Differential Equations Of Equilibrium Research Articles

Variable angle tow-steered curvilinear fibres-based rotating arbitrarily layered composite beams- a coupled vibration of chordwise-flapwise-torsional motions by higher-order beam theory

In the present work, variable angle tow-steered curvilinear fibres-based straight composite rotating beam is studied for its coupled vibrational behaviours from different beam planes by a higher-order beam formulation satisfying plane stress situation and combining with finite element methodology. The structural theory is extended to include all the vibrational motions including chord- and flap-wise motions, axial and torsional vibrations of beam. The centrifugal force stiffening and gyroscopic effects stemming from rotational motions are taken into the formulation. The generic differential equations of equilibrium for the proposed structural theory are evolved adopting virtual dynamic work, Hamilton’s principle and a suitable C1 continuity based elemental equations are gotten. The newly constructed finite element formulation is tested for solved problems that are known in the literature. Based on in-depth analysis, vibration features of curvilinear fibre-based rotating generally layered composite beam including coupled dynamic motions in terms of vibrational modes and their frequencies pertaining to chord- and flap-wise, axial, torsional motions are detailed presuming variable angles along the curvilinear fibre path at the centre and layer edge, short and long beams, hub-radius, beam cross-section, and high rotational speed effects.

A new set of explicit solutions for postbuckling behaviour of anisotropic shear deformable laminated cylindrical shells with general imperfection distribution and thermal loading

Thermal postbuckling analysis is presented for shear-deformable anisotropic laminated cylindrical shell with general imperfection distribution under several typical temperature fields are analyzed, including uniform temperature, tent-like linear distribution, axial and circumferential quadratic parabolic temperature fields over the shell surface and through the shell thickness. On this basis, the general temperature distribution can be obtained based on the three-dimensional steady-state heat transfer equation. At the same time, based on the high order shear deformation theory including anisotropic coupling effect, geometrically nonlinear large deflection displacement-strain relationship, the general initial geometric imperfections obtained by measurement and analytical means are introduced, and the thermal buckling equilibrium differential equation of anisotropic shear deformable laminated cylindrical shell with initial imperfections is established. A two-step perturbation-Galerkin integral method is used to obtain explicit solutions for the path development direction of post-thermal buckling equilibrium. The results of parameter analysis show that the material properties and temperature dependence, temperature distribution, lay-up and geometric parameters such as radius-to-thickness ratio have important effects on the thermal buckling load and postbuckling equilibrium path. The present method provides an effective theoretical prediction and analysis method for the design of the carrying capacity of submarine pipelines.

Seismic response of permafrost slopes supported by the thermal anchor pipe frame: Theoretical analysis and shaking table test verification

To study the seismic response of the thermal anchor pipe frame supporting permafrost slope system, the coupled dynamic interaction model of the permafrost slope with the thermal anchor pipe frame in thawing and freezing periods was established by taking into account the stratification characteristics of the permafrost slope in different typical seasons, respectively. Based on D'Alembert's principle and Dirac's function, the coupled vibration equilibrium differential equations are given and the seismic response of the system is analyzed for different periods. The reliability of the proposed model was further verified through shaking table tests. The results show that the acceleration response at different positions during both freezing and thawing periods is similar to the initial loading seismic wave shape. The peak acceleration and velocity during the thawing period is greater than that during the freezing period. The shear force and bending moment of the frame column are distributed alternately positive and negative along its height. The incremental dynamic shear force and bending moment are greater during the thawing period. However, the peak bending moments and shear forces are greater during the freezing period when the combined effects of frost and earthquakes. The peak bending moment occurs at the position of the first layer of the thermal anchor pipe, and the shear force suddenly changes near the thermal anchor pipe. In seismic design, attention should be paid to checking the internal force of the frame column during the freezing period to prevent plastic bending failure. The results contribute to further understanding the dynamic interaction mechanism of the thermal anchor pipe frame permafrost slope system, which has important practical significance for its seismic resistance.

