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Free-Body Diagrams to Three-Dimensional Force Systems
The first step in solving three-dimensional equilibrium problems,
as in the case of two-dimensions, is to draw a free-body diagram.
Before we can do this, however, it is first necessary to discuss the
types of reactions that can occur at the supports.
Support Reactions. The reactive forces and couple moments
acting at various types of supports and connections, when the
members are viewed in three dimensions, are listed in Table 1–2. It is
important to recognize the symbols used to represent each of these
supports and to understand clearly how the forces and couple
moments are developed. As in the two-dimensional case:
• A force is developed by a support that restricts the translation
of its attached member;
• A couple moment is developed when rotation of the attached
member is prevented.
For example, in Table 1–2, item (4), the ball-and-socket joint
prevents any translation of the connecting member; therefore, a force
must act on the member at the point of connection. This force has
three components having unknown magnitudes, Fx, Fy, Fz. Provided
these components are known, one can obtain the magnitude of force,
2
F = F + F + F , and the force’s orientation defined by its
2
2
z
y
x
coordinate direction angles α, β, γ, Eqs. 1-6. Since the connecting
member is allowed to rotate freely about any axis, no couple moment
is resisted by a ball-and-socket joint. It should be noted that the single
bearing supports in items (5) and (7), the single pin (8), and the single
hinge (9) are shown to resist both force and couple-moment
components. If, however, these supports are used in conjunction with
other bearings, pins, or hinges to hold a rigid body in equilibrium and
the supports are properly aligned when connected to the body, then
the force reactions at these supports alone are adequate for supporting
the body. In other words, the couple moments become redundant and
are not shown on the free-body diagram. The reason for this should
become clear after studying the examples which follow.

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