On Saint-Venant'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 self-equilibrated end loading for a class of geometries based upon polar coordinates. For a curved plane beam, an eigen-equation is derived, whose roots determine the rates of decay and degenerate to the Papkovitch-Fadle solution for the plane strain strip when the beam centre-line 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 anti-plane elastic wedge, subjected to self-equilibrated loading on the inner or outer arcs, radial variation of stress is affected by a combination of free-edge 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 Saint-Venant’s principle (S.V.P) ceases to be applicable for wedge angles 2α > π for symmetric loading and anti-plane 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 Element-Transfer Matrix Method is developed for determination of decay rates of self-equilibrated end loading for frameworks and continuum prismatic beam of arbitrary cross-section. 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 vice-versa. 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 eigen-and principal vectors then forms a similarity matrix which transforms the original transfer matrix into Jordan canonical form. Both bi-orthogonality and symplectic adjoint orthogonality properties of the eigenvectors allow modal decomposition of an arbitrary end load. As a by-product 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 cross-section 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 ill-conditioning. Accuracies of all the developed methods are found to be very good when compared with available exact solutions.