Abstract: In this paper we considered the Neumann problem for
the fourth order differential equation. First we define the weighted Sobolev space
2 Wα and generalized solution for this equation. Then we consider the existence and uniqueness of the generalized solution,
as well as give the description of the spectrum and of the domain of definition of the corresponding operator.
Abstract: In the traditional theory of non-uniform torsion the
axial displacement field is expressed as the product of the unit twist
angle and the warping function. The first one, variable along the
beam axis, is obtained by a global congruence condition; the second
one, instead, defined over the cross-section, is determined by solving
a Neumann problem associated to the Laplace equation, as well as for
the uniform torsion problem.
So, as in the classical theory the warping function doesn-t punctually
satisfy the first indefinite equilibrium equation, the principal aim of
this work is to develop a new theory for non-uniform torsion of
beams with axial symmetric cross-section, fully restrained on both
ends and loaded by a constant torque, that permits to punctually
satisfy the previous equation, by means of a trigonometric expansion
of the axial displacement and unit twist angle functions.
Furthermore, as the classical theory is generally applied with good
results to the global and local analysis of ship structures, two beams
having the first one an open profile, the second one a closed section,
have been analyzed, in order to compare the two theories.