# Applied Displacements And Lagrange Multipliers
Applied displacements are where we add a constraint to the model such that the displacement of certain nodes are fixed in given directions. We apply this displacement constraint by use of Lagrange multipliers.
The basic equations for a linear static analysis are
The applied displacements are applied using Lagrange multipliers. The basic concept is that the structure matrix can be augmented to enforce a displacement condition. The applied displacement can be related to the displacement vector through:
Expanding the matrix equation gives
Solving this equation gives the Lagrange multipliers, which can then be used in
to solve for the displacements.
In these situations, a number of forces have to be added to the system to ensure the specified displacement conditions are met. The extra forces that need to be applied to the system are given by the Lagrange multipliers and are the extra terms in the augmented load vector.