Displacement Formula Research Articles

Analysis and Calculation of Cross-contact Force between Strands of Multi-layer Conductor

Abstract As a key component of the transmission line, the inter-layer cross-contact force of the internal strand is an important factor affecting the mechanical properties of the conductor. In this paper, the cross-contact characteristics between the strands of multi-layer conductors are analyzed. According to the relationship between the contact force and the contact distance between the strands, the finite element simulation of different material strands of different conductors at different cross angles is carried out. Through the analysis, it can be seen that the cross angle between the strands and the material of the strands have a significant impact on the contact force. Then, based on the existing Hertz contact theory model, the relationship between the contact force and the contact distance between the strands is modified, and the contact force and contact distance models suitable for different cross angles of the conductor strands are obtained, that is, the modified Hertz contact model. According to the simulation results, combined with the modified Hertz contact model, the contact force coefficient between the strands is fitted, and the calculation formula of contact force and contact displacement is proposed. The formula can be applied to the calculation of contact force at any intersection angle between strands. The calculation formula provides theoretical support for the accurate structural calculation of the conductor, which is helpful to the analysis and calculation of the contact friction between the strands.

Analysis of the three‐dimensional elasticity theory problem for a radially inhomogeneous cylinder

AbstractIn this paper an axisymmetric problem of elasticity theory for a radially inhomogeneous cylinder of small thickness is studied. It is assumed that homogeneous mixed boundary conditions are specified on lateral surfaces of the cylinder, and boundary conditions that leave the cylinder in equilibrium are specified on the ends of the cylinder. It is considered that elastic moduli are arbitrary continuous functions of a variable along the radius of the cylinder. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thinness of the cylinder walls. To determine its solution, the method of asymptotic integration is used, based on three iteration processes. All solutions of the equilibrium equation are constructed, satisfying the specified homogeneous boundary conditions on lateral surfaces. It is obtained that the solution of the problem consists of a penetrating solution and a solution of the nature of the boundary layer. An analysis of stress‐strain states corresponding to different types of solutions is carried out. Based on the constructed asymptotic solutions, a connection is found between the solutions obtained by the equation of elasticity theory and by the applied theory of shells. It is shown that the penetrating solution determines the internal stress‐strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the principal vector of stress acting in a cross section perpendicular to the axis of the cylinder.The first terms of asymptotic expansions of the penetrating solution in a small parameter coincide with solutions in the applied theory of shells. New classes of solutions are defined as the character of a boundary layer, which are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in the applied theories of shells. The first terms of the asymptotic expansion of the solution, having the nature of a boundary layer, are equivalent to the Saint‐Venant edge effect in the theory of heterogeneous plates. Asymptotic formulas for displacement and stresses are constructed, allowing one to calculate the three‐dimensional stress‐strain state of a radially heterogeneous cylinder of small thickness with any predetermined accuracy. Based on the obtained asymptotic expansions, one can estimate the applicability areas of applied theories and construct a refined applied theory for radially heterogeneous cylindrical shells.

Experimental and numerical investigation on normal strength and high strength RC arches subjected to underwater explosions

As a common structure in engineering, reinforced concrete (RC) arch may be subjected to explosion load in its life cycle. However, there are few studies on the blast resistance of RC arch against underwater explosion. A normal strength reinforced concrete (NSRC) arch and a high strength reinforced concrete (HSRC) arch were designed, and two sets of underwater explosion tests were carried out. Multi-media coupling models of the two RC arches subjected to underwater explosions were established based on the Arbitrary-Lagrangian-Eulerian (ALE) algorithm. The propagation characteristics of shock wave, the acceleration response and damage evolution of the arches were experimentally and numerically investigated. The reliability of the numerical method was verified by comparing the numerical results with the test results. Employing the verified numerical method, the effects of standoff distance and explosive weight on the dynamic response and damage distribution of the NSRC and HSRC arches were further studied. Based on extensive numerical simulations, the prediction formulas for the peak displacement of the given NSRC and HSRC arches due to specific underwater explosions were proposed. The results show that the arch vault and both sides of hance are the most vulnerable to failure when suffering the explosions centrally above the arch. The blast resistance of the HSRC arch is much stronger than that of NSRC arch. The failure modes of RC arches can be summarized as: arch vault cracking, global bending failure, combination of global bending failure and local damage, combination of global bending failure and penetration failure.

