Math: Three Dimensional Force System
SUPPORT FOR RIGID BODIES SUBJECTED TO THREE DIMENSIONAL FORCE SYSTEM
The first step in solving three-dimensional equilibrium problems, as in the case of two dimensions, is to draw a free-body diagram of the body (or group of bodies considered as a system)..
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 5–2. 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 by each support. As in the two-dimensional case, a force is developed by a support that restricts the translation of the attached member, whereas a couple moment is developed when rotation of the attached member is prevented. For example, in Table 5–2, 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, FxFyFz Provided these components are known, one can obtain the magnitude of force. and the force’s orientation defined by the coordinate direction angles Eqs. 2–7.* 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 support 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 may be 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
The first step in solving three-dimensional equilibrium problems, as in the case of two dimensions, is to draw a free-body diagram of the body (or group of bodies considered as a system)..
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 5–2. 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 by each support. As in the two-dimensional case, a force is developed by a support that restricts the translation of the attached member, whereas a couple moment is developed when rotation of the attached member is prevented. For example, in Table 5–2, 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, FxFyFz Provided these components are known, one can obtain the magnitude of force. and the force’s orientation defined by the coordinate direction angles Eqs. 2–7.* 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 support 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 may be 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