Elastic Beam Theory Research Articles

Optomechanical Modeling and Validation of a Distributed Bragg Reflector Force Sensor With Drift and Temperature Compensation

Distributed Bragg reflector fiber laser with the dual-polarization mode is commonly used as a sensing element in optical sensors. Lateral force on such a fiber induces birefringence and results in beating frequency generation in it. Also, the change in the magnitude of the lateral force is correlated to the shift in the beating frequency. This article presents a multi-physics optomechanical model for a fiber laser force sensor based on the birefringence phenomenon. To this end, a theoretical model for the sensing principle was developed through employing the elastic beam theory, Hertzian contact mechanics, and optical birefringence principle. Based on the linearity of the developed optomechanical model to the lateral force and ambient temperature variation, a multi-linear regression calibration for the force sensor was proposed and experimentally validated. Also, the temperature-induced drift in the beating frequency was compensated through exponential adjustment. Moreover, the calibration results showed a relatively small sensitivity for temperature sensing with respect to the external force (cross-talk). This phenomenon was predicted with the theoretical model. Verification of the calibration revealed a root-mean-square error of 0.12°C and 0.04 N for temperature and force sensing, respectively. Furthermore, the validation study showed a root-mean-square error of 0.04± 0.03 N and zero hysteresis for the developed sensor. Moreover, the theoretical sensitivity to external force was similar to the experimental results. For future applications, the compliance of the sensor specifications with the requirements of three surgical procedures was confirmed through comparison with the available literature.

Theoretical method for an elastic infinite beam resting on a deformable foundation with a local subsidence

Local subsidence induced by underneath erosion or excavation is usually encountered by buried pipelines and tunnels, which in turn may cause distress to the existing pipelines and tunnels. A semi-analytical method is proposed for this kind of problem by simplifying the existing pipeline or tunnel as an elastic infinite Euler-Bernoulli beam and the ground as a deformable foundation with a local subsidence. The beam-foundation system is subject to a symmetrical distribution load. Closed-form solutions for the beam responses, including deflection and bending moment, are derived based on the elastic foundation beam theory by expanding the load expression as a Maclaurin series. The proposed method and solutions are validated by a series of 3D FEM numerical simulation and model test results. Sensitivity analyses are then carried out to investigate the influence of different parameters. The results show that the beam response is highly dependent on the length of local subsidence and the beam flexural rigidity. Moreover, comparations between the calculation results based on the Pasternak model and the Winkler model that can be degenerated from the former are also presented to illustrate the rationality of the proposed method in predicting the responses of the foundation beam in this typical problem.

A new hyperbolic-polynomial higher-order elasticity theory for mechanics of thick FGM beams with imperfection in the material composition

A drawback to the material composition of thick functionally graded materials (FGM) beams is checked out in this research in conjunction with a novel hyperbolic-polynomial higher-order elasticity beam theory (HPET). The proposed beam model consists of a novel shape function for the distribution of shear stress deformation in the transverse coordinate. The beam theory also incorporates the stretching effect to present an indirect effect of thickness variations. As a result of compounding the proposed beam model in linear Lagrangian strains and variational of energy, the system of equations is obtained. The Galerkin method is here expanded for several edge conditions to obtain elastic critical buckling values. First, the importance of the higher-order beam theory, as well as stretching effect, is assessed in assorted tabulated comparisons. Next, with validations based on the existing and open literature, the proposed shape function is evaluated to consider the desired accuracy. Some comparative graphs by means of well-known shape functions are plotted. These comparisons reveal a very good compliance. In the final section of the paper, based on an inappropriate mixture of the SUS304 and Si3C4 as the first type of FGM beam (Beam-I) and, Al and Al2O3 as the second type (Beam-II), the results are pictured while the beam is kept in four states, clamped–clamped (C–C), pinned–pinned (S–S), clamped-pinned (C–S) and in particular cantilever (C–F). We found that the defect impresses markedly an FGM beam with boundary conditions with lower degrees of freedom.