GRILLAGE ANALYSIS OF CELLULAR PLATE STRUCTURESOF VARYING DEPTH WITH INCL

A static analysis of cellular plate structures of linearly varying depth is presented by using thesimplified grillage method. The cellular plate structure in this study consists of top and bottomflange plates (non-parallel) separated by longitudinal webs having varying depth. A variationalformulation of thin-walled beam element with linearly varying depth has been carried out for thederivation of the stiffness matrix of the grillage members for various warping restraint effects. Theeffect of warping restraint in stiffness is included by two methods, firstly; the implicit method(through the torsional constant) and secondly; by the explicit method (as independent warpingdegree of freedom). Two differer,t approaches are used to derive the torsional stiffness matrix,namely:by the assumed displacement field and by the differential equation of equilibrium. Theeffect of secondary shear stresses from the warping restraint is also considercd.The varying cross section properties of the eguivalent grillage members are derived from theproperties of the actual cellular plate structure.Several numerical examples and applications are presented . The results are compared with thoseobtained from the flat plate / shell elements of MSCA.{ASTRAN program. Acceptable agreement isobtained.

Axisymmetric Pressing Problem of Discrete Metal Materials

The paper presents an analytically closed solution to the problem of axisymmetric pressing of discrete metal materials by the method of jointly solving the differential equations of equilibrium of the metal and the plasticity conditions of the porous body, taking into account all pressing factors without exception: the type and properties of the charge, loading conditions, porosity, temperature, friction, etc. The purpose of this work is to develop the foundations of the engineering theory of pressure processing of discrete materials using the example of solving the problem of axisymmetric pressing of structurally inhomogeneous metal chips in a movable closed matrix. The basis for constructing a physical and mathema-tical model of the process is the idealized case of uniform compaction of a porous body with the subsequent determination of the lateral pressure coefficient corresponding to the actual degree of compaction at various stages of loading. The resulting equation for the relationship between the stress tensor components and the yield stress and relative compaction density represents the cylindrical Mises plasticity condition, which in the limit at zero porosity transforms into the plasticity condition for compact metals. The boundary value problem is solved for tangential stresses, taking into account the magnitude and direction of action of contact friction forces, which in their physical nature do not differ from the friction forces in the depth of the pressed material. The physico-mathematical model makes it possible to calculate the stress fields and density of the body according to the coordinates of the deformation zone, as well as energy-power parameters (pressure, force, work of deformation) provided that three structural and rheological characteristics are determined: the yield strength, relative compression and the degree of deformation compaction. Due to the fact that the problem is solved in relation to bodies of rotation in a general form and in a general formulation, the solution itself should be considered as methodological for any axisymmetric loading scheme.

Elastodynamic Green's functions for sandwich panels with aluminum foam core and transversely isotropic face sheets using potential functions method

Green's functions of stress and displacement due to wave propagation in a sandwich panel consisting of transversely isotropic face-sheets and aluminum foam core are developed in the present paper. Dynamic partial differential equations of equilibrium for the considered face-sheets and core are expressed in cylindrical coordinates (r, θ, z), and converted into two separate equations using two unknown potential functions. Then, by writing the potential functions as Fourier series in the circumferential direction and using Hankel transform in the radial direction, an analytical solution to the potential functions in Hankel transformed space is developed. Using the boundary conditions and continuity of the problem as well as the place of application of harmonic forces, Green's functions for displacements and stresses in the frequency domain are derived using Hankel's inverse integral transform. These functions are expressed as complex integrals and calculated numerically. To verify the methodology and results, a comparison study is performed for particular cases showing a high level of the accuracy. Green's functions obtained from this paper will give information about the loading models or materials causing the maximum normal stress, shear stress, vertical and radial displacements. In addition, based on the results, with an increase in the core thickness compared to that of face-sheets in a Sandwich panel, an increment in Green's functions of stress and displacement is observed in both real and imaginary parts.

