Beam Centerline Research Articles

Equivalently analytical solution for the large deformation of slender beams under follower loads: a second-order ANCF approach

Follower loads interact with the object's configuration, and the two are unknown variables in geometrically nonlinear analysis. This work aims to solve the large deformation of slender beams under follower loads via the Bernoulli–Euler beam element of absolute nodal coordinate formulation. The second-order approximate function of the beam centerline is utilized to model the beam configuration and the Frenet frame. The elastic force vector and its tangent stiffness matrix are formulated following the decoupling elastic line approach. The constraint of adjacent elements is imposed using Lagrange multipliers. The exact kinematics characteristics of typical follower loads are demonstrated; sequentially, the external force vectors and the load stiffness matrices are rigorously derived based on the configuration and the Frenet frame modelled by the second-order approximate function. Benchmark problems of straight and curved beams under different follower loads are solved, wherein the accuracy and efficiency of the proposed approach are validated as compared to relevant analytical and numerical solutions. It is found that the proposed element yields equivalently analytical nodal solutions, including position, rotation and curvature; considerably fewer proposed elements are used as compared with other finite element methods; the load stiffness matrix of the bending moment raises computational efficiency of the proposed element; the obtained solutions are more accurate than that of the B22 element of ABAQUS when solving pressure loading cases. By further extending the proposed approach, it can enable the unveiling of the spatio-temporal evolution characteristic of slender structures and follower loads from the equivalently analytical perspective.

Optimal Design of a Six‐Dimensional Force Sensor with Three‐Beam Structure

A six‐dimensional force sensor with a three‐beam structure was designed, and its strain and stress distribution under various load conditions were analyzed based on finite element simulation technology. According to the strain distribution, the Wheatstone bridge scheme of the sensor is designed, and the output voltage and sensitivity formulas in each direction are deduced. The optimal patch location was determined according to the strain distribution on the centerline of the strain beam. Finally, the size parameters of the elastic body were optimized based on the response surface method and multiobjective genetic algorithm, which greatly improved the sensitivity of the sensor in all directions. It can provide a useful guidance for the optimal design of the six‐dimensional force sensor with a three‐beam structure.

Linear Dynamic Analysis of a Spatially Curved Bernoulli-Euler Beam Subjected to a Moving Load

<p>This paper considers the dynamic analysis of a spatially curved Bernoulli-Euler beam subjected to a moving load. The isogeometric approach is used for the spatial discretization of the weak form of the equation of motion. Both the reference geometry and the solution space are represented using the same NURBS basis functions that guarantee an accurate description of the beam centerline. The time integration is done by the explicit technique. The presented formulation is validated by the comparison with the existing results from the literature for the curved beam subjected to a constant load moving with a constant velocity. In addition, the influence of the moving load velocity on the dynamic response of a spatially curved beam has been investigated.</p>

The Chevron Effect: Reserve Strength of Existing Chevron Frames

Recently an analysis model has been developed to address the large shear forces (the so-called “chevron effect”) that can develop in the connection regions of chevron-braced frames. These shear forces (and the corresponding moments) are the result of the application of the brace forces at the beam flange, eccentric to the beam centerline. Prior to the presentation of these methods, such forces were not generally considered in design, without apparent incident. Sabelli and Saxey presented an alternative model that determines substantially higher resistance in these connections. Both models resolve the shear and moment within the connection region such that forces outside that region are consistent with those determined using a centerline model. Greater resistance can be determined if the flexural strength of braces and the beam outside the connection region are used to resist a portion of the chevron moment. This paper presents a complete plastic-mechanism strength of the chevron frame with yielding of the beam web due to the local shear forces. This complete plastic mechanism strength can confirm the adequacy of existing designs that did not consider the chevron effect.

A Concise and Geometrically Exact Planar Beam Model for Arbitrarily Large Elastic Deformation Dynamics.

