Elastic Beam Model Research Articles

Determination of the bridging law for mode I delamination via elastic restraint beam model and equivalent crack method

This article presented a simple procedure to determine the Mode I bridging law for both unidirectional and multidirectional composite laminates. It was based on an ‘Elastic Restrained Beam Model’ which assumed that the boundary conditions at the end of the cracked arms of the double cantilever beam (DCB) are elastically restrained by torsion springs. In this way, the closed-form solution of the DCB has been derived, without the need for a crack-length correction used in the corrected beam theory (CBT). We also derived very accurate data reduction formulas for the energy release rate (GI) and the opening displacement at the pre-crack tip (δ*). Then, the bridging law was obtained by numerical differentiation of GI with respect to δ* using the J-integral approach. Two major advantages of the proposed procedure are that it does not require the measurement of the crack length and separation δ* during the test. The derived bridging laws were implemented into the cohesive zone model (CZM) for numerical simulations and the results were proved to have a great agreement with experimental results, which validated that the proposed procedure is suitable for the determination of Mode I bridging law.

Construction of the bending model of micropolar elastic thin beams with a circular axis and its implementation using finite element method

This paper considers the problem of transition from the system of two-dimensional equations of the micropolar (moment) theory of elasticity in a thin curved area to the one-dimensional system of equations describing deformation of the micropolar elastic thin beam with a circular axis. During the transition process, Timoshenko's hypotheses generalized to the micropolar case are applied. As a result, the applied model (with independent fields of displacements and rotation) of a micropolar elastic thin beam with a circular axis has been constructed. It is shown that the model includes the law of conservation of energy, energy theorems and variation principles. All main functionals for the model of the micropolar elastic thin beam with a circular axis are obtained from the functional of the two-dimensional micropolar theory of elasticity, containing only the first derivatives of displacements and rotations. The finite element method (FEM) is taken to study the boundary problems of statics and dynamics of applied model of the micropolar elastic thin beam with a circular axis. The basic concepts and stages of the FEM are formulated: the discretization, the selection of basic nodal unknowns, the approximation of the solution, and the construction of the basic FEM equations. The finite-element solutions of some problems of statics and problems on natural vibrations of beams with a circular axis are considered according to the micropolar theory of elasticity. A comparative analysis with similar problems of beams with a circular axis according to the classical theory of elasticity is carried out. Based on the results, some effective properties of beams with a circular axis have been established when considering their deformations in the context of the micropolar theory of elasticity.

Analysis of Deformation and Stress Characteristics of Anchored‐Frame Structures for Slope Stabilization

In recent years, anchored‐frame structures are widely being used in road slopes for stabilization and improvement. The technology of frame structure with anchors is becoming more and more mature, but the pertinent theory lags behind the application. While more attention is being paid to the control of deformation, there is still no uniform solution to the calculation of deformation in the anchored‐frame structures. According to the classical laterla earth pressure theory and static equilibrium, this paper improves the calculation method of lateral earth pressure and derives the calculation formula of slope‐induced lateral earth pressure. At the same time, based on the elastic foundation beam model, the columns and beams are treated as a whole system, and the appropriate elastic frame beam model is established. The formula of the deformation and bending moments for the columns and beams in the anchored‐frame structures are derived. Additionally, the calculated results based on the abovementioned newly derived formulas are compared with those of finite element simulations for a simulated case study. The results of simulation and analytical calculation are basically consistent, which prove the feasibility of the new analytical method.

Flexural wave propagation analysis of single-walled carbon nanotubes using molecular structural mechanics approach

In the present study, the flexural wave propagation along single-walled carbon nanotubes (SWCNT) is investigated using molecular structural mechanics approach. The dispersion curves associated to the wave propagation characteristics (i.e., phase velocity versus wave number) of (5,5) and (10,10) armchair test tubes are obtained and compared with the results of the Molecular Dynamics (MD), non-local elastic Timoshenko beam model (NT) and non-local Euler beam model (NE). In addition to the advantages of the present approach such as low computational cost against MD technique and its independency to any extra parameters which have to be experimentally evaluated such as unknown length scales in NT and NE methods, the results show good agreement with the MD results, especially at the high wave number (small wavelength) regimes, which the NT and NE techniques fail to achieve an acceptable predictions.

