3D Timoshenko Research Articles

A fiber beam element based on plastic and damage models for prestressed concrete structures

Modeling of prestressed concrete beams and their degrading processes are relevant issues in the assessment of existing structures and infrastructures, particularly for bridges. This work focuses on the formulation and validation of a fiber beam finite element for prestressed concrete structural elements that accounts for nonlinear material behavior and time-dependent phenomena. The element is a 3D Timoshenko beam that follows a force-based approach. A fiber discretization of the cross-section is used to model concrete, steel reinforcements and prestressing cables based on damage and plasticity for concrete, and plasticity for steel through three-dimensional constitutive laws. The steel fibers are fully bonded. Strain-softening in the concrete requires an appropriate regularization procedure. The tendons are treated as additional fibers with prestressing applied as initial strain of the relevant fibers. No additional discretization is required except for the description of the tendons’ profile. Time-dependent phenomena such as creep, shrinkage and cable relaxation are taken into account. The proposed model is implemented in the OpenSees computational framework to carry out specific validation tests and demonstrate the element potential.

Estimation of the Mechanical Parameters for a Reduced Coupled Flexural–Torsional Beam Model of a Tall Building by a Sub-Structure Approach

The use of equivalent beam models to estimate the dynamical characteristics of complex tall buildings has been investigated by several authors. The main reason is the structural response estimation to stochastic loads, such as wind and earthquake, using a reduced number of degrees of freedom, which reduces the computational costs and therefore gives the designer an effective tool to explore a number of possible structural solutions. In this paper, a novel approach to calibrate the mechanical and dynamical features of a complete 3D Timoshenko beam, i.e., describing bending, shear and torsional behavior, is proposed. This approach is based on explicitly considering the sub-structures of the tall building. In particular, the frames, shear walls and lattice sub-systems are modeled as equivalent beams, constrained by means of rigid diaphragms at different floors. The overall dynamic features of the tall building are obtained by equating the deformation energy of an equivalent sandwich beam with that of the selected sub-structures. Finally, the 3D Timoshenko equivalent beam parameters are calibrated by minimizing a suitable function of modal natural frequencies and static displacements. The closed form modal solution of the equivalent beam model is used to obtain the response to stochastic loads.

Buckling of tower buildings on elastic foundation under compressive tip forces and self-weight

Buckling of uniformly and not uniformly compressed tower buildings, resting on Winkler type soil, is investigated. An equivalent beam is introduced, able to capture the essential behavior of the building. It is a 3D Timoshenko beam, modeled in the framework of a direct approach, whose constitutive law is derived via a homogenization procedure, which includes the effect of the column prestress. The continuous model is discretized via finite differences, and a linear bifurcation analysis is carried out by solving an algebraic eigenvalue problem. Numerical results are shown for sample problems, aimed at detecting the structural behavior, and illustrating the role of some mechanical parameters. Results supplied by the equivalent beam model are compared with those derived by finite element analyses, carried out on three-dimensional frames.

Free vibration analysis of 3D non-uniform beam: the rigid segment approach

This paper presents the free vibration analysis of 3D non-uniform Euler-Bernoulli and Timoshenko beams by using the rigid segment method. In the first step, the rigid segment model of the free-free beam with the constant cross section was presented. In this model the elastic beam was discretized by three rigid segments which are connected by elastic joints with six degrees of freedom. The absolute coordinates of the rigid segments in accordance to the inertial frame were used which further led to the appearance of redundant coordinates. For this reason, the Lagrange equations of the first kind was used to describe the motion of rigid segments. After elimination of the Lagrange multipliers, differential equations of motion expressed in independent coordinates were obtained. Finally, the non-uniform beam is approximately described with the number of uniform beams. The differential equations of the whole non-uniform beam were obtained by summing all of the previously obtained differential equations of the uniform beams. The precision of the proposed algorithm is verified by numerical examples.

