A higher order triangular plate finite element using Airy functions

Abstract

The present paper describes the formulation of a new moderately thick plate bending triangular finite element based on Mindlin–Reissner plate theory. It is called a Great Triangular Moderately Thick Plate Finite Element, or GTMTPFE. The formulation is based on the strain approach, on solution of Airy’s function and on the analytical integration in the construction of the stiffness matrix. The strengths associated with this approach consist of: • automatic verification of equilibrium conditions and kinematic compatibility conditions, • the enrichment of the degrees of the interpolation polynomials of displacements, strains and constraints (refinement p), • the consideration distortions sections related to Poisson effects, • the treatment of blocking phenomena related to transverse shear. In general, this approach results in a competitive, robust and efficient new moderately thick plate finite element. This is visible, on the one hand, through its stability against patch tests (constant twists, state of constants moments, transverse shear locking phenomenon, isotropy test). This is visible, through its good response to the patch tests to which it is subjected (constant torsions, state of constant moments, phenomenon of blocking in transverse shears, isotropy test). As has excellent convergence to the reference solution. Thus, it exhibits better performance behavior than other existing plate elements in the literature, particularly for moderately thick plates and for thin plates (L/h ratio greater than 4).

A four-node rectangular plate finite element using Airy functions with transverse shear

“…Recently, a new moderately thick plate bending triangular finite element based on Mindlin-Reissner plate theory. have been developed by Himeur et al (2020).The formulation is based on the strain approach, on solution of Airy's function. In this paper, the objective of this contribution is therefore the formulation of a new rectangular finite element for thin plate bending based on the strain approach Using Airy's Function.…”

“…The geometric and mechanical data used are length L 5 10m; width b 5 1.0 m; Young's modulus E ¼ 1:2 3 10 6 N/ m 2 and Poisson's ratio ν 5 0.0. This example was treated by Himeur et al (2020) to test the performance of the bending elements (triangular plate finite element).…”

“…Recently, a new moderately thick plate bending triangular finite element based on Mindlin–Reissner plate theory. have been developed by Himeur et al . (2020).The formulation is based on the strain approach, on solution of Airy's function.…”

“…The strain technique serves as the foundation for the formulation of the proposed rectangle element, abbreviated RPFEUAF (rectangular plate finite element using Airy function). Bi-harmonic polynomials chosen from the solutions provided by the Airy function development are used to derive interpolation functions of strains, and therefore displacements and stresses fields, from the equilibrium conditions (Himeur et al ., 2020; Himeur et al ., 2015). Their creation makes use of Pascal's triangle.…”

Techniques for element formulation and quadtree-based triangular mesh generation for strain-based finite elements

“…In short, many researchers have presented new formulations for the strain-based elements that satisfy both compatibility and equilibrium conditions [20] , [21] , [24] , [25] , [26] , [27] , [28] , [29] , [30] , [31] , [32] , [33] . However, only a handful of them (such as [20] , [21] , 25 , 33] ) have described the techniques used to obtain the functions that satisfy both conditions, causing the techniques used in the remainder to be left unknown. This paper introduces two new techniques that are applicable to strain-based finite elements.…”