Equations of Equilibrium

When a body is in static equilibrium, no translation or rotation occurs in any direction (neglecting cases of constant velocity). Since there is no translation, the sum of the forces acting on the body must be zero. Since there is no rotation, the sum of the moments about any point must be zero.
In a two-dimensional space, these conditions can be written:

where Fx and Fy are the sum of the components of the forces in the direction of the perpendicular axes x and y, respectively, and M is the sum of the moments of all forces about any point in the plane of the forces.
Figure 3.7a shows a truss that is in equilibrium under a 20-kip (20,000-lb) load. By Eq. (3.11), the sum of the reactions, or forces RL and RR, needed to support the truss, is 20 kips. (The process of determining these reactions is presented in Art. 3.29.) The sum of the moments of all external forces about any point is zero. For instance, the moment of the forces about the right support reaction RR is

(Since only vertical forces are involved, the equilibrium equation for horizontal forces does not apply.)
A free-body diagram of a portion of the truss to the left of section AA is shown in Fig.
3.7b). The internal forces in the truss members cut by the section must balance the external force and reaction on that part of the truss; i.e., all forces acting on the free body must satisfy the three equations of equilibrium [Eq. (3.11)].
For three-dimensional structures, the equations of equilibrium may be written

The three force equations [Eqs. (3.12a)] state that for a body in equilibrium there is no
resultant force producing a translation in any of the three principal directions. The three
moment equations [Eqs. (3.12b)] state that for a body in equilibrium there is no resultant moment producing rotation about any axes parallel to any of the three coordinate axes. Furthermore, in statics, a structure is usually considered rigid or nondeformable, since the forces acting on it cause very small deformations. It is assumed that no appreciable changes in dimensions occur because of applied loading. For some structures, however, such changes in dimensions may not be negligible. In these cases, the equations of equilibrium should be defined according to the deformed geometry of the structure (Art. 3.46).


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