Timoshenko Beam Research Articles

Effect of connection and grouting level on mechanical behavior of pipe roof

ABSTRACT Pipe roof support suffers from engineering defects such as connection and insufficient grouting in tunnel engineering. To analyze the influence of connection and grouting level on the mechanical behavior of pipe roof support during tunnel construction, a series of pure bending tests were conducted to measure the mechanical property of pipe roof, and quantitatively analyze the influence of the connection and grouting level on the bearing capacity and bending stiffness of the pipe roof. In addition, the double shear mechanical analysis model of pipe roof based on the Timoshenko beam theory and the Pasternak elastic foundation theory was established by considering the shear deformation of pipe roof, the supporting effect of surrounding rock of tunnel face and the delayed effect of shotcrete for primary support. Then, the solution methods for pipe roof deflection and internal force were derived based on this model. Based on an engineering case, the deflection and internal force distribution, as well as the effect of connection and grouting levels on the pipe roof were discussed. Corresponding construction suggestions were proposed and had achieved good results in the case. This research can provide reference for the design and construction of pipe roof.

Low frequency bandgap and high stiffness of innovative auxetic metamaterial with negative thermal expansion

Achieving multifunctional integration is crucial in mechanical metamaterials to meet the requirements of structures and equipment serving in complex environments across various engineering fields. In this study, we propose two novel bi-material and tri-material metamaterials based on the original edge-shared quadrangle (EQ/EQO) metamaterial, which exhibit negative thermal expansion (NTE), negative Poisson's ratio (NPR), and bandgap characteristics. The effective thermoelastic properties, including coefficient of thermal expansion (CTE), Poisson's ratio (PR), and Young's modulus, are compared and analyzed through theoretical formulation based on Timoshenko beam theory and numerical simulation using continuum elements with Abaqus. Additionally, finite element analysis is employed to calculate the energy band structures of these metamaterials based on Bloch's theorem, while the transmission characteristic curve verifies the validity of the obtained band structure. We systematically discuss the bandgap characteristics, coupling relationship between bandgaps and thermoelastic properties, as well as the influence of geometrical parameters on both bandgaps and thermoelastic properties. Our results demonstrate that the effective thermoelastic properties of these metamaterials are primarily determined by their frame structure's geometrical parameters, particularly the incline angle. Furthermore, for bi-material metamaterials, their specific stiffness closely relates to their bandgaps; however, for tri-material metamaterials, only the central local resonant unit determines their bandgap behavior. Notably, planar and 3D tri-material metamaterials can concurrently achieve a low-frequency bandgap along with high stiffness as well as NTE/NPR behavior. This work presents a new design strategy for optimizing multifunctional metamaterials' bandgaps while incorporating NTE behavior.

Choosing between commercially pure titanium and Ti-6Al-4V gyroid structures for orthopedic applications: An analysis through Timoshenko beam theory, the Gibson-Ashby model and experimental methods

For decades, research has focused on using gyroid lattice structures to prevent stress-induced osteopenia and improve osseointegration by promoting cell adhesion and transition, particularly in cases where implant removal is not desired after complete healing. Despite these advantages, in-depth research is still needed for innovative biomedical implants to reach optimum functionality. Although commercially pure titanium (Cp-Ti) and Ti6Al4V are both popular materials for standalone bone implants, they are not likely paired within the same application due to mechanical property differences. This study proposes a case-specific workflow to support orthopedic implant design considerations, by analytically and experimentally investigating the mechanical response of additively manufactured lattice gyroid structures in both Cp-Ti and Ti6Al4V with different porosity rates. The experimental results of both materials indicate a significant decrease in the elastic modulus against two employed analytical methods, Gibson-Ashby model and Timoshenko beam theory. This is linked to inherent manufacturing-induced stress concentrations on the physical gyroids. The experimental outcome presents itself significantly closer to the predictions of Timoshenko beam theory compared to Gibson Ashby method. Both gyroid titanium structures are loyal to Timoshenko theory, displaying a combination of multi deformation behaviors instead of one-way deformation. The deformation response of Ti6Al4V showed a 45⁰ shear band while Cp-Ti exhibited smoother deformation during compression. Higher porosity rates led to an increase in ductility due to extended horizontal plateau regions in both Cp-Ti and Ti6Al4V lattice structures. The results provide valuable comparative insight and considerations on the material choice for gyroid lattice structures within bone implants.