P Y Curves of Laterally Loaded Piles Near Earth Slopes

Design of piles under lateral loads using numerical analysis is a time-consuming process, requiring competent geotechnical engineers who can accurately model the soil profile and construction sequence. Therefore, most engineers have resorted to the p-y method that is a less time-consuming process in both the modeling and running time. Contrary to the numerical analysis method, the p-y method doesn’t require the burden of constructing a complicated 3D model. This method simply uses the relation between the soil resistance per unit length (p) and the lateral deformation (y) to deduce the straining actions on the pile, bending moment, and shear forces, which govern the structural design. However, the simplicity of this method comes with its shortcomings. The p-y method, for instance, cannot directly take into account the effect of earth slopes on the laterally loaded piles, and its results are somewhat approximate. A well-instrumented case study from the Caltrans site at Oregan State University is analyzed in this research. The studied case consists of a laterally loaded single vertical pile embedded in a cohesive soil layer near an earth slope of 2H:1V. A three dimensional numerical model of the case study is constructed, utilizing the finite element code, Plaxis 3D 2020. The p-y curves of the loaded piles were back-calculated from the numerical model using the elastic beam theory by performing the differentiation of the shear force acting on the pile along the full height of the earth slope. Normalized p-y curves were obtained to determine the p-multiplier, a factor that helps convert the p-y relation of a pile in leveled ground to that of a pile near earth slopes. Overall, it was found that the p-multiplier ranges between (0.4-0.8), (0.6-0.83), (0.8-0.95), and (0.98-1) for piles located at a distance of 0D, 2D, 4D, and 8D respectively from the crest of the earth slope, for various target depths. A parametric study for the effect of the distance of the pile from the crest of the slope, as well as the slope inclination, on the p-y curves was conducted. The curves were constructed for a single pile located at distances of 0D,2D,4D, and 8D from the crest of the earth slope. The performed study revealed that the p-multiplier, at a target depth of 1m, measured from the top of the pile, for the studied slope inclinations, ranges between (0.3-0.45) for the pile at a distance of 0D, (0.76-0.8) at a distance of 2D, (0.82-0.93) at a distance of 4D and (0.98-1) at a distance 8D. Analysis results showed that the effect of slope inclination diminishes when the pile is placed at a distance 8D from the crest or farther. These values can be implemented into p-y curves software, such as LPILE, to determine the straining actions required for design of a laterally loaded pile near sloping ground.

Multiquadric radial basis function approximation method and asymptotic numerical method for nonlinear and linear static analysis of single-walled carbon nanotubes

In this paper, a numerical algorithm combining the meshless collocation technique based on the globally supported Multiquadric Radial Basis Function (MQRBF) approximation method and the Asymptotic Numerical Method (ANM) is developed to investigate the nonlinear and linear static behaviors of Single-Walled Carbone Nanotubes (SWCNTs) based on a Nonlocal Continuum Beam Model (NCBM). The NCBM which accounts for the effects of small scale, shear deformation and geometric nonlinearity of von-Kármán type is constructed by implementing the Nonlocal Elasticity Theory of Eringen (NET) into the First-Order Shear Deformation Elastic Beam Theory (FOSDEBT) and used to model the SWCNTs as a Continuous Elastic Nanobeam (CEN). Based on the Total Lagrangian Formulation (TLF) the nonlocal nonlinear governing equations of SWCNT are elaborated in a quadratic matrix strong form. These equations are then solved numerically by applying the proposed algorithm. This algorithm permits to transform the quadratic nonlocal nonlinear governing equations into a sequence of linear ones shared the same tangent stiffness matrix which are then solved numerically by the MQRBF approximation method, to get the whole branch of solution a continuation procedure is required. In order to highlight the reliability and efficiency of the present numerical algorithm, comparative studies are conducted.

A semi-analytical approach for simulating oblique kinematic distress of offshore pipelines due to submarine landslides

Deep water offshore pipelines are usually directly laid on the seabed, a fact that makes them vulnerable to geohazards, such as submarine landslides and debris flows. The intersection angle between the pipeline and the moving sediments can vary taking into account the pipeline route and the unstable nature of deep seabed. The aim of the current study is to examine the distress of a seabed-laid offshore natural gas pipeline subjected to combined lateral and axial kinematic distress due to an oblique submarine landslide or a debris flow. For this purpose, a new semi-analytical model is developed combining the finite-difference method and the elastic-beam theory. Firstly, the proposed model is compared with a finite-element model for various intersection angles, as well as with previous analytical models considering exclusively lateral loading. Subsequently, various combinations of soil resistance forces and loading conditions that affect the examined problem are investigated. Realistic input data were taken from the offshore section of the high-pressure natural gas pipeline TAP (Trans Adriatic Pipeline). Finally, useful conclusions are drawn regarding the applicability and efficiency of the proposed approach to accurately represent the pipeline response under these loading conditions.

