Deformation Formula Research Articles

Modified approach for predicting seismic-induced deformation of landfills considering strength parameters of GMB-GCL interface within the liner system

This study presents an innovative methodology for predicting seismic-induced sliding displacement, a key determinant in evaluating the seismic stability of landfills. The novelty of this research lies in the incorporation of the softening characteristics of the geosynthetic interface within the liner system, a factor that has been largely overlooked in previous studies. A dynamic stability analysis was performed on a landfill using the ABAQUS software, with an emphasis on the impact of coupled parameters, particularly the strength of the interface. The results highlight PGA (Peak Ground Acceleration), PGV (Peak Ground Velocity), Ia (Arias Intensity), and residual interface shear strength (μ) as effective predictors. The study further identifies the combination of PGA and μ as the optimal parameter pairing for predicting the seismic permanent deformation of the landfill. A multi-dimensional data regression approach was employed to propose a calculation formula for seismic permanent deformation, taking into account liner damage. To enhance the seismic design methodology for landfills, the probability density function was integrated into the study. This innovative approach provides a new perspective on seismic stability assessment in landfill engineering and designs.

Normal contact stiffness modeling and contact characteristic research of scraped surfaces considering the truncated cone high points

In order to study the normal contact characteristics of scraped surface, a normal contact stiffness model of scraped surface was proposed considering truncated cone high points, where scraped surface was equivalent as tandemly connected truncated cone high points with asperity. The high points parameters such as the percentage of high points (POP) and the height of high points (HOP) were obtained by lapping-image processing and high points height histogram, respectively. Through high points elastic deformation formula and fractal asperity elastic-plastic deformation formula, the normal contact stiffness model of scraped surface is established. Results show that increasing fractal dimension and reducing contact gap by lapping, while decreasing HOP and increasing POP by scraping can improve the contact characteristics of scraped surface.

Design and development of a flexible-rigid Triglide sorting manipulator

Flexible manipulators have been widely applied in various tasks involving grasping and manipulation, and their unique characteristics in terms of lightweight design and adaptability make them particularly suitable for tackling complex spatial pick-and-place operations. In this work, a Triglide flexible manipulator is presented and developed, based on the parallel Biglide mechanism and dual leaf springs in parallelogram structure, which features adjustable stiffness by changing the transverse configuration of the leaf springs. To analyze the deflections of the leaf springs efficiently, a segmented approach is deployed, to calculate large deflection within the plane of the leaf-springs parallelogram structure. The adopted approach uses classical bending deformation formulas from mechanics of materials. Consequently, the static models of the manipulator are derived for analyzing bending deformation and workspace by means of numerical calculations. Finite element analysis is adopted to investigate the mechanical characteristics of the manipulator. A prototype of the variable stiffness manipulator is built, and an experimental setup is established for static testing. The obtained results align with the previously observed mechanical characteristics. The main advantages of the flexible robotic manipulator lie in its simple structure, small footprint, and simple kinematic model for control.

Anti-progressive collapse performance and design method of novel T-stub connections

To address the insufficient anti-collapse capacity of T-stub connections due to premature failure, this study proposes a novel T-stub connection (NTC) by adding four V-shaped laminated plates. Monotonic loading tests are conducted on the T-stub web and V-shaped laminated plate to analyze the influence of geometric parameters on their deformation and load-bearing capacity. The working mechanism of NTC is investigated through the validated numerical models by analyzing the failure mode, impact force history, deformation pattern, internal force development, and energy dissipation. The research results indicate that, compared to the T-stub connection, the NTC exhibits two-phase failure characteristics, which begin with the fracture of the bolt-hole in the T-stub web, followed by the gradual straightening and eventual fracture of the V-shaped laminated plates. NTC undergoes three stages under impact loads: peak, oscillation, and platform stages. The addition of V-shaped laminated plates enhances the ductility of NTC by 93 %−130 % and its energy dissipation capacity by 189 %−235 %. This significant enhancement in deformation and energy dissipation capabilities can be attributed to the addition of the V-shaped laminated plates. The deformation and bearing capacity calculation formulas for both the T-stub web and V-shaped laminated plate are provided. Additionally, a design method for NTC is proposed based on theoretical analysis, and its rationality is verified through numerical examples involving three different sets of beam and column sections.

