Compatibility Equations Research Articles

Panel-Pin Model for Kinematic and Equilibrium Analysis of Rigid Origami

In this study, the mathematical model referred to as the panel-pin model is proposed for analyzing the infinitesimal-deformation mechanism and the large-deformation equilibrium path of a rigid origami composed of rigid faces (panels) and subjected to deformation only at its crease lines. The panel-pin model represents a rigid origami as a structure of rigid panels pin-connected at the vertices and has the following advantages: 1) consistent formulation of compatibility equations for any type of the rigid origami structure; 2) systematic computation of vertex displacements, folding angles, and their derivatives; and 3) ease of accounting for gravity acting on the panels. The infinitesimal mechanism of the panel-pin model is studied according to the standard procedure for mechanisms and linkages, and an equilibrium path is traced as the trajectory of equilibrium points obtained by minimizing the total potential energy of the model with a rotational spring along each crease line. The numerical examples with multiple degrees of freedom of the mechanism show that the multiple equilibrium paths can be obtained by assigning the initial imperfection, and this is also confirmed by the physical model.

Nonlinear vibroacoustic response and sound transmission loss analysis of functionally graded doubly curved shallow shells

This paper presents the semi-analytical nonlinear vibroacoustic and sound transmission loss behaviors of functionally graded doubly curved shallow. The model is developed by considering the simple power law scheme of material distribution in the thickness direction, under thermal load and incident oblique plane sound wave, as well as Donnell’s nonlinear shallow shell theory. By defining the stress function in terms of the force resultants and applying differential operators to these resultants, the coupled compatibility equation and nonlinear equation of motion are obtained only with regard to the stress function and displacement components. The methods of inverse differential operator and average form approach are then utilized to solve the particular and homogeneous solutions of the stress function associated with different boundary conditions. Based on this solution and the use of the Galerkin method with trigonometric mode shape functions, the nonlinear partial differential equation of motion is turned into the Duffing equation. Next, the method of multiple scales is implemented to capture the frequency of the shell corresponding to transverse motion. Finally, the nonlinear vibration and acoustic responses of shells are studied by considering the variation of important parameters such as aspect ratio, dimensionless amplitude, volume fraction power of functionally graded material, phase portrait, sound transmission loss, velocity, average mean square velocity of drive point, and the sound power level of the shells.

Cracking time and prediction model of low-alloy steel reinforced seawater sea-sand concrete based on DIC technology

The cracking of seawater sea-sand concrete (SSC) cover due to the corrosion of low-alloy steel (LS) is a critical factor affecting its serviceability and durability. To analyze the cracking behavior of the SSC cover, a thick-walled cylinder model was developed in this study. The cylinder was divided into multiple coaxial rings of equal thickness, and damage variables were introduced to determine the extent of damage for each cracked ring based on the position of the rust expansion crack front. A predictive model for the cracking time of SSC cover was established by using the displacement compatibility equation at the interface of reinforcing steel and cover concrete, in combination with Faraday's law. To verify the accuracy of the model, the Digital Image Correlation (DIC) technique was utilized to determine the timings of two pivotal stages: the initial cracking and complete cracking of the cover. The comparison between the predictive outcomes and experimental results showed an error range of 4 %–10 %, indicating that the predictive model is relatively accurate.

Application of modified first order shear deformation theory to vibration analysis of SSSS and CCSS thick anisotropic rectangular plates

