Chapter 9 – Axisymmetric Elements

1 Chapter 9 Axisymmetric ElementsLearning Objectives To review the basic concepts and theory ofelasticity equations for Axisymmetric behavior. To derive the Axisymmetric element stiffnessmatrix, body force, and surface traction equations. To demonstrate the solution of an axisymmetricpressure vessel using the stiffness method. To compare the finite element solution to an exactsolution for a cylindrical pressure vessel. To illustrate some practical applications ofaxisymmetric ElementsIntroductionIn previous chapters, we have been concerned with line or one-dimensional Elements (Chapters 2 through 5) and two-dimensional Elements (Chapters 6 through 8). In this Chapter , we consider a special two-dimensional element called the Axisymmetric element is quite useful when symmetry with respect to geometry and loading exists about an axis of the body being analyzed.

2 Problems that involve soil masses subjected to circular footing loads or thick-walled pressure vessels can often be analyzed using the element developed in this 7/8117 Chapter 9 - Axisymmetric Elements1/66 Axisymmetric ElementsIntroductionWe begin with the development of the stiffness matrix for the simplest Axisymmetric element, the triangular torus, whose vertical cross section is a plane then present the longhand solution of a thick-walled pressure vessel to illustrate the use of the Axisymmetric element equations. This is followed by a description of some typical large-scale problems that have been modeled using the Axisymmetric ElementsDerivation of the Stiffness MatrixIn this section, we will derive the stiffness matrix and the body and surface force matrices for the Axisymmetric , before the development, we will first present some fundamental concepts prerequisite to the understanding of the 7/8117 Chapter 9 - Axisymmetric Elements2/66 Axisymmetric ElementsDerivation of the Stiffness MatrixAxisymmetric Elements are triangular tori such that each element is symmetric with respect to geometry and loading about an axis such as the z , the z axis is called the axis of symmetry or the axis of revolution.

3 Each vertical cross section of the element is a plane triangle. The nodal points of an Axisymmetric triangular element describe circumferential ElementsDerivation of the Stiffness MatrixIn plane stress problems, stresses exist only in the x-y plane. In Axisymmetric problems, the radial displacements develop circumferential strains that induce stresses r , , zand rzwhere r, , and z indicate the radial, circumferential, and longitudinal directions, respectively. Triangular torus Elements are often used to idealize the Axisymmetric system because they can be used to simulate complex surfaces and are simple to work with. CIVL 7/8117 Chapter 9 - Axisymmetric Elements3/66 Axisymmetric ElementsDerivation of the Stiffness MatrixFor instance, the Axisymmetric problem of a semi-infinite half-space loaded by a circular area (circular footing)

4 Can be solved using the Axisymmetric element developed in this ElementsDerivation of the Stiffness MatrixFor instance, the Axisymmetric problem of a domed pressure vessel can be solved using the Axisymmetric element developed in this 7/8117 Chapter 9 - Axisymmetric Elements4/66 Axisymmetric ElementsDerivation of the Stiffness MatrixFor instance, the Axisymmetric problem of stresses acting on the barrel under an internal pressure ElementsDerivation of the Stiffness MatrixFor instance, the Axisymmetric problem of an engine valve stem can be solved using the Axisymmetric element developed in this 7/8117 Chapter 9 - Axisymmetric Elements5/66 Axisymmetric ElementsDerivation of the Stiffness MatrixFor instance, an Axisymmetric specimen loaded under ElementsDerivation of the Stiffness MatrixAn Axisymmetric 7/8117 Chapter 9 - Axisymmetric Elements6/66 Axisymmetric ElementsDerivation of the Stiffness MatrixBecause of symmetry about the z axis, the stresses are independent of the coordinate.

5 Therefore, all derivatives with respect to vanish, and the displacement component v (tangent to the direction), the shear strains r and zand the shear stresses r and zare all ElementsDerivation of the Stiffness MatrixConsider an Axisymmetric ring element and its cross section to represent the general state of strain for an Axisymmetric problem. CIVL 7/8117 Chapter 9 - Axisymmetric Elements7/66 Axisymmetric ElementsDerivation of the Stiffness MatrixThe displacements can be expressed for element ABCD in the plane of a cross-section in cylindrical coordinates. We then let u and w denote the displacements in the radial and longitudinal directions, ElementsDerivation of the Stiffness MatrixThe side AB of the element is displaced an amount u, and side CD is then displaced an amount u + ( u/ r) in the radial direction.

6 The normal strain in the radial direction is then given by:rur CIVL 7/8117 Chapter 9 - Axisymmetric Elements8/66 Axisymmetric ElementsDerivation of the Stiffness MatrixThe strain in the tangential direction depends on the tangential displacement v and on the radial displacement u. However, for Axisymmetric deformation behavior, recall that the tangential displacement v is equal to zero. Axisymmetric ElementsDerivation of the Stiffness MatrixThe tangential strain is due only to the radial displacement. Having only radial displacement u, the new length of the arc AB is (r + u)d , and the tangential strain is then given by: rud rdrd ur CIVL 7/8117 Chapter 9 - Axisymmetric Elements9/66 Axisymmetric ElementsDerivation of the Stiffness MatrixConsider the longitudinal element BDEF to obtain the longitudinal strain and the shear strain.