Advanced Thick FGM Plates-Cylindrical Shells in Linear Unsteady Supersonic Flow

Introduction: Thick functionally graded (FG) plates-cylindrical shells with third-order shear deformation theory (TSDT) of displacement models under thermal vibration in linear unsteady supersonic flow are investigated by using the generalized differential quadrature (GDQ) method. Method: The nonlinear coefficient term of TSDT models and the advanced nonlinear shear correction coefficient expression are used to study the thermal vibration of thick FG plates and cylindrical shells in linear unsteady supersonic flow. At the intersection of FG plates-cylindrical shells, the displacements, stress resultants, and moment resultants could be reasonably assumed in the continuity condition for the differential volume in linear unsteady supersonic flow, respectively. Result: The dynamic equilibrium differential equations of thick FG plates and cylindrical shells in linear unsteady supersonic flow, respectively in terms of partial derivatives of displacements and shear rotations subjected to partial derivatives of thermal loads, mechanical loads consist of the linear supersonic aerodynamic pressure and inertia terms can be presented. Two parametric effects including environment temperature and functionally graded material (FGM) power law index on the thermal stress and center displacement of thick FG plates-cylindrical shells in linear unsteady supersonic flow are investigated. Conclusion: The linear unsteady supersonic flow over the outer surface of thick FG platescylindrical shells provides the additional aerodynamic effect on the values of displacements and stresses.

A quasi-3D hyperbolic formulation for the buckling study of metal foam microplates layered with graphene nanoplatelets-embedded nanocomposite patches with temperature fluctuations

The work analyzes the buckling behavior of size-dependent microplates with a metal foam core, covered by graphene nanoplatelets (GNPs)-embedded nanocomposite patches. Microplates rest on a bi-parameter elastic substrate, and they are immersed in a thermal environment to observe the effect of temperature fluctuations on their elastic buckling performance. All material properties for each layer of the microstructure are thickness-dependent. A novel quasi-3D shear and normal (Q-3D S-N) hyperbolic theory is here proposed to describe the kinematic relations, accounting for the transverse normal strain. At the same time, a modified couple stress theory (MCST) is employed to account for the size dependence of the mechanical behavior due to the presence of a material length-scale parameter. Using the energy method and virtual work principle, the differential equilibrium equations are derived and solved analytically, where solutions are verified against the existing literature. The study focuses on the impact of different parameters on the normalized critical buckling load (NCBL). Based on the results from the systematic investigation, it is found that the addition of GNPs to the microplate enhances its stiffness, leading to increased values of NCBL, which in turn reduce for an increased imperfection ratio.

A mesh-free Hermite-type approach for buckling analysis of functionally graded polygonal thin plates

This work aims to establish a developed Meshless process which uses the point interpolation and the Hermite-type point interpolation methods for instability analysis of functionally graded material (FGM) thin polygonal plates. The structure’s material characteristics should gradually change according to the thickness direction. The minimization principle is used to obtain the governing nonlinear static equilibrium differential equations and related boundary conditions. To obtain the discrete weak form, point interpolation method (PIM) is used to approximate membrane components, while Hermite-type point interpolation method (HtPIM) ones are used for flexural components. The critical buckling loads and the associated eigenmodes are obtained by transforming the resulting equations into an eigenvalues problem. To evaluate the validity and effectiveness of the presented methodology, several examples concerning specimens of FGM rectangular and polygonal plates of various geometric planar shapes (L, T and C) subjected to uniaxial and biaxial in-plan loads are studied. The outcomes are contrasted with those of the finite element approach and with some data from the literature. Based on the analysis of the comparative study, the proposed methodology presents a very good agreement with the validated and previously published numerical results.

Predicting monotonic lateral load-displacement behavior of offshore flexible piles in saturated MCC soils: An application of the cavity expansion theory

This paper proposes an application of the cavity expansion theory to predict the load-displacement response of monotonically laterally loaded piles and the effective stress distribution of saturated modified Cam-clay (MCC) soil surrounding the pile. This approach incorporates geometric relationship, equilibrium equation, and boundary conditions to derive governing equations. These equations are subsequently solved to obtain monotonic p-y curves, which encompass the elastic, elastoplastic, and critical processes. To determine the load-displacement behavior and effective stress distribution of the soil, the proposed approach combines the deflection equilibrium differential equation with the finite-difference method. The accuracy and effectiveness of the approach are validated through a comparison with a Finite Element Method (FEM) simulation. Parametric studies are also presented to investigate the effects of overconsolidation ratio, in-situ coefficient, and mean effective stress. The validity and capacity of the proposed model are further demonstrated using model pile tests. The results affirm that this approach can well predict the monotonic load-displacement response of the pile and effectively captures important phenomena observed in pile tests.

