Calculation of the stress state of a noncircular helicoidal tube based on the tensor theory of shells

2021 ◽  
Author(s):  
A.G. Sorokina ◽  
V.A. Kirsanova ◽  
V.F. Fomicheva
Keyword(s):  
Strain Energy ◽  
Nabla Operator ◽  
Angular Coordinate ◽  
Total Potential ◽  
Tensor Theory ◽  
Displacement Vectors ◽  
Helicoidal Surface ◽  
Deformed State ◽  
Set Of Points ◽  
Theory Of Shells

The purpose of the study was to develop an algorithm for calculating helical-symmetric shells with a closed contour in oblique Gaussian coordinates. The twist and length of the shell were taken unchanged. The method is based on the representation of the generating contour of the helicoidal surface by a discrete set of points with the replacement of differentiation along the angular coordinate by finite differences. The unknown were the displacement vectors at the indicated points of the contour. Due to the helicoidal symmetry, the differentiation of vector quantities with respect to the helical coordinate was replaced by vector multiplication. The tensor of deformations and the tensor of the parameters of the change in curvature were calculated using the nabla operator, represented in oblique Gaussian coordinates. Integration over the contour coordinate was replaced by summation over discrete points. The tensors found, which characterize the deformed state, were used to calculate the strain energy of one period of the helicoidal shell, and then the total potential of the mechanical system was compiled. The unknown displacements were determined by minimizing the total potential, taking into account the constraints that prohibit the displacement of the shell as a rigid whole. The study gives a numerical example of the application of the developed approach.

Buckling of Multiple-Bay Ring-Reinforced Cylindrical Shells Subject to Hydrostatic Pressure

1953 ◽  
Vol 20 (4) ◽  
pp. 469-474
Author(s):  
W. A. Nash
Keyword(s):  
Cylindrical Shell ◽  
Hydrostatic Pressure ◽  
Strain Energy ◽  
Elastic Strain Energy ◽  
Middle Surface ◽  
Elastic Buckling ◽  
Elastic Instability ◽  
Total Potential ◽  
External Forces ◽  
Work Done

Abstract An analytical solution is presented for the problem of the elastic instability of a multiple-bay ring-reinforced cylindrical shell subject to hydrostatic pressure applied in both the radial and axial directions. The method used is that of minimization of the total potential. Expressions for the elastic strain energy in the shell and also in the rings are written in terms of displacement components of a point in the middle surface of the shell. Expressions for the work done by the external forces acting on the cylinder likewise are written in terms of these displacement components. A displacement configuration for the buckled shell is introduced which is in agreement with experimental evidence, in contrast to the arbitrary patterns assumed by previous investigators. The total potential is expressed in terms of these displacement components and is then minimized. As a result of this minimization a set of linear homogeneous equations is obtained. In order that a nontrivial solution to this system of equations exists, it is necessary that the determinant of the coefficients vanish. This condition determines the critical pressure at which elastic buckling of the cylindrical shell will occur.

scholarly journals Calculation of a beam on an inhomogeneous basis

2018 ◽  
Vol 251 ◽  
pp. 04023
Author(s):  
Nikolaj Atarov
Keyword(s):  
Numerical Calculation ◽  
Power Law ◽  
Modulus Of Elasticity ◽  
Strain Energy ◽  
Deformation Modulus ◽  
Bending Moment ◽  
System Of Differential Equations ◽  
Total Potential ◽  
Modulus Of Deformation ◽  
Symmetric Load

The questions of analytical calculating the beam on the base with the modulus of deformation (modulus of elasticity), which is changed by thickness of the layer by power law, was deals in the article. The purpose of work was receiving finite expressions for reactive pressure of the basе on a beam and internal efforts in a beam when using model of the base with two characteristics (coefficient of subgrade resistance). A system of differential equations second-order with respect to the displacements of points the surface of a layer with a continuously changing modulus of elasticity was obtained based on the minimum of the total potential strain energy. The calculation of the rigid beam on the base on the action of the symmetric load was performed and the formulas for the reactive pressures of the base were got. Numerical calculation is executed and the analysis of influence of change of the module of deformation of a layer by the amount of reactive pressure and the bending moment in a beam was given. It is shown that with increase in the module of deformation on layer thickness the basе with two characteristics on nature of work is approaching vinklerovsky.

