An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams

The theoretical results relevant to the vibration modes of Timoshenko beams are here used as benchmarks for assessing the correctness of the numerical values provided by several finite element models, based on either the traditional Lagrangian interpolation or on the recently developed isogeometric approach. Comparison of results is performed on both spectrum error (in terms of the detected natural frequencies) and on the l2 relative error (in terms of the computed eigenmodes): this double check allows detecting for each finite element model, and for a discretization based on the same number of degrees-of-freedom, N, the frequency threshold above which some prescribed accuracy level is lost, and results become more and more unreliable. Hence a quantitative way of measuring the finite element performance in modeling a Timoshenko beam is proposed. The use of Fast Fourier Transform is finally employed, for a selected set of vibration modes, to explain the reasons of the accuracy decay, mostly linked to a poor separation of the natural frequencies in the spectrum, which is responsible of some aliasing of modes

Vibration Analysis of Timoshenko Beams using Isogeometric Analysis

In this paper, the finite free-form beam element is formulated by the isogeometric approach based on the Timoshenko beam theory to investigate the free vibration behavior of the beams. The non-uniform rational B-splines (NURBS) functions which define the geometry of the beam are used as the basis functions for the finite element analysis. In order to enrich the basis functions and to increase the accuracy of the solution fields, the h-, p-, and k-refinement techniques are implemented. The geometry and curvature of the beams are modelled in a unique way based on NURBS. All the effects of the the shear deformation, and the rotary inertia are taken into consideration by the present isogeometric model. Results of the beams for non-dimensional frequencies are compared with other available results in order to show the accuracy and efficiency of the present isogeometric approach. From numerical results, the present element can produce very accurate values of natural frequencies and the mode shapes due to exact definition of the geometry. With higher order basis functions, there is no shear locking phenomenon in very thin beam situations. Finally, the benchmark tests described in this study are provided as future reference solutions for Timoshenko beam vibration problem

Studies on knot placement techniques for the geometry construction and the accurate simulation of isogeometric spatial curved beams

The present paper investigates the use of different knot placement techniques for isogeometric analysis of spatial curved beams, to enhance analysis results in cases when geometries are given in terms of data points. Focusing on analysis-aware modeling for structural static and vibration simulations of spatial free-form curved beams, the knot placement techniques based on uniformly spaced knots as well as on De Boor’s and Piegl and Tiller’s algorithms are studied. For this purpose, an isogeometric formulation for linear Euler–Bernoulli beams based on the Euler–Rodriguez transformation rule is implemented. Different case studies and numerical examples are presented and the results are validated against “overkill” solutions computed with a commercial finite element software. The results show that the De Boor’s knot placement algorithm typically leads to better approximation errors and is therefore the suggested strategy for this kind of problems

Isogeometric analysis of plane-curved beams

A curved beam element based on the Timoshenko model and non-uniform rational B-splines (NURBS) interpolation both for geometry and displacements is presented. Such an element can be used to suitably analyse plane-curved beams and arches. Some numerical results will explore the effectiveness and accuracy of this novel method by comparing its performance with those of some accurate finite elements proposed in the technical literature, and also with analytical solutions: for the cases where such closed-form solutions were not available in the literature, they have been computed by exact integration of the governing differential equations. It is shown that the presented element is almost insensitive to both membrane- and shear-locking, and that such phenomena can be easily controlled by properly choosing the number of elements or the NURBS degree

Intrinsically Selective Mass Scaling with Hierarchic Structural Element Formulations

[EN] Hierarchic shear deformable structural element formulations possess the advantage of being intrinsically free from transverse shear locking, that is they avoid transverse shear locking a priori through reparametrization of the kinematic variables. This reparametrization results in shear deformable beam, plate and shell formulations with distinct transverse shear degrees of freedom. The basic idea of selective mass scaling within explicit dynamic analyses is to scale down the highest frequencies in order to increase the critical time step size, while keeping the low frequency modes mostly unaffected. In most concepts, this comes at the cost of nondiagonal mass matrices. In this contribution, we present first investigations on selective mass scaling for hierarchic formulations. Since hierarchic structural formulations possess distinct transverse shear degrees of freedom, they offer the intrinsic ability for selective scaling of the high frequency shear modes, while keeping the bending dominated low frequency modes mostly unaffected. The proposed instrinsically selective mass scaling concept achieves high accuracy, which is typical for selective mass scaling schemes, but in contrast to existing concepts it retains the simplicity of a conventianl mass scaling method and preserves the diagonal structure of a lumped mass matrix. As model problem, we study frequency spectra of different isogeometric Timoshenko beam formulations for a simply supported beam. We discuss the effects of transverse shear parametrization, locking and mass lumping on the accuracy of results.This work has been partially supported by the Deutsche Forschungsgemeinschaft (DFG) under grant OE 728/1-1. This support is gratefully acknowledged.Oesterle, B.; Trippmacher, J.; Tkachuk, A.; Bischoff, M. (2022). Intrinsically Selective Mass Scaling with Hierarchic Structural Element Formulations. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 99-108. https://doi.org/10.4995/YIC2021.2021.12418OCS9910

Isogeometric analysis for functionally graded microplates based on modified couple stress theory

Analysis of static bending, free vibration and buckling behaviours of functionally graded microplates is investigated in this study. The main idea is to use the isogeometric analysis in associated with novel four-variable refined plate theory and quasi-3D theory. More importantly, the modified couple stress theory with only one material length scale parameter is employed to effectively capture the size-dependent effects within the microplates. Meanwhile, the quasi-3D theory which is constructed from a novel seventh-order shear deformation refined plate theory with four unknowns is able to consider both shear deformations and thickness stretching effect without requiring shear correction factors. The NURBS-based isogeometric analysis is integrated to exactly describe the geometry and approximately calculate the unknown fields with higher-order derivative and continuity requirements. The convergence and verification show the validity and efficiency of this proposed computational approach in comparison with those existing in the literature. It is further applied to study the static bending, free vibration and buckling responses of rectangular and circular functionally graded microplates with various types of boundary conditions. A number of investigations are also conducted to illustrate the effects of the material length scale, material index, and length-to-thickness ratios on the responses of the microplates.Comment: 57 pages, 14 figures, 18 table

Effects of parameterization and knot placement techniques on primal and mixed isogeometric collocation formulations of spatial shear-deformable beams with varying curvature and torsion

We present a displacement-based and a mixed isogeometric collocation (IGA-C) formulation for free-form, three-dimensional, shear-deformable beams with high and rapidly-varying curvature and torsion. When such complex shapes are concerned, the approach used to build the IGA geometric model becomes relevant. Although IGA-C has been so far successfully applied to a wide range of problems, the effects that different parameterization and knot placement techniques may have on the accuracy of collocation-based formulations is still an unexplored field. To fill this gap, primal and mixed formulations are used combining two parameterization methods (chord-length and equally spaced) with two knot placement techniques (uniformly spaced and De Boor). With respect to the space-varying Frenet local frame, we derive the strong form of the governing equations in a compact form through the definition of two matrix operators conveniently used to perform first and second order derivatives of the vector fields involved in the formulations. This approach is very efficient and easy to implement within a collocation-based scheme. Several challenging numerical experiments allow to test the different considered parameterizations and knot placement techniques, revealing in particular that with the primal formulation an equally spaced parameterization is definitively the most recommended choice and it should always be used with an approximation degree of, at least, , although some caution must be adopted when very high Jacobians and small curvatures occur. The same holds for the mixed formulation, with the difference that is enough to yield accurate results