The study of generalized continuum models through the numerical investigation of discrete systems considered as an approximation to a homogenized continuum limit is nowadays a well-known research ...approach in mechanics. In the present paper, a system constituted by a large number of beams interconnected via ideal hinges, called here a pantographic sheet, is considered, and some numerical simulations concerning the static and dynamic analysis of the system are presented and discussed. The observed behavior significantly differs from what one would expect from ordinary first gradient continuum models. Moreover, interesting application possibilities entailed by the specific characteristics of the structure, and in particular by the strong non-linear behavior of the mechanical variables, are discussed.
•In the paper by dell׳Isola, Della Corte, Giorgio and Scerrato "Pantographic 2D sheets: discussion of some numerical investigations and potential applications" a numerical study of a (pantographic) mechanical system constituted by two arrays of parallel fibers, mutually connected by frictionless hinges, is performed. The fibers are modeled as Euler beams, and static and dynamic numerical results are provided. Several cases of wave propagation are examined, including internal discontinuities acting as a damper, crash of waves from opposite directions and waves originated by a system of self-equilibrated forces.
A collocation technique based on the use of Bernstein polynomials to approximate the field variable is assessed in Boundary Value Problems (BVPs) of beams with governing non-linear differential ...equations. The BVPs are transformed into unconstrained optimization problems by means of an extended cost function which leverages the properties of the Bernstein basis to enforce the boundary conditions. The minimization of the squared error cost function is conducted by means of the quasi-Newton Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. The method is tested in benchmarks of various types of non-linearities, including materials with Ludwick stress–strain curves, follower loads and beams on Winkler foundation. The approach is compared with Isogeometric collocation (IGA-c) and straightforward (pseudospectral) Bernstein collocation in terms of performance and computational effort. Moreover, the accuracy and convergence of the method is discussed to ease its successful application to other non-linear beam problems.
•Bernstein polynomials and Broyden–Fletcher–Goldfarb–Shanno optimization are combined to solve non-linear beam boundary value problems.•Number of collocation points and unknown parameters are decoupled.•Boundary conditions are enforced via an extended cost function with penalty parameter.•The numerical performance is compared with Isogeometric collocation and standard Bernstein collocation.
In the paper we present a new finite-element formulation for the dynamic analysis of geometrically exact three-dimensional beams. We limit our studies to implicit time-integration schemes and ...possible approaches for increasing their robustness and numerical stability. In contrast to standard displacement-rotation based approach we present here a spatial and temporal discretization based on velocities and angular velocities. To describe the rotational degrees of freedom quaternions are used. The time-integration scheme and the governing equations of the three-dimensional beam are modified accordingly. In the numerical implementation the Galerkin-type discretization is employed to obtain the finite-element formulation of the problem. The result of our studies is simple, but accurate, efficient and robust numerical model.
•We present velocity-based approach for dynamic analysis of three-dimensional beams.•Spatial and temporal discretization are based on additive quantities.•Rotational degrees of freedom are handled using quaternion algebra.•A special care is taken in deriving discrete kinematic compatibility equations.
In this article a non-linear model for dynamic analysis of rotating thin-walled composite beams is introduced. The theory is deduced in the context of classic variational principles and the finite ...element method is employed to discretize and furnish a numerical approximation to the motion equations. The model considers shear flexibility as well as non-linear inertial terms, Coriolis' effects, among others. The clamping stiffness of the beam to the rotating hub is modeled through a set of spring factors. The model serves as a mean deterministic basis to the studies of stochastic dynamics, which are the objective of the present article. Uncertainties should be considered in order to improve the predictability of a given modeling scheme. In a rotating structural system, uncertainties are present due to a number of facts, namely, loads, material properties, etc. In this study the uncertainties are incorporated in the beam-to-hub connection (i.e. the connection angle and the springs) and the rotating velocity. The probability density functions of the uncertain parameters are derived employing the Maximum Entropy Principle. Different numerical studies are conducted to show the main characteristics of the uncertainty propagation in the dynamics of rotating composite beams.
