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wiki/concepts/Plane Stress and Plane Strain Elements.md

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+---
+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]]"
+---
+
+# 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]]
+