تجزیه و تحلیل حساسیت و بهینه سازی از اجزا خرپائی/باریکه ای از دلخواه مقطع II. تنش برشی

This paper presents a general approach for detailed analysis and design optimization of arbitrary cross-sections of truss/beam built up structures. The approach allows arbitrary shape parametrization of 2-D cross-sections, as long as the coordinates of the contour vertices and their velocities are available, and is well suited for integration with existing CAD modelers and FEM analyzers. It leads to an inexpensive 2-D size/shape optimization in an alternative to costly 3-D shape optimizations, virtually impossible for real-life built up structures. Any composite multi-contour cross-section is first discretized with elementary triangles. Direct integration on the surface, using closed-form formulas, allows computation of the cross-section axial properties. Numerical integration on the boundary, along the line segments used to describe the contour, allows the computation of the shear properties. The power-series method is used to obtain the equilibrium equations and their governing linear warping system. The design sensitivities are calculated by the direct differentiation method requiring only backward substitutions on the triangular stiffness matrix. Numerical tests extensively verify the accuracy and the practical use of the formulation and implementation.

Although truss/beam like components are extensively used in civil and mechanical engineering structural applications, only very few authors have addressed the topic of design optimization of arbitrary cross-sections. Apostol and Santos [1] and [2] described an arbitrary parametric cross-section through geometrical shape variables, and expressed each vertex coordinate as a linear combination of these shape variables. The cross-section axial properties were computed and differentiated analytically with respect to the selected shape variables. These properties, representing finite element input quantities, were used as performance measures in the optimization model. However, previous research on design optimization of arbitrary cross-sections, has been focused on axial properties and stresses. Still, it is well known that the shear stresses induced by transverse shear loads and especially by torsion can reach significant values for space frames and therefore they can neither be neglected, nor poorly approximated. The analysis of the torsion problem may be performed by the semi-inverse method proposed by Saint-Venant by the mid of the last century. The method is exact, provided that a warping function satisfying a differential equation and certain boundary conditions is known. A comprehensive description of the semi-inverse method can be found in Timoshenko [3]. One of the first papers addressing cross-section optimization of elastic bars under torsion seems to belong to Banichuk [4], although shear stresses were not considered. Pilkey and Liu [5] avoided finite or boundary elements for the computation of the warping function by using the direct integration of the boundary integral equation instead. Their approach to the solution of the torsion problem was used by Schramm and Pilkey [6] for structural optimization in conjunction with shape description via B-splines. The optimization contained the torsion moment of inertia and the cross-section area only, whereas the transverse shear, the shear center and the shear stresses were not taken into consideration. The same authors extended the previous research to thin-walled beam theory, Schramm et al. [7], describing the shape with non-uniform rational B-splines (NURBS), Schramm and Pilkey [8] and [9]. A variational approach based on power-series, was introduced by Mindlin [10] for the computation of the warping function. This approach was further improved by Kosmatka [11] and [12], and it was selected in this paper as the basis for the cross-section analysis and shape optimization. Kosmatka used exact Gauss-quadrature integration for higher order polynomials, for numerical evaluation of the integrals on a surface, according to Dunavant [13]. In this paper, the power-series approach used by Kosmatka is used for numerical evaluation of the warping function. However, the Green's formula is used to convert the surface integrals into contour integrals, see Press et al. [14]. The direct differentiation method is used to differentiate the equilibrium equations with respect to cross-section shape design parameters. The design sensitivities are finally used to demonstrate the validity of the approach for numerical shape optimization of truss/beam arbitrary cross-sections. Although two design parametrization approaches are developed and compared in this paper, the current method allows the use of any general parametrization like those provided by parametric and associative CAD systems.

A general approach for detailed analysis and design optimization of cross-sections of arbitrary shape of truss/beam built up structures is proposed. The approach allows for both direct and indirect parametric definition of an arbitrary cross-section, and may be easily integrated with the parametrization supported by state-of-the-art CAD systems. The truss/beam components are subjected to axial and transversal forces, and to bending and torsion moments. Both shear properties and stresses are considered in both the analysis and the optimization process. The power-series method is used for the solution of the shear problem, and boundary integration is proposed for carrying-out the integrations required for the solution of the warping equations. A method for computing the design sensitivities of the shear properties and stresses with respect to cross-section design parameters is developed. This method uses direct differentiation of the warping equations and only requires backward substitutions on the triangular stiffness matrix, making it a quite attractive alternative for structural shape optimization of truss/beam cross-sections of built up structures. A numerical example illustrates the applicability of the method to design optimization of a real-life cross-section.