Exact solutions for thermal stresses in a rotating thick. Given that the thick cylinder with variable thickness is divided into n homogenous disks, n sets of differential equations with constant coefficients. A semianalytical solution for elastic analysis of rotating. Moment of inertia for uniform objects live physics. Elastoplastic stress analysis in rotating disks and pressure vessels made of functionally graded materials latin american journal of solids and structures 2016 819834 chan 1999. Tutorial 11 thick walled cylinders and rotating discs. Rotating cylinders series rt hydraulic rotating cylinders n continuous 500 rpm capability n 1,500 psi pressure rating n flush and flange mountings n exclusive coupling sealing system n nitrotechardened coupling housing and stem n 4. Deformations and stresses in rotating fgm pressurized thick. A hollow thick walled cylinder with the internal radius of a 100 cm, the outer radius of b 115 cm is considered, which is rotating around the zaxis at the constant angular velocity of. This generates large amounts of stresses and strains on the material of the rotor disc. The disk has free boundary condition and rungekutta numerical method was used for numerical solution.
Analysis of twodimensional functionally graded rotating. Obtaining the moment of inertia of the full cylinder about a diameter at its end involves summing over an infinite number of thin disks at different distances from that axis. Rahimi mechanical engineering department, tarbiat modares university, tehran, iran. When the initial condition is a rotating cylinder with the. Deformations and stresses in rotating fgm pressurized. Elastic analysis of rotating thick cylindrical pressure vessels. Stresses in thick walled cylinders thick walled cylinders have an average radius less than 20 times the wall thickness. Thickwalled cylinder with external pressure of 5330 psi. Stress for thick walled cylinders using lames equations my. First, using the results presented in the chapter 2, we will explore the field of velocities in the cylinder container rotating with an angular velocity. Failure theories and stress concentrations example 1 with solution for cylinders 7. Rotating discs 2019 other related documents eg262 worked examples 8 eg262 worked examples 4 exam 2016, questions 5. These notes relate to the stresses and strains existing in thick walled cylinders when rotated at speed they are generally applicable to design of flywheels.
Keywordsadomian decowalled mposition method, thick cylinder, rotating disk, axisymmetric deformation. Rotating discs and cylinders materials engineering. Strengthening and design of the linear hardening thickwalled. The subscript r indicates an acceleration in the radial direction and the minus sign indicates that the particles are accelerating towards the centre of the disc. Deformations and stresses in rotating fgm pressurized thick hollow. For this purpose, a rotating cylindrical shell with variable thickness is divided into homogenous disks. Pdf stress intensity factor of radial cracks for rotating.
For the purpose of analysis, thick walled cylinder can be considered to consist of a series of thin rings figure 1a. Consider a thick walled cylinder with open ends as shown above. Calculation of the hoop burst speed for rotating discs in high speed turbomachinery, the discs which support the blades are highly loaded structural members. The strengthening and design of the thick walled cylinders is done using the rotational autofrettage method. Thick cylinders subjected to internal and external pressure. Lopez department of mathematics and earth systems science center, the pennsylvania state university. His solution very logically assumed that a thick cylinder to consist of series of thin cylinders such that each exerts pressure on the other.
Stress analysis thick cylinder pdf document fdocuments. Tutorial 11 thick walled cylinders and rotating discs 1. Following that, the set of non corresponding author, email. Mechanical principles thin walled vessels and thick walled. Behrooz hassani analysis of twodimensional functionally.
Here, the exact solution obtained from the previous part of the study will be followed up by an example. Consider a thin disc rotating with constant angular velocity. It will have two cases plane stress case 0or as a plain strain case 0 2. Series rt hydraulic rotating cylinders series rt hydraulic rotating cylinders n continuous 500 rpm capability n 1,500 psi pressure rating n flush and flange mountings n exclusive coupling sealing system n nitrotechardened coupling housing and stem n 4. Lopez department of mathematics and earth systems science center, the pennsylvania state university, university park, pennsylvania 16802 received 26 april 1996. Determine the maximum shear stress in the cylinder. The development of the expression for the moment of inertia of a cylinder about a diameter at its end the xaxis in the diagram makes use of both the parallel axis theorem and the perpendicular axis theorem. Johnson school of mechanical, aerospace and civil engineering.
