Closed-form Analytical Solution Research Articles

An extended separation-of-variable method for free vibrations of orthotropic rectangular thin plate assemblies

Rectangular plate assemblies are widely used in various engineering fields, but there are not closed-form solutions for them with arbitrary homogeneous boundary conditions. Hence, this work is to develop an extended separation-of-variable method to achieve closed-form analytical solutions for the vibrating problems of orthotropic rectangular thin plate assemblies. In this method, the mode function of each plate is in an explicit separation-of-variable form, and the frequencies corresponding to two-direction eigenfunctions are mathematically independent of each other. This method can deal with all homogeneous boundary conditions. The two-direction eigenvalue equations can be solved simultaneously, and the obtained closed-form eigensolutions satisfy the Rayleigh’s principle exactly. Numerical experiments validate the accuracy of the present solutions, and parametric study is conducted.

Abundant analytical soliton solutions and Evolutionary behaviors of various wave profiles to the Chaffee–Infante equation with gas diffusion in a homogeneous medium

The generalized exponential rational function method is used in this work to obtain a variety of rich families of analytical soliton solutions and to exemplify the dynamics of solitonic structures of the (2+1)-dimensional ChaffeeInfante (CI) equation. This CI equation is a well-known reaction–diffusion equation that can describe the physical process of mass transport and particle diffusion and has been widely used in fluid dynamics, electromagnetic wave fields, high-energy physics, fluid mechanics, coastal engineering, and ion-acoustic waves in plasma physics, optical fibers, and other disciplines. The GERF method is successfully used to derive closed-form analytic solutions to the CI equation, including rational, exponential, trigonometric, and hyperbolic function solutions. The CI equation has indeed been observed to have bright and dark solitons, singular and combined singular soliton profiles, periodic oscillating nonlinear waves, and kink-wave profiles. Furthermore, three-dimensional graphical representations are presented to depict the dynamical behavior of the achieved solutions. We can better understand the dynamical properties and structures of these solutions by using these three-dimensional postures. This technique could be used to solve a wide variety of higher-dimensional nonlinear equations confronted in mathematical physics and applied sciences.

Rapid stability analysis of serrated end mills using graphical-frequency domain methods

Serrated tools result in an irregular distribution of chip thickness and varying time delays along their flutes that disturb the regenerative chatter mechanism and maximize productivity in rough machining operations. The chatter stability simulation with such tools is complex due to the delay differential equations with multiple and distributed delays. Although the time domain numerical integration methods and the semi-discretization method are accurate, they are computationally expensive and hence inhibit the optimal design of serrated cutters. Despite the advantages that serrated tools offer, there has been little attention paid to rapidly predict stability lobes diagram for these tools to help instruct their optimal designs. To address this need, new graphical-frequency methods are presented to solve for stability of serrated tools with arbitrary geometry. The graphical method averages the number of teeth in cut to make the system time-invariant and results in a closed-form analytical solution. It reduces the simulation time by up to ∼99% compared to the semi-discretization method. Generalized solutions to the classical zero-order approximation and the multi-frequency methods are also presented by using the Nyquist stability criterion. The use of the zero-order approximation method and the multi-frequency method reduces simulation times by up to ∼97% and ∼79% respectively compared to the semi-discretization method. These are significant improvements over reports in the previous literature. Stability predictions are validated experimentally and benchmarked with high-fidelity time domain methods. The presented methods are general and can be applied to any end mill with non-uniform pitch and helix angles as well.

Buried pipeline subjected to static pipe bursting underneath: a closed-form analytical solution

The present study proposes a closed-form analytical solution for investigating the behaviour of buried adjacent pipeline subjected to ground heave induced from static pipe bursting underneath. The governing differential equation of the adjacent pipe is derived based on the concept of a beam (both Euler–Bernoulli's and Timoshenko's beam) resting on a Pasternak foundation. A Fourier series is used to solve the governing differential equation. The proposed solution is validated with the results obtained from earlier full-scale experimental works and finite-element-based numerical study. Further, a parametric study is conducted to investigate the influence of relative pipe–soil stiffness and pipe–pipe intersection angles on the adjacent pipe response during the pipe bursting process. It is perceived that normalised maximum pipe deformation reduces and normalised maximum pipe bending moment increases as the value of the relative pipe–soil stiffness increases. Further, for the present case, the maximum pipe curvature is obtained at a 35° intersection angle. Hence, it is understood that a 90° intersection angle is not always the critical criterion. The study will be useful in the preliminary design stage for obtaining quick and relevant results without performing rigorous numerical or experimental study.