Abstract
Euler-Bernoulli beams are distributed parameter systems that are governed by a non-linear partial differential equation (PDE) of motion. This paper presents a vibration control approach for ...such beams that directly utilizes the non-linear PDE of motion, and hence, it is free from approximation errors (such as model reduction, linearization etc.). Two state feedback controllers are presented based on a newly developed optimal dynamic inversion technique which leads to closed-form solutions for the control variable. In one formulation a continuous controller structure is assumed in the spatial domain, whereas in the other approach it is assumed that the control force is applied through a finite number of discrete actuators located at predefined discrete locations in the spatial domain. An implicit finite difference technique with unconditional stability has been used to solve the PDE with control actions. Numerical simulation studies show that the beam vibration can effectively be decreased using either of the two formulations.
The computation of analytic and numerical beam based quantities are derived for full 3D representation of the quadrupoles magnetic field, which can be computed by finite element code or measured. The ...impact of this more accurate description of the non homogeneity of the field is estimated on beam based observables and non linear correctors strengths, and compared with the less accurate models in the case of HL-LHC.
This study investigates an analytical beam model to improve the analyses of welded thin plates with welding-induced curved distortions. The model addresses the rigidity of a butt-welded joint and its ...effect on plate bending and structural stress by including a rotational spring at the welded end. The spring rotational stiffness, ka, is replaced by the fixity factor, ρa. The validity of the model is based on the assumption of small displacement and moderate rotation of the mid-plane of the welded plate. Using the Finite Element Analysis of a two-dimensional model, a semi-analytical method for the fixity factor computation is developed. Compared with the numerical analysis, the beam model showed a maximum error of 3% in deflection and hot-spot structural stress. Results suggest that the fixity factor is mainly dependent on the width of the weld bead and the far-end constraint. The introduction of ρa can improve the analytical solution by 9% in the evaluation of the hot-spot structural stress. Neglecting the non-ideal joint rigidity may lead up to 54% underestimation in terms of fatigue life, when the S–N curve slope, m, equals 5. However, the relevance of ρa decreases for increasing geometric slenderness of the welded plates.
•A non-linear beam theory models the mechanics of welded thin-plates with curvature.•Rotational spring at beam end accounts for weld rigidity effect on plate response.•Non-ideal constraints improve analytical structural stress assessment near welds.•A semi-analytical procedure provides fixity factors over 0.9 for butt-welded joints.•The fixity factor for butt-welds depends on weld width and far-end constraint.
Mechanical properties of open-porous materials are often described by constructing a cellular network with beams of constant cross sections as the struts of the cells. Such models have been applied ...to describe, for example, thermal and mechanical properties of aerogels. However, in many aerogels, the pore walls or the skeletal network is better described as a pearl-necklace, in which the particles making up the network appear as a string of pearls. In this paper, we investigate the effect of neck sizes on the mechanical properties of such pore walls. We present an analytical and a numerical solution by modeling these walls as corrugated beams and study the subsequent deviations from the classical scaling theory. Additionally, a full numerical model of such pearl-necklace-like walls with concave necks of varying sizes are simulated. The results of the numerical model are shown to be in good agreement with those resulting from the computational one.
The paper presents a formulation of the geometrically exact three-dimensional beam theory where the shape functions of three-dimensional rotations are obtained from strains by the analytical solution ...of kinematic equations. In general it is very demanding to obtain rotations from known rotational strains. In the paper we limit our studies to the constant strain field along the element. The relation between the total three-dimensional rotations and the rotational strains is complicated even when a constant strain field is assumed. The analytical solution for the rotation matrix is for constant rotational strains expressed by the matrix exponential. Despite the analytical relationship between rotations and rotational strains, the governing equations of the beam are in general too demanding to be solved analytically. A finite-element strain-based formulation is presented in which numerical integration in governing equations and their variations is completely omitted and replaced by analytical integrals. Some interesting connections between quantities and non-linear expressions of the beam are revealed. These relations can also serve as useful guidelines in the development of new finite elements, especially in the choice of suitable shape functions.
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