The potential of large elastic deformations in control applications, e.g., robotic manipulation, is not yet fully exploited, especially in dynamic contexts. Mainly because essential geometrically exact continuum models are necessary to express these arbitrarily large deformation dynamics, they typically result in a set of nonlinear, coupled, partial differential equations that are unsuited for control applications. Due to this lack of appropriate models, current approaches that try to exploit elastic properties are limited to either small deflection assumptions or quasistatic considerations only. To promote further exploration of this exciting research field of large elastic deflection control, we propose a geometrically exact, but yet concise a beam model for a planar, shear-, and torsion-free case without elongation. The model is derived by reducing the general geometrically exact the 3D Simo–Reissner beam model to this special case, where the assumption of inextensibility allows expressing the couple of planar Cartesian parameters in terms of the curve tangent angle of the beam center line alone. We further elaborate on how the necessary coupling between position-related boundary conditions (i.e., clamped and hinged ends) and the tangent angle parametrization of the beam model can be incorporated in a finite element method formulation and verify all derived expressions by comparison to analytic initial value solutions and an energy analysis of a dynamic simulation result. The presented beam model opens the possibility of designing online feedback control structures for accessing the full potential that elasticity in planar beam dynamics has to offer.

BEAM ELEMENT UNDER FINITE ROTATIONS

The present work focuses on the 2-D formulation of a nonlinear beam model for slender structures that can exhibit large rotations of the cross sections while remaining in the small-strain regime. Bernoulli-Euler hypothesis that plane sections remain plane and perpendicular to the deformed beam centerline is combined with a linear elastic stress-strain law.The formulation is based on the integrated form of equilibrium equations and leads to a set of three first-order differential equations for the displacements and rotation, which are numerically integrated using a special version of the shooting method. The element has been implemented into an open-source finite element code to ease computations involving more complex structures. Numerical examples show a favorable comparison with standard beam elements formulated in the finite-strain framework and with analytical solutions.

Design for Local Member Shear at Brace and Diagonal-Member Connections: Full-Height and Chevron Gussets

Large local member shear forces develop in beams in chevron-braced frames due to the delivery of brace forces to beam flanges, which are at a distance from the beam centerline (Fortney and Thornton, 2015, 2017; Hadad and Fortney, 2020). Using the “lower bound theorem” (Thornton, 1984), Sabelli and Arber (2017) developed design methods to address this local member shear by optimizing the internal stress distribution and thus maximizing the resistance utilized in design. This paper further develops those design methods for chevron beams and extends them to gusset connections at columns.

Large Bending Deformation of a Cantilevered Soft Beam under External Load: The Applicability of Inextensibility Assumption of the Centerline

Background: Soft materials, including elastomers and gels, are pervasive in biological systems and technological applications. Despite the rapid developments of soft materials in the recent decades, it is still challenging to theoretically model and predict the large-deformation behaviors of soft structures. Objective: The goal of this work is to give a general theoretical model to investigate the large deformation of a cantilevered soft beam under various loads. In particular, the applicability of the inextensibility assumption of the beam centerline is explored. Methods: The governing equations of the soft beam system are derived according to the principle of minimum potential energy. In order to investigate the large deformation of the soft beam, the curvature of the beam centerline is exactly considered and the Yeoh model is utilized to account for the hyperelasticity of the soft beam. The derived ordinary differential equations are discretized by the Galerkin method and then solved by the iterative algorithm. Results: Based on the proposed theoretical model, large bending deformations of the cantilevered soft beam are analyzed for various types of external loads, including uniformly distributed force, tipend concentrated force, and non-uniformly distributed force. Different values of the amplitude of the external loads are considered and fruitful deformed configurations are presented. Conclusion: The proposed model is able to study the large deformation of the soft beam effectively. The inextensibility assumption of the beam centerline is applicable when the amplitude of the external load is relatively small. When the amplitude of the external load is sufficiently large, the extension of the centerline needs to be considered.

Asymptotic Behavior of Stable Structures Made of Beams

In this paper, we study the asymptotic behavior of an varepsilon -periodic 3D stable structure made of beams of circular cross-section of radius r when the periodicity parameter varepsilon and the ratio {r/varepsilon } simultaneously tend to 0. The analysis is performed within the frame of linear elasticity theory and it is based on the known decomposition of the beam displacements into a beam centerline displacement, a small rotation of the cross-sections and a warping (the deformation of the cross-sections). This decomposition allows to obtain Korn type inequalities. We introduce two unfolding operators, one for the homogenization of the set of beam centerlines and another for the dimension reduction of the beams. The limit homogenized problem is still a linear elastic, second order PDE.

