Discretization of the Timoshenko Beam problem by thep and theh-p versions of the finite element method

Abstract

Summary.In this paper the discretization of the Timoshenko Beam problem by the p and the h -p versions of the finite element method is considered. Optimal error estimates are established. The locking phenomenon disappears as the thickness of the beam decreases.

On Locking and Robustness in the Finite Element Method

“…IVO BABUgKA AND MANIL SURI This problem (along with various extension processes for it) has been analyzed in the context of locking in [1], [9]. We will distinguish between two cases.…”

“…The most ideal remedy to locking is to employ a method that is robust, i.e., one which is more or less uniformly convergent for all t. Various robust methods (particularly mixed methods) have been proposed and analyzed in the context of locking; see for example [7] for the plate problem, [1], [9] for a beam problem, [6], [14] for Poisson ratio locking, among others.…”

“…These problems have a well-defined associated limit problem for t-* to, satisfying certain properties. In 5, we present a simplified analysis for the beam problem which was analyzed in 1], [9]. In [6], it was shown that a necessary condition for the absence of locking is the satisfaction of a suitable approximability condition for this limit problem.…”

Elementos finitos con acciones repartidas equivalentes de cualquier orden. Aplicación a los modelos de vigas de Timoshenko y Bernoulli-Euler

“…Este enfoque para el modelo de Timoshenko se ha seguido en (18) empleando polinomios del mismo grado para aproximar el desplazamiento y el giro, pero con un enfoque diferente al utilizado en este trabajo, al no considerar dicho autor la exactitud nodal, que evita además el fenómeno de bloqueo en la solución, ni emplear tampoco la proyección de la acción original sobre el espacio de soluciones del problema homogéneo. …”

Locking and robustness in the finite element method for circular arch problem

“…These are the topics of the current paper. At this point we should mention that for a closely related problem, the Timoshenko beam, the investigation is quite complete (see [1], [9] and [10]). …”