Modelling of non-linear elastic constitutive relationship and numerical simulation of rocks based on the Preisach–Mayergoyz space model

SUMMARY The existence of pores, cracks and cleavage in rocks results in significant non-linear elastic phenomena. One important non-linear elastic characteristic is the deviation of the stress–strain curve from the linear path predicted by Hooke's law. To provide a more accurate description of the non-linear elastic characteristics of rocks and to characterize the propagation of non-linear elastic waves, we introduce the Preisach–Mayergoyz space model. This model effectively captures the non-linear mesoscopic elasticity of rocks, allowing us to observe the stress–strain and modulus–stress relationships under different stress protocols. Additionally, we analyse the discrete memory characteristics of rocks subjected to cyclic loading. Based on the Preisach–Mayergoyz space model, we develop a new non-linear elastic constitutive relationship in the form of an exponential function. The new constitutive relationship is validated through copropagating acousto-elastic testing, and the experimental result is highly consistent with the data predicted by the theoretical non-linear elastic constitutive relationship. By combining the new non-linear elastic constitutive relationship with the strain–displacement formula and the differential equation of motion, we derive the non-linear elastic wave equation. We numerically solve the non-linear elastic wave equation with the finite difference method and observe two important deformations during the propagation of non-linear elastic waves: amplitude attenuation and dispersion. We also observe wave front discontinuities and uneven energy distribution in the 2D wavefield snapshot, which are different from those of linear elastic waves. We qualitatively explain these special manifestations of non-linear elastic wave propagation.

Elasticity solutions for functionally graded beams with arbitrary distributed loads

This paper derives the exact general elasticity solution for functionally graded rectangular beams subjected to arbitrary normal and tangential loads and with arbitrary end constraints. The general solution consists of bending moments and their integrals and derivatives, along with load-independent function sequences of the longitudinal coordinate. The method for determining function sequences has been established based on the stress function method. General solution formulas for stresses, strains and displacements have been derived and used to solve explicit special solutions for six cases involving concentrated forces, uniformly loads, and quadratically distributed loads with different displacement constraints scenarios. The results obtained are compared with existing exact solutions and those of Euler–Bernoulli and Timoshenko beams, and the errors of the latter two are analysed.

Relief displacement of airborne objects

ABSTRACT The increasing availability of unoccupied aircraft systems (UAS, also referred to as drones) has led to their use in taking vertical aerial photographs at relatively small spatial scales. These photographs can be used to measure the distances between objects appearing in the photographs. However, relief displacement can cause an object above or below ground level to appear at a point in a vertical aerial photograph that is not directly in-line with the object’s actual location, causing a measurement error. A UAS was used in this study as a photographed airborne object because its location and altitude could be controlled. We were interested in predicting the horizontal distance of the UAS’s appearance from the centre of a vertical aerial photograph. Predictions of the location of the photographed UAS’s appearance in vertical aerial photographs over both level and sloped surfaces matched measured appearance distances within 0.06–0.48 m. This study shows that the relief displacement formulas typically used to compute the height of a vertical structure appearing in a vertical aerial photograph can additionally be used to compute the actual location of an airborne object (e.g., a flying UAS, bird, bat) if the object’s altitude is known or can be estimated.

An optimized MOPSA algorithm for highly accurate deterministic lateral displacement microfluidic sorting chip design

Abstract Deterministic lateral displacement (DLD) sorting is a passive sorting technology. Due to the advantages of small volume and high accuracy, the DLD devices have been used in many applications. The critical diameter (Dc) value of the DLD device depends on the geometric structure of the internal post array. However, the existing empirical formula cannot match all types of post-arrays. Achieving the desired Dc value typically involves multiple iterative processes, leading to increased labor and time costs. In this paper, we propose an optimized trajectory prediction algorithm based on the original MOPSA, which considers the impact of fluidic pressure on particle trajectories. The method is to stretch the 2D physical field space into 3D, the particle expands from a 2D circle into a 3D ball on the pressure analysis. The net pressure on the particle is calculated, and the displacement of the particle is predicted by combining Newton’s second law and the displacement formula. A DLD device with Dc=10.6 μm was designed using this optimized algorithm and fabricated. It is verified by separation of 10 μm and 11 μm polystyrene particles. The test results were consistent with the simulation results. Compared with the empirical formula and the original MOPSA, this optimized algorithm can improve the prediction accuracy of particle trajectories in DLD devices.

