Beam Theory Research Articles

A semi-analytical method for determining the mode-I delamination R-curve and fiber bridging traction-separation law of Double-double laminates

Delamination, as one of the most prevalent failure modes of composites, was significantly influenced by the interface angles. In contrast to the limited combinations of interface angles found in traditional laminates, DD laminates, with unique sequence [±Φ/±Ψ]rT and diverse interface angles , urgently required extensive experiments and high-precision simulation analyses to delve into the complexities and unique properties of their interlayer performance. However, the calculation process of current method for determining the R-curve and the bridging traction-separation law was time-consuming. This study proposed a simplified semi-analytical method based on the Euler–Bernoulli beam theory that only required recording initial crack length, final crack length, initial compliance and final compliance. The method validation showed that the deviation between the results obtained from the semi-analytical method and the MBT method averaged no more than 2%. In addition, a tri-linear cohesive zone model based on the delamination fracture micro-mechanism was developed combining semi-analytical method. Both the initial load and ultimate load errors between the simulation and experimental results were within 5N, showing the accuracy of the model and the semi-analytical method. This method provided a simple and effective way to study the delamination propagation behaviors of DD laminates.

Stud-concrete interactional model of shear connectors in steel-concrete composite structures

A theoretical calculation method, named stud-concrete interactional model, of load-slip relationship of stud shear connectors in composite structures is presented. This model has been developed based on the elastic foundation beam theory and considers the different stress states of the concrete around studs during the damage process. Two important stud-concrete interaction parameters, called equivalent spring stiffness and yielding slip, are introduced for the first time to quantify the shear contribution of concrete and the relative slip between stud and concrete, respectively. The range of yielding slip has been determined based on a force-flow model and a finite element model. The proposed stud-concrete interactional model has been utilized to derive the load-slip formula considering the elastic, elastoplastic, and plastic stages of stud shear connectors. Experimental results show that the proposed formula are valid and outperform existing formulas. This paper offers new insight into the shear behavior of stud shear connectors.

Petrov-Galerkin method for small deflections in fourth-order beam equations in civil engineering

Abstract This study explores the Petrov–Galerkin method’s application in solving a linear fourth-order ordinary beam equation of the form u ″ ″ + q u = f u^{\prime\prime} ^{\prime\prime} +qu=f . The equation entails two distinct boundary conditions: pinned–pinned conditions on u u and u ′ u^{\prime} , and clamped–clamped conditions on u u and u ″ {u}^{^{\prime\prime} } . To satisfy these boundary conditions, we have built two sets of basis functions. The explicit forms of all spectral matrices were reported. The nonhomogeneous boundary conditions were easily handled using perfect transformations, ensuring the numerical solution’s accuracy. Detailed analysis of the method’s convergence was studied. Some numerical examples were presented, accompanied by comparisons with other existing methods in the literature.

Full-Range Moment–Curvature Relationships for Beams Made of Low-Hardening Aluminium Alloys

Aluminium alloys are characterised by a rounded stress–strain relationship, with no sharply defined yield point. For example, aluminium alloy grades 6061-T6, 6082-T6, and 7075-T6 exhibit low-hardening response, which is close to linear elastic-linear plastic hardening characteristics. Commonly, the behaviour of aluminium alloys is described by Ramberg–Osgood (RO) one-dimensional constitutive relationship in the format of strain in terms of stress. In the case of low-hardening response, an alternative Richard–Abbott (RA) relationship of stress as a function of strain can be used. Both relations are analytically irreversible, but the RA is more appropriate for use in slender beams theory. In the present study, we use the latter function to derive moment as an explicit function of curvature for the sectional relation of beams. Since the obtained relation is expressed via special functions, we also propose its close approximation, which is more useful for practical purposes. It is uncomplicated and reasonably accurate compared to available models. The predictive capabilities of the new moment–curvature models developed in this article are verified with experimental results available in the literature for beams tested under four-point and three-point bending. In the case of four-point beams, predictions show very good agreement with experiments, while for three-point bending of beams, higher discrepancies are observed.

Mechanical Behavior of Multiple Edge‐Cracked Nanobeams by Taking Into Account the Multiple Cracks Effects

ABSTRACTBy exploiting the stress‐driven model, within the Euler–Bernoulli beam theory, a novel nonlocal analytical model is proposed in order to simulate the mechanical behavior of multiple edge–cracked nanobeams by taking into account the multiple cracks effects. According to the present model, the nanobeam is split in correspondence with each of the edge cracks, thus obtaining beam segments, connected to each other by means of massless elastic rotational springs. Firstly, the proposed model is validated by considering experimental data available in the literature, related to bending tests on two cantilever microbeams, each of them containing a single edge crack (i.e., ). Then, the model is employed to simulate a bending test on a cracked cantilever microbeam containing two edge cracks (i.e., ) and a parametric study is performed by varying both the crack depth, the distance between cracks, and the characteristic length of the material in order to investigate the influence of such parameters on the microbeam mechanical response.

