--- title: "Plane stress versus plane strain" subject: "Computational Techniques in Civil Engineering" chapter: 3 key: "semester_8/computational_techniques_in_civil_engineering:plane-stress-vs-plane-strain" source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/plane-stress-vs-plane-strain site: BCT Engineering --- # Plane stress versus plane strain Computational Techniques in Civil Engineering · Chapter 3: Elasticity in solids - Asked in: 2079 Jestha, 4+4 marks; 2078 Chaitra, 5 marks; 2078 Kartik, 4 marks; 2070 Bhadra, 5 marks (asked 4 times) - Video: [Plane Stress vs Plane Strain Conditions Simple explanation with Example](https://www.youtube.com/watch?v=VWNLtTB7mJs) — Civil Engineering & SHM - 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-vs-plane-strain - Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-3 ## Question Differentiate between plane stress and plane strain problems with suitable examples, conditions and necessary figures. ## Short answer Plane stress applies to thin plates loaded in their plane: sigma_z=0 while epsilon_z is nonzero, example a gusset plate. Plane strain applies to long bodies with constant transverse load: epsilon_z=0 while sigma_z is nonzero, examples a dam or tunnel. The constitutive matrix uses different E and nu relations. ## Answer #### Definitions **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 $\varepsilon_z=-\dfrac{\nu}{E}(\sigma_x+\sigma_y)\neq0$. **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 $\sigma_z=\nu(\sigma_x+\sigma_y)\neq0$. #### Comparison | Point | Plane stress | Plane strain | |---|---|---| | Geometry | Thin plate, thickness $\ll$ other dimensions | Long body, length $\gg$ cross-section | | Loading | In the plane of the plate, uniform over thickness | Perpendicular to the length, constant along it | | Zero components | $\sigma_z=\tau_{xz}=\tau_{yz}=0$ | $\varepsilon_z=\gamma_{xz}=\gamma_{yz}=0$ | | Non-zero out-of-plane | $\varepsilon_z=-\nu(\sigma_x+\sigma_y)/E$ | $\sigma_z=\nu(\sigma_x+\sigma_y)$ | | $[D]$ matrix | $\dfrac{E}{1-\nu^2}\begin{bmatrix}1&\nu&0\\\nu&1&0\\0&0&\frac{1-\nu}{2}\end{bmatrix}$ | $\dfrac{E}{(1+\nu)(1-2\nu)}\begin{bmatrix}1-\nu&\nu&0\\\nu&1-\nu&0\\0&0&\frac{1-2\nu}{2}\end{bmatrix}$ | | Examples | Deep beam, gusset plate, plate with a hole, shear wall | Gravity dam, retaining wall, tunnel, long embankment, strip footing | The plane strain matrix is obtained from the plane stress one by replacing $E\to E/(1-\nu^2)$ and $\nu\to\nu/(1-\nu)$. #### Figures ```text Plane stress Plane strain thin plate long dam / tunnel ->+------+<- ||||||||||||| load ->| |<- +-----------+ z (long) ->+------+<- ==============> no strain t small, sz=0 ez = 0 ``` In plane stress the plate is free to change thickness; in plane strain each slice of unit thickness of the long body is prevented from changing length by neighbouring slices. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which set of components is zero in plane stress? A. epsilon_z, gamma_xz, gamma_yz B. sigma_x, sigma_y, tau_xy C. sigma_z, tau_xz, tau_yz D. epsilon_x, epsilon_y, gamma_xy Answer: C. In plane stress the through-thickness stresses sigma_z, tau_xz and tau_yz vanish. In plane strain epsilon_z, gamma_xz and gamma_yz vanish instead. (Key point: Zero components) 2. In plane strain, what is sigma_z? A. Zero B. nu(sigma_x + sigma_y), not zero C. -nu(sigma_x + sigma_y)/E D. E epsilon_z Answer: B. Because epsilon_z = 0, sigma_z = nu(sigma_x + sigma_y) is non-zero. In plane stress it is epsilon_z = -nu(sigma_x + sigma_y)/E that is non-zero. (Key point: Out-of-plane quantity) 3. Which pairing is correct? A. Plane stress: long gravity dam; plane strain: thin gusset plate B. Plane stress: tunnel; plane strain: deep beam C. Both apply only to thin plates D. Plane stress: thin plate, deep beam or shear wall; plane strain: long dam, retaining wall or tunnel Answer: D. Plane stress suits thin bodies loaded in their plane; plane strain suits long prisms with loading perpendicular to the length and constant along it. (Key point: Geometry and examples) 4. How is the plane strain [D] obtained from the plane stress [D]? A. Replace E by E/(1 - nu^2) and nu by nu/(1 - nu) B. Replace E by E(1 - nu^2) and nu by nu(1 - nu) C. Set nu = 0 D. Transpose the matrix Answer: A. Substituting E -> E/(1 - nu^2) and nu -> nu/(1 - nu) in the plane stress matrix gives the plane strain matrix. (Key point: Relating the D matrices) ## 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 - [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 - [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 - [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 ## Related in other subjects - [Plane stress and plane strain differences](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Aplane-stress-plane-strain-difference) — Finite Element Method, chapter 6 - [Compute D matrix for plane problems](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Aelasticity-matrix-plane-stress-strain) — Finite Element Method, chapter 6 - [Deadlock; necessary conditions with examples](https://www.bctengineer.com/question.md?key=semester_6%2Foperating_system%3Adeadlock-necessary-conditions-examples) — Operating System, chapter 7 - [Deadlock conditions and why all necessary](https://www.bctengineer.com/question.md?key=semester_6%2Foperating_system%3Adeadlock-conditions-why-necessary) — Operating System, chapter 7