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.

The occurrence and mechanism of hysteresis between axial deformation and short‐circuit electromagnetic force during the vibration of power transformer windings

AbstractMechanical stability is one of the core capabilities of power transformers. External short‐circuit accidents are the main cause of winding instability. International Electrotechnical Commission 60076‐5 standard recommends a method to calculate the short‐circuit strength of power transformer windings by comparing the stress within windings under the effect of the maximum electromagnetic force with the critical stress of the winding. This method assumes that the maximum deformation will be produced by the maximum electromagnetic force, which corresponds to the first peak of the waveform. However, owing to the interactions between disks during the vibration process, the maximum deformation may occur after the occurrence of the maximum electromagnetic force. The hysteresis phenomenon between disk deformation and electromagnetic force is studied. The definition of the hysteresis phenomenon during the vibration process is first demonstrated, and the mechanism of the hysteresis phenomenon is investigated. The vibration model is established. By decoupling analysis, the conditions for the formation of hysteresis are proposed, and the mechanism of the hysteresis phenomenon is validated by the experiment, which is conducted on a winding sample. In the deformation formula, the term that determines the time‐varying characteristic is found. The waveform‐determining term is the difference between the two cosine components, whose frequencies are the natural vibration frequency and the electromagnetic force frequency. When the two frequencies are close, the maximum deformation lags behind the maximum force, and the hysteresis phenomenon occurs.

Theoretical, numerical, and experimental investigation on second-order Bezier curve flexure hinges

This paper proposes the second-order Bezier curve flexure hinge. The flexure hinge can obtain different notch profiles by changing the position of the control points. Firstly, the compliance and precision of the flexure hinge are modeled based on the force deformation formula of the cantilever beam in material mechanics. Compared with the simulation data of ANSYS Workbench, the modeling error is less than 7%. Secondly, the influence of geometric parameters on compliance and precision is analyzed. The flexure hinge is compared with circular, elliptical, and parabolic hinges. The results show that the second-order Bezier curve flexure hinges have a higher compliance precision ratio. In addition, the use of second-order Bezier curve flexure hinges provides better performance than the use of traditional circular and elliptical flexure hinges in bridge amplification mechanisms. Finally, a flexure hinge was manufactured and measured. The experimental results show that the error between the experimental and modeling values is 8.76%. Both the simulation and experimental results verified the effectiveness of the model.

A Novel Deformation Analytical Solution and Constitutive Model for Fractured Rock Masses

In order to study the deformation and stability of a fractured rock mass, existing research suggests that fracture deformation, which is usually obtained by evaluating the equivalent deformation modulus, dominates the deformation of a fractured rock mass. However, this parameter is difficult to obtain in theory and practice, which limits the application of rock mass deformation analysis methods. In order to calculate the deformation of fractured rock masses, the mass of the rock is regarded as a sponge-like material, and it is assumed that the deformation of a water-saturated fractured rock mass under external force load is approximately equal to the net flow of fracture water. Based on this assumption, firstly, the relationship between rock mass deformation and fracture flow is studied through a single-fracture rock mass model. The hydraulic properties of the fracture are characterized by the permeability coefficient, and the fracture deformation in the rock mass is equivalent to the fracture flow. The fracture deformation calculation formula is derived from the fracture hydraulics calculation formula, and this is compared with the measured data. The rationality of the calculation formula was verified. On this basis, the calculation formula for rock mass deformation, including multiple groups of fracture surfaces, is proposed, and the stress-strain constitutive relationship of a complex rock mass is established. The correctness of the calculation method was verified by comparing it with other theoretical calculation results.

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.

