The Timoshenko beam element includes the effects of axial, bending, and shear deformation. It has six degrees of freedom at each node. The element can have these actions released at either end so long as an under-restrained system is not produced.

References

1. Fang, W. "EN234: Three-dimensional Timoshenko beam element undergoing axial, torsional and bending deformations." December 16, 2015.
2. Yilmaz, Murat. “A Practical Approach to Implement Releases and Partial Fixities in Finite Elements Using Already Existing Stiffness Equations.” Dicle University Journal of Engineering, 13:3 (2022). Pages 571-578.

Element Geometry

The Timoshenko beam has two nodes and is formulated in a local x, y & z coordinate system.

Degrees of Freedom

The Timoshenko beam contains six degrees-of-freedom (DoFs) at each node:

1. x-translation
2. y-translation
3. z-translation
4. rotation about x
5. rotation about y
6. rotation about z

There are six nodes defined for this formulation. These nodes are numbered from 1 to 6 in a counter-clockwise direction, bottom surface first, then top surface. This element formulation does not contain rotational degrees of freedom.

The local stiffness matrix is formulated with the following degree-of-freedom order:

$\begin{bmatrix}u_1 & v_1 & w_1 & \theta_{1,x} & \theta_{1,y} & \theta_{1,z} & u_2 & v_2 & w_2 & \theta_{2,x} & \theta_{2,y} & \theta_{2,z} \end{bmatrix}$

Where:

$u_k=$ x-translation at node $k$