Three-dimensional Elastic Beam Research Articles

Numerical analysis of the mechanical behavior of a ring-spinning triangle using the Finite Element Method

The purpose of this study is to investigate the quantitative relationships between the mechanical performance of a ring-spinning triangle and the spinning parameters by using the Finite Element Method (FEM). The constituent fibers in the ring-spinning triangle are considered as three-dimensional elastic beam elements with tensile, compressive, torsional, and bending capabilities. The accuracy of the proposed FEM model in calculating the fiber tension distribution of the spinning triangle has been validated by comparing it with earlier models and experimental data. Compared with the earlier models, some important theoretical factors ignored previously, including the fiber torsional strain and the frictional contact of fibers with the bottom roller, are considered in the current model. With the input of various spinning parameters, some essential factors of the spinning triangle, including the fiber tension distribution and fiber torsion distribution, are numerically examined and their quantitative relationships are discussed in detail.

Atomistic finite element model for axial buckling of single-walled carbon nanotubes

An atomistic finite element model is developed to study the buckling behavior of single-walled carbon nanotubes with different boundary conditions. By treating nanotubes as space-frame structures, in which the discrete nature of nanotubes is preserved, they are modeled using three-dimensional elastic beam elements for the bonds and point mass elements for the atoms. The elastic moduli of the beam elements are determined via a linkage between molecular mechanics and structural mechanics. Based on this model, the critical compressive forces of single-walled carbon nanotubes with different boundary conditions, geometries as well as chiralities are obtained and then compared. It is indicated that at low aspect ratios, the critical buckling load of nanotubes decreases considerably with increasing aspect ratios, whereas at higher aspect ratios, buckling load slightly decreases as the aspect ratio increases. It is also indicated that increasing aspect ratio at a given radius results in the convergence of buckling envelops associated with armchair and zigzag nanotubes.

Vibrational analysis of single-walled carbon nanotubes using beam element

Vibrational analysis of single-walled carbon nanotubes (SWCNTs) is performed using a finite element method (FEM). To this end, the vibrational behavior of bridge and cantilever SWCNTs with different side lengths and diameters is modeled by three-dimensional elastic beams and point masses. The beam element elastic properties are calculated by considering mechanical characteristics of the covalent bonds between the carbon atoms in the hexagonal lattice. The mass of each beam element is assumed as point masses at nodes coinciding with the carbon atoms. Implementing the atomistic simulation approach, the natural frequencies of zigzag and armchair SWCNTs are computed. It is observed that the findings are in good agreement with the molecular structural mechanics data available in the literature. Then, the computational results are adopted to develop a predictive equation to propose a quick tool for estimating natural frequencies of the SWCNTs with different boundary conditions and geometrical parameters.

Convergence of equilibria of three-dimensional thin elastic beams

A convergence result is proved for the equilibrium configurations of a three-dimensional thin elastic beam, as the diameter h of the cross-section goes to zero. More precisely, we show that stationary points of the nonlinear elastic functional E , whose energies (per unit cross-section) are bounded by Ch , converge to stationary points of the Γ-limit of E/h . This corresponds to a nonlinear one-dimensional model for inextensible rods, describing bending and torsion effects. The proof is based on the rigidity estimate for low-energy deformations by Friesecke, James, and Muller [4] and on a compensated compactness argument in a singular geometry. In addition, possible concentration effects of the strain are controlled by a careful truncation argument.

Finite element modeling of single-walled carbon nanotubes with introducing a new wall thickness

A three-dimensional finite element (FE) model for armchair, zigzag and chiral single-walled carbon nanotubes (SWCNTs) is proposed. By considering the covalent bonds as connecting elements between carbon atoms, a nanotube is simulated as a space frame-like structure. Here, the carbon atoms act as joints of the connecting elements. To create the FE models, nodes are placed at the locations of carbon atoms and the bonds between them are modeled using three-dimensional elastic beam elements. Using Morse atomic potential, the elastic moduli of beam elements are obtained via considering a linkage between molecular and continuum mechanics. Also, a new wall thickness ( = bond diameter) equal to 0.1296 nm is introduced. In order to demonstrate the applicability of FE model and new wall thickness, the influence of tube wall thickness, diameter and chirality on the Young's modulus of SWCNTs is investigated. It is found that the choice of wall thickness significantly affects the calculation of Young's modulus. For the values of wall thickness used in the literature, the Young's moduli are estimated which agree very well with the corresponding theoretical results and experimental measurements. We also investigate the dependence of elastic moduli on diameter and chirality of the nanotube. The larger tube diameter, the higher Young's modulus of SWCNT. The Young's modulus of chiral SWCNTs is found to be generally larger than that of armchair and zigzag SWCNTs. The presented results demonstrate that the proposed FE model and wall thickness may provide a valuable tool for studying the mechanical behavior of carbon nanotubes and their application in nano-composites.

