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Nearly Singular Integrands In The Axisymmetric Finite Element Formulation

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1998 Annual Conference


Seattle, Washington

Publication Date

June 28, 1998

Start Date

June 28, 1998

End Date

July 1, 1998



Page Count


Page Numbers

3.421.1 - 3.421.11

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Paper Authors

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John D. Clayton

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Joseph Rencis University of Arkansas

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NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract

Session 1268

Nearly Singular Integrands in the Axisymmetric Finite Element Formulation John D. Clayton1, Joseph J. Rencis2 Georgia Institute of Technology/Worcester Polytechnic Institute


The formulation and explicit integration of the stiffness matrix for the two-node one-dimensional washer element are examined. An example problem is presented to illustrate the effectiveness of using various numerical integration methods for obtaining the element stiffness matrix when nearly singular integrations (for elements very close to the axis) are involved. Numerical examples are given for the two-node washer axisymmetric elasticity element. Errors in nodal displacements from the finite element solutions are compared for different integration methods. Integrating nearly singular axisymmetric washer elements the authors find that using a few sampling points with regular Gauss quadrature is inadequate and recommend new guidelines for numerically integrating these elements’ stiffness matrices.


Axisymmetric finite element (FE) models may be used to represent three-dimensional (3-D) structures exhibiting symmetry about a central axis of rotation. For conventional axisymmetric elements to be acceptable for modeling a structure, the body’s geometry, loading, boundary conditions, and material properties must all be independent of the T coordinate. Three common types axisymmetric elasticity elements include the two-node washer, the three-node triangle, and the general (distorted) four-node quadrilateral. Structures commonly modeled using axisymmetric elasticity elements include thick-walled pressure vessels, soil masses subjected to circular footing loads, and flywheels rotating at constant angular velocities.

The stiffness matrix for general axisymmetric elasticity elements is of the following form1: KE = ³³³ BT D B r dr dT dz (1) V

where B is the kinematic matrix relating element strains to element nodal displacements (H = B uE), D is the material law (Hooke’s law in this case) relating element stresses to element strains (V = D H), superscript T denotes the transpose operation, and V is the element volume. In axisymmetric elements the hoop strain Hr is not constant; it is a function of 1/r that varies with radial position in the element. For this reason, B contains 1/r terms, as does the BT D B r product.

Prior to proceeding further with the axisymmetric problem, the designation of integrals as regular, singular2, or nearly singular3,4 must be explained. A regular (or proper) integral has finite lower and upper limits of integration, and its integrand remains finite over these limits. A singular (or improper) integral may have at least one of its integration limits as infinite (rf) or 1 Graduate Student, George W. Woodruff School of Mechanical Engineering 2 Associate Professor of Mechanical Engineering

Clayton, J. D., & Rencis, J. (1998, June), Nearly Singular Integrands In The Axisymmetric Finite Element Formulation Paper presented at 1998 Annual Conference, Seattle, Washington.

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