Lateral buckling of pipelines due to internal pressure: A geometrically nonlinear finite element analysis

This work presents an analysis of the lateral buckling of pipelines triggered by internal pressure. Geometrical imperfection amplitudes and friction of pipeline-soil are studied concerning their influence on the critical load and on the post-buckling configuration. To perform the analyses, numerical models are developed using geometrically-exact 3D Timoshenko beam finite elements, which may undergo large displacements and finite rotations. The interaction between the pipeline and the soil is simulated through a contact model. Geometrical imperfections are addressed by imposing prescribed displacements and considering uneven surfaces to represent the soil. The latter, by the way, allow to input concomitant vertical and lateral imperfections to the models. Varying vertical and lateral imperfection amplitudes and soil friction coefficients, the results show that all the parameters together influence the lateral buckling concerning critical loads and shape, order and amplitude of post-buckling configurations of the pipeline. Comparisons with analytical models are also presented.

An efficient multibody dynamic model of three-dimensional meshing contacts in helical gear-shaft system and its solution

The dynamics of helical gear-shaft systems are characterized by three-dimensional (3D) meshing contacts that have significant variations in the location and size of the contact area, resulting in noise that is unavoidably transmitted to the gearbox through the shaft. Accurate and efficient predictions of the dynamic behaviors of helical gear and shaft are indispensable in reliable and cost-effective gearbox design. Available analytical methods, though computationally feasible, cannot consider multi-point contacts and uneven tooth-load distribution. In contrast, the finite element (FE) method provides a high-fidelity approach to compute the dynamic behaviors of a general gear-shaft system at high expenses of computation. This paper aims to establish a high-efficiency multibody dynamic model for 3D contacts in helical gear-shaft systems, in which the helical gear is pertinently represented under the framework of Arbitrary Lagrangian Eulerian formulation and the shaft is discretized by 3D Timoshenko beam elements. The computational efficiency is greatly improved through the following four steps. First, the low-frequency approximation technique is adopted to reduce the degrees of freedom (DOFs) resulting from the fixed boundary normal modes. Second, under the framework of ALE formulation, only the FE nodes of three meshing tooth-faces are defined as boundary nodes. Then, the dynamic equations and Jacobian matrix are simplified by ignoring the inertial forces associated with deformation. Finally, a two-step algorithm is adopted to accelerate the contact detection process. The accuracy and efficiency of the proposed method are demonstrated through five numerical tests with correlation to commercial nonlinear finite element software.

Equivalent Timoshenko linear beam model for the static and dynamic analysis of tower buildings

• Development of a general framework for evaluation of the equivalent beam stiffness . • Proposal of an innovative, constitutive linear law for the equivalent model. • Identification of a measure for the flexural-to-shear ratio in building deformation. • Derivation of motion equations generalizing the classic 3D Timoshenko model. • Easy-to-use procedure to regain deformation and stress state on the single element. A continuous Timoshenko linear beam model immersed in a three-dimensional space is introduced to study the static and dynamic behavior of tower buildings. A schematization of the building as a periodic system with rigid floors connected by deformable elements (columns and shear walls) is considered. The rigid floors are endowed with six degrees of freedom (three displacements and three rotations). The constitutive equations of the equivalent beam (coarse model) are identified from a discrete model of the three-dimensional frame (fine model) via a homogenization procedure. A complete linear constitutive law is obtained, with axial force coupled with bending and shear force coupled with torsion. The first aim is to investigate the relative importance of the macro-shear and macro-bending contributions to the deformation of the building. Then, the ability of the coarse model to reproduce the local stress distribution of the fine model is checked. Finally, the representativeness of the coarse model for the detection of the natural frequencies of the fine model is analyzed.

