On state-space elastostatics within a plane stress sectorial domain: the wedge and the curved beam

Abstract

The plane stress sectorial domain is analysed according to a state-space formulation of the linear theory of elasticity. When loading is applied to the straight radial edges (flanks), with the circular arcs free of traction, one has the curved beam; when loading is applied to the circular arcs, with the flanks free of traction, one has the elastic wedge. A complete treatment of just one problem (the elastic wedge, say) requires two state-space formulations; the first describes radial evolution for the transmission of the stress resultants (force and moment), while the second describes circumferential evolution for determination of the rates of decay of self-equilibrated loading on the circular arcs, as anticipated by Saint-Venant’s principle. These two formulations can be employed subsequently for the curved beam, where now radial evolution is employed for the Saint-Venant decay problem, and circumferential evolution for the transmission modes. Power-law radial dependence is employed for the wedge, and is quite adequate except for treatment of the so-called wedge paradox; for this, and the curved beam, the formulations are modified so that ln r takes the place of the radial coordinate r. The analysis is characterised by a preponderance of repeating eigenvalues for the transmission modes, and the state-space formulation allows a systematic approach for determination of the eigen- and principal vectors. The so-called wedge paradox is related to accidental eigenvalue degeneracy for a particular angle, and its resolution involves a principal vector describing the bending moment coupled to a decay eigenvector. Restrictions on repeating eigenvalues and possible Jordan canonical forms are developed. Finally, symplectic orthogonality relationships are derived from the reciprocal theorem.

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