Dynamical Modeling and Dynamic Characteristics Analysis of a Coaxial Dual-Rotor System

The dual-rotor structure is the primary excitation source of aero-engine vibration. It is essential to study the dynamical model and analyse the dynamic characteristics such as critical speed and the unbalanced response for rotor system dynamics. The coaxial dual-rotor support scheme of an aero-engine was introduced in the previous work, and the physical model with a high-speed flexible inner rotor of a sizeable length-diameter ratio has been designed. Then a finite element dynamic model based on the Timoshenko beam elements and rigid body kinematics of the dual-rotor system is modelled, with the Newmark-β method and Newton-Raphson method used for the numerical calculation to study the dynamic characteristics of the system. Three different simulation models, including beam-based FE (1D finite element) model, solid-based FE (3D finite element) model, and TM (transfer matrix) model, were designed to study the characteristics of mode and the critical speed characteristic of the dual-rotor system. The unbalanced response of the dual-rotor system was analysed to study the influence of mass unbalance on the rotor system. The effect of different disc unbalance phases and different speed ratios on the dynamic characteristics of the dual-rotor system were investigated in detail. The experimental result shows that the beam-based FE model is effective and suitable for studying the dual-rotor system. Conflict of Interest Statement The authors declare no conflicts of interest.

Iterative method for large-scale Timoshenko beam models assessed on commercial-grade paperboard

AbstractLarge-scale structural simulations based on micro-mechanical models of paper products require extensive numerical resources and time. In such models, the fibrous material is often represented by connected beams. Whereas previous micro-mechanical simulations have been restricted to smaller sample problems, large-scale micro-mechanical models are considered here. These large-scale simulations are possible on a non-specialized desktop computer with 128GB of RAM using an iterative method developed for network models and based on domain decomposition. Moreover, this method is parallelizable and is also well-suited for computational clusters. In this work, the proposed memory-efficient iterative method is numerically validated for linear systems resulting from large networks of Timoshenko beams. Tensile stiffness and out-of-plane bending stiffness are simulated and validated for various commercial-grade three-ply paperboards consisting of layers composed of two different types of paper fibers. The results of these simulations show that a linear network model produces results consistent with theory and published experimental data

Nonlinear vibration analysis of geometrically imperfect FGM Timoshenko beams with porosity and elastically restrained ends in thermal environments

This paper aims to analyze the combined influences of pore imperfection, initial geometric imperfection, elasticity of tangential constraints of ends, elastic foundations and elevated temperature on the nonlinear vibration of functionally graded material (FGM) beams. The pores are assumed to distribute into FGM according to even and uneven patterns and effective properties of porous FGM are estimated using a modified rule of mixture. Motion equations of geometrically imperfect beams are established within the framework of Timoshenko beam theory incorporating von Kármán nonlinearity and foundation interaction. Analytical solutions are assumed to satisfy simply supported and clamped conditions of beam ends and Galerkin method is applied to derive a nonlinear time ordinary differential equation. The frequencies of nonlinear free vibration are determined employing fourth-order Runge-Kutta numerical integration scheme. Parametric studies are carried out to assess numerous effects on both linear and nonlinear frequencies. The results reveal that frequency nonlinearity is more significant for beams with tangentially restrained ends, especially at elevated temperatures. It is also found that geometric imperfection increases the linear frequency and lowers ratio of nonlinear-to-linear frequencies.

