Abstract
The present paper combines an effective beam theory with a simple and accurate numerical technique opening the door to the prediction of the structural behavior of planar beams characterized by a continuous variation of the cross-section geometry, that in general deeply influences the stress distribution and, therefore, leads to non-trivial constitutive relations. Accounting for these peculiar aspects, the beam theory is described by a mixed formulation of the problem represented by six linear Ordinary Differential Equations (ODEs) with non-constant coefficients depending on both the cross-section displacements and the internal forces. Due to the ODEs’ complexity, the solution can be typically computed only numerically also for relatively simple geometries, loads, and boundary conditions; however, the use of classical numerical tools for this problem, like a (six-field) mixed finite element approach, might entail several issues (e.g., shear locking, ill-conditioned matrices, etc.). Conversely, the recently proposed isogeometric collocation method, consisting of the direct discretization of the ODEs in strong form and using the higher-continuity properties typical of spline shape functions, allows an equal order approximation of all unknown fields, without affecting the stability of the solution. This makes such an approach simple, robust, efficient, and particularly suitable for solving the system of ODEs governing the non-prismatic beam problem. Several numerical experiments confirm that the proposed mixed isogeometric collocation method is actually cost-effective and able to attain high accuracy.
Highlights
Non-prismatic structural elements are widely used in several engineering fields, e.g., large span structures, mechanical components, aeronautical applications [1,2]
The paper is structured as follows: In Section 2, we summarize the differential equations governing the beam model under investigation and provide some highlights on the beam model properties
This modeling strategy is substantially different from the one usually adopted for curved beams that are formulated with respect to a curvilinear coordinate running along the beam centerline [33,34], but has the following main advantages: (i) it avoids complications coming from the handling of a curvilinear coordinate; (ii) it allows to obtain linear and simple Ordinary Differential Equations (ODEs); and (iii) in numerical discretization, it reduces the mapping between real and reference elements to a trivial scaling function
Summary
Non-prismatic structural elements are widely used in several engineering fields, e.g., large span structures, mechanical components, aeronautical applications [1,2]. Recent papers have highlighted that smooth changes of the crosssection geometry lead to non trivial stress distributions that deeply influence the whole beam behavior [10,11,12] In this context, the use of prismatic beam Ordinary Differential Equations (ODEs) assuming that the cross-section area and inertia are variable parameters results in a non-effective approach, as noticed since the ’60s of the past century [13]. The numerical results discussed in [14] and the application of the proposed model to several engineering problems [1,4] have already been shown the highly improved performances of the adopted beam theory in terms of ability in predicting stress, strain, displacements, and stiffness Even if it is a 1D theory ( representing a simplification of the 2D problem), the above mentioned beam model is still represented by a system of ODEs which requires a numerical treatment to obtain solutions in non-trivial cases.
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