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From the natural continuum to the elastic spring cell in three and two dimensions -- a flexibility approach

I St Doltsinis (Faculty of Aerospace Engineering and Geodesy, University of Stuttgart, Stuttgart, Germany)

Engineering Computations

ISSN: 0264-4401

Article publication date: 14 June 2019

Issue publication date: 15 August 2019




The employment of spring cell substitutes for the numerical analysis of solids and structures in place of finite elements has occasioned research on the subject with regard to both, the applicability of existing approaches and the advancement of concepts. This paper aims to explore in the context of linear elasticity the substitution of the simplex tetrahedral element in space and the triangle in the plane by corresponding spring cells deduced on a flexibility basis using the natural formalism.


The natural formalism is characterized by the homogeneous definition of strain and stress along the lines connecting nodes of the simplex tetrahedron and the triangle. The elastic compliance involves quantities along the prospective spring directions and offers itself for the transition to the spring cell. The diagonal entities are interpreted immediately as spring flexibilities, the off-diagonal terms account for the completeness of the substitution. In addition to the isotropic elastic material, the concept is discussed for anisotropic elasticity in the plane.


The natural point of view establishes the spring cell as part of the continuum element. The simplest configuration of pin-joined bars discards all geometrical and physical cross effects. The approach is attracting by its transparent simplicity, revealing deficiencies of the spring cell and identifying directly conditions for the complete substitution of the finite element.

Research limitations/implications

The spring cell counterparts of the tetrahedral- and the triangular finite elements allow employment in problems in three and two dimensions. However, the deficient nature of the approximation requires attention in the design of the discretization lattice such that the conditions of complete finite element substitution are approached as close as possible.

Practical implications

Apart from plane geometries, triangular spring cells have been assembled to lattice models of space structures such as membrane shells and similar. Tetrahedral cells have been used, in modelling plates and shell structures exhibiting bending stiffness.


The natural formalism of simplex finite elements in three and two dimensions is used for defining spring cells on a flexibility basis and exploring their properties. This is a novel approach to spring cells and an original employment of the natural concept in isotropic and anisotropic elasticity.



Doltsinis, I.S. (2019), "From the natural continuum to the elastic spring cell in three and two dimensions -- a flexibility approach", Engineering Computations, Vol. 36 No. 5, pp. 1676-1698.



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