This research presents a vibration analysis of a thick anisotropic rectangular plate using modified first shear deformation theory. Modified first shear deformation theory, which is not built upon the classical plate theory, was used to develop the kinematic equations and constitutive relations of a deformed section of thick anisotropic rectangular plate from which the generalized stress equations were determined. Using the assumptions of the theory, the strain energy and external work equations were formulated, and by employing the principles of total minimum potential energy, the total potential energy functional of a thick anisotropic plate was developed. Minimization of the total potential energy functional with respect to the deflection function (w) and with respect to the shear rotations (Φx) and (Φy) respectively, resulted in the governing differential equation and the two compatibility equations of the plate. The displacement functions that satisfy the governing and compatibility equations were obtained by solving the governing and compatibility equations. From the general displacement function, the peculiar deflection equations (shape functions) were obtained for the boundary conditions considered, which are simply supported on its four edges (SSSS), clamped on two adjacent edges, and simply supported on the other two (CCSS). Using the displacement equation and the equation for rotation in the x-direction (Φx) and equation for rotation in y-direction (Φy) the direct governing and two direct compatibility equations were obtained, from which coefficients that enable the formula for calculating fundamental natural frequencies to be obtained. For the boundary condition analyzed in this work, the stiffness coefficients (kR, kQ, kRQ, kq, kRRQ , kRQQ , kNR, kNQ, kNRQ and kλ ) were computed and used in determining the fundamental natural frequency parameter values at various span to depth ratios (5,10,20, 25 and 100), aspect ratios (1 to 2 at the increment of 0.1) and angle of fibre orientation (0o, 15o, 45o). The solutions of this study were compared with those from previous researchers. The fundamental natural frequency parameter values obtained in this study were compared with the work of Reddy (1984) for 0o angle of fibre orientation at span to depth ratios of 5,10, 20, 25, 50 and 100 at an aspect ratio of 1. The percentage difference values were 6.073%, 3.197%, 1.132%, 0.788%, 0.255%, and 0.112%, respectively. These differences revealed the closeness of the results of this present study to the results of Reddy (1984). This shows that the present theory provides good and acceptable solutions to the vibration problems of thick anisotropic rectangular plates.

Stiffness analysis of nutation face gear considering roughness

PurposeThe purpose of this paper is to calculate the meshing stiffness of nutation face gear considering the roughness, establish the calculation method of time-varying meshing stiffness of rough tooth surface and analyze the influence of roughness, load and other factors on the meshing stiffness of tooth surface.Design/methodology/approachThe Weierstrass–Mandelbrot (W-M) function in the Majumdar–Bhushan model is used to characterize the rough contact line of the tooth surface, the normal height and radius of the micro convex body are calculated and the contact flexibility of the contact point of the tooth surface is obtained. The contact flexibility and the bending shear deformation flexibility obtained previously are substituted into the improved deformation compatibility equation for iterative calculation, and the time-varying meshing stiffness of the nutation face gear considering the roughness is obtained.FindingsCompared with ABAQUS finite element simulation results, it is found that the meshing stiffness curve of rough tooth surface is more gentle than that of smooth tooth surface, the meshing stiffness value is smaller and the meshing stiffness change is smaller at the position where the number of gear teeth coincide changes.Originality/valueIn the process of calculating contact deformation, the fractal theory W-M function is used to characterize the contact line of the rough nutation face gear, and the deformation coordination condition considering roughness is improved. Therefore, the method of time-varying meshing stiffness considering roughness can obtain more accurate results, which provides theory and data for the subsequent dynamics analysis of the nutation face gear transmission.

On resonance behavior of sandwich conical shell

Forced vibration is one of the essential behaviors of structures to predict the mechanical properties of various systems. Magneto-rheological (MR) or electro-rheological (ER) materials, due to many possible control-based applications, have gotten much attention in the oscillating system. Thus, in the present work, for the first time, the forced vibration of a conical shell made of three layers with an electrorheological (ER) core and reinforced composite with GPLs is investigated. The formulations are derived from the principle of Hamilton. With the aid of compatibility equations, the interfaces between the core and composite face sheets are simulated. The coupled hyperbolic differential quadrature method (HDQM) and Runge-Kutta solution procedure is employed to solve the coupled governing equations of the current system. The computational method’s accuracy is confirmed by comparing computed results with those available in the literature. The results show that the number, weight fraction, thickness, and pattern of composite layers, electric/magnetic field, and geometry of the conical panel have an important role in the central deflection and phase plot of the current sandwich structure.