Finite element modeling of single-walled carbon nanotubes

A three-dimensional finite element (FE) model for armchair, zigzag and chiral single-walled carbon nanotubes (SWCNTs) is proposed. The model development is based on the assumption that carbon nanotubes, when subjected to loading, behave like space-frame structures. The bonds between carbon atoms are considered as connecting load-carrying members, while the carbon atoms as joints of the members. To create the FE models, nodes are placed at the locations of carbon atoms and the bonds between them are modeled using three-dimensional elastic beam elements. The elastic moduli of beam elements are determined by using a linkage between molecular and continuum mechanics. In order to evaluate the FE model and demonstrate its performance, the influence of tube wall thickness, diameter and chirality on the elastic moduli (Young's modulus and shear modulus) of SWCNTs is investigated. The investigation includes armchair, zigzag and chiral SWCNTs. It is found that the choice of wall thickness significantly affects the calculation of Young's modulus. For the values of wall thickness used in the literature, the obtained values of Young's modulus agree very well with the corresponding theoretical results and many experimental measurements. Dependence of elastic moduli to diameter and chirality of the nanotubes is also obtained. With increased tube diameter, the elastic moduli of the SWCNTs increase. The Young's modulus of chiral SWCNTs is found to be larger than that of armchair and zigzag SWCNTs. The presented results demonstrate that the proposed FE model may provide a valuable tool for studying the mechanical behavior of carbon nanotubes and their integration in nano-composites.

A co-rotational finite element formulation for buckling and postbuckling analyses of spatial beams

A consistent co-rotational finite element formulation and numerical procedure for the buckling and postbuckling analyses of three-dimensional elastic Euler beam is presented. All coupling among bending, twisting, and stretching deformations for a beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. However, the third-order terms, which are relevant to the twist rate and curvature of the beam axis, are also considered. An incremental–iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A bisection method of the arc length is proposed to find the buckling load. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method and to investigate the effect of third-order terms on the buckling load and postbuckling behavior of three-dimensional beams.

Validation of a finite element model of the Kirschner-Ehmer external skeletal fixation system

Abstract Objective To determine the validity of finite element analysis (FEA) as a means of examining biomechanical properties of the Kirschner-Ehmer external skeletal fixation system. Sample Population 10 paired tibiae harvested from skeletally mature dogs weighing between 30 and 38 kg immediately following euthanasia for reasons unrelated to musculoskeletal disease. Procedure A gap fracture was created in each bone; fragments were stabilized with 3 frame configurations (type I, type II, and type III), using enhanced-profile threaded pins. Each bone-frame construct was tested, using a materials testing machine in 3 modes of testing: axial compression (AC), mediolateral (ML) bending, and craniocaudal (CC) bending, for a total of 9 tests/bone. The elastic limit of the constructs was not exceeded during testing. Mean stiffness values were determined from load-displacement curves. A finite element model of each construct was created, using three-dimensional elastic beam elements, and stiffness values were calculated, using FEA. Correlations between experimental and FEA data then were determined. Results Significant differences in stiffness were seen among all 3 constructs in CC bending and AC, with stiffness increasing with construct complexity. No significant difference in ML bending stiffness was seen between type-II and type-III constructs; however, both were significantly stiffer than the type-I constructs. The experimental and FEA stiffness data were strongly correlated (AC, r = 0.994; ML bending, r = 0.998; CC bending, r = 0.985). Conclusions and Clinical Relevance Strong correlations among experimental and FEA data indicate that FEA is a valid method of comparing stiffness of Kirschner-Ehmer external skeletal fixation constructs. (Am J Vet Res 1999;60:615–620)

A consistent finite element formulation for linear buckling analysis of spatial beams

