ES-140 Three-Dimensional Equilibrium

For three dimensional equilibrium the two vector equations are...



These become six scalar equations...

Note: The subscript for M indicates the axis about which moments are to be taken. As in two dimensional equilibrium, any six independent equations may be used.


For instance, six independent moment equations could be used.

In three dimensional equilibrium, considerable algebraic manipulation may be saved by initially writing moment equations about a point through which as many as possible unknowns pass.


Consider this example...

If the tensions in the cables and the reaction at a ball and socket are to be found, taking moments about point A will considerably reduce the work load. The tensions in the cables will appear as three unknowns in three equations. The additional force equations of equilibrium may then be used to solve for Ax, Ay and Az, the reactions at the ball and socket joint, with only one unknown per equation.

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