Four-Node “Locking-Free” Elements: Capturing Analytical Stresses in Pure Bending: For Two Orthogonal Directions

Abstract

We can exactly reproduce linear axial stresses without shear (To satisfy equilibrium point-wise, the (u, v) components of a typical displacement vector must be coupled through the Poisson’s ratio ν. ), in four-node (plane) elements. Neither their geometrical shapes—including concavity—nor the Poisson’s ratio ν pose any impediment. This locking-free condition can only be achieved when the neutral axis, under pure bending, is aligned along one of the two pre-selected directions.

Here, an element shape function, which consists of two quadratic polynomials in (x, y), is a linear combination of Rayleigh mode vectors that satisfy point-wise equilibrium. The unique coefficients for each shape function involve the element nodal coordinates and the Poisson’s ratio.

Following Clough’s guidelines, (Clough, Comput Struct 12:361–370, 1980) especially the Quotation III of the Introduction, the entries in the element stiffness matrix and the (equivalent) nodal forces (to substitute for a given spatial profile of the boundary traction) are determined as virtual work quantities (Note that for structural mechanics problems, the virtual work principle is absolutely equivalent to the Ritz variational formulation (Ritz, J Reine Angew Math 135:1–61, 1908; Wendroff, Math Comput 19(90):218–224, 1965), making the notion of variational crimes (Strang, Variational crimes in the finite element method. In: Aziz AK (ed) Mathematical foundations of the finite element method with application to partial differential equations. Proceedings Symposium, University of Maryland, Baltimore. Academic, New York, pp 689–710, 1972; Gander and Wanner, SIAM Rev 54(4), 2012) completely irrelevant here.) (by exact integration).

This Rayleigh modal approach models:

• (1) compressible media (The conventional finite element method with nodal degrees-of-freedom prevails.), with the Poisson’s ratio in the range − 1 < ν < 1∕2

• (2) and in Chap. 8, incompressible media (Where the eight nodal displacements cannot be arbitrarily prescribed.), with ν = 1∕2

In pure bending, which is associated with an arbitrarily oriented neutral axis, let us reiterate that quadrilateral plane elements, with eight degrees-of-freedom, will invariably yield non-zero shear stresses. There is not an adequate number of degrees-of-freedom to guarantee point-wise equilibrium with shear-free linear stresses. In that sense, the element cannot be entirely defect-free (MacNeal, Finite Elem Anal Des 5(1):31–37, 1989). This problem is elaborated in Chap. 9