Mr.Dongyu Liu,

I want to know some more details about you designed a new 2d finite element..the 2-d finite element usually 3- node or 6-node triangle, quadrilateral or rectangular element is used to simulate 2-d problem. in this process have you developed any new kind of element if it is so then what is the shape and minimum number of node need to define the element. What do you mean locking ? is it like not satisfying the equilibrium condition??..

Hi Dongyu Liu

I suppose that you are asking about the "shear locking" phenomenon possibly exists in beam and frame structures.

To clarify the phenomenon and to identify whether the newly-developed 2D FEM element suffers from "shear locking" or not, we should consider Euler-Bernoulli beams and Timoshenko beams as follows:

1. Euler-Bernoulli beams (EB):

- Nodal variables: deflection and slope

- Main assumption: the plane normal to the neutral axis before deformation remains normal to the neutral axis after deformation. Thus, the slope will be the first derivative of deflection in terms longitudinal coordinate.

- Method to obtain finite element matrix equation: Galerkin's method

- Shape functions: Hermitian shape functions, which are based on a general cubic polynomial.

The exact solution for the EB deflection is also a cubic function. So, the Hermitian shape functions can result in the exact solution.

2. Timoshenko beam (Timo):

- Nodal variables: deflection and slope (however, in comparison with EB, the slope is no longer be the first derivative of deflection due to the exist of transverse shear deformation).

- Main assumption: the plane normal to the beam axis before deformation does not remain normal to the beam axis after deformation any longer.

- Method to obtain finite element matrix equation: energy method

- Shape function: any shape function can be used to interpolate the nodal variables as long as the inter-element continuity of both the deflection and slope is satisfied (Langrange interpolation functions).

The element stiffness matrix of Timo has 02 components: 01 bending stiffness term and 01 shear stiffness term. The bending stiffness term is obtained using the exact integration (full integration) from the bending straining energy. The shear stiffness term is obtained using the reduced integration technique.

Talking about the exact (full) integration or the reduced integration is talking about number of integration point in Gauss-Legendre quadrature for isoparametric elements. These integration point quantity can be determined based on the order of polynomials or order of derivative in the formulation of element stiffness matrix.

For element bending stiffness, the more the integration point, the higher the accuracy but the computational cost. Therefore, we should use the minimum integration points for the bending stiffness, this number can be determined when the increase of integration point lead to the convergence of the bending stiffness matrix.

For element shear stiffness, there is a limit for the number of integration point.

No. of shear integration point = no. of bending integration point (minimum) - 1

"Shear locking" would happen when you use the full integration for the shear stiffness term. (refer to Fig.1.shear_locking_EB_Timo). Therefore the "shear locking" phenomena is a human-related phenomena, in other words, it depends on the numerical integration technique you choose to compute the element stiffness matrix, it is not an inherent characteristic of the newly-developed 2D FEM element.

From the attached figure we see that, the "shear locking" phenomenon extremely stiffen the beam, in other words, the shear stiffness is over-estimated.

In physical sense, as the beam become thinner, the bending strain energy is more significant than the shear energy (shear stiffness is negligibly small). However, when calculate the Timo beam using exact (full) integration for the shear stiffness matrix, we will see the reverse phenomenon (shear stiffness is irrationally large/over-estimated).

Finally, we can conclude that:

i) The existence of "shear locking" phenomenon depend on you selection of number of integration points for the element shear stiffness matrix. You should use the reduced integration technique to when calculating the shear stiffness matrix to prevent shear locking which occurs with exact (full) integration when the ratio of beam length to beam thickness is large.

ii) And the phenomenon only exist in Timoshenko-beam element type (transverse shear deformation is considered).

Hope it helps.

Kind regards,

MC Trinh

If you have developed your own element, simply check it through benchmarks (you may find them online for shear locking,membrane locking and volumetric locking) .

Elements those are prone to locking usually distinguish themselves in extreme conditions (ie. thickness is approaching to zero,one dimension is getting small relatively to the others) by gaining artificial strength (which wasn't supposed to).

If you have ways to calculate strain energy , you can simply calculate artificial strain energy and compare it with strain energy stored in your system in order to check locking phenomena mentioned above in your system, All of them can be identified via artificial strain energy.

Regards, İsmail.

Further Notes:

If your elements are reduced in one or more dimensions (ie shells and beams) you need to check both membrane locking which occurs due to coupling of in-plane deformations and out-of plane deformations,Or Shear Locking which occurs because of coupling between out-of plane deformations. Both arose from integration took place and incompatibility of shape functions' degrees and Gauss points used to evaluate them.

For continuum elements the locking usually appears due to the fact that shape functions being unable to exhibit kinematics which you are trying to impose to the system (ie bending of 3D linear continuum element exhibits locking due to artificial rotations detected at Gauss points) For volumetric locking the main reason that you are trying to use incompressible or nearly incompressible materials while not providing a constraint for element itself to satisfy incompressibility causes abrupt increases in strain energy due to calculation of bulk modulus which yields to infitinity.