Three stress-based triangular elements
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ORIGINAL ARTICLE
Three stress‑based triangular elements Mohammad Rezaiee‑Pajand1 · Arash Karimipour1 Received: 26 September 2018 / Accepted: 26 April 2019 © Springer-Verlag London Ltd., part of Springer Nature 2019
Abstract To obtain proper stresses, three new triangular elements are formulated in this study. First, a complementary energy functional is used within an element for the analysis of plane problems. In this energy manifestation, the Airy stress function will be applied as a functional variable. Then, some basic analytical solutions are assigned for the stress functions. These trial functions are matched with each element number of degrees of freedom. The result is a number of equations with anonymous constants. Subsequently, according to the principle of minimum complementary energy, the unknown constants can be expressed in terms of displacements. Finally, this system can be rewritten in terms of the nodal displacement. In this way, three new triangular elements are formulated. To validate the results, extensive numerical studies are performed. The findings clearly demonstrate accuracies of structural displacements as well as stresses. Keywords Triangular element · Plane problem · Accurate stress · Finite-element method · Airy stress function · Variational techniques List of symbols AT6 Accurate triangular element t Thickness of the element U Displacement vector along element boundaries 𝜇 Poisson’s ratio Dij Elastic modulus l and m Direction cosines of the outer normal 𝛱C∗ Complementary energy within the element 𝜎 The stress vector of the element C Elastic flexibility matrix 𝜑 Airy stress function Cij Elastic compliances ui Nodal displacements belong to x VC∗ Complementary energy along the element boundaries T Surface force vector on the element boundaries E Young’s modulus qe Elemental nodal displacement vector x′ and y′ Axes of material symmetry
* Mohammad Rezaiee‑Pajand [email protected] 1
vi ( ) Nodal displacements belong to y Ni0 𝜉1 , 𝜉2 Shape function
1 Introduction Finite-element methods implement different elements to solve a great variety of problems. Displacement techniques are the common ways to formulate a new element. The weakness of this scheme is inaccurate stresses. There are several ways to remedy this fault and improve the responses. In one of the studies, an 8-node element was selected, and the Airy function was utilized to establish a new element by Fu et al. [1]. The performance of this element was carefully evaluated by the researchers, which showed some improvements. Previously, Stricklin et al. [2] explained some weaknesses of 8-node element in analysis of a cantilever beam. In another study, Lee and Bathe [3] studied different influences achieved by 8-node and 12-node and Lagrangian (Q9, Q16) elements using numerical analyses over different meshes. Their obtained results showed that displacement outcomes, when 8-node element was used, were more accurate than when 12-node was selected. Moreover, Q9 and Q16 could reach much higher order of the complet
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