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Question: Beam Element in ANSYS Theory Reference

L2020's picture

Hi
I have read some descriptions about beam element in chapter 14 of Element Library of ANSYS Theory Reference(ANSYS Help),it is mentioned that the stiffness matrix of a beam element in element coordinates is a 12x12 matrix that its elements are calculated in the same way as przemieniecki's book(Przemieniecki, J. S., Theory of Matrix Structural Analysis, McGraw-Hill, New York (1968))
Unfortunately I didn't find this book in the library,if any body has read this book,if it is possible guide me how przemieniecki have reached to such stiffness matrix?

Sincerely yours

 

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ramdas chennamsetti's picture

R. Chennamsetti, Scientist, R&D Engineers, India

In ANSYS, beam element actually represents a frame element. There is no separate frame element. 

They add one rotational (with respect to its own axis => torsion) and one axial translational DOF to a 3D beam element which has two translations (deflections) and two rotational (bending) DOF at each node. 

So, at each node - original beam has four DOF (two translations & two rotations). Add above mentioned two more DOF. This gives six DOF = three translations and three rotation. There are two nodes in each element => 12 DOF.

- Ramdas. 

Laleh: 

Following ODE are solved bi Przemieniecki to get the column "i" of the stiffnes matrix by applwing unit diplacements at "i" DOF and zero at all other than "i" DOF´s (Formal definition of the stiffness matrix)

Torsion or twist: T(x)=GJ(dΘ/dx)

Axial: P(x)=EA(du/dx)

Bending (two planes): M(x)=EI(d2v/dx2)

Shear: Q(x)=GAs(dv/dx)

Where T(x), P(x), M(x) and Q(x) are calculated from the end forces that appear when the unit "i" and zero to all other than "i" displacement are applied to the beam.

Working out the polynomials obtained by solving above ODE´s to eliminate the integration constants you should get the end forces that make up the "i" column of the stiffness matrix. 

Note that above equations are uncoupled meaning that proper local coordinate system is to be selected to make those equations applicable (Principal section axis and so on).

I am not sure if this clarify your question so, please feel free of asking for more clear of more complete information. Either case, I think that a Dover version (relatively unexpensive) could be found by entering in Amazon, Barnes & Noble and the like).

Regards, 

Carlos

L2020's picture

Your answer was clear and very helpful for me,now I understand the procedure that przemieniecki have done.

Thanks alot 

Laleh Fatahi

http://laleh.fatahi.googlepages.com

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