Hello,
I would like to compute de the stress tensor of a Timoshenko beam at its Gauss points, to be able to implement an elastoplastic law in my finite element calculations.
Firstly,I know the displacement field at any point of my beam thanks to the relation u(x) = N(x) U, where U is the matrix of degrees of freedom at the nodes of my beam tU = (ux1, uy1 , uz1, θx1, θy1, θz1, ux2, uy2, uz2, θx2, θy2, θz2)
Then, I took as an expression of N the form given in this article https://www.researchgate.net/publication/236659875_Shape_functions_of_three-dimensional_Timoshenko_beam_element#fullTextFileContent , which corresponds to a Timoshenko model.
I deduce the deformations for small strains with ε = 1/2 (grad(u) +tgrad(u)), I obtained the equation shown in the picture.
I then apply Hooke's law to find the stress.
I then obtain that for a traction test (ux2 = constant, the other components of U are zero), the displacement field and the strain tensor are constant on my beam in particular along a cross-section, with only εxx non-zero, on the other hand the stress tensor has non-zero components other than σxx.
I conclude that my model shows that the cross sections are non-deformable, with therefore additional "virtual" forces, which prevent the beam subjected to traction along x, from being refined along y and z in accordance with the Poisson effect . On the other hand, I would like to have a "natural" behavior where the beam is refined according to y and z.
Do you have any articles for this?
Thanks a lot
Hi Arthur Boldron
I understand that you want to compute the stress tensor of a Timoshenko beam at its Gauss points, and implement an elastoplastic law in your finite element calculations. I searched the web for some articles that might be relevant to your problem, and here are some of them:
- Elastoplastic Timoshenko beams by M. Brokate and J. Sprekels. This article derives a Timoshenko type elastoplastic beam equation by dimensional reduction from a general 3D system with von Mises plasticity law. It consists of two second-order hyperbolic equations with an anisotropic vectorial Prandtl–Ishlinskii hysteresis operator.
- Accurate Timoshenko Beam Elements For Linear Elastostatics and LPB Problems by A. Ibrahimbegovic and E. Wilson. This article presents a new formulation of Timoshenko beam elements that can accurately capture the linear elastostatic and large plastic bending behavior of beams. The formulation is based on the principle of virtual work and the use of consistent tangent stiffness matrices.
- A viscoelastic Timoshenko Beam Model: Regularity and Numerical Approximation by A. Mielke, R. Rossi, and G. Savaré. This article studies a viscoelastic Timoshenko beam model that incorporates both shear and rotational inertia effects. The article proves the existence and uniqueness of solutions, as well as their regularity properties. It also proposes a numerical scheme based on finite differences and finite elements, and analyzes its convergence.
https://core.ac.uk/display/289298451
Article Accurate Timoshenko Beam Elements For Linear Elastostatics a...
Article A viscoelastic Timoshenko Beam Model: Regularity and Numeric...
https://civil-terje.sites.olt.ubc.ca/files/2020/01/Timoshenko-Beams.pdf
Technical Report Finite Element Analysis of a 4-Bar Mechanism with Incorporat...
- Badges
- Science topic