Patterns for gridshells with negligible geometrical torsion at nodes

Abstract Curved envelope structural building envelopes have been quite popular in architecture in the past decades, and pose many challenges in their design, manufacturing and planning. In gridshells, a popular structural morphology for curved structure, designers will often strive to orient beams such that their top face is parallel to the envelope surface. However, this tends to induce geometrical torsion along the beam centerline, which complexifies significantly the manufacturing of the connection nodes or of the beams themselves. It is well known that such issue can be avoided by aligning beams with principal curvature directions of the envelope surface, thus yielding a quadrangular paneling. In this article, we study how other types of patterns (non-quadrangular) can be used to design torsion-free grid-shells. Based on asymptotic considerations, we derive a set of geometrical rules which, if fulfilled by a pattern, insure that a surface can be covered by this pattern with negligible torsion and limited deviation of beams from surface normals. A wide variety of patterns fulfill these rules, offering interesting possibilities for the design of curved architectural envelopes (Figure 1) is shown. As these rules are based on first order asymptotic analysis, we perform global validation on case studies. One main application is for structures in which face planarity is not necessary, for example ones cladded with ETFE cushions.

Failure investigation of nuclear grade POCO graphite target in high energy neutrino physics through numerical simulation

Abstract In many high energy physics neutrino beamlines, targets are made up of isotropic POCO graphite grade for production of neutrinos for high energy particle physics research and are bombarded with highly energetic pulsed proton beam. The pulsed proton beam with a small beam size creates thermal stress waves as well as radiation damage such as displacement damage, void formation, swelling, and gas formation amongst others. As a result, after a few years of operation one of the longest operated targets appears to have undergone bulk swelling and fractured along the beam centerline. A complex interaction of dynamic loading due to beam, material degradation due to irradiation are thought to have caused such a failure. Here, we present a numerical simulation to capture the combined effects of swelling and the dynamic effect of thermal stress waves due to beam heating, to explain the failure of target. An empirical formula has been developed to take into account swelling as a function of temperature and proton fluence and has been implemented in a commercial finite element code. Extensive X-ray diffraction (XRD) were performed on graphite samples from the failed target to understand the lattice parameter changes due to irradiation and build a valid empirical model. Simulation results show that swelling plays a major role in elevating the stress state in the material exceeding the failure strength while the pulsed beam heating also introduces fatigue loading that would explain brittle fracture emanating from inside of the material and propagating outward. Such empirical formulations will help in identifying the threshold values of critical parameters to extend the service life of future multi-megawatt neutrino targets and targets in other accelerator environments.

Coupled Dynamics of Cable-Harnessed Structures: Experimental Validation

The experimental study and model validations for the coupled dynamics of a cable-harnessed beam structure are presented. The system under consideration consists of multiple pretensioned cables attached along the length of the host beam structure positioned at an offset distance from the beam centerline. Analytical model presented by the coupled partial differential equations (PDEs) for various coordinates of vibrations is found, and the displacement frequency response functions (FRFs) obtained for both Euler–Bernoulli and Timoshenko-based models are compared to those from the experiments for validation. The results are shown to be in very good agreement with the experiments.

Resonant vibrations of FG viscoelastic imperfect Timoshenko beams

An investigation is performed on the viscoelastic nonlinear vibrations of functionally graded imperfect Timoshenko beams. The internal viscosity is incorporated using the Kelvin–Voigt scheme. Beam's centerline stretching is the cause of geometric nonlinearities. Material property distributions follow the Mori–Tanaka model. Shear deformation and rotary inertia are incorporated using the Timoshenko theory. A slight curvature is included to account for a geometric imperfection. The coupled axial/transverse/rotational motion model is developed using Hamilton's energy/work/energy loss. Galerkin's method is used, without neglecting the longitudinal inertia/displacement, and a large dimensional discretized/truncated model is obtained. Numerical integrations, using a continuation-based technique, are employed for force/frequency diagrams. Viscosity, material gradient index, and imperfection effects on the system vibrations are investigated.

Quasi-static cyclic testing and analytical investigation of steel plate shear walls with different post-tensioned beam-to-column rocking connections

