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corotational formulation for 3D beam


I am trying to implement the corotational formulation for three-dimensional beam elements given by Crisfield (Non-linear Finite Element Analysis of Solids and Structures, vol. 2, 1997). At a certain point the linear stiffness matrix K, linking the local nodal generalized forces q_il and the local displacements p_l, is needed. I suppose it contains terms such as 4EIy/L0, 2EIy/L0 (the same for Iz), EA/L0, and GJ/L0. However, I don't manage to write it down. What I don't understand is why 7 local degrees of freedom are considered (3 rotations for node 1, 3 for node 2, axial displacement) and not only 6, since torsion should, in my mind, be linear along the beam element and only require one associated degree of freedom (as for axial deformation).

Could anyone help me ?

Thank you,


Hi frederic

I have worked with MAC's element. there are 12 d.o.f in other words non linear torsion is not considered

hi Frederic

the local d.o.f are given on page 214  for one node  

eq  17.1  a for the three translations

eq  17.1 b  for the three  rotations

for the element the number of freedoms are then 2*(3=3)=12

in his paper, computer methods in applied mechanics and engineering 81(1990) pg 131-150

MAC gives the elastic stiffnesss matrix for the rotational freedoms in eq.31

and then augments it later with the translation freedoms to obtain the standard tangent stiffness matrix in eq. 60



Hello sir,

          This is Abhishk here.I am a PG student doing M-tech in Structural Enginnering.

I am doing project on 3d vibration analysis of orthotropic plate,please send me tutorial for modelling for my topic.


hi buddy

send me more details i ll help you out IN SIMULATION if you want to do it it in IDEAS NX or ABAQUS. 

Thank you ikalo,

I've obtained the article by Crisfield (1990). The implementation in MATLAB gives good results for the the same two first test cases as MAC.

Hi fredo

I am glad it is working o.k.

I think the last example in the paper needs an arc length algorithm to obtain post buckling equilibrium path.



I wonder if my implementation of 3D corotational beam elements works fine. When testing it on the cantilever beam subject to a moment at its free end, I find the error surprisingly high... (see below)


Crisfield, in his paper (1990), seemingly had no problem with this, even when using only 5 elements.


Could someone tell me if something goes wrong with my results ?


Thank you,




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