Application of layerwise HSDT and fracture analysis in the ultimate strength of composite plates with delamination in bending

To obtain the flexural response and the load-carrying capacity of composite plates with delamination, an efficient closed-form analytical solution is presented. Layerwise Higher-order Shear Deformation Theory and Rayleigh-Ritz approximation technique are applied. Fracture analysis is used to obtain the load-carrying capacity and primary steps of delamination propagation. The results include continuous functions. Furthermore, a considerable experimental study has been performed with specimens of various geometries and delamination types. The experimental investigation includes the fabrication procedure, tension experiments to obtain material properties and three-point bending tests of the delaminated plates.

An Analytical Solution for Seismic Interaction between Urban River-Canyon Topography and Nearby Buildings

The interaction between urban river-canyon topography and the river-side building is investigated by using a whole analytic model of a semicircle river-canyon and a shear wall supported by a semicircle rigid foundation embedded in a homogenous half-space. The closed-form analytical solution for system response is presented based on the wave function expansion method. The analysis focuses on the effects of the canyon-building interaction on system response. The strength of the interaction between the river-canyon topography and the building changes periodically as the distance between the canyon and the structure increases, leading to the interaction having beneficial or harmful effects on the building’s seismic response. The foundation peak response of the building can be amplified by about 10%, and the peak of the building relative response can be amplified by about 40%. The distribution of canyon-structure spacing with strong or weak interaction is closely related to the dynamic characteristics of the building and the incident angle of the wave. When designing buildings along the river, the building and canyon should be analyzed as a whole model to determine whether the location of the building is in a position with strong interaction with the river-canyon. The model in this paper may be useful for obtaining insight into the effects of canyon-structure interaction and interpreting the observed response in buildings and seismic response estimation in general.

Experimental, Analytical and Numerical Studies of Interfacial Bonding Properties between Silane-Coated Steel Fibres and Mortar

A systematic investigation of the effects of silane coatings on steel fibre–mortar interfacial bond properties was conducted, combining pullout tests, analytical solutions, and meso-scale FE simulations. Nine silane coatings were tested, and their effects were evaluated by 30 single fibre pullout tests. They were found to increase the peak force and energy consumption up to 5.75 times and 2.48 times, respectively. Closed-form analytical solutions for pullout load, displacement, and interfacial stress distribution during the whole pullout process were derived based on a tri-linear bond-slip model, whose parameters were calibrated against the pullout tests. Finally, the calibrated bond-slip models were used to simulate the pullout tests and complex failure of multi-fibre specimens in mesoscale finite element models. Such an approach of combining pullout tests, analytical solutions, and mesoscale modelling provides a reliable way to investigate the effects of fibre–mortar interfacial properties on the mechanical behaviour of steel fibre reinforced concrete members in terms of structural strength, stiffness, ductility, and failure mechanisms.

A Relativistic and Electromagnetic Correction to the Ramo–Shockley Theorem

The classical Ramo–Shockley (RS) theorem gives the current induced on perfect conductors by the motion of nearby charges, assuming nonrelativistic motion of those charges in electrostatic fields. This article illustrates how relativistic and electromagnetic effects modify RS in some simple examples. Specifically, we present explicit, closed form analytic solutions of Maxwell’s equations for the induced current distribution on perfectly conducting plates due to the motion of a line charge moving parallel and perpendicular to the plates. The results have been verified by several methods, including particle-in-cell simulations. They are compared with the classical electrostatic theory used to derive RS. New insights into the limitation and validity of RS are provided. Electromagnetic shocks are explicitly calculated in closed form when the line charge strikes a parallel plate transmission line.

Radiative Non-Coaxial Rotating Flow for Viscous Fluid over Accelerated Disk with MHD and Porosity Effects

An analytical solution to analyze the effects of radiation, magnetic, and permeability in an accelerating non-coaxial rotation phenomenon is not yet reported in the previous studies. Therefore, a radiative mixed convection flow for non-coaxial rotating MHD viscous fluid in a porous medium past an accelerated disk is studied. The fluid motion in this problem is induced by two sources which are rotating and buoyancy force. The dimensional coupled differential equations subjected to initial and accelerated boundary conditions are transformed to the dimensionless equations by utilizing appropriate dimensionless variables. The Laplace transform technique is applied to generate the closed form analytical solution for this problem. The impacts of Prandtl number, Grashof number, radiation, magnetic, porosity, and accelerated parameters on the temperature and velocity fields are illustrated graphically. The velocity and temperature profiles satisfy both the initial and boundary conditions, and the present results are found in accordance to the published work. The velocity is improved with the assistance of acceleration, radiation and porosity, while the implementation of magnetic field causes the opposite effect. Increasing radiation leads to the growth of the thermal boundary layer as well as reducing the heat transmission rate. This result can significantly contribute to the designing of heating systems because the imposition of radiation able to sustain an environment for a specific temperature. The obtained analytical solution can be used to check the correctness of the solution obtained from the numerical and experimental studies.

