---
type: concept
title: "Finite Element Thermal Stress Analysis"
complexity: intermediate
domain: computational-mechanics
created: 2026-05-29
updated: 2026-05-29
address: c-000070
aliases:
  - thermal stress finite element analysis
  - thermal strain load vector
  - temperature-induced stress
tags:
  - concept
  - finite-element-method
  - thermal-stress
  - coupled-field
status: current
related:
  - "[[Finite Element Heat Transfer and Field Problems]]"
  - "[[Finite Element Load Vector Assembly]]"
  - "[[Plane Stress and Plane Strain Elements]]"
  - "[[Axisymmetric Finite Elements]]"
  - "[[Displacement-Based Finite Element Formulation]]"
sources:
  - "[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]"
---

# Finite Element Thermal Stress Analysis

## Definition

Finite element thermal stress analysis computes stresses caused by temperature-induced strains, especially when expansion or contraction is constrained.

## How It Works

The source treats thermal strain as an initial strain contribution. For a uniform temperature change in an isotropic material, thermal strain is proportional to the coefficient of thermal expansion and the temperature change. The constitutive relation is written in terms of mechanical strain minus thermal strain, so the thermal contribution enters the finite element equations as an equivalent nodal force vector.

The same idea is applied to one-dimensional bars, plane stress and plane strain elements, and axisymmetric triangular elements. If the structure is free to expand, thermal strain may produce displacement without stress. If constraints or material incompatibility prevent free expansion, thermal stresses appear.

## Why It Matters

Thermal loading is not just another external force. It changes the strain state inside the element and can create stress only through constraint, incompatibility, or temperature gradients. Treating it as an equivalent nodal contribution keeps the global equation format compatible with the displacement formulation.

## Connections

- [[Finite Element Heat Transfer and Field Problems]] can supply the temperature distribution.
- [[Finite Element Load Vector Assembly]] explains the equivalent nodal force interpretation.
- [[Plane Stress and Plane Strain Elements]] and [[Axisymmetric Finite Elements]] provide common structural discretizations for thermal stress.

## Sources

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