Wellbore stability in a depleted reservoir by finite element analysis of coupled thermo-poro-elastic units in an oilfield, SW Iran

Sustaining wellbore stability in depleted reservoirs is a crucial concern. With production from hydrocarbon reservoirs, the reservoir's pore pressure is reduced over time, and the reservoir is depleted since field development is one of the main purposes for oil companies. Heavy mud weight in depleted reservoir caused fracture due to reduced fracture gradient, and low mud weight caused blow out in high-pressure zone or well collapse due to shale beds that required high mud weight to prevent collapse. Considering geomechanics and coupled equilibrium equation, continuity equation, Hook’s law, compatibility equation, Darcy’s law, and thermal relation, the Thermo-poro-elastic equation was derived in this research. A finite element method has been designed to execute the fully coupled thermo-poro-elastic non-linear models. The finite element model was validated by analogizing it to the available analytical solutions for the thermo-poro-elastic wellbore troubles in shale. The non-linear thermal-poro-elasticity finite element model was used to analyze wellbore stability in a depleted limestone reservoir during drilling. The numerical results showed that a decrease drilling fluid’s temperature (cooling) causes to increase in the potential for tensile failure and reduces the potential of shear failure. Due to the depletion reservoir, the potential of tensile failure increased than shear failure, so heating the drilling fluid could cause wellbore stability in the depleted reservoir. Furthermore, based on the numerical results, it may be supposed that the drilling fluid’s temperature is one of the essential elements in the wellbore stability analysis in depleted reservoirs.

Research on load distributions for ball screw feed system with assembly errors

This paper proposes a load distribution model for the ball screw feed system with the assembly errors of the guide rails, which is used to predict the variation law of the load distributions for the ball screws and carriage-guide rail systems. Firstly, under the action of the contact loads of the ball screws, the bending deformation of the screw axis corresponding to each ball is derived. Secondly, a static model of the machine tool bearing system with the assembly errors is established. Then, the load distribution model of the ball screw feed system is proposed based on the load balance and deformation compatibility equation, and the solution process of the proposed model including judging the ball-raceway contact state is given. In addition, the results are compared with those of the existing models to demonstrate the validity of the proposed model. Finally, the relation between the curve shape of the load distribution and the bending of the screw shaft axis is explored, and the variation law of the load distributions for the feed system under the effects of the axial, radial loads, and assembly errors is discussed.

Numerical Study on Slender Circular Reinforced Concrete Column Clamped with Steel Sheet

Various methods to enhance the performance of circular reinforced concrete column have been widely investigated. However, most researches used internal strengthening method such as intensifying steel reinforcement. Others utilized carbon fiber reinforced polymer (CFRP), glass fiber and aramid to retrofit either reinforcement or concrete surface. Few studies incorporated steel to cover concrete surface but neglecting clamp pressure between steel and concrete surface. This study proposes a steel sheet strap strengthening method by initial clamping stress on steel sheets as an external confining technique through experimental works and in-depth numerical analysis. The experimental investigation includes nine steel sheet strengthened reinforced concrete specimens of 20 MPa subjected to axial concentric and axial eccentric. Retrofitting devices will be fabricated from BJ37 steel. Steel sheet straps in each specimen will be subjected to varying initial clamp pressure by fastening bolts. Applied load, lateral displacement and vertical and horizontal deformation will be measured by the instrumentation. Experimental works will observe failure mode, force displacement, stress and ductility. Section analysis based on experimental results will define compatibility equations for numerical stage. Strengthening model from experimental stage will also be analyzed in numerical study with inhouse software 3DNLFEA. F-D curve and moment-curvature loading behavior and ductility will be obtained from numerical analysis. It is believed that the proposed strengthening mechanism with its geometrical and stress variations could enhance the section, member and structural ductility of circular reinforced column against vertical and lateral load. Furthermore, the developed analytical model could well predict the behavior of the proposed clamp mechanism for practical applications.