Elastic-Plastic Analytical Solution for SEMI’s Horizontal Brace with a Circumferential Through Crack under Tension and Bending

The elastic-plastic behavior of semi-submersible’s horizontal brace with a circumferential through crack which lies at its boundary was studied. Both tension and bending were considered to investigate the closed-form analytical solution. The results indicate that the tensile plastic zone and crack tip opening displacement (CTOD) on the cracked section increase sharply after a smoothly increment when loads became larger. The cracked horizontal brace with a greater initial circumferential through crack has a larger tensile plastic zone and earlier compressive plastic zone appearance on the cracked section. Compared with the load of tension, the bending load has larger effect on the plastic zones of the cracked section and CTOD of the crack. Introduction Since the first semi-submersible platform (SEMI) named as “Ocean Driller” which was designed for drilling wells in the ocean, the SEMIs have gained popularity in the recent decades with ongoing development in oil and natural gas exploration in deepwater. Because of the high risk of environment pollution and casualties of operators which a destruction accident of SEMI might bring, making sure the structure is safe during its service life has become the most important task of SEMI designers and operators. Although the safety design standards for SEMI structures are quite strict, cracks inescapably are initiated during their service life. According to the destruction accidents happened before, Moan (2009) and Zaron et al. (2015) found that the cracks often occur at the horizontal braces which function as the supporting structures in SEMIs and bear complex loads. The presence of such cracks at critical locations can compromise the safety of the braces and then can cause serious disaster eventually. Because of the initial fracture of a horizontal brace, e.g., the accident of Alexander Kielland platform, the loads were transferred to the other braces and led them break because of the overload (see Colin et al. [2014]).

An analytical model for the behavior of I-shaped beam to cylindrical column connections at room temperature and high temperatures

PurposeIn this paper, a closed-form analytical solution for the prediction of moment-rotation and the rotational stiffness-rotation curves of I-shaped beam to cylindrical column connections, commonly used on offshore platforms, at room and elevated temperatures, are presented.Design/methodology/approachAn analytical solution for the prediction of moment-rotation and the rotational stiffness-rotation curves of I-shaped beam to cylindrical column connections is presented. The results of this model are compared with those of a non-linear coupled mechanical-thermal finite element model and small-scale experimental tests previously provided by the authors.FindingsIn this paper, a closed-form analytical solution for the prediction of moment-rotation and the rotational stiffness-rotation curves of I-shaped beam to cylindrical column connections, commonly used on offshore platforms, at room and elevated temperatures, is presented. The required yield and plastic moments in this model are provided as an extension to Roark's relationships. The results of this model are compared with those of a non-linear coupled mechanical-thermal finite element model and small-scale experimental tests previously provided by the authors. A reasonable agreement has been found between the analytical model results and the experimental/numerical modeling results.Originality/valueThis article is extracted from the author’s doctoral thesis, and all its achievements belong to the authors of the article.

Modelling the seasonal variations of soil temperatures in the Arctic coasts

The soil temperature within the Arctic coasts within the continuous permafrost is not widely measured; the temporal and spatial resolutions of the measured temperature observations are relatively high. In this study, we examined the methods to interpolate, hindcast and forecast temperature measurements within the active layer and shallow permafrost when the temperature measurements at the surface or near the surface are available. The temperature variations along the year are periodic, and hence attempts are made to express the seasonal variations with a combination of periodic function (Fourier components); which are used as boundary conditions to reach the analytical solutions. The temperature measurements from surface to about 10 metre of depths at the Baydaratskaya Bay, Kara Sea are available. We adopted a data-driven model based on simplified analytical closed-form solution derived from the boundary conditions. The parameters of the solution are calibrated from the field measurements and validated with field observations. The model then can be used to hindcast and forecast temperature at any points within the soil.

