Curved Timoshenko Beam Research Articles

Simultaneous solution of implicitly defined curved, linear Timoshenko beams in two‐dimensional bulk domains

AbstractA novel formulation of curved Timoshenko beams which are defined by all level sets of a scalar function in a two‐dimensional bulk domain is presented, including a corresponding finite element method for the simultaneous solution of these beams. The required geometrical setup and differential operators are introduced and the model is defined in strong form. The weak form for the simultaneous solution involves the co‐area formula. A crucial aspect is how the difference vector, describing the rotational deformation in this model, is discretized wherefore two different approaches are shown in this paper. Higher‐order convergence studies in the residual errors confirm that the proposed method is valid and achieves expected convergence rates.

Implementation of variable cross-section curved beam in train-turnout dynamic interactions

The abundance of variable cross-section curved rails in railway turnouts emphasizes the necessity of intricately modeling them, which facilitates a more accurate evaluation of train-turnout interactions. This study presents a general formulation for analyzing both free and forced vibrations of a variable cross-section curved Timoshenko beam and its implementation in train-turnout dynamic interactions. First, the natural frequencies and mode shapes for in-plane and out-of-plane free vibrations of the beam are determined through eigenvalue analysis, taking into careful consideration the characteristics of variable cross-section and curvature. Then, the forced vibration solution is derived using modal superposition and orthogonality. Furthermore, comparative analyses using finite element method (FEM) validate the natural frequencies and dynamic responses of a beam under various boundary conditions, confirming the reliability and accuracy of the proposed method. Finally, the developed beam model is then applied to simulate the switch rail and point rail under train-turnout interactions, revealing the differences from existing methods that modeled these components as uniform cross-section straight beams. Numerical analyses provide new insights by comparing wheel-rail forces and rail acceleration. Considering curve and variable cross section characteristics could contribute to a more accurate evaluation of train-turnout dynamic interactions.

Numerical characterisation of FRP curved reinforcement produced by filament winding

Previous studies have demonstrated that an automated filament winding process can conveniently produce shear reinforcement for concrete structures. The method allows the creation of geometrically complex elements suitable for materials optimisation. This is in addition to the advantages of non-metallic reinforcement. On the contrary, the biggest drawback is the anisotropy properties of fibrous materials. Concentrations of stresses and material defects characterise the reinforcement’s curved portions; thus, material efficiency reduction is expected. Generally, the strength of pultruded FRP stirrups is estimated at about 30-40% of the tensile strength in the direction of the fibres. Still, experimental studies have shown that efficiency can be increased due to the manufacturing process and cross-section aspect ratio. This study aims to identify a mechanical model for predicting the bent strength of FRP stirrups fabricated by the filament winding technique. This is here formalised as a non-linear equilibrium problem of a curved Timoshenko beam on an elastic foundation. Numerical results agree with the experimental data available from previous studies, assessing the model’s suitability.

Effect of 3D printing geometrical monocline on behaviors of curved Timoshenko beams based on isogeometric analyses

It has been demonstrated that quite a few 3D printable materials are geometrically monoclinic. In this paper, the two-scale stochastic isogeometric analyses of 3D printed curved beams are carried out to investigate effect of geometrical monocline on global behavior of a curved Timoshenko beam. The two-scale isogeometric variational principle, basis functions, and corresponding governing equations are investigated. A comparison of monoclinic, orthotropic, and isotropic structures under specific loading is made using the proposed method as well as scale model experiments. Results demonstrate that the anisotropic material feature considerably affects global behavior of 3D printing beam. Uncertainty of material property is quantified through Monte Carlo simulations for several representative 3D printed structures under distributed loading. It is indicated that porosity might be the main cause of the material anisotropic property of representative volume element (RVE).

Zero-thermal-expansion metamaterial with broadband vibration suppression

Although conventional metamaterials possess extraordinary properties, they cannot meet the practical requirements of engineering structures subjected to both temperature fluctuations and vibration excitations. To bridge this research gap, this paper proposes a novel dual-functional metamaterial with zero thermal expansion and broadband vibration suppression. Inspired by the thermal mismatch and band gap effects, we present an innovative design strategy that incorporates star-shaped re-entrant lattices and locally rudder-shaped struts with two-phase materials. To effectively guide the coupled design, a theoretical model accounting for stretching-bending deformations is established to predict the thermoelastic behavior of the metamaterial. Moreover, a parameterized dynamic model using the spectral element method is developed to study the vibration characteristics. Particularly, the spectral formulation of curved Timoshenko beams is derived, including in-plane and out-of-plane vibrations. The comparisons between theoretical predictions and finite element simulations validate the accuracy of analytical models in characterizing thermal deformations and vibration responses. Finally, the case studies shed light on the underlying formation mechanisms of zero thermal expansion and multiple band gaps, while also unveiling the effects of geometric parameters on the performance of such dual-functional metamaterials. Our innovative design strategy and analytical methodology not only offer appealing alternatives for engineering applications but also push the boundaries of metamaterial properties by enabling the transition from single- to dual-functionality.