scholarly journals A Finite Circular Arch Element Based on Trigonometric Shape Functions

2007 ◽  
Vol 2007 ◽  
pp. 1-19 ◽  
Author(s):  
H. Saffari ◽  
R. Tabatabaei
Keyword(s):  
Equilibrium Equation ◽  
Displacement Function ◽  
Trigonometric Function ◽  
Polar Coordinates ◽  
Element Formulation ◽  
Shape Functions ◽  
Total Potential ◽  
Matrix Operations ◽  
Displacement Vectors ◽  
Locking Phenomena

The curved-beam finite element formulation by trigonometric function for curvature is presented. Instead of displacement function, trigonometric function is introduced for curvature to avoid the shear and membrane locking phenomena. Element formulation is carried out in polar coordinates. The element with three nodal parameters is chosen on curvature. Then, curvature field in the element is interpolated as the conventional trigonometric functions. Shape functions are obtained as usual by matrix operations. To consider the boundary conditions, a transformation matrix between nodal curvature and nodal displacement vectors is introduced. The equilibrium equation is written by minimizing the total potential energy in terms of the displacement components. In such equilibrium equation, the locking phenomenon is eliminated. The interesting point in this method is that for most problems, it is sufficient to use only one element to obtain the solution. Four examples are presented in order to verify the element formulation and to show the accuracy and efficiency of the method. The results are compared with those of other concepts.

scholarly journals Stress-Deformed State of a Packing Ring with Eccentric Holes

2020 ◽  
Vol 74 (4) ◽  
pp. 287-292
Author(s):  
Ilman Hasanov ◽  
Ibrahim Abbasov ◽  
Nurlan Gurbanov
Keyword(s):  
Composite Materials ◽  
Symbolic Method ◽  
Non Linear ◽  
Deformed State ◽  
Packing Ring ◽  
Anisotropic Bodies ◽  
Concrete Solution ◽  
Applied Fields ◽  
Sealing Materials ◽  
Theory Of Shells

AbstractCurrently, various variants of physical and geometrical non-linear calculation of anisotropic bodies have been developed. In spite of the large and increasing number of studies on the theory of shells there are still many unsufficiently developed problems important both in scientific and applied fields, for example, development of practically convenient methods for calculating of anisotropic sealing composite materials weakened by eccentric holes under the influence of local loadings. Stress-deformed state of a packing ring with eccentric holes of sealing materials was studied. In composite materials, the Hooke’s equation was used for this purpose. Also, using Lurie’s symbolic method, the concrete solution of the sealing problem with eccentric holes was obtained.

scholarly journals Strengths and strain energies of volcanic edifices: implications for eruptions, collapse calderas, and landslides

2012 ◽  
Vol 12 (7) ◽  
pp. 2241-2258 ◽  
Author(s):  
A. Gudmundsson
Keyword(s):  
Potential Energy ◽  
Magma Chamber ◽  
Strain Energy ◽  
Constant Load ◽  
Total Potential Energy ◽  
Shield Volcano ◽  
Total Potential ◽  
Fracture Formation ◽  
Constant Displacement ◽  
Elastic Mismatch

