Error through the constitutive relation for beam and C0 plate finite elements: static and vibration analyses Original Research Article

Applications of the so-called ‘error through the constitutive relation’ are presented for the static and modal analyses of beams and plates. The error associated to the performed computation is estimated from the lack of verification, by two admissible fields, of the constitutive relation. One is an admissible stress resultant field and the other is the generalised strain given by the finite element analysis. Admissible stress resultants are derived from the finite element stress solution following two main ideas. Firstly, loads and moments ensuring the element equilibrium are calculated locally using an ‘on average’ equilibrium property of the finite element stress solution. Secondly, the stress resultants are determined within a set large enough to allow an optimisation in the way of rendering the error estimation more accurate. For the normal mode analyses, an error estimation is made for each eigenvector associated to a frequency within a given range. An automatic procedure is described in case of beam structures that ensures a given accuracy for all modes in a given frequency range. A set of examples is presented wherein the estimated error is compared to the error obtained from the analytical solution. Theses examples show that the beam and plate analyses are areas well adapted to the method.