Simulation and structural optimization of 3d Timoshenko beam networks based on fully analytic network solutions

This article is concerned with the efficient and accurate simulation and optimization of linear Timoshenko beam networks subjected to external loads. A solution scheme based on analytic ansatz-functions known to provide analytic solutions for the deformation and rotation of a single beam with given boundary data is extended to the full network. It is demonstrated that the analytic approach is equivalent to a finite element (FE) method where only one element with a suitably chosen shape function per beam is required. The solution of the FE-type system provides analytic solutions at the nodes, from which the solutions along the beams can be reconstructed. Consequently analytic solutions for the network can be computed by a numerical scheme without applying a spacial discretization. While the assembly of the local stiffness matrices is slightly more expensive compared to an FE model using, e.g., linear ansatz-functions, the complexity of the solution of the FE-system is not. This is particularly interesting for topology and material optimization problems formulated on the network. In order to demonstrate the efficiency of the approach a numerical comparison to the case of linear ansatz-functions is provided followed by a series of studies with topology and multi-material optimization problems on networks.

Effect of Non-Uniform Torsion on Elastostatics of a Frame of Hollow Rectangular Cross-Section

Abstract In this paper, results of numerical simulations and measurements are presented concerning the non-uniform torsion and bending of an angled members of hollow cross-section. In numerical simulation, our linear-elastic 3D Timoshenko warping beam finite element is used, which allows consideration of non-uniform torsion. The finite element is suitable for analysis of spatial structures consisting of beams with constant open and closed cross-sections. The effect of the secondary torsional moment and of the shear forces on the deformation is included in the local finite beam element stiffness matrix. The warping part of the first derivative of the twist angle due to bimoment is considered as an additional degree of freedom at the nodes of the finite elements. Standard beam, shell and solid finite elements are also used in the comparative stress and deformation simulations. Results of the numerical experiments are discussed, compared, and evaluated. Measurements are performed for confirmation of the calculated results.

Structural optimization with fatigue and ultimate limit constraints of jacket structures for large offshore wind turbines

The purpose of the research presented in this paper is to develop and implement an efficient method for analytical gradient-based sizing optimization of a support structure for offshore wind turbines. In the jacket structure optimization of frame member diameter and thickness, both fatigue limit state, ultimate limit state, and frequency constraints are included. The established framework is demonstrated on the OC4 reference jacket with the NREL 5 MW reference wind turbine installed at a deep water site. The jacket is modeled using 3D Timoshenko beam elements. The aero-servo-elastic loads are determined using the multibody software HAWC2, and the wave loads are determined using the Morison equation. Analytical sensitivities are found using both the direct differentiation method and the adjoint method. An effective formulation of the fatigue gradients makes the amount of adjoint problems that needs to be solved independent of the amount of load cycles included in the analysis. Thus, a large amount of time-history loads can be applied in the fatigue analysis, resulting in a good representation of the accumulated fatigue damage. A reduction of 40 % mass is achieved in 23 iterations using the CPLEX optimizer by IBM ILOG, where both fatigue and ultimate limit state constraints are active at the optimum.

Dynamics of 3D Timoshenko gyroelastic beams with large attitude changes for the gyros

This work is concerned with the theoretical development of dynamic equations for undamped gyroelastic beams which are dynamic systems with continuous inertia, elasticity, and gyricity. Assuming unrestricted or large attitude changes for the axes of the gyros and utilizing generalized Hooke׳s law, Duleau torsion theory, and Timoshenko bending theory, the energy expressions and equations of motion for the gyroelastic beams in three-dimensional space are derived. The so-obtained comprehensive gyroelastic beam model is compared against earlier gyroelastic beam models developed using Euler–Bernoulli beam models and is used to study the dynamics of gyroelastic beams through numerical examples. It is shown that there are significant differences between the developed unrestricted Timoshenko gyroelastic beam model and the previously derived zero-order restricted Euler–Bernoulli gyroelastic beam models. These differences are more pronounced in the short beam and transverse gyricity cases.

A full structural mechanical description of beam elements for rotors in electrical machines

Purpose – The purpose of this paper is to get a more consistent finite element description for three-dimensional (3D) Timoshenko beam elements. It extends the common description of beam elements by modifying the shape functions and considers the warping of the cross-section due to torsion. Design/methodology/approach – The paper builds mainly on a finite element description of 3D Timoshenko beam elements. The implementation of high-order shape functions for torsion is done by adding a seventh degree of freedom to the system. Findings – The results reveal that for some beams, depending on their physical dimensions, the warping of the cross-section has large influence. In comparison to a conventional FE program, the extended finite element description considers the warping and yields more accurate results. Practical implications – An application of the extended finite element description is done with an implementation of the code in MATLAB. The static and dynamic behavior of a rotor in an electrical machine...