Abstract. Natural hazards associated with volcanic edifices depend partly on how fracture resistant the edifices are, i.e. on their strengths. Observations worldwide indicate that large fluid-driven extension fractures (dikes, inclined sheets), shear fractures (landslides), and mixed-mode fractures (ring dikes and ring faults) normally propagate more easily in a basaltic edifice (shield volcano) than in a stratovolcano. For example, dike-fed eruptions occur once every few years in many basaltic edifices but once every 102-3 yr in many stratovolcanoes. Large landslides and caldera collapses also appear to be more common in a typical basaltic edifice/shield volcano than in a typical stratovolcano. In contrast to a basaltic edifice, a stratovolcano is composed of mechanically dissimilar rock layers, i.e. layers with mismatching elastic properties (primarily Young's modulus). Elastic mismatch encourages fracture deflection and arrest at contacts and increases the amount of energy needed for a large-scale edifice failure. Fracture-related hazards depend on the potential energy available to propagate the fractures which, in turn, depends on the boundary conditions during fracture propagation. Here there are two possible scenarios: one in which the outer boundary of the volcanic edifice or rift zone does not move during the fracture propagation (constant displacement); the other in which the boundary moves (constant load). In the former, the total potential energy is the strain energy stored in the volcano before fracture formation; in the latter, the total potential energy is the strain energy plus the work done by the forces moving the boundary. Constant-displacement boundary conditions favor small eruptions, landslides, and caldera collapses, whereas constant-load conditions favor comparatively large eruptions, landslides, and collapses. For a typical magma chamber (sill-like with a diameter of 8 km), the strain energy change due to magma-chamber inflation is estimated at the order of 1014 J (0.1 PJ). For comparison, the surface energy needed to form a typical feeder dike is of the same order of magnitude, or 1014 J. There are several processes besides magma-chamber inflation that may increase the strain energy in a volcano before eruption. Thus, during a typical unrest period with magma-chamber inflation, the added strain energy in the volcano is large enough for a typical feeder dike to form. An injected dike, however, only reaches the surface and becomes a feeder if it is able to propagate through the numerous layers and contacts that tend to deflect or arrest dikes. The strong elastic mismatch between layers that constitute stratovolcanoes not only encourages fracture arrest, but also the storage of more strain energy (than in a typical basaltic edifice/shield volcano) before fracture formation and failure. It is thus through producing materials of widely different mechanical properties that stratovolcanoes become strong and resilient.

Consideration of interlaminar strain–energy continuity in composite plate analysis using improved higher order theory

2018 ◽  
Vol 42 (3) ◽  
pp. 211-221
Author(s):  
Temesgen Takele Kasa
Keyword(s):  
Composite Laminates ◽  
Strain Energy ◽  
Variational Calculus ◽  
Order Theory ◽  
Higher Order ◽  
Refined Theory ◽  
Total Potential ◽  
Lagrange Multiplier Technique ◽  
Plate Analysis ◽  
Higher Order Theory

The main goal of this paper is to suggest an improved higher order refined theory for analysing perfectly bonded stacked composite laminates with the usual lamination configurations. The analysis incorporates continuous flexural and in-plane displacements at the interfaces. Furthermore, the transverse shear stress is continuous and constrained with the Lagrange multiplier technique by introducing 14 new unknown variables that are expressed in terms of the interfacial strain energy, which is assuming to be continuous throughout the thickness of the laminate. To determine the newly introduced flexural and in-plane unknown variables, the total potential energy is minimised using variational calculus. The numerical results are compared with those from existing reliable published papers. In general, the proposed approach is sufficient for analysing laminate structures with the required accuracy.

The wustite-spinel interface

1987 ◽  
Vol 45 ◽  
pp. 268-269
Author(s):  
S.R. Summerfelt ◽  
C.B. Carter
Keyword(s):  
Solid State ◽  
Solid State Reaction ◽  
Strain Energy ◽  
Matrix Material ◽  
Lattice Misfit ◽  
Solid State Reactions ◽  
Oxygen Sublattice ◽  
Large Particles ◽  
Small Lattice ◽  
Coherent Interfaces

The wustite-spinel interface can be viewed as a model interface because the wustite and spinel can share a common f.c.c. oxygen sublattice such that only the cations distribution changes on crossing the interface. In this study, the interface has been formed by a solid state reaction involving either external or internal oxidation. In systems with very small lattice misfit, very large particles (>lμm) with coherent interfaces have been observed. Previously, the wustite-spinel interface had been observed to facet on {111} planes for MgFe2C4 and along {100} planes for MgAl2C4 and MgCr2O4, the spinel then grows preferentially in the <001> direction. Reasons for these experimental observations have been discussed by Henriksen and Kingery by considering the strain energy. The point-defect chemistry of such solid state reactions has been examined by Schmalzried. Although MgO has been the principal matrix material examined, others such as NiO have also been studied.