Fgm, hollow circular cylinder, rotating, thermal stresses, elasticity. As with the rotating flow associated with discs, the proximity of a surface can significantly alter the flow structure. A thick cylinder of internal diameter 160 mm is subjected to an internal pressure of 40 nmm2. The radial and hoop stresses at any radius r in a disc of uniform thickness rotating. Flow between a stationary and a rotating disk shrouded by a.
Hassan zafarmand,mehran kadkhodayan 9 have foundthe nonlinear elasticity solution of functionally graded nanocomposite rotating thick disks. When the wall is thicker than this, it is regarded as a thick wall and it is treated differently as described later. Let the cylinder to be subjected to internal pressure and external pressure. Functionally graded rotating thick cylinder with variable thickness 16. The general equations to calculate the stresses are. Let the cylinder is subjected to internal pressure p i and outer pressure p o. Research article a semianalytical solution for elastic. Heat and mass transfer together with hybrid nanofluid flow. May 24, 2017 whereas for pressurised cylinders the loading is through applied radial boundary tractions, for a rotating disc, the loading is a radial body force which increases linearly with radius. The thick cylinder is divided into disk form layers form with their thickness corresponding to the thickness of the cylinder. Thus hoop and radial stresses in rotating thick cylinders can be obtained. It was assumed that the materials plastic deformation follows an isotropic strainhardening rule based on. The lift of a cylinder executing rotary motions in a uniform flow.
All of the cylindrical shell properties except the poisson ratio are considered exponential function along the radial direction. Based on the frobenius series method, stresses analysis of the functionally graded rotating thick cylindrical pressure vessels fgrtcpv are examined. With r equal to zero the above equations will yield infinite stresses whatever the speed of. The textbook problem of stress in perfectly elastic disks or cylinders is solved in standard texts 1,2,3,4,5,6.
Highreynolds number flow past a rotating cylinder with. Using disk form multilayers, a semianalytical solution has been derived for determination of displacements and stresses in a rotating cylindrical shell with variable thickness under uniform pressure. The primary assumption is that the cylinders are not subject to internal or external prssure. Thus hoop and radial stresses in rotating thick cylinders can be obtained from the equations for rotating discs provided that poissons ratio u is replaced by ul u, e. The cylinder of a hydraulic ram is 254 mm internal diameter and 50 mm thick. This article utilizes the average stress method to obtain the stress intensity factor of rotating solid and hollow disks cylinders containing a radial crack. Elastic analysis of functionally graded rotating thick. Mechanically driven circulation in a rotating cylinder. In a cylinder with inside diameter 200 mm radius 100 mm and outside diameter 400 mm radius 200 mm there is a pressure 100 mpa relative to the outside pressure. Axisymmetric deformation analysis of thickwalled cylinders. Pdf stress analysis of rotating composite solid disc. Temperature and thickness effects on thermal and mechanical. Results are shown for both of the plane stress and plane strain assumptions and are validated against the known data introduced in.
Rotating a hori zontal line around the x axis gives a cylinder. Rotating rings discs and cylinders 7 1 thin rotating ring or cylinder let us consider a thin ring or cylinder as shown below 3 2 2. The rotational autofrettage gives more uniform stress distribution than the pressure autofrettage. Pdf elastic analysis of rotating thick cylindrical. Twodimensional functionally graded rotating thick disks 455 fig. From a thick walled cylinder, we get the boundary conditions. The rotating disk cylindrical and spherical vessel was made of a ceramicmetal functionally graded material, i. If the objectvessel has walls with a thickness greater than onetenth of the overall diameter, then these objects can be assumed to be thick walled. Find the maximum hoop stress in the material and the radial and hoop stress at a point in the barrel 19 mm from the inner surface when the fluid pressure inside the cylinder is 9. The vessel is considered in both plane stress and plane strain conditions.
Goldstein 1938 refers to several historical papers on both rotating spheres and cylinders, crediting magnus 1 853 with the first laboratory experiments examining the lift on a rotating cylinder. The cylinders are not rotating about their centre of mass, so we must use the parallel axis theorem. Rotating disks rotating disks mounted on solid shafts. Rotating volumes with the cylindershell method jakes math. Flow between a stationar y and a rotat ing disk shrouded by a co rotating cylinder j. It is loaded by internal pressure pi and external pressure po as seen below.