Analysis and Measurement of Adhesive Behavior for Gecko‐Inspired Synthetic Microwedge Structure

AbstractSynthetic microwedge adhesive inspired by gecko has great advantages in its prominent adhesion property and controllability. Clarifying the adhesion mechanism is significant for designing structure and enhancing performance. However, previous investigations have not given a thorough exposition in the adhesion mechanism of microwedge structure, especially, the complex relationship among shear load, contact area, and normal adhesion. In this paper, aiming to clarify the relationship mentioned above, a novel microwedge adhesive representation combined profile‐constructed elastic beam model with a peel zone radius exact‐solved cylindrical Johnson–Kendall–Roberts (JKR) model is proposed. First, the contact area can be obtained by reconstructing principal profile to the curved neutral surface based on the elastic beam model under different shear loads. Then the cylindrical JKR model with exact‐solved fitting radius of peel zone is formulated to establish the relationship between shear load and normal adhesion of a single microwedge. Further, the effective contact between neighboring microwedges is considered to constrain the single microwedge deformation. Moreover, the microwedge array model takes into account the influence of normal movement, which is expressed with a correction factor based on comparison of theoretical analysis and measurement results. Finally, the model is verified with the simulation and the experimental measurement of two customized apparatuses.

A case study on the height of a water-flow fracture zone above undersea mining: Sanshandao Gold Mine, China

In undersea gold mines, the development of a water-flow fracture zone and its connection with the aquifer may cause massive water and sand inrush disasters. In this study, approaches including theoretical analysis, numerical simulation and field detection are employed to identify the development height of the water-flow fracture zone caused by undersea mining in the Xinli Zone of the Sanshandao Gold Mine to ensure mining safety. An improved Winkler elastic foundation beam model, considering the coupled influences of seawater pressure and backfill support, was established to calculate the height of the water-flow fracture zone. The result demonstrates that the height of the water-flow fracture zone depends on the elastic modulus of the overburden strata and the compression modulus of the filling material. Then, an experimental study utilizing a custom-made apparatus is conducted to obtain the Winkler foundation compression characteristics of the filling material used in the gold mining operation. The theoretical analyses are confirmed by numerical simulations and show that the height of the water-flow fracture zone decreases with the increase in mining level because the loads from overburden weight decreases with the mining depth. The theoretical analysis, numerical simulation and field detection present that the height of the mining-induced water-flow fracture zone is 39 m, 37 m, and 40.5–45 m, respectively, after mining at the − 135 m level. These values are reasonably consistent, suggesting that the proposed theoretical and numerical models and the utilized field detection method can provide valuable information for determining the overburden stability of an undersea mineral seam and improving mining safety.

Vertical Load and Settlement at the Foot of Steel Rib with the Support of Feet‐Lock Pipe in Soft Ground Tunnel

In soft ground tunnels, feet‐lock pipes have been widely used to decrease the concentration of load and settlement at the foot of steel ribs. This paper presents an analytical method to predict the vertical load and settlement at the foot of steel ribs with the support of the feet‐lock pipe. First, the mechanical model of a steel rib and feet‐lock pipe combined structure involving the ground reaction at the foot was proposed. In this model, the deformation compatibility among the steel rib, the feet‐lock pipe, and the ground at the tunnel foot were considered, and an elastic foundation beam model with double parameter for the feet‐lock pipe was proposed. Then, based on the proposed mechanical model, the analytical equations for predicting the vertical load and settlement at the foot of the steel rib were derived using structural analysis and beam theory on elastic foundation. The predicted vertical loads and settlements were validated by comparing with the results of field measurements and the Winkler foundation beam model for the feet‐lock pipe, and the results show that the feet‐lock pipe can effectively reduce the load acting on the ground, where the steel rib was installed, and finally improve the stability of the tunnel structure.