An isogeometric Timoshenko curved beam element with an enhanced representation of concentrated loads

Isogeometric analysis directly employs CAD geometry representation. This approach is very convenient for the analysis of curved beams, since the arbitrarily-curved beams can be analysed with no geometry approximation errors (as far as the CAD model is concerned). CAD models typically use NURBS basis functions, which also introduce higher continuity along an entire computational domain. Although this continuity can be beneficial for analysis in many aspects, it also causes oscillations in numerical solutions when concentrated load is applied inside a domain. This paper extends approach introduced for straight beams based on basis extension to the generally curved Timoshenko beams using adaptive basis enrichment. Extension functions are proposed and the superior performance over standard IGA is documented.

Coupling effects of manufacturing error and flexible ring gear rim on dynamic features of planetary gear

Manufacturing errors widely exist in and deteriorate the dynamic property of planetary gear train (PGT). To solve this problem, the ring gear is often designed with a thin rim to compensate for the effects of manufacturing errors via the elastic deflections of the rim. Existing dynamic models of the PGT only consider the effects of either the elasticity of the rim of the ring gear or the manufacturing errors. The coupling effects of manufacturing errors and the flexible ring gear are ignored. To understand the dynamic behaviors of the PGT better, a dynamic model of the PGT coupled with typical manufacturing error and flexible ring gear is developed in this study. The tooth contact analysis of the ring-planet mesh, which is calculated based on the potential energy method and uniformly curved Timoshenko beam theory, is studied using the influence coefficient method. A numerical algorithm is proposed to solve the integrated dynamic equations of the PGT. Calculated results show that the dynamic features of the PGT are complex, and the load sharing characteristic is improved when the flexible ring gear is incorporated.

Model konačnih elemenata kružno zakrivljene Timošenkove grede za analizu vibracija u ravni

Curved beams are used so much in the arches and railway bridges and equipments for amusement parks. There are few reports about the curved beam with the effects of both the shear deformation and rotary inertias. In this paper, a new finite element model investigates to analyze In-Plane vibration of a curved Timoshenko beam. The Stiffness and mass matrices of the curved beam element was obtained from the force-displacement relations and the kinetic energy equations, respectively. Assembly of the elemental property matrices is simple and without need to transformation matrix because of using the local polar coordinate system. The natural frequencies of curved Euler-Bernoulli beam with large thickness are not sufficiently accurate. In this case, using the curved Timoshenko beam element is necessary. Moreover, the influence of vibration absorber is discussed on the natural frequencies of the curved beam.

Improved Strategy of Two‐Node Curved Beam Element Based on the Same Beam’s Nodes Information

The curved beam with a great initial curvature is the typical structure and applied widely in real engineering structures. The common practice in the current literature employs two‐node straight beam elements as the elementary members for stress and displacement analysis, which needs a large number of divisions to fit the curved beam shape well and increases computational time greatly. In this paper, we develop an improved accurate two‐node curved beam element (IC2) in 3D problems, combining the curved Timoshenko beam theory and the curvature information calculated from the same beam curve. The strategy of calculating the curvature information from the same bean curve in the IC2 beam element and then transferring the curvature information to the two‐node straight beam element can greatly enhance the accuracy of the mechanical analysis with no extra calculation burden. We then introduce the finite element implementation of the IC2 beam element and verify by the complex curved beam analysis. By comparison with simulation results from the straight two‐node beam element in the MIDAS (S2‐MIDAS) and the three‐node curved beam element adopted in the ANSYS (C3‐ANSYS), the simulation results of the typical quarter arc examples under constant or variable curvature show that the IC2 beam element based on curved beam theory is a combination of efficiency and accuracy. And, it is a good choice for analysis of complex engineering rod structure with large initial curvature.

A hybrid tyre model for in-plane dynamics

A ring model for pneumatic and non-pneumatic tyres is presented. In this model, the tread is discretised by means of lumped masses, interconnected by curved Timoshenko beams, ideally assumed massless and unloaded, and linked to the rim by spring-damper elements taking into account the sidewall properties. The equations of motion are derived via Hamilton's variational principle. The hybrid continuous-discrete approach allows to overcome the need of employing expensive methods to solve the governing equations of the systems, since they are split in two different sets. As a result, a closed-form solution for the displacement field of each beams can be provided. A qualitative analysis is carried out for different values of the elastic modulus of the tread. Results for both the normal and the tangential interactions are coherent with the physical behaviour of the real tyre. Finally, the model is validated and compared to already existing ones, showing the best overall agreement with experimental data.

