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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

f=Ku\mathbf{f} = \mathbf{Ku}

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:

α=eTu\boldsymbol{\alpha} = \mathbf{e}^{T}\mathbf{u}

Where the e\mathbf{e} matrix has a value of 1 for the degree of freedom that is constrained. We can then form an augmented system equation

{fα}=[KeeT0]{uλ}\begin{Bmatrix} \mathbf{f} \\ \boldsymbol{\alpha} \\ \end{Bmatrix} = \begin{bmatrix} \mathbf{K} & \mathbf{e} \\ \mathbf{e}^{\mathbf{T}} & \mathbf{0} \\ \end{bmatrix}\begin{Bmatrix} \mathbf{u} \\ \boldsymbol{\lambda} \\ \end{Bmatrix}

Where λ\boldsymbol\lambda are the Lagrange multipliers used to enforce a constraint condition and α\boldsymbol\alpha are the applied displacements. For a structural problem the Lagrange multipliers are the forces that need to be applied to the system to endure that the displacement condition is met.

Expanding the matrix equation gives

u=K1(feλ)α\mathbf{u} = \mathbf{K}^{- 1}\left( \mathbf{f} - \mathbf{e}\boldsymbol{\lambda} \right)\boldsymbol{\alpha}
α=eT(K1(feλ))\boldsymbol{\alpha} = \mathbf{e}^{T}\left( \mathbf{K}^{- 1}\left( \mathbf{f} - \mathbf{e}\boldsymbol{\lambda} \right) \right)


(eTK1fα)=(eTK1e)λ\left( \mathbf{e}^{T}\mathbf{K}^{- 1}\mathbf{f} - \boldsymbol{\alpha} \right) = \left( \mathbf{e}^{T}\mathbf{K}^{- 1}\mathbf{e} \right)\boldsymbol{\lambda}

Solving this equation gives the Lagrange multipliers, which can then be used in

u=K1(feλ)\mathbf{u} = \mathbf{K}^{- 1}\left( \mathbf{f} - \mathbf{e}\boldsymbol{\lambda} \right)

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.

feλ\mathbf{f} - \mathbf{e}\boldsymbol{\lambda}