TY - JOUR

T1 - A fundamental class of stress elements in lower bound limit analysis

T2 - Lower bound elements

AU - Makrodimopoulos, Athanasios

N1 - Publisher Copyright:
© 2020 The Author(s).

PY - 2020

Y1 - 2020

N2 - There is a major restriction in the formulation of rigorous lower bound limit analysis by means of the finite-element method. Once the stress field has been discretized, the yield criterion and the equilibrium conditions must be applied at a finite number of points so that they are satisfied everywhere throughout the discretized structure. Until now, only the linear stress elements fulfil this requirement for several types of loads and structural conditions. However, there are also standard types of problems, like the one of plates under uniformly distributed loads, where the implementation of the lower bound theorem is still not possible. In this paper, it is proven for the first time that there is a class of stress interpolation elements which fulfils all the requirements of the lower bound theorem. Moreover, there is no upper restriction of the polynomial interpolation order. The efficiency is examined through plane strain, plane stress and Kirchhoff plate examples. The generalization to three-dimensional and other structural conditions is also straightforward. Thus, this interpolation scheme which is based on the Bernstein polynomials is expected to play a fundamental role in future developments and applications.

AB - There is a major restriction in the formulation of rigorous lower bound limit analysis by means of the finite-element method. Once the stress field has been discretized, the yield criterion and the equilibrium conditions must be applied at a finite number of points so that they are satisfied everywhere throughout the discretized structure. Until now, only the linear stress elements fulfil this requirement for several types of loads and structural conditions. However, there are also standard types of problems, like the one of plates under uniformly distributed loads, where the implementation of the lower bound theorem is still not possible. In this paper, it is proven for the first time that there is a class of stress interpolation elements which fulfils all the requirements of the lower bound theorem. Moreover, there is no upper restriction of the polynomial interpolation order. The efficiency is examined through plane strain, plane stress and Kirchhoff plate examples. The generalization to three-dimensional and other structural conditions is also straightforward. Thus, this interpolation scheme which is based on the Bernstein polynomials is expected to play a fundamental role in future developments and applications.

KW - Bernstein polynomials

KW - higher-order stress elements

KW - limit analysis

KW - lower bound

UR - http://www.scopus.com/inward/record.url?scp=85097951596&partnerID=8YFLogxK

U2 - 10.1098/rspa.2020.0425

DO - 10.1098/rspa.2020.0425

M3 - Article

AN - SCOPUS:85097951596

SN - 1364-5021

VL - 476

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

IS - 2243

M1 - 0425

ER -