Semi-analytical modeling and vibration analysis of bolted frame structure under non-uniform pretension

Most of the frame structures in mechanical equipment are connected by bolts, and the non-uniform pretension among the bolts will have an obvious impact on the vibration characteristic parameters of the frame structure. In this paper, semi-analytical modeling and vibration analysis of a frame structure considering non-uniform pretension of bolts are carried out based on the Timoshenko beam theory and Rayleigh-Ritz method. Specifically, a method is proposed to decompose the frame structure into several substructures and model them respectively, and further the coupling springs are applied among the substructures. Through the vibration test of a beam lap structure with a bolted joint, the analytical expressions between bolt pretension and bolted joint stiffness and damping parameters are obtained. Moreover, the method of simulating the non-uniform pretension among each bolted joint is proposed. Considering the external factors such as reinforcing rib structures and flexible rod exciter, and a complete semi-analytical dynamic model of the bolted frame structure introducing the non-uniform pretension is further established. A case study is carried out on a frame structure that contains four connecting bolts with non-uniform pretension. The natural characteristics of the frame structure are solved by using the developed modeling method, and the rationality of the analysis results is verified by experiments. Based on this, the increasing trends of the resonant frequencies and the resonant response of the frame structure under different pretension conditions are analyzed from two aspects of experiment and simulation.

Analysis of nonlinear wave propagation within architected materials consisting of nonlinear Timoshenko beam structural elements

In the present work, a full nonlinear Timoshenko beam employing nonlinear shape functions is developed. The extended Hamilton principle is employed for deriving the differential equations of motion and the associated boundary conditions. The general form of the boundary conditions is then utilized to determine the static solution of the beam motion. Using this solution for the deformation and rotation of the beam, the nonlinear shape functions of the beam are identified, which leads to the linear and nonlinear mass and stiffness matrices of the Timoshenko beam element. The nonlinear dispersion diagram incorporating the non-linear corrections is obtained using the Linstedt–Poincaré perturbation method. An analysis of the effect of internal transverse shear and bending on the nonlinear dispersion characteristics of wave propagation in two-dimensional periodic network materials made of nonlinear Timoshenko beams is done. The formulated theory shows that the percentage of correction factor of the nonlinear kinematics versus the linear dynamical behavior is inversely proportional to the frequency amplitude. The shear and extension modes are shown to have the higher effect in the non-linear correction term in comparison to the flexural mode.

Theoretical Solutions for Forecasting the Response of the Existing Pipeline Induce by Tunneling underneath

In order to accurately and efficiently assess the impact of tunnel excavation on overlying existing pipeline, an analytical method is proposed to solve this problem. First, the vertical free displacement of the surrounding soil due to tunnel excavation can be derived by the Loganathan formula. Next, the overlying existing pipeline can be treated as a Timoshenko beam resting on the Vlasov foundation model, and the influence of the surrounding soil on the both sides of the existing pipeline is taken into consideration. Finally, an analytical solution for the longitudinal deformation of the existing pipeline can be obtained by using the integral method. Case analysis results demonstrate that the calculated results of this method closely in line with measured data. Compared to the degenerate analytical solution given by this method, the result from this method is more consistent with the measured data. Further parameter studies show that the volume loss rate, diameter of new tunnel, skew angle, and vertical distance between tunnel and pipeline are significant factors affecting the existing pipeline response due to tunneling underlying.

Application of PISA method for monopile design in layered soil

The foundation design for offshore wind turbines (OWTs) is critical to ensuring their structural integrity and long-term stability. Large-diameter monopiles are widely used for OWTs owing to their proven effectiveness. The conventional design approach, derived from the oil and gas industry’s p-y model, has limitations when applied to monopiles for OWTs. Consequently, a novel design method based on the PISA joint industry research project was developed, which extends beyond the traditional lateral soil resistance (p-y curves) and incorporates additional soil reaction components from shaft friction, pile base shear, and base moment. Unlike the simplistic rule-based conventional approaches, the proposed method necessitates site-specific 3D finite element (FE) simulations for precise calibration of soil reaction curves. Although successfully applied in European offshore wind farms, the PISA method’s performance in regions with distinct seabed compositions, such as China, remains to be verified. Therefore, this paper presents a comprehensive case study conducted at the Xiangshan wind farm, a representative site characterized by layered soil. The study employed in-situ investigations (cone penetration test, borehole) and laboratory tests (direct simple shear, bender element) to delineate the seabed soil profile and derive essential soil parameters for accurate foundation design. To facilitate the design, an Abaqus-based design tool was developed, which automated the extraction of soil reaction curves from the 3D FE simulations, calibration of PISA springs, and execution of one-dimensional beam-spring analyses employing Timoshenko beam theory. Comparative analyses were performed on the computed responses of piles, encompassing load-deflection curves, pile deflections, and bending moment profiles. The outcomes from the 3D FE simulations, uncalibrated rule-based PISA model, and calibrated PISA model were compared to validate the PISA method for monopile design in layered soil conditions.