Deformation of solid earth by surface pressure: equivalence between Ben-Menahem and Singh’s formula and Sorrells’ formula

SUMMARY Atmospheric pressure changes on Earth’s surface can deform the solid Earth. Sorrells derived analytical formulae for displacement in a homogeneous, elastic half-space, generated by a moving surface pressure source with speed $c$. Ben-Menahem and Singh derived formulae when an atmospheric P wave impinges on Earth’s surface. For a P wave with an incident angle close to the grazing angle, which essentially meant a slow apparent velocity $c_a$ in comparison to P- ($\alpha ^{\prime }$) and S-wave velocities ($\beta ^{\prime }$) in the Earth ($c_a \ll \beta ^{\prime } \lt \alpha ^{\prime }$), they showed that their formulae for solid-Earth deformations become identical with Sorrells’ formulae if $c_a$ is replaced by $c$. But this agreement was only for the asymptotic cases ($c_a \ll \beta ^{\prime }$). The first point of this paper is that the agreement of the two solutions extends to non-asymptotic cases, or when $c_a /\beta ^{\prime }$ is not small. The second point is that the angle of incidence in Ben-Menahem and Singh’s problem does not have to be the grazing angle. As long as the incident angle exceeds the critical angle of refraction from the P wave in the atmosphere to the S wave in the solid Earth, the formulae for Ben-Menahem and Singh’s solution become identical to Sorrell’s formulae. The third point is that this solution has two different domains depending on the speed $c$ (or $c_a$) on the surface. When $c/\beta ^{\prime }$ is small, deformations consist of the evanescent waves. When $c$ approaches Rayleigh-wave phase velocity, the driven oscillation in the solid Earth turns into a free oscillation due to resonance and dominates the wavefield. The non-asymptotic analytical solutions may be useful for the initial modelling of seismic deformations by fast-moving sources, such as those generated by shock waves from meteoroids and volcanic eruptions because the condition $c / \beta ^{\prime } \ll 1$ may be violated for such fast-moving sources.

Consolidation effects on pipe-soil interaction due to tunneling

Existing studies on soil-pipe interaction due to tunneling mainly focus on short-term responses. However, in areas with high water tables and low permeability soil, long-term ground movement and associated pipe responses may occur due to dissipation of excess pore pressure generated during tunnel construction. In this paper, a Winkler solution with time-varying subgrade modulus and the corresponding greenfield soil displacement formula are developed to investigate the tunneling effects on existing pipelines. The pipe is considered as an infinite Euler beam of finite width resting on a poroelastic half-space, and adhesion and drainage effects between the pipe and soil are considered using bounding techniques. The greenfield consolidation settlement is evaluated using a modified Gaussian curve. The findings indicate that the subgrade modulus decreases while greenfield soil displacement increases during the consolidation process. The time-dependent behavior of the subgrade modulus is governed by the drainage condition at the pipe-soil interface, whereas the greenfield soil displacement is primarily influenced by the drainage condition at the tunnel-soil interface. The study reveals that the bonded contact condition, hydraulic boundary condition, and displacement constraint conditions all influence the bending moment of the pipe.

