---
type: concept
title: "Plane Stress and Plane Strain Elements"
complexity: intermediate
domain: computational-mechanics
created: 2026-05-29
updated: 2026-05-29
address: c-000066
aliases:
  - plane stress elements
  - plane strain elements
  - constant strain triangle
  - CST element
  - linear strain triangle
  - LST element
tags:
  - concept
  - finite-element-method
  - continuum-elements
  - plane-stress
  - plane-strain
status: current
related:
  - "[[Finite Element Method]]"
  - "[[Displacement-Based Finite Element Formulation]]"
  - "[[Finite Element Modeling and Convergence Checks]]"
  - "[[Isoparametric Finite Elements]]"
  - "[[Finite Element Load Vector Assembly]]"
sources:
  - "[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]"
source_refs:
  - source: "[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]"
    raw_path: ".raw/AFirstCourseInTheFiniteElementMethod/"
    raw_files:
      - "AFirstCourseInTheFiniteElementMethod_033.md"
      - "AFirstCourseInTheFiniteElementMethod_001.md"
      - "AFirstCourseInTheFiniteElementMethod_043.md"
      - "AFirstCourseInTheFiniteElementMethod_037.md"
    md_indices:
      - 33
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    match: "heuristic-heading-keyword"
    confidence: high
---

# Plane Stress and Plane Strain Elements

## Definition

Plane stress and plane strain elements are two-dimensional continuum finite elements used when a three-dimensional body can be idealized by behavior in a representative plane.

## How They Work

Plane stress assumes the out-of-plane normal and shear stresses are negligible, which is appropriate for thin plates loaded in their plane. Plane strain assumes the out-of-plane normal strain and shear strains are negligible, which is appropriate for long bodies whose geometry and loading do not vary significantly along the length.

The textbook develops the constant-strain triangular element as the simplest plane element. Each node carries in-plane displacement components, and the element uses a linear displacement field that produces constant strain over the triangle. It then introduces the linear-strain triangle as a higher-order alternative and compares element behavior.

## Why It Matters

Plane elements are the first continuum step beyond line elements. They expose key modeling issues that remain important in larger finite element work: element shape quality, stress recovery, compatibility along edges, boundary traction conversion, and convergence under mesh refinement.

## Connections

- [[Finite Element Modeling and Convergence Checks]] gives the practical checks needed before trusting plane element results.
- [[Isoparametric Finite Elements]] generalizes the plane element construction to quadrilateral and higher-order mappings.
- [[Finite Element Thermal Stress Analysis]] reuses plane stress and plane strain constitutive matrices with thermal strain terms.

## Sources

- [[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]