Investigation of Passive Pile Groups’ Responses Induced by Combined Surcharge-Induced and Excavation-Induced Horizontal Soil Loading

Excessive lateral forces and the deformation of pile groups caused by adjacent surcharge loading and excavation can potentially lead to the failure or collapse of nearby pile foundations. This paper presents a simplified analytical method to investigate the lateral displacement of passive pile groups caused by combined surcharge-induced and excavation-induced horizontal soil loading and validates it by utilizing two published centrifuge tests and numerical modeling. Firstly, the passive load resulting from surcharge-induced horizontal soil loading is determined using the improved formula of Boussinesq and the theory of local plastic deformation. The additional horizontal force exerted on the pile axis subjected to excavation-induced horizontal soil loading is also taken into account using the Mindlin solution. Furthermore, the shielding effect between pile groups is also proposed to analyze the response of a laterally loaded pile group due to excavation unloading. Secondly, according to the Pasternak foundation model of soil–pile interaction, the equilibrium differential equations are solved by the finite difference method. Finally, parametric analyses are undertaken to evaluate the behavior of different constraints on the pile head and surcharge procedure. The results demonstrate that the capped-head pile groups can provide greater restraint compared to the free-head groups for the lateral displacement of pile groups. The front pile experiences greater forces than the rear one in passive pile groups as a result of the smaller soil pressure on the rear pile. Additionally, increasing the horizontal stiffness of the capped head with fully fixed ends can significantly decrease the horizontal deformation of the pile groups.

Thermoelastic creep evolution in a variable thickness functionally graded piezoelectric rotating annular plate considering convection and radiation heat transfer

The thermoelastic time-dependent creep progress of a rotating annular plate with nonuniform thickness made of functionally graded piezoelectric material considering convection heat transfer and solar radiation was investigated. The nonlinear differential equation of heat transfer considering conduction, convection, and solar radiation was solved using the differential transformation method. Then, the differential equation of equilibrium for the annular plate containing creep strains was derived. Firstly, the differential equation was solved analytically for the zero time . Then, the stress and strain rates were achieved using Norton’s law with Prandtl–Reuss equations. Finally, the time-dependent creep stresses and deformation redistributions were obtained employing an iterative method.

Face sheet defects in plastic forming of aluminum foam sandwich panel: theoretical models, numerical and experimental studies

In this paper, the theoretical models of face wrinkling and dimpling defects occurred in the plastic forming of aluminum foam sandwich panel (AFSP) were constructed and the influences of AFSP parameters on defect were investigated to predict and avoid the surface forming defects. The wrinkling and dimpling behaviors of shaped AFSPs were analyzed at different scales first, and the analytical models of critical wrinkling and dimpling stresses of face sheet for the forming of AFSP were developed using energy method and differential equilibrium equation method respectively. Finite element simulations of curved AFSPs forming were performed using macroscopic and mesoscopic AFSP models, and plastic forming experiments of spherical and saddle-shaped AFSPs with different core relative densities were conducted. The surface profiles of simulated parts and the face wrinkles and dimples in experimental parts indicated that the established theoretical models can predict the occurrence of face sheet defects in curved AFSPs forming accurately. Furthermore, the effects of geometric and material parameters of AFSP on face wrinkling suggested that the seriousness of face wrinkling can be reduced by decreasing the thicknesses of face sheet and core, and the wrinkling can be suppressed when the elastic Poisson's ratio of core is close to 0 and 0.5.