Third-order geometric stiffness formulation for improved mesh convergence of thin and wide spatial beams in the generalized strain beam formulation

AbstractThis paper presents the stiffness formulation of a beam element with the relevant third-order nonlinear geometric effects for relatively wide and thin rectangular beams, in particular when loaded in the plane and simultaneously deformed out of the plane. The element is initially straight in its undeformed configuration. The formulation is based on Timoshenko beam theory with nonuniform torsion and Wagner effects. The derivation is carried out by means of the Hellinger–Reissner variational principle with custom interpolation functions. The element is incorporated into the generalized strain beam formulation for multibody systems. Numerical simulations of precision flexure mechanisms show that the use of a single third-order element per flexible member can already yield adequate performance, at a significant reduction of the necessary degrees of freedom and the computation time, compared with using multiple second-order elements in the generalized strain beam formulation.

Large deformation theory of thin steel cantilever beams under free end load cases

Large deformation theory of thin steel cantilever beams under free end load cases

Deformation and Fracture Mechanisms of Thick Hard Roofs in Upward Mining Coalfaces: A Mechanical Model and Its Validation

The safety and efficiency of underground coal mining are threatened by thick hard roofs characterized by large overhang areas, problematic spontaneous caving, and high dynamic load upon their breakage. In this study, a mechanical model of the bearing capacity of thick hard roofs in upward mining coalfaces associated with mining activities is built based on bending theories for beams with single generalized displacement and the elastic foundation beam theory. Using this method, we analyze the deformation and fracture mechanisms of a thick hard roof during upward mining. We further derive the mechanical equations of rotational angle, bending moment, shear force, and deflection of the free overhang and coal-bearing zone in the thick hard roof and an equation for calculating the limiting span. The mechanical behaviors of the thick hard roof bearing state are analyzed under different parameters. The results show that the foundation coefficient, roof thickness, and angle of upward mining have little influence on the roof bending moment but are positively correlated to the limiting span. Roof load and overhang length have a significant influence on the roof bending moment. They are negatively and positively correlated with the limiting span, respectively. Finally, a case study is performed on the Ш601 upward mining coalface in the Zhuzhuang Coal Mine. The distribution characteristics of the bending moment of the thick hard roof at different extraction stages are analyzed. At each stage, the limiting spans of the thick hard roof upon breaking were calculated as 13.18, 18.82, and 22.50 m, respectively, being close to the on-site measured periodic weighting lengths of 13.33, 19.33 m, and 22.67 m. This close fit proves the feasibility and accuracy of the developed mechanical model. The present study offers theoretical guidance for estimating the weighting length of thick hard roofs in coalfaces and for engineering technology control in similar scenarios.

Uncertainty quantification in the modal analysis of aircraft stiffeners: A Perturbation Technique approach in SFEM

Structural members, such as stiffeners, are crucial in aeronautical structures, providing essential dynamic stiffness. However, variability introduced by manufacturing and assembly processes, as well as material inconsistencies, can lead to uncertainties in structural performance. It is crucial to incorporate these uncertainties into structural analysis to ensure reliable design. This paper utilizes the Stochastic Finite Element Method (SFEM) to address uncertainties in typical structural members used in aeronautics. The Perturbation Technique, based on Taylor series expansions, was employed to model uncertainties in aircraft stiffeners. The study focuses on natural frequencies and modal analysis to evaluate the impact of uncertainties on beams with hat and Z sections, commonly used as stiffeners in aircraft panels. These stiffeners were modeled using the Timoshenko beam theory, and sensitivity analysis was performed to identify key contributors to variability. The perturbation parameter was validated through Monte Carlo simulations. Sensitivity analysis, employing gradient-based methods, identified significant factors affecting variability in natural frequencies. A different perturbation parameter was necessary based on the stiffener's geometry: thickness variations required a perturbation parameter on the order of 10−3, whereas dimensions changes in the flange and height required parameters on the order of 10−2. These results underscore the importance of choosing appropriate perturbation magnitudes to avoid inaccuracies in the deterministic frequency response. Once a perturbation parameter is established, it can be applied to similar regions, ensuring the robustness of the SFEM methodology in analyzing the dynamic response of aeronautical structural reinforcements.

