# Shear Locking

## 22 Jun 2021 Shear Locking

When using solid elements in FEA (https://www.fea-solutions.co.uk/solid-elements/), they can be found to be unrealistically stiff when evaluating bodies which are subject to bending loads. This results in unrealistic low deflections, and consequently in unrealistic low stresses.

This happens because in Continuum Mechanics, which is the underlying mathematical model used for solid elements, only translational Degrees Of Freedom (DOF) but no rotational DOF are defined at the nodes.

Ideally, a block of material will become curved when it is put under a pure bending load. However, a single first order element (https://www.fea-solutions.co.uk/element-order/) cannot have a curved shape when subject to bending. Its lower surface will experience a compressive stress, whilst the top surface will have tensile stresses. This will introduce an artificial shear stress, which means the strain energy of the element is generating shear, rather than bending, deformation. The element will then become overly stiff or ‘locked’ under this bending load.

There are ways in which shear locking can be prevented:
– Increasing the number of elements across the wall thickness.
– Using hexagonal elements (or quadrilateral elements in 2D) instead of tetrahedral elements (or triangular elements in 2D).
– Using second order elements.

Using more than one element across the thickness of a body subjected to bending allows the curvature to be modelled more realistically.

Hexagonal elements (or quadrilateral elements in 2D) provide much more realistic results than tetrahedral elements (or triangular elements in 2D) due to the superior element definition they are based on.

Likewise, second order elements have more nodes in a single element than first order elements. This allows them to deform into concave or convex shapes more realistically if they are subjected to bending.

The above methods should be used together to most effectively eliminate any likelihood of shear locking occurring in the FEA.

Poor tetrahedral meshes might result in stresses of just 30% or lower of the realistic value.

Shear locking is an issue relevant only to solid elements. Surface elements (https://www.fea-solutions.co.uk/surface-elements/) and line elements (https://www.fea-solutions.co.uk/line-elements/) are not affected by this issue, because their mathematical models support rotational DOFs.

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