The Explicit Expression of Internal Forces in Prismatic Membered Structures*

摘要

Analytic expressions for member forces in linear elastic redundant trusses have recently been given by the author. It was shown that the internal forces in a truss are the ratios of two multilinear homogeneous polynomials in the longitudinal stiffnesses of the elements of the structure. The order of the polynomials is equal to the number of nodal degrees of freedom of the structure. The number of terms of each polynomial is equal to the number of statically determinate stable substructures that can be derived from the original structure. It was shown that coefficients of the polynomials can be computed through the equilibrium equations and by enforcing global compatibility of deformations. This paper generalizes these results to the case of linear elastic structures, composed of uniform prismatic elements that have extensional, flexural, and torsional stiffness. This is done by replacing each bi-modal bending element with a unimodal moment element and a unimodal shear element. This allows the representation of deformation of the elements by six uncoupled basic deformation patterns in the case of structures in space, thus paving the way for truss-type analysis for general structures that are composed of uniform prismatic elements. As a result, multilinear polynomials that appear in expressions for stress resultants are the products of axial, bending, and torsional stiffnesses of subsets of the original structure. The number of terms appearing in the polynomials renders the exact analytic expressions intractable for practical engineering structures. However, the construct of the analytic equations may constitute a basis for writing approximate expressions for member forces in frames, explicitly in terms of rigidities of components of the structure. This paper describes such an approximate expression, with a reduced number of terms in the polynomials. The theory is illustrated with numerical examples of analysis of planar structures