Self-Centering Steel Plate Shear Walls (SC-SPSW) have been recently proposed to help achieve possible immediate occupancy following an earthquake for buildings located in regions of high seismicity. For this purpose, like other recently developed low-damage systems, SC-SPSWs seek to: (1) eliminate damage to the surrounding gravity frame of the seismic force resisting system; (2) provide frame recentering, in order to return the building back to its pre-earthquake near vertical alignment, and; (3) concentrate hysteretic energy to replaceable elements, providing rapid and practical reparability.A testing program was conducted investigating the quasi-static cyclic behavior of SC-SPSW systems detailed with three different beam-to-column connections and two SPSW infill configurations. Specimens consisted of one-third scaled single-bay three-story frames. Results are presented for SC-SPSWs detailed with three different post-tensioned (PT) beam-to-column rocking connections that: either (1) rock about the top and bottom beam flanges; (2) rock about the top beam flanges only referred to as the NewZ-BREAKSS connection, or; (3) rock about the beam centerline. The latter two connections were investigated to mitigate floor system damage due to PT boundary frame expansion (i.e., beam-growth) that occurs with the first mentioned rocking connection detail. Experimental and numerical results are presented for both a solid infill web plate and infill web strip configuration. Results presented show that solid infill web plates provide some additional strength and energy dissipation due to their strength in compression and affects frame recentering for quasi-static loading conditions of progressive increasing displacement cycles. In contrast, infill web strips are essentially tension-only and have the benefit that they are not susceptible to tearing from the boundary frame and does not affect frame recentering. Numerical results show that the compression strength of the solid infill web plate in these experiments was approximately 10 percent that of the yield strength of the infill web plates in tension. Furthermore, lessons learned from this testing program are presented that show that for a multiple actuator configuration along the height of a multi-story SPSW specimen, a top level displacement control with force control of the lower actuators is necessary to avoid undesired actuator interaction along the frame heights.

Isogeometric shape optimization of nonlinear, curved 3D beams and beam structures

Straight beams, rods and trusses are common elements in structural and mechanical engineering, but recent advances in additive manufacturing now also enable efficient freeform fabrication of curved, deformable beams and beam structures, such as microstructures, metamaterials and conformal lattices. To exploit this new design freedom for applications with nonlinear mechanical behavior, we introduce an isogeometric method for shape optimization of curved 3D beams and beam structures. The geometrically exact Cosserat rod theory is used to model nonlinear 3D beams subject to large deformations and rotations. The initial and current geometry are parameterized in terms of NURBS curves describing the beam centerline and an isogeometric collocation approach is used to discretize the strong form of the balance equations. Then, a nonlinear optimization problem is formulated in order to optimize the positions of the control points of the NURBS curve that describes the beam centerline, i.e., the geometry or shape of the beam. To solve the design problem using gradient-based algorithms, we introduce semi-analytical, inconsistent analytical and fully analytical approaches for calculation of design sensitivities. The methods are numerically validated and their performance is investigated, before the applicability and versatility of our 3D beam shape optimization method is illustrated in various numerical applications, including optimization of an auxetic 3D metamaterial.

Estimated radiation risk of cancer from dental cone-beam computed tomography imaging in orthodontics patients

BackgroundRadiation dose evaluation is important to cone-beam computed tomography (CBCT) for routine orthodontic treatment planning, especially for a significant proportion of children in orthodontic patients. This study evaluated the patient radiation dose and estimated the radiation cancer risk on dental CBCT according to the calculations by the Monte Carlo simulation method.MethodsThe dental CBCT scanner evaluated in this project was the i- CAT® (Imaging Sciences International Inc., PA, U.S.A.) device. Organ doses and effective doses were calculated by using personal computer-based Monte Carlo simulation (PCXMC 2.0 Rotation) software. The cancer risk resulting from the exposure to ionizing radiation was estimated by using the BEIR VII (Biologic Effects of Ionizing Radiation VII) report model, and the risk of exposure-induced death (REID) was assessed by PCXMC 2.0 Rotation software.ResultsThe largest contribution to the organ dose and effective dose at Zref 83 cm positioned in the dental CBCT x-ray beam centerline was from the salivary glands (738.29μGy, 7.38 μSv). The different organ doses showed the maximum values at the different Zref locations, and the largest contribution to the organ dose and effective dose of all simulated positions was from the thyroid (928.77μGy, 37.5 μSv). The REID values in the 10-year olds (22.6 × 10− 7, female; 19 × 10− 7, male) were approximately double than those in 30-year olds (10.4 × 10− 7, female; 8.88 × 10− 7, male) for all cancers. The highest change during age range from 10 to 30 was shown in breast cancer of females.ConclusionsAlthough individual cancer risk estimates as a function of gender and age are small, the concern about the risks from dental CBCT is related to the rapid increase in its use for orthodontic practice, especially in children patients.

