Clough’s Plane Triangular Elements for Linear Elastic Problems: Virtual Work Substituting Variational Principle

Abstract

Clough (Proceedings, 2nd conference on electronic computation, A.S.C.E. structural division, Pittsburgh, PA, pp 345–378, 1960) introduced the term finite element with his classical displacement formulation of plane strain triangles. These are plane elements with constant stress distributions. Analogous plane stress triangular elements can be similarly formulated. The corresponding (The triangle is the simplest polygon in $\mathfrak{R}^{2}$ and the tetrahedron is its three-dimensional counterpart.) three-dimensional cases of tetrahedral elements do not pose any conceptual difficulty. To emphasize this natural extension, we first review the three-dimensional field equations of continuum mechanics, and then formulate the element stiffness matrix for triangular domains.

An important feature of triangular elements is that their shape functions are linear polynomials in the physical (x, y) coordinate variables. This renders the stress and strain fields to be constant within an element. Hence, the point-wise equilibrium is always satisfied unconditionally. Thus, it does not matter even if the linear elasticity formulations are coupled vector field problems, the shape function vectors with one zero component still qualify to be admissible functions. (Courant (Bull Am Math Soc 49(1):1–29, 1943) emphasized this concept starting from his Sect. II as he focused on the Rayleigh–Ritz Method, just above his equation number (12).) Hence the “uncoupling (This is elaborated for four-node elements in Sect. 5.4.)” of the shape function vectors, à la Courant scalar field problems (Courant, Bull Am Math Soc 49(1):1–29, 1943), still persists even for coupled vector field problems of elasticity (It will be stated in Eqs. (5.7) and (5.8) that the independence of displacement vectors under a single nodal displacement is standard even for four-node elements that violate point-wise equilibrium.) when spatial discretization with triangles is invoked. (Similar conclusions can be drawn when trapezoidal elements are used for three-dimensional linear elasticity problems.) Assuming the other parts of a finite element computer program to be without any flaw, any linear stress field on arbitrary domains will be exactly reproduced irrespective of meshing details. (In Chap. 6 we analyze this concept (the patch test) in depth.) From the theoretical standpoints, this observation raises a valid question as to whether other elements with more nodes will have such a property that brings the finite element method close to very reliable approximation of problems with boundaries of arbitrary shapes.

Clough employed the physical concepts of virtual work to identify the entries of stiffness matrices and nodal forces to be virtual work quantities. The variational principle (though mathematically elegant) was not essential because, unlike Courant, Clough was not addressing the abstract class of elliptic boundary value problems for which the “rigid” and “natural” boundary conditions always emerge from the associated variational principle. It is important to state that for physical problems, where there is a notion of energy, (due to the self-adjointness property of the partial differential equations of mathematical physics) the principle of virtual work is congruent with the variational formulation of Ritz (vide Ritz, J Reine Angew Math 135:1–61, 1908; Gaul, In: GAMM: Proceedings in applied mathematics and mechanics. Springer, Rotterdam, 2011).