Shear Locking in Finite Element Method

In finite element analysis, use of lower order elements is widely accepted to reduce the model size and computational time. Lower order elements like Q4 and H8 have linear interpolation functions which are numerically integrated using Gauss integration formulas.

Due to linear interpolation functions and numerical integration, the displacements computed in the element are lower than the actual values. In bending dominated problems large shear stresses are produced when specific combination of element geometry and full numerical integration is used in lower order elements. Performance of lower order elements is affected by aspect ratio where L>>h causes increased stiffness in bending mode. This non-physical high stiffness (due to numerical integration) in the element requires more energy to produce deformations. This phenomenon is termed as Shear locking.

As an example, we consider a cantilever beam of length 254 mm and height 20 mm in SCIFESOL.

Cantilever Beam

Using analytical solution procedures from theory of elasticity we get the displacement and stresses.

Analytical Results

The beam is meshed with two layers of Q4 elements having aspect ratio 3.62

Mesh

The beam is having a load of 10 N at the tip.

Boundary Conditions

Using pure displacement formulation, the Q4 elements predict results which are in ~50% error.

Displacement in Standard Q4

Large shear stresses having magnitude of ~10MPa are reported which confirms shear locking in standard Q4 elements having bad aspect ratios.

Shear Stress in Standard Q4

Now to prevent shear locking in standard Q4 elements we can use Enhanced Assumed Strain formulation which is available in all commercial software’s.

In SCIFESOL we can activate EAS method using element properties dialog box.

Element Properties Dialog box

After re-running the model with EAS mode activated, we observe that the shear stress and displacement results are improved and compare well with the analytical results. In both analysis cases we have used full integration rule (2x2) for Q4 elements.