Experimental and numerical study of locking steel plate damper for seismic resistance

In this paper, a new metal energy dissipator named locking steel plate damper (LSPD) is proposed for structural seismic applications. The damper is easy to fabricate and convenient to install and replace. The LSPD consists of two locking steel plates with energy dissipating steel rings and an outer shell plate for peripheral restraint and steel ring reset. Experiments were conducted using a proposed static cyclic displacement loading regime, which loaded the two specimens. The results of experiment showed that the damper possessed excellent energy dissipation capability. Thirteen comparison models were designed by varying the radius, thickness and width of the ring and FE simulations were carried out. By comparing the simulation results of the 13 models, the effects of the steel ring design parameters on the energy dissipation capacity of the damper are summarized. Finally, the suggested formulas for the yield displacement and the practical formulas for the initial stiffness of the LSPD are presented through theoretical derivation and parametric analysis. The average error between theoretical and simulated values for the 13 models was 2.11%, with a maximum error of 7.33%, which provide reliable suggestions for the design and fabrication of the damper.

A method of liquid refractive index measurement based on image correlation coefficient

According to the difference of speckle image caused by the axial displacement due to the refraction of a transparent medium, the liquid refractive index is measured by using the image correlation coefficient. An optical measurement system based on laser speckle is built into the experiment. Firstly, the CCD camera is adjusted to move along the optical axis in air, and multiple speckle images at different positions are collected, which are correlated with the speckle image at zero position to obtain the image correlation coefficients, and then the exponential calibration relation between the correlation coefficient and the axial displacement can be obtained. Secondly, two speckle images are collected when the laser speckle spreads respectively through the liquid to be measured and water as the internal standard, and the correlation coefficient between them is calculated. Finally, according to the above calibration relation, the corresponding axial displacement between the liquid to be measured and water can be derived, and then the refractive index of the liquid to be measured is calculated using the axial displacement formula. The results show that this method can realize the measurement of liquid refractive index, and the optical structure is simple and easy to operate.

Construction of homogeneous solutions of the torsion problem for a radially inhomogeneous transversely isotropic cylinder

The torsion problem for a radially inhomogeneous transversely isotropic cylinder of small thickness was investigated by the method of asymptotic integration of elasticity theory equations. It is assumed that the side part of the cylinder is stress-free, and boundary conditions are set at the ends of the cylinder, leaving the cylinder in equilibrium. The elastic moduli are thought to be arbitrary continuous functions of the variable along the cylinder radius. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thin-walledness of the cylinder. Homogeneous solutions are built, i.e. any solutions of the equilibrium equation satisfying the condition of no stresses on the side surfaces. It is shown that the solution of the torsion problem consists of a penetrating solution and a boundary layer character solution similar to Saint-Venant's edge effect in the theory of inhomogeneous plates. The penetrating solution determines the internal stress-strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the torsional moments of stresses acting in the cross-section perpendicular to the cylinder axis. Solutions having the boundary layer character are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in applied shell theories. Asymptotic formulas for displacement and stresses are built, which make it possible to calculate the three-dimensional stress-strain state of a radially inhomogeneous transversely isotropic cylinder of small thickness. Based on the obtained asymptotic expansions, it is possible to assess the applicability of applied theories and build a refined applied theory for radially inhomogeneous cylindrical shells

Study on the effect of tunneling and secondary grouting on the deformation of existing pipelines in bedrock raised strata

Shield crossing the bedrock raised strata easily leads to large deformation of the soil, resulting in deformation and damage of the upper pipelines. To investigate the law of pipeline deformation caused by shield underpassing in bedrock raised strata, the changes in the action range of the additional thrust of the cutter head (p1), the friction resistance of the shield shell (p2), and the additional grouting force (p3) were analysed. A convergence model of the excavation face suitable for bedrock raised strata was proposed. The calculation formula of soil displacement at the axis of the pipeline was derived by using Mindlin’s solution and stochastic medium theory. The solutions of the vertical displacement, bending moment and strain of the pipeline were obtained by the energy variational method and the principle of minimum potential energy. Pipeline deformation analysis and reliability verification were carried out by an engineering case in Hangzhou, China. The influence of bedrock distribution and secondary grouting behind the pipe wall on pipeline deformation was studied. The results show that the calculated results are similar to the measured data with a high degree of coincidence, and the degree of agreement is higher compared to the existing method. When the shield tunneling direction is well controlled, the settlement and bending moment of the pipeline caused by the shield crossing the raised bedrock stratum will be reduced, and the reduction increases with increasing bedrock length and intrusion depth. The secondary grouting behind the pipe wall can reduce the settlement of the pipe line, but it is necessary to reasonably control the three parameters of grouting amount per unit length (Vinj), grouting ring length (Lh), and grouting ring angle (δ). Lh should not be too long and can be controlled below 30 m, δ should be controlled within 60°, and Vinj can be adjusted according to the deformation size of the pipeline to prevent excessive or insufficient correction.