Smooth size design for the natural frequencies of curved Timoshenko beams using isogeometric analysis

Curved thick beams with smoothly variable cross-section size are very common in practical engineering problems. Designing the variable size based on the traditional finite element methods often leads to non-smooth solutions. To guarantee the smoothness of size distribution, an isogeometric analysis (IGA)-based design approach is proposed in this work to optimize the cross-section size of curved Timoshenko beams for natural frequencies. Due to the geometric exactness and high-order continuity of IGA, there is high accuracy of natural frequency prediction and sensitivity analysis for design optimization. It is found that small numbers of design variables lead to parameterization-dependent solutions, while large numbers of design variables induce design fluctuation. To avoid the undesirable design fluctuation, a stability transformation method-based K-S aggregation constraint scheme is proposed to regularize the size distribution and also achieve the stable convergence of optimal solutions. Multiple design studies including the deterministic and reliability-based optimization problems are performed to demonstrate the applicability and effectiveness of the proposed approach.

On the locking free isogeometric formulations for 3-D curved Timoshenko beams

Locking-free isogeometric formulations of 3-D curved Timoshenko beams are studied. In particular, the global B¯ projection method and a mixed formulation are examined and compared, showing equivalently optimal convergence with regard to displacement solutions. These two formulations are proved to be equal when the beam cross-sectional properties are uniform. For beams with non-uniform cross-sectional properties, the mixed formulation is superior in terms of stress recovery. In addition to these two methods, an alternative locking-free formulation, a C0 NURBS element with selectively reduced integration is suggested in this study, making use of the traditional selective reduced integration (SRI) rule designed for Lagrangian elements. This SRI C0 NURBS element is simple to implement, preserves the sparsity of the global stiffness matrix and requires fewer quadrature point evaluations. Most importantly, due to the use of NURBS basis functions, the exact curve geometry is preserved with all three locking-free isogeometric elements. Benchmark problems and illustrative examples with complex curved geometries are examined for a detailed investigation of the considered locking-free elements.

Dynamic response of a curved railway track subjected to harmonic loads based on the periodic structure theory

In this study, the authors have analysed the dynamic response of a curved railway track that is subjected to moving and non-moving harmonic loads. The track is considered as a curved Timoshenko beam supported by periodically spaced discrete fasteners. The displacement and rotation of the curved rail are expressed as the superposition of track modes in the frequency domain. Periodic structure theory is applied to the equations of motion of the curved track, allowing the dynamic response of the track to be calculated efficiently in a reference cell. The effect of the stiffness and damping of the fasteners, the fastener spacing and the radius of curvature on the mobility and decay rate of the track are analysed for non-moving loads on the rail head. The vibration of the rail due to moving loads is also discussed. It is found that the dynamic response of a curved rail with a large radius has the same characteristics as that of a straight track. However, the dynamic response of the track is significantly affected when the radius of curvature becomes small. The radius affects the mobility; it also has an effect on the track decay rate below 2000 Hz and the velocity of the rail in the vertical direction when the radius is smaller than about 15 m and for the lateral direction when it is less than about 30 m. Moreover, the curvature has a significant influence on the vertical/lateral cross mobility, the magnitude of which increases as the radius is reduced. When the radius is larger than 10 m, the amplitude of the lateral vibration under a moving vertical load and the vertical response to a moving lateral load are inversely proportional to the radius.

Free vibration of deep curved FG nano-beam based on modified couple stress theory

Vibration analysis of deep curved FG nano-beam has been carried out based on modified couple stress theory. Material properties of curved Timoshenko beam are assumed to be functionally graded in radial direction. Governing equations of motion and related boundary conditions have been obtained via Hamilton's principle. In a parametric study, influence of length scale parameter, aspect ratio, gradient index, opening angle, mode number and interactive influences of these parameters on natural frequency of the beam, have been investigated. It was found that, considering geometrical deepness term leads to an increase in sensitivity of natural frequency about variation of aforementioned parameters.

Dynamic response of curved Timoshenko beams resting on viscoelastic foundation

Curved beams' dynamic behavior on viscoelastic foundation is the subject of the current paper. By rewritten the Timoshenko beams theory formulation for the curved and twisted spatial rods, governing equations are obtained for the circular beams on viscoelastic foundation. Using the complementary functions method (CFM), in Laplace domain, an ordinary differential equation is solved and then those results are transformed to real space by Durbin's algorithm. Verification of the proposed method is illustrated by solving an example by variating foundation parameters.

