Displacement Formula Research Articles

On the Development of Refined Plate Theory for Static Bending Behavior of Functionally Graded Plates

This work gives information about the development of refined plate theory to study the static bending behavior of functionally graded material (FGM) plates. The significant advantage of our proposed theory is that only one unknown variable exists in its displacement formula and governing equation. To illustrate the accuracy and effectiveness of this theory, an analytical approach based on the Navier solution is employed to obtain the solution for static bending of simply supported FGM plates. A good agreement for static bending of FGM plates with other literature results has been instituted. This work also investigates the deflection, in-plane normal, and shear stresses of sinusoidally loaded FGM rectangular plates with four simply supported edges. The influence of some parameters on the bending performance of FGM plates is also carefully considered.

A method of constitutive experiment for accurate description of tube performance under rotary-draw bending

Establishing accurate constitutive relations is an important prerequisite for improving the springback prediction accuracy of rotary-draw bending of tubes. In this paper, the stress loading paths are studied for rectangular tube and composite circular tube in rotary-draw bending processes. It is found that there is reverse loading with tangential/circumferential stress for both the rectangular tube and the base/covered tube of composite circular tube. And the reverse loading is in the plastic deformation range. Two constitutive experimental schemes are designed according to the above research results: (1) The constitutive experiment is designed according to the variation of stress peak points of the tube with the bending process and the yield state of the material. The calculation method of the loading force F is given, too. (2) The displacement of the beam of the tensile tester is determined according to the corresponding strain during stress fluctuations, and an example is given to illustrate how the displacement of the beam is calculated according to the strain corresponding to each trajectory point. The calculation formula of the beam displacement during repeated loading, unloading and reverse loading is developed, too.

Enlightening the role of critical parameters for the determination of the tensile strength by means of the Circular Semi Ring test

Analytic formulae for stresses and displacements are provided for the Circular Semi Ring under either compression or tension. The respective expressions for the tensile strength for both tests are obtained and compared to existing ones. In addition, it is shown that smooth contact between the specimen and the loading device should be preferred. The suggested tests seem to remove some drawbacks of existing methods for obtaining the tensile strength of brittle materials: Fracture is achieved at points where the stress field is purely tensile and, also, very low external force is required thus avoiding premature fracturing at the specimen-loading device contact region. From the theoretical point of view the tension test appears to be more advantageous regarding the very low fracture load and the fact that the maximum tensile stress is higher in absolute value from the respective compressive one.

Stress effect of the interface between buried pipeline and sandy soil layer in a cold environment

The interaction between buried pipe and soil is an important aspect of disaster prevention and mitigation. In this paper, interfacial stress tests between room-temperature or frozen sand and pipeline were performed to study the interfacial stress problems experienced by a buried pipeline in sand. According to the basic assumptions, a calculation model for the mechanical stress of the buried pipeline was established. The experimental data were used to verify the correctness of the theoretical calculation model and the stress–displacement formula of the buried pipeline interface was fitted. The stress–displacement curve of the interface between the pipe and the room-temperature soil was compared with the stress–displacement curve of the interface between the pipe and the frozen soil. The results show that the interfacial stress value calculated by the theoretical model of pipe–soil interfacial stress is consistent with experimental values. The stress at the pipe–soil interface after the sand was frozen is much larger than the stress at the pipe–soil interface when the sand is at room temperature. Frozen soil has a great influence on the pipe–soil interaction. This theoretical formula can be used to analyze the interfacial stress problem for the pipe–soil coupling and lay a foundation for further studies of the influence of the permafrost–pipeline interaction on the stress and strength of the pipeline.

