Dear Sacheen,

Membrane locking

Membrane locking does only occur in curved beam and shell elements. The term describes a
stiffening effect that occurs if pure bending deformations (“inextensional bending”) are accompanied
by parasitic membrane stresses. It is sometimes confused with shear locking and
volumetric locking because these effect the membrane part of shell elements. However, they
are completely different phenomena.

As membrane locking is associated with the curvature of a structure it only occurs if the elements
are actually curved. For instance in the analysis of a cylindrical shell with four-node
elements there is no membrane locking when the mesh is aligned to the edges of the shell,
because the individual elements are flat. Linear triangles are always free from membrane
locking because they are always flat, regardless of the shape of the shell. Quadratic and biquadratic
elements usually show strong membrane locking in any situation.

Shear locking

Shear locking can occur in 2d and 3d solid elements as well as shell elements. The effect is
significant only if there is a certain (in-plane) “bending” deformation of the structure.

From a mathematical point of view, shear locking is not existent. Looking at the corresponding
differential equation, there is no ill-conditioning or “small parameter”. Actually, the critical
parameter in the case of shear locking is the aspect ratio of the element (i.e. no property of
the underlying mathematical problem itself). This can be understood most easily with the help
of an analogy to the Timoshenko beam element. The aspect ratio of a 2d solid element has the
same effect on the stiffness matrix as the length-to-thickness ratio in the beam element.

Further information

Martin J. Gross
www.matfem.de