Effect of different roller end-flange constructions on the fatigue life of the cylindrical roller bearings: A novel flange deformation formula

This paper performs the fatigue life analysis of the radial Cylindrical Roller Bearings (CRBs) with consideration of various roller end-flange shapes such as toroidal-toroidal, spherical-toroidal, and spherical-conical. A novel formula for each flange contact deformation in cylindrical roller bearing with different roller end-flange geometry including toroidal geometry is developed. Inner ring misalignment angle and radial deflection results obtained from the present study are verified with numerical and experimental results from the literature. The results approach to that of literature when the flange is changed from toroidal geometry to conical geometry. Using formulas developed in the present study, the effect of various roller end-flange geometries on the bearing life is investigated for different external loadings. It is observed that the bearing life increases when the roller end and flange are changed to toroidal geometry.

Deformations of a three-dimensional model of a trihedral double lattice rod tower

Introduction. The calculation of deformations of spatial structures, as a rule, is carried out numerically on the basis of the finite element method. With the development of computer mathematical systems for regular systems an opportunity to obtain analytical solutions appeared. Such solutions can be used as test ones to evaluate numerical solutions and for preliminary calculation of a structure at design stage. The task has been set to derive deformation dependences of a spatial truss under various loads from its size and number of panels.
 
 Materials and methods. The truss of the tower structure is statically determined. The calculation of force values in the bars is performed by cutting out nodes in a Maple symbolic mathematics system. The Maxwell – Mohr formula is used to determine the displacement of a node at the top of the frame, considering only the longitudinal deformations of the rods. From the generalization of a series of analytical solutions for trusses with consecutively increasing number of panels onto an arbitrary number of panels the sought formula is derived by the induction method.
 
 Results. An algorithm of derivation of formulas for the deflections of a building along two mutually perpendicular horizontal axes under the action of lateral uniformly distributed nodal loads is presented. Effort distribution patterns are obtained for the truss rods and analytical stress dependences of some rods on the number of panels are found. Linear asymptotics of the solutions for the number of panels and the points of extremum are found and computed.
 
 Conclusions. The considered model of the spatial statically determined tower truss allows deriving exact formulas for deformations under the action of various loads within the limits of the adopted model. It is possible to use derived formulas for a preliminary estimation of designed structure and to apply as test ones for numerical calculations. The extremum points and analytical expressions for asymptotes allow the derived formulas to be used for solving stiffness optimization problems of a structure.

Study on the Influence of Foundation Pit Excavation on the Deformation of Adjacent Subway Tunnel in the Affected Area of Fault Zones

In order to understand and mitigate the deformation issues caused by the excavation of foundation pits within the influence area of fault zones, ensuring the safety of the subway system and supporting the sustainable development of the city, this paper takes the foundation pit project near the Shenzhen Metro Line 5 in the F1322 fault zone influence area as the research background, and uses theoretical analysis, numerical simulation and on-site measurement and other research methods to carry out in-depth research on the deformation problems of the adjacent subway tunnel caused by the unloading of the foundation pit excavation. The research results demonstrate that by considering the collaborative deformation effect between retaining piles and excavation sidewall soil under the spatial impact of foundation pit excavation and using the Mindlin solution to calculate the additional stress on the sidewall and bottom of the foundation pit, a deformation calculation formula for the tunnel has been established. The impact characteristics of different geological parameter variations on the horizontal and vertical displacements of the tunnel have been analyzed. The sensitivity analysis of maximum tunnel displacement to different geological parameters reveals that the most sensitive factor is the elastic modulus, followed by the internal friction angle, while the cohesion has the least influence. By fitting the data, it has been found that the maximum deformation of the tunnel under composite strata excavation exhibits a good linear relationship with the index bH/L and a prediction formula for the maximum tunnel deformation under this geological condition has been developed. Therefore, the research findings of this study can be utilized to assess the impact of foundation pit excavation on the deformation of subway tunnels under similar geological conditions. They can also be employed to formulate construction monitoring plans and risk management strategies, contributing to the safety of subway systems and the sustainable development of cities.