Analytical solutions for the mechanical behaviors of a hard roof subjected to any form of front abutment pressures

This study derived the analytical solutions for the deflection, bending moment and shear force of a hard roof under an arbitrary front abutment pressure for both the initial and periodic fracture problems using the elastic foundation beam model. With the assumption that the overburden pressure is constant in a small section, the roof subjected to arbitrarily distributed pressures is divided into a number of sufficiently small sections, whose closed-form solutions can be obtained easily. The proposed analytical solutions are validated by comparisons with those calculated using a model reported in the literature, in which the overburden pressure is assumed to follow a normal distribution. The parametric studies on the foundation stiffness, front abutment pressure and support resistance show that: (1) with the decrease of elastic foundation stiffness, the deflection and the maximum bending moment significantly increase, and the shear force significantly changes in the unmined region; (2) as the peak point position moves inside the coal wall, the deflection of the goaf roof, as well as the maximum bending moment and shear force, obviously decrease; (3) with the increase of support resistance, the maximum bending moment and deflection significantly decrease, and the shear force significant decreases in the hydraulic support region; (4) when the coal seam is soft and peak position of the front abutment pressure is near the coal wall, initial fracture occurs at the midspan; otherwise, it occurs inside the coal wall. For the periodic fracture condition, the fracture position is closer to the coal wall when the coal seam is harder, and the fracture length increases when the peak position of front abutment pressure moves inside the coal wall.

The Asymptotic Expansion Load Decomposition elastoplastic beam model

SummaryA new higher‐order elastoplastic beam model is derived and implemented in this paper. The reduced kinematic approximation is based on a higher‐order elastic beam model using the asymptotic expansion method. This model introduces new degrees of freedom associated to arbitrary loads as well as eigenstrains applied to the beam. In order to capture the effect of plasticity on the structure, the present elastoplastic model considers the plastic strain as an eigenstrain imposed on the structure, and new degrees of freedom are added on the fly into the kinematics during the incremental‐iterative process. The radial return algorithm of J2 plastic flow is used. Because of the constant evolution of beam kinematics, the Newton‐Raphson algorithm for satisfying the global equilibrium is modified. An application to a cantilever beam loaded at its free extremity is presented and compared to a three‐dimensional reference solution. The beam model shows satisfying results even at a local scale and for a significantly reduced computation time.

Finite element model of ballasted railway with infinite boundaries considering effects of moving train loads and Rayleigh waves

This paper proposes a three-dimensional model incorporating finite element (FE) meshes with infinite element (IE) boundaries for ballasted railways. Moving train loads are simulated with sliding motions of moving elements which have hard contact feature at the interface with supporting rails. Dynamic responses of ballasted railway under different train speeds are investigated in time domain and frequency domain to identify the predominant frequency and critical speed. Rayleigh wave (R-Wave) propagation is simulated using the combined FE-IE model to determine the velocity of R-Wave in the layered embankment model and its relationship with the critical speed of the ballasted railway. The proposed model is successfully validated against the results of Euler-Bernoulli Elastic Beam (E-BEB) model.

Extendable chord for improved helicopter rotor performance

Extendable blade sections are investigated as a method for reducing rotor power and improving helicopter performance. A validated helicopter power prediction method, based on an elastic beam model is utilized. The static extendable chord can deliver a rather small power reduction in hover, and significant power savings at high speed flight, however, the cruise power is increased. In hover, the active chord is best deployed in the middle part of the blade, and just inboard of the tip at high speed flight. The increase in chord length can lead to power savings at high speed flight but the benefits decrease in other speeds. The 1/rev dynamically extendable chord can lead to an overall power reduction over the speed range of a helicopter. The best deployment location is at the blade tip, which is different from the statically extendable chord. It is best extended out in the retreating side, and retracted back in the advancing. The power reduction by the 1/rev dynamically extendable chord increases with the increase in the length of the chord extension and take-off weight of the helicopter. Generally, a lower harmonic extendable chord can save more power than one actuated at higher harmonics. The dynamic chord can reduce more power than the corresponding static chord.

Elastic Beam Model and Bending Analysis of Silver Nanowires

In this study, bending analysis of silver (Ag) modeled nanowires has been carried out for six-various boundary conditions. Silver nanowires have a great importance for Nano-electro-mechanical systems (NEMS) technology. The displacement, rotation of cross-section and bending moment values of elastic beam models of silver nanowires under uniform load have been calculated. Numerical results have been presented as graphics and tables. The influence of boundary conditions to deformation and bending moment has been discussed. As the boundary conditions become rigid, the values of displacement and cross-sectional rotation under uniform load reduce.

