scholarly journals A New Solution to Well-Known Hencky Problem: Improvement of In-Plane Equilibrium Equation

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 653
Author(s):  
Xue Li ◽  
Jun-Yi Sun ◽  
Zhi-Hang Zhao ◽  
Shou-Zhen Li ◽  
Xiao-Ting He
Keyword(s):  
Closed Form ◽  
Equilibrium Equation ◽  
Circumferential Stress ◽  
Closed Form Solution ◽  
Form Solution ◽  
Circular Membrane ◽  
Equilibrium Equations ◽  
Power Series Method ◽  
Out Of Plane ◽  
Plane Equilibrium

In this paper, the well-known Hencky problem—that is, the problem of axisymmetric deformation of a peripherally fixed and initially flat circular membrane subjected to transverse uniformly distributed loads—is re-solved by simultaneously considering the improvement of the out-of-plane and in-plane equilibrium equations. In which, the so-called small rotation angle assumption of the membrane is given up when establishing the out-of-plane equilibrium equation, and the in-plane equilibrium equation is, for the first time, improved by considering the effect of the deflection on the equilibrium between the radial and circumferential stress. Furthermore, the resulting nonlinear differential equation is successfully solved by using the power series method, and a new closed-form solution of the problem is finally presented. The conducted numerical example indicates that the closed-form solution presented here has a higher computational accuracy in comparison with the existing solutions of the well-known Hencky problem, especially when the deflection of the membrane is relatively large.

scholarly journals A Revisit of the Boundary Value Problem for Föppl–Hencky Membranes: Improvement of Geometric Equations

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 631 ◽  
Author(s):  
Yong-Sheng Lian ◽  
Jun-Yi Sun ◽  
Zhi-Hang Zhao ◽  
Xiao-Ting He ◽  
Zhou-Lian Zheng
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Boundary Value ◽  
Closed Form Solution ◽  
Form Solution ◽  
Circular Membrane ◽  
Computational Accuracy ◽  
Power Series Method ◽  
Membrane Problem ◽  
Geometric Equation

In this paper, the well-known Föppl–Hencky membrane problem—that is, the problem of axisymmetric deformation of a transversely uniformly loaded and peripherally fixed circular membrane—was resolved, and a more refined closed-form solution of the problem was presented, where the so-called small rotation angle assumption of the membrane was given up. In particular, a more effective geometric equation was, for the first time, established to replace the classic one, and finally the resulting new boundary value problem due to the improvement of geometric equation was successfully solved by the power series method. The conducted numerical example indicates that the closed-form solution presented in this study has higher computational accuracy in comparison with the existing solutions of the well-known Föppl–Hencky membrane problem. In addition, some important issues were discussed, such as the difference between membrane problems and thin plate problems, reasonable approximation or assumption during establishing geometric equations, and the contribution of reducing approximations or relaxing assumptions to the improvement of the computational accuracy and applicability of a solution. Finally, some opinions on the follow-up work for the well-known Föppl–Hencky membrane were presented.

scholarly journals Closed-Form Solution for Circular Membranes under In-Plane Radial Stretching or Compressing and Out-of-Plane Gas Pressure Loading

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1238
Author(s):  
Bin-Bin Shi ◽  
Jun-Yi Sun ◽  
Ting-Kai Huang ◽  
Xiao-Ting He
Keyword(s):  
Closed Form ◽  
Plane Stress ◽  
Large Deflection ◽  
Closed Form Solution ◽  
Form Solution ◽  
Gas Pressure ◽  
Circular Membrane ◽  
Pressure Loading ◽  
Out Of Plane ◽  
Large Deflection Problem

The large deflection phenomenon of an initially flat circular membrane under out-of-plane gas pressure loading is usually involved in many technical applications, such as the pressure blister or bulge tests, where a uniform in-plane stress is often present in the initially flat circular membrane before deflection. However, there is still a lack of an effective closed-form solution for the large deflection problem with initial uniform in-plane stress. In this study, the problem is formulated and is solved analytically. The initial uniform in-plane stress is first modelled by stretching or compressing an initially flat, stress-free circular membrane radially in the plane in which the initially flat circular membrane is located, and based on this, the boundary conditions, under which the large deflection problem of an initially flat circular membrane under in-plane radial stretching or compressing and out-of-plane gas pressure loading can be solved, are determined. Therefore, the closed-form solution presented in this paper can be applied to the case where the initially flat circular membrane may, or may not, have a uniform in-plane stress before deflection, and the in-plane stress can be either tensile or compressive. The numerical example conducted shows that the closed-form solution presented has satisfactory convergence.