Saturated Boundary Feedback Stabilization for a Nonuniform Timoshenko Beam With Disturbances and Uncertainties

Saturated Boundary Feedback Stabilization for a Nonuniform Timoshenko Beam With Disturbances and Uncertainties

Theoretical and Numerical Investigation of the Effective Mechanical Properties of an Arc Star-Shaped Auxetic Honeycomb

Background: Auxetic honeycombs have attracted a lot of attention due to their excellent properties, including lightweight, and outstanding impact resistance and energy absorption. Methods: This study focuses on a new type of arc star-shaped honeycomb (ASSH) by replacing the tip angles of the classical star-shaped honeycomb (SSH) with curved edges. The theoretical expressions of the effective Poisson’s ratio and Young’s modulus are deduced by using the Timoshenko beam theory and energy method. Furthermore, the numerical model is also established through the Finite Element Method (FEM); then, both the analytical and computational approaches are adopted to conduct parametric analysis. Results: It was found that the effective mechanical properties obtained by theoretical analysis and the FEM are consistent with each other, and ASSH bears tunable effective Poisson’s ratio and Young’s modulus under varying geometric configurations. Conclusion: The present work may provide some guidance for the design and analysis of future auxetic honeycombs.

Longitudinal analysis model for segment lining uplift during shield tunnelling considering shearing dislocation of circumferential joints

During shield tunnelling, longitudinal deformation occurs in the shield tunnel structure owing to the buoyancy generated by the synchronous slurry, which has adverse effects on the assembly quality and even the structural integrity of the segmental ring structure. However, the current longitudinal beam-like model for analysing the uplift response during shield tunnel construction adopts the Euler–Bernoulli beam to describe the shield tunnel. This model does not consider the shearing dislocation effect, causing errors in the structural longitudinal deformation analysis. In this context, this study proposes a shield tunnelling uplift model based on the Timoshenko beam theory, which can effectively consider the shearing dislocation effect, aiming to evaluate the longitudinal response of shield tunnels more accurately. A Pasternak two-parameter foundation with springs reflecting the resistance of the overburden with different thicknesses was adopted to simulate the supporting effects of the synchronous grouting layer and surrounding soil layer composite on the tunnel, considering the influences of both the shearing effect of the foundation and tunnel buried depth on the shield tunnel uplift. The dynamic buoyancy coefficient, determined through an inverse analysis of the field measurement data, was introduced to accurately calculate the magnitude of the dynamic buoyancy generated by the flow of synchronous slurry. The proposed model was validated by comparing its results with actual measurements. The uplift characteristics of the shield tunnel during construction were obtained by using the proposed model in an actual project. It was found that the longitudinal analysis model for shield tunnel uplift based on the Euler–Bernoulli beam theory underestimates the uplift and overestimates the internal forces generated during shield tunnelling.

Pseudo-static Winkler springs for longitudinal underground structures subjected to shear waves

ABSTRACT Shear waves in underground longitudinal structures impose deformations that need to be accounted for in structural design. Simplified approaches for such structures typically consider the Winkler foundation model, assuming Euler-Bernoulli beam theory, which neglects shearing-induced distortions, especially significant when shear waves are a dominant component of the seismic motion. In order to overcome these limitations, Timoshenko beam models have been proposed in the literature. These approaches however depend on an appropriate determination of the ground springs. Existing analytical formulations often assume plane-strain conditions, inadequate for representing low frequencies and thus are not directly applicable for pseudo-static interaction analyses. The present paper develops analytical solutions to determine transverse and rotation Winkler springs for structures subjected shear waves. The proposed springs overcome the drawbacks of plane-strain models and can be construed as a generalisation of them. The springs are obtained as a function of the seismic wavelength and the ground-structure stiffness contrast. Results obtained are validated against solutions from the literature and numerical results from a full 3D finite-element model. A non-dimensional parametric study is also presented, that allow an expedited evaluation of ground springs for practical applications.

