# 3.1: Equilibrium of Structures

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- 42951

Engineering structures must remain in equilibrium both externally and internally when subjected to a system of forces. The equilibrium requirements for structures in two and three dimensions are stated below.

3.1.1 Equilibrium in Two Dimensions

For a structure subjected to a system of forces and couples which are lying in the *xy* plane to remain at rest, it must satisfy the following three equilibrium conditions: \[\sum F_{x}=0 ; \sum F_{y}=0 ; \sum M_{z}=0\]

The above three conditions are commonly referred to as the equations of equilibrium for planar structures. \(\Sigma F_{X}\) and \(\Sigma F_{Y}\) are the summation of the \(x\) and \(y\) components of all the forces acting on the structure, and \(\Sigma M_{Z}\) is the summation of the couple moments and the moments of all the forces about an axis \(z\), perpendicular to the plane *xy* of the action of the forces.

3.1.2 Equilibrium in Three Dimensions

A structure in three dimensions, that is, in a space, must satisfy the following six requirements to remain in equilibrium when acted upon by external forces: \[\begin{array}{l}

\sum F_{x}=0 ; \sum F_{y}=0 ; \sum F_{z}=0 \\

\sum M_{x}=0 ; \sum M_{y}=0 ; \sum M_{z}=0

\end{array}\]