Equilibrium in Three Dimensions

In general, the conditions for the equilibrium of a rigid body is.

and

For three dimensional problems, the resultant can be decomposed into three components. Imply

Three translational and three rotational motion are needed for determining a three dimensional structure is in static equilibrium state or not.

In other words, the possible forces and moments due to an applied action or a reaction in a three dimensional structure are  three rectangular forces and three rectangular moments, or one resultant force and one moment in space.

Since there are only six equations obtained from the equilibrium equations of a rigid body in three dimenstion, no more than six unknowns can be determined by the system of six equations.

Reactions at Supports and Connections

In order to construct the free body diagram for analysing the equilibrium of rigid body in three dimensions, the types of reactions at supports and connections should be evaluated first. The types of reactions at supports and connections can be divided into three types:

1. Reactions equivalent to three force components and three couple components

   For a fixed support, no translational motion and rotation motion is allowed for the free body to move and thus the free body is fully constrained.

   

2. Reactions equivalent to three force components and two couple components

   For an axial thrust and radial load support or connection, the rotational motion in one dimension is enabled by equipping with a fictionless hinge or pin, only two couple is reacted by the support or connection on the free body. But, the translational motion is stopped by the reaction force of the hinge support.
   However, when using this type of support or connection in the design, the other two couples may not always exist in normal operation.

   

   

3. Reactions equivalent to three force components and one couple component

   For an universal joint, the rotational motion in two dimensions is enabled by equipping with a pair of hinges oriented at 90 degree to each other, only one couple is reacted by the support or connection on the free body. But, the translational motion is stopped by the reaction force of the hinge support.

   

4. Reactions equivalent to three force components

   For a ball and socket joint, the free rotational motion in three dimensions is enabled by equipping with a ball in a frictionless socket, or a free end on a rough surface, no couple is reacted by the support or connection on the free body. But, the translational motion is stopped by the reaction force of the socket support or the friction force generated by the rough surface.

   

5. Reactions equivalent to two force components and two couple components couple components

   For a radial load support or connection, the rotational motion in one dimension is enabled by equipping with a fictionless hinge or pin, only two couple is reacted by the support or connection on the free body. Because there is no axial thrust, the axis is free to move on the axial direction, and only two translational motions are stopped by the reaction force of the hinge support.
   However, when using this type of support or connection in the design, the other two couples may not always exist in normal operation.

   

6. Reactions equivalent to two force components

   For a free roller support or connection, the rotational motion in one dimension is enabled by equipping with a roller with fictionless hinge or pin on rough surface or a rail, and the free rotational motion on other two dimensionsuonly two couple is allowed by the free end design. And therefore there is no couple reacted by the support or connection on the free body. Because there is a roller, the free translation is allowed along the rolling direction of the roller, and only two translational motion are stopped by the reaction force of the roller support on the rought surface or the rail.

   

7. Reactions equivalent to one force components

   1. For a ball support or connection

      For a ball support, simulart to the ball and socket joint, the free rotational motion in three dimensions is enabled by equipping with a free spherial end on a surface as a point contact or a free end on a frictionless surface, no couple is reacted by the support or connection on the free body. Besides, the two translational motions in horizontal or lateral and longitudinal direction also also free to motion. Only the translational motion in vertial or normal direction is stopped by the reaction force of the supporting  surface.

      

   2. For free short cable support or connection

      For a cable, the only possible reaction force is the tension of the cable. Therefore a free short cable provides only one constraint along the cable to the free body.