Model updating of a shear‐wall tall building using various vibration monitoring data: Accuracy and robustness

SummaryAcceleration measurements are often used for model updating of civil engineering structures, especially in the case of seismic monitoring. It is yet unclear if accelerations alone would generate an accurate and robust finite‐element (FE) model. This study examines this notion and analyzes the possibility of using other vibration monitoring data for model updating of shear‐wall tall buildings. This study compares the accuracy and robustness of the FE models being optimized via accelerations, roof displacement, wall rotations, interstory drift ratios, and the linear combination of these measurements. A numerical case study is analyzed using Timoshenko beams for modeling the lateral vibration of a benchmark 42‐story building under seismic excitations. Results show that the acceleration response of the examined building is mostly governed by its higher vibration modes. Depending on the characteristics of ground motions, using accelerations alone may generate an FE model biased towards higher‐order modes without effectively capturing the lower‐order modes. For instance, the first modal frequency of the updated FE model could be 12.0% lower than the true value, and the reconstructed displacement and rotation responses are noticeably inaccurate. Employing multi‐source monitoring data for model updating, for example, the combinations of roof displacement and acceleration measurements, could reduce the normalized root‐mean‐square errors in displacements by more than 70%. This study also quantifies the robustness of the FE model under various measurement noise levels and 50 pairs of earthquake records. Finally, the effects of multi‐source data on FE model updating are validated via experiments on a 7‐story shear wall building. Analysis reveals that a more accurate and robust FE model can be determined via a combination of accelerations and top displacement than via acceleration alone.

Vibration analysis of a sandwich Timoshenko beam reinforced by GOAM/CNT with various boundary conditions using VIM

In the present study, a sandwich beam reinforced by carbon nanotubes (CNTs)/graphene origami auxetic metamaterials (GOAM) and porous cores is studied. Various distributions of CNTs in facesheets and porosity in the core are stated in terms of thickness. The equations of motion for a sandwich beam are derived and solved under various boundary conditions using variational iteration method (VIM), in which, it is a powerful technique with higher convergence with the computational method that gives analytical response to a linear problem. The effect of the types of CNT distributions, porosity coefficient, types of porous cores, porosity parameter, weight fraction and folding degree of GOAM, different geometric dimensions and different B.C.’s on the natural frequency is investigated. The obtained results are compared with the other literature, that there is a good agreement between them. On the other hands, the maximum error percentage of the present work (VIM) with Civalek and Kiracioglu [66] (Discrete singular convolution method) for the first five frequencies is equal to 0.3% according to Table 2, while in Table 3, the maximum error percentage of the present work (VIM) with Yas and Samadi [67] (differential quadrature method) for the first frequency with various boundary conditions and nanocomposite material is equal to 0.22%. In this regard, the present study can be a suitable platform to find the natural frequency and prevent the phenomenon of resonance and increase the life of the structure. Moreover, it is seen that with enhancing of weight fraction of GOAM, frequency increases because increasing the stiffness of structures. The importance of this research is to extract the formulation of sandwich beam structure for GOAM under different boundary conditions and compare the results between them by repeating the changes.

Stability and regularity for double wall carbon nanotubes modeled as Timoshenko beams with thermoelastic effects and intermediate damping

This research studies two systems composed by the Timoshenko beam model for double‐wall carbon nanotubes, coupled with the heat equation governed by Fourier's law. For the first system, the coupling is given by the rotation speed of the vertical filament in the beam from the first beam of Timoshenko and the Laplacian of temperature , where we also consider the damping terms fractionals , , and , where . For this first system, we proved that the semigroup associated to system decays exponentially for all . The second system also has three fractional dampings , , and , with . Furthermore, the couplings between the heat equation and the Timoshenko beams of the double wall carbon nanotubes for the second system are given by the Laplacian of the rotation speed of the vertical filament in the beam of the first beam of Timoshenko and the Lapacian of temperature . For the second system, we prove the exponential decay of the associated semigroup for and also show that this semigroup admits Gevrey classes for , and we finish our investigation proving that is analytic when the parameters .