Improving shape formation under conditions of plane tensile stress

Thin-walled axisymmetric truncated parts made of sheet billets are actively used in rocket and aerospace engineering. Improvement to their shape formation, based on directed material thickness change will ensure the production of parts with minimum thickness variation. This will also enable aviation and space industry enterprises to attain leading positions, as well as reduce labor costs. This work studies the possibility of obtaining thin-walled axisymmetric parts of truncated tapered shape using one of the methods of sheet metal stamping under flat tensile stress conditions (flanging). The mechanism was identified and the analysis of the stress-strain state of the billet during deformation was carried out. This takes into account the minimizing of the difference between the specified and technologically possible thicknesses. A mathematical model was developed to consider the shaping method based on the process of flanging. Theoretical studies were based on the principles of the plastic deformation theory of sheet materials. This was achieved by the following factors: approximate differential equations of force equilibrium; equations of constraint; plasticity conditions; and fundamental constitutive relations under given initial and boundary conditions. The process of flanging was simulated using the LS-DYNA software package with the following initial data of a conical billet made of 12Kh18N10T steel: cone angle 16.4°, thickness Sbillet = 0.3 mm. The aim was to eliminate errors in designing a tool for future implementation of the method on a manufactured die tooling, as well as to confirm the theoretical conclusions on the selection of technological parameters and achieve minimal thickness variation. The steps of computer modeling are presented, indicating the main process parameters such as material model, mechanical characteristics of the workpiece material, type of elements, kinematic loads, conditions of contact interaction of elements with each other, etc.

Lateral-torsional buckling performance of high strength steel rectangular hollow section beam-columns

A new approach namely Plate-Beam theory was employed to perform the investigation into the lateral torsional buckling (LTB) performance of high strength steel (HSS) rectangular hollow section (RHS) beam-columns. Firstly, the total strain energy corresponding to the LTB failure mode of the beam-columns was established, and then the equilibrium differential equation was developed, through the variation of energy functional. Based on the Plate-Beam theory, the equivalent restrained torsional rigidity (warpage constant) that can simultaneously accounted for the secondary warping effect was derived, and the formulation process did not involve any sectoral coordinate operation. The deducing process of the total strain energy of RHS beam-columns was further discussed and recommended to describe the torsion performance of composite and irregular thin-walled members. Based on the FE program verified against test data and theoretical solution, the accuracy of the provisions of HSS RHS beam-columns given in Eurocode3 and GB50017-2017 was assessed. The comparison of numerical program against codified design provisions revealed that the resistance prediction in Eurocode3 was still serviceable for HSS RHS beam-columns, in conjunction with the torsional rigidity according to the Plate-Beam theory.

A refined shear deformation theory of an asymmetric sandwich beam with porous core: Linear bending problem

This paper is devoted to a sandwich beam with an asymmetric structure under uniformly distributed load. The first end of this beam is simply supported, while second end is clamped. The individual nonlinear deformation function of a plane beam cross section – the shear effect with consideration of the classical shear stress formula is analytically developed. Displacements, strains and stresses taking into account the individual nonlinear deformation function for successive layers of this beam are elaborated in detail. The system of two differential equation of equilibrium of this beam, based on the principle of stationary potential energy, is obtained. This system of equations is precisely solved analytically, according to the theory of differential equations. The individual nonlinear deformation functions, shear stresses and deflections of two cases of sample beams are determined in detail: the first with variable core porosity in the direction of the beam depth, the second with constant core porosity – standard sandwich structure. The calculations results specified in Tables and graphically presented in Figures.

A semi-analytical form-finding method of the 3D curved cable considering its flexural and torsional stiffnesses in suspension bridges