Analysis and Prediction of the Factors Influencing Postconstruction Surface Deformation of Pipe-Jacking Tunnel in Soft Clay Strata in China

During the postconstruction operation period, the engineering structure and operation safety of a pipe-jacking tunnel are more likely to be threatened by the surrounding environment and groundwater when it is located in a soft clay stratum. Surface settlement data related to the construction of 24 rectangular pipe-jacking tunnels in a soft clay stratum in China were collected and analyzed to determine the effect of pipe-jacking tunnel engineering on its deformation based on factors such as structural section form, geotechnical condition, groundwater condition, and buried depth. The postmaximum surface settlement and postformation loss rate were obtained, the correlation coefficients between the postsurface deformation of soft clay strata and various influencing factors were calculated using a software for correlation analysis, and the variation rules for each influencing factor were explored in depth. Subsequently, multiple linear regression methods were used to predict the maximum surface deformation of the pipe-jacking tunnel project in soft clay strata during the postoperation period, and actual engineering cases were verified and analyzed. The results show that the relative buried depth of the pipe-jacking tunnel in soft clay stratum has the greatest influence on the postmaximum ground settlement, followed by the section area of the pipe-jacking tunnel, and the relative height coefficient of groundwater level has the least influence. Under the reduction coefficient of 0.937 − 0.846, a new prediction formula for surface settlement deformation after the construction of the soft clay layer was derived. The error between the formula and the measured data was 9.05%, which can be used as a reference for controlling design parameters to predict the settlement after construction of pipe-jacking engineering.

An empirical formula for the accumulative deformation of saturated clay under different cyclic loading frequencies

Road and subway tunnels in soft clay soils subjected to traffic loading often cause lager settlement. It can pose danger to the surrounding buildings and structures. Hence, it is necessary for engineers to correctly predict the long-term deformation of saturated clay foundations under cyclic loading. In this paper, a series of dynamic triaxial tests were carried out on Nanning clay to investigate the effect of loading frequency and cyclic stress on axial strain accumulation of clay under undrained condition. Results show that loading frequency has a great effect on the axial strain accumulation of clay. Based on the experimental results, an empirical formula for calculating the accumulated deformation of saturated clay under cyclic loading was presented.

Giant flexoelectric effect with liquid crystal dimer CB7CB between concentric cylinders

ABSTRACT Liquid crystal dimer CB7CB (1″,7″-bis(4-cyanobiphenyl-4′-yl)heptane) is confined between two concentric cylinders with the liquid crystal director initially aligned axially along the concentric cylinder. When a non-uniform radial electric field perpendicular to the concentric cylinder axis is applied, the director is gradually changed as the applied voltage increases. We obtain a threshold voltage analytical formula for the angular deformation of the director between two concentric cylinders. In addition, the effect of different flexoelectric coefficients on the deformation of the director between two concentric cylinders under the radial electric field was also investigated. The results show that the flexoelectric coefficients will determine how the director will be deformed. It provides a more accurate theoretical analysis for predicting the coupling effects of dielectric effect and giant flexoelectric effect on the director orientation distribution between two concentric cylinders under a non-uniform radial electric field.

Estimation and Adjustment of Interface Stiffnesses for Machine-Tool

This study adopts a practical approach to establish the estimation and adjustment methods of the interface stiffness for the machine-tool through the finite element analysis (FEA). First, numerical and experimental modal analysis (FMA and EMA) for each single subsystem of the machine-tool structure are performed. Then, the parameters obtained from EMA are used as the objective criterion function, and the FMA is conducted iteratively to solve the material Young’s modulus and Poisson’s ratio for each single subsystem structure, in which the geometrical model is simplified and FE mesh convergence is performed to ensure the quality and efficiency of the numerical analysis. Next, by considering that the machine-tool is assembled by subsystem stacking and that loading on each contact surface is deduced, the FE method is used to calculate the deformation of each contact interface after the single subsystem is stacked successively. The initial value of the interface stiffness estimated by the deformation formula in the mechanics of material is utilized as the initial condition for the iterative calculation in FMA. The changes of the modal parameters are observed in the analysis and the interface, which have a significant impact on the natural frequency variations of the whole machine-tool is selected as the main adjustment object. Then, the adjustment method proposed in this study is applied repeatedly to modify this interface stiffness. The results show that footing interface stiffnesses play the most important role that intensively affects the numerical analysis results of modal parameter. After the repeated adjustments of the interface stiffnesses of the footing, the error of the natural frequency of the whole machine-tool is less than 5%, which is calculated from the comparison between EMA and FMA results. It indicates that the proposed adjustment method in this study for footing interface stiffness determination has a valuable reference in practical use.