Natural frequencies and stability for Z-shaped fluid-conveying pipes with constrained-layer damping – Semi-analytic modeling with experimental validation

As the common fluid-conveying component, pipe structures are susceptible to excessive vibration due to internal fluid and external excitation. Constrained-layer damping (CLD) is commonly applied to suppress structural vibrations. However, effective modeling and vibration suppression analysis for Z-shaped fluid-conveying pipes are lacking to explore novel dynamic behaviors. This paper presents a dynamic model for analyzing the three-dimensional vibration of Z-shaped fluid-conveying pipes treated by partial CLD based on the semi-analytical method. Utilizing Timoshenko beam theory and calculating the centrifugal, Coriolis, and inertial force of fluids, the energy of straight and curved fluid-conveying pipes treated by CLD is derived. Based on the Lagrangian energy function, the equation of motion is obtained by using the Rayleigh-Ritz method. The improved artificial spring set is deployed to model the interaction of pipe sections and the non-uniform constraint distribution of clamps. Modal parameters are obtained by solving characteristic equations. The proposed method is verified by comparisons with references and modal experiment results. Frequency parameters, damping effects, and stability are analyzed under various fluid, geometric, and CLD parameters. In conclusion, a novel dynamic model is provided for analyzing the vibration and stability of the Z-shaped fluid-conveying pipe treated by CLD, which lays a foundation for vibration suppression research of fluid-conveying pipe systems.

Inverse Differential Quadrature Based Model for Static Behaviour of Variable Stiffness Curved Composite Beams

Recent advancements in fibre placement technologies have expanded the potential applications of variable stiffness curved composite beams in industries such as aerospace, automotive, and naval engineering. Accurate solution techniques for examining these beams, especially those composed of advanced composite materials, are indispensable. In view of this demand, this study proposes a new high-order computational tool that combines the higher-order accuracy of the emerging inverse differential quadrature method (iDQM) and the simple kinematics of the Timoshenko beam theory for efficient and accurate prediction of the static behaviour of both constant stiffness and variable stiffness curved beam structures. This novel application of iDQM to curved beam analysis is leveraged upon its excellent potential to mitigate differentiation-induced errors by using the so-called indirect approximation strategy. Simple procedures for implementing different orders of iDQM models are presented to analyse curved beam problems, and are independently benchmarked against closed-form Navier's solutions, as well as numerical solutions obtained through the differential quadrature method (DQM) and finite element method (FEM), demonstrating excellent spectral accuracy. Furthermore, the iDQM scheme offers outstanding potential in recovering transverse shear stress, achieving superior accuracy over lower-order FEM in approximating higher-order derivatives. Remarkably, iDQM predictions for variable stiffness curved beams exhibit satisfactory agreement with the Strong Unified Formulation, achieving over 98% computational efficiency. Finally, convergence analysis of iDQM solutions reveal up to three orders of improved accuracy and faster convergence rates compared to the DQM, constituting a new benchmark for curved beam analysis.

Free Vibration of Axially Loaded Functionally Graded Porous Cracked Beams

This study employed a method of analysis to examine the free oscillation behavior of functionally graded (FG) porous-cracked beams under axial loads and varied boundary conditions. The cracked beam system was composed of interconnecting beam segments kept together by massless rotating springs. Based on the Timoshenko beam theory (TBT) or Euler–Bernoulli theories (EBT), each segment was sectionally flexible. Using a power-law function, mechanical features were expected to gradually change along with the height of the beam. Two different pore distributions, even and uneven, were also explored. Subsequently, Hamilton’s theory was used to derive the equations of kinematic motion for FG-cracked porous beams, and the transfer matrix (TMM) approach was used to get the determinantal equation. A parametric analysis was performed to evaluate how cracks, porosity distribution, slenderness ratio, volume fraction index, boundary conditions, and axial load affect the dynamic characteristics of FG beams. The results revealed that the suggested analytical approach provided several findings that were similar to the existing analytical outcomes in the literature. Moreover, the computer analysis demonstrated that porosity, especially when there were cracks, significantly affected the eigenfrequencies of FG beams. Hence, this study opened the door for possible applications in a variety of engineering settings by offering some crucial insights into the complex dynamics of FG porous-cracked beams.