scholarly journals Application of exponential functions in weighted residuals method in structural mechanics. Part 3: infinite cylindrical shell under concentrated forces

2021 ◽  
Vol 5 (2) ◽  
pp. 165-176
Author(s):  
Igor Orynyak ◽  
Yulia Bai ◽  
Anastasiia Hryhorenko
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Fundamental Problem ◽  
Weighted Residual Method ◽  
Exponential Functions ◽  
Analytical Treatment ◽  
Concentrated Force ◽  
Exact Relation ◽  
Analytical Technique ◽  
Weighted Residuals

Solution for cylindrical shell under concentrated force is a fundamental problem which allow to consider many other cases of loading and geometries. Existing solutions were based on simplified assumptions, and the ranges of accuracy of them still remains unknown. The common idea is the expansion of them into Fourier series with respect to circumferential coordinate. This reduces the problem to 8th order even differential equation as to axial coordinate. Yet the finding of relevant 8 eigenfunctions and exact relation of 8 constant of integrations with boundary conditions are still beyond the possibilities of analytical treatment. In this paper we apply the decaying exponential functions in Galerkin-like version of weighted residual method to above-mentioned 8th order equation. So, we construct the sets of basic functions each to satisfy boundary conditions as well as axial and circumferential equilibrium equations. The latter gives interdependencies between the coefficients of circumferential and axial displacements with the radial ones. As to radial equilibrium, it is satisfied only approximately by minimizations of residuals. In similar way we developed technique for application of Navier like version of WRM. The results and peculiarities of WRM application are discussed in details for cos2j concentrated loading, which methodologically is the most complicated case, because it embraces the longest distance over the cylinder. The solution for it clearly exhibits two types of behaviors – long-wave and short-wave ones, the analytical technique of treatment of them was developed by first author elsewhere, and here was successfully compared. This example demonstrates the superior accuracy of two semi analytical WRM methods. It was shown that Navier method while being simpler in realization still requires much more (at least by two orders) terms than exponential functions.

Method of Spling Weight and Residual Value for Analyzing Bending of Laminated Skew Plates under Complex Stress State

2011 ◽  
Vol 295-297 ◽  
pp. 2661-2665
Author(s):  
Ping An Shi
Keyword(s):  
Boundary Conditions ◽  
Present Method ◽  
Composite Plate ◽  
Least Squares Method ◽  
Laminated Composite ◽  
Complex Stress State ◽  
Laminated Composite Plate ◽  
Weighted Residual Method ◽  
Nonlinear Bending ◽  
Weighted Residuals

In order to meet the needs of boundary conditions and eliminate the residual of governing equations, the weighted residuals method is combined with least squares method solve the nonlinear bending problem of symmetrically angle-lay orthotropic laminated composite plate, and the double fifth B-spline is taken as trial function to seek an efficient method for large deflection of plate. The analytical solution of bending problems of symmetrically angle-lay orthotropic laminated composite plate with clamped boundary conditions is obtained by using the weighted residual method. The results from the present method are in good agreement with those derived from other methods, and the present method has the advantages of simple principle, high calculation efficiency and easy to satisfy boundary conditions as well, superiority to finite element method. In addition, the design of computer program is simple and it is easy to be programmed.

Free vibration analysis of three-layer thin cylindrical shell with variable thickness two-dimensional FGM middle layer under arbitrary boundary conditions

2021 ◽  
pp. 109963622110204
Author(s):  
Xue-Yang Miao ◽  
Chao-Feng Li ◽  
Yu-Lin Jiang ◽  
Zi-Xuan Zhang
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Volume Fraction ◽  
Ritz Method ◽  
Admissible Function ◽  
Free Vibration Analysis ◽  
Middle Layer ◽  
Two Dimensional ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions

In this paper, a unified method is developed to analyze free vibrations of the three-layer functionally graded cylindrical shell with non-uniform thickness. The middle layer is composed of two-dimensional functionally gradient materials (2D-FGMs), whose thickness is set as a function of smooth continuity. Four sets of artificial springs are assigned at the ends of the shells to satisfy the arbitrary boundary conditions. The Sanders’ shell theory is used to obtain the strain and curvature-displacement relations. Furthermore, the Chebyshev polynomials are selected as the admissible function to improve computational efficiency, and the equation of motion is derived by the Rayleigh–Ritz method. The effects of spring stiffness, volume fraction indexes, configuration on of shell, and the change in thickness of the middle layer on the modal characteristics of the new structural shell are also analyzed.