Statistical Coulomb interactions in multi-beam SEM

Statistical Coulomb interactions in conventional scanning electron microscopy mostly affect the probe size via energy spread and virtual source broadening in the emitter vicinity. However, in a multi-beam probe forming system such as a multi-beam scanning electron microscopes (MBSEM), the trajectory displacement due to interactions in the whole column can give a contribution to the final probe size. For single-beam systems, this can be expressed using approximate formulae for the total trajectory displacement in a beam segment (Jansen’s theory) or by integrating contributions of infinitesimally thin beam slices (the slice method). We build on Jansen’s theory of statistical Coulomb interactions and develop formulae for the trajectory displacement in a multi-beam system. We also develop a more precise semi-analytical result using the slice method. We compare both approaches with a Monte Carlo simulation and show a good agreement with the results of the slice method. Finally, we discuss the implications of our results for the optical design of multi-beam SEM. In a multi-beam with probe size dominated by Coulomb interactions, an increase in the number of beamlets does not necessarily provide an increase of throughput, because the probe size is limited by the total current. Furthermore, we disprove the notion of “the fewer the crossovers — the less the Coulomb interactions” by showing the quadratic dependence of trajectory displacement on segment length.

Landslide displacement prediction based on variational mode decomposition and WA-GWO-BP model

Many models have been widely used in landslide displacement prediction. However, few studies have proposed quantitative prediction formulas. Thus, the variational mode decomposition (VMD) theory was applied to decompose the “step-like” displacement of landslides into trend displacement, periodic displacement, and random displacement. Then, a novel prediction model based on wavelet analysis (WA) and a back-propagation neural network (BPNN) optimized by the grey wolf optimizer (GWO) algorithm was proposed (the GWO-BP model) to obtain a prediction formula. In this model, a polynomial function was first used to predict the trend displacement. All the hidden periods of periodic displacement were calculated using the WA method, and a trigonometric function was applied to predict the periodic displacement. In addition, based on an analysis of the grey relational degree (GRD), the main triggering factors, which can affect the random displacement, were determined. Then, the mathematical connection between random displacement and triggering factors was obtained with the GWO-BP model. Finally, all the predicted values were superposed to achieve the prediction cumulative displacement based on the time series model. The Outang landslide in the Three Gorges Reservoir area, China, was taken as an example, and the displacement data of monitoring sites MJ01 and MJ02 from December 2010 to December 2016 were selected for analysis. The results indicated that the root mean square errors (RMSE) between the real displacement values and the prediction values obtained using the formula were 14.79 mm and 12.59 mm, respectively. The correlation coefficient R values were 0.99 and 0.93, respectively. This model can be used to obtain the landslide displacement formula and provide a solid basis for developing early warning systems for landslides.

Stress and strain intensity factors within angle area of the boundary

Stress and strain state of the designs and structures within the area (zone) with angle cutouts, boundary cuts are characterized by emergence of the stress concentration areas and requires assessment of strength and reliability of facilities that is an actual task of the engineering practice. The theoretical study of stress and strain state within angle cutouts of the boundary area is reduced to research of singular solutions for the elastic problem with power singularity. Moreover, definition of stress or strain concentration in the non-regular point of the area boundary loses its significance. The stress and strain state within angle cutout top of the plane area boundary, which is recorded with use of intensity factor, is considered in the article. Two approaches to obtaining of formula for displacements and stresses within non-regular point of the plane area boundary with use of the stress and strain intensity factor are given. A difference in formula of stresses and displacement obtained for the limit values of stress and strain determine the practical significance of the work during experiments and determination of critical values for stress and strain.

Static analysis of tall buildings based on Timoshenko beam theory

In this paper, the continuum model, which is known as Kwan model, has been presented for the analysis of tall buildings that have been as an appropriate approximation of the overall behavior of the structure. Tall building was modeled as a cantilever beam and analyzed with the assumption of flexural behavior based on Euler–Bernoulli Beam Theory, then the displacement of floors was calculated. o consider the shear lag effects in the overall displacement of the structure, Timoshenko’s beam model has been considered and related relations were extracted. The lateral displacement formulas obtained and calculated for the framed tube system modeled by Kwan’s method. To verify the results, numerical models were created in software (ETABS) and statically were analyzed for lateral loading. Finally the results were compared with those obtained by computer analysis and the corresponding diagrams were presented. At the end, the shape factor formula has been developed to improve the results of the Timoshenko’s theory.