Hamel's field variational integrator for simulating dynamics of thin-walled geometrically exact beams with warping effects

Using discrete variational principle, a Hamel's field variational integrator (HFVI) for simulating dynamics of thin-walled geometrically exact beams with warping effects is originally proposed. Firstly, a spacetime independent interpolation way for thin-walled beam elements is established on SE(3) × R space by using Hamel's formalism. Then, a new discretization formulation of convective velocity and convective strains on Lie algebra is proposed by using frame operators, which helps to reduce the system's nonlinearity. After that, the variations of discrete convective velocity and discrete convective strain are decoupled from the cross-section configuration variables. This way can avoid the computation of nonlinear exponential map's tangent map and reduce the computation complexity significantly. According the Hamiltonian form of proposed HFVI, the warping variables and discrete convective strains can be independently solved by using discrete compatibility equations on Lie algebra. Finally, the efficiency and accuracy of HFVI is validated by six static and dynamic examples.

An Improved Incremental Procedure for the Ground Reaction Based on Hoek-Brown Failure Criterion in the Tunnel Convergence-Confinement Method

The purpose of this study is to introduce the Hoek–Brown nonlinear failure criterion into the convergence–confinement method (CCM), which is based on the linear failure criterion, and implant it into the ground reaction curve (GRC) in this theory. Based on consistent and rigorous theoretical analysis and according to the mechanical relationships, the principle of equilibrium, the material constitutive law, and the strain/displacement compatibility equation are used to deduce the closed-form analytical solution of the stress/displacement of the rock mass in the plastic region and to complete the simulation and interpretation of the nonlinear behavior of surrounding rock due to the advancing excavation of a circular tunnel. In this paper, confinement loss is regarded as the concept of increment in numerical analysis, and the calculation process and analysis steps of the explicit analysis method (EAM) are proposed. This method can not only realize the calculation of analytical solutions but can also be performed using simple spreadsheets. In addition, to verify the feasibility of this calculation method, this study uses published data to verify the validity of the analytical solution implemented by the incremental procedure and to study the influence of nonlinear failure criteria on the GRC, especially the mechanical behavior at the intrados of the tunnel and the distributions of tress/displacement on the cross-section of the tunnel. A comparison between the published results and the proposed solution in this study reveals consistent and favorable trends.

Design and Computational Analysis of an Axisymmetric Contoured Supersonic Propulsion Nozzle

The design of supersonic axisymmetric nozzles to a required exit Mach number necessitates applying the Method of Characteristics (MoC) that reduces the partial differential equations system of hyperbolic form governing such internal flow-fields to two ordinary differential equations known as the characteristic and the compatibility equations. These relationships are then transformed into finite difference equations and solved using an appropriate predictor-corrector algorithm.The present study applies the MoC to the design of a contoured nozzle that accelerates the combustion gases to supersonic velocities. The selected configuration has a throat constituted of two circular arcs with the one downstream attached to a 2nd-order polynomial profile at the attachment point. The solution, in terms of the static pressure along the centerline and the wall is obtained. It was found that the flow expands gradually without any instability or possibility of shock waves. A gradual expansion represented by a decrease in pressure is noticed immediately downstream of the throat, the flow tending to expand gradually from the throat to the exit. Along the wall, the expansion is important immediately downstream of the throat while it tends to stabilize downstream to reach the ambient pressure at the outlet. In terms of performance, the thrust coefficient developed shows the great expansion process within the divergent supersonic section. The configuration is also found to develop a great specific thrust.

