Engineering with Computers

, Volume 33, Issue 3, pp 647–667 | Cite as

A numerical study on the symmetrization of tangent stiffness matrix in non-linear analysis using corotational approach

  • W. A. Da Silva
  • W. T. M. Silva
  • F. Evangelista Junior
Original Article


In the corotational kinematics (EIRC), the movement is decomposed in deformational and rigid body components using projection operators. Large rigid body translations and rotations, and infinitesimal deformation tensors are adopted. In this way, a non-symmetric stiffness matrix is obtained for the finite element. Literature points that this matrix may be symmetrized in the usual way and the original non-symmetric matrix tends to be in a symmetric form as equilibrium is approached. This paper performed symmetrization studies using Frobenius norm and Absolute Maximum Coefficient of the anti-symmetric part of stiffness matrix in several numerical analyses with high non-linearity solved with an incremental-iterative scheme. An element independent corotational (EICR) kinematics is used to analyze non-linear spatial frames with 3D Euler–Bernoulli beam elements. The results show that this symmetrization not always occurs and depends significantly on the magnitude of displacements and finite-element mesh refinement. The authors demonstrated that, in some cases, the symmetrization would only take place with very refined meshes when the discretized structure approaches the continuum. In this way, the current literature and general mathematical proof of that symmetrization have to be restated.


Geometrically non-linear analysis Corotational approach Symmetrization of stiffness matrix 



The authors are thankful to the Graduate Program in Structures and Civil Construction (PECC) of the University of Brasília (UnB) for the technical support; and to the National Council for the Scientific and Technological Development (CNPq) for the financial support of this research.


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Copyright information

© Springer-Verlag London 2016

Authors and Affiliations

  • W. A. Da Silva
    • 1
  • W. T. M. Silva
    • 2
  • F. Evangelista Junior
    • 2
  1. 1.Departamento de Engenharia CivilUniversidade Federal de Goiás, UFGCatalãoBrazil
  2. 2.Departamento de Engenharia CivilUniversidade de Brasília, UnBBrasíliaBrazil

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