---
title: "Plane stress, strain, axisymmetric with constitutive"
subject: "Computational Techniques in Civil Engineering"
chapter: 3
key: "semester_8/computational_techniques_in_civil_engineering:plane-stress-strain-axisym-constitutive"
source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/plane-stress-strain-axisym-constitutive
site: BCT Engineering
---

# Plane stress, strain, axisymmetric with constitutive

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

- Asked in: 2074 Bhadra, 2+2+2 marks
- Video: [Lec 14 : Mathematical formulation plane stress plane strain](https://www.youtube.com/watch?v=59RxQtSsnD0) — NPTEL IIT Guwahati
- 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/plane-stress-strain-axisym-constitutive
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-3

## Question

Describe the plane stress, plane strain and axisymmetric problems with their examples and constitutive relations to be used for stress analysis problems.

## Short answer

Plane stress: thin plate loaded in its plane, sigma_z=0, D=E/(1-nu^2)[[1,nu,0],[nu,1,0],[0,0,(1-nu)/2]]. Plane strain: long prism, eps_z=0, D=E/((1+nu)(1-2nu))[[1-nu,nu,0],[nu,1-nu,0],[0,0,(1-2nu)/2]]. Axisymmetric: solid of revolution with symmetric loads, fields in r,z, hoop strain u/r; D has four components with nu/(1-nu) coupling. Examples: thin plate with hole, dam, footing.

## Answer

#### Plane stress
**Plane stress:** a thin plate loaded only by forces in its own plane, so the stresses through the thickness are zero: $\sigma_z=\tau_{xz}=\tau_{yz}=0$. The stress state is two-dimensional, $\{\sigma\}=\{\sigma_x,\sigma_y,\tau_{xy}\}^T$, although the strain $\varepsilon_z=-\dfrac{\nu}{E}(\sigma_x+\sigma_y)\neq0$.
Examples: thin plate with a hole, gusset plate, deep beam, shear wall. Constitutive relation:
$$
\begin{Bmatrix}\sigma_x\\\sigma_y\\\tau_{xy}\end{Bmatrix}=\frac{E}{1-\nu^2}\begin{bmatrix}1&\nu&0\\\nu&1&0\\0&0&\frac{1-\nu}{2}\end{bmatrix}\begin{Bmatrix}\varepsilon_x\\\varepsilon_y\\\gamma_{xy}\end{Bmatrix}
$$

#### Plane strain
**Plane strain:** a long prism (length much larger than cross-section) with uniform cross-section and load that does not vary along its length and is perpendicular to it. Movement along the axis is prevented, so $\varepsilon_z=\gamma_{xz}=\gamma_{yz}=0$, but $\sigma_z=\nu(\sigma_x+\sigma_y)\neq0$.
Examples: gravity dam, retaining wall, long tunnel, strip footing. Constitutive relation:
$$
\begin{Bmatrix}\sigma_x\\\sigma_y\\\tau_{xy}\end{Bmatrix}=\frac{E}{(1+\nu)(1-2\nu)}\begin{bmatrix}1-\nu&\nu&0\\\nu&1-\nu&0\\0&0&\frac{1-2\nu}{2}\end{bmatrix}\begin{Bmatrix}\varepsilon_x\\\varepsilon_y\\\gamma_{xy}\end{Bmatrix}
$$

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

## Check yourself

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

1. Which [D] has the factor E/(1 - nu^2)?
   A. Plane strain
   B. Plane stress
   C. Axisymmetric
   D. 3D solid
   Answer: B. Plane stress [D] = E/(1 - nu^2)[1 nu 0; nu 1 0; 0 0 (1 - nu)/2]. Plane strain uses E/((1 + nu)(1 - 2 nu)). (Key point: Plane stress vs plane strain matrices)

2. How large is the axisymmetric [D] matrix and what stresses does it relate?
   A. 3 x 3 for sigma_x, sigma_y, tau_xy
   B. 6 x 6 for all 3D stresses
   C. 4 x 4 for sigma_r, sigma_z, sigma_theta, tau_rz
   D. 2 x 2 for sigma_r, sigma_z
   Answer: C. Axisymmetry has four non-zero stresses: sigma_r, sigma_z, sigma_theta and tau_rz. (Key point: Axisymmetric D size)

3. Which example correctly matches its problem type?
   A. Gravity dam: plane stress
   B. Deep beam: plane strain
   C. Circular footing: plane stress
   D. Circular footing or pile: axisymmetric; gravity dam: plane strain
   Answer: D. Gravity dams are long prisms (plane strain); circular footings and piles are bodies of revolution (axisymmetric); deep beams and shear walls are plane stress. (Key point: Examples)

## 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 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
- [Axisymmetric; plane stress strain-displacement](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Aaxisym-strain-disp-plane-stress) — 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×
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- [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

## Related in other subjects

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