Statics is the branch of mechanics that analyzes bodies at rest or in equilibrium. The first condition of equilibrium states that the sum of all external forces acting on a body must equal zero, written as \(\Sigma F = 0\). The second condition requires that the sum of all moments about any chosen point also equals zero, \(\Sigma M = 0\). Together, these two conditions provide the three scalar equations (in two dimensions) or six (in three dimensions) needed to solve for unknown forces and reactions in a structure.
To apply these equations, engineers draw a free body diagram (FBD), which is a sketch of the body isolated from its surroundings with all external forces and moments clearly indicated. When the reactions of a structure can be found using the equations of equilibrium alone, the structure is called statically determinate. The type of support influences the number and direction of reaction forces available. A roller support provides only one reaction force normal to the surface on which it rests, while a pin support provides two reaction forces, both horizontal and vertical.
A key concept in statics is the moment of a force about a point, defined as \(M = F \times d\), where \(F\) is the force magnitude and \(d\) is the perpendicular distance from the point to the line of action of the force. Two equal, opposite, and non-collinear forces form a couple, which produces a pure rotational effect with no net translational force. The centroid of a body is the geometric center where the body would balance perfectly if supported there. The moment of inertia, expressed as \(I = \int y^2\,dA\), measures a cross-section's resistance to bending, and the parallel axis theorem \(I = I_c + A d^2\) allows the moment of inertia about any parallel axis to be found from the value about the centroidal axis.