During the configuration transformation, the 3D curved main cable's torsion increased the construction control's complexity in the suspension bridge with the spatial cable system. The calculation of the torsional angle of the main cable must consider the cable's flexural and torsional stiffnesses. However, most form-finding methods adopted the assumption of the ideal flexible cable, failing to obtain the torsional angle of the main cable. Hence, this study proposed a semi-analytical form-finding method for the 3D curved cable that considered the main cable's flexural and torsional stiffnesses. This method was based on the Kirchhoff-Love rod theory, which provided a feasible framework for analyzing the torsion in the main cable. Firstly, the Euler angles were incorporated to facilitate the parameterization of finite rotation and the transformation of static equilibrium differential equations. Then the governing equations of the target configuration were established using the geometric compatibility conditions and the static equilibrium conditions. Moreover, the equations were numerically solved through the finite difference method and the Levenberg-Marquardt method. Finally, four examples were employed to verify the feasibility and accuracy of the proposed method, including two cantilever beams and two suspension bridges with curved 3D cable and 2D cable, respectively. The results of the case study demonstrated that the torsion in the main cable was primarily caused by bidirectional bending rather than internal torque. This torsion was observed under the influence of the cable's self-weight and non-eccentric hanger forces. Furthermore, it was observed that the maximum torsion did not occur at the mid-span point. The position of the inflection point was found to be highly sensitive to the orientation constraints imposed at the fixed end of the cable.

Creep of Concrete in Shell Structures: Nonlinear Theory.

The creep of concrete is one of the main problems threatening concrete structural development and the stability and safety of structures. However, the nonlinear theory is the key to solving the problem of taking into account the physical and mechanical properties of concrete creep in shell structures. To create such a theory, the original shell is replaced by a continuous equivalent elastic shell. To determine the stress-strain state of the structure, the equations of nonlinear creep and crack growth are derived, and a deformation model of the section is created. The behavior of the structure at all stages of the life cycle is investigated by solving the solving systems of differential equations of equilibrium, motion, and perturbation of the equivalent shell. The values of the ratios of dependence of long-term and short-term critical loads on deformations, forces, cracks, etc., are given. The accuracy of the solution of the developed nonlinear theory is compared with the linear theory of concrete creep as well as experimental data. The results show that, according to the linear theory, for the values for the short term and long term, up to 56% and up to 39% of critical loads are overestimated, respectively. The creep process in practical engineering can be effectively controlled by the results of the proposed theory.

Evaluation of Computational Analysis for Stiffened Slab Using Finite Strip Method

A rectangular slab stiffened by ribs oriented perpendicular to the support of the slab as structurally equivalent to an orthotropic plate has been analyzed using the finite strip method. The analytic formulation was obtained from the governing equation of motion of the plate using the classical plate theory to an arbitrary boundary and initial condition. The stiffness formulation from total internal energy was considered. The force-displacement equation technique by applying stiffness and uniformly distributed loading was used to obtain the modal solution and hence, displacement at an arbitrary point of the plate for the assumed harmonic function. Approximate stresses following the displacement are also obtained. The comparative solution of the equilibrium differential equation is obtained by means of convergent series expansions of levy plates and the application of Timoshenko and Woinowsky-Krieger averaging of bending rigidity, utilizing suitable orthogonal displacement functions. A computer procedure based on the finite strip method is given through the application of numerical examples, which shows that the procedure provides good results, especially with low discretization and computational effort.

Dynamic modelling of an armoured face conveyor considering the curved chains

PurposeDue to the structural layout, mining process, and working environment, curved chains such as horizontal and vertical bends inevitably exist in the armoured face conveyor (AFC). With the increasing power, conveying capacity, and distance of the AFC, the dynamic influence of these curved chains should be highly emphasized. This paper establishes a dynamic model of the AFC by multi-body system theory and finite segment method, in which the curved chains can be fully considered.Design/methodology/approachThe scraper chains are firstly grouped into the straight, horizontal bend, vertical convex and concave bend sections. Each bend section running in a circle is simplified as an ideal arc. Through solving its differential equilibrium equation and using Newton's second law, its running resistance is derived. Then the grouped chains are discretized into finite control elements according to the Kelvin model, and the governing equation of each control element is established. The dynamic model of the AFC is obtained by assembling these equations, and the corresponding simulation model is developed by using MATLAB/Simulink.FindingsCase studies with real scenarios are provided, and simulations are carried out. The results show that the running resistance contributed by the curved chains is larger than the traditional empirical value.Originality/valueThe work in this paper helps the dynamic performance design of AFC, with a deep understanding of the curved chains.