Approximate Formulas for Cylindrical Shell Free Vibration Based on Vlasov’s and Enhanced Vlasov’s Semi-Momentless Theory

2018 ◽  
Author(s):  
Igor Orynyak ◽  
Yaroslav Dubyk
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Middle Surface ◽  
Axial Direction ◽  
Mode Shapes ◽  
Natural Mode ◽  
Transverse Deflection ◽  
Approximate Formulas

Simple approximate formulas for the natural frequencies of circular cylindrical shells are presented for modes in which transverse deflection dominates. Based on the Donnell-Mushtari thin shell theory the equations of motion of the circular cylindrical shell are introduced, using Vlasov assumptions and Fourier series for the circumferential direction, an exact solution in the axial direction is obtained. To improve the results assumptions of Vlasov’s semimomentless theory are enhanced, i.e. we have used only the hypothesis of middle surface inextensibility to obtain a solution in axial direction. Nonlinear characteristic equations and natural mode shapes, are derived for all type of boundary conditions. Good agreement with experimental data and FEM is shown and advantage over the existing formulas for a variety of boundary conditions is presented.

scholarly journals Strain Growth in a Finite-Length Cylindrical Shell Under Internal Pressure Pulse

2017 ◽  
Vol 139 (2) ◽  
Author(s):  
Qi Dong ◽  
Q. M. Li ◽  
Jinyang Zheng
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Internal Pressure ◽  
Finite Length ◽  
Pressure Pulse ◽  
Elastic Response ◽  
Local Dynamic ◽  
Growth Phenomenon ◽  
Modal Coupling ◽  
Elastic Responses

Strain growth is a phenomenon observed in the elastic response of containment vessels subjected to internal blast loading. The local dynamic response of a containment vessel may become larger in a later stage than its response in the earlier stage. In order to understand the possible mechanisms of the strain growth phenomenon in a cylindrical vessel, dynamic elastic responses of a finite-length cylindrical shell with different boundary conditions subjected to internal pressure pulse are studied by finite-element simulation using LS-DYNA. It is found that the strain growth in a finite-length cylindrical shell with sliding–sliding boundary conditions is caused by nonlinear modal coupling. Strain growth in a finite-length cylindrical shell with free–free or simply supported boundary conditions is primarily caused by the linear modal superposition, possibly enhanced by the nonlinear modal coupling. The understanding of these strain growth mechanisms can guide the design of cylindrical containment vessels.

Dynamic thermo-elastic response of a discrete multi-layered cylindrical shell subjected to transient thermal loading

2008 ◽  
Vol 222 (4) ◽  
pp. 549-561 ◽  
Author(s):  
J Y Zheng ◽  
X D Wu ◽  
Y J Chen ◽  
G D Deng ◽  
Q M Li ◽  
...  
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Thermal Loading ◽  
Elastic Response ◽  
Elastic Part ◽  
Dynamic Part ◽  
Layered Cylindrical Shell ◽  
Stress Boundary Conditions ◽  
Transient Thermal ◽  
Stress Boundary

Explosion containment vessels (ECVs) are used to fully contain the effects of explosion events. A discrete multi-layered cylindrical shell (DMC) consisting of a thin inner cylindrical shell and helically cross-winding flat steel ribbons has been proposed, which has obvious advantages of fabrication convenience and low costs. The applications of ECVs are closely associated with blast and thermal loads, and thus, it is important to understand the response of a DMC under transient thermal load in order to develop a design code and operation procedures for the use of DMC as ECV. In this paper, a mathematical model for the elastic response of a DMC subjected to thermal loading due to rapid heating is proposed. Based on the axisymmetric plane strain assumption, the displacement solution of the dynamic equilibrium equations of both inner shell and outer ribbon layer are decomposed into two parts, i.e. a thermo-elastic part satisfying inhomogeneous stress boundary conditions and a dynamic part for homogeneous stress boundary conditions. The thermo-elastic part is solved by a linear method and the dynamic part is determined by means of finite Hankel transform and Laplace transform. The thermo-elastic solution of a DMC is compared with the solution of a monobloc cylindrical shell, and numerical results are presented and discussed in terms of winding angle and material parameters.

scholarly journals Modified Fourier–Galerkin Solution for Aerospace Skin-Stiffener Panels Subjected to Interface Force and Mixed Boundary Conditions

Materials ◽  
2019 ◽  
Vol 12 (17) ◽  
pp. 2794
Author(s):  
Renluan Hou ◽  
Qing Wang ◽  
Jiangxiong Li ◽  
Yinglin Ke
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Beam Theory ◽  
Mixed Boundary Conditions ◽  
Combined Loading ◽  
Maximum Deviation ◽  
Concentrated Force ◽  
Stiffened Panels ◽  
Deformation Equation ◽  
Functional Boundary

Aeronautical stiffened panels composed of thin shells and beams are prone to deformation or buckling due to the combined loading, functional boundary conditions and interface forces between joined parts in the assembly processes. In this paper, a mechanical prediction model of the multi-component panel is presented to investigate the deformation propagation, which has a significant effect on the fatigue life of built-up structures. Governing equations of Kirchhoff–Love shell are established, of which displacement expressions are transformed into Fourier series expansions of several introduced potential functions by applying the Galerkin approach. This paper presents an intermediate quantity, concentrated force at the joining interface, to describe mechanical interactions between the coupled components. Based on the Euler–Bernoulli beam theory, unknown intermediate quantity is calculated by solving a 3D stringer deformation equation with static boundary conditions specified on joining points. Compared with the finite element simulation and integrated model, the proposed method can substantially reduce grid number without jeopardizing the prediction accuracy. Practical experiment of the aircraft panel assembly is also performed to obtain the measured data. Maximum deviation between the experimental and predicted clearance values is 0.193 mm, which is enough to meet the requirement for predicting dimensional variations of the aircraft panel assembly.

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