Analytical Study of the Snap-Through and Bistability of Beams With Arbitrarily Initial Shape

Abstract We derive the snap-through solution and the governing snapping force equations for an arbitrarily preshaped beam deflected under a mid-length lateral point force. The exact solution is obtained based on the classical theory of elastic beams as a superposition of the initial shape and the modes of buckling. Two kinds of solutions are identified depending on the axial force level. The two solutions, bifurcation conditions, bistability conditions, and the snapping force equations are derived and discussed. The snap-through and snapping force solutions are then calculated for two common beam initial shapes, the curved (first buckling shape) and the inclined one (V-shape). In both cases, explicit expressions are obtained describing the snap-through behavior. The analytical modeling results show excellent agreement with finite element simulations. The comparison between the two cases shows a similar snap-through behavior qualitatively, while several differences and similarities are noticed quantitatively.

Strata Movement Model of Filling Coal Mining Based on Two-Parameter Elastic Foundation

To avoid the discontinuity defect of the Winkler elastic foundation beam model, a two-parameter elastic foundation model of Pasternak reflecting the continuity of the waste foundation was applied in the regular analysis of strata movement in waste filling coal mining. A two-parameter elastic foundation beam strata movement mechanical model was established. The formulae to calculate the shear force, bending moment, deflection and foundation reaction force of roof strata were obtained using elastic foundation beam theory. The two-parameter elastic foundation beam model and the Winkler elastic foundation beam model were applied to an engineering example to analyze the influence of different waste foundation coefficient k and shear stiffness Gp on the force and deformation of roof strata respectively, and the results were compared accordingly. The results show that the shear stiffness has a certain influence on the force state and subsidence of the roof strata, when waste foundation coefficient is smaller, the difference between the two elastic foundation models is greater. Considering the continuity of the waste filling material, the subsidence of the roof beam obtained by the two-parameter foundation model is smaller than that by the Winkler model. The prediction of the development height of water flowing fractured zone and surface subsidence by the new model is more in line with the engineering practice.

Euler–Bernoulli theory accurately predicts atomic force microscope cantilever shape during non-equilibrium snap-to-contact motion

We prove that the Euler–Bernoulli elastic beam theory can be reliably used to describe the dynamics of an atomic force microscope cantilever during the far from equilibrium snap-to-contact event. In conventional atomic force microscope operation, force-separation curves are obtained by post-processing voltage versus time traces produced by measuring one point on the cantilever close to the hanging end. In this article, we assess the validity of the Euler–Bernoulli equation during the snap-to-contact event. The assessment is based on a direct comparison between experiment and theory. The experiment uses Doppler vibrometry to measure displacement versus time for many points along the long axis of the cantilever. The theoretical algorithm is based on a solution of the Euler–Bernoulli equation to obtain the full shape of the cantilever as a function of time. The algorithm uses as boundary conditions, experimentally obtained information only near the hanging end of the cantilever. The solution is obtained in a manner that takes into account non-equilibrium motion. Within experimental error, the theory agrees with experiment indicating that the Euler–Bernoulli theory is appropriate to predict the cantilever kinematics during snap-to-contact. Since forces on the tip can be obtained from the instantaneous shape of the cantilever, this work should allow for computation of tip-sample forces during the snap-to-contact event from a conventional force-distance measured input.