Fully isogeometric modeling and analysis of nonlinear 3D beams with spatially varying geometric and material parameters

We present a fully isogeometric modeling and simulation method for geometrically exact, nonlinear 3D beams with spatially varying geometric and material distributions, both along the beam axis and through its cross-section. The approach is based on the modeling of 3D beams using the Cosserat rod theory and the numerical discretization using B-Spline and NURBS parameterizations in an isogeometric collocation method. Transversally varying material constitutions are represented using non-homogeneous, functionally graded beam cross-section definitions such as laminates and continuously graded cross-sections. Furthermore, to model the axial variation of material and geometry, we introduce the parameterization of cross-section properties as spline curves along the beam centerlines. This fully isogeometric modeling and analysis concept, which is based on spline parameterizations of initial beam centerline curves, kinematic unknowns and axially varying material and geometric parameters, has various practical applications enabled by advances in manufacturing technology, including multi-material 3D printing and advanced manufacturing of composites with automated fiber placement. We verify and demonstrate the modeling and simulation approach using several numerical studies and highlight its practical applicability.

Free vibrations of rotationally restrained nonhomogeneous circular beams by means of the Green function

A one-dimensional linear model is developed to tackle the free vibrations of nonhomogeneous circular beams. The material distribution can depend symmetrically on the cross-sectional coordinates. It can be either continuous (functionally graded materials) or constant in the layers that constitute the beam (multi-layered material). The end supports are identical rotationally restrained pins by means of linear rotational springs. The extensibility of the beam centerline is incorporated into the mechanical model. We determine the Green function matrix of the related problem in closed form. With this in hand, a numerical technique (based on the Fredholm theory) is used to tackle the vibratory problem. Thus, it is possible not only to get the eigenfrequencies but also to find out how the spring stiffness affects the dynamic behavior. Graphical results contribute to the better understanding of the issue.

A Review on Stress and Deformation Analysis of Curved Beams under Large Deflection

The paper presents a review on large deflection behavior of curved beams, as manifested through the responses under static loading. The term large deflection behavior refers to the inherent nonlinearity present in the analysis of such beam system response. The analysis leads to the field of geometric nonlinearity, in which equation of equilibrium is generally written in deformed configuration. Hence the domain of large deflection analysis treats beam of any initial configuration as curved beam. The term curved designates the geometry of center line of beam, distinguishing it from the usual straight or circular arc configuration. Different methods adopted by researchers, to analyze large deflection behavior of beam bending, have been taken into consideration. The methods have been categorized based on their application in various formats of problems. The nonlinear response of a beam under static loading is also a function of different parameters of the particular problem. These include boundary condition, loading pattern, initial geometry of the beam, etc. In addition, another class of nonlinearity is commonly encountered in structural analysis, which is associated with nonlinear stress-strain relations and known as material nonlinearity. However the present paper mainly focuses on geometric nonlinear analysis of beam, and analysis associated with nonlinear material behavior is covered briefly as it belongs to another class of study. Research works on bifurcation instability and vibration responses of curved beams under large deflection is also excluded from the scope of the present review paper.

Geometrically Exact Finite Element Formulations for Slender Beams: Kirchhoff–Love Theory Versus Simo–Reissner Theory

The present work focuses on geometrically exact finite elements for highly slender beams. It aims at the proposal of novel formulations of Kirchhoff-Love type, a detailed review of existing formulations of Kirchhoff-Love and Simo-Reissner type as well as a careful evaluation and comparison of the proposed and existing formulations. Two different rotation interpolation schemes with strong or weak Kirchhoff constraint enforcement, respectively, as well as two different choices of nodal triad parametrizations in terms of rotation or tangent vectors are proposed. The combination of these schemes leads to four novel finite element variants, all of them based on a C1-continuous Hermite interpolation of the beam centerline. Essential requirements such as representability of general 3D, large-deformation, dynamic problems involving slender beams with arbitrary initial curvatures and anisotropic cross-section shapes or preservation of objectivity and path-independence will be investigated analytically and verified numerically for the different formulations. It will be shown that the geometrically exact Kirchhoff-Love beam elements proposed in this work are the first ones of this type that fulfill all the considered requirements. On the contrary, Simo-Reissner type formulations fulfilling these requirements can be found in the literature very well. However, it will be argued that the shear-free Kirchhoff-Love formulations can provide considerable numerical advantages when applied to highly slender beams. Concretely, several representative numerical test cases confirm that the proposed Kirchhoff-Love formulations exhibit a lower discretization error level as well as a considerably improved nonlinear solver performance in the range of high beam slenderness ratios as compared to two representative Simo-Reissner element formulations from the literature.