M of couple of force, Fa  can be expressed as:

 

Although the couple of force, Fa  can be resolved into three couple components of rectangular force, Fx Fy and Fz, the moments of the  three couple components,  M1, M2 and M3  may not be the rectangular couple components because the plane containing the force couple may be at an angle to  the coordinate planes. However, the vector sum of  M1, M2 and M3 is always equal to moment  M.

The rectangular components Mx, My and Mz  of moment M can be determined by resolving the moment arm vector ra into rectangular components, imply

 

Couple Addition

For the addition of couples at their intersection line, forces at the point of application can be represnted by a resultant force. The resultant forces R and -R of force couples of forces  Fa and Fb also form a couple of force R.

 

Since all force couples can be translated to the same points of application with common moment arm, the resultant moment M of force couples can be expressed as:

 

Therefore the resultant moment M of force couples is the vector sum of moments   Ma and Mb  of forces   Fa and Fb.