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

# Axisymmetric; plane stress strain-displacement

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

- Asked in: 2072 Asoj, 4+6 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-strain-disp-plane-stress
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-3

## Question

Explain the term axisymmetric problem with examples. Derive the strain-displacement and constitutive relationships that exist in a plane stress problem for isotropic material.

## Short answer

An axisymmetric problem is a solid of revolution with loading and supports symmetric about its axis, so fields depend only on r,z and tangential displacement is zero; examples are circular footings, piles and pressure vessels. For plane stress, strain-displacement gives eps_x=du/dx, eps_y=dv/dy, gamma_xy=du/dy+dv/dx; constitutive relation is sigma=E/(1-nu^2)[[1,nu,0],[nu,1,0],[0,0,(1-nu)/2]]epsilon, with eps_z=-nu(sigma_x+sigma_y)/E.

## Answer

#### Axisymmetric problem
**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.

#### Plane stress: strain-displacement relations
Let a point move by $u$ in $x$ and $v$ in $y$. For small displacements
$$
\varepsilon_x=\frac{\partial u}{\partial x},\quad \varepsilon_y=\frac{\partial v}{\partial y},\quad \gamma_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}
$$
In matrix form
$$
\{\varepsilon\}=\begin{bmatrix}\partial/\partial x&0\\0&\partial/\partial y\\\partial/\partial y&\partial/\partial x\end{bmatrix}\begin{Bmatrix}u\\v\end{Bmatrix}
$$
(derived from the change in length of $dx$ and $dy$ and the change in the right angle of a small rectangle). The out-of-plane strain $\varepsilon_z=-\nu(\sigma_x+\sigma_y)/E$ is not needed in the analysis.

#### Plane stress: constitutive relations
Start from Hooke's law for an isotropic material in 3D:
$$
\varepsilon_x=\frac{1}{E}[\sigma_x-\nu(\sigma_y+\sigma_z)],\quad \varepsilon_y=\frac{1}{E}[\sigma_y-\nu(\sigma_x+\sigma_z)],\quad \gamma_{xy}=\frac{\tau_{xy}}{G},\ G=\frac{E}{2(1+\nu)}
$$
For plane stress put $\sigma_z=0$:
$$
\varepsilon_x=\frac{\sigma_x-\nu\sigma_y}{E},\quad \varepsilon_y=\frac{\sigma_y-\nu\sigma_x}{E},\quad \gamma_{xy}=\frac{2(1+\nu)}{E}\tau_{xy}
$$
and $\varepsilon_z=-\nu(\sigma_x+\sigma_y)/E$. In matrix form $\{\varepsilon\}=[C]\{\sigma\}$:
$$
\begin{Bmatrix}\varepsilon_x\\\varepsilon_y\\\gamma_{xy}\end{Bmatrix}=\frac{1}{E}\begin{bmatrix}1&-\nu&0\\-\nu&1&0\\0&0&2(1+\nu)\end{bmatrix}\begin{Bmatrix}\sigma_x\\\sigma_y\\\tau_{xy}\end{Bmatrix}
$$
Inverting the first two equations: from $E\varepsilon_x=\sigma_x-\nu\sigma_y$ and $E\varepsilon_y=\sigma_y-\nu\sigma_x$, solve for $\sigma_x=\dfrac{E}{1-\nu^2}(\varepsilon_x+\nu\varepsilon_y)$ and $\sigma_y=\dfrac{E}{1-\nu^2}(\varepsilon_y+\nu\varepsilon_x)$. Also $\tau_{xy}=G\gamma_{xy}=\dfrac{E}{2(1+\nu)}\gamma_{xy}$. Hence
$$
\begin{Bmatrix}\sigma_x\\\sigma_y\\\tau_{xy}\end{Bmatrix}=\frac{E}{1-\nu^2}\begin{bmatrix}1&\nu&0\\\nu&1&0\\0&0&\dfrac{1-\nu}{2}\end{bmatrix}\begin{Bmatrix}\varepsilon_x\\\varepsilon_y\\\gamma_{xy}\end{Bmatrix}
$$
i.e. $\{\sigma\}=[D]\{\varepsilon\}$ with the plane stress matrix $[D]$ above.

## Check yourself

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

1. What is gamma_xy in terms of displacements u and v?
   A. du/dx + dv/dy
   B. du/dy + dv/dx
   C. du/dx - dv/dy
   D. u/y + v/x
   Answer: B. The shear strain is the change in the right angle: gamma_xy = du/dy + dv/dx. (Key point: Plane stress strain-displacement)

2. What is the role of epsilon_z = -nu(sigma_x + sigma_y)/E in plane stress analysis?
   A. It is zero
   B. It is non-zero but not needed in the analysis
   C. It is needed as a third degree of freedom
   D. It equals epsilon_x
   Answer: B. Thickness strain exists but is not required to find the in-plane displacements and stresses. (Key point: Out-of-plane strain)

3. Which condition makes a problem axisymmetric?
   A. Loads and supports symmetric about the axis of a body of revolution, so fields depend on r and z only
   B. Plate thickness much smaller than other dimensions
   C. Length much larger than cross-section
   D. Material is anisotropic
   Answer: A. Body of revolution with axis-symmetric loading and supports means the fields do not depend on theta. (Key point: Axisymmetric problem)

## Related in Computational Techniques in Civil Engineering

- [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
- [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
- [Axisymmetric: constitutive, strain-displacement](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Aaxisym-constitutive-strain-disp) — Computational Techniques in Civil Engineering, chapter 3, asked 2×
- [Plane stress constitutive; axisymmetric problems](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Aplane-stress-constitutive-axisym) — Computational Techniques in Civil Engineering, chapter 3
- [Constitutive relation for 2D isotropic material](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Aconstitutive-2d-isotropic) — Computational Techniques in Civil Engineering, chapter 3
- [Plane stress/strain; 3D equilibrium equation](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Adefine-plane-stress-3d-equilibrium) — Computational Techniques in Civil Engineering, chapter 3

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