Study of Large Deflection in Nano-Beams Using the Nonlocal Elasticity Theory

A nonlocal elastic beam model is developed by incorporating Eringen’s nonlocal constitutive equation into the large deflection beam theory for a nano-cantilever Euler–Bernoulli beam. The equilibrium equations are solved in an iterative manner using the shooting method. Deformed configurations of beam for different values of nonlocal parameter are plotted, and the final length of the beam after deflection is obtained. It is demonstrated that the beam scale has a significant impact on its flexural behavior under the large deflection, such that the nano-beam deflection can be either larger or smaller than the classical theory, depending on the loading level.

Plasmon-enhanced optical bending and heating on V-shaped deformation of gold nanorod

The plasmon-enhanced optical bending and heating on the V-shaped deformation of a straight gold nanorod (GNR), irradiated by a linear polarized light at the longitudinal surface plasmon resonance, are studied theoretically to explain the finding in previous experiment. Multiple multipole method is employed to calculate the optical load and heating numerically, and an elastic beam model is used to analyze the bending moment and stress in the GNR theoretically. According to our analysis, we think, first, the plasmonic heating softens the GNR to reduce the yield strength of gold, and the non-uniform optical load induces a maximum bending moment at the middle cross section of a freestanding GNR. Then an irreversible breakpoint of the plastic hinge at the middle of GNR is developed to form a V-shaped GNR. The photothermal deformation of V-shaped GNR involving multidisciplinary interplay is worth for further investigation.

A theoretical 4-stage shear model for single-lap torqued bolted-joint with clearances

In this paper, a theoretical 4-stage shear model for a single-lap torqued bolted-joint with clearance is proposed. The deformation mechanism of the bolted-joint indicates that the contact status is the key factor for the nonlinearity of the load-displacement curve. On the basis of the contact status, the expression of this model and its parameters are proposed. Compared with the previous Tri-linear model, this model is improved on three aspects. First, the 1st stage of the model has been improved by using the partial slip theory. Second, an additional 2nd stage is considered for taking into account the asynchronous slip of the interfaces between components and between bolt head and component. Third, the 4th stage of this model is derived from the elastic foundation beam model and the Cai-Shan Liu’s approximate contact model. Moreover, the 4-stage shear model is validated by 3-D detailed FE-models. The results show that the 4-stage model can correctly describe the shear behavior, especially the nonlinearity of the load-displacement curve in the 1st stage and the 4th stage. In addition, the effects of clearance and bolt length on the joint stiffness in each stage are presented.

Adaptive control of complex systems with unknown dynamics and input constraint: Applied to a chaotic elastic beam

SummaryOwing to the limitations of system identification and modeling techniques, there is usually some unknown dynamics in the mathematical models of the complex systems. In addition, external perturbations can affect the chaotic systems' responses and may destroy the desired control purpose. Consideration of such uncertain dynamics and external fluctuations in control applications is important in research and practice. On the other hand, because of the limited operation of control actuators, most of the practical implementations of control systems are forced with some input constraints. Therefore, this paper investigates the control problem of uncertain autonomous and/or nonautonomous complex chaotic systems in the presence of input saturation. The upper bounds of the unknown dynamics, modeling uncertainties, external perturbations, and the parameters of the saturation function are assumed to be unknown in advance. To make a fast control response, an adaptive nonsingular terminal variable structure controller is proposed to assure the finite‐time stability of the equilibrium states. Rigorous stability analysis is performed to prove the correct performance of the designed control algorithm. Numerical simulations on the unified system and a chaotic elastic beam model are developed to demonstrate the usefulness of the introduced adaptive control strategy. It is worth to notice that the derived adaptive nonsmooth sliding mode approach is general and it can be easily adopted for controlling of a wide class of uncertain MIMO nonlinear systems.