Elimination of self-straining in isogeometric formulations of curved Timoshenko beams in curvilinear coordinates

Isogeometric formulations of curved Timoshenko beams in curvilinear coordinates often result in self-straining of membrane and shear strains, due to the discretization of the local displacement field. Self-straining means a failure in exact representation of rigid body motions, which consequently deteriorates response quality. To overcome the difficulty of self-straining, we propose an invariant formulation that discretizes the global displacement field. It turns out that the approximated membrane, shear, and bending strain measures are invariant regardless of initial geometry in the proposed formulation. For effective applications to any arbitrarily curved structures and locking-free formulations to alleviate membrane and shear locking, the proposed invariant formulation is combined with selective reduced integration (SRI) and B̄ projection method. Numerical examples demonstrate the effectiveness and applicability of the proposed invariant formulation, which gives much more accurate results together with both SRI and B̄ projection method, in comparison to the existing isogeometric formulations.

Order reduction method for locking free isogeometric analysis of Timoshenko beams

The order reduction (OR) method is proposed to provide an effective and efficient locking-free formulation in the isogeometric analysis of plane curved Timoshenko beams. The shear and membrane strains are represented by the reduced order B-spline basis functions, and the locking phenomenon can be prevented. The modified strains are equivalent to the original ones in the sense of the least square approximation. Different order reduction strategies are discussed, and it is suggested to use multiple sets of lower order B-spline basis functions to improve the accuracy. The theoretical analysis of the formulation and the computational cost is carried out to assess the efficiency of the OR method. The performance of the proposed method is tested by the straight beam, curved beams with constant and varied curvatures problems, and compared with other methods such as the selective and reduced integration method, the local discrete shear gap method, and the local B̄ method (without strain smoothing). The numerical results show that the proposed method is competitive, which leads to an efficient locking free formulation.

Low-fidelity 2D isogeometric aeroelastic analysis and optimization method with application to a morphing airfoil

Low-fidelity isogeometric aeroelastic analysis has not received much attention since the introduction of the isogeometric analysis (IGA) concept, while the combination of IGA and the boundary element method in the form of the potential flow theory shows great potential. This paper presents a two-dimensional low-fidelity aeroelastic analysis and optimization framework consisting of a closely coupled isogeometric potential flow model and isogeometric curved Timoshenko beam model combined with a boundary layer model. Application of the framework to the optimization of the landing performance for an active morphing airfoil demonstrates the potential of the isogeometric aeroelastic framework.

An analytical solution for out-of-plane deflection of a curved Timoshenko beam with strong nonlinear boundary conditions

This paper presents a simple method for finding the analytical solution for nonlinear boundary problems. The shifting function method is applied to developing the static deflection of an out-of-plane curved Timoshenko beam with nonlinear boundary conditions. Considering the out-of-plane motion of a uniform curved Timoshenko beam of constant radius R and a doubly symmetric cross section, three coupled governing differential equations are derived via Hamilton’s principle. After some simple arithmetic operations, the curved beam system can be decomposed into a complete sixth-order ordinary differential characteristic equation and the associated boundary conditions. An example is given to illustrate the analysis and show that the proposed method performs very well for problems with strong nonlinearity.

Fault feature analysis of planetary gear system with tooth root crack and flexible ring gear rim

Planetary gear transmission has a wide application in different areas due to its advantages such as compactness, large torque-to-weight ratio, reduced noise and vibrations. However, its dynamic responses are much more complex due to the complicated structures and relative motions, which make it difficult in the fault feature extractions at the view point of fault detection. Better understanding on the dynamic features of a planetary gear transmission and the corresponding internal excitation sources will benefit the fault feature extractions. In this paper, an analytical model for mesh stiffness calculation is developed based on the potential energy principle and uniformly curved Timoshenko beam theory, which enables exploring the effects of the tooth root crack fault and the flexible ring gear rim on the dynamic responses. Based on the developed model, the frequency spectrum structures of the planetary gear transmission can be revealed and analyzed theoretically in the presence of tooth crack and flexible ring gear. A case study is performed to demonstrate the effectiveness of the developed model, where the tooth root cracks are seeded in a tooth of the sun, planet, and ring gears. The simulated results indicate that the complicated modulation phenomenon can be observed where the causes of different frequency components can be revealed. This study is expected to be able to give some theoretical guidance on the identification of vibration sources for planetary gear transmissions.