Equilibrium equations in polar coordinates 2d youtube. Fourier analysis in polar and spherical coordinates. In the ansys program, plane42 is a 4 node isoparametric element with 2 degrees of freedom at each node. Chapter 3 formulation of fem for twodimensional problems 3. Because the plate is thin, the stress distribution may be very closely approximated by assuming that the foregoing is likewise true throughout the plate.
We have to consider nonlinear terms due to developing large slopes in system. Lecture l5 other coordinate systems in this lecture, we will look at some other common systems of coordinates. Spherical polar coordinates in spherical polar coordinates we describe a point x. Module 4 boundary value problems in linear elasticity. The reference point analogous to the origin of a cartesian coordinate system is called the pole, and the ray from the pole in the reference direction is the polar axis. We will present polar coordinates in two dimensions and cylindrical and spherical coordinates in three dimensions. Angles in autocad programs angles measure sets the direction for angle 0. Plane elasticity in polar coordinates elasticity in. Compare between methods of entering coordinate system.
Because the plate is thin, the stress distribution may be very closely approximated by assuming that the foregoing is likewise true through. We will discuss two major methods for solving elasticity equations. The equations of linear elasticity 15 governing equations in spherical polar coordinates x 1 x rsin cos. Stress functions relate six stresses to fewer functions defined in such a manner. We shall, as a condition of the problem, take the body force and and. Compare between methods of entering coordinate system absolute, relative, polar in autocad lubna zaghlul bashir technology university, baghdad, iraq email address. The wave equation on a disk changing to polar coordinates example physical motivation consider a thin elastic membrane stretched tightly over a circular. Introduction to polar coordinates polar coordinates involve the geometry of circles. Both categories of problems use the same elements defaults to plane stress. Two dimensional problems in polar coordinate system in any elasticity problem the proper choice of the coordinate system is extremely important since this choice establishes the. This would be tedious to verify using rectangular coordinates.
To specify points in space using spherical polar coordinates, we first choose two convenient, mutually perpendicular reference directions i and k in the picture. But there is another way to specify the position of a point, and that is to use polar coordinates r. Introduction to polar coordinates mit opencourseware. We will look at polar coordinates for points in the xyplane, using the origin 0. We consider the twodimensional plane with principal stress directions coinciding with. Direct method for deriving equilibrium equations in solid continuous systems. Polar coordinates are another way of describing points in the plane. Governing equations in cylindrical polar coordinates.
From elasticity to equations mathematical institute universiteit. Spherical polar coordinates university of sheffield. For the love of physics walter lewin may 16, 2011 duration. About entering 2d polar coordinates autocad lt 2018. Here is a set of practice problems to accompany the polar coordinates section of the parametric equations and polar coordinates chapter of the notes for paul dawkins calculus ii course at lamar university. Hence, nonlinear governing equations in cartesian system. Instead of giving x and y coordinates, well describe the location of a point by. Two dimensional problems in polar coordinate system in any elasticity problem the proper choice of the coordinate system is. A transform elastic equations from xy form to polar form. Applied elasticity in engineering materials technology. These lecture notes present the theory of applied elasticity.
Nov 16, 2010 equilibrium equations in polar coordinates. Elasticity problems in polar coordinates 10 i main topics a motivation b cartesian approach c transformation of coordinates d equilibrium equations in polar coordinates e biharmonic equation in polar coordinates f stresses in polar coordinates ii motivation a many key problems in geomechanics e. General solution of elasticity problems in two dimensional. Compatibility equation for plane elasticity in terms of polar coordinates.
Recall that in practice, for example for finite element techniques, it is usual to use curvilinear coordinates but we wont go that far we illustrate the solution of laplaces equation using polar coordinates kreysig, section 11. To this end, first the governing differential equations discussed in chapter 1 are expressed in terms of polar coordinates. Antiplane shear problems are nice and simple, but we rarely load a solid so as to cause antiplane shear deformation. However the material it covered is tangentially related to the topic of fluid mechanics it concerns another branch of continuum mechanics, the study of stress and strain in solid bodies.
Jul 22, 2017 for the love of physics walter lewin may 16, 2011 duration. This was a 4th year module which has now been amalgamated with other modules. The special form of this solution function allows us to. Chapter 3 formulation of fem for twodimensional problems. Me340a theory and applications of elasticity lecture notes. Jun 01, 2019 pdf on jun 1, 2019, charles chinwuba and others published solution of elasticity problems in two dimensional polar coordinates using mellin transform find, read and cite all the research you. Solution technique for partial differential equations. Fem formulation of 2d elasticity plane stressstrain displacement approximation strain and stress approximation derivation of element stiffness matrix and nodal load vector assembling the global stiffness matrix. The stress fields that solve any 2d elasticity problem in 2d polar coordinates become. Stress components in terms of airy stress function f fr.