Overcoming stretching and shortening assumptions in Euler–Bernoulli theory using nonlinear Hencky’s beam models: Applicable to partly-shortened and partly-stretched beams

This paper addresses the challenges of the Euler–Bernoulli beam theory regarding shortening and stretching assumptions. Certain boundary conditions, such as a cantilever with a horizontal spring attached to its end, result in beams that partly shorten or stretch, depending on the spring stiffness. The traditional Euler–Bernoulli beam model may not accurately capture the geometrical nonlinearity in these cases. To address this, nonlinear Hencky’s beam models are proposed to describe such conditions. The validity of these models is assessed against the nonlinear Euler–Bernoulli model using the Galerkin method, with examples including cantilever and clamped-clamped configurations representing shortened and stretched beams. An analysis of a cantilever with a horizontal spring, where stiffness varies, using the nonlinear Hencky’s model, indicates that increasing horizontal stiffness stiffens the system. This analysis reveals a transition from softening to linear behavior to hardening near the second resonance frequency, suggesting a bifurcation point. Despite the computational demands of nonlinear Hencky’s models, this study highlights their effectiveness in overcoming the inherent assumptions of stretching and shortening in Euler–Bernoulli beam theory. These models enable a comprehensive nonlinear analysis of partly shortened or stretched beams.

Nonlinear structural responses of the buried oval steel pipelines crossing strike-slip faults

The permanent ground deformation may lead to bending failure or fractures of the buried steel pipelines. In practical applications, the buried pipelines may also exhibit ovality defects due to soil pressure, gravity of the pipeline, and misalignment of joints, which have the potential to diminish the load-bearing capacity of the buried pipelines. This paper introduces a comprehensive analytical methodology to examine the dynamic behavior of pipelines with different initial ovality defects subjected to strike-slip faults. A modified permissible lateral displacement function is introduced to determine the strain distribution of the deformed pipelines. One may predict the yield displacement of the oval pipelines accurately based on the yield theory of classical beams and the static equilibrium method. Then, the accuracy of the proposed method was validated by developing a finite element model and other results. Good agreement was reached characterized by the yield displacement and the yield strain, respectively. Finally, a parametric analysis was conducted on the effects of different ovality, steel grades, diameter-to-thickness ratios, fault inclination angles, and soil properties on the displacement and strain distributions of the deformed pipeline.

Modeling mechanical behavior of buried pipe suffering from frost-heaving force based on the Winkler Elastic Foundation Beam Theory

Modeling mechanical behavior of buried pipe suffering from frost-heaving force based on the Winkler Elastic Foundation Beam Theory

Structural dynamic characteristics of vertical axis wind and tidal current turbines

Vertical axis turbines have received great attention in both offshore wind and tidal current energy communities considering their advantages of economic design and unidirectional operation. However, their commercialization process is rather slow compared with the development of horizontal axis turbines due to technical challenges resulting from higher loads from unsteady aero/hydrodynamic forces, centrifugal forces and gravity of the structures. These are mainly because, while the inherent unsteady tangential pulls and fatigue loads intensify the structural vibration, aggravating structural safety and fatigue life, the relationship between the geometric parameters and the structural responses of blades remains unclear. Therefore, in order to clarify this issue, in this study, we focus on the modal and transient analysis of vertical axis turbines using multi-body dynamics, geometrically exact beam theory and detached eddy simulation. We find that the natural frequency of the rotor is directly related to the ratio of blade length and arm length. The effect of arms cannot be ignored in the structural analysis of the turbine. Moreover, an increase of blade length intensifies the deformation and vibration of the turbine’s structure, making the turbine more prone to fatigue failure. This article is expected to extend the existing knowledge of wind and tidal current turbines and provide a reference for choosing proper design parameters and developing suitable-capacity vertical axis turbines.

Vibration response of nanobeams subjected to random reactions

Nanobeams composed of materials exhibiting flexoelectric properties have been successfully used in advanced technological equipment, including electronic circuits and very sensitive sensors, owing to their remarkable unique effects. Therefore, it is essential to determine their mechanical behavior as a practical need. This study employs an analytical methodology to provide a precise solution to the vibration issue of nanobeams under random stationary loads. Under such circumstances, the nanobeam is influenced by both the temperature and moisture conditions simultaneously. Additionally, the beam is supported by a viscoelastic foundation that considers both the viscous resistance parameter and the elastic parameter. Analytical formulations are derived by integrating classical beam theory with nonlocal strain gradient theory in order to elucidate the impact of size effects on nanobeams. This research also demonstrates the reliability verification, which clearly validates the accuracy of the calculation formula used in this work. This study examines the impact of material parameters, viscoelastic foundation, temperature, and moisture on the displacement spectrum at the middle and entire length of the beam. The study also explores how the flexoelectric effect decreases the displacement in the nanobeam. Subsequently, we provide scientifically derived findings that have significant relevance for building practical nanobeam specifications.

