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A structure that is initially at rest and remains at rest when subjected to a system of forces and couples is said to be in a state of static equilibrium. If a structure is in equilibrium, then all its members and parts are also in equilibrium. Thus, a structure or one of its members in equilibrium should maintain a balance of force and moment.
Practically all structures are arranged in a three-dimensional configuration of structural elements. For a space structure subjected to three-dimensional systems of forces and couples, the conditions of zero resultant force and zero resultant couple can be expressed as
∑Fx=0 ∑Fy=0 ∑Fz=0
∑Mx=0 ∑My=0 ∑Mz=0 .....(1.1)
These six equations are called equations of equilibrium of space structures. These are the necessary and sufficient conditions for the equilibrium of a space structure.
Since the principal load-carrying elements of most structures are substantially in a two-dimensional stress condition, the structural analyst may isolate and treat such elements as two-dimensional structures,
so-called planar structures; structures that may be considered to lie in a plane that also contains the lines of action of all the forces acting on the structure. Thus, for a planar structure lying in the xy plane and subjected to a coplanar system of forces and couples, the above requirements for equilibrium reduce to
∑Fx=0 ∑Fy=0 ∑Fz=0 .....(1.2)
These three equations are referred to as the equations of equilibrium of planar structures. Here, the first two equations of equilibrium, EF = 0, Fy = 0, represent, respectively, the algebraic sum of x and y components of all the forces are zero, thereby indicating that the resultant force acting on the structure is zero. The third equation, EM, 0, represents the algebraic sum of moments of these force components about an axis perpendicular to the XY plane (z-axis) and the moments of any couples acting on the structure is zero, thereby indicating that the resultant couple acting on the structure is zero.
Notes 1. When a space structure is acted upon by a concurrent system of loads and reactions, it is impossible for the resultant:
Effect of the system to be a couple, for the line of action of all the forces of a concurrent system, intersect in a common point. Therefore the equations of equilibrium to be satisfied are
∑Fx=0 ∑Fy=0 ∑Fz=0 .....(1.3)
2.similarly for a planar structure subjected to concurrent coplanar force system, the above requirements for equilibrium reduce to
∑Fx=0 ∑Fy=0 ........(1.4)