An Extended Separation-of-Variable Method for Free Vibration of Rectangular Mindlin Plates

For rectangular thick plates with non-Levy boundary conditions, it is important to explore analytical free vibration solutions because the classical inverse and semi-inverse exact solution methods are not applicable to this category of problems. This work is to develop an extended separation-of-variable (SOV) method to find closed-form analytical solutions for the free vibration of rectangular Mindlin plates with arbitrary homogeneous boundary conditions. In the extended SOV method, characteristic differential equations and boundary conditions in two directions are obtained by employing the Rayleigh principle and the assumption that the mode functions are in the SOV form, and two transcendental eigenvalue equations are achieved through boundary conditions. But these two eigenvalue equations cannot be solved simultaneously since there are two equations and only the natural frequency is the unknown variable. Considering this, the second assumption in this method is that the natural frequencies corresponding to two-direction mode functions are independent of each other in the mathematical sense, thus there are two unknowns in two transcendental eigenvalue equations, and the closed-form solutions for plates with arbitrary boundary conditions can be obtained non-iteratively. From the physical sense, the natural frequencies pertaining to different direction mode functions should be the same, and this conclusion is validated analytically and numerically. The present natural frequencies and mode shapes agree well with those obtained by other analytical and numerical methods. Especially, for the plates with at least two opposite sides simply supported, the present solutions are exact.

Free and forced vibration of double beam with arbitrary end conditions connected with a viscoelastic layer and discrete points

The Double-Beam System with Intermediate Supports composed of two parallel beams, arbitrary intermediate supports, and an optional interconnecting viscoElastic Layer (DBS-ISEL) has many engineering applications. However, few publications have developed the exact and explicit mode shape solutions of free vibrations of the DBS-ISEL by far, and the forced vibrations of the DBS-ISEL are not systematically treated. In this work, the free and forced vibrations of the general configuration of the DBS-ISEL under the general boundary conditions are systematically attacked. An analytical solution in closed-form is formulated based on the orthogonality conditions of the mode shape solutions. A general numerical solution is also presented. Numerical examples are demonstrated to show that the proposed solution agrees well with the literature. For the damped vibrations, the asymptotic deflections of the two beams converge to the static deflections. The parametric study shows that the intermediate elastic support could effectively modulate the deflections of the beams. This research is beneficial to the designing, monitoring and maintaining of the DBS-ISEL.

Full-range behavior of FRP-to-concrete bonded joints subjected to combined effects of loading and temperature variation

Concrete structures strengthened with externally bonded fiber-reinforced polymer (FRP) composites are likely to experience significant temperature variations (i.e., thermal loadings) due to daily and seasonal temperature changes or possible fire exposure during service life. Such thermal loadings may lead to changes in the mechanical properties of components (including FRP, bonding adhesive and concrete substrate) and induce interfacial thermal stress due to the thermal incompatibility between FRP and concrete. A limited bond length is usually adopted to investigate the bond behavior (e.g., the local bond-slip characteristics) of the FRP-to-concrete interface under temperature variations. This paper presents an analytical study for the first time to predict the full-range deformation behavior of a mechanically loaded FRP-to-concrete bonded joint with a limited bond length under temperature variations. The solution is in a closed-form manner and has enabled isolation of the interfacial thermal stress effect from the temperature-induced changes of material properties, thereby leading to an accurate calibration of the bond-slip characteristics and the interfacial fracture energy. The closed-form analytical solution has been verified by comparing the analytical predictions with the existing experimental and numerical results in the literature. To further understand the mechanism of the FRP-to-concrete interface under combined thermal and mechanical loadings, the effects of thermal stress and bond length on the interfacial responses and the full-range deformation behavior of the FRP-to-concrete interface are comprehensibly analyzed.

Static plastic analysis of metallic sandwich beam with functionally graded core

Optimal design of density gradient is a vibrant research topic in the realm of sandwich structure due to its potential benefits of enhancing mechanical performance. Herein, we propose a general yield criterion for functionally graded sandwich structures that accounts for the continuously varying yield strength distribution and the interaction of plastic bending and stretching in large deflection regime. The yield criterion facilitates the derivation of closed-form analytical solutions that predict the quasi-static response of fully clamped functionally graded sandwich beam subject to transverse loading. The analytical solutions are quantitatively validated by the finite element (FE) simulations and allow for parametric studies to investigate the effect of gradient layout on the load-bearing capability. We find that the symmetric gradient layout with hourglass-shaped density distribution outperforms the uniform counterpart in load-bearing, whereas the trapezoidal gradient layout fails to provide enhanced mechanical performance.