A closed form solution for bending/stretching analysis of functionally graded circular plates under asymmetric loading using the Green function

2010 ◽  
Vol 224 (6) ◽  
pp. 1153-1163 ◽  
Author(s):  
A R Saidi ◽  
A Naderi ◽  
E Jomehzadeh
Keyword(s):  
Green Function ◽  
Closed Form ◽  
Volume Fraction ◽  
Plate Theory ◽  
Closed Form Solution ◽  
Form Solution ◽  
Functionally Graded ◽  
Transverse Displacement ◽  
Circular Plates ◽  
Equilibrium Equations

In this article, a closed-form solution for bending/stretching analysis of functionally graded (FG) circular plates under asymmetric loads is presented. It is assumed that the material properties of the FG plate are described by a power function of the thickness variable. The equilibrium equations are derived according to the classical plate theory using the principle of total potential energy. Two new functions are introduced to decouple the governing equilibrium equations. The three highly coupled partial differential equations are then converted into an independent equation in terms of transverse displacement. A closed-form solution for deflection of FG circular plates under arbitrary lateral eccentric concentrated force is obtained by defining a new coordinate system. This solution can be used as a Green function to obtain the closed-form solution of the FG plate under arbitrary loadings. Also, the solution is employed to solve some different asymmetric problems. Finally, the stress and displacement components are obtained exactly for each problem and the effect of volume fraction is also studied.

scholarly journals A Closed-Form Solution for the Boundary Value Problem of Gas Pressurized Circular Membranes in Contact with Frictionless Rigid Plates

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1017
Author(s):  
Dong Mei ◽  
Jun-Yi Sun ◽  
Zhi-Hang Zhao ◽  
Xiao-Ting He
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Boundary Value ◽  
Closed Form Solution ◽  
Form Solution ◽  
Gas Pressure ◽  
Circular Membrane ◽  
Rigid Plate ◽  
Pressure Loading ◽  
Numerical Example

In this paper, the static problem of equilibrium of contact between an axisymmetric deflected circular membrane and a frictionless rigid plate was analytically solved, where an initially flat circular membrane is fixed on its periphery and pressurized on one side by gas such that it comes into contact with a frictionless rigid plate, resulting in a restriction on the maximum deflection of the deflected circular membrane. The power series method was employed to solve the boundary value problem of the resulting nonlinear differential equation, and a closed-form solution of the problem addressed here was presented. The difference between the axisymmetric deformation caused by gas pressure loading and that caused by gravity loading was investigated. In order to compare the presented solution applying to gas pressure loading with the existing solution applying to gravity loading, a numerical example was conducted. The result of the conducted numerical example shows that the two solutions agree basically closely for membranes lightly loaded and diverge as the external loads intensify.

scholarly journals Analytic Simulations of Packaging Cardboards with Non-Rigid Adhesives

2019 ◽  
Vol 7 (6) ◽  
pp. 01-15
Author(s):  
R. Hussein
Keyword(s):  
Boundary Conditions ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Equilibrium Equations ◽  
The Real ◽  
Usual Assumption ◽  
Perfect Bonding

The understanding of the cardboard performance is necessary to the design of packaging containers and the protection of their contents for safe deliveries. The use of adhesives is unavoidable in the manufacturing of the cardboards. Like all materials, the adhesives have finite stiffness but when used in the literature, they are assumed perfectly rigid. This study changes this assumption by using the real properties of adhesives. A closed-form solution for cardboard panelsassembled withnon-rigid adhesives, and subjected to edgewise loading is presented. The solution satisfies the equilibrium equations of the layers, the compatibility equations of stresses and strains at the interfaces, and the boundary conditions. To investigate the effects of the finite values of adhesivestiffness on the responses, numerical evaluations are conducted. The results obtained have shown that the adhesive stiffness has a strong effect on the performance. Beyond a certain level of stiffness, the usual assumption of perfect bonding used in classical theories is acceptable. This could provide an answer to what constitutes perfect bonding in terms of the ratio of the fluted layer, or simply flute, stiffness to the bonding stiffness.

scholarly journals Closed-Form Solution of a Peripherally Fixed Circular Membrane under Uniformly Distributed Transverse Loads and Deflection Restrictions

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Teng-fei Wang ◽  
Xiao-ting He ◽  
Yang-hui Li
Keyword(s):  
Approximate Solution ◽  
Closed Form ◽  
Closed Form Solution ◽  
Axisymmetric Deformation ◽  
Form Solution ◽  
Circular Membrane ◽  
Rigid Plate ◽  
Membrane Stress ◽  
Transverse Loads ◽  
First Time

The problem of axisymmetric deformation of a peripherally fixed and uniformly loaded circular membrane under deflection restrictions (by a frictionless horizontal rigid plate) was analytically solved, where the assumption of constant membrane stress adopted in the existing work was given up, and a closed-form solution of this problem was presented for the first time. The numerical analysis shows that the closed-form solution presented here has higher calculation accuracy than the existing approximate solution.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations
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