---
type: concept
title: "Axisymmetric Finite Elements"
complexity: intermediate
domain: computational-mechanics
created: 2026-05-29
updated: 2026-05-29
address: c-000067
aliases:
  - axisymmetric element
  - triangular torus element
  - body of revolution finite element
tags:
  - concept
  - finite-element-method
  - continuum-elements
  - axisymmetric-analysis
status: current
related:
  - "[[Plane Stress and Plane Strain Elements]]"
  - "[[Isoparametric Finite Elements]]"
  - "[[Finite Element Thermal Stress Analysis]]"
  - "[[Finite Element Modeling and Convergence Checks]]"
sources:
  - "[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]"
source_refs:
  - source: "[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]"
    raw_path: ".raw/AFirstCourseInTheFiniteElementMethod/"
    raw_files:
      - "AFirstCourseInTheFiniteElementMethod_044.md"
      - "AFirstCourseInTheFiniteElementMethod_043.md"
      - "AFirstCourseInTheFiniteElementMethod_045.md"
      - "AFirstCourseInTheFiniteElementMethod_082.md"
    md_indices:
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    match: "heuristic-heading-keyword"
    confidence: high
---

# Axisymmetric Finite Elements

## Definition

Axisymmetric finite elements model bodies of revolution when the geometry, material behavior, boundary conditions, and loading are symmetric about an axis.

## How They Work

The simplest axisymmetric element is a triangular ring, or triangular torus, formed by rotating a triangular cross section around the axis of symmetry. The unknowns are radial and axial displacements in the cross section, but the strain state includes radial, axial, circumferential, and shear components.

The stiffness and load terms include the circumferential integration effect, commonly appearing through a radius-weighted area integral. This lets a two-dimensional mesh represent a three-dimensional body of revolution such as a thick pressure vessel, circular footing problem, or axisymmetric solid.

## Why It Matters

Axisymmetric elements are efficient when their assumptions hold. They avoid the cost of a full 3D mesh while retaining the hoop strain and hoop stress behavior that plane stress or plane strain idealizations would miss.

## Connections

- [[Plane Stress and Plane Strain Elements]] are also 2D idealizations, but they do not represent circumferential strain.
- [[Finite Element Thermal Stress Analysis]] includes an axisymmetric thermal strain case.
- [[Isoparametric Finite Elements]] generalizes the same cross-section mapping idea to higher-order or quadrilateral axisymmetric elements.

## Sources

- [[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]