Title:

On SaintVenant's principle and the state transfer matrix method

Three exact solutions are considered, within the context of the linear mathematical theory of elasticity, pertaining to the decay of selfequilibrated end loading for a class of geometries based upon polar coordinates. For a curved plane beam, an eigenequation is derived, whose roots determine the rates of decay and degenerate to the PapkovitchFadle solution for the plane strain strip when the beam centreline radius of curvature approaches infinity; this shows that the decay rates are largely insensitive to the beam curvature except for very small inner radius. For the plane and antiplane elastic wedge, subjected to selfequilibrated loading on the inner or outer arcs, radial variation of stress is affected by a combination of freeedge stress interference and the convergent or divergent geometry. When the load is applied to the inner arc, the two effects act in concert in which case decay is assured; when the load is applied to the outer arc, the two effects act in opposition, and SaintVenant’s principle (S.V.P) ceases to be applicable for wedge angles 2α > π for symmetric loading and antiplane deformation, and 2α > 257º for asymmetric loading. It is concluded that the crack tip stress singularity, which is at the heart of Linear Elastic Fracture Mechanics, is attributable to the failure of S.V.P. for just one particular eigenmode for the wedge angle 2α = 2 π. A Finite ElementTransfer Matrix Method is developed for determination of decay rates of selfequilibrated end loading for frameworks and continuum prismatic beam of arbitrary crosssection. Nodal displacements and forces on either side of a repeating cell are considered as state variables and are related by a cell transfer matrix. Assuming consecutive state vectors to be related by a constant multiple λ leads directly to an eigenvalue problem; the decay factors, λ, are the eigenvalues of the symplectic transfer matrix. Eigenvalues occurs as reciprocal pairs (that is, if λi is an eigenvalue then so is 1/ λi) according to whether decay is from left to right or viceversa. The multiple eigenvalues λi = 1/ λi = 1 are associated with the rigid body eigenvectors and their related principal vectors which describe the force transmission modes. The matrix of eigenand principal vectors then forms a similarity matrix which transforms the original transfer matrix into Jordan canonical form. Both biorthogonality and symplectic adjoint orthogonality properties of the eigenvectors allow modal decomposition of an arbitrary end load. As a byproduct of the method, it is possible to determine exact ‘continuum’ beam properties of the framework, which is useful in preliminary design work. The method is applied to the important case of a beam of rectangular crosssection for a wide range of aspect ratios. The Transfer Matrix Method is modified to become the Force or Displacement Transfer Matrix Method, which has the advantage of reducing in size the original transfer matrix by one half, and overcomes numerical illconditioning. Accuracies of all the developed methods are found to be very good when compared with available exact solutions.
