scholarly journals Gol'denveizer Problem of Elastic Torus

2021 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Differential Equation ◽  
Complex Form ◽  
Toroidal Shell ◽  
Membrane Theory ◽  
Real Form ◽  
Element Analysis ◽  
Numerical Studies ◽  
Axial Forces ◽  
Ode System ◽  
Theory Of Shells

The Gol'denveizer problem of a torus can be described as follows: a toroidal shell is loaded under axial forces and the outer and inner equators are loaded with opposite balanced forces. Gol'denveizer pointed out that the membrane theory of shells is unable to predict deformation in this problem, as it yields diverging stress near the crowns. Although the problem has been studied by Audoly and Pomeau (2002) with the membrane theory of shells, the problem is still far from resolved within the framework of bending theory of shells. In this paper, the bending theory of shells is applied to formulate the Gol'denveizer problem of a torus. To overcome the computational difficulties of the governing complex-form ordinary differential equation (ODE), the complex-form ODE is converted into a real-form ODE system. Several numerical studies are carried out and verified by finite-element analysis. Investigations reveal that the deformation and stress of an elastic torus are sensitive to the radius ratio, and the Gol'denveizer problem of a torus can only be fully understood based on the bending theory of shells.

scholarly journals Small Symmetrical Deformation of Thin Torus with Circular Cross-Section

2021 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cross Section ◽  
Circular Cross Section ◽  
Complex Form ◽  
Real Form ◽  
Element Analysis ◽  
Numerical Studies ◽  
Ode System ◽  
Symmetrical Deformation

By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, a complex-form ordinary differential equation (ODE) for the small symmetrical deformation of an elastic torus is successfully transformed into the well-known Heun's ODE, whose exact solution is obtained in terms of Heun's functions. To overcome the computational difficulties of the complex-form ODE in dealing with boundary conditions, a real-form ODE system is proposed. A general code of numerical solution of the real-form ODE is written by using Maple. Some numerical studies are carried out and verified by both finite element analysis and H. Reissner's formulation. Our investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell.

scholarly journals Small Symmetrical Deformation of Thin Torus with Circular Cross-Section

2020 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Differential Equation ◽  
Cross Section ◽  
Circular Cross Section ◽  
Complex Form ◽  
Real Form ◽  
Element Analysis ◽  
Numerical Studies ◽  
Ode System ◽  
Symmetrical Deformation ◽  
Maple Code

By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, a complex-form ordinary differential equation (ODE) for the small symmetrical deformation of an elastic torus is successfully transformed into the well-known Heun's ODE, whose exact solution is obtained in terms of Heun's functions. To overcome the computational difficulties of the complex-form ODE in dealing with boundary conditions, a real-form ODE system is proposed. A general code of numerical solution of the real-form ODE is written by using Maple. Some numerical studies are carried out and verified by finite element analysis. Our investigations show that the mechanics of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell. A general Maple code is provided as essential part of this paper.

scholarly journals Small Symmetrical Deformation of Thin Torus with Circular Cross-Section

2021 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cross Section ◽  
Circular Cross Section ◽  
Complex Form ◽  
Real Form ◽  
Element Analysis ◽  
Numerical Studies ◽  
Ode System ◽  
Symmetrical Deformation

By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, a complex-form ordinary differential equation (ODE) for the small symmetrical deformation of an elastic torus is successfully transformed into the well-known Heun's ODE, whose exact solution is obtained in terms of Heun's functions. To overcome the computational difficulties of the complex-form ODE in dealing with boundary conditions, a real-form ODE system is proposed. A general code of numerical solution of the real-form ODE is written by using Maple. Some numerical studies are carried out and verified by both finite element analysis and H. Reissner's formulation. Our investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell.

scholarly journals Experimental and Numerical Studies on the Structural Performance of a Double Composite Wall

2021 ◽  
Vol 11 (2) ◽  
pp. 506
Author(s):  
Sun-Jin Han ◽  
Inwook Heo ◽  
Jae-Hyun Kim ◽  
Kang Su Kim ◽  
Young-Hun Oh
Keyword(s):  
Shear Strength ◽  
Precast Concrete ◽  
Structural Performance ◽  
Behavioral Characteristics ◽  
Element Analysis ◽  
Numerical Studies ◽  
Shear Performance ◽  
Flexural Analysis ◽  
Composite Wall ◽  
Current Code