Fracture modes of brittle junctions under shear

Friction and wear of brittle materials during sliding contact are often associated with the fracture-induced failure of interlocking asperities under shear loading. Using systematic in-situ experiments we disclosed two dominating fracture failure modes associated with the aspect ratio of the junction: (I) small aspect ratio junctions fail antisymmetrically by stable propagation of two subsurface cracks, (i.e. hand-shaking mechanism), causing the roughness creation on both surfaces and formation of particles, (II) large aspect ratio junctions fail asymetrically by unstable propagation of a single crack. Numerical simulations revealed that the transition between these two failure modes is dictated by a competition between shear versus bending induced stress state. Combining the linear elastic fracture mechanics and Timoshenko beam theory, a simple theoretical model is developed, predicting the transition in the critical force associated with each mode as a function of junction’s aspect ratio and fracture properties. This understanding opens new venues towards developing a fracture model for surface asperities and physics-based predictive models for wear of brittle solids.

Study on the Horizontal Axis Deviation of a Small Radius TBM Tunnel Based on Winkler Foundation Model

During the construction stage of the small radius TBM (tunnel boring machine) interval, the improper control of the boring parameters and the boring posture can cause the horizontal axis deviation of the shield tunnel. In order to address this issue, the TBM segments lining structure of the small radius interval is simplified as the continuous circular curved beam based on the longitudinal equivalent continuous model and Winkler elastic foundation beam theory. The theoretical model is solved through the transfer matrix method, and its applicability is verified by comparing it with the field monitoring data. It is found that the horizontal axis deviation of the completed tunnel increases with the total jack thrust, and the lateral displacement tends to be stable when the distance between the ring and the tail is far. The horizontal axis deviation has a negative relationship with the thrust difference or path difference when the jack thrust in the outside of the shield curve is larger than that of inside the shield curve. The horizontal axis deviation has a positive relationship with the thrust difference or path difference when the jack thrust in the outside of the shield curve is smaller than that of inside the shield curve.

Characteristics of Roof Ground Subsidence While Applying a Continuous Excavation Continuous Backfill Method in Longwall Mining

Activities of traditional longwall mining will result in ground subsidence and therefore cause issues such as damages to buildings and farmlands, water pollution and loss, and potential ecological and environmental problems in the mining region. With advantages of the longwall backfill mining method, as well as the room and pillar mining method, a continuous excavation and continuous backfill (CECB) method in longwall mining is recommended to effectively control the ground subsidence. In this method, mining roadways (MRs) are initially planned in a panel, and then they are excavated and backfilled in several stages until the whole panel is mined out and backfilled. According to the geologic conditions of an underground coal mine, and the elastic foundation beam theory, a mechanical model was built to study the subsidence of the roof while using this new mining method. In addition, methods to calculate roof subsidence in various stages in CECB were also provided. The mechanical parameters of backfilling materials, which were used in the theoretical calculation and the numerical analysis for mutual check, were defined through analyzing the stability conditions of the coal pillars and the filling bodies. The control effect for the ground subsidence of using the newly proposed mining method was analyzed based on both simulation results and site monitoring results, including the ground subsidence, horizontal displacement, tilt, curvature and horizontal strain. This research could provide suggestions to effectively control ground subsidence for a mine site with similar geologic conditions.

Evaluation on out-of-plane shear stiffness and ultimate capacity of perfobond connector

Perfobond shear connectors (PBLs) are increasingly applied in steel-concrete composite structures. The out-of-plane (lateral) shear behavior of PBLs is still unknown to researchers and designers, although the connectors undertake considerable out-of-plane shear forces in some applications. Thus, push-out tests with three specimens were first conducted to investigate the lateral shear performance of PBLs and further used to validate the corresponding numerical models. The test results show that perforated rebars and concrete dowels are irrelevant to the lateral shear capacity, while they can reduce the separations and improve the ductility. Secondly, parametric FEA (finite element analysis) models for the push-out tests were built and validated based on the test results. The plastic strains in concrete mainly develop at the regions close to the fillet welds. Further, 144 extended FEA models with varying height, length, and thickness of perfobond plates and concrete compressive strength were conducted to reveal the lateral shear mechanism of PBLs. By enlarging the plate height, the lateral shear capacity slightly increases, while the shear stiffness remains as constant. The shear capacity and stiffness increase with the plate length and thickness, as well as the concrete compressive strength. Finally, based on the existing shear capacity equations for channel and angle connectors, and the theory of elastic foundation beams, the expressions for the lateral shear capacity and stiffness of PBLs were put forward. The proposed equations agree with the results of the parametric study.

