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 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 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 On Symmetrical Deformation of Toroidal Shell with Circular Cross-Section

2020 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Differential Equation ◽  
Cross Section ◽  
Circular Cross Section ◽  
Gauss Curvature ◽  
Toroidal Shell ◽  
Symmetrical Deformation ◽  
Deformation Equation ◽  
Bending Capacity ◽  
Heun’S Equation ◽  
Geometric Study

By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, the complicated deformation equation of toroidal shell is successfully transferred into a well-known equation, namely Heun's equation of ordinary differential equation, whose exact solution is obtained in terms of Heun's functions. The computation of the problem can be carried out by symbolic software that is able to with the Heun's function, such as Maple. The geometric study of the Gauss curvature shows that the internal portion of the toroidal shell has better bending capacity than the outer portion, which might be useful for the design of metamaterials with toroidal shell cells.

Determination of collapse moment of different angled pipe bends with initial geometric imperfection subjected to in-plane bending moments

2021 ◽  
pp. 095440622199640
Author(s):  
Manish Kumar ◽  
Pronab Roy ◽  
Kallol Khan
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Circular Cross Section ◽  
Initial Imperfection ◽  
Bend Angle ◽  
Bending Moments ◽  
Element Analysis ◽  
Geometric Imperfection ◽  
Initial Geometric Imperfection

From the recent literature, it is revealed that pipe bend geometry deviates from the circular cross-section due to pipe bending process for any bend angle, and this deviation in the cross-section is defined as the initial geometric imperfection. This paper focuses on the determination of collapse moment of different angled pipe bends incorporated with initial geometric imperfection subjected to in-plane closing and opening bending moments. The three-dimensional finite element analysis is accounted for geometric as well as material nonlinearities. Python scripting is implemented for modeling the pipe bends with initial geometry imperfection. The twice-elastic-slope method is adopted to determine the collapse moments. From the results, it is observed that initial imperfection has significant impact on the collapse moment of pipe bends. It can be concluded that the effect of initial imperfection decreases with the decrease in bend angle from 150∘ to 45∘. Based on the finite element results, a simple collapse moment equation is proposed to predict the collapse moment for more accurate cross-section of the different angled pipe bends.

scholarly journals Validation of Welding Simulations Using Thermal Strains Measured with DIC

2011 ◽  
Vol 70 ◽  
pp. 129-134 ◽  
Author(s):  
Maarten De Strycker ◽  
Pascal Lava ◽  
Wim Van Paepegem ◽  
Luc Schueremans ◽  
Dimitri Debruyne
Keyword(s):  
Residual Stresses ◽  
Cross Section ◽  
Circular Cross Section ◽  
Welding Process ◽  
Weld Bead ◽  
Image Correlation ◽  
Steel Tubes ◽  
Element Analysis ◽  
Strain Evolution ◽  
The Cross

Residual stresses can affect the performance of steel tubes in many ways and as a result their magnitude and distribution is of particular interest to many applications. Residual stresses in cold-rolled steel tubes mainly originate from the rolling of a flat plate into a circular cross section (involving plastic deformations) and the weld bead that closes the cross section (involving non-uniform heating and cooling). Focus in this contribution is on the longitudinal weld bead that closes the cross section. To reveal the residual stresses in the tubes under consideration, a finite element analysis (FEA) of the welding step in the production process is made. The FEA of the welding process is validated with the temperature evolution of the thermal simulation and the strain evolution for the mechanical part of the analysis. Several methods for measuring the strain evolution are available and in this contribution it is investigated if the Digital Image Correlation (DIC) technique can record the strain evolution during welding. It is shown that the strain evolution obtained with DIC is in agreement with that found by electrical resistance strain gauges. The results of these experimental measuring methods are compared with numerical results from a FEA of the welding process.

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.

Finite Element Analysis on the Blast Resistant Performance of Multibarrel Tube-Confined Concrete Column in Different Cross-Sections

2013 ◽  
Vol 721 ◽  
pp. 545-550
Author(s):  
Sai Wu ◽  
Jun Hai Zhao ◽  
Er Gang Xiong
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Detonation Wave ◽  
Cross Section ◽  
Cross Sections ◽  
Failure Modes ◽  
Circular Cross Section ◽  
Blast Load ◽  
Element Analysis ◽  
The Impact

Based on the finite element analysis software ANSYS/LS-DYNA, this paper numerically analyzed the dynamic performance of MTCCCs with different cross sections under blast load, followed by the study and comparison on the differences of the detonation wave propagation and failure modes between the columns in circular cross section and square cross section. The results show: The blast resistant performance of the circular component is more superior than the square component for its better aerodynamic shape that can greatly reduce the impact of the detonation wave on the column; The main difference of the failure modes between the circular and square cross-sectional components under blast load lies in the different failure mode of the outer steel tube. The simulation results in this paper can provide some references for the blast resisting design of MTCCCs.

scholarly journals Numerical Studies on the Stiffness of Arc Elliptical Cross-section Helical Spring Subjected to Circumference Force

Mechanika ◽  
2021 ◽  
Vol 27 (4) ◽  
pp. 327-334
Author(s):  
Yuan WANG ◽  
Qingchun WANG ◽  
Zehao SU
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Cross Section ◽  
Circular Cross Section ◽  
Elliptical Cross Section ◽  
Helical Spring ◽  
Element Analysis ◽  
Elliptical Cross ◽  
Stiffness Property ◽  
Regression Formula

 Due to its excellent properties, elliptical cross-section helical spring has been widely used in automobile industry, such as valve spring, arc spring used in Dual Mass Flywheel and so on. Existing stiffness formulae of helical spring remain to be tested, and stiffness property of elliptical cross-section arc spring has been little studied. Hence, study on the stiffness of elliptical cross-section helical spring is significant in the development of elliptical cross-section helical spring. This paper proposes a method to study the stiffness property of elliptical cross-section helical spring that the experiment design method is adopted with finite element analysis. Firstly, the finite element analysis method was used to verify the cylindrical (circular cross-section and elliptical cross-section) springs. Then, the regression formula was designed and derived compared with the reference springs’ stiffness formulae by experimental design. Last, regression formula was verified with existing experiment data. The novelty in this paper is that simulation technology of arc spring was investigated and a stiffness regression equation of arc elliptical cross-section spring was obtained using orthogonal regression design, with significance in wide use of the arc elliptical cross-section helical spring promotion. 

Numerical Investigation into Rockburst Prevention Technology in Deep-Buried Tunnels with Circular Cross-Section

2011 ◽  
Vol 368-373 ◽  
pp. 1192-1195
Author(s):  
Xin Yu Wang ◽  
Zhu Shan Shao ◽  
Yu Ming Cui
Keyword(s):  
Finite Element Analysis ◽  
Cross Section ◽  
Circular Cross Section ◽  
Surrounding Rock ◽  
Dynamic Failure ◽  
Construction Process ◽  
Underground Construction ◽  
Element Analysis ◽  
The Finite Element Analysis ◽  
Two Measures

Rockburst is a common dynamic failure phenomenon in deep-buried underground construction process, which greatly threatened the safety of equipment and person. In the paper, combined with the engineering of Xinjiang Xiabandi hydraulic and the Russense’s rockburst criteria, the finite element analysis are employed to study the effect of advance dig-hole and softening of surrounding rock. The results of the study indicate that the two measures can effectively reduce the intensity of rockburst level in the construction of deep-buried tunnel.

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