Analysis of 3D Timoshenko frames having geometrical and material nonlinearities

A three-dimensional Timoshenko beam is formulated based on the force method and the Hellinger–Reissner variational principle. Material nonlinearities are taken into account by the fiber discretization scheme. The element kinematic is adopted such that makes it appropriate for large displacement analysis, but deformations can only be moderately large. Like the other force-based elements, the proposed beam is first formulated in a basic coordinate system without rigid body motions. Then, the resulted flexibility matrix and resisting force vector are transformed to the global coordinates using a corotational scheme. This finite element analysis is locking-free, because of the exact force interpolation functions. In addition, a 3D extension of the “Curvature and Shear Based Displacement Interpolation” or CSBDI, proposed by Jafari et al., is presented in the paper [10]. To facilitate the numerical implementation of the element, a clear algorithm for state determination of the element is presented. Various numerical examples evaluate the accuracy and efficiency of the suggested element.

A new approach for dynamic modeling of n-viscoelastic-link robotic manipulators mounted on a mobile base

In this article, the mathematical modeling of a robotic system composed of n flexible links and a mobile platform has been considered. Most of the mechanical systems including the nonholonomic constraints are analyzed by Lagrangian formulation and its associated “Lagrange multipliers.” Eliminating these variables from the obtained equations is a time-consuming and cumbersome task. So, the Gibbs–Appell formulation and the assumed mode method are used to make the derivation of motion equations easier. Also, to model the system thoroughly and accurately, the dynamic interactions between the manipulator and the mobile platform and the coupling effects arising simultaneously from large motions and small deflections are taken into consideration. The links (assumed as 3D Timoshenko beams) undergo tension-compression, torsion and spatial bending (in two directions), while the effects of internal and external damping (as dissipative forces) are also considered in the mathematical modeling. A systematic approach is developed based on the derived formulation to establish the dynamic equations of motion. In order to alleviate the computational complexity of the suggested method, all the mathematical operations are carried out through \(3\times 3\) and \(3\times 1\) matrices only. Finally, this method is applied to a mobile manipulator with two flexible links to demonstrate the ability of the proposed method in deriving the equations of motion of such complex systems.

Nonlinear Vibration Analysis of an Elastic Rotor Supported on Angular Contact Ball Bearings Considering Six Degrees of Freedom and Waviness on Balls and Races

Nonlinear vibration analysis of an elastically deformable shaft supported on two lubricated angular contact ball bearings is reported herein considering six-degrees of freedom (6-DOF) and waviness on races and balls. This is an extension work of the investigation published by the authors Babu, C. K., Tandon, N., and Pandey, R. K., 2012, “Vibration Modeling of a Rigid Rotor Supported on the Lubricated Angular Contact Ball Bearings Considering Six Degree of Freedom and Waviness on Balls and Races,” ASME J. Vib. Acoust., 134, p. 011006. Elastic deformation of shaft, frictional moment, and waviness on races and balls have been incorporated in the model for the vibration investigations of rotor's CG. Two noded 3D Timoshenko beam element having 6-DOF has been employed in the computation of the shaft's deformation. Governing equations with appropriate boundary conditions have been solved using 4th order Runge–Kutta method. It is observed that vibration amplitude enhances considerably after incorporating the elastic deformation in comparison to the amplitude achieved using rigid rotor model approach. Moreover, the influence of outer race's radial waviness is large on the amplitudes of vibrations in comparison to radial waviness of inner race. However, it is worth noting here that in case of rigid rotor model the presence of radial waviness on inner race yields high amplitudes of vibrations.