A High-Order Meshless Energy-Preserving Algorithm for the Beam Equation

A High-Order Meshless Energy-Preserving Algorithm for the Beam Equation

Detection of Crack Location and Crack Depth in Leaf Spring

The objective of this work is to find the crack location and crack depth in leaf spring of an army jeep. The spring is subjected to heavy jerks and vibration during the military operation. Therefore it is necessary to find the crack in a leaf spring. Crack in a leaf spring introduces local flexibility that would affect the vibration response of the leaf spring. Main problem is to detect existence of a crack together with its location and depth in the leaf spring .We used a natural frequency a basic criteria for crack detection. Analytical calculations are done to find out natural frequency of leaf spring by using Euler's beam theory. Ansys 16.0 is used to find out natural frequency along with experimentation on FFT analyzer. Now, We propose a method based on some set of fuzzy rules obtained from the information supplemented by Numerical Analysis. Fuzzy controller use comprises of three input variables (first, second and third natural frequency) and two output variables (relative crack length and relative crack depth) are generated with Triangular MF.

Modeling Method of Meshing Stiffness of High Speed Spur Gears

As one of the most important dynamic excitation sources, the gear time-varying mesh stiffness is the key parameter of gear system dynamic model. So how to calculate the gear mesh stiffness accurately is of great importance. Based on Euler beam theory, this paper proposes a primitive calculation algorithm (PCA) to calculate the time-varying mesh stiffness of spur gears. A simplified gear model is developed based on nonlinear strain-displacement relationships, and a finite element program is used to simulate the gear's meshing stiffness. The rationality of this method was verified through finite element analysis. The results indicate that mesh stiffness plays a crucial role in controlling mesh deformation. In view of this, this study lays a theoretical foundation for further analysis of the vibration and noise of gears.

A Higher-Order Theory for Nonlinear Dynamic of an FG Porous Piezoelectric Microtube Exposed to a Periodic Load

This paper investigates the nonlinear dynamic deflection, natural frequency, and wave propagation in functionally graded (FG) porous piezoelectric microscale tubes under periodic load, hygrothermal conditions, and an external electric field. The piezoelectric material used to make the smart microtubes has pores that may be smoothly changed or uniformly distributed over the tube wall. Here, three types of porosity distribution are taken into consideration. The nonlinear motion equations are constructed using a novel shear deformation beam theory and the modified couple stress theory (MCST). The nonlinear motion equations are solved using the fourth-order Runge–Kutta technique and the Galerkin approach. The effects of various geometric parameters, porosity distribution type, porosity factor, periodic load amplitude and frequency, material length scale parameter, moisture, and temperature on the nonlinear dynamic deflection, natural frequency, and wave frequency of FG porous piezoelectric microtubes are explored through a number of parametric investigations.

Output performance analysis of a piezoelectric micro nuclear battery powered by radioisotopes

Abstract This paper investigates the output performance (output voltage, electrical energy output, power output) of a piezoelectric micro nuclear battery powered by radioisotopes. The theoretical formulations are base on Euler-Bernoulli beam theory and include the effects of load nonlinearity due to radioactive source. By employing extended Hamilton’s principle, the nonlinear electromechanical Lagrange equations are derived then solved by using Runge-Kutta method to obtain the dynamic output response. The results based on present electromechanical dynamic model are validated through direct comparisons with the results in the open literatures. The effects of initial gap, length of the piezoelectric layer, thickness of the collector and load impedance on the output voltage, energy output and power output of the piezoelectric micro nuclear battery are discussed in detail.

Free vibration of flexible two links manipulator with variable cross-sectional areas

This paper demonstrated the free vibration of a planar two-link flexible manipulator with harmonic drivers and non-uniformity in the cross-section of links. A dynamic model has been developed that incorporates flexibilities in both link and joint experiencing harmonic vibration, with joint flexibility modeled as torsional spring-inertia components combination. Two links were modeled based on Von Karman’s nonlinear strain relations and theory of the Euler-Bernoulli beam. The nonlinear governing equations were derived using the extended Hamilton’s principle for planar two-link flexible manipulators with variable cross-sections for each link. The coefficients of the obtained partial differential equations were as a function of spatial coordinates and were reduced to the ordinary differential equations for a family of cross-section geometries with exponentially varying widths of rectangular sections of each link. The frequency equation was obtained and analytical solutions of the vibration were achieved for different exponential shapes of each link (widening and narrowing beam for each link). Natural frequencies and mode shapes were presented for varying cross-section areas of the links. The effects of the end masses, payload, beam mass density, flexural rigidity ratio, and joint frequency were investigated on the mode shapes and natural frequencies when the power of the exponential function of the cross-sectional areas of two links was varied. The numerical results were verified with experimental results.