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

On the whole spectrum of Timoshenko beams. Part I: a theoretical revisitation

The problem of free vibrations of the Timoshenko beam model is here addressed. A careful analysis of the governing equations allows identifying that the vibration spectrum consists of two parts, separated by a transition frequency, which, depending on the applied boundary conditions, might be itself part of the spectrum. For both parts of the spectrum, the values of natural frequencies are computed and the expressions of eigenmodes are provided. This allows to acknowledge that the nature of vibration modes changes when moving across the transition frequency. Among all possible combination of end constraints which can be applied to single-span beams, the case of a simply supported beam is considered. These theoretical results can be used as benchmarks for assessing the correctness of the numerical values provided by several numerical techniques, e.g. traditional Lagrangian-based finite element models or the newly developed isogeometric approach

On the whole spectrum of Timoshenko beams. Part II: Further applications

The problem of free vibrations of the Timoshenko beam model has been addressed in the first part of this paper. A careful analysis of the governing equations has shown that the vibration spectrum consists of two parts, separated by a transition frequency, which, depending on the applied boundary conditions, might be itself part of the spectrum. Here, as an extension, the case of a doubly clamped beam is considered. For both parts of the spectrum, the values of natural frequencies are computed and the expressions of eigenmodes are provided: this allows to acknowledge that the nature of vibration modes changes when moving across the transition frequency. This case is a meaningful example of more general ones, where the wave-numbers equation cannot be written in a factorized form and hence must be solved by general rootfinding methods for nonlinear transcendental equations. These theoretical results can be used as further benchmarks for assessing the correctness of the numerical values provided by several numerical techniques, e.g. finite element models

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

Sardinia radio telescope finite element model updating by means of photogrammetric measurements

The 64 m diameter Sardinia Radio Telescope (SRT), located near Cagliari (Italy), is the world’s second largest fully steerable radio telescope with an active surface. Among its peculiarities is the capability of modifying the configuration of the primary mirror surface by means of electromechanical actuators. This capability enables, within a fixed range, balancing of the deformation caused by external loads. In this way, the difference between the ideal shape of the mirror (which maximizes its performance) and the actual surface can be reduced. The control loop of the radio telescope needs a procedure that is able to predict SRT deformation, with the required accuracy, in order to reduce deviation from the ideal shape. To achieve this aim, a finite element model that can accurately predict the displacements of the structure is required. Unfortunately, the finite element model of the SRT, although very refined, does not give completely satisfactory results, since it does not take into account essential pieces of information, for instance, thermal strains and assembly defects. This paper explores a possible update of the finite element model using only the benchmark data available, i.e. the photogrammetric survey developed during the setup of the reflecting surface. This updating leads to a significant reduction in the differences between photogrammetric data and results of the numerical model. The effectiveness of this tuning procedure is then assessed

Are higher-gradient models also capable of predicting mechanical behavior in the case of wide-knit pantographic structures?

The central theme of this study is to investigate a remarkable capability of a second-gradient continuum model developed for pantographic structures. The model is applied to a particular type of this metamaterial, namely the wide-knit pantograph. As this type of structure has low fiber density, the applicability of such a continuum model may be questionable. To address this uncertainty, numerical simulations are conducted to analyze the behavior of a wide-knit pantographic structure, and the predicted results are compared with those measured experimentally under bias extension testing. The results presented in this study show that the numerical predictions and experimental measurements are in good agreement; therefore, in some useful circumstances, this model is applicable for the analysis of wide-knit pantographic structures

Structural Evaluation of Typical Historical Masonry Vaults of Cagliari: Sensitivity to Bricks Arrangements

Masonry vaults have a great diffusion in the historical architectural heritage: in this work, their structural behavior is investigated. Attention is focused on lowered sail vaults composed by several brick arrangements, a typical nineteenth-century masonry vault which have great diffusion in Cagliari (Sardinia). The target is evaluating the role played by bricks arrangement in their mechanical behavior. A series of rigorous laser scanner surveys have been performed in order to obtain the effective geometry both at macro-level – the vault shape – and at micro-level – brick patterns. A NURBS (Non-Uniform Rational B Spline) representation of the geometry is adopted and adaptive upper bound limit analyses are performed. NURBS entities, which are common in commercial CAD packages, have the great advantage to describe complex geometries such as curved elements, with very few elements. An upper bound limit analysis formulation is adopted, in which the NURBS elements forming the mesh are idealized as rigid bodies with dissipation allowed only along interfaces. The mesh constituted by few NURBS elements is progressively adjusted through a genetic algorithm in order to minimize the live load multiplier. Limit analysis is performed initially to determine the collapse multiplier of vertical loads, to assess the load bearing capacity of the vault, then attention is focused on differential settlements, that may be a serious hazard for this structural typology

Energy Dissipating Devices in Falling Rock Protection Barriers

Rockfall is a phenomenon which, when uncontrolled, may cause extensive material damage and personal injury. One of the structures used to avoid accidents caused by debris flows or rockfalls is flexible barriers. The energy dissipating devices which absorb the energy generated by rock impact and reduce the mechanical stresses in the rest of the elements of the structure are an essential part of these kinds of structures. This document proposes an overview of the performance of energy dissipating devices, as well as of the role that they fulfil in the barrier. Furthermore, a compilation and a description of the dissipating elements found in the literature are proposed. Additionally, an analysis has been performed of the aspects taken into account in the design, such as experimental (quasi-static and dynamic) tests observing the variation of the behaviour curve depending on the test speed and numerical simulations by means of several finite element software packages