Effect of homogeneous generalized thermoelasticity on semiconductor layer under magnetic field on Green and Naghdi model without energy dissipation

This paper aims to explore the impact of viscosity and time on the spread of thermoelastic waves within a uniform and isotropic three-dimensional medium subject to a thermal load on its surface. This study utilizes the temperature-rate-dependent thermoelasticity based on the GN model, specifically applying the GN II model of generalized thermoelasticity, which does not account for energy dissipation. The normal mode analysis technique is employed to address the non-dimensional coupled field equations, yielding precise formulas for displacement, stress, temperature distribution, and strain. This issue is further illustrated by graphically depicting the field variables for a material similar to copper alongside the corresponding results. Comparative analyses of numerical data, with and without considering viscosity effects, suggest that the wave propagation speed will be limited.

The stress state of a plate with several holes under the action of axial tensile load

The purpose of the research is to develop a numerical method for determining the stress state of plates with several holes. The research approach is based on the development of the superposition method for singular solutions. The study objective is to solve a practical problem of determining the stress concentration coefficients in a plate with two holes of the same diameter. The study uses the numerical method of elasticity theory in the form of a superposition of four special solutions, each of which lies entirely within the corresponding hole contour. The resulting contour integrals in the displacement formula are replaced by their approximate values using the M-point trapezoidal formula or the Gaussian quadrature formula. Equating to zero the expression in variations of unknown constants (the principle of minimum potential strain energy), we have obtained a system of linear algebraic equations 4MN of unknown distribution densities. After determining the constants using known formulas, the stress-strain state of the plate with holes can be calculated. The error of the developed method is no more than 0.6% compared to the known solution.

Impacts of Surface Deformation Induced by Underground Mining of Metal Mines on Above-Ground Structures: A Case Study

The surface deformation caused by underground mining seriously affects the normal life and personal safety of local residents and also causes unfavorable factors for the safe and efficient exploitation of underground resources. While the study of surface deformation caused by underground mining of metal mines requires a large amount of measured data as support, the measured data is particularly scarce, which severely hinders the study of surface deformation caused by underground mining. In this paper, in order to study the impact of underground mining on surface structures in metal mines, we take the Fu Lao Zhuang Iron Mine in Anhui Province, China, as the research object and put forward a comprehensive measurement method based on the flat plate beam theory. Using empirical formulas combined with the methods of thickness-to-span ratio and the relaxation coefficient, etc., we carry out numerical simulation calculations for the displacement of the surface triggered by the mining of the ore body by using FLAC3D software. We calculate the maximum inclination deformation, curvature, and horizontal deformation values of the ground surface by referring to the displacement and deformation with reference to the displacement and deformation formula; the maximum tilt deformation, curvature, and horizontal deformation values of the ground surface are calculated, and finally, the permissible values of the design specifications are combined to make a judgment. The research results of this paper put forward the prerequisite for improving the surface deformation induced by underground mining.

Failure analysis of sandwich beams under three-point bending based on theoretical and numerical models

Abstract This article presents a comprehensive study on the failure behavior of foam core sandwich beams under three-point bending using theoretical analysis and finite element methods. A displacement formula for the foam sandwich beam is derived, considering the shear deformation of the foam core. Based on this formula, the deflection is obtained using energy and Rayleigh–Ritz methods. The failure loads of face yielding, core shearing, and indentation are combined to construct a failure mechanism map. The proposed theoretical model is then compared with existing theoretical analyses, demonstrating higher prediction accuracy. To investigate nonlinear damage and size effects, a series of finite element analyses is conducted. The results suggest that increasing the face sheet thickness has a greater impact on the ultimate load capacity, while the foam core thickness is more effective in enhancing bending stiffness.