An analytical nonlinear dynamic model for linear guide with carriage flexibility

Accurate estimation of vibration characteristics of linear guide is imperative for sophisticated structural performance of linear motion equipment. It has been illustrated that flexible model considering carriage deformation is a more precise description of realistic system. However, few dynamic model of carriage flexibility is found for vibration analysis when complex external load is applied. This paper proposes an analytical model to analyze the nonlinear vibration behavior of flexible linear guide system under multidirectional harmonic excitations. The outward deformation of flexible carriage skirt region is calculated by the classical deformation formula of vertical loaded cantilever beam. A lumped dynamic model accounting for segmented restoring force is proposed by coupling the vertical, lateral and rolling displacement and outward deformation. Dynamic responses of rigid and flexible models are compared, and the condition that neglects carriage flexibility is obtained. Numerical simulation exhibits abundant and fascinating nonlinear phenomena when the effects of external excitation, preload and carriage weight are discussed. Additionally, excitation experiments are performed to verify the proposed model. The developed theoretical model is promising for structure optimization of linear guide.

The Deformation and Shear Vortex Width of Flexible Vegetation Roots in an Artificial Floating Bed Channel

As an ecological measure to deal with river water quality problems, artificial floating islands have been widely used all over the world, but the research on root deformation and shear vortex width under the action of artificial floating islands is rare. In this paper, the relationship between the deformation of vegetation roots and parameters of vegetation roots under different hydrodynamic conditions is experimentally studied. The results show that the Cauchy number (Ca) value gradually increases with the increase of velocity, and that the smaller the diameter is, the greater the Ca value is. The value of the buoyancy number (B) will increase with the increase of root length and will decrease with the increase of root diameter. The corresponding deformation formula of flexible root systems under hydrodynamic conditions is obtained, and has high simulation accuracy. Based on theoretical analysis and machine learning, a formula for the shear vortex width of flexible vegetation is established, δ=0.361+0.0738lelαCdf. The research results can provide a theoretical basis for hydrodynamic and solute transport in artificial floating island channels.

Universal deformation formula, formality and actions

In this paper we provide a quantization via formality of Poisson actions of a triangular Lie algebra $(\mathfrak{g},r)$ on a smooth manifold $M$. Using the formality of $E$-polydifferential operators for Lie algebroids $E$ over $M$, we obtain a deformation quantization of $M$ together with a quantum group $\mathscr{U}{\rho\hbar}(\mathfrak{g})$ and a map of associated DGLA's. This motivates a definition of quantum action in terms of $L\_\infty$-morphisms which generalizes the well-known definition given by Drinfeld.

Analysis of the protective effect of setting isolation piles outside the foundation pit on the underpass tunnel side

Excavating and unloading a foundation pit cause deformation or even destruction of any existing adjacent tunnel. In construction projects, isolation piles are often set between the foundation pit and tunnel to protect the existing tunnel. The aim of the present study was to investigate the horizontal displacement law of an existing tunnel outside the isolation piles as a result of foundation pit excavation, so as to analyze the protective effect of isolation piles on the existing tunnel. The unloading effect of the sidewalls, the unloading effect of the base of the foundation pit, and the deflection deformation of the isolation piles were considered. A three-dimensional mechanical calculation model for foundation pits, isolation piles, and existing tunnels was established. Based on the Mindlin’s solution, the calculation formula of the additional stress caused by the unloading of the foundation pit was deduced. Based on the energy variation method, the calculation formula for the deflection deformation of the isolation piles was derived. Using the mirror image method, the calculation formula for the horizontal displacement of the existing tunnel caused by the deformation of the isolation pile was derived. The influence of different spacing distances between isolation piles on the horizontal displacement of an existing tunnel was studied. The research results showed that the calculation results in this study were consistent with the actual engineering data available and numerical simulation results. It was concluded that the installation of isolation piles could effectively reduce the horizontal displacement of tunnels and fulfil a beneficial protective role, and that the protective effect gradually increases as the distance between isolation piles is reduced.