Timoshrnko beam mass and stiffness matrices?

[BaharSh]

Can someone please guide me to a reference where I can find finite element matrices (stiffness matrix and consistent mass matrix) for a Timoshenko beam?

 

[corus]

Try books by Timoshenko

corus

 

[Dinosaur]

BaharSh,

Forgive my ignorance, but . . .

Except that a Timeshenko beam clearly has something to do with the famous author, what makes a Timeshenko beam different from any other sort of ideal rod?

Ed

 

[BaharSh]

Dinosaur,

A Timoshenko beam model accounts for shear deformation. Therefore it is more suitable for thick beams than the Euler-Bernoulli model where deforamtion is only due to axial strain.


Bahar

 

[GBor]

Algor's beam inputs include an allowance for shear areas along the strong and weak axes. This is an accounting for a Timoshenko beam. Their documentation should have some information, but it will be specific to their software and will probably not include specifics about the stiffness matrix.

Spyrakos put out a book called, "Finite Element Modeling" in the mid-90's, I think. The FEA information is out of date and he never discusses the specific modifications to the stiffness matrix based on the shear areas, but he does address the impact of ignoring the shear areas and discusses the shear areas in terms of the impact to shear deformation.

 

[cloche]

ANSYS element type BEAM186 is a timoshenko beam. I looked in the help but I didn't find out the matrix of this beam. Anyway, here are some references:

S. Timoshenko. Strength of Material, Part I, Elementary Theory and Problems. 3rd Edition. D. Van Nostrand Co., Inc.. New York, NY. 1955.

S. Timoshenko. Strength of Material, Part II, Elementary Theory and Problems. 3rd Edition. D. Van Nostrand Co., Inc.. New York, NY. 1956.

S. Timoshenko and J. N. Goodier. Theory of Elasticity. 3rd Edition. McGraw-Hill Book Co. Inc.. New York, NY. 1970

S. Timoshenko and S. Woinowsky-Knieger. Theory of Plates and Shells. McGraw-Hill Book Co. Inc.. New York, NY. 1959.

 

[GBor]

Those are good references, but do not have FEA information since most of them were written before the concept of finite elements surfaced.

 

[cloche]

Sure, GBor.
I figure out that one must also look at a theoretic FEM book, where the process of building the element matrices is described. Then, it's just a "mixing" matter btw elasticity theory and FE theory... Not so easy, I fear...
As regards FE theory, I've got a very good book from Prof. B. Atzori, Univ. of Padua (Italy), but I realy fear it's only in Italian, and moreover not distributed on the market.

 

[Dinosaur]

BaharSh,

Thanks for clearing that up for me.

What you need is in a book called "Theory of Matrix Structural Analysis" by J.S.Przemieniecki, pp. 70-82.
My copy of the book is copyright 1968, 1985 and published by Dover. ISBN 0-486-64948-2.

Good Luck - Ed