| Paper description |
A beam problem is though classical but not well stated theoretically at present. Different
from previous publications, this paper begins with definitions of the generalized displacements.
With two assumptions and the shear stress free condition, the axial displacement is first
mathematically expanded into two terms and then expressed as an orthogonal form in terms of
the generalized displacements. Based on the orthogonal form, the generalized stresses are
defined, and the uncoupled constitutive relations are then derived for beam problems after the
generalized strains are properly measured. The principle of virtual work is proposed and the
variationally consistent higher-order beam theory is eventually established. With these
preliminaries, the finite element method is readily formulated like a three-dimensional elastic
problem, and then validated through typical examples. The results show that, while accurately
simulating the deflection, the higher-order beam element can capture the effect of clamped end
and load jump via smoothly modeling the warping of cross section by using a locally refined
mesh. It is straightforward to extend the current work to modern beam structures by taking into
account the effect of nonlocal elasticity, small scales and material heterogeneities. |
(创新港)