Analysis of ice shelf flexure and its InSAR representation in the grounding zone of the southern McMurdo Ice Shelf

Abstract. We examine tidal flexure in the grounding zone of the McMurdo Ice Shelf, Antarctica, using a combination of TerraSAR-X repeat-pass radar interferometry, a precise digital elevation model, and GPS ground validation data. Satellite and field data were acquired in tandem between October and December 2014. Our GPS data show a horizontal modulation of up to 60 % of the vertical displacement amplitude at tidal periods within a few kilometres of the grounding line. We ascribe the observed oscillatory horizontal motion to varying bending stresses and account for it using a simple elastic beam model. The horizontal surface strain is removed from nine differential interferograms to obtain precise bending curves. They reveal a fixed (as opposed to tidally migrating) grounding-line position and eliminate the possibility of significant upstream bending at this location. The consequence of apparent vertical motion due to uncorrected horizontal strain in interferometric data is a systematic mislocation of the interferometric grounding line by up to the order of one ice thickness, or several hundred metres. While our field site was selected due to its simple boundary conditions and low background velocity, our findings are relevant to other grounding zones studied by satellite interferometry, particularly studies looking at tidally induced velocity changes or interpreting satellite-based flexure profiles.

Analytical model for the load transmission law of rock bolt subjected to open and sliding joint displacements

The present study focuses on the load transmission law along the rock bolt and the efficacy for different anchored conditions. Shear displacement along the joint face is considered on the basis of traditional rock bolt stress calculation method with regard to joint open displacement. Combining the elastic foundation beam model and the rock bolt pull-out model, an analytical solution for the stress distribution along the rock bolt through a single joint face is derived. The performance of the proposed analytical solution is compared with an experimental result and a previous study. In addition, joint displacement, joint location, and dip angle of the joint face are set as variables to further explore the corresponding anchored effect. The results indicate that larger joint displacement along the joint face tend to cause a distinct increase of the shear and compressive stress along the rock bolt. When the joint face moves inward, the interaction force between the rock bolt and the surrounding rock mass decreases, suggesting that the strength of the rock bolt is not taken full use. Moreover, for a certain joint displacement and joint location, larger anchored angle between the rock bolt and the joint face contributes to better reinforcement effect.

Safety monitoring of continuously welded pipeline subjected to ground deformation using wireless micro-electro-mechanical system inclinometers

Ground deformation, which is a common consequence of surface loads, construction activities, subterranean cavitations, soil subsidence, and so on, poses a serious threat on buried pipelines in the vicinity. Aiming at the influence of ground deformation on continuously welded underground pipelines, this article presents a real-time technique for pipeline health monitoring using wireless micro-electro-mechanical system inclinometers. Based on the multi-point inclination measurements, a strategy is proposed to achieve the two-level target: (1) detection and localization of the abnormal event and (2) estimation of pipeline flexural deflection in virtue of the elastic foundation beam model and cubic spline interpolation. Numerical studies are carried out to investigate the applicable conditions of the strategy and the influence of sensor placement and measurement error on the estimation accuracy. Field testing is finally conducted on a 24-m buried steel pipeline to validate the effectiveness of the proposed method for safety monitoring of the pipeline subjected to ground deformation.

On the interpretation of ice-shelf flexure measurements

ABSTRACTTidal flexure in ice shelf grounding zones has been used extensively in the past to determine grounding line position and ice properties. Although the rheology of ice is viscoelastic at tidal loading frequencies, most modelling studies have assumed some form of linear elastic beam approximation to match observed flexure profiles. Here we use density, radar and DInSAR measurements in combination with full-Stokes viscoelastic modelling to investigate a range of additional controls on the flexure of the Southern McMurdo Ice Shelf. We find that inclusion of observed basal crevasses and density dependent ice stiffness can greatly alter the flexure profile and yet fitting a simple elastic beam model to that profile will still produce an excellent fit. Estimates of the effective Young's modulus derived by fitting flexure profiles are shown to vary by over 200% depending on whether these factors are included, even when the local thickness is well constrained. Conversely, estimates of the grounding line position are relatively insensitive to these considerations for the case of a steep bed slope in our study region. By fitting tidal amplitudes only, and ignoring phase information, elastic beam theory can provide a good fit to observations in a wide variety of situations. This should, however, not be taken as an indication that the underlying rheological assumptions are correct.