The field of tensions of a rotating anisotropic disc of a variable thickness loaded with undistracted forces on the outer contour

The work gives a solution of the plane elasticity problem for rotating polar-orthotropic annular disks of a variable thickness. The disk is loaded with a system of equal focused forces on the outer contour applied evenly along the rim and symmetric concerning the diameter. The disk is seated with an interference fit on the flexible shaft so that a constant contact pressure acts on the interior contour. The stresses and deformations arising in such a rotating anisotropic annular disk will be non-axisymmetric. A conclusion of a fourth-order partial differential equation for the effort function is drawn. Its general solution is searched out in the form of a Fourier series of cosines with even numbers. As a result, an infinite system of ordinary differential equations is solved for the coefficients of the series. These differential equations correspond to the linear Volterra integral equations of the 2 nd kind, which are solved using resolvents. Constants of integration are determined from the border conditions. Expressions for the stress components are written through the effort function by the well-known formulas. We find the components of the displacement vector in the disk by the integration of the Hooke’s law equations for the polar-orthotropic plate. We calculate the deformation components in a ring anisotropic disk by Cauchy differential relations if we know the displacements. The solved formulas for stresses, deformations and displacements completely describe the stress-deformed state in a rotating polar-orthotropic disc of variable thickness with a system of focused forces on the outer contour. The results of the work can be used in the design of working disks of turbomachines and turbo compressors, as well as rotors of centrifugal stands.

Analysis of jet oscillations with acoustic radiation in the recorder by direct aeroacoustic simulations.

To elucidate the selection mechanism of a predominant mode in acoustic radiation from the recorder, the jet oscillations predicted by direct aeroacoustic simulations are analyzed based on a proposed formula for hydrodynamic and acoustic jet displacements. The displacements are well represented by the formula, taking into account the non-zero initial amplitude around the windway exit and variations of oscillation center with streamwise position for jet displacement. The analysis is applied to the jet oscillations in two different recorders with a straight- and an arch-shaped windway, where the shift of the predominant mode from the first to the second mode occurs for the straight-shaped recorder at a lower jet velocity compared to the arch-shaped recorder. The analytical results present the influence of the recorder shape on the amplification rate of the hydrodynamic jet displacement, the acoustic feedback effects and the phase relation between the hydrodynamic jet displacement and acoustic pressure in the resonator, along with the jet offset to the edge. Compared to the arch-shaped recorder, the convex curve of the amplification rate with the non-dimensional frequency based on the windway height for the straight-shaped recorder is located in the higher-frequency region, which contributes to the predominance of the higher mode.

Simplified Calculation of Pylon Top Displacement of Multi-Pylon Suspension Bridge

The long-span multi-pylon suspension bridge is the subject of growing interest. Under live load, the longitudinal deflection of the mid-pylon is an important control parameter for a multi-pylon suspension bridge design. It is important to establish a simplified method of calculating pylon deflection for the preliminary design and selection of a multi-pylon suspension bridge. Based on deflection theory and deformation compatibility condition, considering main cable horizontal constraints and the interaction among pylons and girders with the methods of equivalent stiffness and moment distribution, the simplified calculation formulas of pylon displacement at the top for three-, four-, and five-pylon suspension bridges are derived. The validity of the formulas are verified by model experiment, real bridge testing, and finite-element analysis. For a floating system, the error between the simplified formula and the finite element method is less than 10%, and that of the model experiment is within 25%. For a consolidation system, the error between the simplified formula and the finite element method is within 16%, and that of the real bridge testing is less than 11%. As the number of pylons increases, the simplified formulas tend to be less accurate.

Study on Synthesis Stiffness of Inner Supported Deep Excavation in Soft Soil Area

The horizontal displacement of retaining structure is particularly associated with synthesis stiffness of support system. The synthesis stiffness of support system is highly interrelated with three parts including the retaining wall, the inner brace and the soil at bottom of excavation. In order to estimate the lateral movement of retaining wall, a synthesis stiffness formula is been proposed. The proposed model concludes a variety of factors, such as horizontal flexural rigidity of retaining wall, compressional stiffness of inner brace, average vertical support spacing, average horizontal support spacing, depth of wall penetration and stiffness of the soil below final excavation depth. Comparing with the Clough synthesis stiffness, more factors is been considered. Comparing with the MVSS synthesis stiffness, the factor of soil below final excavation is been defined exactly. After that, finite element method is employed to calculate the maximal lateral displacement of the retaining wall. And a good relationship is established between maximal horizontal displacement and synthesis stiffness formula. Finally, the calculations are used to compare with the observed data from several deep excavations, the result validates the applicability of proposed model. The research provides a new method for describing the comprehensive stiffness of support system of foundation pit. It is convenient to estimate the maximum horizontal displacement of the retaining structure at the design stage and to select the support system quickly.