Kinematic and dynamic analysis of a 4-DOF over-constraint parallel driving mechanism with planar sub-closed chains

AbstractIn this paper, a new over-constrained parallel driving mechanism (PDM) with planar sub-closed chains is proposed. First, the number of over-constraints on the PDM is calculated. Then, an analysis is conducted as to the kinematics of the hybrid manipulator, including positions, velocities, and accelerations of all bodies. Furthermore, the Newton–Euler approach is taken to deduce the kinematic formula of each link and the formula of inertial force at the center of mass. However, it remains difficult to solve the equation since the number of equations is smaller than that of unknown variables. To solve this problem, the screw theory is applied in the present study to analyze the cause of over-constraints, with the link’s elastic deformation introduced as the supplement of deformation compatibility equations. Moreover, the actuation forces and constrained forces/moments are calculated simultaneously. Finally, the dynamic model is verified through simulation and experimentation. The proposed modeling approach provides a fundamental basis for the structural optimization and friction force computation of the over-constrained PDM.

Constraining Dark Boson Decay Using Neutron Stars

Inspired by the well-known anomaly in the lifetime of the neutron, we investigated its consequences inside neutron stars. We first assessed the viability of the neutron decay hypothesis suggested by Fornal and Grinstein within neutron stars, in terms of the equation of state and compatibility with observed properties. This was followed by an investigation of the constraint information on neutron star cooling that can be placed on the decay rate of the dark boson into standard model particles, in the context of various BSM ideas.

Deep learning framework for multiscale finite element analysis based on data-driven mechanics and data augmentation

In this study, a deep learning framework for multiscale finite element analysis (FE2) is proposed. To overcome the inefficiency of the concurrent classical FE2 method induced by the repetitive analysis at each macroscopic integration points, the distance-minimizing data-driven computational mechanics is adopted for the FE2 analysis. Macroscopic strain and stress data are directly assigned to the material points of the macroscopic finite element models without constitutive model. Here, the macroscopic problems are solved with the offline macroscopic material genome database, without solving the microscopic problems simultaneously. The novelty of the proposed approach lies in using a deep neural network to enable adaptive sampling points without prior knowledge of the specific mechanical problem. The proposed data augmentation framework updates the sampling points gradually using the distance minimization algorithm with the mechanistic constraints, including the equilibrium and compatibility equations. Particularly, the deep neural network plays a crucial role as a guide in the phase space sampling process, facilitating the efficient use of sparse data. Thus, this method shows high feasibility and significantly improves the computational efficiency of offline computing.

Prying model and mechanism for circular all-FRP flange integrally formed with tube

This paper focuses on prying action of integrally-formed circular all-FRP flanges intended for assembly of FRP members. The connection is manufactured by prepreg hand-work layup and compression moulding process. Tensile tests have been performed on three connections of different flange thickness (2.2 mm, 3.0 mm, 3.8 mm). The influences of flange thickness and bolt preloads on prying force have been investigated. Corresponding displacements and strains on tube and flange plate were also analyzed. Then, an analytical model for prying force as well as displacement, strain and stress fields is developed based on the lamination theory of circular shells and plates and the compatibility equations. Quite good agreement is obtained when predicting prying effect factor and fairly good agreement is achieved for flange separation and strains. The experimental and theoretical results indicate that prying force has negative correlation with flange thickness when flange is thick enough, but contrary when flange is very thin. Further theoretical analysis shows that reducing bolt stiffness by BFRP bolts will significantly reduce prying force, by the maximum 39.06% for the connection with 23 mm-thick flange.

Research of the deformability of the cantilever axis of the satellite of a multi-satellite planetary gear and its influence on the load distribution over power flows

A mathematical model of the axis of the satellite, cantilevered in the cheek of the carrier of the planetary gear, is presented. The axis of the satellite is presented in the form of a beam on an elastic base. The influence of the flexibility of the axle and the drive elements mated with it on the distribution of the load over power flows is considered. The Leibniz formulas and the Euler method were used to obtain the model. The coefficient of uneven load distribution over power flows is obtained by solving a system of displacement compatibility equations, including deformations of the wheel teeth, rolling bearing, satellite axle and mating parts, taking into account the initial non-adherence of the teeth due to gear manufacturing errors. The load capacity of a planetary gear with cantilever axes of satellites is analyzed depending on the number of power flows. Keywords: planetary gear, deformation, cantilever axle, load distribution. fplehanov@list.ru, alexandrsun4009@gmail.com