Video Graphics and High-Voltage Electron Microscopy For Analysis of Microtubule Distributions In Fixed Cells

1990 ◽  
Vol 48 (1) ◽  
pp. 524-525
Author(s):  
Richard Mcintosh ◽  
David Mastronarde ◽  
Kent McDonald ◽  
Rubai Ding
Keyword(s):  
Electron Microscopy ◽  
Cross Sections ◽  
Cell Shape ◽  
Video Camera ◽  
Computer Memory ◽  
High Voltage Electron Microscopy ◽  
Thin Sections ◽  
Voltage Electron ◽  
Morphogenetic Event ◽  
Set Of Points

Microtubules (MTs) are cytoplasmic polymers whose dynamics have an influence on cell shape and motility. MTs influence cell behavior both through their growth and disassembly and through the binding of enzymes to their surfaces. In either case, the positions of the MTs change over time as cells grow and develop. We are working on methods to determine where MTs are at different times during either the cell cycle or a morphogenetic event, using thin and thick sections for electron microscopy and computer graphics to model MT distributions.One approach is to track MTs through serial thin sections cut transverse to the MT axis. This work uses a video camera to digitize electron micrographs of cross sections through a MT system and create image files in computer memory. These are aligned and corrected for relative distortions by using the positions of 8 - 10 MTs on adjacent sections to define a general linear transformation that will align and warp adjacent images to an optimum fit. Two hundred MT images are then used to calculate an “average MT”, and this is cross-correlated with each micrograph in the serial set to locate points likely to correspond to MT centers. This set of points is refined through a discriminate analysis that explores each cross correlogram in the neighborhood of every point with a high correlation score.

Study of the strained-deformed state of a nonunion fracture model of the distal femoral metaepiphysis in different variants of its fixation

2011 ◽  
Vol 0 (4) ◽  
pp. 72
Author(s):  
Grigoriy Golka ◽  
Anton Bilostotskiy ◽  
Igor Subbota ◽  
Valeriy Sukhoveckiy ◽  
Oleg Fadeev
Keyword(s):  
Fracture Model ◽  
Deformed State

Modeling of Strain Energy Harvesting in Pneumatic Tires Using Piezoelectric Transducer

2014 ◽  
Vol 42 (1) ◽  
pp. 16-34 ◽  
Author(s):  
Ali E. Kubba ◽  
Mohammad Behroozi ◽  
Oluremi A. Olatunbosun ◽  
Carl Anthony ◽  
Kyle Jiang
Keyword(s):  
Energy Harvesting ◽  
Strain Energy ◽  
Fiber Composite ◽  
Experimental Tests ◽  
Evaluation Study ◽  
Monitoring Systems ◽  
Monitoring Device ◽  
Optimum Location ◽  
Piezoelectric Fiber Composite ◽  
Piezoelectric Fiber

ABSTRACT This paper presents an evaluation study of the feasibility of harvesting energy from rolling tire deformation and using it to supply a tire monitoring device installed within the tire cavity. The developed technique is simulated by using a flexible piezoelectric fiber composite transducer (PFC) adhered onto the tire inner liner acting as the energy harvesting element for tire monitoring systems. The PFC element generates electric charge when strain is applied to it. Tire cyclic deformation, particularly at the contact patch surface due to rolling conditions, can be exploited to harvest energy. Finite element simulations, using Abaqus package, were employed to estimate the available strain energy within the tire structure in order to select the optimum location for the PFC element. Experimental tests were carried out by using an evaluation kit for the energy harvesting element installed within the tire cavity to examine the PFC performance under controlled speed and loading conditions.

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