Fractional Thermoelasticity Model of a 2D Problem of Mode-I Crack in a Fibre-Reinforced Thermal Environment

A model of fractional-order of thermoelasticity is applied to study a 2D problem of mode-I crack in a fibre-reinforced thermal environment. The crack is under prescribed distributions of heat and pressure. The normal mode analysis is applied to deduce exact formulae for displacements, stresses, and temperature. Variations of field quantities with the axial direction are illustrated graphically. The results regarding the presence and absence of fiber reinforcement and fractional parameters are compared. Some particular cases are also investigated via the generalized thermoelastic theory. The presented results can be applied to design different fibre-reinforced isotropic thermoelastic elements subjected to the thermal load in order to meet special technical requirements.

Rotational motions of the Ms7.0 Jiuzhaigou earthquake with ground tilt data

Observation and research of rotational motions induced by earthquake has been ignored in the past decades due to the lack of understanding of important ground rotational motions and practical difficulty involved in directly measuring the rotational components. Currently, there is a paucity of methods to directly measure rotational motions in China. The present study attempts to discuss the possibility of obtaining seismic rotation from ground tilt data. The rotational motions generated by shear dislocation source are derived based on the displacement formula of elastic wavefields in the homogenous space. The characteristics of the three components of translational and rotational motions in near, intermediate, and far fields are compared and analyzed. Based on the corresponding relationship between the rotation and ground tilt on free surface boundary, a method to indirectly construct the rotational motions in the layered half space is obtained. Finally, the theoretical rotational motions generated by the M s7.0 earthquake in Jiuzhaigou, Sichuan Province, on August 8, 2017 are calculated. The results indicate that the rotational motions generated by the Jiuzhaigou earthquake are concentrated in the vicinity of the epicenter (approximately 150 km) and mainly distributed in the area perpendicular to the fault strike. The existence of free surface boundary attenuates the total energy of the rotational field and especially in the direction of the dominant energy. The ground tilt data is compared with rotational motions in the layered half space, and the results indicate that theoretical rotations and actual ground tilt data are essentially consistent, especially in the far field region. Thus, it is possible to obtain horizontal components of rotation from ground tilt data, and this provides more information for a complete description of the motion characteristics of an underground medium.

Long-range propagation through inhomogeneous turbulent atmosphere: analysis beyond phase screens

Many applications rely on the propagation of electromagnetic waves through extended regions of the atmosphere over which the refractive index can vary in a complex manner. Gradients and curvature of the mean refractive index profile result in ray bending and the associated phenomena of mirages, atmospheric lensing, and wave trapping in parabolic cavities. Stochastic refractive index fluctuations due to turbulence cause a random displacement of the trajectory and give rise to the wander, or spot dancing, of a propagating optical beam. In this paper we model these features of the refractive index profile locally and describe propagation through the corresponding regions. We derive formulas for the mean ray path that capture the effects of both atmospheric turbulence and variations in the mean refractive index profile, including the non-paraxial effects associated with the bending of the guiding ray path. We also give formulas for the mean-squared transverse displacement of a ray from the mean trajectory, which can provide for example an estimate of the magnitude of the beam wander due to turbulence.

STRESS INTENSITY FACTOR CALCULATION FOR AUTOFRETTAGED TUBE TEST-SPECIMENS USING FINITE ELEMENT METHOD

The residual stresses, which are generated due to an autofrettage process, in a cracked autofrettaged tube may cause partial crack tip closure. In such a case, the superposition of the stress intensity factor due to both mechanical loading and residual stresses will not produce accurate results. It is, therefore, important to consider the crack closure effect in the calculation of the stress intensity factor. This has been done, in this paper, by using contact elements on the crack faces in order to prevent overlapping. In a first step, the residual hoop stresses for different degrees of autofrettage are calculated from a two-dimensional axisymmetric non-linear elastic-plastic finite element model. Elastic perfectly plastic material behaviour is assumed. The finite element results are in good agreement with the analytical solution found in the literature. For the stress intensity factor calculation, three different autofrettaged test-specimens are considered, namely ring, C-shaped and split ring. Crack length to specimen width ratio between 0.1 and 0.8 and several degrees of autofrettage are considered. Contact elements on the crack faces are used in order to prevent the overlapping and account for any possible partial crack surface closure. A spring and gap in series define the contact element. The stress intensity factor is calculated from the opening displacements behind the crack tip using the two-point displacement formula and the one-point formula.