Axisymmetric conditions are assumed for the twodimensional functionally graded disk. Deformations and stresses in rotating fgm pressurized thick hollow cylinder under thermal load m. Jul 01, 2020 they will be used for an approximate calculation of circulation in a rotating cylinder with a braking lid and in a fixed casing with a rotating disk. Elastoplastic stress analysis in rotating disks and. Strengthening and design of the linear hardening thick. Stresses in a thick walled pressurized cylinder for plane stress. The approach involves finding an expression for a thin disk at distance z from the axis and summing over all such disks. Thick cylinders the problem of determination of stresses in a thick cylinders was first attempted more than 160 years ago by a french mathematician lame in 1833. Highreynolds number flow past a rotating cylinder with and.
For instance, in the turbine section of an aircraft gas turbine engine, the rotor disc frequently turns at speeds in the region of 0 r. Stress and strain analysis of rotating fgm thick truncated. Rotating cylinders, annuli, and spheres flow over rotating cylinders is important in a wide number of applications from. Cones and spheres and circular cylinders are solids of revolution. Nov 24, 2011 the rotation of a fast moving disc or cylindrical member can set up axial, circumferential and radial stresses. If it is subject to an internal pressure of 150 mpa, determine the radial and tangential stress distributions and show the results on a plot using a spreadsheet.
Consider a cylinder of mean diameter d, wall thickness t and length l. The kinetic energy in discs is considerable and whilst designed to avoid burst failure, if this occurs the surrounding structure is often not strong enough to contain the disc fragments, e. Pdf on aug 28, 2019, alemayehu adde y published stress analysis of rotating composite solid disc find, read and cite all the research you need on researchgate. Elasticity solutions of twodimensional functionally graded rotating annular and solid disks with variable thickness are presented. Pdf strengthening and design of the linear hardening. The modulus of elasticity, the thermal coefficient of. Pdf elastic analysis of rotating thick cylindrical pressure. Analytical solutions what is materials technology mate. Hoop stress, 1 radial stress, 2 from a thick walled cylinder, we get the boundary conditions. Consider a typical ring located at a radius r having a. If a circle is moved away from the axis, rotation produces a torus a doughnut.
Pdf strengthening and design of the linear hardening thick. May 20, 2020 the heat transfer with the fluid flow over the surface of a rotating disk has been receiving extraordinary attention from researchers due to its numerous applications in engineering and aeronautical sciences, such as thermal energy generating systems, gas turbine rotators, rotating machinery, geothermal industry, chemical processes, medical equipment, and computer storage. Elastoplastic stress analysis in rotating disks and pressure. The centre of each rod is rdisk rcyl from the axis of rotation at the centre of the object. Flow between a stationary and a rotating disk shrouded by. The lift of a cylinder executing rotary motions in a. Material particles are subjected to a centripetal acceleration a r 2 r. Due to the existence of shear stress in the thick cylindrical shell. Calculation of the hoop burst speed for rotating discs. This is a special case of a torus for a 0 see below, as well as of a thick walled cylindrical tube with open ends, with r 1 r 2 and h 0.
A cylinder is regarded as thin walled when the wall thickness t is less than 120 of the diameter d. For a rotating disc with variable thickness tr the equation of motion in radial. It may be assumed, for a thin disc, that the stress in the axial direction is zero. May 01, 2017 the residual stress distribution is defined explicitly for thick cylinders with linear strain hardening behavior.
Keywordsadomian decomposition method, thick walled cylinder, rotating disk, axisymmetric. Mechanically driven circulation in a rotating cylinderdisk. Material properties vary through both the radial and axial directions continuously. The moments of inertia for a a disk or a cylindrical rod are is. It is assumed that the cracks are located radially at center, internal or external radius of the geometry. A rotating cylinder moving in a uniform stream experiences a force normal to the direction of motion. Accepted january, 2009 using the infinitesimal theory of elasticity, closed form solutions for onedimensional steadystate. If the allowable stress in the material is 120 nmm2, find the thickness required. Consider a thick walled cylinder having an inner radius a.
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