In this study, experiments and numerical analyses were carried out to examine the flexural and shear performance of a double composite wall (DCW) manufactured using a precast concrete (PC) method. One flexural specimen and three shear specimens were fabricated, and the effect of the bolts used for the assembly of the PC panels on the shear strength of the DCW was investigated. The failure mode, flexural and shear behavior, and composite behavior of the PC panel and cast-in-place (CIP) concrete were analyzed in detail, and the behavioral characteristics of the DCW were clearly identified by comparing the results of tests with those obtained from a non-linear flexural analysis and finite element analysis. Based on the test and analysis results, this study proposed a practical equation for reasonably estimating the shear strength of a DCW section composed of PC, CIP concrete, and bolts utilizing the current code equations.

scholarly journals Design and Analysis of a Novel Axial-Radial Flux Permanent Magnet Machine with Halbach-Array Permanent Magnets

Energies ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 3639
Author(s):  
Rundong Huang ◽  
Chunhua Liu ◽  
Zaixin Song ◽  
Hang Zhao
Keyword(s):  
Permanent Magnet ◽  
Permanent Magnets ◽  
Torque Ripple ◽  
Axial Flux ◽  
Element Analysis ◽  
Torque Density ◽  
Axial Forces ◽  
Halbach Array ◽  
Radial Flux ◽  
High Torque

Electric machines with high torque density are needed in many applications, such as electric vehicles, electric robotics, electric ships, electric aircraft, etc. and they can avoid planetary gears thus reducing manufacturing costs. This paper presents a novel axial-radial flux permanent magnet (ARFPM) machine with high torque density. The proposed ARFPM machine integrates both axial-flux and radial-flux machine topologies in a compact space, which effectively improves the copper utilization of the machine. First, the radial rotor can balance the large axial forces on axial rotors and prevent them from deforming due to the forces. On the other hand, the machine adopts Halbach-array permanent magnets (PMs) on the rotors to suppress air-gap flux density harmonics. Also, the Halbach-array PMs can reduce the total attracted force on axial rotors. The operational principle of the ARFPM machine was investigated and analyzed. Then, 3D finite-element analysis (FEA) was conducted to show the merits of the ARFPM machine. Demonstration results with different parameters are compared to obtain an optimal structure. These indicated that the proposed ARFPM machine with Halbach-array PMs can achieve a more sinusoidal back electromotive force (EMF). In addition, a comparative analysis was conducted for the proposed ARFPM machine. The machine was compared with a conventional axial-flux permanent magnet (AFPM) machine and a radial-flux permanent magnet (RFPM) machine based on the same dimensions. This showed that the proposed ARFPM machine had the highest torque density and relatively small torque ripple.

scholarly journals An Improved Analytical Solution for Process-Induced Residual Stresses and Deformations in Flat Composite Laminates Considering Thermo-Viscoelastic Effects

Materials ◽  
2018 ◽  
Vol 11 (12) ◽  
pp. 2506 ◽  
Author(s):  
Chao Liu ◽  
Yaoyao Shi
Keyword(s):  
Differential Equation ◽  
Finite Element ◽  
Analytical Solution ◽  
Residual Stresses ◽  
Composite Laminates ◽  
Composite Structures ◽  
Numerical Models ◽  
Element Analysis ◽  
Current Time ◽  
Viscoelastic Effects

Dimensional control can be a major concern in the processing of composite structures. Compared to numerical models based on finite element methods, the analytical method can provide a faster prediction of process-induced residual stresses and deformations with a certain level of accuracy. It can explain the underlying mechanisms. In this paper, an improved analytical solution is proposed to consider thermo-viscoelastic effects on residual stresses and deformations of flat composite laminates during curing. First, an incremental differential equation is derived to describe the viscoelastic behavior of composite materials during curing. Afterward, the analytical solution is developed to solve the differential equation by assuming the solution at the current time, which is a linear combination of the corresponding Laplace equation solutions of all time. Moreover, the analytical solution is extended to investigate cure behavior of multilayer composite laminates during manufacturing. Good agreement between the analytical solution results and the experimental and finite element analysis (FEA) results validates the accuracy and effectiveness of the proposed method. Furthermore, the mechanism generating residual stresses and deformations for unsymmetrical composite laminates is investigated based on the proposed analytical solution.

