Boundary Conditions, Displacements, and Force Recovery

After the global system has been assembled, boundary conditions are imposed, unknown nodal displacements are solved, and support reactions and member forces are recovered.

Free and constrained degrees of freedom

Partition the displacement vector into unknown free coordinates $\{d_f\}$ and prescribed coordinates $\{d_c\}$. For a fixed support component, the corresponding $d_c$ is zero; a prescribed support movement may give a nonzero value.

Partitioned equations

After reordering coordinates,

$$\begin{bmatrix}K_{ff}&K_{fc}\\K_{cf}&K_{cc}\end{bmatrix}\begin{Bmatrix}d_f\\d_c\end{Bmatrix}=\begin{Bmatrix}F_f\\F_c\end{Bmatrix}.$$

The free-coordinate equations give

$$K_{ff}d_f=F_f-K_{fc}d_c.$$

Support reactions

Once all displacements are known, reactions at constrained coordinates are recovered from the complete equilibrium equation. Unknown reactions should not be treated as prescribed external loads before the displacement solution.

Element displacements

For each member, its components are extracted from the global displacement vector to form $\{d_e\}$ and transformed to local coordinates:

$$\{u_e\}=[T]\{d_e\}.$$

Member-end forces

Equivalent nodal actions and initial strains must be included consistently with the adopted element formulation. For the simplest unloaded element,

$$\{q_e\}=[k_e]\{u_e\}.$$

The components of $\{q_e\}$ provide local axial forces, shears, and end moments.

Verification

Check global equilibrium of applied loads and reactions, joint equilibrium, satisfaction of prescribed displacements, matrix symmetry where applicable, and consistency of local member-force sign conventions.