Examples of plots of the equilibrium values of the torque versus the. Polar coordinates use a distance and an angle to locate a point. Polar coordinates mctypolar20091 the x,y coordinates of a point in the plane are called its cartesian coordinates. Civl 78117 chapter 6 plane stressplane strain stiffness equations part 1 1181. Airy solved the 2d elasticity problems in plane polar coordinates in a stressbased formulation by finding airy stress potential harmonic functions, r of the space coordinates r, that identically satisfied the differential equations of equilibrium.
These systems include gravitational fields, which obey the inversesquare law, as well as systems with point sources, such as radio antennas. Cartesian coordinates are used, which are appropriate for geometries which are have straight boundaries. Radially asymmetric systems may also be modeled with polar coordinates. General solution of elasticity problems in two dimensional polar. However, the coriolis acceleration we are discussing here is a real acceleration and which is present when rand both change with time. Then a number of important problems involving polar coordinates are solved. Just as professor jerison loves the number zero, the rest of mit loves circles. Pdf two dimensional problems in polar coordinate system. Laplaces equation in the polar coordinate system as i mentioned in my lecture, if you want to solve a partial differential equation pde on the domain whose shape is a 2d disk, it is much more convenient to represent the solution in terms of the polar coordinate system than in terms of the usual cartesian coordinate system. Barber department of mechanical engineering, university of michigan, usa. A half space subject to periodic traction on the surface an elastic material occupies a half space, x0. The equilibrium equations in cylindrical coordinates are. Classical mechanics lecture notes polar coordinates.
On the surface of the material, x0, the traction vector is prescribed. The basic theory includes the definition of stress, strain and elastic energy. Slager from elasticity to equations bachelorscriptie, 26 juni 2009 scriptiebegeleider. Specifically, it is used in the modeling of thin structures that react elastically to external forces.
Vector elds in polar coordinates university of sheffield. With dynamic input, you can specify absolute coordinates. The assumption of inplane deformation is more useful. In mathematics, the polar coordinate system is a twodimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.
We shall see that these systems are particularly useful for certain classes of problems. Elasticity equations in polar coordinates see section 3. The goal of the class is to provide an introduction to the theory of elasticity, plasticity and fracture and their applications. In cylindrical polar 2d coordinates, we have rotated a. Introduction to elasticitypolar coordinates wikiversity. To demonstrate how a 2d formulation works well use the following steady, ad equation. Use absolute polar coordinates when you know the precise distance and angle coordinates of the point.
Abstract in this work, the mellin transform method was used to obtain solutions for the stress field components in two dimensional 2d elasticity problems in terms of plane polar coordinates. Then a number of important problems involving polar coordinates are. Elasticity equations in cylindrical polar coordinates 1. Systems with a radial force are also good candidates for the use of the polar coordinate system.
The airy stress function for specific twodimensional plane conditions is computed and the stresses and displacements at a given point can be found using mathematica. The subject of elasticity is concerned with the determination of the stresses and displacements in a. With both cartesian and polar coordinates, can enter absolute coordinates based on the origin 0,0, or relative coordinates based on the last point specified. In mathematics, the biharmonic equation is a fourthorder partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of stokes flows. In cartesian coordinates systems, derivative with respect to orthogonal directions is equal to volume derivative as is shown in eq. Polar coordinates, parametric equations whitman college. Absolute polar coordinates are measured from the ucs origin 0,0, which is the intersection of the x and y axes. We consider the two dimensional plane with principal stress directions coinciding with. In mathematics, a spherical coordinate system is a coordinate system for threedimensional space where the position of a point is specified by three numbers. In quantum mechanics, the angular momentum is associated with the operator, that is defined as for 2d motion the angular momentum operator about the. Schematic of generic problem in linear elasticity or alternatively the equations of strain compatibility 6 equations, 6 unknowns, see. The laplacian in polar coordinates trinity university. Direct method for deriving equilibrium equations in solid. Here, the twodimensional cartesian relations of chapter 1 are recast in polar coordinates.
591 1211 1100 1200 1296 581 1146 1081 197 1078 1207 1498 1437 1603 1501 573 566 324 925 635 530 1384 386 1108 1338 914 107 387 847 68 824 1332 1094 1113 598 1007