Shear Locking & Hourglass in the Finite Element Method

Compact reference explaining the causes, mathematics and practical fixes used in FEM for shear locking and hourglass modes.

Introduction

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.

Shear locking is a numerical pathology that occurs in first-order (linear) bending-dominated elements, especially:

  • 4-node bilinear quadrilateral (Q4)

  • 3-node triangular (T3)

  • 2-node beam elements with linear shape functions

When bending occurs, these elements artificially accumulate shear strain, making the element too stiff i.e the displacements computed in the element are lower than the actual values.

As a result:

  • Bending deflections are much smaller than the theoretical value.

  • Stress results in bending become incorrect.

Performance of lower order elements is also affected by aspect ratio of the elements where L>>h which causes increased stiffness in bending mode.

How to Fix or Avoid Shear Locking

 

✔ 1. Use Reduced Integration with Hourglass Control

Example: Q4 → Q4R

  • Relaxes shear constraints

  • Removes artificial stiffness

✔ 2. Use Higher-Order Elements

Example:

  • Q4 → Q8 or Q9

  • Beam2 → Beam3

Higher-order interpolation captures curvature better.

✔ 3. Use Selective Reduced Integration (SRI)

  • Only shear terms are reduced

  • Prevents hourglassing

✔ 4. Use Mixed Formulations

Example:

  • MITC (Mixed Interpolation of Tensorial Components) shell elements.

  • Assumed natural strain (ANS) elements.

These treat shear strain separately.

Hourglass Control in Reduced-Integration Elements

Hourglass control is a stabilization method used in reduced-integration finite elements such as 4-node quadrilaterals and 8-node hexahedra. Because these elements use fewer integration points, some strain modes are not captured, producing zero-energy deformation patterns known as hourglass modes.

These modes cause the mesh to distort without resistance, leading to numerical issues such as spurious oscillations, unrealistic deformations, and checkerboarding.

Hourglass control suppresses these unwanted modes by adding a small amount of artificial stiffness or damping. This preserves the advantages of reduced integration (better performance and less locking) while maintaining stability.

  • Prevents zero-energy distortions
  • Improves numerical stability
  • Maintains reduced integration efficiency
  • Used in explicit dynamics and solid elements

Hourglass Mode Diagram

Case Study

This excersise is performed in SCIFESOL scilab solver package

 

We consider a cantilever beam of length 254 mm and height 20 mm in SCIFESOL.

shear

Test quadrilateral element

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

Analytical Results

Analytical Results

 

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

Mesh

Mesh

 

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

Boundary Conditions

Boundary Conditions

 

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

Displacement in Standard Q4

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.

Displacement in Standard Q4

Shear Stress in Standard Q44

 

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

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.

Displacement in EAS Q4

Displacement in EAS Q4

Shear Stress in EAS Q4

Shear Stress in EAS Q4

 

Results Summary

Results Summary

Hourglass Control in CalculiX

The C3D8R in CalculiX is an 8-node, linear reduced-integration brick element with 1 integration point. It is efficient for large, 3D simulations, avoids volumetric/shear locking, but is prone to hourglassing (spurious zero-energy modes) and may require finer meshes to accurately capture bending and stress concentrations

Recommended Elements in CalculiX

Element Shear Locking Hourglass Risk Recommendation
C3D8 Severe shear locking No ❌ Avoid for bending
C3D8R No shear locking Yes (requires control) ✔ Good but watch hourglassing
C3D8I No shear locking No ✅ Best for bending
C3D20/C3D20R Almost none R-element may hourglass ✔ Excellent
S4 Some shear locking No ✔ Good
S4R No shear locking Hourglass possible ✔ Good but check hourglass energies

Practical Summary

Shear locking comes from an incompatible shear strain field (linear interpolation), and Reduced Integration fixes it but introduces hourglass modes which are then corrected by adding artificial hourglass stiffness using stabilization matrices.

 

Issue Cause Fix
Shear locking Overconstrained shear strain field Reduced integration, SRI, MITC
Hourglass modes Reduced integration removes constraints → zero-energy modes Hourglass control (Flanagan–Belytschko + stiffness or viscosity)
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