Free vibration analysis of inflatable beam made of orthotropic woven fabric

The free vibration of inflatable beams was studied using the dynamic stiffness method. A 3D Timoshenko beam with a homogeneous orthotropic woven fabric (OWF) was considered. Using the usual total Lagrangian form of the virtual work principle, the model took the geometric nonlinearities and the inflation pressure follower force effect into account. The nonlinear equilibrium equations were then linearized around the prestressed reference configuration. The exact dynamic stiffness matrix was developed by directly solving the governing differential equations of a 3D loaded inflatable beam in a free vibration. The effects of the inflation pressure, fabric mechanical properties and the boundary conditions on the natural frequencies and mode shapes of the inflatable beams were demonstrated. The proposed model was validated favorably through its comparison with a 3D thin shell finite element model and an isotropic fabric model found in the literature.

A new 3D Timoshenko finite beam element including non-uniform torsion of open and closed cross sections

In this paper, a new linear-elastic 3D Timoshenko finite beam element, which allows consideration of warping torsion (W-beam), is presented. It is suitable for the analysis of spatial structures consisting of beams with constant open and hollow cross-sections (HCS). The analogy between 2nd-order beam theory (with axial tension) and torsion (including warping) is used for the formulation of equations for non-uniform torsion. The effect of the secondary torsional moment and of the shear forces on the deformations is included in the local finite beam element stiffness matrix. The warping part of the first derivative of the twist angle is considered as an additional degree of freedom at the nodes of the finite elements. This degree of freedom represents the part of the twist angle that is caused by the bimoment. Results of the numerical experiments are discussed, compared, and evaluated. Measurements were performed for confirmation of some of the calculated results. They have shown the importance of the inclusion of warping in stress-deformation analyses of HCS beams.

Exact solution of 3D Timoshenko beam problem: problem-dependent formulation

Problem-dependent interpolation functions for displacements and rotations are obtained from the exact analytical solution of the 3D Timoshenko beam problem by introducing a full set of boundary conditions. The developed methodology allows us to derive a new solution that coincides with the classical result of the engineering beam theory. In addition, the proposed interpolation enables exact strain recovery at any point within the problem domain.

Numerical buckling analysis of an inflatable beam made of orthotropic technical textiles

This paper is devoted to the linear eigen and nonlinear buckling analysis of an inflatable beam made of orthotropic technical textiles. The method of analysis is based on a 3D Timoshenko beam model with a homogeneous orthotropic woven fabric. The finite element model established here involves a three-noded Timoshenko beam element with C0-type continuity for the transverse displacement and quadratic shape functions for the bending rotation and the axial displacement. In the linear buckling analysis, a mesh convergence test on the beam critical load was carried out by solving the linearized eigenvalue problem. The stiffness matrix in this case is generally assumed not to be a function of displacements, while in the nonlinear buckling problem, the tangent stiffness matrix includes the effect of changing the geometry as well as the effect of the stress stiffening. The nonlinear finite element solutions were investigated by using the straightforward Newton iteration with the adaptive load stepping for tracing the load–deflection response of the beam. To assess the effect of geometric nonlinearities and the inflation pressure on the stability behavior of inflatable beam: a simply supported beam was studied. The influence of the beam aspect ratios on the buckling load coefficient was also pointed out. To check the validity and the soundness of the results, a 3D thin-shell finite element model was used for comparison. For a further validation, the results were also compared with those from experiments at low inflation pressures.

Experimental Modeling Using Pseudo Mode Shape Method

The foundations of most large industrial machines are complicated in configuration and shape that result in difficulties of modal testing and finite element modeling. Pseudo Mode Shape Method (PMSM) needs only the measurements of frequency response functions at the joint interfaces of the substructure and the mother structure to develop the equivalent dynamic matrices (called the pseudo matrices) of mass, damping, and stiffness of the substructure, which greatly simplifies the modeling procedure of the complicated substructure. Experimental validation of PMSM was conducted by modeling the foundation of a rotor-bearing-foundation system. The foundation is regarded as the substructure and modeled by PMSM. The rotor is the mother structure and modeled by finite element method using 3D Timoshenko beam elements. The effects of rigid body modes of PMSM in this experiment are also investigated.