---
title: "Axisymmetric: constitutive, strain-displacement"
subject: "Computational Techniques in Civil Engineering"
chapter: 3
key: "semester_8/computational_techniques_in_civil_engineering:axisym-constitutive-strain-disp"
source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/axisym-constitutive-strain-disp
site: BCT Engineering
---

# Axisymmetric: constitutive, strain-displacement

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

- Asked in: 2077 Chaitra, 5 marks; 2070 Magh, 5 marks (asked 2 times)
- Video: [Strain-Displacement Relations in FEM | Strain Displacements matrix for Constant Strain Triangle_FEA](https://www.youtube.com/watch?v=FkixEVhOF_w) — 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-constitutive-strain-disp
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-3

## Question

What do you understand by axisymmetric problem? Explain with examples and write the constitutive relations and strain-displacement relations for axisymmetric condition.

## Short answer

Axisymmetric problems describe bodies of revolution with symmetric loading, reduced to r-z plane. Displacements u (radial) and w (axial) give ε_r = ∂u/∂r, ε_z = ∂w/∂z, ε_θ = u/r, γ_rz = ∂u/∂z+∂w/∂r. The constitutive matrix [D] = (E(1-ν)/[(1+ν)(1-2ν)]) [[1, ν/(1-ν), ν/(1-ν), 0]; [ν/(1-ν), 1, ν/(1-ν), 0]; [ν/(1-ν), ν/(1-ν), 1, 0]; [0,0,0,(1-2ν)/(2(1-ν))]].

## Answer

**Axisymmetric problem:** a body of revolution about an axis (the $z$-axis) whose loading and supports are also symmetric about that axis. All quantities are independent of the circumferential angle $\theta$, so the 3D problem reduces to a 2D problem in the $r$-$z$ plane. The circumferential displacement is zero ($v_\theta=0$), the shear strains $\gamma_{r\theta}=\gamma_{z\theta}=0$, and the non-zero components are $\varepsilon_r,\varepsilon_z,\varepsilon_\theta,\gamma_{rz}$.

Examples: circular footing, pile or well under axial load, pressure vessel and cylindrical tank, water tank, shaft, tunnel/borehole under uniform pressure, thick-walled cylinder under internal pressure, soil sample in a triaxial test.

**Strain-displacement relations** (displacements $u$ along $r$ and $w$ along $z$):
$$
\{\varepsilon\}=\begin{Bmatrix}\varepsilon_r\\\varepsilon_z\\\varepsilon_\theta\\\gamma_{rz}\end{Bmatrix}=\begin{Bmatrix}\dfrac{\partial u}{\partial r}\\[2mm]\dfrac{\partial w}{\partial z}\\[2mm]\dfrac{u}{r}\\[2mm]\dfrac{\partial u}{\partial z}+\dfrac{\partial w}{\partial r}\end{Bmatrix}
$$
The hoop strain $\varepsilon_\theta=u/r$ arises because a ring at radius $r$ stretches from circumference $2\pi r$ to $2\pi(r+u)$.

**Constitutive relation** $\{\sigma\}=[D]\{\varepsilon\}$ with $\{\sigma\}=\{\sigma_r,\sigma_z,\sigma_\theta,\tau_{rz}\}^T$:
$$
[D]=\frac{E(1-\nu)}{(1+\nu)(1-2\nu)}\begin{bmatrix}1&\dfrac{\nu}{1-\nu}&\dfrac{\nu}{1-\nu}&0\\[2mm]\dfrac{\nu}{1-\nu}&1&\dfrac{\nu}{1-\nu}&0\\[2mm]\dfrac{\nu}{1-\nu}&\dfrac{\nu}{1-\nu}&1&0\\[2mm]0&0&0&\dfrac{1-2\nu}{2(1-\nu)}\end{bmatrix}
$$

```text
        z
        ^   |p|
        |   v v v      axis of symmetry
   -----+---------
   |    |   r ->  |   Only the r-z half-plane
   |    |         |   is analysed.
   -----+---------
```

## Check yourself

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

1. What defines an axisymmetric problem?
   A. A thin plate loaded in its plane
   B. A body of revolution with loading and supports symmetric about the axis, so quantities are independent of theta
   C. Any problem with a symmetric cross-section only
   D. A long prism with constant section
   Answer: B. A solid of revolution with axis-symmetric loading and supports reduces to a 2D problem in the r-z plane. (Key point: Axisymmetric definition)

2. What is the hoop (circumferential) strain epsilon_theta in terms of radial displacement u?
   A. du/dr
   B. u/r
   C. du/dz
   D. w/r
   Answer: B. A ring at radius r stretches from 2 pi r to 2 pi (r + u), so epsilon_theta = u/r. (Key point: Hoop strain)

3. Which stress and strain components are non-zero in axisymmetry?
   A. Only sigma_r and sigma_z
   B. sigma_r, sigma_z, sigma_theta, tau_rz and the matching strains
   C. All six components
   D. Only tau_rz and gamma_rz
   Answer: B. Circumferential displacement and the theta shears vanish, leaving epsilon_r, epsilon_z, epsilon_theta, gamma_rz (and corresponding stresses). (Key point: Non-zero components)

## 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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- [Conditions for axisymmetric stress](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Aaxisym-stress-conditions) — Computational Techniques in Civil Engineering, chapter 3

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