What Does It Mean for a Structure to Be “in Equilibrium”?

A structure is in equilibrium when the resultant force and resultant moment acting on it are both zero. Therefore, it has no linear or rotational acceleration and remains stationary under the applied loads.

Conditions for Structural Equilibrium

For a two-dimensional structural system, three conditions must be satisfied:

$$\sum F_x = 0$$ $$\sum F_y = 0$$ $$\sum M = 0$$

Here, $F_x$ and $F_y$ are the horizontal and vertical forces, while $M$ represents the moment, or turning effect of a force about a point.

Applied loads are balanced by support reactions. These reactions transfer forces from the structure into its supports and foundations, creating a continuous load path.

Structural equilibrium concerns the whole structure, while internal stresses still develop within its members. Depending on the loading and geometry, members may experience tension, compression, bending, torsion, or shear. Equilibrium alone does not guarantee that a structure is safe: material strength and deformation must also be considered.

Worked Example

A simply supported beam is $4\,\text{m}$ long and carries a central downward load of $1000\,\text{N}$. Let the upward reactions be $R_A$ and $R_B$.

Taking moments about support A:

$$R_B(4) - 1000(2) = 0$$

Therefore:

$$R_B = 500\,\text{N}$$

Using vertical force equilibrium:

$$R_A + R_B - 1000 = 0$$

So $R_A = 500\,\text{N}$. The upward reactions total $1000\,\text{N}$, balancing the load, and the moments also balance. Because the load is central and the beam and supports are symmetric, the equal reactions are also physically reasonable.

A common misconception is that equilibrium means no forces act on the structure. In fact, several forces may act, but their resultant force and moment must equal zero.

Exam Technique

For A3.2 HL questions, draw a clear free-body diagram, label loads, distances, and reactions, then show all three equilibrium conditions where relevant. Include units and state whether moments are clockwise or anticlockwise. If the load is off-centre, do not assume equal reactions; calculate each reaction by taking moments about one support.