# Euler-Bernoulli beam element versus continuum beam element

I am using OpenSees to model a simply supported beam with a point load in the middle. The model is in consistent units. The beam is made up of bilinear quad elements. I have used 30 elements along the length of the beam and 1 element along the height of the beam.

The material is isotropic with the following properties:

$E = 80000$

$\nu = 0.0$

The loading $P = -10$. The model is 2 dimensions and 2 DOF's per node, although the software allows for input of element thickness in the 3rd dimension. The elements are $1$ unit by $1$ unit and the image does not reflect this (not to scale).

From basic structural mechanics the deflection at the base middle of the beam is given by

$$u_{max} = \frac {PL^3}{48EI}$$

which in this case equals $-0.84$ units.

Running a static analysis on the numerical model however yields a result of $-0.56$ units. This is very different.

Why does the beam modelled by continuum elements not reflect the true displacement for such a simple problem? What could be going wrong here?

• Are you refering to 2D or 3D elements as continuum elements? Jun 10 '17 at 15:53
• The domain is 2D with 2DOF per node. But the software allows the user to input an element thickness even for 2D (I'm not sure why) see opensees.berkeley.edu/wiki/index.php/Quad_Element Jun 10 '17 at 16:05

## 1 Answer

The standard, displacement formulation quadrilateral is notoriously bad at representing bending behavior, especially with only one element through the thickness of the beam. This is often referred to as "shear locking", so named because when the element shape functions attempt to represent pure bending of the beam, they also produce large, non-physical in-plane shear stresses.

Most production structural analysis FE codes have special 4-node quadrilateral elements that have better behavior in this case. Alternatively, the standard 8-node and 9-node quadrilateral elements and the 6-node triangle element have good performance for bending problems like this one.

If you are interested in the reason for this problem and ways to develop better quadrilateral elements for bending problems, take a look at these notes, Felippa FEM Notes.

If you are really interested in more details, a history and description of different element designs to deal with shear locking is contained in this book by MacNeal, Finite Elements: Their Design and Performance.

• Interesting. I tried constructing another geometry made of Tri31 (opensees.berkeley.edu/wiki/index.php/Tri31_Element) elements but these did not perform very well either. Granted I only put 8 elements for a total beam length of 4 units and height of 1 unit. However I still feel like for the right element type this should be enough. In the following list of provided elements (opensees.berkeley.edu/wiki/index.php/Element_Command) which continuum 2D element type would you recommend? Please note I am only interested in continuum type elements and not 1D beam type elements. Jun 10 '17 at 20:57
• The 3-node triangle (aka constant stress triangle) is rarely used for stress analysis because its performance is so poor. Sometimes mesh generators use a few of these elements to handle difficult transitions. The documentation on the OpenSees page you refer to is very limited. You might try the "enhanced strain quad" to see if that helps. Your expectation for the required mesh density for this problem is not unreasonable. Which is exactly why NASTRAN, Abaqus, ANSYS, etc include better-behaving 4-node quads. Jun 10 '17 at 21:28
• enhancedQuad is a bit better, provided enough elements are used to discretize the beam. I've found for maybe 10 elements or less the results become nonsensical again.. Jun 11 '17 at 8:26
• I took a closer look at the OpenSees docs. I see that one of the goals of the SSPQuad element is to eliminate shear locking (opensees.berkeley.edu/wiki/index.php/SSPquad_Element) by using reduced integration. They specifically mention beam bending problems. So that element is probably worth a try. Jun 11 '17 at 11:10