Reaction

Background

Reaction are forces that are developed at the fixed (restrained) Node. Figure below shown schematically in blue arrows the external forces exserted on the model, while the red arrows represents the reactions forces.

The Reaction are computed once the solution has converged. Depending on the type of analysis the reaction are computed at

• StaticAnalysis::UpdateDomain()
  Compute the reaction forces considering internal and external forces. This process is performed independently in ech processor.
• DynamicAnalysis::UpdateDomain()
  Compute the reaction forces considering inertial, viscous, internal and external forces. This process is performed independently in ech processor as well.

In order to guarantee continuity in the solution, if parallel processor are used, the Reaction are collected using:

• Analysis::ReducedParallelReaction()
  This function adds the reaction vector across processor and distribute the solution.

Once the Analysis is done, and the state vectors (displacements, velocities, and accelerations) are known, the Reaction can be computed assembling all force contribution in the model. The process depend on the Analysis, thus:

• For StaticAnalysis

  \[ \mathcal{R} = \sum_\textrm{e = 1}^{\textrm{N}_\textrm{e}} {\mathbf{A}^\textrm{e}}^\top \mathbf{F}^\textrm{e}_{\textrm{int}} \mathbf{A}^\textrm{e} - \mathbf{F}_{\textrm{ext}}\,. \]

• For DynamicAnalysis

  \[ \mathcal{R} = \sum_\textrm{n = 1}^{\textrm{N}_\textrm{n}} {\mathbf{A}^\textrm{n}}^\top \textbf{M}^\textrm{n} \ddot{\textrm{U}}^\textrm{n} \mathbf{A}^\textrm{n} + \sum_\textrm{e = 1}^{\textrm{N}_\textrm{e}} {\mathbf{A}^\textrm{e}}^\top \left( \textbf{M}^\textrm{e} \ddot{\textrm{U}}^\textrm{e} + \textbf{C}^\textrm{e} \dot{\textrm{U}}^\textrm{e} + \mathbf{F}^\textrm{e}_{\textrm{int}} \right) \mathbf{A}^\textrm{e} - \mathbf{F}_{\textrm{ext}}\,, \]

where \(\mathbf{A}^\textrm{n} : \mathbb{R}^{\textrm{N}_\textrm{dof}^\textrm{n}} \to \mathbb{R}^{\textrm{N}_{\textrm{total}}}\) is the nodal assembly operator, \(\mathbf{A}^\textrm{e} : \mathbb{R}^{\textrm{N}_\textrm{dof}^\textrm{e}} \to \mathbb{R}^{\textrm{N}_{\textrm{total}}}\) is the element assembly operator, \(\mathbf{M}^\textrm{e}, \mathbf{C}^\textrm{e}, \mathbf{K}^\textrm{e}\) are the Element mass, damping, and stiffness respectively, \(\mathbf{M}^\textrm{n}\) is the point masses applied to the Node, \(\mathbf{F}_{\textrm{ext}}\) the vector of external forces, \(\mathbf{F}^\textrm{e}_{\textrm{int}}\) the Element internal forces, and \(\textrm{N}_\textrm{n}\) are the number of Node in the model, and \(\textrm{N}_\textrm{e}\) the number of Element in the model.

Attention

The vector \(\mathcal{R}\) only has non-zero values where there is fixed support.

By construction, this gives the total reaction forces developed at the supports.

REFERENCE:

• Bathe K. Jurgen, "Finite Element Procedures", Chapter 4: pages 187-188, Prentice-Hall, 1996.