Performance of a Hermitian element for a beam with rotational constraints

The behaviour of a Hermitian two-node element based on the Bernoulli beam equation is examined. The assumed constraints generate rotation-dependent distributed moments. It is shown that for these moments a potential exists, and that a rigid translation is the only rigid body mode of the element. The analysis of the Bernoulli equation demonstrates that very large values of α = e/EI enforce the condition w,x = 0, resulting in displacements equal to zero. The element is examined for two types of constraints. The first type of constraint, diminishing rotations of a beam (α < 0), yields regular solutions which, however, seem to have a non-differentiality near the end of the beam. A special procedure is developed to evaluate analytical solutions for long beams or stiff constraints, for which computer accuracy is exceeded. For the second type of constraint, enlarging rotations of a beam (α > 0), a highly oscillatory nature of the solution is proven.