---
title: "Conditions for axisymmetric stress"
subject: "Computational Techniques in Civil Engineering"
chapter: 3
key: "semester_8/computational_techniques_in_civil_engineering:axisym-stress-conditions"
source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/axisym-stress-conditions
site: BCT Engineering
---

# Conditions for axisymmetric stress

Computational Techniques in Civil Engineering · Chapter 3: Elasticity in solids

- Asked in: 2071 Bhadra, 4 marks
- Video: [Axi-symmetric Problems in FEM I finite elements analysis I Axisymmetric Problems in FEM Explained](https://www.youtube.com/watch?v=8lImesNTLm4) — Mahesh Gadwantikar
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/axisym-stress-conditions
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-3

## Question

What are the conditions at which axisymmetric stress exists? Write the stress-strain relations for axisymmetric condition.

## Short answer

Conditions: body is a solid of revolution, loads, supports and material are symmetric about the axis, so quantities depend only on r,z. Stress-strain: sigma = E(1-nu)/((1+nu)(1-2nu))*[1,a,a,0; a,1,a,0; a,a,1,0; 0,0,0,(1-2nu)/(2(1-nu))] epsilon, with a=nu/(1-nu); strains eps_r=du/dr, eps_z=dw/dz, eps_theta=u/r, gamma_rz=du/dz+dw/dr.

## Answer

#### Conditions for axisymmetric stress
Axisymmetric stress exists when all of the following hold:
1. The body is a **solid of revolution** about an axis (the $z$-axis).
2. The **loads** (surface and body forces) are symmetric about the axis, so they do not depend on the angle $\theta$.
3. The **supports/boundary conditions** are symmetric about the axis.
4. The material is also axisymmetric (isotropic, or the same in every radial plane).

As a result displacement, strain and stress are functions of $r$ and $z$ only, the tangential displacement is zero, and $\gamma_{r\theta}=\gamma_{z\theta}=0$, $\tau_{r\theta}=\tau_{z\theta}=0$. The non-zero stresses are $\sigma_r$, $\sigma_z$, $\sigma_\theta$ and $\tau_{rz}$. Examples: circular footing, pile, water tank, pressure vessel.

#### Stress-strain relations
$$
\begin{Bmatrix}\sigma_r\\\sigma_z\\\sigma_\theta\\\tau_{rz}\end{Bmatrix}=\frac{E(1-\nu)}{(1+\nu)(1-2\nu)}\begin{bmatrix}1&\frac{\nu}{1-\nu}&\frac{\nu}{1-\nu}&0\\\frac{\nu}{1-\nu}&1&\frac{\nu}{1-\nu}&0\\\frac{\nu}{1-\nu}&\frac{\nu}{1-\nu}&1&0\\0&0&0&\frac{1-2\nu}{2(1-\nu)}\end{bmatrix}\begin{Bmatrix}\varepsilon_r\\\varepsilon_z\\\varepsilon_\theta\\\gamma_{rz}\end{Bmatrix}
$$
with $\varepsilon_r=\partial u/\partial r$, $\varepsilon_z=\partial w/\partial z$, $\varepsilon_\theta=u/r$, $\gamma_{rz}=\partial u/\partial z+\partial w/\partial r$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which is NOT a condition for axisymmetric stress?
   A. Body is a solid of revolution
   B. Loads are symmetric about the axis
   C. Supports are symmetric about the axis
   D. The body must be a thin plate
   Answer: D. Axisymmetry needs a solid of revolution with symmetric loads, supports and material; thin plate is a plane stress idea. (Key point: Conditions)

2. What follows for the tangential displacement and shear components?
   A. They are all non-zero
   B. Tangential displacement is zero and gamma_r theta = gamma_z theta = 0, tau_r theta = tau_z theta = 0
   C. Only sigma_z is zero
   D. epsilon_theta is zero
   Answer: B. Symmetry removes the tangential displacement and theta shears, leaving sigma_r, sigma_z, sigma_theta and tau_rz. (Key point: Resulting zero components)

3. What is the common factor in the axisymmetric [D] matrix?
   A. E/(1 - nu^2)
   B. E(1 - nu)/((1 + nu)(1 - 2 nu))
   C. E/(2(1 + nu))
   D. E nu/((1 + nu)(1 - 2 nu))
   Answer: B. [D] = E(1 - nu)/((1 + nu)(1 - 2 nu)) times a matrix with 1 on the diagonal, nu/(1 - nu) off-diagonal and (1 - 2 nu)/(2(1 - nu)) for shear. (Key point: Stress-strain matrix)

## Related in Computational Techniques in Civil Engineering

- [Plane stress, strain, axisymmetric with constitutive](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Aplane-stress-strain-axisym-constitutive) — Computational Techniques in Civil Engineering, chapter 3
- [Plane strain, axisymmetric; plane stress constitutive](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Aplane-strain-axisym-constitutive-plane-stress) — Computational Techniques in Civil Engineering, chapter 3
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- [Plane stress versus plane strain](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Aplane-stress-vs-plane-strain) — Computational Techniques in Civil Engineering, chapter 3, asked 4×

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