Testing of insulated sandwich panels with GFRP shear connectors

This paper focused on the panel which sandwiches an Expanded Polystyrene (EPS) layer between two Steel Reinforced Concrete Layers (SRCLs). The EPS layer provides thermal insulation in the panel while the two exterior SRCLs are connected by Glass Fiber Reinforced Plastic (GFRP) shear connectors to form the composite action and hence achieve a higher flexural resistance in the panel. Four types GFRP shear connectors which consist of diagonal web members, webs with circular perforations, webs with slotted perforations and solid webs, respectively, were considered in this research. To experimentally address the flexural responses of the panels with the four types of GFRP shear connectors, four full-scale experimental specimens were constructed and tested. The test results show that generally any of the four GFRP shear connectors can enable the panel to exhibit satisfactory flexural performance although the GFRP shear connectors with the solid webs tend to increase the ultimate flexural resistance of the panel. Beyond the experimental work, an analysis model based on the transformed cross-section approach and elastic beam theory was developed for capturing the flexural response of the panel up to the cracking limit state. Moreover, an analysis model based on the assumed strain and stress diagrams was established to compute the ultimate flexural resistance of the panel.

A novel amplification ratio model of a decoupled XY precision positioning stage combined with elastic beam theory and Castigliano's second theorem considering the exact loading force

Abstract A novel mathematical model which can precisely calculate the amplification ratio of a bridge-type amplification mechanism has been proposed. The elastic beam theory and the Castigliano’s second theorem were utilized to establish the mathematical model with consideration of the deformation of input beams, output beam, connecting beams and flexure hinges. The proposed model was compared with existing theoretical models and finite element model (FEM), and the results validated that the proposed model featured a better performances and was the most proximal model to the FEM analysis. With the interval of the adjacent flexible hinges changed, the error at the peak of amplification ratio with the FEM was limited to 6.76%, which was lower than most of the typical existing models. To further verify the effectiveness and expansibility of the proposed method, a decoupled XY precision positioning stage was investigated. Considering the influence of the loading force exerted on amplification mechanism, the displacement amplification ratio model of the positioning stage was established, and the results were further proved through the finite element simulation. The experimental results presented that the amplification ratios for X-direction and Y-direction motions are 5.83 and 5.71, the measured workspace of the stage is 174.9 μm × 171.3 μm, and the cross-coupling error was evaluated to less than 3%.

A new theoretical method for calculating front abutment stress during coal mining

AbstractThe calculation of abutment stress has always been the most important topic in safe underground mining and design. In this paper, based on the formation mechanism of abutment stress, combined with the mechanical properties of actual strata and the theory of elastic foundation beams, a structure model of overlying strata with an elastic foundation is established, and a new theoretical calculation method of abutment stress is proposed. The new calculation method takes into account the interaction of the overlying key strata and the effect on the transmission of the front abutment stress and then analyzes the stress patterns of multilayered key strata under the combined action of the bending moment, shear force, and nonuniform load and its transmission to the underlying strata. In addition, based on field observations of panel 10 305 in the Xinglongzhuang coalmine, the influence range, peak position, and variation trend of the front abutment stress before and after breaking of the overlying key strata are studied and verified. The results show that the influence range of the abutment stress remains unchanged after KS1 breaks (eg, from 0 to 180 m), and its peak position shifts forward from 8 to 19 m, while the influence range (0‐180 m) and its peak position (6‐8 m in front of coal wall) remain unchanged after KS2 breaks; the abutment stress decreases, and the maximum decrease is 3.9 MPa in the range of 0‐22 m ahead of the coal wall after KS1 breaks, while the maximum decrease in the abutment stress is 2.1 MPa in the range of 0‐71 m ahead of the coal wall after KS2 breaks. The results of the new calculation method are basically consistent with the field observations and the existing theoretical results, showing that the method has good adaptability and reliability and can provide a new theoretical basis for on‐site design and construction practices.

