Michell Structures

Algebraic Formulation for Moderately Thick Elastic Frames, Beams, Trusses, and Grillages within Timoshenko Theory

“…The use of the finite element method has definitely dominated for many years, but it is based on the approximation of displacement fields [1,2]. The direct algebraic formulation in which the matrix form of the solving system of displacement equations is the product of three matrices with diagonal constitutive matrix [3,4] is a specific mathematical problem within the theory of frames. This form is used, among others, in topological optimization [4,5], analysis of systems with large uncertainties [6], the use of convex sets techniques [7], plastic range of the analysis [8], and shakedown of elastic-plastic structures [9].…”

“…The direct algebraic formulation in which the matrix form of the solving system of displacement equations is the product of three matrices with diagonal constitutive matrix [3,4] is a specific mathematical problem within the theory of frames. This form is used, among others, in topological optimization [4,5], analysis of systems with large uncertainties [6], the use of convex sets techniques [7], plastic range of the analysis [8], and shakedown of elastic-plastic structures [9]. Potential applications of the formulation as well as of the present paper can be easily extended for the gridwork and framework methods in plate theories [10], including moderately thick plates.…”

“…Less known is the work of Lewiński [3] who generalized the truss formulation for any flat or spatial bar structure within the framework of Bernoulli-Euler's thin bar theory. The equations are recently reminded in the monograph of Michel structures [4], important from the point of view of topological optimization in which diagonal constitutive matrix is essential.…”

“…The aim of this work is to generalize the formulation [4] into the theory of moderately thick frames within the Timoshenko theory [14,15], while maintaining the diagonal form of the matrix of elasticity coefficients. According to the authors' best knowledge, such a formulation is not known in the literature.…”

“…Each group of equations is built for the whole structure, not for single bars, which avoids the procedures of building local matrices and their aggregation to a global form. A special case of derived equations is the formulation for thin frames given by Lewiński [3,4] and algebraic equations of the truss theory [3,4,14,15]. The key advantage of the presented formulation is that the obtained stiffness matrices are identical to those of the finite element method; however, their process of preparation does not require approximation of the displacement fields and the diagonal constitutive matrix is applied.…”