PSO-Based Soft Lunar Landing with Hazard Avoidance: Analysis and Experimentation

The problem of real-time optimal guidance is extremely important for successful autonomous missions. In this paper, the last phases of autonomous lunar landing trajectories are addressed. The proposed guidance is based on the Particle Swarm Optimization, and the differential flatness approach, which is a subclass of the inverse dynamics technique. The trajectory is approximated by polynomials and the control policy is obtained in an analytical closed form solution, where boundary and dynamical constraints are a priori satisfied. Although this procedure leads to sub-optimal solutions, it results in beng fast and thus potentially suitable to be used for real-time purposes. Moreover, the presence of craters on the lunar terrain is considered; therefore, hazard detection and avoidance are also carried out. The proposed guidance is tested by Monte Carlo simulations to evaluate its performances and a robust procedure, made up of safe additional maneuvers, is introduced to counteract optimization failures and achieve soft landing. Finally, the whole procedure is tested through an experimental facility, consisting of a robotic manipulator, equipped with a camera, and a simulated lunar terrain. The results show the efficiency and reliability of the proposed guidance and its possible use for real-time sub-optimal trajectory generation within laboratory applications.

Free-edge crack onset induced by thermal loading

In composite laminates the layer-wise material mismatch induces highly localized stress concentrations at the free edges. This is referred to as free-edge effect. As a consequence interlaminar crack initiation may occur in addition to the typically considered intralaminar failure modes. The stresses do not necessarily have to be caused by mechanical loads. Also temperature differences may induce significant interlaminar stresses. In the present study, delamination onset in symmetric angle-ply laminates caused by thermal stresses is investigated by means of the coupled stress and energy criterion within the framework of finite fracture mechanics on the basis of the well-known generalized plane strain state. Using this relatively young criterion, both failure load and the length of the initiated crack can be predicted. Besides the intrinsic material properties in terms of strength and fracture toughness, only the interlaminar stresses as well as the released energy during crack initiation are required for the evaluation of the coupled criterion. Since no analytical closed-form solution of the free-edge effect is known, the required field quantities are obtained by employing a finite element model. Effects as the influence of the thickness and the orientation of the individual layers on the failure temperature are studied. For that purpose, a dimensional analysis is performed allowing to scale the results of a certain boundary value problem to self-similar configurations yielding a significant reduction of the numerical effort. It is revealed that especially for thick angle-ply laminates with large fiber orientation angles and temperature differences as they may occur in aeronautical applications interlaminar crack initiation can be the predominant failure mode. For validation purposes the results obtained by the coupled criterion are compared to the widely-used cohesive zone model.

Prediction of the acceleration and stopping manoeuvres of a bare hull surface combatant by closed-form solutions and CFD

In this study, the acceleration and stopping manoeuvres of a bare hull surface combatant have practically been predicted by closed-form (analytical) solutions derived from basic mechanical principles. They have also been simulated directly by an unsteady RANS (Reynolds-averaged Navier-Stokes) solver with overset grid structure to compare the results with those of closed-form approaches. A well-known surface combatant model DTMB5415 has been chosen for validation. The instantaneous drag forces of the model ship in the acceleration and stopping manoeuvres have been provided by both CFD and Holtrop's methods. It has been observed that the results by analytical solutions have a good match with those of the manoeuvring simulations based on unsteady RANS solver. Successful predictions have also been made consistent with the results obtained when experimental resistance values are used in analytical solutions. It has been concluded that the closed-form solution methods yield practically satisfactory results in a very short time when a fast resistance prediction method, such as Holtrop's method or CFD, is used.

Closed-form solution for shock wave propagation in density-graded cellular material under impact

Density-graded cellular materials have tremendous potential in structural applications where impact resistance is required. Cellular materials subjected to high impact loading result in a compaction type deformation, usually modeled using continuum-based shock theory. The resulting governing differential equation of the shock model is nonlinear, and the density gradient further complicates the problem. Earlier studies have employed numerical methods to obtain the solution. In this study, an analytical closed-form solution is proposed to predict the response of density-graded cellular materials subjected to a rigid body impact. Solutions for the velocity of the impinging rigid body mass, energy absorption capacity of the cellular material, and the incident stress are obtained for a single shock propagation. The results obtained are in excellent agreement with the existing numerical solutions found in the literature. The proposed analytical solution can be potentially used for parametric studies and for effectively designing graded structures to mitigate impact.