Experimental and Numerical Investigation of Mode I and Mode II Interlaminar Behavior of Ultra-Thin Chopped Carbon Fiber Tape-Reinforced Thermoplastics

This manuscript describes an experimental and numerical study conducted to investigate two representative cases of delamination in ultra-thin chopped carbon fiber tape-reinforced thermoplastics (UT-CTT): delamination in a double cantilever beam (DCB, mode I) and delamination in an end-notched flexure (ENF, mode II). In examples of mode I delamination, unstable crack growth resulted in sawtooth-like load–displacement curves after the initial stage of increasing load, while some specimens displaying mode II delamination exhibited stable crack propagation. The crack surfaces of DCB and ENF UT-CTT specimens were observed by a three-dimensional measurement macroscope and a scanning electron microscope. The values of mode I and mode II interlaminar fracture toughness were obtained based on linear elastic fracture mechanics and beam theory in reference to JIS K 7086 standard, respectively. Furthermore, numerical analysis utilizing the surface-based cohesive zone model based on the triangular traction–separation law was used to predict the delamination behavior in UT-CTT. The minimum and maximum values of critical strain energy release rate were used to define the range of load–displacement curves corresponding to unstable crack propagation under mode I, while the parameters including interlaminar shear strength and friction were taken into account for the predictions of delamination propagation under mode II. The numerically predicted load–displacement curves showed good correlation with the theoretical and experimental results. The research provides interlaminar mechanical properties for damage modeling and numerical simulation method to express fracture behaviors of UT-CTT structures.

Calculation Model of Supporting System for Tunnel Under Shallow and Weak Surrounding Rock Considering the Synergistic Effects

In order to give full play to the bearing capacity of the various components of a tunnel supporting system under shallow and weak surrounding rocks, a study is conducted on the synergistic mechanism of the tunnel supporting system in these conditions. In this paper, the synergistic mechanism of the tunnel supporting system is investigated by considering each function of the supporting equipment. Then, a theoretical model is proposed to calculate the overall bearing capacity of the supporting system for the tunnel under the shallow-buried weak surrounding rocks. The ultimate calculation formula for the supporting system, which includes advanced pipe roof, leading conduit, steel arch and feet-lock anchor bolt, is obtained on the basis of the structural mechanics and elastic foundation beam theory. To validate the proposed model, the interacting forces and deformation of each support structure component are calculated respectively. Next, the calculation results are compared with those of the load-structure method and strata-structure method. Statistical analysis shows that the deformations obtained by the three methods are consistent, and that the results of each method are rather similar. It indicates that the theoretical method proposed is reliable and applicable.

Study on Damage of Shallow Foundation Building caused by Surface Curvature Deformation in Coal Mining Area

Coal mining can lead to various forms of surface deformation, the damage caused by curvature deformation is the most extensive, which has become the key to restricting the development and utilization of coal resources under villages. Therefore, it is very important to master the deformation and destruction mechanism of buildings under surface curvature deformation for the rational exploitation of coal resources under villages. In this paper, the deformation and damage of buildings caused by curvature deformation are studied with theoretical analysis and numerical simulation, aiming at the shallow foundation building in rural China, and the following objectives are achieved: 1) Based on the theory of elastic foundation beam, the longitudinal and transverse walls are considered as a whole and analyzed, and a mathematical model of the cooperative deformation in building walls under surface curvature deformation is established. 2) The failure mechanism of masonry buildings is analyzed, including the distribution of internal stress and the form of wall cracks. 3) The influence of various factors on the internal stress changes of buildings is discussed. 4) The results of numerical simulation of wall stress distribution are similar to theoretical analysis, and the influence of doors and windows on the additional stress distribution is analyzed emphatically. This study can provide important guidance for the optimization of coal mining design under villages and the protection of village buildings.

Strain-based multimode integrating sensing for a bridge-type compliant amplifier

This paper presents a compact strain-based multimode sensing approach, in which sensing units are monolithically combined in a compound compliant bridge-type amplification mechanism. Under the same strain gauges configuration, this approach can realize different-mode precision sensing for output displacement, input displacement and input force measurement. Based on elastic beam theory and the modified geometric analysis method, an output/input displacement-sensing model only relating to the moment, and an input force-sensing model only relating to the axial force are analytically derived. These analytic sensing models indicate their good linear sensing performance, which is validated experimentally. The related experiments and analysis indicate that it has excellent heat-insulating performance, and their measurement accuracies can reach the tens of nanometer level for displacement sensing and the sub-Newton level for force sensing in void output loads, which fulfill the requirements of most of the nano-positioning stages.