scholarly journals A Review on Numerical Analysis of RC Flat Slabs Exposed to Fire

2020 ◽  
Vol 27 (1) ◽  
pp. 1-5
Author(s):  
Hanadi Naji ◽  
Nibras Khalid ◽  
Mutaz Medhlom
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Numerical Analysis ◽  
Bending Moment ◽  
Mechanical And Thermal Properties ◽  
Vertical Deflection ◽  
Element Analysis ◽  
Flat Slabs ◽  
Numerical Studies ◽  
The Finite Element Analysis

This paper aims at presenting and discussing the numerical studies performed to estimate the mechanical and thermal behavior of RC flat slabs at elevated temperature and fire. The numerical analysis is carried out using finite element programs by developing models to simulate the performance of the buildings subjected to fire. The mechanical and thermal properties of the materials obtained from the experimental work are involved in the modeling that the outcomes will be more realistic. Many parameters related to fire resistance of the flat slabs have been studied and the finite element analysis results reveal that the width and thickness of the slab, the temperature gradient, the fire direction, the exposure duration and the thermal restraint are important factors that influence the vertical deflection, bending moment and force membrane of the flat slabs exposed to fire. However, the validation of the models is verified by comparing their results to the available experimental date. The finite element modeling contributes in saving cost and time consumed by experiments.

scholarly journals Displacement Formulations for Deformation and Vibration of Elastic Circular Torus

2021 ◽  
Author(s):  
bohua sun
Keyword(s):  
Free Vibration ◽  
Gaussian Curvature ◽  
Turning Point ◽  
Closed Form Solution ◽  
Complex Form ◽  
Form Solution ◽  
Element Analysis ◽  
Displacement Boundary ◽  
Engineering Mechanics ◽  
Maple Code

The formulation used by most of the studies on an elastic torus are either Reissner mixed formulation or Novozhilov's complex-form one, however, for vibration and some displacement boundary related problem of a torus, those formulations face a great challenge. It is highly demanded to have a displacement-type formulation for the torus. In this paper, I will carry on my previous work [ B.H. Sun, Closed-form solution of axisymmetric slender elastic toroidal shells. J. of Engineering Mechanics, 136 (2010) 1281-1288.], and with the help of my own maple code, I am able to simulate some typical problems and free vibration of the torus. The numerical results are verified by both finite element analysis and H. Reissner's formulation. My investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell, and also reveal that the inner torus is stronger than outer torus due to the property of their Gaussian curvature. Regarding the free vibration of a torus, our analysis indicates that both initial in u and w direction must be included otherwise will cause big errors in eigenfrequency. One of the most intestine discovery is that the crowns of a torus are the turning point of the Gaussian curvature at the crown where the mechanics' response of inner and outer torus is almost separated.

Theoretical Analysis of Activation Energy Effect on Prandtl–Eyring Nanoliquid Flow Subject to Melting Condition

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ikram Ullah ◽  
Rashid Ali ◽  
Hamid Nawab ◽  
Abdussatar ◽  
Iftikhar Uddin ◽  
...  
Keyword(s):  
Differential Equation ◽  
Activation Energy ◽  
Convective Flow ◽  
Skin Friction Coefficient ◽  
System Of Equations ◽  
Energy Effect ◽  
Melting Condition ◽  
Ode System ◽  
Physical Importance ◽  
Controlling Variables

Abstract This study models the convective flow of Prandtl–Eyring nanomaterials driven by a stretched surface. The model incorporates the significant aspects of activation energy, Joule heating and chemical reaction. The thermal impulses of particles with melting condition is addressed. The system of equations is an ordinary differential equation (ODE) system and is tackled numerically by utilizing the Lobatto IIIA computational solver. The physical importance of flow controlling variables to the temperature, velocity and concentration is analyzed using graphical illustrations. The skin friction coefficient and Nusselt number are examined. The results of several scenarios, mesh-point utilization, the number of ODEs and boundary conditions evaluation are provided via tables.

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