A consistent co-rotational finite element formulation and numerical procedure for the linear buckling analysis of three-dimensional elastic Euler beam is presented. A mechanism for generating configuration dependent conservative moment is proposed and the corresponding load stiffness matrix is derived. It is assumed that the prebuckling displacements and rotations of the structure and the corresponding deformations of the elements are linearly proportional to the external loading. The prebuckling rotations of the structure are fixed axis rotations or small rotations, and the effect of the prebuckling displacement on transformation matrix for the coordinates transformation can be ignored. All coupling among bending, twisting and stretching deformations for beam element is considered by consistent linearization of the fully geometrically nonlinear beam theory. An inverse power method for the solution of the generalized eigenvalue problem is used to find the buckling load and buckling mode. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

A finite element structural dynamics model of a beam with an arbitrary moving base—Part II: Numerical examples and solutions

A general lumped mass finite element structural dynamics model was developed for a three-dimensional elastic beam with an arbitrary and large base movement in a previous paper. The six degrees of freedom of the base movement can incorporate either a prescribed arbitrary motion of the base or as coupling of the beam with other substructures. The beam can be pretwisted and have a mass centre offset from the elasticity centre in order to model some specific structures such as helicopter blades. Large deflections and small strains of the beam were assumed so that the geometrical nonlinearities are included. The centrifugal stiffness terms caused by the large base movement were specifically considered so that the dynamics model is applicable for both nonlinear and linear analyses. This dynamics model is expected to provide a general and fundamental element for rotating beams and beam-like multibody structures. In this paper, it is applied to solve both an eigenvalue problem of a spinning beam and to simulate the dynamic response of a space-based robotic manipulator arm with complex time-varying base motions. The comparison of the present results with existing solutions verifies the adequacy and the modelling flexibility of the present model. Some significant deficiencies existing in previous models are confirmed. An approximation in a linear dynamics model and its effect on the dynamic response are discussed in detail.

A finite element structural dynamics model of a beam with an arbitrary moving base—Part I: Formulations

A general lumped mass finite element structural dynamics model was developed for a three-dimensional elastic beam with an arbitrary and large base movement. The six degrees of freedom of the base movement can incorporate either a prescribed arbitrary motion of the base or as coupling of the beam with other substructures. The beam can be pretwisted and have a mass centre offset from the elasticity centre in order to model some specific structures such as helicopter blades. The equations of motion were derived using the virtual work principle. Large deflections and small strains of the beam were assumed so that the geometrical nonlinearities are included. The centrifugal stiffness terms caused by the large base movement were specifically considered so that the dynamics model is applicable for both nonlinear and linear analyses. This dynamics model is expected to provide a general and fundamental element for rotating beams and beam-like multibody structures. Numerical examples are presented to validate the model and demonstrate its great modelling flexibility in a companion paper.

Measurement and modeling of in situ lead stiffness of surface mounted packages

An approach to the experimental determination of in situ lead stiffness has been used to measure selected surface mount component (SMC) lead stiffness under simulated service conditions. Good agreement was obtained between the experimental data and three-dimensional elastic beam finite element analysis (FEA). The agreement was achieved through the use of a stiffness matching point, which provided insight into the level of constraint provided by the solder joint at the lead foot. When adequate solder was available, the solder served as a built-in boundary. When there was insufficient solder creep relaxation in the solder because of the high level of stress in the joint permitted rotation of the lead and the solder served as a pinned joint. Location of the stiffness matching point was found to be approximately 0.08 mm (0.003 in) below the solder fillet on the lead for the lateral stiffness component and about 0.15 mm (0.006 in) above the land for the transverse stiffness component.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On a consistent theory, and variational formulation of finitely stretched and rotated 3-D space-curved beams

This paper deals with finite rotations, and finite strains of three-dimensional space-curved elastic beams, under the action of conservative as well as nonconservative type external distributed forces and moments. The plausible deformation hypothesis of “plane sections remaining plane” is invoked. Exact expressions for the curvature, twist, and transverse shear strains are given; as is a consistent set of boundary conditions. General mixed variational principles, corresponding to the stationarity of a functional with respect to the displacement vector, rotation tensor, stress-resultants, stress-couples, and their conjugate strain-measures, are stated for the case when conservative-type external moments act on the beam. The momentum-balance conditions arising out of these functionals, either coincide exactly with, or are equivalent to, those from the “static method”. The incremental variational functionals, governing both the Total and Updated Lagrangian incremental finite element formulations, are given. An example of the case of the buckling of a beam subject to axial compression and non-conservative type axial twisting couple, is presented and discussed.