7 The element displaces by amounts u and w in the radial and longitudinal directions at point element displaces additional amounts:( w/ z)dzalong line BE and ( u/ r)dralong line EF. Axisymmetric ElementsDerivation of the Stiffness MatrixFurthermore, observing lines EFand BE, we see that point F moves upward an amount ( w/ r)drwith respect to point Eand point B moves to the right an amount ( u/ z)dzwith respect to point E. The longitudinal normal strain is given by:zwz The shear strain in the r-zplane is:rzuwzr CIVL 7/8117 Chapter 9 - Axisymmetric Elements10/66 Axisymmetric ElementsDerivation of the Stiffness MatrixSummarizing the strain-displacement relationships gives: The isotropic stress-strain relationship, obtained by simplifying the general stress-strain relationships, is:rzrzuu w uwrr z zr 100100112001 The procedure to derive the element stiffness matrix and element equations is identical to that used for the plane-stress in Chapter 6.

8 An Axisymmetric solid is shown discretized below, along with a typical triangular element. Derivation of the Stiffness MatrixAxisymmetric ElementsStep 1 - Discretize and Select Element TypesCIVL 7/8117 Chapter 9 - Axisymmetric Elements11/66 The procedure to derive the element stiffness matrix and element equations is identical to that used for the plane-stress in Chapter 6. Derivation of the Stiffness MatrixAxisymmetric ElementsStep 1 - Discretize and Select Element TypesThe stresses in the Axisymmetric problem are:Derivation of the Stiffness MatrixAxisymmetric ElementsThe element displacement functions are taken to be:Step 2 - Select Displacement Functions12 3(, )urz a ar az 45 6(, )wrz a ar az The nodal displacements are: iiijjjmmmuwduddwduw CIVL 7/8117 Chapter 9 - Axisymmetric Elements12/66 Derivation of the Stiffness MatrixAxisymmetric ElementsThe function u evaluated at nodeiis:Step 2 - Select Displacement Functions12 3(, )iiiiur za ar az The general displacement function is then expressed in matrix form as.

9 12 345 6iaarazaaraz 12345610000001aaarzarzaa Derivation of the Stiffness MatrixAxisymmetric ElementsBy substituting the coordinates of the nodal points into the equation we can solve for the a's:Step 2 - Select Displacement Functions123111iiijjjmmmurzaurzaurza 1axu 456111iiijjjmmmwrzawrzawrza 1axw CIVL 7/8117 Chapter 9 - Axisymmetric Elements13/66 Derivation of the Stiffness MatrixAxisymmetric ElementsPerforming the inversion operations we have:Step 2 - Select Displacement Functions1211iijjmmrzArzrz 2ij m jm i mi jArz z rz z rz z where Ais the area of the triangle 11[]2ijmijmijmxA Derivation of the Stiffness MatrixAxisymmetric ElementsStep 2 - Select Displacement Functions11[]2ijmijmijmxA ijmjmi jmimjrzzrz zrr jmimijmijimrz zrz zr r mij ijm i jm jirz zrz zr r ijmCIVL 7/8117 Chapter 9 - Axisymmetric Elements14/66 Derivation of the Stiffness MatrixAxisymmetric ElementsStep 2 - Select Displacement Functions12312ijm iijm jijmmauauAau The values of amay be written matrix form as:45612ijm iijm jijm mawawAaw Derivation of the Stiffness MatrixAxisymmetric ElementsStep 2 - Select Displacement FunctionsExpanding the above equations.

10 1231aurzaa Substituting the values for ainto the above equation gives: 112ijm iijm jijmmuurzuAu CIVL 7/8117 Chapter 9 - Axisymmetric Elements15/66 Derivation of the Stiffness MatrixAxisymmetric ElementsStep 2 - Select Displacement FunctionsWe will now derive the u displacement function in terms of the coordinates rand z. 112iij jmmiij jmmiij jmmuu uurzuuuAuu u Multiplying the matrices in the above equations gives: 1(, )2ii ii j j j jurzrz urz uA mm mmrzu Derivation of the Stiffness MatrixAxisymmetric ElementsStep 2 - Select Displacement FunctionsWe will now derive the w displacement function in terms of the coordinates rand z. 112iij jmmiij jmmiij jmmww wwrzwwwAww w Multiplying the matrices in the above equations gives: 1(, )2ii i i j j j jwrzrzwrzwA mm mmrzw CIVL 7/8117 Chapter 9 - Axisymmetric Elements16/66 Derivation of the Stiffness MatrixAxisymmetric ElementsStep 2 - Select Displacement FunctionsThe displacements can be written in a more convenience form as:(, )iij jmmurz Nu Nu N u where: 12iiiiNrzA (, )iij jmmwrz Nw Nw Nw 12mmmmNrzA 12jjjjNrzA Derivation of the Stiffness MatrixAxisymmetric ElementsStep 2 - Select Displacement FunctionsThe elemental displacements can be summarized as.