A size-dependent quasi-3D model for bending and buckling of porous functionally graded curved nanobeam

In this work, we provide an analysis of the bending and buckling properties of functionally graded (FG) nanobeams with trigonometric function-dependent porosity distribution. Herein, a high-order curved nanobeam model considering both shear deformations and thickness stretching effect is established. Two-phase local/nonlocal stress-driven gradient theory is utilized to capture small-scale effects. Hamilton's principle is adopted to obtain the governing equation and boundary conditions, and the analytical solutions, such as exact formulas for lateral displacement and critical buckling load are obtained by employing Navier's method for simply supported boundary conditions. Furthermore, the effects of local volume fraction, nonlocal parameter, material length scale parameter, shear deformation, material distribution, and porosity distribution pattern in stress-driven models are analyzed in detail. The parametric studies indicate that the thickness stretching effect has a different influence on bending and buckling. In addition, the thickness stretching effect and the opening angle show different coupling mechanisms during bending and buckling. On the other hand, the higher-order buckling mode leads to a more significant scale effect and thickness tensile effect, but reduces the influence of the opening angle. Finally, we find that due to the coupling effect between the pore distribution pattern and the orientation bias of the FG nanobeam, the variation of the material gradient index leads to a different order of the dimensionless critical buckling loads predicted by the three pore distributions.

Corrections to the Theory of Elastic Bending of Thin Plates for 2D Models in the Reissner Approximation

Elastic bending stresses and strains in the lithosphere are typically calculated based on the Kirchhoff–Love (thin) plate theory. The criterion for its applicability is the smallness of plate thickness to length ratio. In oceanic plates, due to the buoyant force of the mantle, the main deformations are not uniformly distributed along the plate but are concentrated in the vicinity of the subduction zone. Therefore, the effective length of the bending part of the plate is a few fractions of the actual length, and the criterion of a thin plate is partially violated. In this paper, we analyze the possibility of applying thick plate bending equations. The existing variational theories of 3D bending of thick plates are much more complicated than the Kirchhoff–Love theory, as they involve solving three differential equations instead of one, and have limited application due to their complexity. Since geophysical applications frequently use 2D models, in this paper we analyze in detail the potential and accuracy of the thick plate bending theory for 2D models. After the conversion to the 2D plane strain and plane stress approximation, the original 3D Reissner thick plate bending equations are written out in the form similar to the Kirchhoff equations with additive corrections and are supplemented with the explicit formulas for longitudinal displacement. The comparison of the analytical solutions of the 2D Reissner equations with the exact solutions shows that the 2D approximation only provides a correction for the plate bending function. However, this correction refines the Kirchhoff–Love theory by almost an order of magnitude. At the same time, the solution of the equations in this case turns out to be almost as simple as the solution of the thin plate equations.

Loading Rate of Fracture Toughness Tests in Transition Regime

Abstract ASTM E1921-21, Standard Test Method for Determination of Reference Temperature, T0, for Ferritic Steels in the Transition Range, contains a limitation on the stress intensity factor loading rate (K˙) for the quasistatic loading. The specimens should be loaded at a rate such that K˙ during the initial elastic portion is between 0.1 and 2 MPa√m/s. However, based on recent improvements in the test technique, it should be possible to narrow the loading rate range in order to reduce variability and have more comparable results between laboratories. For all users to be able to achieve the reduced range, clear guidelines are needed, which are not yet available. The Belgian Nuclear Research Centre (SCK CEN) has extensive experience in fracture toughness testing and has constructed a large database on which statistical studies were performed. This paper illustrates, using the SCK CEN database, that Wallin’s theoretical displacement formulas can only be used in the case in which the compliance of the testing setup is taken into account. An improved equation is proposed, guidelines for implementing the equation are drawn up, and these are verified using the database. It is clear that the methodology is sufficiently robust to provide an average loading rate within the suggested range of 0.5 and 2 MPa√m/s. This methodology could thus be integrated into the testing standard ASTM E1921-21.