Surface Topography Analysis of Point Contact Cutting Considering Vibration

Precision manufacturing engineering requires not only excellent materials but also precise machining methods. The point contact cutting is widely applied to precision turning, milling, grinding, and so on. However, the machining principle and the surface topography formation process of the point contact cutting are not well described mathematically. To solve the problem, we proposed a new surface topography analysis method considering vibration in this paper. First, based on the geometry topology theory, we reduced the dimension of the point contact cutting process. Moreover, the surface topography displacement formula was derived. Finally, the relationships between the influence factors (the shape of the cutting tools, the cutting parameters, and vibration) and the surface topography displacement were analyzed and explored. The surface topography analysis method proposed in this paper provides the theoretical foundation for designing, controlling, and optimizing the point contact cutting process.

Influence of the driving frequency and equivalent parameters on displacement amplitude of electrostatic linear comb actuator

A new method determining the equivalent dynamic parameters such as stiffness, vibrating mass, and air damping factor in motion direction of shuttle (i.e. in y-direction) is proposed, thence the differential motion equation of shuttle is established and solved to achieve a typical displacement formula. Simulation and experimental results show that the change of ELCA' displacement is inappreciable while the range of driving frequency up to 27 Hz (error of 10% with driving voltage is a square wave). Moreover, the range of driving frequency for the ELCA can be extended up to 1 kHz with displacement amplitude error of 10% while the shape of driving voltage is a harmonic sine wave.

Laboratory Investigation on Hydrodynamic Performance of an Innovative Aeration Device with a Wave-Driven Heaving Buoy

Coastal seawater quality is of significance for the environment, ecology and fisheries. In recent years, the hypoxia or anoxia problems of bottom seawater aggravated due mainly to the seawater stratification and eutrophication. This paper addresses an innovative aeration device with a wave-driven heaving buoy to enhance the dissolved oxygen concentration for bottom water. A series of physical experiments was conducted to investigate its hydrodynamic performance and air flow rate. The response amplitude of heaving components and the average value of air flow rate were examined with the related parameters, including incident wave height, incident wave steepness and aeration depth. It was found that with increasing incident wave height, the average heaving displacement and the average air flow rate increase respectively. With the increase of incident wave steepness, the relative value of average heaving displacement increases obviously for high wave period scenarios, it increases slightly for small wave period scenarios in comparison and the relative value of air flow rate increases evidently. With the increase of aeration depth, the average heaving displacement and the average air flow rate decrease respectively. With the increase of relative aeration depth, the relative value of average heaving displacement and the relative value of air flow rate decrease respectively. In addition, the dimensional analysis and the least squares methods were used to obtain the prediction formulas for the average heaving displacement and the average air flow rate, and they agreed well with the related experimental data.

A Generalized Structure-Dependent Semi-Explicit Method for Structural Dynamics

In this paper, a novel generalized structure-dependent semi-explicit method is presented for solving dynamical problems. Some existing algorithms with the same displacement and velocity update formulas are included as the special cases, such as three Chang algorithms. In general, the proposed method is shown to be second-order accurate and unconditionally stable for linear elastic and stiffness softening systems. The comprehensive stability and accuracy analysis, including numerical dispersion, energy dissipation, and the overshoot behavior, are carried out in order to gain insight into the numerical characteristics of the proposed method. Some numerical examples are presented to show the suitable capability and efficiency of the proposed method by comparing with other existing algorithms, including three Chang algorithms and Newmark explicit method (NEM). The unconditional stability and second-order accuracy make the novel methods take a larger time-step, and the explicitness of displacement at each time-step succeeds in avoiding nonlinear iterations for solving nonlinear stiffness systems.