If you consider a beam with thickness t under combined shear and bending deformations, the portion of the internal energy associated with the bending deformation is proportional with t^3, but the portion related to shear deformations is proportional with t. Now if we reduce the thickness t, the value of t^3 approaches zero much faster than t and as a result, all the strain energy of the beam will come from shear deformation. This is not correct because for a thin beam it is always the bending deformations which provide most part of the energy. So the shear deformation beam can lead to wrong results for very thin beam. This is called shear locking.
Membrane locking happens when you model a curved surface of with flat (facet) shell elements. In this case because the elements are not in the same plane, bending vector of one element will have a projection on the next element which acts like an in-plane moment or drilling moment. Again in-plane deformations energy is proportional to t but bending energy is proportional to t^3 and the same problem happens when t becomes very small, i.e. membrane resistance of the next element stops bending deformation on this element. This is called membrane locking.
Hello all, in fact i'm studing the transverse shear locking for 3D elements (3D solid-shell elements) by analysing the subspace of null transverse shear strains applied for 2D elements but ,till now, not for 3D elements (so far as i know), but it's very difficule to know the basic components for the null transverse shear deformations, someone can help me ?? thanks in advance.
Shear Locking vs. Membrane Locking
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
In reply to Shear Locking vs. Membrane Locking by Martin J. Gross
"Linear triangles are
"Linear triangles are always free from membrane locking because they are always flat, regardless of the shape of the
shell."
Is the above correct for finite deformations? Why?
Shear Locking vs. Membrane Locking
If you consider a beam with thickness t under combined shear and bending deformations, the portion of the internal energy associated with the bending deformation is proportional with t^3, but the portion related to shear deformations is proportional with t. Now if we reduce the thickness t, the value of t^3 approaches zero much faster than t and as a result, all the strain energy of the beam will come from shear deformation. This is not correct because for a thin beam it is always the bending deformations which provide most part of the energy. So the shear deformation beam can lead to wrong results for very thin beam. This is called shear locking.
Membrane locking happens when you model a curved surface of with flat (facet) shell elements. In this case because the elements are not in the same plane, bending vector of one element will have a projection on the next element which acts like an in-plane moment or drilling moment. Again in-plane deformations energy is proportional to t but bending energy is proportional to t^3 and the same problem happens when t becomes very small, i.e. membrane resistance of the next element stops bending deformation on this element. This is called membrane locking.
Peyman
subspace analysis for transverse shear locking in 3D elements.
Hello all, in fact i'm studing the transverse shear locking for 3D elements (3D solid-shell elements) by analysing the subspace of null transverse shear strains applied for 2D elements but ,till now, not for 3D elements (so far as i know), but it's very difficule to know the basic components for the null transverse shear deformations, someone can help me ?? thanks in advance.