Application of a Nonlinear Deformation Model to the Analysis of Complete Moment-Curvature Diagrams of Reinforced Concrete Elements with Reinforcement A500

This article suggests a calculation procedure for the parameters of complete moment-curvature deformation diagrams with a down leg that can be used for the calculations of the multistorey monolithic building frameworks taking into account the specifics of reinforced concrete. To construct complete deformation diagrams for bending elements, a non-linear deformation model is used that is based on using the physical diagrams of the concrete and reinforcements to calculate the inner moments of cross-sections at all loading stages via balance and strain compatibility equations. Using this model, the authors researched complete moment-curvature deformation diagrams for the elements with different percentages of longitudinal tensile reinforcement that changes the bending properties of the structure. We used grade А500 steel as the reinforcement. The parameters of the tensile reinforcement diagram were determined in the tension testing of samples with constant deformation rates. The performed computer tests to calculate the moment-curvature diagram parameters for reinforced concrete beams reinforced with grade А500 efficiency rods and conventional grade A400 reinforcement rods with different longitudinal reinforcement congestion rates helped the authors assess the bending properties of the elements under conditions approaching their destruction and determine their efficient use in the calculations of building frameworks taking into account the redistribution of forces.

Nonlinear thermo-elastic analysis of curved panels

The target of this paper is to assess the size-dependent geometrical nonlinear static characteristics of curved panels (CPs) enriched by nano-additives incorporating thermal effects. The size-dependency of the analyses is considered by developing the basic equations based on the modified couple stresses theory (MCST). In order to achieve the fundamental equations of compatibility and equilibrium, the transverse shear strain components for fairly thick and also thin micro-shells are considered on the basis of the 1th-order transversely shear deformation theory (FSDT). In order to find the response of the achieved set of nonlinear basic differential equations in form of a closed-form pressure-deflection relationship, an analytical approach is utilized on the basis of the Galerkin method. The achieved close-form relation representing the nonlinear equilibrium path of proposed curved micro-panels is examined by various examples from the present references. Finally, the introduced closed-form relationship is utilized for carrying out an extensive parametric numerical study incorporating the impacts of material length-scale parameter (MLSP) and the characteristics of geometry and material on the geometrical nonlinear path of equilibrium of the proposed micro-shell panels.

A New Calculation Method of Force and Displacement of Retaining Wall and Slope

On the basis of the force and safety factor analysis of traditional retaining walls, a new analytical method of force and displacement of the slope is suggested, the numerical theoretical solution of the stress distribution of the sliding body can be obtained under the conditions, which the stresses distribution are satisfied with the differential equilibrium equations, the boundary conditions, the compatibility equation and the macroscopic equilibrium equations. The interface stresses between the sliding body and the retaining wall is continuous, and the theoretical solution of retaining wall stress distributions can be obtained, while the stress differential equilibrium equation, the compatibility equation, macroscopic force, and moment balance equations are satisfied. The strain and displacement solutions can be obtained by using Duncan Chang and Hooke constitutive equations for the slope and the retaining wall, respectively. The transfer station of landfill in the Guandukou Town of Badong County is taken as an example, the results of the sliding body and retaining wall analysis show: The stress and strain solutions of a slope and a retaining wall can be obtained by the proposed method. The anti-slip force of the retaining wall calculated by the method in this paper contains the positive pressure and shear force along the contact surface and varies with the deformation of the slope, in addition, the numerical theoretical solution of the retaining wall shows that the retaining wall shape and material can be optimized according to the calculation results. It is feasible for the proposed analysis method of slope with retaining wall design to be run many years.