On Saint-Venantʹs principle in pin-jointed frameworks

Numerical experiments performed by Hoff 50 years ago on the application of Saint- Venant’s principle to frameworks are extended, and his conclusions qualified. Guided by these results, a general treatment is formulated employing a state variable approach ; this leads directly to an eigenvalue problem in which the decay rates are the eigenvalues of the cell transfer matrix. Unity eigenvahles correspond to the transmitting modes of rigid body displacement and the stress resultants of tension, bending moment, shearing force and twisting moment. Consideration of the associated eigenvectors and principal vectors gives exact values for the equivalent “continuum” beam properties of the framework, such as equivalent cross-sectional area, Poisson’s ratio, second moment of area, shear coefficient and torsion constant.